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5ebf19dc7e9f6afdefcb4462f2c9afbe749c1d32 53146 F20101109_AAAWDV lightcap_c_Page_032.pro e90aea957f9bd015b344cfd646a4c4df c77cbfeae116f1759dc8cf721fe059e683407b8c F20101109_AAAVYP lightcap_c_Page_017.tif cf72e0669429fa2dac221abd79184fbd 90bc02f696fbcd2a547cb24f103f471e30dce6b2 MEASUREMENT AND CONTROL ISSUES IN A NOVEL DYNAMIC RADIOGRAPHIC IMAGING SYSTEM By CHRIS ALAN LIGHTCAP A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008 2008 Chris Alan Lightcap To Kerry Robertson, and all my friends and family who have helped me reach where I am today. ACKNOWLEDGMENTS I would like to thank my advisor, Dr. Scott Banks, for all of his support and dedication for the project. I thank my committee, Dr. B.J. Fregly, Dr. Warren Dixon, Dr. Tony Schmitz, and Dr. Manuel Bermudez, for their help and advice. I thank J.D. Yamokoski, Samuel Hamner and Jessica Allen for their various contributions to my dissertation. Last but not least, I thank Kerry for her loving support. TABLE OF CONTENTS page A CK N O W LED G M EN T S ................................................................. ........... ............. ..... L IS T O F T A B L E S .................................................................................8 LIST OF FIGURES .................................. .. ..... ..... ................. .9 A B S T R A C T ......... ....................... ............................................................ 1 1 CHAPTER 1 INTRODUCTION ............... .............................. ............................ 12 Parameter Identification: Building a Physical Model..........................................................14 Controller Development: Designing an Intelligent System .................................................15 Measurement Analysis: Assessing the Imaging Performance .............................................16 2 PARAMETER IDENTIFICATION ......................................................... ..............18 In tro d u ctio n ......................................................... ................... ................ 1 8 G eom etric and Flexibility C alibration ........................................................................ ... ... 19 Literature Review .............. ................. ........... ........................... 19 M eth o d s .............. ...... ...... ...................... ..................... ................2 0 Experim ental setup and procedure ........................................ .......................21 Twolevel geometric and flexibility calibration ....................................................24 M onte C arlo sim u lation ......................................... .............................................2 7 E xperim ental assessm ent................................................ ....................................28 Results .........................................28 C conclusions ....................... .............................................. 3 1 Identification of Dynam ic Characteristics........................................................... ... .......... 32 Literature R eview .................. ..................................... ........... ...............32 R igidLink FlexibleJoint M odel......................................................... ................ 34 Leastsquares parameter identification...................... .... ....................... 35 Trajectory optim ization ................ .................... ...................... ...............36 E stim action of Link Position........................ ...... ............... .................. ............... 38 Experim ental Setup and Procedure ........................................... ........... ............... 40 Experimental Results .............. .......... ........ ............... 41 C o n c lu sio n s ..............................................................................4 6 3 NONLINEAR CONTROL AND ESTIMATION............... .................... .................48 Introduction ......... .............. ......................................... ............................48 L iteratu re R ev iew .......................................................... ................. 4 9 Extended K alm an Filter Observer .......................................................... ............... 53 T he E extended K alm an F ilter .......................................................... ...........................53 RigidLink FlexibleJoint Robot (RLFJ) Model .................................. ...............54 Measurement Model ...................................................... ............. 57 O bservability in N onlinear System s.................................................................... ...... 59 RigidLink FlexibleJoint Control ................................................ .............................. 62 T theoretical B background ......................................................................... ................... 62 Control Objective ............................................................63 Error System for Filtered Tracking Error.............................................. .................. 65 Partial Stability Analysis for Filtered Tracking Error................. ............................66 Auxiliary Control and Backstepping Error System.............................. ............... 67 Partial Stability Analysis for Backstepping Error System ...........................................69 C controller Stability R esult............................ ..................... ................. ............... 70 C losedL oop Stability R esult ................................................ .............................. 71 S im u latio n ......... ..............................................................................................7 2 Experim ent .............. .......................................... ................. 76 RealTime Implementation Issues.......................... .................... 77 Experim mental Results ........... ............. .. ... ........... .. ...... .... ........79 C conclusion .......... .... ..... ......... ............................................82 4 M EA SU REM EN T A N A LY SIS ..................................................................... ..................84 Introduction ........................................................84 L iteratu re R ev iew .............................................................................84 U uncertainty A analysis ................................................................... 86 Im aging G eom etry ......... ....... ....... ........................................................ 86 Covariance of Relative Motion ........ ......... ............... ........ 89 Endeffector tracking error ............ ............ ................ 90 M easurem ent uncertainty ................................................................................. ..91 Im ag e S h arp n ess ............. ........ ...................................................... ........................ .. 9 5 Theoretical Im prove ent .................................. ........................................... 96 E xperim mental M ethods......................................................................................... 99 E xperim mental R results ......... ............ ........................................... .......................... 100 Im age R registration ......................................................................................... 102 M methods ........................................................... ............. .....................102 Results ................. ............ .. ............................. 103 C conclusion ........ .. ......... .. ....... ............ .... .. ....................... .. ............. 105 5 CONCLUSION................ ..... .. .......... ........... ............... .. 108 APPENDIX A OBSERVABILITY M ATRIX .............. ................ .................................. ............... 10 B OBSERVER DESIGN FOR TWOLINK ROBOT ..................... ................... .........115 L IST O F R EFE R E N C E S ............................................................................................ 124 6 B IO G R A PH IC A L SK E T C H ............................................................................... ............... ..... 132 LIST OF TABLES Table page 21. DenavitHartenberg parameters and joint stiffness coefficients...................... ..........29 22. Parameter identification results for link dynamics.................................... ........................42 23. Parameter identification results for motor dynamics ......... .................................43 31. Sim ulation results w without payload ............................................... ............................. 75 32. Sim ulation results w ith payload ................................................. ................................ 75 33. Comparison of numerical integration methods .......... ........................................... 77 34. Experim ental results without payload .............................................................................. 81 35. E xperim mental results w ith payload .............................................................. .....................81 41. Kinematic errors for Blurred and Corrected image groups ............................. ...............105 42. Kinematic differences between Blurred and Corrected image groups..............................105 LIST OF FIGURES Figure page 11. Dynamic radiographic imaging system .............................................. .................. 14 21. Solution sequence for geometric and flexibility calibration .......... ......................... 21 22. Experimental setup of the PA106CE and CMM ........................................ ...............22 23. Tooling ball apparatus .............. ............... ......................................... .................23 24. Experimental test path ............................... ... ............................ 23 25. Contour plot of mean position error ................................................. ....................... 29 26. Mean position error before and after calibration ....................................... ...............30 27. Endeffector position error at novel configurations and loads ............................................30 28. Sam ple identification trajectory...................................................................... ..................38 29. M option capture markers on PA 106CE .................................................... ..................39 210. E ndeffector w eight rack ...................................................................... .. ........................40 211. C coordinate system definitions ................................................................ ..... ...................44 212. M measured and predicted m otor torque .................................................... ...................45 213. Stiffness torque and joint deflection........................................ ............... ............... 45 31. Link position tracking errors in simulation while loaded.....................................................74 32. Link position estimation errors in simulation while loaded ....................................... 74 33. M itsubishi PA 106CE robot w ith m arkers .............................................................................76 34. Experimental link position tracking errors while loaded............................. ...............80 35. Joint stiffness coefficients from experiment while loaded ...............................................80 41. Imaging configuration of robotic imaging system ........... ............................ ..............89 42. Norm of translation part of manipulator Jacobian .......................................... .............94 43. Norm of rotational part of manipulator Jacobian ..............................................................95 44. Spatial frequency response of motion blurred images...........................................................98 9 45. Relationship between relative motion and image contrast ...............................................98 46. Experimental setup of dualrobot imaging system..... ............................................... .........100 47. Relative contrast in blurred and blurred/corrected images.............................. ..............101 48. Image sharpening from LucyRichardson algorithm ................................... ... ..................101 49. U distorted radiographic im ages........... ................. ............................ ............... 104 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MEASUREMENT AND CONTROL ISSUES IN A NOVEL DYNAMIC RADIOGRAPHIC IMAGING SYSTEM By Chris Alan Lightcap May 2008 Chair: Scott A. Banks Major: Mechanical Engineering A novel dynamic radiographic imaging system, one which tracks the patient during an activity, can provide measurements of natural skeletal motion. It is believed that accurate measurements of skeletal kinematics can improve both the treatment and diagnosis of musculoskeletal conditions. However, the measurement and control obstacles of such a proposed system have not been overcome. Using heavy radiographic equipment in combination with light weight manipulators leads to joint deflections and degraded tracking performance. Furthermore, relative motion between imaging components leads to motion blur and degraded measurement performance. Novel static and dynamic identification methods are developed and executed in order to ascertain kinematic, inertial, frictional, and stiffness properties for modelbased observer and controller methods. An extended Kalman filter and adaptive reduced parameterization Lyapunovbased controller are developed in order to control the robotic imaging system. The effect of motion blur on 3D to 2D image registration is also explored, given the anticipated relative motion between components. CHAPTER 1 INTRODUCTION Each year, an estimated 1 of 7 Americans is wounded by a musculoskeletal disease or injury [1]. In the year 2004, The American Academy of Orthopedic Surgeons reported that over 150 million physician visits were due to musculoskeletal conditions [2]. These numbers are expected to increase. The elderly are primarily effected, and the number of individuals over 50 years of age is expected to double between the years 1990 and 2020 [3]. The current methods of treatment have limitations. Advances in the field of musculoskeletal research are becoming increasingly more important with time as the number of cases increase. Effective treatments will help relieve pain for patients and help reduce medical costs. Our society spends an estimated $254 billion each year as a result of musculoskeletal conditions [4]. Therefore, improvements in the treatment and diagnosis are critical to keep pace with our growing and aging society. This rising awareness of musculoskeletal conditions and their impact has contributed to the World Health Organization announcing the creation of "The Bone and Joint Decade". In 2002, President Bush followed their lead by appointing the years ranging from 20022011 as the National Bone and Joint Decade. The goal of these declarations has been to raise awareness about musculoskeletal disorders and promote research and development for their diagnosis and treatment [5]. Since then, it is no coincidence that 55 national governments and more than 150 organizations worldwide have recognized this global initiative. The sobering reality is that more and more people will be suffering from these musculoskeletal conditions over time and we must be prepared to diagnose and treat patients with greater effectiveness. The current methods for diagnosis of musculoskeletal conditions have limitations and must be improved upon. A typical examination for a general musculoskeletal injury begins with the examiner interviewing the patient to collect information, such as, cause of injury, abnormal pain, and other symptoms. Next, a medical doctor administers a physical examination to test the range of motion and function of the joint. If further action is needed, the doctor asks the patient to remain still for a collection of static radiographic or magnetic resonance images to be captured. The problem surfaces when, after these examinations, there is still not enough information and the results are inconclusive. In this case, observation of the patient while performing a dynamic activity, one which requires function of the ailing joint, could provide this missing information. It is logical to investigate the in vivo kinematics of the dysfunctional joint, given the dynamic nature of the musculoskeletal system, yet, this is not common practice. It would be appealing to confirm the results of a past examination or to test a new hypothesis with measurements of skeletal motion. However, there is no imaging modality currently in place to accomplish this goal. The next best measurement tool to estimate skeletal motion is a motion capture system. Yet, even with the measurement of skinmounted motion capture markers one cannot ascertain skeletal motion without artifacts from layers of soft tissue and muscle. The solution is a dynamic radiographic imaging platform, which is designed to have the x ray hardware mounted on two robotics arms. Mounting the radiographic equipment on fast, accurate, and lightweight robotic arms is a novel idea. Yet, before achieving this goal, there await challenges and obstacles that are largely unresolved. Such a system would require accurate tracking performance to meet imaging constraints and acquire useful and meaningful images. It would also require accurate imaging performance to establish an accurate measurement of skeletal motion. The following dissertation discusses the challenges in designing such a system, and respective procedures or methods to overcome them. The subsequent pages in this chapter, will introduce the challenges behind parameter identification, controller development and measurement analysis, in that order. Figure 11. Dynamic radiographic imaging system Parameter Identification: Building a Physical Model An accurate physical model of the robotic imaging system will improve both measurement and tracking capabilities. A dynamic model can serve as a backbone for understanding system performance. The accuracy of imageregistration will depend upon the accuracy of endeffector motion for both robots. The endeffector motion can be estimated with a modelbased extended Kalman filter, but knowledge of geometric and dynamic properties are needed beforehand. A dynamic model can also be used in the development of modelbased control algorithms in order to achieve superior tracking performance. The model can be used as a definite truth or an initial guess in many modelbased control algorithms. Increased knowledge of system dynamics will lead to better control and tracking performance. However, errors in the system model will degrade system performance, and therefore an accurate and reliable identification method is essential. For these reasons, identification of dynamic parameters, such as inertia, mass, friction, and flexibility, is critical for both improved imaging and tracking performance. The theoretical development and experimental results of the static and dynamic identification processes are explained in Chapter 2, "Parameter Identification". Controller Development: Designing an Intelligent System A robotic imaging system requires superior tracking ability, in order for the xray source and flatpanel detector to be properly aligned as a patient exercises his or her joint. This requirement provides a rationale for development and evaluation of a high performance controller for this application. The Mitsubishi PA106CE manipulator possesses the typical nonlinear behavior seen in robotic systems having revolute geometry. In addition, the harmonic drive transmissions, which provide a lightweight alternative to the geared transmissions, have undesirable intrinsic properties, such as nonlinear flexibility, frictional losses, kinematic error, and resonant vibration. Adding to the problem, joint resolvers only record the input angle of rotation, or motor position, and not the true output angle of rotation, or link position. The true position is deflected on account of the transmission flexibility. The sum of these problems results in a highly complex and nonlinear system and a challenging control problem. The methods classified as linear control are attractive because they exhibit advantages such as global exponential stability and guaranteed transient performance. A linear system embodies familiar properties which facilitate the design process and instill comfort. The stability of a linear system is determined from the roots of the characteristic equation or the eigenvalues of the state space matrix. The response of a linear system to a sinusoidal input is also a sinusoidal. And most importantly, the transient response of a linear system can be determined analytically. These properties allow for one to design controllers with rootlocus and frequencydomain methods. However, nonlinear systems do not possess these desirable characteristics. They can have more than one equilibrium point and exhibit behaviors such as limit cycles and chaos. Often, the time response of a nonlinear system won't have an analytic solution. The linear control methods described may lead to loss of controllability, destabilization, and undesirable transient performance in a nonlinear system. The nonlinear nature of the system encourages a design based on Lyapunov stability analysis. These methods have had success with highly nonlinear systems. The addition of an observer for nonlinear state estimation fits well into the design process. The text within Chapter 3, "Nonlinear Control and Estimation", highlights the design of a rigidlink flexiblejoint controller with link position observer. Measurement Analysis: Assessing the Imaging Performance The ultimate goal of the proposed radiographic imaging system is the measurement of skeletal motion. It is clear that the success of the project depends greatly upon the accuracy and precision of this measurement. As a result, performance of the imaging system in simulation and in a realtime tracking experiment is evaluated and discussed. The principal challenge in estimating motion from radiographic images stems from the dynamic nature of the imaging system. Since the xray source and flatpanel detector are moving relative to the patient, and each other, it is likely that the recorded images will contain motion blur. Fortunately, past research has shown that one can improve contrast in an image when the relative motion between objects is known. However, uncertainties in their motion estimates will lead to left over or residual motion blur. Furthermore, motion blur will most likely degrade the quality of the imageregistration algorithm, but the extent of this damage is unknown. It may be possible to improve registration accuracy though motionbased deblurring methods, known as image deconvolution. In Chapter 4, "Measurement Analysis", an estimate for residual motion blur in the system is determined, based on the uncertainties in endeffector and tracking object velocities. In addition, the performance of image deconvolution given the estimated relative motion is evaluated through an experiment and compared to results from simulation. As a last step, the effects of motion blurred and blurred/corrected images on modelbased imageregistration are determined. CHAPTER 2 PARAMETER IDENTIFICATION Introduction The imaging performance of a dynamic radiographic imaging system will be dependent upon accurate endeffector motion estimation. Joint resolver measurements are highly precise and lead to excellent positional repeatability for consistent loading conditions. However, high repeatability does not guarantee accurate positioning performance. Accurate knowledge of link geometries and joint flexibilities are necessary for error compensation. It is essential to have an effective calibration procedure to determine the kinematic and dynamic parameters of an assembled robot, since these parameters are specific to a given manipulator. The Mitsubishi PA106CE is a six degree of freedom manipulator equipped with AC servo motors and harmonic drive transmissions. The large gear ratio and compact design of the harmonic drive makes it ideal for many robotic applications. However, these transmissions are known to exhibit nonlinear flexibility and frictional losses, kinematic error, and resonant vibration, making it difficult to estimate the true endeffector position from joint position alone [6], [7]. In addition, geometric modeling errors, common to all robotic manipulators, further increase endeffector positioning error. Link lengths, twist angles, and joint offsets reported by the manufacturer are only accurate to manufacturing and assembly tolerances. Small errors in these DenavitHartenberg (DH) parameters can propagate to create significant errors in end effector position [8]. A flexible robot model and an effective calibration procedure are required to characterize geometric uncertainties and joint flexibilities. The tracking performance of a dynamic radiographic imaging system will be dependent upon accurate model identification. Precise and reliable identification methods are well documented for rigidlink rigidjoint (RLRJ) robots. However, the Mitsubishi PA106CE is assembled with harmonic drive transmissions and is known to deflect under load. Identification must account for these effects in order to develop an accurate physical model. The methods for robot manipulators with flexible joints are still in development. The complexity of the identification process is reduced by separating the procedure into a static and dynamic identification tasks. Geometric and Flexibility Calibration This section discusses the design and execution of a twolevel nonlinear optimization algorithm for calibration of geometric and flexibility parameters in a serial manipulator, using measurements from a coordinatemeasuring machine (CMM). This method offers greater simplicity than [9] because it does not require computation of the generalized Jacobian matrix. In addition, a Monte Carlo simulation of positioning error was performed based solely on joint resolver uncertainties, establishing the absolute bestcase robot positioning accuracy. The fidelity of the model was assessed from an experimental comparison of robot positioning accuracy, for a novel set of configurations and endeffector loads, before and after the calibration procedure. The generalization of this method, and its extension to other serial manipulators, is also discussed. Literature Review Many researchers have implemented leastsquares or nonlinear optimization techniques to calibrate robotic manipulators. Taghirad et al. developed a model of harmonic drive stiffness through a controlled loading experiment [10]. They defined joint flexibility from the relationship between joint torque and joint deflection measurements. This method could not be implemented on the Mitsubishi PA106CE because there is no direct measurement of joint deflection. Kennedy and Desai modeled the stiffness component of joint torque on a Mitsubishi PA106CE as a function of the applied load [11]. However, their research did not establish the physical model between the applied load and joint deflection. The inclusion of both geometric and nongeometric errors appears in [12] and [13], but these authors used a set of generalized error functions and did not identify physically meaningful parameters. A physical relationship is desirable for 1) extrapolation of points outside the calibrated workspace, and 2) a kinematic and dynamic model used in modelbased control algorithms. Khalil et al. recently developed a nonlinear leastsquares method to calculate physically meaningful geometric and flexibility parameters [9], yet the results from this paper provided only a best case maximum global error of 14.1 mm. An overview of methods for robot calibration is discussed in [14]. Methods The computational resources needed for the solution of an optimization problem greatly increase with the addition of more design variables [15]. Hence, a multilevel optimization method is employed to reduce the number of design variables in a single optimization and increase the speed of convergence (Fig. 21). First, a flexible geometric model is proposed by an outerlevel optimization. Next, an innerlevel optimization determines the bestfit homogeneous transformation between the CMM and robot coordinate systems. This innerlevel optimization promptly converges to a solution given a fixed set of DH parameters and flexibility coefficients, making it unnecessary to burden the parent optimization algorithm with the homogeneous transformation. Next, using this model, the deflected position and orientation are computed from measured joint torques (via motor current) and compared to their respective CMM measurement. The final model is realized upon numerical convergence of this twolevel optimization. YES ____ 5. Final Solution Figure 21. Solution sequence for geometric and flexibility calibration Experimental setup and procedure A Mitsubishi PA106CE robot was positioned on a 0.6 m steel frame table so that its end effector could span the entire workspace of a Brown and Sharpe, MicroVal PFX454 touch trigger probe CMM (Fig. 22). The CMM operates by probing the surface of a spherical tooling ball and calculating the leastsquares estimate of the centroid of the ball. A coordinate system was defined from the location of three tooling balls in the CMM workspace [16]. It had a calibrated measurement uncertainty of 12.1 [tm over its full work volume. To facilitate CMM measurement, a toolingball apparatus was constructed and mounted to the endeffector of the robot (Fig. 23). It was comprised of two parallel aluminum plates, separated by four 31 cm aluminum rods; the mass was 2.4 kg. The extension of tooling balls into the CMM workspace reduced the chance of impingement between the manipulator and the CMM table. The backplate was attached to the endeffector of the robot, and the frontplate contained a set of three tooling balls. The transformation between the CMM coordinate system (CMM) and I auserdefined tooling ball coordinate system (BALL), cT was determined from measurements of the location of the tooling balls. The robot was commanded to move along a Cartesian pattern of uniformlydistributed points in the CMM working volume (Fig. 4). At each static configuration, the CMM measured the position and orientation of the tooling ball coordinate system. The experiment was repeated 10 times for each of three loading scenarios: {0, 22, and 44} N loads. External loads were provided by additional weights lowered onto the weight rack and secured with a set screw and shaft collar. The external load was transmitted through the endeffector of the robot and therefore did not create deflection in the rods (Fig. 23). Maximum deflection of the apparatus from its own weight was 0.030 mm. The approximate touch force of the CMM was 0.2 N, creating a maximum deflection of less than 0.001 mm from contact. These deflections were negligible compared to the robot positioning error from uncertain geometric parameters and joint compliance. Figure 22. Experimental setup of the PA106CE and CMM \\ eiuht rack , Tooling balls measured via CMM a' 31cm Figure 23. Tooling ball apparatus  Geometric and Flexibility Calibration  Experimental Assessment 26 e '  14 1000 800 0 Y(mm) 600 100 X (mm) Figure 24. Experimental test path Bolted to end effector *25 * 12 *.8 *24 * 13 o .  19 *18 700 600 500 400 200 100 Twolevel geometric and flexibility calibration A flexible geometric model was defined from a set of 30 design variables (Table 21). It consisted of departures in link lengths, twist angles, joint offsets, and joint angles from their nominal values, represented by 6a,, 6a,, Id,, 6,, and 6,, respectively [16], and cubic spring coefficients for joints 2, 3, and 5, represented by K,,n., and Kcub,, (joint 1 connects the base of the robot to the first link). The flexibility in the harmonic drive transmission was modeled with the torsional spring model S= KI (q, q)+ Kcb (qm q)3, (2.1) where q(t) e 9V" and q, (t) e 91" denote the link and motor position, respectively, Ki,, and Kcub e 9n" are constant, diagonal, positivedefinite matrices comprised of the linear and cubic joint stiffness coefficients, respectively, and r(t) 9~" denotes the joint torque. The departure in the Hayati parameter [17], labeled as 6&2, was substituted for d2 to eliminate the illconditioned transformation that exists between two parallel joints (i.e. joints 2 and 3). Nominal geometric parameters for the Mitsubishi PA106CE geometry were acquired from the manufacturer [18] and are represented in Table 21. Joints 1, 4, and 6 were subject to negligible torques in the tested configurations, and as a result, stiffness coefficients were not estimated for them. Link flexibility was not taken into account in this model based on previous research with the Mitsubishi PA106CE [9]. A twolevel, LevenbergMarquardt nonlinear leastsquares optimization was implemented in MATLAB (Mathworks, Natick, MA) to solve for model parameters. The outerlevel optimization sought to find a flexible geometric model, consisting of 24 link parameters and 6 stiffness coefficients that satisfied X = arg min fot,,, (2.2) x (x, fouter 2 m (. rmax =1 1=1 where 2 2 2 >2 2 ,tol ,tol dto [,,,o} 8,to and = [a, 5 d, S/, 50, Ki,. Kcu, ], where x,, x., m and ,m represent the estimated and measured endeffector positions and orientations (parameterized as angular rotations), respectively, in the robot's coordinate system, and N= 27 is the number of robot configurations. The estimated location of the tool frame is a function of geometric and flexibility model contained in X. Position errors were normalized with respect to the total length of the manipulator, rax = 1 m, therefore, an orientation error of one radian corresponded to a distance error of one unit at full extension. The remaining terms in the cost function, which represent errors in estimated DH parameters, were normalized with respect to the approximate tolerances, a,to = 0.25 mm, a,,tl = 1x103 rad, d,,o = 0.25 mm, f,,o = x103 rad, and ,,tol = 5x104 rad, and then scaled with the weighting factor = 1.6x104. Large E increased the cost associated with DH parameter error, and thus forced the optimization algorithm to account for location errors without modifications to the geometric model. Instead, the optimization method adjusted torsional spring coefficients to minimize the given cost function. The values for ,, m and ,, were determined from the position and orientation of the end effector in the robot coordinate system, as measured by the CMM, BASKET BASKET Cca T BALLOT (2.3) EE,M CM1v BALL EE,M where 7"T is the transformation between CMM and tooling ball coordinate systems, BTLT is the known constant transformation between endeffector and tooling ball coordinate systems, and BAET is the transformation between robot and CMM coordinate systems. The transformation BAl'T was calculated from the CMM measurement of the tooling balls, BALT was determined from a separate measurement of the toolingball apparatus, and BAET was computed from the innerlevel optimization. For each iteration in the outerlevel optimization, defined in (2.2), an innerlevel optimization was performed using the updated flexible geometric model. The second optimization algorithm searched for the six robot coordinate system parameters that satisfied Y = arg min fnner (2.4) 1 M where f nner = Z ( x + ( Y, max ]=1 ]=1 and [ = basec 7. 7z1 where base pcm o is the origin of the CMM coordinate system in the robot coordinate system, x , Y, and y7 are a set of bodyfixed angular rotations describing the orientation of the CMM coordinate system with respect to the robot coordinate system, and M= 8 is the number of robot configurations. The measured location of the tool frame, obtained from (2.3), is a function of robot coordinate system parameters contained in Y. The subscriptj has been introduced to differentiate from configurations in the outerlevel subscriptt i). The total kinematic model has 30 parameters, consisting of 24 link parameters and six robot coordinate system parameters, and is minimum, complete, and parametrically continuous (MCPC) [19]. Link parameters were determined through the outer optimization loop, while robot coordinate system parameters were determined in the inner optimization loop. The optimization was repeated with different initial guesses to mitigate the problem of finding local minima. The initial seed for the innerlevel optimization, provided in (2.4), was estimated from endeffector measurements along a straight line. The robot followed a linear path along its x and yaxes allowing for leastsquares bestfit approximations to BASE and BASEThe zaxi and reformulated yaxis were determined from CMM _CVM CAMlv C BASE XBASEX BASE (2.5) and y = ZBXxABASE Z X E. (2.6) The last step was taken to ensure orthogonality between all measured axes. The initial seed of Euler angle rotations was determined from the rotation matrix c"R [16]. cM V R C] C] C] I  ( 2.7) BASE = [R BASE Y BASE BASE] 2 Monte Carlo simulation Joint resolver uncertainties cannot be reduced through calibration, and therefore, they determine the lower bound for endeffector positioning uncertainties. A Monte Carlo simulation using the six joint positions as simulation variables was implemented to determine the lower bound for robot position errors. For the PA106CE, the manufacturer reported a +0.64 arc min (0.011 deg) error in the angular position measurement of each link [20]. The probability distribution of each uncertain joint position was assumed to be Gaussian with a standard deviation equal to the reported bounds of that parameter. A forward kinematic analysis was performed for 100,000 cycles at each of the 200 uniformlyselected endeffector positions in a given plane. The experiment was repeated for each crosssectional plane located between z = 400 mm to z = 700 mm in 100 mm increments. Experimental assessment The fidelity of the flexible geometric model was examined with a collection of 10 points not included in the data used for model calibration (Fig. 24). The new set of data included configurations as far as 100 mm from the original control volume and loads up to 90 N, twice the original maximum load. The configuration order and loading order were selected arbitrarily. A comparison was made between the full model and a linear stiffness model (Kcub = 0). Results The Monte Carlo simulation determined a mean endpoint position error of 0.197 mm + 0.096 mm in the xyplane of z = 500 mm (Fig. 25). Results from parallel planes at z = 300 mm and z = 700 mm deviated by less than 0.001 mm. The mean/peak values of position error prior to geometric and flexibility calibration were 0.58/1.22 mm, 1.02/1.50 mm, and 1.80/2.45 mm for the unloaded, 22 N loaded, and 44 N loaded cases, respectively (Fig. 26). The standard deviations of position error, within one configuration, did not exceed 0.014 mm, 0.024 mm, and 0.023 mm, for each loading case, respectively. After optimization of robot geometry and joint stiffnesses, the mean/peak values of position error decreased to 0.32/0.66 mm, 0.28/0.58 mm, and 0.33/0.71 mm, respectively. Mean orientation error, defined as the mean L2 norm of the differences in Euler angle rotations (xyz) between measurements, was reduced from 1.2 deg, 1.2 deg, and 1.1 deg to 0.05 deg, 0.04 deg, and 0.03 deg, in each of the three loading cases, respectively. Linear joint stiffnesses identified by the twolevel optimization were 1.98x104 Nm/rad, 1.27x104 Nm/rad, and 2.56x103 Nm/rad for joints 2, 3, and 5, respectively. The cubic joint stiffnesses were 3.35x107 Nm/rad3, 1.46x107 Nm/rad3, and 3.83x107 Nm/rad3, respectively. Mean positioning errors from uncalibrated configurations decreased from 1.77 mm to 0.55 mm with the linear stiffness model, and to 0.56 mm with the nonlinear stiffness model (Fig. 27). The peak value of position error decreased from 4 mm in the uncalibrated model to 0.92 mm and 1.04 mm with linear and nonlinear stiffness models, respectively. There was at least a 75% reduction in peak positioning error with both models. Mean orientation errors decreased from 2.4 deg to 1.2 deg with both models. 1000 XYPlane of Z=500 mm 900 Position Error in Millimeters S800 0 700 600 .7 500 400 0 100 200 300 400 500 600 700 YCoordinate of Robot (mm) Figure 25. Contour plot of mean position error Table 21. DenavitHartenberg parameters and joint stiffness coefficients Joint No. 6ai (mm) ati (rad) 6di (mm) 68i (rad) 6fi (rad) Kl(Nm/r KcubN3) (Nm/rad) (Nm/rad3) 1 0.33 4.9x104 0.09 5.9x103 n/a oo oo 2 0.46 1.2x104 n/a 7.8x103 1.0x103 1.98x104 3.35x108 3 1.09 2.3x103 1.26 2.9x103 n/a 1.27x104 1.46x108 4 0.48 2.1x104 0.22 8.1x103 n/a oo oo 5 0.30 3.7x105 0.10 1.5x103 n/a 2.56x103 3.84x108 6 3.44 7.1x104 0.17 1.3x102 n/a oo oo k 3 S2 1 0 0 o P0 0 P_ 0\  Without Model  With Linear Stiffness Model "......" With Nonlinear Stiffness Model E4 E w3 C o 2 0 1 \ 0 1 2 3 4 5 6 Position Number 7 8 9 10 Figure 27. Endeffector position error at novel configurations and loads 5 10 15 20 25 Position Number (a) Before Calibration Unloaded .. 22 N Loaded  44 N Loaded 5 10 15 20 25 Position Number (b) After Calibration Figure 26. Mean position error before and after calibration ,z , ~2. Z.r4 h Conclusions The 30 parameter flexible geometric model has been demonstrated to improve static positioning accuracy for the PA106CE robot. Endeffector positioning errors decreased substantially (Fig. 26), showing 50% to 80% reductions in mean position errors across all robot configurations. Measured positioning errors (0.3 mm) differed by less than 0.1 mm from their theoretical minimum determined from Monte Carlo simulation. DH parameters were all within reasonable manufacturing and assembly tolerances for the PA106CE. The large value for 6a6 is most likely from error in measuring the transformation between endeffector and tooling ball coordinate systems, and in assembling the tooling ball apparatus on the endeffector. The superior performance of the model while extrapolating outside the calibrated configurations is a good indication of its strength. The linear stiffness model performed surprisingly well in the experimental assessment (Fig. 26). Nonlinearities that exist in harmonic drive transmissions are most prominent at small operating loads [6], and the joint torques in this experiment, for the most part, fall outside of this region. It may be worth considering a linear stiffness model given its simplicity and similar performance. The method described in this section can be generalized to any serial manipulator represented by DH parameters. The stiffness parameters returned from the optimization of a perfectly rigidjoint manipulator would approach infinity. Accuracy standards for medical robot systems depend upon the application. Radiation therapy is one of the most demanding applications and some believe positioning errors greater than 0.5 mm increase patient risk [21]. Robotic applications in orthopaedic surgery typically seek to achieve positioning errors less than 1 mm or 1 degree for bone cutting [22]. It appears these demanding requirements can be met by the PA106CE utilizing the geometric and flexibility calibration method described here. The combination of a detailed physical model and its accompanying control design will provide an accurate measurement and control system suitable for demanding dynamic applications with the PA106CE robot arm. The identification of mass, inertial, and frictional properties are necessary to complete the dynamic model of the PA106CE, and are discussed in the next section. Identification of Dynamic Characteristics This section discusses a novel method for dynamic parameter identification of a rigidlink flexible joint robot model. The procedure is divided into two parts. First, the rigidlink flexible joint robot dynamics are parameterized in a new form to exclude the stiffness torque. Unknown parameters are estimated from a rich set of motion data. Link position is estimated with groups of motion capture markers on the surface of the robot. Second, the stiffness torque is reconstructed from the identified parameters, and a torsional spring model is deduced from the data. This method offers a better conditioned system of equations for weaklyflexible systems, and more flexibility in the choice of torsional spring model. The twostep identification method is experimentally evaluated on a Mitsubishi PA106CE. Literature Review Typically, parameter identification of robot manipulators is performed in the frequency domain or timedomain. Frequencydomain identification methods reduce the robot dynamics to a pair of second order linear differential equations under the small angle assumption. Parameters are identified through nonlinear optimization with a cost function defined as the sum of the squared differences between predicted and measured outputs of the system at different frequencies [23]. Design variables can be chosen as the polynomial coefficients which define the system transfer function or the physical parameters of the robot. Li et al. reported a frequencydomain technique that combined the finite element method with experimental modal analysis to identify stiffness and damping coefficients [24]. Link dynamics were excited with mechanical impulses and the responses were compared to an analytic model to determine robot parameters. The objective function was defined as the sum of the squared error between the natural frequencies obtained from the analytical model and those obtained from the experiment. Frequencydomain methods provide good estimates for parameters in a linear model, but fail to identify nonlinearities, such as mass center and Coulomb friction parameters. Alternatively, the dynamic model of the robot can be obtained through direct experimentation. Taghirad et al. developed a model of harmonic drive stiffness through a controlled loading experiment of a onelink manipulator [10]. AlbuSchaffer et al. proposed the identification of joint flexibilities prior to assembly [25]. They derived a linearized form of the flexiblejoint dynamics under the small angle assumption and satisfied this condition by fixing the motor position with an electromagnetic brake. They excited the link dynamics with mechanical impulses and measured the response of the system. Nonlinear optimization was used to determine flexibility and damping parameters from the impulse response. Identification in the timedomain is most often performed with linear leastsquares methods. A solution is achieved through linear parameterization of the robot dynamics into a regressor matrix, containing functions of the known joint displacement, and an array of unknown parameters [26]. An overdetermined system of equations is formed with sampled data and solved with leastsquares techniques. Desired trajectories for identification can be determined by selecting appropriate optimization criteria. Researchers often minimize the condition number of the regressor matrix, or maximize its minimum singular value, thus improving the robustness of the numerical solution to measurement noise [27]. Calafiore et al. minimized the covariance matrix of parameter estimates by maximizing the Fisher information matrix [28]. The design variables in both cases are the Fourier coefficients or polynomial coefficients defining the proposed robot traj ectory. Pham et al. proposed a linear parameterization of the flexiblejoint dynamics and explained two methods for parameter identification [29]. The first method requires augmenting the sensing capabilities of the robot with additional joint encoders. The latter identification method compensates for unmeasurable link position, but the link masses and Coulomb friction parameters are lost. Pham et al. did not discuss trajectory optimization for improved parameter estimation. RigidLink FlexibleJoint Model The nlink rigidlink flexiblejoint (RLFJ) robot dynamics can be expressed as M(q)q + Vm (q, q)q + G(q) + F(q) + K(q q,) = 0, (2.8) Jq, + Bq4 + K(q, q) = u, (2.9) where q(t), q(t), q(t) e 91" and q, (t), 4, (t), q (t) e 9" denote the position, velocity, and acceleration of the link and motor angle, respectively, M(q) e 9~" represents the positive definite inertia matrix, V (q, 4) e 9 "" represents the centripetalCoriolis matrix, G(q) e 9"and F(q) e 9" denote the gravitational and frictional effects of the link dynamics, respectively, K,J,B e 9j" are constant, diagonal, positivedefinite matrices representing joint stiffness, motor inertia, and motor viscous friction, respectively, and u(t) 9~" denotes the motor torque [30]. Accordingly, the onelink RLFJ robot dynamics can be written as I q mglx sin(q)+ mgly cos(q) + c sign(q) + v + k(q q) = 0, (2.10) J,,, + b,,, +k(q,,, q) = u, (2.11) where q(t), (t), q(t) e 91 and q4(t), (t), ~, (t) e 91 denote the position, velocity, and acceleration of the link and motor angles, respectively, and I, m, c, v, k, J, b 9 are the unknown parameters representing link inertia, link mass, Coulomb and viscous friction, joint stiffness, motor inertia, and motor viscous friction, respectively. Leastsquares parameter identification The onelink RLFJ robot dynamics, given by (2.10) and (2.11), can be linear parameterized as the following expression: I mgl mgl, q sin(q) cos(q) sign(q) 4 q q,, 0 (2.12)c 0 0 0 0 0 q q q4, v Lu] k J b To identify elements in the regressor matrix, motor positions, link positions, and motor torques are recorded as the link follows a desired trajectory. The unknown model parameters are then determined by forming an overdetermined system of equations from the measured data. The problem can be simplified into the following expression and solved with leastsquares methods. Y(q, q,~, q,,q,,, q)0 = x, (2.13) where 0= (Y Y) Yx. The moment of inertia obtained from a singlejoint experiment is a composite of all links between the endeffector and the moving joint. Individual link inertias can be obtained in a sequential manner using the parallel axis theorem and group inertia properties. The formulation of the system of equations in (2.12) is problematic for a robot with weak joint elasticity. In this scenario, the joint deflections are small compared to the link and motor kinematics, resulting in a poorlyscaled and illconditioned matrix. The condition number can be reduced by rewriting the onelink dynamics to eliminate the stiffness term as follows: Iq mgl, sin(q) + mgly cos(q) + c sign(q) + viq + Jq, + bq, =u, (2.14) The dynamics are then parameterized as I mglx mgl, [q sin(q) cos(q) sign(q) q q iim m] c =U. (2.15) v J b The formulation in (2.15) is better posed for a robot with weak joint elasticity. The omission of joint deflection from the regressor matrix eliminates the scaling problem. Joint stiffnesses are determined subsequent to the identification of inertial and frictional model parameters. The stiffness torque, or torque transmitted through the torsional spring, is determined from the following expression. r, = Iq mgl sin(q) + mgl cos(q) + csign(q) + vq (2.16) = K(q, q) A suitable stiffness model can be selected by examining the relationship between joint deflection and stiffness torque obtained in (2.16). Trajectory optimization In the dynamic identification of RLRJ manipulators, the desired link trajectory is carefully selected to sufficiently excite all dynamics. The optimization criteria are commonly chosen as the condition number of the regressor matrix or the determinant of the Fisher information matrix. The case of RLFJ robot dynamics is no different. An optimal excitation trajectory can be determined from the constrained nonlinear optimization: mincond(Y(q, 4q, q, q, jm)), (2.17) s.t. qm < qm(t) < qmax l max< L(t) < qmax where the regressor matrix is now a function of both the motor position and link position. The excitation trajectory, or motor position, can be parameterized as a finite Fourier series [27]. q,(t)= q ,+ a" si(noft) b cos(nw(t) (2.18) n1 no>f nOf The design variables are the amplitudes of the fundamental frequency and N 1 harmonics. The fundamental frequency should be chosen near the natural frequency of the system if the motor command cycle and joint position sampling frequencies will permit. In the case of RLFJ robot dynamics, the regressor matrix is a function of both motor position and link position. However, link position cannot be parameterized independent of motor position because it is a function of the robot dynamics. It should be determined with an approximation of the robot model and forward dynamics. The link acceleration can be estimated from an approximation of the robot dynamics, S= I (mglx sin(q) mglycos(q) csign() vq k(q q)), (2.19) and, given the initial state of the robot, can be integrated to obtain link positions and velocities. An exciting trajectory for link 2 of the Mitsubishi PA106CE robot is shown in Fig. 28. 0.4 0.3 0 0 S0.2 0.1 0 0 1 2 3 4 5 Time (seconds) Figure 28. Sample identification trajectory Estimation of Link Position Typical robotic systems are assembled with one joint resolver or encoder for each link, and therefore have insufficient sensing hardware for a RLFJ model. To overcome this limitation, motion capture markers are fixed to the external surface of the robot to provide an estimate of the link position (Fig. 29). The axis of rotation and respective center of rotation for each link can be determined from a sufficiently rich set of marker data. The analytic solution for an axis of rotation obtained in [31] minimizes the cost function P N pE l [(k ; iP)] (2.20) p=\ k=\ where ni is a unit vector in the direction of the axis of rotation, fP is the observed position of markerp in frame k, and iin is the center of rotation of markerp. An unbiased leastsquares solution for a point on the axis of rotation is obtained in [32] given the cost function E = I 9 2 I (r 2, (2.21) p=1 k=1 where in is a point on the axis of rotation, and rP is the radius of the circle mapped out by markerp. After the position of each marker relative to its neighboring axis of rotation is known, link position is determined through nonlinear optimization [33]. The cost function is defined as (2.22) E = min P k( , q p=l where vp is the measured position of markerp in frame k, and vp (q) is the predicted marker position given an angle of rotation q. The initial guess for the first frame is selected as the initial motor position. Subsequent frames are seeded with the preceding link position. Figure 29. Motion capture markers on PA106CE Figure 210. Endeffector weight rack Experimental Setup and Procedure A Mitsubishi PA106CE robot was mounted on a steel frame table with a height of 0.6 m (Fig. 29). The table was attached to the floor with a set of four bolts and internallythreaded concrete anchors. The robot was positioned in the center of the viewing volume of an eight camera passivemarker motion camera system (Motion Analysis Corp., Santa Rosa, CA). The residual measurement error obtained through calibration was on the order of 0.3 mm, with a standard deviation of 0.1 mm. Clusters of three or four reflective markers were positioned on the exterior surface of four links. Realtime measurement and control of the robot and motion capture system was achieved with a Linux kernel patched with RTAILXRT, and the OROCOS and ORCA software libraries [34], [35]. An endeffector apparatus was designed and constructed to support additional load on the robot endeffector (Fig. 210). A 44.5 N load was attached to the device to excite robot inertial and stiffness dynamics. A description of each link coordinate system is included in Fig. 211. The robot was commanded to follow, for each joint, the singlejoint trajectory obtained from minimization of the regressor matrix condition number. The remaining links were fixed with electromagnetic brakes. Motor position was recorded with joint resolvers, with a reported 10 arc min electric error and 22 arc min R/D (ResolvertoDigital) converter error [20], leading to an overall accuracy of 0.64 arc min (0.011 deg) for the angular position of the motor (gear ratio = 50). Motor torque was determined from the input current and the motor constant. Link position was estimated from the measurement of motion capture markers on the surface of the robot. The sampling frequency of the motion capture system was 250 Hz. Leastsquares and nonlinear optimization algorithms were performed in MATLAB (Mathworks, Natick, MA). The experiment was repeated 1Ox for each link. The coefficient of variation (CV) of an estimated parameter, a measure of its fidelity, was defined as the ratio of the standard deviation to the sample mean of the population. The sample mean of each parameter and its CV were recorded for each experiment. Linear stiffness coefficients were determined for each experiment after the identification of inertial, mass center, and frictional parameters. The r2 correlation coefficients were also determined for each experiment. Experimental Results The rootmeansquare (RMS) error in the estimation of marker radius r was not greater than 0.3 mm for all measured trials. The RMS error in reconstructed marker position from joint angle optimization was between 0.5 and 1.1 mm for all measured trials. Model parameters for the link and motor dynamics, and their coefficients of variation, are reported in Table 22 and Table 23. Coulomb friction torques were equal to 9.08, 9.28, 5.81, 1.48, 1.84, and 0.87 Nm forjoints 1 through 6, respectively. Mean RMS errors between measured and reconstructed torques equaled 2.48, 19.33, 6.55, 0.56, 1.37, and 0.33 Nm, for joints 1 though 6, respectively. Measured and predicted torques for joint 2 are shown in Fig. 2 11. The joint stiffnesses were determined from linear regressions of the joint deflection and corresponding reconstructed stiffness torque (Fig. 212). The estimated parameters ranged from 4.78x102 in joint 6 to 1.19x104 in joint 2 (Table 22). The r2 correlation coefficients for joints 1 through 6 were equal to 0.66, 0.64, 0.54, 0.84, 0.71, and 0.55. Table 22. Statistic Mean CV (%) Mean CV (%) Mean CV (%) Mean CV (%) Mean CV (%) Mean CV (%) Parameter identification results for link dynamics I (kg m2) c (Nm) v (Nm s) mglx (Nm) 1.13 0.64 6.15 9.62 2.27 0.50 0.03 0.68 0.16 0.68 0.08 0.43 9.08 1.75 9.28 11.07 5.81 0.67 1.48 6.50 1.84 4.96 0.87 0.95 9.88 1.87 18.38 6.82 5.79 0.87 1.45 7.02 1.75 5.13 0.92 0.70 1.52 1.78 111.64 4.47 8.33 0.93 < 0.01 n/a* 0.26 1.34 0.10 6.12 * not applicable for parameter estimates which equal zero. Joint mgly (Nm) 2.08 3.59 7.80 16.89 8.56 0.64 0.18 33.81 0.63 8.01 < 0.01 n/a* Joint 1 2 Table 23. Parameter identification results for motor dynamics Statistic b (Nm s) J (kg m2) k (Nm/rad) Mean 10.05 1.06 5.98 x 103 CV (%) 1.75 0.64 3.86 Mean 23.28 12.57 1.19x 104 CV (%) 6.15 5.33 4.74 Mean 6.35 2.48 4.60 x 103 CV (%) 0.58 0.18 2.90 Mean 1.46 0.03 7.09 x 102 CV (%) 7.33 3.72 5.05 Mean 1.78 0.16 7.72 x 102 CV (%) 5.39 0.44 5.17 Mean 0.96 0.07 4.78 x 102 CV (%) 0.83 0.36 0.74 error (Nm) 2.48 2.38 19.33 3.72 6.55 1.12 0.56 8.83 1.37 3.76 0.33 1.21 Z4, Xs  Y4, Zs \4, Y5 Z3 Y3 X2 SY, Z2 Figure 211. Coordinate system definitions I ~~T i 1 I u rl ~~I" S Measured Torque  Model Torque 400 300 200 100 0 100 200 300 400 0 150 100 50 0 50 100 150 200 0.015 0.01 0.005 0 0.005 Joint Deflection (rad) Figure 213. Stiffness torque and joint deflection 0.01 2 4 6 8 Time (seconds) Figure 212. Measured and predicted motor torque STorqueDeflection Data  Linear Approximation I ." rtf. Conclusions A rigidlink flexiblejoint model accurately represents the dynamic behavior of the Mitsubishi PA106CE. The largest RMS residuals occurred in the identification of joint 2, corresponding to less than 8% of the amplitude of motor torque. The remaining joints all have residuals proportional to their motor torque. CV percentages for all friction and inertial parameter estimates were below 12%. Mass center estimates had slightly higher CV percentages because of smaller magnitudes. The resulting joint stiffnesses are similar in magnitude to estimates obtained through static calibration [36]. Principal sources of error in the identification process are measurement and synchronization errors in the joint angle, input torque, and motion capture measurements. The remaining differences between the predicted and actual motor torque result from unmodeled dynamics. An example is positiondependent friction arising from manufacturing errors and misalignment of the harmonic drive components during assembly [6]. In addition, there was a disturbance torque acting on the actuated link from adjacent links. An electromagnetic brake will fix motor position, but the harmonic drive may flex due to adjacent link motion. A better identification might be possible with a complete nlink identification method. Furthermore, errors in the stiffness estimate will arise from nonlinearities in the harmonic drive transmission, such as soft windup and hysteresis which occur in regions of low applied torque [7]. The advantage of separating the identification process into two steps is clear. The condition number of the reformulated regressor matrix is smaller, leading to improved robustness to noise in the regressor matrix and torque measurements. More importantly, one has more insight and flexibility in determining the torsional spring model from the computed stiffness torques. It presents no additional complexity to fit a cubic spring model to the measured joint deflection and stiffness torque. However, a cubic element in the regressor matrix would damage the identification process because of its much smaller size compared to other regressor elements. Furthermore, one can represent the torsional spring stiffness with non linearparameterizable phenomenon, such as hysteresis [37]. This effect is common in harmonic drives, and it is unidentifiable through traditional linear leastsquares identification. Using motion capture for link position measurement is an important element of the identification method. Several researchers, e.g. Khalil [9], have used motion capture as part of their identification strategy, but the present study is an improvement upon previous work. In the last decade, there have been significant improvements in the accuracy, robustness, realtime measurement capabilities, and sampling frequency of motion capture systems. These improvements and lower costs make motion capture systems an increasingly attractive option for identification and realtime control of robotic systems. The next chapter highlights the development of a modelbased control algorithm using this rigidlink flexiblejoint model. The principal challenge in implementing such a controller is the measurement of link position in realtime. In this case, an extended Kalman filter integrating joint resolver and motion capture measurements is used to estimate link position in realtime. CHAPTER 3 NONLINEAR CONTROL AND ESTIMATION Introduction Endeffector positioning accuracy and reliable control strategies are critical for high performance medical robot applications, such as the dynamic radiographic imaging system. However, the nonlinear flexibilities and high frictional losses found in the harmonic drive transmissions of the Mitsubishi PA106CE make this objective a challenging task. Harmonic drive systems can be modeled with joint flexibility in order to reduce modeling error and improve control, but incorporating this phenomenon in the dynamic model requires link positions and velocities, which are not frequently measured on commercial systems. A solution is to design an observer to estimate the link kinematics. In linear control systems, the controller and observer are designed separately since their error dynamics are independent. But, there is no separation of controller and observer designs in nonlinear systems. The interdependence of these two components greatly increases the complexity of the control problem. Furthermore, it is desirable to have the ability to change the endeffector tool, and as a result change the mass of the payload, and still meet performance requirements. In addition, operating conditions, such as ambient temperature and duration of operation, will influence parameters in the physical model, e.g. joint friction. The dynamic radiographic imaging system must have acceptable performance despite changes in the dynamic model. This contribution of this section is the design, implementation, and evaluation of a novel observercontroller combination for accurate estimation and control of a RLFJ manipulator. The EKF expands upon previous work by incorporating 1) flexiblejoint dynamics into the process model, and 2) positions of retroreflective markers into the measurement model. The markers can be positioned ad hoc, on any link of the robot, and with no apriori local position information needed. The state vector in this derivation contains motor and link kinematics, unknown dynamic and kinematic parameter estimates, and local marker positions. The adaptive reduced parameterization RLFJ controller contrasts previous work by utilizing EKFbased estimates of link positions/velocities and motor positions/velocities. Literature Review There exists a wealth of literature focused on the development of controllers for flexible joint manipulators, but the majority of researchers have not investigated the need for such a controller in industrial robots. Often a flexiblejoint controller is implemented on a twolink manipulator in a controller environment, where its measurement capabilities are augmented with a second joint encoder or strain gages to measure the joint deflection. This option is not often available to the enduser of an assembled robot. Although many have addressed the problem of unmeasurable states in the design of their controller, their typical choice of estimated states does not coincide with the states of the PA106CE which are unmeasurable, i.e. link positions and velocities. This section will discuss the breadth of literature devoted towards the development of adaptive rigidlink flexiblejoint robot controllers, emphasizing those with nonlinear observers to account for unmeasurable states in the system. Early research by Nicosia et. al derived the Lagrangian form of the flexiblejoint robot dynamics [38]. This is the most commonly accepted flexiblejoint model, where the motor and link dynamics are coupled through a torsional spring. The first controllers for flexiblejoint robot systems used feedback linearization methods to achieve stability [30], [39]. The drawback of this class of controllers is a requirement of link acceleration and jerk measurements, and exact modelknowledge of robot dynamics. The communities studying nonlinear control research improved upon this result with singular perturbation and integral manifold techniques for flexiblejoint control [40], [41]. Their design calls for a separation of the robot dynamics into slow and fast counterparts, where each component is accounted for separately. Yet these methods did not account for uncertainty in the model, required a weakflexibility assumption, and did not ensure global asymptotic tracking. Further research by Ghorbel and Spong [42] and Khorasani [43] extended the traditional singular perturbation method to account for parametric uncertainty in the model, hence, creating one of the first adaptive flexiblejoint controllers. A new control technique, called backstepping, changed the landscape of adaptive flexible joint control [44], [45], [46], [47], [48]. The advantages of these methods were threefold. The control engineer could now construct globally asymptotic controllers that account for parametric uncertainty, while eliminating the restrictions on the magnitude of joint flexibility, and the requirements of link acceleration and link jerk measurements [49]. The general idea is to partition the dynamics into a series of cascaded subsystems, where the desired state of one subsystem becomes the control inputs to another. The objective is to design the control input and auxiliary signals to achieve the desired tracking performance. The literature focused on the development of controllers for flexiblejoint robots is extensive, with many researchers using the backstepping design methodology. Dawson et al. designed an exact modelknowledge controller with global asymptotic link position tracking based on a rigidlink electricallydriven (RLED) manipulator [50]. Their stability result was extended to a robust controller for a RLFJ manipulator to account for uncertainties in the model dynamics in [51], [52]. Lozano et. al. investigated the case of parametric uncertainties in the dynamics of an nlink RLFJ robot model and designed an adaptive controller with global asymptotic stability [53]. Brogliato et. al. expanded on this result and compared their adaptive controller to an exactmodel knowledge controller [54]. A survey of RLFJ controllers and others is found in [49]. These designs all require complete knowledge of robot kinematics, also known as fullstate feedback. It is well understood that a commercial robot may not have complete knowledge of link and motor kinematics. Therefore, some RLFJ controllers estimate one or more states of the robot model, also known as partialstate feedback. Nicosia et al. began investigating the estimation of states in flexiblejoint robots in [55], and outlined the development of two observers. In the first observer, it was assumed that motor position and elastic displacement are known, while in the second observer, it was assumed that link positions and motor velocities were known. In both cases, local asymptotic stability was achieved given exact modelknowledge of robot dynamics and link position feedback. Tomei expanded upon this result with an observer to estimate the motor positions and velocities given a measurement of link positions and velocities [56]. Some researchers have studied the outputfeedback control problem, where both motor and link velocities are either unmeasurable or assumed to be corrupted by noise. Nicosia and Tomei designed a semiglobal exact modelknowledge controller without measurements of link and motor velocities [57]. Lim et al. was able to show semiglobal exponential tracking for the case of known model dynamics [58]. Similarly, Arteaga demonstrated semiglobal asymptotic stability with a modelbased observer to estimate link and motor velocities [59]. Lim et al. extended their previous result to a system with parametric uncertainties in [60]. In one of the most recent developments in outputfeedback tracking control, Dixon et al. developed a globally adaptive outputfeedback tracking controller for the flexiblejoint dynamics, based on the link velocity filter found in traditional robot systems [61]. Chatlatanagulchai et al. designed a neural network observer for estimation of link positions and robot parameters [62], and used this estimate in conjunction with a variable structure controller. The computer simulation of a two link flexiblejoint robot showed good results. A recent paper published by Kim describes the stability result of an adaptive flexiblejoint tracking controller without overparameterization, the problem that occurs when n xp parameters must be updated forp unknown robot parameters of an nlink robot [63]. All references thus far require the measurement of link position. Until recently, it was believed that good tracking performance could not be achieved without the direct measurement of states which define the tracking error, i.e. link position [60]. Abdollahi et al. successfully implemented a neural network observer to estimate link positions and velocities in [64]. The stability of the observer coupled with a PD controller was demonstrated in a computer simulation. However, the controller does not take advantage of both link and motor kinematics and there are no experimental results. This section describes an EKF to estimate link and motor kinematics in a RLFJ manipulator. Gourdeau and Schwartz utilized an EKF for estimation of joint positions and dynamic parameters in a rigidlink rigidjoint (RLRJ) robot model for a directdrive twolink robot [65]. Timcenko and Kircanski proposed a Kalman filter to estimate the driving torque in a RLFJ robot model and used a simple feedforward/feedback model for control [66]. Lertpiriyasuwat et al. applied an EKF to a twolink flexiblelink rigidjoint (FLRJ) robot model and augmented the measurement of endeffector position to the system model [67]. Lertpiriyasuwat and Berg combined realtime endeffector position measurements from a laser tracker sensor with joint positions measurements to estimate endeffector position and orientation in a RLRJ robot [68]. However, this method lacked the ability to utilize marker position measurements from any location on the robot. None of these filters estimate link kinematics, motor kinematics, local marker positions, or dynamic parameters in a RLFJ manipulator. Extended Kalman Filter Observer The Extended Kalman Filter An observer should utilize knowledge of plant and measurement dynamics, statistics of the process and measurement noise, and an initial condition, to yield an estimate of the state of the system. The wellknown Kalman filter is an optimal estimator for linear systems [69], with respect to the minimum error covariance, under the assumption that the plant and measurement dynamics are linear and known, the process and measurement noise are zeromean Gaussian random processes and uncorrelated with each other, and the initial state of the system is a Gaussian random vector with known mean and covariance. The extended Kalman filter (EKF) is an extension of the original filter to nonlinear systems. Although the EKF has been a popular choice among researchers, it has several disadvantages [67]. Foremost, the state estimate is no longer an optimal estimate. The performance of the filter will depend upon the accuracy of the linear approximation. Second, the computational burden and complexity of determining the partial derivatives of the state and measurement functions can be large. Third, the system dynamics must be continuously differentiable, which is especially a concern when modeling static friction. Although the EKF is no longer an optimal estimate, it can be modified to ensure local exponential stability with a prescribed degree of stability [70], known as exponential data weighting in linear systems. The modified form of the extended Kalman filter is summarized in (3.1) (3.4). Given the system and measurement models k(t) = f(x) + G(x)w(t), (3.1) y(t) = h(x) + v(t), (3.2) with process and measurement noise equal to w(t) ~ N(0, Q(t)) and v(t) ~ N(O,R(t)), the state estimate and error covariance update equations are x = f() + K(t)(t) h(i)], (3.3) P(t) = (F(, t) + aln)P(t) + P(t)(F(i, t)+ alJ) + G(C, t)Q(t)G' (, t) K(t)H(t)P(t), (3.4) with F(t) = f (), H(t) = h (), a > 0, and the Kalman gain K(t) = P(t)H (t)R Using the previous update equations, the estimation error, defined as '(t) = x(t) (t), is bounded via the following expression: 4< o ea. (3.5) where AP < P(t) < Ap. The rate of convergence can be controlled with the parameter a > 0. RigidLink FlexibleJoint Robot (RLFJ) Model The nlink RLFJ robot dynamics can be expressed as M(q)q + Vm (q, 4q) + G(q) + F(q) + K(q q,) = 0 (3.6) Jql + Bq + K(q4 q) = u, where q(t), q(t), q(t) e 9 and q, (t), m, (t), qm, (t) e 91" denote the position, velocity, and acceleration of the link and motor angle, respectively, M(q) e 91 represents the positive definite inertia matrix, V (q, 4) e 91n represents the centripetalCoriolis matrix, G(q) e 91" and F(4) e 91" denote the gravitational and frictional effects of the link dynamics, respectively, K, J, and B 91 are constant, diagonal, positivedefinite matrices representing joint stiffness, motor inertia, and motor viscous friction, respectively, and u(t) e 91" denotes the motor torque [30]. As described in [65], one can define a dynamic parameter vector 0 e 9' to allow for changes in the dynamic model of the robot. It contains the unknown or changing mass, inertial, and frictional parameters. The link and motor acceleration can be rewritten as S= M(q,0XK(q, q) Vm(q, 4,0 ) G(q,0) F(q, 0)), (3.7) and qm = J (u B K(q q)). (3.8) The robot can be supplemented with a set of retroreflective markers to improve the accuracy of the link position measurement. Identify the position of the ith marker in the local link coordinate system L, as p>, where i = 1,... m, and the collection of all markers as pL e 93m A kinematic parameter vector e 9c6 can be defined to represent the position and orientation of the robot coordinate system in the motion capture coordinate system. It contains the position of the robot coordinate system origin and the set of XYZ Euler angle rotations between robot and motion capture coordinate systems. It is unnecessary to include kinematic parameters in the state equation if the robot's position and orientation are known a priori and processing power is limited. One can define a state vector x = [q' qT q O'T (pL ) such that S= f(x,u), and M (q, 0)(K(qm q) Vm (q, q, 0)q G(q, 0) F(q, 0)) f(x,u)= qM (3.9) J 1(u Bq K(qm q)) (n+p+3m)xl Uncertainties in the link and motor dynamics are modeled with the random variables wL and w Unknown dynamic and kinematic parameter estimates are modeled as random walk processes, such that 0 = w0, q = w,, and pL = p. The combined uncertainties are grouped together into the random variable w = w w wo w w \, such that = f(x, u) + G(x)w, (3.10) 0 0 0 0 0 1(q, 0) 0 0 0 0 0 0 0 0 0 G(x)= 0 J 0 0 0 (3.11) 0 0 IPara 0 0 0 0 0 Ir.ans 0 0 0 0 0 IGeom A perturbation model is needed for the extended Kalman filter. One can define a state perturbation using a first order Taylor series expansion of the robot model with respect to a nominal trajectory [71]. x2 = f(xo, u), (3.12) =8f (x u) + G(x,)w. (3.13) The partial derivative of the robot dynamics with respect to the states can be written as 0 I 0 0 0 F F21 F, F23 0 F, = F(t)= 0 0 0 I 0 (3.14) 41 0 F F F45 0 0 0 0 0 F =M' q+ q+ +K (3.15) S M q q q(3.16) F = M1 a Vq +V + OF. (3.16) 22 04 0q F2 = M 'K. (3.17) M1 am G OF 8K F2 =M  q+"q+++(qq ) (3.18) 80 o 8 80 8 0 8o 0 o F41= J1K. (3.19) F43 =J1K. (3.20) F44 =J1B. (3.21) J.. U B K. OK F45 = 1 q., + q +K(qB q) (3.22) The robot dynamics must be defined by continuous differentiable functions in order to evaluate the partial derivatives. As a result, the friction model is adapted from Makkar et al. as F(q) = ,1 tanh(22/)+ 73y, (3.23) where 7y, 72, and 73 are positive, constant coefficients representing Coulomb and viscous friction in the joint [72]. Measurement Model One can define a measurement vector h (x) = q. l (p and introduce a random variable v = v v such that y = h(x)+ v, (3.24) where pG (q, n, pL) E 913m represents the position of all markers in the global coordinate system, and vf, v and vT represent the uncertainties in the measurement of motor positions, motor velocities, and marker positions, respectively. One can define a measurement perturbation in the same manner as the state perturbation. yo = h(yo), (3.25) Sy = h() +v. (3.26) dxC) The linearization of the measurement model for the rigidlink flexiblejoint dynamics is as follows: 0 0 0 0 0 0 Oh = H(t)= 0 0 I 0 0 0 (3.27) H31 0 0 0 0 H36 H37 G T 'OG T TG aO aq a T 'PG T aG H 36 1 M. Tp (3.29) S0 0 H37 = 0 0. O (3.30) 0 0 To facilitate the analysis of these partial derivatives, one can use the homogeneous coordinate system transformation p =vect{tT(q)pL1, (3.31) where pL1 is defined as the local marker position in homogeneous coordinates, and f (x) = vect(x) is an operator which eliminates the homogeneous coordinate from the last row of a position vector [73]. The partial derivative of the global marker position with respect to the link position is defined as G vectT(q) (3.32) Fp ~ {T(q) 4 $T(q) 8q 8q = vect 1 vect p 1 L (3.33) Oq aq, I J N A\ SLT(q) 'T L, where =TOTT GT3T. 41 T . Oq2 0 q2 The partial derivative of the global marker position with respect to the kinematic parameter vector is defined as _ vect GT(q) T(q);, (3.34) L= vect (q T(q)p I vect OT(q) T(q)pI (3.35) i L, I OO L, I The partial derivative of the global marker position with respect to the local marker position is defined as apG G R(q)p + pG G P L R(q)p1 + LR(q). (3.36) where LGR(q) is the rotational transformation between global and local coordinate systems. Observability in Nonlinear Systems The determination of observability in nonlinear systems is not as straightforward as the procedure for linear systems. The main differences in the observability of linear systems are 1) the observability matrix is constant, and 2) independent of the control input. Nonlinear systems can exhibit singular inputs that render the statespace realization unobservable [74]. The statespace realization x = f(x) f (3.37) y = h(x)' is said to be locally observable at x, e 91" if there exists a neighborhood U, c 91" ofxo in which every state x # xo e Q is distinguishable from xo. In other words, there must exist a unique mapping between each observation and respective state. For the nonlinear system described in (3.37), the condition for local observability in a neighborhood Uo c 91" containing the origin is rank(O)= n, (3.38) Vh where O= (3.39) VL"1h and Lh is the Lie derivative of the function h(x) with respect to the function f(x). The relationship in (3.38) reduces to the wellknown observability criterion given a linear system. The statespace realization is locally weakly observable (LWO) if the observability rank condition is satisfied at the origin. The observability matrix for the nonlinear system described in (3.6) can be written as 0 0 I 0 0 0 0 0 0 0 I 0 0 0 H31 0 0 0 0 H36 H37 0 0 0 I 0 0 0 F41 0 F43 F44 F45 0 0 A31 H31 0 0 0 A36 A37 O = 31 31 36 37 (3.40) F21 F22 F23 0 F25 0 0 ( 44F41 F41 F44F43 F43 +F44F44 25 0 0 031 032 033 034 035 036 037 Q11 Q12 22F23 F23 15 0 0 Q21 Q22 Q23 Q24 Q25 0 0 Q31 Q32 33 Q34 35 36 Q37 as described in Appendix A. This expression has M8n+ 2m rows and N=4n+6+p+3m columns. For the case in which the number of columns is less than the number of rows, i.e. p + 6 < 4n + 9m, the rank of the matrix can be at most N, the number of states in the extended Kalman Filter. From inspection, the columns are not dependent upon each other. Therefore, the observability matrix has full rank. It may be possible for either F25 or F45 to equal zero, but not at the same time, and the matrices F41 or F43 are linearly dependent upon each other. Neither condition affects the rank of the observability matrix. For the case in which the number of columns is greater than the number of rows, higher order Lie derivatives are required to increase the number of rows in order to achieve an Nrank matrix. The observability matrix, without measurements of marker positions, reduces to the following expression: 0 0 I 0 0 0 0 0 I 0 0 0 0 I 0 F 0 F F F 0= 41 43 44 45 (341) F F F 0 (3.41) F21 F22 F23 0 F25 F44F41 F41 F44 F43 F43 F 44 F44 25 Q11 12 F22F23 F23 15 Q21 Q22 Q23 Q24 Q25 This expression hasM 8n rows andN=4n+p columns. As long asp < 4n, the number of columns will be less than the number of rows and the rank of the matrix can be at most N. From inspection, the columns are not dependent upon each other. As a result the observability rank condition is satisfied, a necessary and sufficient condition for observability in nonlinear systems. RigidLink FlexibleJoint Control Theoretical Background The principal theory behind the Lyapunovbased stability analysis and controller design was first developed by Aleksandr Mikhailovich Lyapunov (18571918) at the turn of the 20th century. He wrote that a continuously differentiable and positivedefinite function with a negativedefinite derivative would converge, along with the states that constitute its value, to zero. It is a simple idea but an extremely powerful design tool. Unfortunately, its application for control system design would not become popular until more than 50 years after his death. It was in 1983 that Artstein [75] and Sontag [76] first introduced the idea of the control Lyapunov function. The idea was to design a control input to ensure the negative definite derivative of a Lyapunov function comprised of the error signal. This method bridged the distance between descriptions of stability and constructive design tools. A breakthrough occurred a few years later with the technique called integrator backstepping. Lyapunovbased methods could now be applied to a much broader class of systems, those which are extended or cascaded systems. The first use ofbackstepping as a design tool occurred in [77]. Soon afterward, an adaptive backstepping method was developed by Kanellakopoulos et al. for nonlinear systems with unknown parameters [78]. Later, Freeman and Kokotovic developed robust backstepping to achieve global stabilization despite the presence of unknown disturbances [79]. In the last decade, the research community has continued to advance the control of nonlinear systems, with update law modifications and progress in non modelbased controllers, such as neural networks and fuzzy logic controllers. A brief understanding of Lyapunovbased stability is necessary for understanding the following controller design. Stability Definition: The equilibrium point x = 0 of x = f(x) is stable if, for each e > 0, there exists 8 > 0 such that x(0) < 8 :> x(t) < e, Vt > 0o. unstable if it is not stable. asymptotically stable if it is stable and 6 can be chosen such that x(0) < 3> limx(t) = 0. Lyapunov Stability Theorem: Let x = 0 be the equilibrium point of k = f(x) and D c 91" be a domain containing x = 0. Let V : D > 91 be a continuously differentiable function such that V() = 0 and V(x) > 0inD {0}, and V(x) < 0 in D, then, the equilibrium point x = 0 is stable. Also, given V(x) < 0 in D {0}, then x = 0 is asymptotically stable. Control Objective An adaptive rigidlink flexiblejoint controller is described below based on the work in [63], [61]. The controller has a reduced parameterization to decrease the total number of integration variables, and assumes that both motor positions and velocities are measured and available for feedback, and that link positions and velocities are estimated in the extended Kalman filter. These modifications significantly reduce the complexity of the controller and make its design unique. The control objective is to design a global fullstate feedback tracking controller for the robot model described in (3.6). The link position tracking error e(t) e 9"is defined as e = q q. (3.42) To facilitate the control design, define a filtered tracking error r,(t) e 9" as r, = + ce + a2ef, (3.43) where ef (t) e 9" is an auxiliary filter variable and a1, a2 > 0. The auxiliary filter variable, introduced to eliminate the dependence of the controller on higher order derivatives in the stability analysis, is defined as ef = ke +p, (3.44) where p = (ka2 +a3)( ke + p)(ka, a2 )e and k,,a3 > 0 Define the joint deflection 3(t) e 9" as the difference between the motor and link position, 3 = q, q. In order to account for constant parametric uncertainties in the robot model, define the parameter estimation error (t) e 9P as S(t) = 0 (t), (3.45) where 0 91e contains constant unknown model parameters and 0(t) e 91' is an estimate of those parameters. Define two subsets of parameters, 0L c 0, and 0, c 0O, representing unknown parameters in the link dynamics (e.g. link mass, inertia, and friction), and motor dynamics (e.g. motor inertia, friction, and joint stiffness), respectively. Define a third subset of parameters 0 c 0, such that K diag(0K )= In The following properties [80] aid in the subsequent development of the adaptive controller. Property 1. The inertia matrix, M(q), is a symmetric, positivedefinite matrix that satisfies the following inequalities m, < 22 < cM(q) < m M 2 V G 91". (3.46) where mi, and m2 are known positive constants. Property 2. A skew symmetric relationship exists between the timederivative of the inertia matrix, AM(q), and the centripetalCoriolis matrix, V (q, ) This enables the useful relationship 1 M(q)Vm((q,q) =O V e 91". (3.47) Property 3. Diagonalization of x e 9" and y 91" is commutative, such that diag(x)y = diag(y)x (3.48) Property 4. The cascaded diagonalization of x e 91" and y 91" can be simplified as diag (diag(x) y) = diag(x) diag (y) (3.49) Error System for Filtered Tracking Error Define the first backstepping error r(t) e 91" as r = u, q,,,, where u, e 91" is an auxiliary control input. It is analogous to the desired joint deflection. Define the desired link regressor matrix Y, (qd, q,, qd ) e R"" as the linear parameterization of the link dynamics YOL = M(qd )(d ) + Vm (qd, d )(d ) + G(qd ) + F(d ) (3.50) The openloop error dynamics for the filtered tracking error are expressed as M(q)rL = M(q)iq M(q)q + aM(q)e + a2M(q) Substituting the link dynamics in (3.7), and the following expressions for the error dynamics, e = r, ae aef, and e = krL + a2e aef, (3.52) (3.53) the filtered tracking error dynamics can be rewritten as M(q)Lr = YOL V (q, )rL + Kq Kul ka,2M(q)rL + (3.54) where X = M(q)~d + Vm (q, 4)(d + ale + c2ef ) + G(q) + F(q) YdOL (a1a2 + a2a3)M(qf 2 2 (q)e + aM(q)rL + K (3.55) The function (t) contains the undesired terms in the Lyapunov derivative. Using the mean value theorem and knowledge of system dynamics, it is possible to place an upper bound on this expression as wX \ + 1 1 + a +a2 a3)AM e + 1a2' 2a') M e 1+a M rL + AK L7 , (3.56) where w = [e ef r ], AK and AM represent the maximum eigenvalues of the inertia and stiffness matrices, respectively, and yo and yl are positive bounding constants [60]. Partial Stability Analysis for Filtered Tracking Error Begin a partial stability analysis for the link position tracking errors and parametric uncertainties by defining a Lyapunov candidate function as V = 1/2eKe + 1/2efKe + 1/2rM(q)r + 1/20LrFL '1 + 1/20KTKFr1 (3.57) Evaluate the time derivative, substitute expressions for the error and filter dynamics in (3.52)  (3.54), and utilize the skew symmetric relationship in Property 2 to yield S= aeTKe 3e f TKef + e KrL k le Kr, + r, (YL + Kq Ku ka2M(q)r + x) S 1 1 (3.58) +o L r + KFr K Design the first auxiliary control and parameter estimate update laws as u1 = e kef + diag [K ] Y +Oq, (3.59) 0 = Y rL, (3.60) and OK = F .li, '(rL)YdL. (3.61) Add and subtract r K diag(OK )YdO ,and substitute (3.59) (3.61) into (3.58) to yield = ae"Ke aefTKef rL ka M(q)rL + rLTX. (3.62) Using Property 1 to bound the inertial term, and given the upperbound for z(t) in (3.56), simplifies the derivative of the Lyapunov function into 12 <_ A 2 K 2 ka2 112 +A'0 III 2+ LM1 I 2 + 223Ill IIK (3.63) +TM fr2 the teermsJ +inK thee77 abo + a l tM whil  The first three terms in the expression above are negative for all time, while the remaining terms are positive for all time. However, using nonlinear damping [81] and designing the control gain k = k, +a, +kK+ + /(2/1y +C IM2)]. (3.64) a2 AM where c = aa + a2a + a2 a 2 one can write a much leaner form of the derivative. S< IK 2 _a3 K l2f 4J2 2kr 212 _ 2 + 4 2 +2 I _I n + 2 (3.65) Auxiliary Control and Backstepping Error System Begin with the backstepping error q = u, qm and differentiate with respect to time to yield S= u qm, (3.66) where u, = e klef + q + diag[O JOL (3.67) Differentiating the expression for the auxiliary control input and utilizing the diagonalization properties yields S= +0 L m, (3.68) where 0, = k12e + k1a3ef + qd + diagK[ Yd L, (3.69) and 02 = k,2 +r ,li, ,'[r; +diag[ K dF Y (3.70) Insert the auxiliary control input u, through addition and subtraction to yield 7 = +f22r +r u. (3.71) Design u, = 10 + kq7 + kn,, 27 to enable the following expression for the dynamics of the first backstepping error: =0 r + ; k,77 k,0 (3.72) Define the second backstepping error as a = u, 4. Obtain the backstepping error dynamics by differentiating with respect to time, and substituting the expression for the motor dynamics. Jr = J, + B4m + K3 u. (3.73) In order to evaluate this expression, determine an expression for the timederivative of the auxiliary control input, such that 1u, = ka2e + ka3e f + qd + diag OK )fd + diag YdL f dk+ L .37 ^=s' ai d )~ "(3.74) + (k, + k f22 + 2k, f212 This simplifies to ui = 03 + QrL, where Q3 = k, (aa2 +2 aa, )e +k1 (a22 32 +f + diag(K dL + (k, + kR22 X '1 q +4k,Q2 ii ,'( L diag(fL ) + 2kf2diag (OK dFLd + (3.75) and Q4 = k(a2 + kla3)+F. lil (YL a >ag(JL)+diag[OK ]TL YT + (k + k~ 22 + 4k,2r 2 i, it(Y, )Liag()r + 2k,n 2F.li .y,, L g (3.76) Define the linear parameterization of the motor dynamics as YMOM = J(23 + 4rL)+Bq, +Ks, (3.77) and substitute this expression into the error dynamics to yield JiC = YMO U. (3.78) Design the control effort u = YM,0 + 7 + kTr to yield the following expression for the second backstepping error dynamics: J~i = Y, r kz. (3.79) Partial Stability Analysis for Backstepping Error System Begin a partial stability analysis for the backstepping errors and parametric uncertainties by defining a second Lyapunov candidate function as V2 = 1/2 'r + 1/2z'J. + 1/2 MF, (3.80) Differentiate with respect to time and substitute expressions for the two backstepping error dynamics to yield V2 = T Q2rL +kl k k,,2 ))+jrT( II krZ), 1TFMO. (3.81) Design the parameter estimate update law 0, = FYM ; such that Vi2 = (2rL k k, f221)+ r' ( k,). (3.82) The Lyapunov derivative can be upperbounded as V2 < k 1 2 k,e 1Qi kn 2 2+ l L (3.83) Using nonlinear damping [81], the expression can be simplified into the following. 2 < k 2 k1 2 4k (3.84) Controller Stability Result Theorem 1. Given the control strategy described above, semiglobal asymptotic link position tracking is obtained in the sense lime(t)= 0, provided that the controller and damping gains are selected to satisfy x(0) 2 + 2 k > 4A [ IT where x= e ef rL 77 ,1 and A = min(cl AK I a,3 K k, k,,,k,). Proof. Define the composite Lyapunov function V(t) = V (t) + V2 (t) the timederivative can be written as follows: + w2 4 2 2 2 2 <al_2 A e2fa3 _AK f e k r L ik, f_ + (w2 +w4 +e + ef 2+ 72 + rL2) kn (3.85) (3.86) (3.87) (3.88) . The upper bound on (3.89) Simplify the expression for the upper bound into V<A x 2+ 2 x 4 , 4k (3.90) and rearrange terms to yield the following expression: < [i 2 1 2 x 2 (3.91) If the condition in (3.86) is satisfied, then the derivative can be further simplified into the form V<P Ix2, (3.92) where /f is some positive bounding constant. From (3.92) it can be seen that the Lyapunov function is a radially unbounded, positive definite function for all z(t) e 9'5" and t, where z(t) = e ef rL r n O OK OM. (3.93) Since its timederivative is negative semidefinite, V(z) e Lm, and as a result z(t) e L Given that the actual parameters of the manipulator are bounded, it can be shown that 0L,0K and 0, e L4. The control efforts uq,u,,and u, and the timederivatives of the error signals, represented in (3.52) (3.54), (3.72), and (3.78), are also bounded for all time, based on the signals that define them. Therefore, x(t) e L4, x(t) e L., and x(t) e L2, based on the preceding information and the structure of (3.92). Use Barbalat's Lemma [82] to show that lim x(t) = 0, tco and as a result lime(t)= 0, (3.94) i.e. semiglobal asymptotic stability for the proposed adaptive controller. ClosedLoop Stability Result The exponential convergence of the EKF has been demonstrated in [70]. Using (3.5), it is possible to approximate the error dynamics as $ = a with a = 1 / E The closedloop controller and observer dynamics when using the EKF state estimate are written as follows. x = f(x,), (3.95) S= g(x, ). (3.96) where =[e e 1 rL, 7 u . The system represented in (3.95) and (3.96) is in standard singularly perturbed form [83]. Asymptotic and exponential stability of controller and observer dynamics, respectively, ensures asymptotic stability of the combined closedloop system. Simulation The EKF observer and corresponding RLFJ controller were implemented in MATLAB and Simulink (The MathWorks, Inc., Natick, MA). The equations of motion and associated differentiation of the six degreeoffreedom model of the PA106CE manipulator were determined with the analytic software program AutoLev (OnLine Dynamics, Inc., Sunnyvale, CA). The desired trajectory of the robot was based on the motion of the knee joint in gait. The position of a marker fixed to the lateral condyle was recorded using a motion capture system (Motion Analysis, Corp., Santa Rosa, CA), and a fourterm Fourier series approximation of the trajectory was determined using nonlinear optimization in MATLAB and transformed into joint space using an inverse kinematics algorithm. Five markers were arbitrarily attached to the virtual PA106CE model, two on link 2, and three on link 6. Inertial, frictional, and stiffness parameters in the simulation model were determined from the dynamic identification experiment in Ch. 2. Random zeromean Gaussian noise with covariances 0.1 mm2 (a = 0.30 mm) and 1x107 rad2 (a = 0.02 deg) were added to the measurement of each marker position and motor position, respectively. In order to limit the number of states in the extended Kalman filter, only mass and inertia properties for link six are included as unknown parameters. The remaining robot parameters, as well as initial conditions for unknown parameters, are corrupted by random Gaussian noise. The standard deviation is chosen such that 99.7% of chosen parameters fall within 10% of their nominal amount (3c = 10% ). The performance of the EKFRLFJ controller was assessed from rootmeansquare (RMS) and steadystate errors for the kneegait trajectory while carrying no payload and a 5 kg payload. For comparison, the same trajectory and loading conditions were executed using a desired compensation adaptationlaw (DCAL) controller [84]. The DCAL controller is an adaptive modelbased controller based on the backstepping design method. It compensates for nonlinearities arising from Coriolis and centrifugal effects, but does not incorporate joint flexibilities into its model. Results. The EKFRLFJ controller and DCAL controller were implemented with the set of gains in Tables IA and IB, respectively. As expected, the tracking errors were smaller for the EKFRLFJ controller in both loading cases, especially for joints 2, and 3, where there was a 71% and 83% reduction in steadystate tracking error. The significant improvement in tracking performance for joints 2 and 3 was most likely a result of the increased deflection in these joints due to a gravitational load (Tables 31 and 32). There was not a significant difference between the maximum and RMS motor torques between controllers. The mean RMS and steadystate link position estimation errors amongst all joints were less than 0.1 deg for both the unloaded and loaded cases. The mass and inertia estimation errors converged to less than 0.1 kg and 0.3 kg m2 in the unloaded case, and 0.1 kg and 0.2 kg m2 in the loaded case. The mean RMS and steady state marker position estimation errors amongst all markers were equal to 1.3 mm and 1.7 mm, respectively. The link position tracking errors and estimation errors are shown in Fig. 31 and 3 2, respectively. 2 0 2 I 6 0 0 c2h ^ 6  Joint 1  Joint 2 Joint 3 Joint 4 Joint 5 Joint 6 80 100 Figure 31. Link position tracking errors in simulation while loaded Joint 1 Joint 2 Joint 3 "' Joint 4  Joint 5 2 Joint6 0 2 Joint 1 20 40 6 Time (sec) 0 80 100 Figure 32. Link position estimation errors in simulation while loaded 20 40 60 Time (sec) 0.40 0 Table 31. Simulation results without payload DCAL Controller RMS Torque (Nm) Max Torque (Nm) RMS Tracking Error (deg) Max Tracking Error (deg) SteadyState Tracking Error (deg) EKFRLFJ Controller RMS Torque (Nm) Max Torque (Nm) RMS Tracking Error (deg) Max Tracking Error (deg) SteadyState Tracking Error (deg) Table 32. Simulation results DCAL Controller Jnt. 1 2 RMS Torque (Nm) 21.8 97.7 Max Torque (Nm) 40.7 126.7 RMS Tracking Error (deg) 0.7 2.2 Max Tracking Error (deg) 3.9 10.9 SteadyState Tracking Error (deg) 0.6 1.5 EKFRLFJ Controller Jnt. 1 2 RMS Torque (Nm) 22.9 102.0 Max Torque (Nm) 49.1 128.5 RMS Tracking Error (deg) 0.3 0.8 Max Tracking Error (deg) 2.9 5.1 SteadyState Tracking Error (deg) 0.3 0.7 with payload 3 22.8 34.2 0.8 3.9 0.8 3 22.4 43.9 0.3 2.2 0.3 Jnt. 1 21.5 34.7 0.6 3.4 0.6 Jnt. 1 22.2 38.1 0.4 2.4 0.5 68.2 93.8 1.6 7.1 1.4 2 71.6 97.3 0.4 2.1 0.4 14.5 24.8 0.7 1.8 1.0 3 13.9 25.7 0.2 1.6 0.2 Experiment A Mitsubishi PA106CE robot was mounted on a steel frame table with a height of 0.6 m. The table was attached to the floor with a set of four bolts and internallythreaded concrete anchors. The robot was positioned in the center of the viewing volume of an eightcamera passivemarker motion camera system (Motion Analysis, Corp., Santa Rosa, CA). The sampling frequency of the motion capture system was set to 250 Hz. Time delays due to image acquisition and processing are on the order of 3 to 4 frames, or 12 16 ms for this setup. The residual measurement error, obtained through calibration of the motion camera system, was on the order of 0.3 mm, with a standard deviation of 0.1 mm. Groups of two and three retroreflective markers were attached to link 2, and the endeffector, respectively. Realtime measurement and control of the robot and motion capture system was attained with a GNU/Linux kernel patched with RTAILXRT, and the help of the Orocos and Orca software libraries [34], [35]. An end effector apparatus was designed and constructed to support additional load on the robot end effector. The extra payload was added in order to excite robot inertial and stiffness dynamics. The experimental trajectory and loading conditions were identical to those in simulation. Figure 33. Mitsubishi PA106CE robot with markers RealTime Implementation Issues The realtime implementation of an EKFRLFJ controller presents many obstacles, such as numerical integration errors, timedelay artifacts, and measurement and synchronization errors. The most challenging obstacle is to achieve an accurate numerical integration given the limited step size and the computational burden in evaluating the partial derivatives of the dynamics. Several integration methods were evaluated on the EKF algorithm, including an explicit Euler method, a modified predictorcorrector Euler method, a 4th order RungeKutta algorithm, and an AdamsBashforth AdamsMoulton method [85]. The host PC contained an Intel Pentium processor running at 3.2 GHz with 3 GB of RAM. The minimum step size required for stability, and the computation time required for one iteration of the EKF were recorded for each method and are displayed in Table 33. Table 33. Comparison of numerical integration methods 4th Order Explicit Euler Modified Euler 4Orr ABAM RungeKutta Execution time (ms) 0.8 1.5 2.9 variable Max. step size (ms) 1.1 1.4 1.6 1.6 All integration methods except the explicit Euler method fail to yield a stable estimate in the time required for one iteration. Additionally, the difference between the expected execution time and the maximum step size is too small to safely account for computation time from other processes. Instability in the numerical integration arises because the link dynamics and motor dynamics exist on two different time scales. Implicit integration methods, which are successful for such numerical systems, require an unpredictable amount of processing time in order to iterate through a rootfinding method. The increased computational time increases the step size, further increasing the computational time. In order to avoid this computational problem, the dynamics were discretized and the state estimate was obtained with a discretetime EKF. Given the continuoustime system and perturbation model x = f(x,,u,), (3.97) S= Fx + Gw. (3.98) one can define the discretetime model k+l =k +rkWk, (3.99) where ,(Ik =eFt. (3.100) The solution for the predicted state estimate can be obtained from the state transition matrix as xk+1 = kXk (3.101) However, since the partial derivatives of the dynamics with respect to the states are known, the predicted state estimate is obtained with firstorder implicit Euler integration. tk+1 (3.102) k+= Xk Jf(Xk)J tk such that xk+l =x + f(xk+1)At. This expression can be rearranged into the form g(xk1) xk+ +f(x)At = 0. (3.103) and solved using the NewtonRaphson method for rootfinding, such that 1x = x,= J l(x)g(x). (3.104) where J i l) g(k+ l)I+ F(xk)At. (3.105) Xk+l The solution for the predicted error covariance matrix is obtained from the state transition matrix and can be written as Pk1 kPkTk +r r (3.106) where FQkFT GQG'At The updated state estimate and error covariance matrix are obtained from the discretetime EKF equations x= x +K(zk Hkxk). (3.107) P, =(I KkHk )Pk (3.108) and Kk = P Hk(HkPH HkR +Rk). (3.109) Experimental Results Initially, the EKFRLFJ controller did not have improved tracking performance in joints 1, 4, and 6, the result of relatively high joint stiffnesses coupled with low stiffness torques. However, there were improvements in the link position tracking errors for joint 2 and 3, stemming from the much larger signaltonoise ratios for estimated joint deflections in heavily loaded joints. In order to take advantage of this observation, a mixed rigidjoint/flexiblejoint model was used for the EKFRLFJ controller. The joints which did not incur significant stiffness torques during the activity, i.e. joints 1, 4, and 6, were defined as rigid. The remaining joints were defined as flexible. The mixed model provided smaller steadystate tracking errors for most joints in the PA10 6CE manipulator (Tables 34 and 35). RMS tracking errors decreased for all joints while loaded, yet there was not a consistent reduction in RMS tracking errors while unloaded. RMS torques were similar between controllers in both loading scenarios, and maximum torques were slightly greater in the EKFRLFJ controller in both loading scenarios. The link position tracking errors and estimated joint stiffness coefficients for the mixed controller are shown in Fig. 34 and Fig. 35, respectively.  Joint 1 Joint 2 Joint 3 Joint 4 Joint 5 SJoint 6 100 Time (sec) Figure 34. Experimental link position tracking errors while loaded 15000  Joint 2  Joint 3 ......... Joint 5 10000 5000    /" ) 50 100 150 Time (sec) Figure 35. Joint stiffness coefficients from experiment while loaded Table 34. Experimental results without payload DCAL Controller RMS Torque (Nm) Max Torque (Nm) RMS Tracking Error (deg) Max Tracking Error (deg) SteadyState Tracking Error (deg) EKFRLFJ Controller RMS Torque (Nm) Max Torque (Nm) RMS Tracking Error (deg) Max Tracking Error (deg) SteadyState Tracking Error (deg) Table 35. DCAL Controller RMS Torque (Nm) Max Torque (Nm) RMS Tracking Error (deg) Max Tracking Error (deg) SteadyState Tracking Error (deg) EKFRLFJ Controller RMS Torque (Nm) Max Torque (Nm) RMS Tracking Error (deg) Max Tracking Error (deg) SteadyState Tracking Error (deg) Jnt. 1 18.9 39.3 1.3 4.6 1.6 Jnt. 1 19.3 47.1 1.1 5.6 1.3 82.3 117.2 1.0 3.7 1.7 2 80.9 128.9 0.8 2.5 1.3 Experimental results Jnt. 1 2 20.2 120.3 39.2 161.9 1.4 1.2 4.8 4.3 1.6 2.2 Jnt. 1 2 20.2 118.4 45.3 196.0 0.9 0.8 6.0 2.9 1.2 1.2 16.9 34.5 0.7 2.4 1.5 3 18.3 53.2 0.8 3.1 1.4 with payload 3 29.7 3 53.6 5 0.9 2 5.0 5 1.7 3 3 31.5 3 61.1 6 0.9 1 3.1 4 1.8 2 5 5.1 11.6 0.8 2.3 1.1 5 5.1 12.5 1.6 4.4 2.1 5 8.8 14.3 3.3 5.5 4.2 5 8.5 14.4 2.3 8.4 3.4 Conclusion The performance of the EKFRLFJ controller, compared to the DCAL controller, was significantly better in simulation. Mean steadystate tracking errors decreased more than 50% for both unloaded and loaded trials. RMS torques were nearly identical between the two controllers, indicating almost equivalent energy expenditure. Performance improvements using the EKF RLFJ controller were not as universal in realtime tracking experiments. Again, RMS torques were almost identical between controllers, but strong improvements in tracking performance only occurred while the endeffector was loaded. Mean steadystate tracking errors decreased 22% while loaded, but increased 4% while unloaded. Reduced performance while unloaded is expected, given that reduced loading conditions lead to reduced excitation of stiffness dynamics. Joints 2, and 5 had the greatest reduction in steadystate tracking errors while loaded (> 50%), and also had the greatest stiffness torques relative to joint stiffnesses. Differences between simulation and experimental results can be attributed to unmodeled dynamics, limited actuation bandwidth, discretization errors, synchronization errors, and measurement noise. Closer investigation of these effects in simulation revealed that parameter error in the EKF model had the greatest effect on simulation performance. Greater accuracy of identification methods, or inclusion of these parameters as states in the EKF model, may improve performance. However, computational resources needed to include more parameters in a real time experiment are beyond our current capabilities. This paper discusses a novel observercontroller combination for accurate estimation and control of a RLFJ manipulator. The EKFRLFJ controller contrasts previous work by not requiring direct measurements of link positions, critical for estimation and control of industrial manipulators with flexible joints, since these manipulators are often assembled without link position resolvers. The tracking performance of the Mitsubishi PA106CE improved with the proposed EKF observer and RLFJ controller while loaded. This observercontroller should provide an even greater improvement, and for any loading conditions, in manipulators with greater joint flexibility. CHAPTER 4 MEASUREMENT ANALYSIS Introduction The paramount objective of the dynamic radiographic imaging system is the measurement of skeletal kinematics. This goal is achieved with modelbased registration of bone or implant models to radiographic images. These methods require knowledge of the 3D geometry and calibration parameters, such as focal length and principal point. Fortunately, it has been shown that kinematic measurements obtained from registration are not strongly influenced by errors in calibration parameters [86]. However, it has been suggested that this measurement is degraded by motion blur, arising from relative motion between the xray source, flatpanel detector, and object, but the magnitude of this effect is unknown. This section discusses the uncertainties in the relative motion of the dynamic radiographic imaging system, their propagation to motion blur, and their influence on image contrast and image registration. Literature Review Motion blur is an imaging artifact introduced by relative motion between the image sensor and object. During image capture, the elements of the image sensor collect photons, in their respective bins, and transform them into electrical signals. The motion of the object relative to the image sensor will project these photons in a line or curve along the image sensor, thus blurring the resulting image. The amount of motion blur increases as the exposure time or relative motion increases, and is more pronounced at higher resolutions. Point spread function (PSF) is the result of a single point of light projected on the image sensor. Motion blur can be artificially generated with accurate knowledge of the PSF, and is often added in the field of computer graphics to generate the feeling of motion. In addition, an image with motion blur can be corrected given accurate knowledge of the blurgenerating PSF. The theoretical foundation behind correcting image blur given relative motion is not a recent development. Motion blur is equivalent to convolution (or multiplication in the frequency domain) between the original image and a PSF, written as g=hf +n, (4.1) wherefis the original image, h is the PSF, n is shot noise, and g is the resulting image. Thus, the simplest method of image correction, inverse filtering, amounts to the inverse discrete Fourier transform of the complex division f .(g) (4.2) f (h) The Weiner filter expands upon the inverse filtering method by incorporating information about the signaltonoise ratio of the image, however, this information must be known a priori [87]. An improvement upon this method is the LucyRichardson algorithm. It yields the maximum likelihood estimate of the original image given unknown Poisson noise statistics [88]. For an insightful review of motion deblurring methods, see [89]. Nonetheless, these methods are not often implemented on physical systems because motion information is scarcely available. In [90], motionbased deblurring methods are revisited given motion estimates based on a secondary image detector with low spatialresolution and high temporalresolution. In the context of the dynamic radiographic imaging system, these additional costs are unnecessary given that we can estimate the motion of the xray source and flatpanel detector from joint resolver and motion capture measurements. However, residual motion blur, or that which is leftover after deconvolution, will remain despite a solid theoretical foundation and reliable experimental data. This residual motion blur may impair image registration [91], [92], and further investigation is needed to establish the extent of this degradation. Much literature discusses the methods for registration of 3D implant or bone models to 2D radiographic images. The most common objective or cost functions can be divided into three categories, featurebased, contourbased, and intensitybased. Using a contourbased metric, Banks and Hodge assessed the difference between the segmented image and the predicted contour with Fourier descriptors [93]. As an alternative, one can count the number of pixels that disagree between two binary images [94]. Mahfouz et al. defined a cost function having both an intensity matching score and a contour matching score in [95]. The intensity matching score and contour matching score are similar to a crosscorrelation between the two images, and the edge detected images [96], in that order. These methods provide a foundation for estimating implant or skeletal kinematics from radiographic images. Uncertainty Analysis Imaging Geometry Motion blur is a direct result of relative motion between the image sensor and the object. It can be characterized as the length in pixels in which a point of light is smeared across the image sensor during a fixed exposure time. The inplane translation and inplane rotation of the object relative to the image sensor will generate the principal motion blur. The outofplane translation and rotation of the object relative to the image sensor will increase or decrease the size of the object in the image plane and therefore generate some motion blur. However, it can be shown that this amount is negligible for typical projection geometries. This section describes the imaging geometry of the robotic imaging system and its connection with motion blur. The relationship between the size of the object on the image sensor and the imaging geometry can be described as Dlm = Dob (4.3) L1 L2 where D,, is the diameter of the object on the image sensor, Dobj is the diameter of the object in the physical world, and L1 and L2 represent the source to sensor distance and object to sensor distance, respectively (Fig 41). The change in size of the object on the image sensor will result from the following expression: A1J)m (L, L2)(L1 L2) Db, (4.4) AD = 2 D (4.4) V (L, L2(L L') ob J where L' = L2 obec At is the new object to sensor distance. Using 10 cm/s and 10 ms as /sensor approximations for the outofplane motion of the object and the exposure time of the image sensor, the change in L2 can be written as L L2 =0.1(0.01)= 10 (4.5) Therefore, the change in the size of the object in the image plane will be approximately 0.1% of the size of the image, resulting in ADm 0. This analysis demonstrates that outofplane translations will not generate motion blur in the dynamic radiographic imaging system. The analysis of motion blur is further complicated because the motion of the xray source relative to the object and image sensor will also generate motion blur (Fig. 41). In order to facilitate the analysis, define an effective object to image sensor velocity which generates the same motion blur as the relative motion between xray source and object, written as L2 V object soure (4.6) /sensor L, /object The effect of xray source motion on motion blur is small compared to the object and sensor motion, and it is in the opposite direction of the natural motion of the object. The outofplane translation of the xray source relative to the object and image sensor will not generate a significant amount of motion blur. Using similar triangles, the size of the object on the image sensor can be written as D L Dob, (4.7) L1 L2 b where D,, is the diameter of the object on the image sensor, Dobj is the diameter of the object in the physical world, and L1 and L2 represent the source to sensor distance and object to sensor distance, respectively (Fig 41). The change in size of the object on the image sensor can be written as (L, L')L2 ADm (L, L2)(L'L2) D (4.8) Vi Z L ALI L2) 0b where L1' = v our bc, At. It can be shown that the change in length is negligible when using object an approximation for the outofplane motion and exposure time, L, L' =0.1(0.01)= 103, (4.9) and therefore the change in the size of the object in the image is negligible, ADm & 0. Using similar arguments, it can be shown that the inplane and outofplane rotational motion of the x ray source will not have an effect on motion blur. In conclusion, the inplane motion of the xray source will lead to an effective inplane object to sensor motion, resulting in the following combined or total inplane relative sensor motion: L2 vobect =obc + object (4.10) /sensor /senso /souce On the other hand, only the inplane rotation of the image sensor relative to the object will lead to rotational image blur, such that sensor sensoeor t doJect 4 / / / / s ourcob object Figure 41. Imaging configuration of robotic imaging system Covariance of Relative Motion An estimate for the relative motion between the target object and the endeffector of the robot is needed in order to simulate motion blur. It can be written as objet,/ /sensor object/ object/ /Sensor /sensor (4.12) where vbent /sensor is the measured relative motion between the target object and endeffector, i.e. the tracking error, and object / ensor is the measurement uncertainty. The covariance of the relative motion can be written as / e oject sensor /sensor Sobec nsor /sensor where e2 /sensor cov (vobjec , and C,2 object, /sensor /v, sensor cov ,Vobect/e /sensor ) object/ objected / sensor (4.11) cov (Vobject (4.13) Endeffector tracking error The relative motion between the target object and the endeffector of the robot was measured in a series of tracking experiments with the Mitsubishi PA106CE. The robot was commanded to follow a path approximated from the location of the femoral condyle of the knee in gait (also used in the previous chapter). The same path was followed at three different fundamental frequencies in order to observe the relationship between the object's velocity and tracking error. The fastpaced trajectory, which resembled the knee joint in a fast walk, exceeded the joint velocity limits of the PA106CE. This scenario might arise for some faster moving activities, and is worth considering in the analysis. The DCAL controller was used in place of the EKFRLFJ controller because it has a much simpler design and produced similar tracking results. In the slowpaced tracking experiment, the DCAL controller produced the following tracking error covariance matrix: 586 23 404 e Zbjec = 23 2829 17 mm2z (4.14) 404 17 817 In the mediumpaced tracking experiment, the DCAL controller yielded the following tracking error covariance matrices: 801 99 464 e obec = 99 6536 44 m 2, (4.15) 464 44 1308 In the fastpaced tracking experiment, the DCAL controller yielded the following tracking error covariance matrices: 71.8 1.6 4.9 e b 1.6 176.3 2.4 .103m /2 (4.16) 4.9 2.4 2.2 The crosscovariance terms are small compared to the diagonal elements, and as a result, the covariance matrices are simplified as diagonal matrices for the remaining analysis. The strongest correlation existed between tracking errors in the x and z directions (r2 = 0.2), however, outof plane motions are neglected in the motion blur simulation. Measurement uncertainty The relative motion between the target object and endeffector is a function of both motions, such that object V object ob sensoglobal (4.17) Therefore, the relative motion measurement uncertainty is dependent upon the uncertainties in target object and endeffector velocities. The law of propagation of uncertainty, which explains the relationship between the statistics of the returning value of a function and the individual uncertainties which constitute its value, can be used to determine the measurement uncertainty. This leads to the following expression: 2 2 2 object ~ object + a sensor lob (4.18) V, /sensor V /globalglobal and 2 = 2 and object object source /sensor Uncertainty in EndEffector Velocity Estimate. The law of propagation of uncertainty can also be used to determine the uncertainty in endeffector velocity. The joint measurement noise covariance matrix can be written as C 2(41) ... C2(41,46) 221 C4 *. (4.19) C 2 (41,46) ... C2(46) J This is equivalent to the diagonal matrix ao = diag(0c2 (4),' ...,2 (46)), because of the independence of angular position measurement noise. Using the Jacobian relationship between joint space and task space velocities, x = J(q) 4, the endeffector measurement noise covariance, .2 can be expressed as a = J(q)'a2 J(qT, (4.20) where J(q) is the manipulator Jacobian. The uncertainties in joint positions are neglected since 2 2 they are small in comparison to those in joint velocities, oa << . The upper bound on the uncertainty in linear and angular velocities can be evaluated from Ss /global < J ( 2 (4.21) and 2 Sen global < Rq) 2 ,2 (4.22) where J, (q) e 913x6 and JR (q) e 913x6 are the translational and rotational parts of the manipulator Jacobian, respectively. The upper bound for the norm of a 2 was determined from the output of the extended Kalman filter during a simple tracking experiment. The robot was commanded to follow the gaitlike path as the estimated error covariance matrix, returned from the extended Kalman filter, was recorded. The error covariance matrix did not have a strong dependence upon the selected trajectory. The upper bounds on the norms of J, (q) and JR (q) were determined from a numerical simulation sampling le4 random configurations inside the workspace of the manipulator (Fig. 42, Fig. 43). As a result, it can be written that 2 sensor 1140mm ad 2.16 X 104rad2 = 281mm2 (4.23) V% 71lbO") S2s (4.23) S Uncertainty in Object Velocity Estimate. The uncertainty in the estimate of the object velocity was obtained from the results of the MATLAB simulation of an extended Kalman Filter for rigidbody pose estimation [97]. The measurement error covariance matrices returned from the filter equaled 32 0.1 0.9 2 objeclo 0.1 32 0.6 m 2, (4.25) 0.9 0.6 34 5.1 0.2 0.5 a 2 ( and object= 0.2 4.4 0.4 deg2 (4.26) g0.5 0.4 5.2 Although small in this example, the differences in angular velocity uncertainties between directions can be attributed to the location of markers in the local coordinate system. It is more difficult to estimate outofplane rotations when markers are placed in a single plane. Covariance of Relative Motion. The uncertainty in the relative motion estimate can be determined using (4.23) (4.26), and (4.18), such that object < 318mm 2, (4.27) and 2,objec" < 7.3 deg2 (4.28) /eSnor S Using (4.15), (4.27), (4.28) and (4.13), the covariance of the relative motion between end effector and object can be determined as cov vo and covOo, O .ey < diag([14.2 17.4 56.9]), (4.30) given the results from the mediumpaced tracking experiment. 1.2 1.8 0 6 *o. ,o *, ** *" :.*** ,. ode .4 o I. 0 dn0 04 0, do o 0.2 0 200 400 600 800 1000 Sample number Figure 42. Norm of translation part of manipulator Jacobian 0.4 0 200 400 600 800 1000 Sample number Figure 42. Norm of translation part of manipulator Jacobian 1.8 1.6 g p 1.4 1.2 0 200 400 600 800 1000 Sample number Figure 43. Norm of rotational part of manipulator Jacobian Image Sharpness The spatial frequency response (SFR), or modulation transfer function (MTF), of an image depicts relative image contrast for a range of spatial frequencies, and is the spatialdomain analog to the (timedomain) frequency response of a mechanical system. The sharpness, or contrast, of an image is most often assessed from its SFR. A slantededge is often used to determine the frequency response because pixels along the slanted edge can be subsampled into multiple bins. This procedure increases the effective resolution of the image and permits observation of frequencies greater than the normal Nyquist frequency. The SFR is then obtained from the Fourier transform of the derivative of the oversampled edge. The spatialdomain behavior of motion blur is analogous to a common lowpass filter, attenuating high frequency content in the image, and can be assessed from the SFR. This section discusses the adverse effect of motion blur on image contrast, and its improvement with image deconvolution, through simulations and experiments. Theoretical Improvement The SFR of an image with a slantededge was determined in three different situations: 1) without motion blur, 2) with motion blur, and 3) with motion blur and image deconvolution. Motion blur was generated from the probability distribution of the expected relative motion, as determined in the preceding uncertainty analysis. It is a result of both source to object motion and sensor to object motion, leading to the following propagation of uncertainties: cov vobje = cov vobect L2 COV oe (4.31) /sensor I/sensor L1 /source) Using approximations for the covariance of the relative motion in (4.29), and the approximation L O 0.1, the total uncertainty in linear motion can be upperbounded as cov rVobj, The total uncertainty in angular motion is small compared to the uncertainty in linear motion and is neglected in the motion blur simulation. Image deconvolution was performed using the damped LucyRichardson deconvolution algorithm [88]. The PSF, used in this algorithm, was defined as the true relative motion plus the measurement uncertainty perturbation term in (4.27): objc < 318mm 2 (4.33) /sensor S Each probability distribution was sampled at 200 positions, and the response of each sample was recorded. The SFR of the slantededge was obtained with the SlantedEdge Analysis Tool in MATLAB [98]. Image sharpness was evaluated from the area under the SFR curve from zero to the Nyquist frequency, such that /= FN (o)) do). (4.34) As expected, there was a significant improvement in image contrast with the motion based deblurring method, despite uncertainties in the precise endeffector and tracking object velocities. The mean [t value increased from 0.43 in the set of blurred images to 0.69 in the set of blurred/corrected images, an increase of more than 60% contrast. The gto value from the static undistortedd) image was 1.04. The mean responses of the blurred and corrected image sets, those which corresponds to their respective mean [t value, are shown in Fig. 44. The blue solid line represents the SFR of the original image, without motion blur. Although the mean t values are much different between the two sets of images, the ranges of contrast are similar, extending from 0.14 1.04 and 0.23 1.34 in the blurred and blurred/corrected images, respectively. In a second simulation, motion blur was added in even increments to the slantededge image. The image contrast, as expected, decreased with an increase in relative object/sensor motion (Fig. 45). The thin red solid line and thick black solid line represent the probability density functions of the tracking error and its estimation error. It can be seen from Fig. 45 that tracking errors will lead to [t values in the range of 0.1 1.0, while measurement uncertainties alone will lead to [ values in the range of 0.5 1.0. SUndistorted Image * Corrected Image ""*"" Blurred Image 0.2 L 0 0.1 0.2 0.3 0.4 Spatial Frequency (cycles/pixel) Figure 44. Spatial frequency response of motion blurred images 1.4 25 Probability Distribution S1.2 of Measurement SUncertainty 20 1 S 15 0.8 Probability Distribution 0.4 of Tracking Error N 5 5 'I. 0.2 o  ^t5 0 0.5 0 Relative Motion (m/s) Figure 45. Relationship between relative motion and image contrast Q 0.8 0.6 a Experimental Methods The improvement in image contrast from motionbased deblurring methods was also evaluated in a dualrobot tracking experiment. For this experiment, two Mitsubishi PA106CE manipulators were positioned in the workspace of an elevencamera optical motion camera system (Motion Analysis Corp., Santa Rosa, CA). Each manipulator was positioned on a 0.6 m tall steel frame table, located 2.4 m from each other. A monochrome video camera (JAI Pulnix, San Jose, CA) and 25 x 20 cm board were attached to the endeffectors of the source (right) and sensor (left) robots (Fig. 46). The ISO 12233 test chart was printed and attached to the board in order for the video camera to capture images of a slantededge during the experiment. The sensor manipulator was commanded to follow a path resembling the path of the kneejoint center during normal gait, while the source robot was commanded to track the endeffector of the sensor robot, such that the ISO test chart would remain in sight for the complete trajectory. Four tracking markers were attached to the board, while an EKF, developed by Halvorsen et al., was used to estimate the location of the target from this set of markers [97]. Four tracking markers were also attached to the surface of the source robot in order to implement a second EKF [Ch. 3]. A set of 20 images was captured along one period of the gaitlike trajectory using an asynchronous trigger and 10 ms exposure time. The position and orientation of the camera and the target object were recorded from the two EKF observers. The SFR of each image, evaluated from the slantededge, was determined before and after image deconvolution. The bestcase SFR was defined from the static, thus undistorted, image of the test chart. Image contrast was evaluated from the area under the SFR curve from 0 2 cycles/pixel, defined as a. Residual motion blur was defined as the percent difference between t in the undistorted image and [t in the corrected image. Figure 46. Experimental setup of dualrobot imaging system Experimental Results As expected the spatial frequency response of each image was improved using motion information acquired from the EKF. The mean [t value increased from 0.11 in the blurred image group to 0.19 in the blurred/corrected image group, an increase of more than 70%. The [to value from the static, undistorted image (0.33) served as a gold standard for comparison. The [t value of each image relative to the static image, expressed as a percentage, is shown in Fig. 47. The greatest and smallest improvements in image contrast were 418% (Image #6) and less than 1% (Image #17). The blurred and blurred/corrected images of the slantededge for these cases are shown in Fig. 48. The mean relative contrast was equal to 34.3% and 58.2% in the blurred and blurred/corrected image groups, respectively. The mean residual motion blur, expressed as a percentage, was equal to 41.8%. 140 Blurred Images 120 B Blurred/Corrected Images S100 Undistorted Image S80 60 U.. I a 60 40 20 0 5 10 15 20 Image Number Figure 47. Relative contrast in blurred and blurred/corrected images A B C m m "I D Figure 48. Image sharpening from LucyRichardson algorithm A) Blurred image #6, B) Blurred/Corrected image #6 (bestcase), C) Blurred image #17, D) Blurred/Corrected image #17 (worstcase) Image Registration It has been shown in the preceding section that motion blur will exist in the acquired images, yet it has not been seen how motion blur affects registration accuracy. This section discusses the impact of motion blur on image registration in the context of the robotic imaging platform, and the potential improvements with image deconvolution. Methods Four different radiographic images were collected from four different studies, representing the range of contrast and resolution that might be expected in typical clinical studies. Image RU is an 800x800 pixel, underexposed radiograph of an uncemented total knee arthroplasty (TKA). Image RC is a 1024x1024, high contrast radiograph of a cemented TKA. Image FC is a 352x288, low contrast fluoroscopic image of a cemented TKA, and image FN is a 512x512, high contrast fluoroscopic image of a natural knee. Twenty randomly blurred images were generated from each of the four original images. Blur statistics were determined by sampling a distribution of relative tracking errors obtained in a series of experiments with a Mitsubishi PA106CE robot during a gait tracking task, described in (4.29). Tracking errors resulted in standard deviations of blur in the horizontal/vertical directions of 4.9/2.4, 6.0/2.9, 1.1/0.5, and 1.4/0.7 pixels for the RU, RC, FC, and FN images, respectively, given a conservative exposure time of 10 ms. Four additional sets of 20 randomly blurred and subsequently deblurred, or deconvolved, images were also generated from each original image. Each blurred image was deconvolved using the damped LucyRichardson algorithm in MATLAB (Mathworks, Natick, MA) and estimated measurement uncertainties, described in (4.27). The JointTrack opensource software program was used to register the implant or bone models to their respective images [99]. Registration was performed using simulated annealing optimization algorithm and an edge plus grayscale cost function [95]. The mean and standard deviation of the image registration error were recorded for each image and used to compare the blurred and blurred/corrected sets of images. A paired ttest was used to determine if the RMS translational and rotational errors were greater in the blurred images compared to the blurred then corrected images (a = 0.05). Results Motion blur generated from typical tracking errors measured in a robotic imaging system did not have a significant impact on image registration accuracy. Both mean anteriorposterior (AP) and superiorinferior (SI) RMS errors for the blurred and blurred/corrected image sets were less than 0.5 mm. The mean flexionextension (FE) RMS errors equaled 0.4 and 0.3 deg. for the blurred and blurred/corrected image sets, respectively, and the mean varusvalgus (VV) and internalexternal (IE) RMS errors were less than 1 deg. Motion blur had the greatest impact in mediallateral (ML) translation where mean RMS errors totaled 2.0 mm for both the blurred and blurred/corrected image sets. Kinematic errors for both image groups are summarized in Table 4 1. Mean RMS errors decreased in the blurred/corrected image group for five of the six degrees of freedom (Table 42). Image deblurring significantly reduced RMS registration errors only for flexionextension. (p < 0.05). 4l Figure 49. Undistorted radiographic images A) RU, B) RC, C) FC, D) FN Table 41. Kinematic errors for Blurred and Corrected image groups (RMS errors) RU RC FC FN Parameter B C B C B C B C AnteriorPosterior Translation (mm) 0.32 0.34 0.77 0.44 0.66 0.58 0.22 0.17 SuperiorInferior Translation (mm) 0.12 0.18 0.44 0.51 0.59 0.33 0.63 0.28 MedialLateral Translation (mm) 1.87 2.61 2.58 2.07 2.66 1.71 0.96 1.95 FlexionExtension (deg) 0.26 0.18 0.77 0.60 0.19 0.18 0.42 0.22 VarusValgus (deg) 0.15 0.18 0.47 0.26 0.26 0.27 0.47 0.15 InternalExternal Rotation (deg) 0.47 0.46 1.23 0.66 1.08 0.38 0.39 0.32 Table 42. Kinematic differences between Blurred and Corrected image groups (RMS errors) Parameter RU RC FC FN mean AnteriorPosterior Translation (mm) 0.02 0.33 0.08 0.05 0.11 SuperiorInferior Translation (mm) 0.06 0.07 0.26 0.35 0.12 MedialLateral Translation (mm) 0.74 0.51 0.95 0.99 0.07 FlexionExtension (deg) 0.08 0.17 0.01 0.20 0.12* VarusValgus (deg) 0.03 0.21 0.01 0.32 0.12 InternalExternal Rotation (deg) 0.01 0.57 0.70 0.06 0.34 * statistically greater than zero (p < 0.05) Conclusion Motion blur, representative of a dynamic robotic imaging system, has a harmful effect on image contrast, as shown in simulations and experiments. However, contrast can be improved with motionbased deblurring methods. Increases in the [t value of more than 60% and 70% were observed once the images were deconvolved using the estimated motion, in simulations and experiments. On the other hand, motion blur, representative of a dynamic robotic imaging system, did not significantly degrade kinematic measurements based upon modelimage registration. Inplane translational and rotational errors were less than 0.5 mm and 1.0, respectively. These errors are within bounds for research and clinically useful kinematic measurements. Errors in the ML direction exceeded those in other directions because of low registration sensitivity to outofplane translations. Mean RMS errors decreased in blur corrected images, but only differences in flexionextension errors were significantly improved (p < 0.05). Typical absolute errors with this cost function and optimization for unblurred images have been reported as 0.1 mm and 0.40 [95]. Additive errors due to expected image blur would result in net RMS errors of 0.6mm and 1.40, which are sufficiently accurate for many research and clinical assessments. Shorter exposure times can directly reduce measurement degradation if motion blur is problematic. The limited impact of motion blur on image registration can be attributed to several factors. First, various registration cost functions are robust to imaging artifacts, such as motion blur, since they do not solely depend on edge information. Second, the simulated annealing (SA) method used in this study is a global optimization method, reducing the chance of spurious solutions from local minima. Lastly, user supervision of modelimage registration safeguards against poor or unrealistic solutions, and reduces the error covariance between images. As an alternative to motionbased deblurring, one might choose to blur rendered images before comparison with acquired images, in order to improve their registration. Blurring velocities could also provide a missing bridge between the kinematics of a single frame and the entire sequence of motion. Modeling joint kinematics as a continuous process would enforce continuous kinematic estimates and likely improve results. Further investigated is warranted to determine the accuracy of this method. Radiographic imaging systems provide superior joint kinematic measurement accuracy compared to optical motion capture systems with skin affixed markers. By definition, DRI systems will be subject to motion blur and potentially significantly degradation of skeletal kinematic measurements. This investigate demonstrates that knee kinematics from modelimage registration are not significantly degraded by motion blur for subject and robot velocities typical of normal gait. CHAPTER 5 CONCLUSION The dynamic radiographic imaging system described in these pages will be one of the first to acquire natural dynamic joint motion. Its results will likely improve the current standards for treatment and diagnosis of musculoskeletal conditions. With a rising need for improved healthcare worldwide, especially for the aging population, this imaging system may develop into a great success. The potential is large, but there are challenges concerning the measurement and control of such as system. The tracking performance of the robotic imaging system is hindered from nonlinearities arising from its revolute geometry, friction and joint flexibilities. In order to account these modeling intricacies, novel calibration and identification methods were developed. Their estimates initialized adaptive modelbased control methods and enabled the PA106CE to meet stringent tracking requirements. The taskspace RMS and steadystate tracking errors of the robotic imaging system were 1.9 cm and 2.3 cm, respectively, allowing the joint to remain in sight of the flatpanel detector for the duration of the activity. The measurement performance of the robotic imaging system is hampered by motion blur arising from the dynamic nature of the system. But, these artifacts can be reduced with accurate tracking and motionbased deblurring methods. RMS tracking errors were less than 9.0 cm/s, providing a manageable amount of motion blur, and the corrected images showed more than 60% improvement in contrast compared to blurred images. Furthermore, it was shown that accurate registration is possible given motion blurred images. RMS inplane translational and rotations errors in motion blurred images were less than 0.5 mm and 1.0 deg. The results in this dissertation indicate that a dynamic radiographic imaging system can acquire clinically useful kinematic measurements. The dualrobot imaging system can provide acceptable tracking errors despite nonlinear flexiblejoint dynamics, and images acquired in such a system can provide accurate motion information despite motion blur. Mobile and flexible radiographic imaging systems are the future of medical imaging, and further development is critical to the field of orthopedics and beyond. APPENDIX A OBSERVABILITY MATRIX As shown in Chapter 3, the state dynamics and partial derivatives can be written as q x = f(x,u)= (A.1) 0 01000 0 I 0 0 0 F21 F22 F23 0 F25 and C= F(t)= 0 0 0 I 0. (A.2) Dx F41 0 F43 F44 F5 0 0 0 0 0 where p =M '(q, )(K(q, q) V (q, q, ) G(q, ) F(4, 0)), (A.3) and a = J1 (u B,, K(qg q)). (A.4) The observation h(x) is based on measurements of motor positions and velocities, and marker positions. The measurement dynamics and partial derivatives can be written as h(x)= [q cT (p)T (A.5) S0 0 1 0 0 0 0 Oh and = H(t) = 0 0 I 0 0 0 (A.6) Hz 31 0 0 0 H36 H37 The first Lie derivative of function h(x) with respect to f(x) is as follows: q 0 0 I 0 0 0 'U p m ^ Lfh = 0 0 I 0 0 0 = (A.7) H,3 0 0 0 0 H36 H37 a H31q 0 where H3 ,= PqG .. (A.8) q Oq ) apG F l 8GTq ) LT(q) and = vect pi .4 ... vect pP\q, .4N (A.9) q L aq, I aq j The gradient of the first Lie derivative with respect to the states can be written as 0 0 0 I 0 0 0 VLfh = F41 0 F43 F44 F45 0 0 (A.10) A31 H31 0 0 0 A36 37 where A31 =q 6 = and A37 =  1 q a7 apL The second Lie derivative of function h(x) with respect to f(x) is as follows: q 0 0 0 I 0 0 0 'U 0 Lfh = F41 0 F43 F44 F45 0 + (1.+11) L h = F41+ F43q+ F44O (A.12) A31q + H31z + A36, The gradient of the second Lie derivative with respect to the states can be written as F21 F2, F,3 o F25 O 0 VL2h = F44F4 F4 F44F43 F43+F44F44 25 0 0 (A.13) f 31 032 033 034 035 036 037 where 025 = 41 3 +4 c + FF45. (A.14) =0 0 0 + 31 A31 31 + H31F21 + A + A36F41 Oq Oq Oq H H 31 032 H31 l+ + H31F22 + a + A6F42. aq =0 0,3 = HF23 36+43 . 034 = A36F44. 035= H31F25 + 36F45. 0 A31 + H31 + A36 0,36 + . 037 =A31 + H31 U+ A36 37 aL dvapL op The third Lie derivative of function h(x) with respect to f(x) is as follows: F23 F44F43 033 F43 F44F44 034 F25 025 035 0 0 037 F214 + F22/ + F234m L3h= F44F41+ 41 + F+F4+ F44 4m F44F44 T31q + e 32e + 33om + 034c The gradient of the third Lie derivative with respect to the states can be written as VLfh = 0Q21 Q"31 F22F23 23 Q33 0 0 0 0 Q36 Q37 (A.15) (A.16) (A.17) (A.18) (A.19) (A.20) (A.21) h 21 L3fh= F44F41 031 (A.22) where (A.23) (A.24) 8F a8F O11 221 22+ O 2 + F22F21 . dq dq aF 8F OF 15 2 FF 22F + 23 m 21 =F41F21 + (F43 +F44F44 )F41. 22 F44 F41 F41 F22 ' * 23 F41F23 + F43 'F44F44 F43 * *24 F44F43 + (F43 F44F44 4 F . (a8F 8F aF 8F 8F43 Q25 44 F41 F44 41 41 +F41F25 F 43 +F44 3 m SF 8F 44 4 3+ 24 4F44 F4+F 4F F4 5 dB~a aodBdB 8q 8q 31 7 217 Q32= 0+ 31 8q 33 m 34 34F 41  ^ + 4^ 1 02 + 32F22 + 0q AO33 8q 0q, 0o ""33 31 32 32F 23m 34 43 Oqm Oqm Oqm 14 31 032 aM a 33 m+ 33 +34F 44 f, a' a3 ia 33. 34 a 35 800+ +O32F25 +0 m + 40 34F 45 a0 a0 00 o 0 831 832 00 q+0+ 0a 33, m+ a34 (A.25) (A.26) (A.27) (A.28) (A.29) (A.30) (A.31) (A.32) (A.33) (A.34) (A.35) (A.36) (A.37) (A.38) 31 32 33 3 + 0 334 37 aL m pL" (A.39) This leads to the observability matrix 0 0 I 0 0 0 0 0 0 0 I 0 0 0 H31 0 0 0 0 H36 H37 0 0 0 I 0 0 0 F4 0 F43 F44 F45 0 0 A31 H31 0 0 0 36 A37 O = 31 31 36 37 (A.40) F F F 0 F O O F21 F22 F 23 0 25 0 0 F44F41 F41 F44F43 F43 +44F44 025 0 0 031 032 033 034 035 036 037 Q11 Q12 F22 F2 F3 Q15 0 0 Q21 Q22 Q23 Q24 Q25 0 0 Q31 Q32 Q33 Q34 Q35 Q36 Q37 where the elements are described in (A. 14) (A.21), and (A.25) (A.39). APPENDIX B OBSERVER DESIGN FOR TWOLINK ROBOT The equations of motion for a twolink manipulator with joint flexibility can be written as M(q)q + Vm (q, q)q + G(q) + F(q) + K(q q ) = 0. (B.1) Jq4, + B4q +K(q, q) = u. (B.2) p, +2pzc, p + pc where M(q) = 32 P2 +p (B.3) P2 + P3C2 P2 V. (q, ) = p3s22 p3s2(41 2 (B.4) P3S24 1 0 P4S1 + ps127 G(q) = ,4S + P5S12 (B.5) P5S12 Sp6 tanh(p7ll)+ p41 (B.6) J =, (B.7) B P4 0 (B.8) and K = 6 (B.9) 0 P17 The coefficients in the inertia matrix, centripetalCoriolis matrix, and gravitational component are based on the mass and inertia properties of the each link, such that P1 = I + m2 + 2 + 2(1 +1 22), (B.10) p, = I + m22, (B.11) p = m21211, (B.12) p4 mgl, m2g2, (B.13) and p5 = mgl2. (B.14) The State Model The partial derivative of the robot dynamics with respect to the states is 0 1 0 0 0 F21 F22 F,23 F, = F(t)= 0 0 0 I 0 .(B.15) F41 0 F43 F44 F45 0 0 0 0 0 Solution for F21: Given the definition for the inertia matrix, the centripetalCoriolis acceleration matrix, and the gravitational and frictional components of the link dynamics, the solution to the partial differential equations contained in F21 can be written as follows: 8M .. 8M .. 8M ..7 q = aq .q2q (B.16) wq _q, =qq where aq = and q2 = such that aq, q p3pcs2A OM .. 0 2ps P3S24q S Fq= (B.17) aq 0 p3sAq, Furthermore, a Fn a '. 4 4, (B.18) aq aqj, aq2 O F, O av p3cq2 12 PCq( p 2c ( 2l +that where q = and q = such that aq, L0] aq P3c2 avq l 0 p3c212 p2 1 (B.19) q 0 p3c2 q1 8G _p4cc+p5c12 p5c12 [_ P4C 2 P5C12 (B.20) q P5CI2 P5C12 The inertia matrix is positive definite for all robot configurations, and therefore, the inverse of the inertia matrix is nonsingular. 1 1 P2 p P3C M 2 p2 c2 p + 2p3c(B.21) PlP2 P2 P c 2 P2 P3C2 A + 23C2 Therefore, Fq + q + + K (B.22) 8q 8q 8q F 1 p p p3c2 plP2 P p PIc L p2 p3c P +2p3C2 J 1 p2 0p 2 1 2 3 2 1 ( B 2 3 ) O 2pss pS 0 pUc, p3c2 (4q +42)42 P4C, +p5C12 P5C12I oPl 0( 0 p3s, 0 P3C24 P5cP2 p5C2 0 7 Solution for F22: The partial derivatives necessary for the solution to F22 can be written as aVm +Vm 2p3s2c2 2p3s2(1 + q2) (B.24) qL 2p3s2q1 O ,8F p6 7Sech (pLq )+ p8 O and F= PP7sech 2(p7)+p8 0 (B.25) 4q 0 p9plosech (pIo2)+Pi. Therefore, 22(t) = M1 aVm + V +F, (B.26) 0q 8q 1 2 P2 P2P3c2 PFp( p PIC2I P3C2 P +2pc, 0 1 ppjloSecli(Pio/2)+P il Solution for F23: The solution to F23 can be written as F23 (t)= M K, 1 P  PP3C2 JP6 0 PIP212 2 pC P3C2 P1 + 2p3C2 P ' p2P p 2 C]L PC 17] Solution for F25: The partial derivatives necessary for the solution to F25 can be written as 8M .. 9 M .. 8M .. 8M .. q = q q ... q , 00 001 002 OOP S2 1, q O0 ao2 i2 and 1 + q21 2c 21 + C22 such that c2q1 1 M .. q1 q42 V0 L0 1 ix2 2 2 00 0 ~ 02 2c2,i + c,2 cj21 2 S2(q + 2)42 s2 s"q1 OG 0 3 sO S 0x61 B0 1x3 0 S12 Ox6 tanh(p47) p6sech2 (P7q ) (B.34) 0 tanh(pio02) P9sech2(plo,02) 2  2psq, 2pzsq24  2p3s2(4 + 2)J +P6P7secli(p74) +P 0 0 (B.27) (B.28) (B.29) whe .. where q 90, (B.30) 0, 8x 0 1 olx8 aF o, 0, o8 0 1 (B.31) (B.32) (B.33) 0 42 K (q K( q,) ao9 01x5 1q qml 0 x15 0 q2 qm2 (B.35) S 1(M.. Oav, .+ G aF F25(t) M 1q+ q+ ++ y 00 0o 0o 0o 1 F 2 PlP2 P2 p P2 p pC 4q1 4 2 c24+2q2 01x81 0 4 +2 c24 018] _lx01x2S2442 S 2(41 +42)42 01x81 S2 S21 01x8, +[ 1x3 o S12 016 01x3 0 s12 6Ol6 P2 p3C2 p, +2p3c2 (B.37) tanlp74) p6secl (p4)4 o o 0 0 tanil/,,, ) p9seci(Pli0%2)4 q,  q, o 0 0 q2 q,2 Solution for F41: The motor inertia matrix is constant, diagonal, and positive definite. The inverse is defined as 1 P13 P12PI3 1 0 0 P12 The solution for F41 can be written as F41(t) = JK, F41(t) 1 1 3 0 lP16 1 Pl 13 6 0 P12P13 0 O P17 P12P13 0 12P7 Therefore, 8K K(q 80 q,) , (B.36) 0', + 015 (B.38) Solution for F43: (B.39) (B.40) The solution for F43 can be written as F43(t)= J'K, 4 ( 1 0 ]Pl6 1 3l6 P12P1z3 0 p, J pJ 17 P 1 zP3 0 P 12P17 Solution for F44: The solution for F44 can be written as F44 (t)= JB, S1 P13 0 P14 0' 1 I3 P14 0 F44(t) (= P121 0 2 P 3 1 l2P zPl3 0 z 2L o p,5 pp 2 p, p,5 Solution for F45: The partial derivatives necessary for the solution to F45 can be written as 0a .. 800 B . o0 K ( 00 0, Oi L 01 1 0, I3 q,1 0,I3 0 q.1 0 Olx4 0 qm2 Olx4' 0 01x2 qm2 Ox2' q,1 q1 0 q) = 5 I OIA qml q Therefore, F4 (t) =J ( sJ SB K 01qm + Oq K + (qm 00 M 00 00 (B.48) p,13qm1 0 P,3qml 0 p13 (qm, q) 0 0 P,2qm2 0 P12,m2 0 P12(qm2 q2)] (B.41) (B.42) (B.43) (B.44) (B.45) (B.46) (B.47) 1 0IXII P12P13 0,,,x The Measurement Model (B.49) The partial derivative of the robot dynamics with respect to the states is 0 0 1 0 0 0 0 Oh = H(t) 0 0 0 0 0 0 (B.50) H, 0 0 0 0 H36 H37 Solution for H31: Define the transformation between the global coordinate system and the base coordinate system of the robot as an xyz Euler angle sequence of rotations, represented as [01 02 03], and a translation [,4 5 6 ], such that C02C3 C2S3 S02 4 G S)ScqCs q+CO.S0 SO!S.S3 +CSOIC03 SCIC02 0 sT CS42C43 + Sc4s3 sC41s42s43 + cS1c43 s41cC42 6 (B.51) 0 0 0 1 The transformation from one link coordinate system to another is a function of the link position. C2 s 0 a1 12T S2C12 C2C12 S12 12S2 (B.52) Ss21 C2S12 C12 CS, 0 0 0 1 where s2 = sin(q2), c2 = cos(q), s12 = sin(a12), c2 = cos(a12), and a, a, and S represent the constant link lengths, twist angles, and joint offsets, respectively [16]. The partial derivative of the transformation between link coordinate systems, with respect to the link positions, is defined as follows: s2 c2 0 0 1T c2C 2C12 0 0 (B.53) q2 c2S12 ss12 0 0 0 0 0 0 The solution for H31 can be written as O'T 1 f i'T 01 TP1 vect\ T T pT vect G TT pL q[ Oq 1 q, H31(t) = (B.54) G 01 1 G 1L 1 vect oTU'T. p vect GTOT 2T L v e c o OQ 2 0 1 O 2 P Mhi p [ 8ql J q2 Solution for H36: The partial derivatives of the global to base transformation matrix, with respect to the kinematic parameters, are defined as 0 cOS0f2C!3 SO S03 sOfsf2cO3 +C 1S3 0 0 C1S02S!3 SOIC03 S!S 2S3 +C 1c30 0 0 0 C1 C02 0 SOIC02 0 0 0  sf2cO S02S4 c2 0 SC 432C c SIC0S43 SOS02 0 crc2 cr4c2SO3 CO1S2 0 0 0 0 o c02s43 c02CO3 0 0  SfOIS43 +cr3cr13 S Sr1S23 COIS 0 0 crsf2S3O3 + SIC0 c1S02c3 SOS3 0 0 0 0 0 0 0 0 0 1 9OT 0 0 0 0 0000 0 0 0 0 OcT 0 0 0 1 905 0 0 0 0 0 0000 GOT 0, a02 (B.55) (B.56) (B.57) (B.58) (B.59) 0000 OcT 0 0 0 0 8056 0 0 0 1 0 0 0 0 The solution to H00can be written as The solution to H36 can be written as vect T( 0T(q) L p 00, LM ... vect ... vect{ OGOT(q) o ,L1 O T(q) Ip OO L 4 PM ^ (B.61) Solution for H37: The solution to H37 is dependent upon rotation from the global to local link coordinate systems. 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[97] K. Halvorsen, T. Soderstrom, V. Stokes, and H. Lanshammar, "Using an extended Kalman filter for rigid pose estimation," J. Biomech. Eng., vol. 127, pp. 475483, 2005. [98] P. D. Burns, "Slantededge MTF for digital camera and scanner analysis," Proc. PICS Conf. IS&T, pp. 135138, 2000. [99] S. Mu, J.D. Yamokoski, and S.A. Banks, JointTrack: An opensource program for measuring skeletal kinematics using modelimage registration. [Online]. Available: https://sourceforge.net/proj ects/j ointtrack/. BIOGRAPHICAL SKETCH Chris A. Lightcap was born in Philadelphia, Pennsylvania. He graduated from Lehigh University in 2004 with a B.S. in mechanical engineering, and received the Cornelius Award for his outstanding merit as an undergraduate student. He moved to Gainesville, Florida in 2004 to begin research in robotics and medical imaging at the University of Florida. He received his M.S. in mechanical engineering and became a Ph.D. candidate in the field of mechanical engineering in 2006. His research interests include orthopaedic biomechanics, medical imaging, and robot assisted surgery. PAGE 1 1 MEASUREMENT AND CONTROL ISSUES IN A NOVEL DYNAMIC RADIOGRAP HIC IMAGING SYSTEM By CHRIS ALAN LIGHTCAP A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008 PAGE 2 2 2008 Chris Alan Lightcap PAGE 3 3 To Kerry Robertson, and all my friends and family who have helped me reach where I am today. PAGE 4 4 ACKNOWLEDGMENTS I would like to thank my advisor, Dr. Scott Banks, for all of his support and dedication for the project. I thank my committee, Dr. B.J. Fregly, Dr. Warren Dixon, Dr. Tony Schmitz, and Dr. Manuel Bermudez, for their help and advice. I thank J.D. Yamokoski, Samuel Hamner and Jessica Allen for their various contributions to my dissertation. Last but not least, I thank Kerry for her loving support. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES................................................................................................................ .........9 ABSTRACT....................................................................................................................... ............11 CHAPTER 1 INTRODUCTION..................................................................................................................12 Parameter Identification: Bu ilding a Physical Model.............................................................14 Controller Development: Designing an Intelligent System....................................................15 Measurement Analysis: Assessi ng the Imaging Performance................................................16 2 PARAMETER IDENTIFICATION.......................................................................................18 Introduction................................................................................................................... ..........18 Geometric and Flexibility Calibration....................................................................................19 Literature Review............................................................................................................19 Methods........................................................................................................................ ...20 Experimental setup and procedure...........................................................................21 Twolevel geometric and flexibility calibration.......................................................24 Monte Carlo simulation............................................................................................27 Experimental assessment..........................................................................................28 Results........................................................................................................................ .....28 Conclusions.................................................................................................................... .31 Identification of Dynamic Characteristics..............................................................................32 Literature Review............................................................................................................32 RigidLink FlexibleJoint Model.....................................................................................34 Leastsquares parameter identification.....................................................................35 Trajectory optimization............................................................................................36 Estimation of Link Position.............................................................................................38 Experimental Setup and Procedure.................................................................................40 Experimental Results.......................................................................................................41 Conclusions.................................................................................................................... .46 3 NONLINEAR CONTROL AND ESTIMATION..................................................................48 Introduction................................................................................................................... ..........48 Literature Review.............................................................................................................. .....49 PAGE 6 6 Extended Kalman Filter Observer..........................................................................................53 The Extended Kalman Filter...........................................................................................53 RigidLink FlexibleJoint Robot (RLFJ) Model.............................................................54 Measurement Model........................................................................................................57 Observability in Nonlinear Systems................................................................................59 RigidLink FlexibleJoint Control..........................................................................................62 Theoretical Background..................................................................................................62 Control Objective............................................................................................................63 Error System for Filtered Tracking Error........................................................................65 Partial Stability Analysis for Filtered Tracking Error.....................................................66 Auxiliary Control and B ackstepping Error System.........................................................67 Partial Stability Analysis for Backstepping Error System..............................................69 Controller Stability Result...............................................................................................70 ClosedLoop Stability Result..........................................................................................71 Simulation..................................................................................................................... ..........72 Experiment..................................................................................................................... .........76 RealTime Implem entation Issues...................................................................................77 Experimental Results.......................................................................................................79 Conclusion..................................................................................................................... .........82 4 MEASUREMENT ANALYSIS.............................................................................................84 Introduction................................................................................................................... ..........84 Literature Review.............................................................................................................. .....84 Uncertainty Analysis........................................................................................................... ...86 Imaging Ge omet ry...........................................................................................................86 Covariance of Relative Motion.......................................................................................89 Endeffector tracking error.......................................................................................90 Measurement uncertainty.........................................................................................91 Image Sharpness................................................................................................................ .....95 Theoretical Improvement................................................................................................96 Experimental Methods.....................................................................................................99 Experimental Results.....................................................................................................100 Image Registration............................................................................................................. ...102 Methods........................................................................................................................ .102 Results........................................................................................................................ ...103 Conclusion..................................................................................................................... .......105 5 CONCLUSION.....................................................................................................................108 APPENDIX A OBSERVABILITY MATRIX..............................................................................................110 B OBSERVER DESIGN FOR TWOLINK ROBOT.............................................................115 LIST OF REFERENCES.............................................................................................................124 PAGE 7 7 BIOGRAPHICAL SKETCH.......................................................................................................132 PAGE 8 8 LIST OF TABLES Table page 21. DenavitHartenberg parameters and joint stiffness coefficients.............................................29 22. Parameter identification results for link dynamics.................................................................42 23. Parameter identification results for motor dynamics..............................................................43 31. Simulation results without payload........................................................................................75 32. Simulation results with payload........................................................................................... ..75 33. Comparison of numeri cal integration methods......................................................................77 34. Experimental results without payload....................................................................................81 35. Experimental results with payload.........................................................................................81 41. Kinematic errors for Blurred and Corrected image groups..................................................105 42. Kinematic differences between Bl urred and Corrected image groups.................................105 PAGE 9 9 LIST OF FIGURES Figure page 11. Dynamic radiographic imaging system..................................................................................14 21. Solution sequence for geometric and flexibility calibration...................................................21 22. Experimental setup of the PA106CE and CMM...................................................................22 23. Tooling ball apparatus.................................................................................................... ........23 24. Experimental test path.................................................................................................... ........23 25. Contour plot of mean position error.......................................................................................29 26. Mean position error before and after calibration....................................................................30 27. Endeffector positio n error at novel confi gurations and loads...............................................30 28. Sample identifi cation trajectory.......................................................................................... ....38 29. Motion capture markers on PA106CE..................................................................................39 210. Endeffector weight rack................................................................................................. .....40 211. Coordinate system definitions............................................................................................ ..44 212. Measured and predicted motor torque..................................................................................45 213. Stiffness torque and joint deflection.................................................................................... .45 31. Link position tracking errors in simulation while loaded.......................................................74 32. Link position estimation errors in simulation while loaded...................................................74 33. Mitsubishi PA106CE robot with markers.............................................................................76 34. Experimental link position tr acking errors while loaded........................................................80 35. Joint stiffness coefficients from experiment while loaded.....................................................80 41. Imaging configuration of robotic imaging system.................................................................89 42. Norm of translation part of manipulator Jacobian..................................................................94 43. Norm of rotational part of manipulator Jacobian...................................................................95 44. Spatial frequency respons e of motion blurred images............................................................98 PAGE 10 10 45. Relationship between rela tive motion and image contrast.....................................................98 46. Experimental setup of dualrobot imaging system...............................................................100 47. Relative contrast in blurre d and blurred/corrected images...................................................101 48. Image sharpening from LucyRichardson algorithm...........................................................101 49. Undistorted ra diographic images.........................................................................................104 PAGE 11 11 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MEASUREMENT AND CONTROL ISSUES IN A NOVEL DYNAMIC RADIOGRAP HIC IMAGING SYSTEM By Chris Alan Lightcap May 2008 Chair: Scott A. Banks Major: Mechanical Engineering A novel dynamic radiographic imaging system, one which tracks the patient during an activity, can provide measurements of natural sk eletal motion. It is believed that accurate measurements of skeletal kinematics can im prove both the treatment and diagnosis of musculoskeletal conditions. However, the measur ement and control obstacles of such a proposed system have not been overcome. Using heavy ra diographic equipment in combination with lightweight manipulators leads to joint deflections an d degraded tracking performance. Furthermore, relative motion between imaging components lead s to motion blur and degraded measurement performance. Novel static and dynamic identific ation methods are developed and executed in order to ascertain kinematic, inertial, frictional, and stiffness properties for modelbased observer and controller methods. An extended Kalman f ilter and adaptive reduced parameterization Lyapunovbased controller are developed in orde r to control the robotic imaging system. The effect of motion blur on 3D to 2D image registration is also explored, given the anticipated relative motion between components. PAGE 12 12 CHAPTER 1 INTRODUCTION Each year, an estimated 1 of 7 Americans is wounded by a musculoskeletal disease or injury [1]. In the year 2004, The American Academy of Orthopedic Surgeons reported that over 150 million physician visits were due to musculoskeletal conditions [2]. These numbers are expected to increase. The elderly are primarily effected, and the numbe r of individuals over 50 years of age is expected to double between the years 1990 and 2020 [3]. The current methods of treatment have limitations. Advances in the fi eld of musculoskeletal research are becoming increasingly more important with time as the num ber of cases increase. Effective treatments will help relieve pain for patients and help reduce me dical costs. Our society spends an estimated $254 billion each year as a result of musculoskeletal conditions [4]. Therefore, improvements in the treatment and diagnosis are critical to keep pace with our growing and aging society. This rising awareness of musculos keletal conditions and their imp act has contributed to the World Health Organization announcing the creati on of The Bone and Joint Decade. In 2002, President Bush followed their lead by appoi nting the years ranging from 20022011 as the National Bone and Joint Decade. The goal of thes e declarations has been to raise awareness about musculoskeletal disorders and promote res earch and development for their diagnosis and treatment [5]. Since then, it is no coincidence that 55 national governments and more than 150 organizations worldwide have recogn ized this global initiative. Th e sobering reality is that more and more people will be suffering from these musc uloskeletal conditions over time and we must be prepared to diagnose and treat pa tients with greater effectiveness. The current methods for diagnosis of musculoskeletal condition s have limitations and must be improved upon. A typical examination for a gene ral musculoskeletal injury begins with the examiner interviewing the patient to collect information, such as, cause of injury, abnormal pain, PAGE 13 13 and other symptoms. Next, a medical doctor administ ers a physical examination to test the range of motion and function of the joint. If further action is needed, the doctor asks the patient to remain still for a collection of static radiographi c or magnetic resonance images to be captured. The problem surfaces when, after these examin ations, there is still not enough information and the results are inconclusive. In this case, observation of the patient while performing a dynamic activity, one which requires function of the ailing joint, coul d provide this missing information. It is logical to investigate the in vi vo kinematics of the dysf unctional joint, given the dynamic nature of the musculoskeletal system, ye t, this is not common practice. It would be appealing to confirm the results of a past ex amination or to test a new hypothesis with measurements of skeletal motion. However, ther e is no imaging modality currently in place to accomplish this goal. The next best measurement tool to estimate skeletal motion is a motion capture system. Yet, even with the measuremen t of skinmounted motion capture markers one cannot ascertain skeletal motion without artifacts from layers of soft tissue and muscle. The solution is a dynamic radiographic imaging platform, which is designed to have the xray hardware mounted on two robotics arms. Mounting the radiographic equipment on fast, accurate, and lightweight robotic arms is a novel idea. Yet, before achieving this goal, there await challenges and obstacles that are largely unresolved. Such a system would require accurate tracking performance to meet imaging constraints and acquire useful and meaningful images. It would also require accurate imaging performan ce to establish an accurate measurement of skeletal motion. The following dissertation discus ses the challenges in designing such a system, and respective procedures or methods to overcom e them. The subsequent pages in this chapter, will introduce the challenges behind parameter identification, contro ller development and measurement analysis, in that order. PAGE 14 14 Figure 11. Dynamic radiographic imaging system Parameter Identification: Building a Physical Model An accurate physical model of the robotic im aging system will improve both measurement and tracking capabilities. A dynamic model can serve as a backbone for understanding system performance. The accuracy of imageregistrati on will depend upon the accuracy of endeffector motion for both robots. The ende ffector motion can be estimated with a modelbased extended Kalman filter, but knowledge of geometric and dynamic properties are needed beforehand. A dynamic model can also be used in the de velopment of modelba sed control algorithms in order to achieve superior track ing performance. The model can be used as a definite truth or an initial guess in many modelbased contro l algorithms. Increased knowledge of system dynamics will lead to better control and tracking performance. However, errors in the system model will degrade system performance, and therefore an accurate and reliable identification method is essential. For these reasons, identifi cation of dynamic parameters, such as inertia, mass, friction, and flexibility, is critical for both improved imaging and tracking performance. PAGE 15 15 The theoretical development and experimental re sults of the static a nd dynamic identification processes are explained in Chapte r 2, Parameter Identification. Controller Development: Designing an Intelligent System A robotic imaging system require s superior tracking ability, in order for the xray source and flatpanel detector to be properly aligned as a patient exercises his or her joint. This requirement provides a rationale for development and evaluation of a high performance controller for this appl ication. The Mitsubishi PA106CE manipulator possesses the typical nonlinear behavior seen in robotic systems ha ving revolute geometry. In addition, the harmonic drive transmissions, which provide a lightweight alternative to the geared transmissions, have undesirable intrinsic properties, su ch as nonlinear flexibility, fric tional losses, kinematic error, and resonant vibration. Adding to the problem, jo int resolvers only record the input angle of rotation, or motor position, and not the true outp ut angle of rotation, or link position. The true position is deflected on account of the transmission flexibility. The sum of these problems results in a highly complex and nonlinear syst em and a challenging control problem. The methods classified as linear control are at tractive because they exhibit advantages such as global exponential stability a nd guaranteed transient performa nce. A linear system embodies familiar properties which facilitate the design process and instill comfort. The stability of a linear system is determined from the roots of the charac teristic equation or the eigenvalues of the statespace matrix. The response of a linear system to a sinusoidal input is also a sinusoidal. And most importantly, the transient response of a linear sy stem can be determined analytically. These properties allow for one to desi gn controllers with rootlocus and frequencydomain methods. However, nonlinear systems do not possess these desirable charac teristics. They can have more than one equilibrium point and exhibit behaviors such as lim it cycles and chaos. Often, the time response of a nonlinear system wont have an analytic solu tion. The linear control methods PAGE 16 16 described may lead to loss of controllabi lity, destabilization, a nd undesirable transient performance in a nonlinear system. The nonlinear nature of the system enc ourages a design based on Lyapunov stability analysis. These methods have had success with highly nonlinear systems. The addition of an observer for nonlinear state estimation fits well into the design process. Th e text within Chapter 3, Nonlinear Control an d Estimation, highlights the design of a rigidlink flexiblejoint controller with link position observer. Measurement Analysis: Assessing the Imaging Performance The ultimate goal of the proposed radiographi c imaging system is the measurement of skeletal motion. It is clear that the success of the project depe nds greatly upon the accuracy and precision of this measurement. As a result, perf ormance of the imaging system in simulation and in a realtime tracking experiment is evaluated and discussed. The principal challenge in estimating motion from radiographic images stems from the dynamic nature of the imaging system. Since the xray source and flatpanel detector are moving relative to the patient, and each other, it is lik ely that the recorded images will contain motion blur. Fortunately, past research has shown that one can improve contrast in an image when the relative motion between objects is known. However, uncertainties in their motion estimates will lead to left over or residual motion blur. Furthermore, motion blur will most likely degrade the quality of the imageregistration algorithm, but the extent of this damage is unkno wn. It may be possible to improve registration accuracy though motionbased deblurring methods known as image deconvolution. In Chapter 4, Measurement Analysis, an estimate for residual motion blur in the system is determined, based on the uncertainties in endeffector a nd tracking object velocities. In addition, the performance of image deconvolution given the esti mated relative motion is evaluated through an PAGE 17 17 experiment and compared to results from simulati on. As a last step, the e ffects of motion blurred and blurred/corrected images on modelba sed imageregistration are determined. PAGE 18 18 CHAPTER 2 PARAMETER IDENTIFICATION Introduction The imaging performance of a dynamic radiogr aphic imaging system will be dependent upon accurate endeffector motion estimation. Join t resolver measurements are highly precise and lead to excellent positiona l repeatability for consistent loading conditions. However, high repeatability does not guarantee accurate positio ning performance. Accurate knowledge of link geometries and joint flexibilities are necessary for error compensati on. It is essential to have an effective calibration procedur e to determine the kinematic and dynamic parameters of an assembled robot, since these parameters are specific to a given manipulator. The Mitsubishi PA106CE is a six degree of fr eedom manipulator equi pped with AC servo motors and harmonic drive transmissions. The large gear ratio and compact design of the harmonic drive makes it ideal for many robotic a pplications. However, these transmissions are known to exhibit nonlinear flexibili ty and frictional losses, ki nematic error, and resonant vibration, making it difficult to estimate the tr ue endeffector position from joint position alone [6], [7]. In addition, geometric modeling errors, common to all robotic manipulators, further increase endeffector positioning error. Link lengths, twist angles, and joint offsets reported by the manufacturer are only accurate to manufacturing and assembly tolerances. Small errors in these DenavitHartenberg (DH) parameters can pr opagate to create sign ificant errors in endeffector position [8]. A flexible robot model and an e ffective calibration procedure are required to characterize geometric uncertain ties and joint flexibilities. The tracking performance of a dynamic radiogr aphic imaging system will be dependent upon accurate model identificati on. Precise and reliable iden tification methods are well documented for rigidlink rigidjoint (RLRJ) robots. However, the Mitsubishi PA106CE is PAGE 19 19 assembled with harmonic drive transmissions and is known to deflect under load. Identification must account for these effects in order to deve lop an accurate physical model. The methods for robot manipulators with flexible joints are still in development. The complexity of the identification process is reduced by separati ng the procedure into a static and dynamic identification tasks. Geometric and Flexibility Calibration This section discusses the design and execu tion of a twolevel nonlinear optimization algorithm for calibration of geometric and flexibil ity parameters in a se rial manipulator, using measurements from a coordinatemeasuring machine (CMM). This method offers greater simplicity than [9] because it does not require computation of the generalized Jacobian matrix. In addition, a Monte Carlo simulati on of positioning error was perf ormed based solely on joint resolver uncertainties, establishing the absolute bestcase robot positioning accuracy. The fidelity of the model was assessed from an experimental comparison of robot positioning accuracy, for a novel set of configurations and e ndeffector loads, before and after the calibration procedure. The generalization of this met hod, and its extension to other se rial manipulators, is also discussed. Literature Review Many researchers have implemented leastsqua res or nonlinear optimi zation techniques to calibrate robotic manipulators. Taghirad et al. developed a model of harmonic drive stiffness through a controlled loading experiment [10]. They defined joint flex ibility from the relationship between joint torque and joint deflection measur ements. This method could not be implemented on the Mitsubishi PA106CE because there is no direct measurement of joint deflection. Kennedy and Desai modeled the stiffness componen t of joint torque on a Mitsubishi PA106CE PAGE 20 20 as a function of the applied load [11]. However, their research did not establish the physical model between the applied load and joint deflection. The inclusion of both geometric a nd nongeometric errors appears in [12] and [13], but these authors used a set of generalized error functions and di d not identify physically meaningful parameters. A physical relationshi p is desirable for 1) extrap olation of points outside the calibrated workspace, and 2) a kinematic and dynamic model used in modelbased control algorithms. Khalil et al. recently developed a nonlinear leastsquares method to calculate physically meaningful geometri c and flexibility parameters [9], yet the results from this paper provided only a best case maximum global erro r of 14.1 mm. An overview of methods for robot calibration is discussed in [14]. Methods The computational resources needed for the solution of an optimization problem greatly increase with the addition of more design variables [15]. Hence, a multilevel optimization method is employed to reduce the number of de sign variables in a single optimization and increase the speed of convergence (Fig. 21). First, a flexible geometric model is proposed by an outerlevel optimization. Next, an innerlevel optimization determ ines the bestfit homogeneous transformation between the CMM and robot coordi nate systems. This innerlevel optimization promptly converges to a solution given a fixed se t of DH parameters and flexibility coefficients, making it unnecessary to burden the parent op timization algorithm with the homogeneous transformation. Next, using this model, the defl ected position and orientation are computed from measured joint torques (via motor current) and compared to their respective CMM measurement. The final model is realized upon numerical co nvergence of this twolevel optimization. PAGE 21 21 Figure 21. Solution sequence for geom etric and flexibility calibration Experimental setup and procedure A Mitsubishi PA106CE robot was positioned on a 0.6 m steel frame table so that its endeffector could span the entire workspace of a Brown and Sharpe, MicroVal PFX454 touch trigger probe CMM (Fig. 22). Th e CMM operates by probing the su rface of a spherical tooling ball and calculating the leastsquares estimate of the centroid of the ball. A coordinate system was defined from the location of thre e tooling balls in the CMM workspace [16]. It had a calibrated measurement uncertainty of 12.1 m over its full work volume. To facilitate CMM measurement, a toolingba ll apparatus was constructed and mounted to the endeffector of the robot (F ig. 23). It was comprised of two parallel aluminum plates, separated by four 31 cm aluminum rods; the ma ss was 2.4 kg. The extension of tooling balls into the CMM workspace reduced the chance of imping ement between the manipulator and the CMM table. The backplate was attached to the endeffector of the r obot, and the frontplate contained a set of three tooling balls. The transformati on between the CMM coordinate system (CMM) and 1. Propose Flexible Geometric Model 2. Homogeneous Calibration 3. Calculate Deformed Position and Orientation 4. Measure Position and Orientation 5. Final Solution NO YES CONVERGED? PAGE 22 22 a userdefined tooling ball coordinate system (BALL), TCMM BALL, was determined from measurements of the location of the tooling balls. The robot was commanded to move along a Ca rtesian pattern of uni formlydistributed points in the CMM working volume (Fig. 4). At each static configura tion, the CMM measured the position and orientation of th e tooling ball coordinate system. The experiment was repeated 10 times for each of three loading scenarios: {0, 22, and 44} N loads. External loads were provided by additional weights lowe red onto the weight rack and secured with a set screw and shaft collar. The external load was transmitted through the endeffector of the robot and therefore did not create deflection in the rods (Fig. 23). Maximum deflection of the apparatus from its own weight was 0.030 mm. The approximate touch force of the CMM was 0.2 N, creating a maximum deflection of less than 0.001 mm from contact. These defl ections were negligible comp ared to the robot positioning error from uncertain geometric parameters and joint compliance. Figure 22. Experimental set up of the PA106CE and CMM PAGE 23 23 Figure 23. Tooling ball apparatus 600 800 1000 100 0 100 200 400 500 600 700 X (mm)3 7 2 1 5 627 21 224 9 10 15 1610 8 3 4 9 26 20 23 Y ( mm ) 11 14 17 5 8 2 25 19 24 12 13 18 6 7 1 Z (mm) Geometric and Flexibility Calibration Experimental Assessment Figure 24. Experimental test path Weight rack Bolted to endeffector Tooling balls measured via CMM 31 cm PAGE 24 24 Twolevel geometric and flexibility calibration A flexible geometric model was defined from a set of 30 design variab les (Table 21). It consisted of departures in link lengths, twist angles, joint offsets, and joint angles from their nominal values, represented by ai, i, di, i, and i, respectively [16], and cubic spring coefficients for joints 2, 3, and 5, represented by Klin,i and Kcub,i (joint 1 connects the base of the robot to the first link). The flexibility in the harmonic drive transmission was modeled with the torsional spring model 3 linmcubm K qqKqq, (2.1) where ()nqt and ()n mqt denote the link and motor position, respectively, Klin and Kcub nn are constant, diagonal, positivedefinite matr ices comprised of the linear and cubic joint stiffness coefficients, respectively, and ()nt denotes the joint torque. The departure in the Hayati parameter [17], labeled as 2, was substituted for d2 to eliminate the illconditioned transformation that exis ts between two parallel joints (i.e. joints 2 and 3). Nominal geometric parameters for the Mitsubishi PA106CE geometry were acquired from the manufacturer [18] and are represented in Table 21. Joints 1, 4, and 6 were subject to negligible torques in the tested configurations, and as a result, stiffness coefficients were not estimated for them. Link flexibility was not taken into account in this model based on previous research with the Mitsubishi PA106CE [9]. A twolevel, LevenbergMarqua rdt nonlinear leastsquares optimization was implemented in MATLAB (Mathworks, Natick, MA) to solve fo r model parameters. The outerlevel optimization sought to find a flexible geometric model, consisting of 24 link parameters and 6 stiffness coefficients that satisfied PAGE 25 25 outerf min arg, (2.2) where 2 2 2 2 2 1 2 1 2 2 max1tol i i tol i i tol i i tol i i tol i i N i m i i N i m i i outerd d a a x x r f and i cub i lin i i i i iK K d a, where ix, m ix, i, and m i represent the estimated and measured endeffector positions and orientations (parameterized as a ngular rotations), respectively, in the robots coordinate system, and N = 27 is the number of robot configurations. Th e estimated location of the tool frame is a function of geometric and flex ibility model contained in Position errors were normalized with respect to the total length of the manipulator, rmax = 1 m, therefore, an orientation error of one radian corresponded to a distance error of one unit at full extension. The remaining terms in the cost function, which represent errors in estimated DH parameters, were normalized with respect to the approximate tolerances, ai,tol = 0.25 mm, i,tol = 1x103 rad, di,tol = 0.25 mm, i,tol = 1x103 rad, and i,tol = 5x104 rad, and then scaled with the weighting factor = 1.6x104. Large increased the cost associated with DH parame ter error, and thus forced the optimization algorithm to account for location er rors without modifications to the geometric model. Instead, the optimization method adjusted torsional spri ng coefficients to minimize the given cost function. The values for m ix, and m i, were determined from the pos ition and orientation of the endeffector in the robot coordinate system, as measured by the CMM, T T T TBALL M EE CMM BALL BASE CMM BASE M EE ,, (2.3) PAGE 26 26 where TCMM BALL is the transformation between CMM a nd tooling ball coor dinate systems, TBALL M EE is the known constant transformation between endeff ector and tooling ball c oordinate systems, and TBASE CMM is the transformation between robot and C MM coordinate systems. The transformation TCMM BALL was calculated from the CMM m easurement of the tooling balls, TBALL EE was determined from a separate measurement of the toolingball apparatus, and TBASE CMM was computed from the innerlevel optimization. For each iteration in the outerlevel optimi zation, defined in (2.2), an innerlevel optimization was performed using the updated flexible geometric model. The second optimization algorithm searched for the six robot coordinate system parameters that satisfied innerf min arg, (2.4) where M j m j j M j m j j innerx x r f1 2 1 2 2 max1 and z y x o cmm basep where o cmm basep_ is the origin of the CMM coordinate system in the robot coordinate system, x y and z are a set of bodyfixed angul ar rotations describing th e orientation of the CMM coordinate system with respect to the robot coordinate system, and M = 8 is the number of robot configurations. The measured location of the tool frame, obtained from (2.3), is a function of robot coordinate system parameters contained in The subscript j has been introduced to differentiate from configurations in the outerlevel (subscript i). The total kinematic model has 30 parameters, consisting of 24 link parameters a nd six robot coordinate system parameters, and is minimum, complete, and parametrically continuous (MCPC) [19]. Link parameters were determined through the outer optimization loop, wh ile robot coordinate system parameters were PAGE 27 27 determined in the inner optimization loop. The opt imization was repeated with different initial guesses to mitigate the problem of finding local minima. The initial seed for the innerlevel optimiza tion, provided in (2.4), was estimated from endeffector measurements along a straight line. The robot followed a linear path along its xand yaxes allowing for leastsquare s bestfit approximations to BASE CMMx and BASE CMMy The zaxis and reformulated yaxis were determined from BASE CMM BASE CMM BASE CMMy x z (2.5) and BASE CMM BASE CMM BASE CMMx z y (2.6) The last step was taken to ensure orthogonality between all measured axes The initial seed of Euler angle rotations was determined from the rotation matrix RCMM BASE [16]. BASE CMM BASE CMM BASE CMM CMM BASEz y x R (2.7)Monte Carlo simulation Joint resolver uncertainties cannot be reduced through calibra tion, and therefore, they determine the lower bound for endeffector posit ioning uncertainties. A Monte Carlo simulation using the six joint positions as simulation variables was implemented to determine the lower bound for robot position errors. For the PA106CE, the manufacturer reported a .64 arc min (0.011 deg) error in the angular position measurement of each link [20]. The probability distribution of each uncertain joint position wa s assumed to be Gaussian with a standard deviation equal to the reported bounds of that parameter. A fo rward kinematic analysis was performed for 100,000 cycles at each of the 200 uniformlyselected endeffector positions in a given plane. The experiment was repeated for e ach crosssectional plane located between z = 400 mm to z = 700 mm in 100 mm increments. PAGE 28 28 Experimental assessment The fidelity of the flexible geometric model was examined with a collection of 10 points not included in the data used for model calibra tion (Fig. 24). The new set of data included configurations as far as 100 mm from the original control volume and loads up to 90 N, twice the original maximum load. The confi guration order and loading order were selected arbitrarily. A comparison was made between the full model and a linear stiffness model (Kcub = 0). Results The Monte Carlo simulation determined a mean endpoint position error of 0.197 mm 0.096 mm in the xyplane of z = 500 mm (Fig. 25) Results from parallel planes at z = 300 mm and z = 700 mm deviated by less than 0.001 mm. The mean/peak values of position error prior to geometric and flexibility calibra tion were 0.58/1.22 mm, 1.02/1 .50 mm, and 1.80/2.45 mm for the unloaded, 22 N loaded, and 44 N loaded ca ses, respectively (Fig. 26). The standard deviations of position error, within one confi guration, did not exceed 0.014 mm, 0.024 mm, and 0.023 mm, for each loading case, respectively. Af ter optimization of robot geometry and joint stiffnesses, the mean/peak values of position error decreased to 0.32/0.66 mm, 0.28/0.58 mm, and 0.33/0.71 mm, respectively. Mean orie ntation error, defined as the mean L2 norm of the differences in Euler angle rotations (xyz) betwee n measurements, was reduced from 1.2 deg, 1.2 deg, and 1.1 deg to 0.05 deg, 0.04 deg, and 0.03 deg, in each of the three loading cases, respectively. Linear joint stiffnesses iden tified by the twolevel optimization were 1.98x104 Nm/rad, 1.27x104 Nm/rad, and 2.56x103 Nm/rad for joints 2, 3, and 5, respectively. The cubic joint stiffnesses were 3.35x107 Nm/rad3, 1.46x107 Nm/rad3, and 3.83x107 Nm/rad3, respectively. Mean positioning errors from uncalibrated c onfigurations decreased from 1.77 mm to 0.55 mm with the linear stiffness model, and to 0.56 mm with the nonlinear stiffness model (Fig. 27). The peak value of position error decreased from 4 mm in the uncalibrated model to 0.92 mm and PAGE 29 29 1.04 mm with linear and nonlinea r stiffness models, respectivel y. There was at least a 75% reduction in peak positioning error with both models. Mean orientation errors decreased from 2.4 deg to 1.2 deg with both models. 0 1 40 1 50 1 50 1 60 1 60 1 70 1 70.180 1 80 1 80 1 90 1 90 1 90 20 20 20 22 10 2 10 2 10 2 1YCoordinate of Robot (mm)XCoordinate of Robot (mm) XYPlane of Z=500 mm Position Error in Millimeters 0 100 200 300 400 500 600 700 400 500 600 700 800 900 1000 Figure 25. Contour plot of mean position error Table 21. DenavitHartenberg paramete rs and joint stiffness coefficients Joint No. ai (mm) i (rad) di (mm) i (rad) i (rad) Klin,i (Nm/rad) Kcub,i (Nm/rad3) 1 0.33 4.9x104 0.09 5.9x103 n / a 2 .46 1.2x104 n / a .8x103 1.0x103 1.98x104 3.35x108 3 1.09 2.3x103 1.26 2.9x103 n / a 1.27x104 1.46x108 4 0.48 2.1x104 0.22 8.1x103 n / a 5 0.30 3.7x105 0.10 1.5x103 n / a 2.56x103 3.84x108 6 3.44 .1x104 0.17 1.3x102 n / a PAGE 30 30 0 5 10 15 20 25 0 1 2 3 Position Number (a) Before CalibrationPosition Error (mm) 0 5 10 15 20 25 0 1 2 3 Position Number (b) After CalibrationPosition Error (mm) Unloaded 22 N Loaded 44 N Loaded Figure 26. Mean position erro r before and after calibration 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 Position NumberPosition Error (mm) Without Model With Linear Stiffness Model With Nonlinear Stiffness Model Figure 27. Endeffector position error at novel configurations and loads PAGE 31 31 Conclusions The 30 parameter flexible geometric model ha s been demonstrated to improve static positioning accuracy for the PA106CE robot. E ndeffector positioning errors decreased substantially (Fig. 26), showing 50% to 80% reductions in m ean position errors across all robot configurations. Measured positioning errors (0.3 mm) differed by less than 0.1 mm from their theoretical minimum determined from Monte Carl o simulation. DH parameters were all within reasonable manufacturing and a ssembly tolerances for the PA106CE. The large value for a6 is most likely from error in measuring the transf ormation between endeff ector and tooling ball coordinate systems, and in assembling the to oling ball apparatus on the endeffector. The superior performance of the model while extrapolating outside the calibrated configurations is a good indication of its strength. The linear stiffness model performed surpri singly well in the experimental assessment (Fig. 26). Nonlinearities that exist in harmonic drive transmissions are most prominent at small operating loads [6], and the joint torques in this experiment, for the most part, fall outside of this region. It may be worth considering a linear s tiffness model given its simplicity and similar performance. The method described in this sectio n can be generalized to any serial manipulator represented by DH parameters. The stiffness para meters returned from the optimization of a perfectly rigidjoint manipulat or would approach infinity. Accuracy standards for medical robot syst ems depend upon the application. Radiation therapy is one of the most demanding applicatio ns and some believe positioning errors greater than 0.5 mm increase patient risk [21]. Robotic applications in or thopaedic surgery typically seek to achieve positioning errors less than 1 mm or 1 degree for bone cutting [22]. It appears these demanding requirements can be met by the PA106C E utilizing the geometric and flexibility calibration method described here. The combina tion of a detailed physical model and its PAGE 32 32 accompanying control design will provide an accurate measurement and control system suitable for demanding dynamic applications with the PA106CE robot arm. The identification of mass, inertial, and frictional propertie s are necessary to complete th e dynamic model of the PA106CE, and are discussed in the next section. Identification of Dynamic Characteristics This section discusses a novel method for dynami c parameter identification of a rigidlink flexible joint robot model. The pr ocedure is divided into two parts. First, the rigidlink flexiblejoint robot dynamics are parameterized in a ne w form to exclude the stiffness torque. Unknown parameters are estimated from a rich set of motion data. Link position is estimated with groups of motion capture markers on the surface of th e robot. Second, the stiffness torque is reconstructed from the identified parameters, a nd a torsional spring model is deduced from the data. This method offers a better conditioned syst em of equations for weaklyflexible systems, and more flexibility in the choi ce of torsional spring model. Th e twostep identification method is experimentally evaluated on a Mitsubishi PA106CE. Literature Review Typically, parameter identification of robot manipulators is performed in the frequencydomain or timedomain. Frequencydomain iden tification methods reduce the robot dynamics to a pair of second order linear differential equati ons under the small angle assumption. Parameters are identified through non linear optimization with a cost f unction defined as the sum of the squared differences between predicted and meas ured outputs of the system at different frequencies [23]. Design variables can be chosen as the polynomial coefficients which define the system transfer function or the phys ical parameters of the robot. Li et al. reported a frequencydomain techni que that combined the finite element method with experimental modal analysis to iden tify stiffness and damping coefficients [24]. Link PAGE 33 33 dynamics were excited with mechanical impulse s and the responses were compared to an analytic model to determine robot parameters. The objective function was defined as the sum of the squared error between the natural frequencie s obtained from the analytical model and those obtained from the experiment. Frequencydomain methods provide good estimates for parameters in a linear model, but fail to identify nonlinearities, such as mass center and Coulomb friction parameters. Alternatively, the dynamic model of the robot can be obtained through direct experimentation. Taghirad et al. developed a model of harmonic drive stiffness through a controlled loading experiment of a onelink manipulator [10]. AlbuSchaffer et al. proposed the identification of joint flexibilities prior to assembly [25]. They derived a linearized form of the flexiblejoint dynamics under the small angle a ssumption and satisfied th is condition by fixing the motor position with an electromagnetic brake. They excited the link dynamics with mechanical impulses and measured the response of the system. Nonlinear optimization was used to determine flexibility and damping pa rameters from the impulse response. Identification in the timedomain is most often performed with linear leastsquares methods. A solution is achieved through linear pa rameterization of the robot dynamics into a regressor matrix, containing functions of the known joint displacement, and an array of unknown parameters [26]. An overdetermined system of equati ons is formed with sampled data and solved with leastsquares techniques. Desired trajectories for identification can be determined by selecting appropriate optimization criteria. Researchers often minimize the condition number of the regressor matrix, or maximize its minimum singular value, thus improving the robustness of the numerical solution to measurement noise [27]. Calafiore et al. minimi zed the covariance matrix of PAGE 34 34 parameter estimates by maximizing the Fisher information matrix [28]. The design variables in both cases are the Fourier coefficients or polyno mial coefficients defining the proposed robot trajectory. Pham et al. proposed a linear parameterization of the flexiblejoint dynamics and explained two methods for parameter identification [29]. The first method requires augmenting the sensing capabilities of the robot with additional joint encoders. Th e latter identification method compensates for unmeasurable link position, but the link masses and Coulomb friction parameters are lost. Pham et al did not discuss trajectory op timization for improved parameter estimation. RigidLink FlexibleJoint Model The nlink rigidlink flexiblejoi nt (RLFJ) robot dynamics can be expressed as 0 ) ( ) ( ) ( ) ( ) ( m mq q K q F q G q q q V q q M (2.8) u q q K q B q Jm m m ) ( (2.9) where nt q t q t q ) ( ), ( ), ( and n m m mt q t q t q ) ( ), ( ), ( denote the position, velocity, and acceleration of the link and mo tor angle, respectively, n nq M ) ( represents the positivedefinite inertia matrix, n n mq q V ) ( represents the centripetalCoriolis matrix, nq G ) (and nq F ) ( denote the gravitational and frictional effects of the link dynamics, respectively, nB J K are constant, diagonal, positivedefinite matrices representing joint stiffness, motor inertia, and motor viscous friction, respectively, and nt u ) ( denotes the motor torque [30]. Accordingly, the onelink RLFJ robot dynamics can be written as 0 ) ( ) ( ) cos( ) sin( m y xq q k q v q sign c q mgl q mgl q I (2.10) PAGE 35 35 u q q k q b q Jm m m ) ( (2.11) where ) ( ), ( ), ( t q t q t q and ) ( ), ( ), ( t q t q t qm m m denote the position, velocity, and acceleration of the li nk and motor angles, respectively, and b J k v c m I , are the unknown parameters representing link inertia, link mass, Coulomb and viscous friction, joint stiffness, motor inertia, and motor viscous friction, respectively. Leastsquares parameter identification The onelink RLFJ robot dynamics, given by (2.10) and (2.11), can be linear parameterized as the following expression: u b J k v c mgl mgl I q q q q q q q q sign q q qy x m m m m0 0 0 0 0 0 0 0 ) ( ) cos( ) sin( (2.12) To identify elements in the regressor matrix, motor positions, link positions, and motor torques are recorded as the link follows a desired tr ajectory. The unknown model parameters are then determined by forming an overdetermined system of equations from the measured data. The problem can be simplified into the following expr ession and solved with leastsquares methods. x q q q q q q Ym m m ) , ( (2.13) where x Y Y YT T 1. The moment of inertia obtained from a singlejoint experiment is a composite of all links between the endeffector and th e moving joint. Individual link inertias can be obtained in a sequential manner using the parallel axis theorem and group inertia properties. PAGE 36 36 The formulation of the system of equations in (2.12) is problematic for a robot with weak joint elasticity. In this scenario, the joint defl ections are small compared to the link and motor kinematics, resulting in a poorlyscaled and illconditioned matrix. The condition number can be reduced by rewriting the onelink dynamics to eliminate the stiffness term as follows: u q b q J q v q sign c q mgl q mgl q Im m y x ) ( ) cos( ) sin(, (2.14) The dynamics are then parameterized as u b J v c mgl mgl I q q q q sign q q qy x m m ) ( ) cos( ) sin(. (2.15) The formulation in (2.15) is better posed fo r a robot with weak joint elasticity. The omission of joint deflection from the regressor matrix eliminates the scaling problem. Joint stiffnesses are determined subsequent to the identification of inerti al and frictional model parameters. The stiffness torque, or torque transmitted through the torsional spring, is determined from the following expression. ) ( ) ( ) cos( ) sin( q q K q v q sign c q mgl q mgl q Im y x s (2.16) A suitable stiffness model can be selected by ex amining the relationship between joint deflection and stiffness torque obtained in (2.16). Trajectory optimization In the dynamic identification of RLRJ manipulators, the desired link trajectory is carefully selected to sufficiently excite all dynamics. Th e optimization criteria are commonly chosen as PAGE 37 37 the condition number of the regressor matrix or th e determinant of the Fisher information matrix. The case of RLFJ robot dynamics is no different An optimal excitation trajectory can be determined from the constr ained nonlinear optimization: )) , ( ( minm mq q q q q Y cond max max max min) ( ) ( q t q q q t q q t sm m (2.17) where the regressor matrix is now a function of both the moto r position and link position. The excitation trajectory, or motor position, can be parameterized as a finite Fourier series [27]. N n f f n f f n o mt n n b t n n a q t q1cos sin ) ( (2.18) The design variables are the amplitudes of the fundamental frequency and N 1 harmonics. The fundamental frequency should be ch osen near the natural frequenc y of the system if the motor command cycle and joint position sa mpling frequencies will permit. In the case of RLFJ robot dynamics, the re gressor matrix is a function of both motor position and link position. However, link position cannot be parameterized independent of motor position because it is a function of the robot dynamics. It should be determined with an approximation of the robot model and forward dyna mics. The link acceleration can be estimated from an approximation of the robot dynamics, ) ( ) ( ) cos( ) sin(1 m y xq q k q v q sign c q mgl q mgl I q (2.19) and, given the initial state of the robot, can be integrated to obtain link positions and velocities. An exciting trajectory for link 2 of the Mitsub ishi PA106CE robot is shown in Fig. 28. PAGE 38 38 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 Time (seconds)Joint Position (rad) Motor Position Link Position Figure 28. Sample id entification trajectory Estimation of Link Position Typical robotic systems are assembled with one joint resolver or encoder for each link, and therefore have insufficient sensing hardware for a RLFJ model. To overcome this limitation, motion capture markers are fixed to the external su rface of the robot to provide an estimate of the link position (Fig. 29). The axis of rotation and respective center of rotation for each link can be determined from a sufficiently rich set of marker data. The analy tic solution for an axis of rotation obtained in [31] minimizes the cost function P p N k p p kn m v E11 (2.20) PAGE 39 39 where n is a unit vector in the direct ion of the axis of rotation, p kv is the observed position of marker p in frame k and pm is the center of rotation of marker p An unbiased leastsquares solution for a point on the axis of rotation is obtained in [32] given the cost function P p N k p p kr m v E11 2 2 2 (2.21) where m is a point on the axis of rotation, and p r is the radius of the circle mapped out by marker p After the position of each mark er relative to its neighboring ax is of rotation is known, link position is determined through nonlinear optimization [33]. The cost function is defined as P p p k p k qq v v E1 2) ( min (2.22) where p kv is the measured position of marker p in frame k and ) ( q vp k is the predicted marker position given an angle of rotation q The initial guess for the first frame is selected as the initial motor position. Subsequent frames are seeded with the preceding link position. Figure 29. Motion capt ure markers on PA106CE PAGE 40 40 Figure 210. Endeffector weight rack Experimental Setup and Procedure A Mitsubishi PA106CE robot was mounted on a st eel frame table with a height of 0.6 m (Fig. 29). The table was attached to the floor wi th a set of four bolts and internallythreaded concrete anchors. The robot wa s positioned in the center of th e viewing volume of an eightcamera passivemarker motion camera system (Mo tion Analysis Corp., Santa Rosa, CA). The residual measurement error obtained through calib ration was on the order of 0.3 mm, with a standard deviation of 0.1 mm. Cl usters of three or four reflec tive markers were positioned on the exterior surface of four links Realtime measurement and c ontrol of the robot and motion capture system was achieved with a Linux kernel patched with RTAILXRT and the OROCOS and ORCA software libraries [34], [35]. An endeffector apparatu s was designed and constructed to support additional load on the robot endeffector (Fig. 210). A 44.5 N load was attached to the device to excite robot iner tial and stiffness dynamics. A desc ription of each link coordinate system is included in Fig. 211. The robot was commanded to follow, for each joint, the singlejoint trajectory obtained from minimization of the regressor matrix c ondition number. The remaining links were fixed PAGE 41 41 with electromagnetic brakes. Motor position was r ecorded with joint resolvers, with a reported arc min electric error and arc min R/D (ResolvertoDigital) converter error [20], leading to an overall accuracy of .64 arc min (0.011 deg) for the angular position of the motor (gear ratio = 50). Motor torque was determined from the input current and the motor constant. Link position was estimated from the measuremen t of motion capture markers on the surface of the robot. The sampling frequency of the moti on capture system was 250 Hz. Leastsquares and nonlinear optimization algorithms were performe d in MATLAB (Mathworks, Natick, MA). The experiment was repeated 10x for each link. The co efficient of variation (CV) of an estimated parameter, a measure of its fidelity, was defined as the ratio of the standard deviation to the sample mean of the population. The sample mean of each parameter and its CV were recorded for each experiment. Linear stiffness coefficients were determined for eac h experiment after the identification of inertial, mass center, and frictional parameters. The r2 correlation coefficients were also determined for each experiment. Experimental Results The rootmeansquare (RMS) error in the estimation of marker radius rp was not greater than 0.3 mm for all measured trials. The RMS erro r in reconstructed marker position from joint angle optimization was between 0.5 a nd 1.1 mm for all measured trials. Model parameters for the link and motor dynamics and their coefficients of variation, are reported in Table 22 and Table 23. Coulomb friction torques we re equal to 9.08, 9.28, 5.81, 1.48, 1.84, and 0.87 Nm for joints 1 through 6, respectively. Mean RMS errors between measured and reconstructed torques equaled 2. 48, 19.33, 6.55, 0.56, 1.37, and 0.33 Nm, for joints 1 though 6, respectively. Measured and predicted torques fo r joint 2 are shown in Fig. 211. The joint stiffnesses were determined from linear regressions of the joint deflection and corresponding reconstructed stiffness torque (Fig 212). The estimated parameters ranged from PAGE 42 42 4.78x102 in joint 6 to 1.19x104 in joint 2 (Table 22). The r2 correlation coefficients for joints 1 through 6 were equal to 0.66, 0.64, 0.54, 0.84, 0.71, and 0.55. Table 22. Parameter identific ation results for link dynamics Joint Statistic I (kg m2) c (Nm) v (Nm s) mglx (Nm) mgly (Nm) Mean 1.13 9.08 9.88 1.52 2.08 1 CV (%) 0.64 1.75 1.87 1.78 3.59 Mean 6.15 9.28 18.38 111.64 7.80 2 CV (%) 9.62 11.07 6.82 4.47 16.89 Mean 2.27 5.81 5.79 8.33 8.56 3 CV (%) 0.50 0.67 0.87 0.93 0.64 Mean 0.03 1.48 1.45 < 0.01 0.18 4 CV (%) 0.68 6.50 7.02 n/a* 33.81 Mean 0.16 1.84 1.75 0.26 0.63 5 CV (%) 0.68 4.96 5.13 1.34 8.01 Mean 0.08 0.87 0.92 0.10 < 0.01 6 CV (%) 0.43 0.95 0.70 6.12 n/a* not applicable for paramete r estimates which equal zero. PAGE 43 43 Table 23. Parameter identification results for motor dynamics Joint Statistic b (Nm s) J (kg m2) k (Nm/rad) error (Nm) Mean 10.05 1.06 5.98 x 103 2.48 1 CV (%) 1.75 0.64 3.86 2.38 Mean 23.28 12.57 1.19 x 104 19.33 2 CV (%) 6.15 5.33 4.74 3.72 Mean 6.35 2.48 4.60 x 103 6.55 3 CV (%) 0.58 0.18 2.90 1.12 Mean 1.46 0.03 7.09 x 102 0.56 4 CV (%) 7.33 3.72 5.05 8.83 Mean 1.78 0.16 7.72 x 102 1.37 5 CV (%) 5.39 0.44 5.17 3.76 Mean 0.96 0.07 4.78 x 102 0.33 6 CV (%) 0.83 0.36 0.74 1.21 PAGE 44 44 Figure 211. Coordinate system definitions X1, Y2 Y1, Z2 Z1, X2 Y4, Z5 X4, Y5 X6 Y3Z3 X3 Y6 Z6 Z4, X5 PAGE 45 45 0 2 4 6 8 10 400 300 200 100 0 100 200 300 400 Time (seconds)Joint Torque (Nm) Measured Torque Model Torque Figure 212. Measured and predicted motor torque 0.015 0.01 0.005 0 0.005 0.01 200 150 100 50 0 50 100 150 Joint Deflection (rad)Stiffness Torque (Nm) TorqueDeflection Data Linear Approximation Figure 213. Stiffness torque and joint deflection PAGE 46 46 Conclusions A rigidlink flexiblejoint model accurately represents the dynamic behavior of the Mitsubishi PA106CE. The largest RMS residuals occurred in the identification of joint 2, corresponding to less than 8% of the amplitude of motor torque. The remaining joints all have residuals proportional to their motor torque. CV percentages for all friction and inertial parameter estimates were below 12%. Mass center estimates had slightly higher CV percentages because of smaller magnitudes. The resulting jo int stiffnesses are similar in magnitude to estimates obtained thro ugh static calibration [36]. Principal sources of error in the identification process are measurement and synchronization errors in the jo int angle, input torque, and mo tion capture measurements. The remaining differences between the predicted a nd actual motor torque re sult from unmodeled dynamics. An example is positiondependent fric tion arising from manu facturing errors and misalignment of the harmonic drive components during assembly [6]. In addition, there was a disturbance torque acting on the actuated link from adjacent links. An electromagnetic brake will fix motor position, but the harmonic drive may flex due to adjacent link motion. A better identification might be possible with a comple te nlink identification method. Furthermore, errors in the stiffness estimate will arise from nonlinearities in the harmonic drive transmission, such as soft windup and hysteresis which occur in regions of low applied torque [7]. The advantage of separating the identification process into two steps is clear. The condition number of the reformulated regressor matrix is smaller, leading to improved robustness to noise in the regressor matrix and torque measuremen ts. More importantly, one has more insight and flexibility in determining the torsional spring m odel from the computed stiffness torques. It presents no additional complexity to fit a cubic spring model to the measured joint deflection and stiffness torque. However, a cubic element in the regressor matrix would damage the PAGE 47 47 identification process because of its much sma ller size compared to other regressor elements. Furthermore, one can represent the torsional spring stiffness with non linearparameterizable phenomenon, such as hysteresis [37]. This effect is common in harmonic drives, and it is unidentifiable through traditional line ar leastsquares identification. Using motion capture for link position measurement is an important element of the identification method. Several researchers, e.g. Khalil [9], have used motion capture as part of their identification stra tegy, but the present study is an im provement upon previous work. In the last decade, there have been significant impr ovements in the accuracy, robustness, realtime measurement capabilities, and sampling fre quency of motion capture systems. These improvements and lower costs make motion capture systems an increasingl y attractive option for identification and realtime control of robotic systems. The next chapter highlights the development of a modelbased control algorithm using this rigidlink flexiblejoint model. The principal challenge in implementing such a controller is the measurement of link position in realtime. In th is case, an extended Kalman filter integrating joint resolver and motion capture measurements is used to estimate link position in realtime. PAGE 48 48 CHAPTER 3 NONLINEAR CONTROL AND ESTIMATION Introduction Endeffector positioning accuracy and reliable control strategies are critical for highperformance medical robot appl ications, such as the dynamic radiographic imaging system. However, the nonlinear flexibilities and high frictional losses found in the harmonic drive transmissions of the Mitsubish i PA106CE make this objective a challenging task. Harmonic drive systems can be modeled with joint flex ibility in order to reduce modeling error and improve control, but incorporating this phe nomenon in the dynamic model requires link positions and velocities, which are not frequen tly measured on commercial systems. A solution is to design an observer to estimate the link kine matics. In linear control systems, the controller and observer are designed separate ly since their error dynamics ar e independent. But, there is no separation of controller and obser ver designs in nonlinear systems. The interdependence of these two components greatly increases the complexity of the control problem. Furthermore, it is desirable to have the ability to change the endeffector tool, and as a result change the mass of the payload, and sti ll meet performance requirements. In addition, operating conditions, such as ambient temperature and duration of operation, will influence parameters in the physical model, e.g. joint friction. The dynamic radiographic imaging system must have acceptable performance desp ite changes in the dynamic model. This contribution of this section is the desi gn, implementation, and evaluation of a novel observercontroller combination for accurate esti mation and control of a RLFJ manipulator. The EKF expands upon previous work by incorporating 1) flexiblejoint dynamics into the process model, and 2) positions of retr oreflective markers into the meas urement model. The markers can be positioned ad hoc, on any link of the robot, and with no a priori local position information PAGE 49 49 needed. The state vector in this derivation co ntains motor and link kinematics, unknown dynamic and kinematic parameter estimates, and local marker positions. The adaptive reducedparameterization RLFJ controller contrasts previous work by util izing EKFbased estimates of link positions/velocities and motor positions/velocities. Literature Review There exists a wealth of lite rature focused on the development of controllers for flexiblejoint manipulators, but the majority of research ers have not investigated the need for such a controller in industr ial robots. Often a flexiblejoint co ntroller is implemented on a twolink manipulator in a controller environment, where its measurement capabilit ies are augmented with a second joint encoder or strain gages to measur e the joint deflection. Th is option is not often available to the enduser of an assembled r obot. Although many have ad dressed the problem of unmeasurable states in the design of their contro ller, their typical choice of estimated states does not coincide with the states of the PA106CE which are unmeasurable, i.e. link positions and velocities. This section will discuss the breadth of literature devoted towards the development of adaptive rigidlink flexiblejoint robot controllers, emphasizing t hose with nonlinear observers to account for unmeasurable states in the system. Early research by Nicosia et. al derived the Lagrangian form of th e flexiblejoint robot dynamics [38]. This is the most commonly accepted fl exiblejoint model, where the motor and link dynamics are coupled through a torsional spring. The first controllers for flexiblejoint robot systems used feedback lineariza tion methods to achieve stability [30], [39]. The drawback of this class of controllers is a requirement of li nk acceleration and jerk measurements, and exact modelknowledge of robot dynamics. The communities studying nonlinear control research improved upon this result with singular perturbation and integral manifold techniques for flexiblejoint control [40], [41]. Their PAGE 50 50 design calls for a separation of th e robot dynamics into slow and fast counterparts, where each component is accounted for separately. Yet these methods did not account for uncertainty in the model, required a weakflexibili ty assumption, and did not ensu re global asymptotic tracking. Further research by Ghorbel and Spong [42] and Khorasani [43] extended the tr aditional singular perturbation method to account fo r parametric uncertainty in the model, hence, creating one of the first adaptive flexiblejoint controllers. A new control technique, called backstepping, changed the landscape of adaptive flexiblejoint control [44], [45], [46], [47], [48]. The advantages of these methods were threefold. The control engineer could now constr uct globally asymptotic controllers that account for parametric uncertainty, while eliminating the restrictions on the magnitude of joint flexibility, and the requirements of link accelerati on and link jerk measurements [49]. The general idea is to partition the dynamics into a series of cascaded subsystems, where the desired state of one subsystem becomes the control inputs to another. The objective is to design the control input and auxiliary signals to achieve th e desired tracking performance. The literature focused on the development of controllers for flexiblejoint robots is extensive, with many researchers using the backstepping design methodology. Dawson et al. designed an exact modelknowledge controller w ith global asymptotic link position tracking based on a rigidlink electrical lydriven (RLED) manipulator [50]. Their stability result was extended to a robust controller fo r a RLFJ manipulator to account for uncertainties in the model dynamics in [51], [52]. Lozano et. al. investig ated the case of parametric uncertainties in the dynamics of an nlink RLFJ robot model and de signed an adaptive controller with global asymptotic stability [53]. Brogliato et. al. expanded on th is result and compared their adaptive controller to an exactm odel knowledge controller [54]. A survey of RLFJ controllers and others PAGE 51 51 is found in [49]. These designs all requi re complete knowledge of robot kinematics, also known as fullstate feedback. It is well understood that a commercial robot may not have complete knowledge of link and motor kinematics. Therefore, some RLFJ contro llers estimate one or more states of the robot model, also known as partialstat e feedback. Nicosia et al. began investigating th e estimation of states in flexiblejoint robots in [55], and outlined the development of two observers. In the first observer, it was assumed that motor position and elastic displacement are known, while in the second observer, it was assumed that link positio ns and motor velocities were known. In both cases, local asymptotic stability was achieved given exact modelknowledge of robot dynamics and link position feedback. Tomei expanded upon this result with an observ er to estimate the motor positions and velocities given a measurement of link positions and velocities [56]. Some researchers have studied the outputf eedback control problem, where both motor and link velocities are either unmeas urable or assumed to be corr upted by noise. Nicosia and Tomei designed a semiglobal exact modelknowledge c ontroller without measurements of link and motor velocities [57]. Lim et al. was able to show se miglobal exponential tracking for the case of known model dynamics [58]. Similarly, Arteaga demonstrated semiglobal asymptotic stability with a modelbased observer to estimate link and motor velocities [59]. Lim et al. extended their previous result to a system with parametric uncertainties in [60]. In one of the most recent developments in outputfeedback tr acking control, Dixon et al. developed a globally adaptive outputfeedback tracking c ontroller for the flexiblejoi nt dynamics, based on the link velocity filter found in traditional robot systems [61]. Chatlatanagulchai et al. designed a neural network observer for estimation of link positions and robot parameters [62], and used this estimate in conjunction with a variable structur e controller. The comput er simulation of a two PAGE 52 52 link flexiblejoint robot showed good results. A recent paper published by Kim describes the stability result of an adaptive flexiblejoint track ing controller without ov erparameterization, the problem that occurs when n x p parameters must be updated for p unknown robot parameters of an n link robot [63]. All references thus far require the measurement of link position. Until recently, it was believed that good tracking performance could not be achieved without the direct measurement of states whic h define the tracking error, i.e. link position [60]. Abdollahi et al. successfully implemented a neur al network observer to estimate link positions and velocities in [64]. The stability of the observer coupled with a PD controller was demonstrated in a computer simulation. However, the controller does not take advantage of both link and motor kinematics and there are no experimental results. This section describes an EKF to estima te link and motor kinematics in a RLFJ manipulator. Gourdeau and Schwartz utilized an EKF for estimation of joint positions and dynamic parameters in a rigidlink rigidjoint (R LRJ) robot model for a directdrive twolink robot [65]. Timcenko and Kircanski proposed a Kalman filter to estimate the driving torque in a RLFJ robot model and used a simple f eedforward/feedback model for control [66]. Lertpiriyasuwat et al. applied an EKF to a twolink flexiblelink rigidjo int (FLRJ) robot model and augmented the measurement of endeffector position to the system model [67]. Lertpiriyasuwat and Berg combin ed realtime endeffector pos ition measurements from a laser tracker sensor with joint positions measurements to estimate endeffector position and orientation in a RLRJ robot [68]. However, this method lacked the ability to utilize marker position measurements from any location on the robot. None of thes e filters estimate link kinematics, motor kinematics, local marker pos itions, or dynamic parameters in a RLFJ manipulator. PAGE 53 53 Extended Kalman Filter Observer The Extended Kalman Filter An observer should utilize knowledge of plant an d measurement dynamics, statistics of the process and measurement noise, and an initial condition, to yield an estimate of the state of the system. The wellknown Kalman filter is an optimal estimator for linear systems [69], with respect to the minimum error c ovariance, under the assumption that the plant and measurement dynamics are linear and known, the process and measurement noise are zeromean Gaussian random processes and uncorrelated with each othe r, and the initial state of the system is a Gaussian random vector with known mean and covariance. The extended Kalman filter (EKF ) is an extension of the or iginal filter to nonlinear systems. Although the EKF has been a popular choice among researchers, it has several disadvantages [67]. Foremost, the state estimate is no longer an optimal estimate. The performance of the filter will de pend upon the accuracy of the linea r approximation. Second, the computational burden and complexity of determin ing the partial derivativ es of the state and measurement functions can be large. Third, the system dynamics must be continuously differentiable, which is especially a concern when modeling static friction. Although the EKF is no longer an optimal estimat e, it can be modified to ensure local exponential stability with a pr escribed degree of stability [70], known as exponential data weighting in linear systems. The modified form of the extended Kalman filter is summarized in (3.1) (3.4). Given the system and measurement models ) ( ) ( ) ( ) ( t w x G x f t x (3.1) ) ( ) ( ) ( t v x h t y (3.2) PAGE 54 54 with process and measurement noise equal to ) ( 0 ~ ) ( t Q N t w and ) ( 0 ~ ) ( t R N t v, the state estimate and error covariance update equations are ) ( ) ( ) ( ) ( x h t y t K x f x (3.3)) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (t P t H t K t x G t Q t x G I t x F t P t P I t x F t PT T n n (3.4) with x x f t F ) ( x x h t H ) ( 0 and the Kalman gain 1()()()TKtPtHtR Using the previous update equations, the estimation error, defined as ) ( ) ( ) ( t x t x t is bounded via the following expression: 0()P t Pte (3.5) where () P PPt The rate of convergence can be controlled with the parameter 0 RigidLink FlexibleJoi nt Robot (RLFJ) Model The n link RLFJ robot dynamics can be expressed as 0 ) ( ) ( ) ( ) ( ) ( m mq q K q F q G q q q V q q M (3.6) u q q K q B q Jm m m ) ( where nt q t q t q ) ( ), ( ), ( and n m m mt q t q t q ) ( ), ( ), ( denote the position, velocity, and acceleration of the link and mo tor angle, respectively, n nq M ) ( represents the positivedefinite inertia matrix, n n mq q V ) ( represents the centripetalCoriolis matrix, nq G ) (and nq F ) ( denote the gravitational and frictional effects of the link dynamics, respectively, K, J, and Bn are constant, diagonal, positivedefinite ma trices representing joint stiffness, motor inertia, and motor viscous friction, respectively, and nt u ) ( denotes the motor torque [30]. PAGE 55 55 As described in [65], one can define a dynamic parameter vector p to allow for changes in the dynamic model of the robot. It co ntains the unknown or changing mass, inertial, and frictional parameters. The link and mo tor acceleration can be rewritten as , ,1q F q G q q q V q q K q M qm m (3.7) and q q K q B u J qm m m 1. (3.8) The robot can be supplemented with a set of retroreflective mark ers to improve the accuracy of the link position measurem ent. Identify the position of the ith marker in the local link coordinate system Li as iL ip where i = 1,m and the collection of all markers as m Lp3 A kinematic parameter vector 6 can be defined to represent th e position and orientation of the robot coordinate system in the motion capture coor dinate system. It contains the position of the robot coordinate system origin and the set of XYZ Euler angle rotations between robot and motion capture coordinate systems. It is unnecessary to include kinematic parameters in the state equation if the robots position and orientati on are known a priori and processing power is limited. One can define a state vector T T L T T T m T m T Tp q q q q x such that u x f x, and 1 3 1 10 , ,m p n m m m m mq q K q B u J q q F q G q q q V q q K q M q u x f (3.9) Uncertainties in the link and motor dynamics are modeled with the random variables Lw and M w Unknown dynamic and kinematic parameter estimates are modeled as random walk PAGE 56 56 processes, such that w w and L P p w The combined uncertainties are grouped together into the random variable T T P T T T M T Lw w w w w w such that w x G u x f x ) ( ) ( (3.10) Geom Trans ParamI I I J q M x G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) (1 1. (3.11) A perturbation model is needed for the extended Kalman filter. One can define a state perturbation using a first order Taylor series expansion of the robot model with respect to a nominal trajectory [71]. ) (0 0 0u x f x (3.12) w x G x x u x f x ) ( ) (0 0 0 (3.13) The partial derivative of the r obot dynamics with respect to the states can be written as 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) (45 44 43 41 25 23 22 21F F F F I F F F F I t F x f (3.14) K q G q q V q q M M Fm 1 21. (3.15) q F V q q V M Fm m 1 22. (3.16) PAGE 57 57 K M F1 23 (3.17) m mq q K F G q V q M M F 1 25. (3.18) K J F1 41 (3.19) K J F1 43 (3.20) B J F1 44 (3.21) q q K q B q J J Fm m m 1 45. (3.22) The robot dynamics must be defined by conti nuous differentiable functions in order to evaluate the partial derivatives. As a result, the friction model is adapted from Makkar et al. as q q q F 3 2 1tanh ) ( (3.23) where 1, 2, and 3 are positive, constant coefficients representing Coulomb and viscous friction in the joint [72]. Measurement Model One can define a measurement vector T T TTG mmhxqqp and introduce a random variable T T T Tv v v v3 2 1, such that v x h y ) (, (3.24) where 3,,GLmpqp represents the position of all markers in the global coordinate system, and Tv1, Tv2, and Tv3 represent the uncertainties in the measurement of motor positions, motor velocities, and marker positions, respectively. One can define a measurement perturbation in the same manner as the state perturbation. PAGE 58 58 ) (0 0y h y (3.25) 0() hy yxv x (3.26) The linearization of the measurement model for th e rigidlink flexiblejoint dynamics is as follows: 37 36 310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) ( H H H I I t H x h. (3.27) T T G M T Gq p q p H 1 31. (3.28) T T G M T Gp p H 1 36. (3.29) ML M G M L Gp p p p H 0 0 0 0 0 011 1 37. (3.30) To facilitate the analysis of these part ial derivatives, one can use the homogeneous coordinate system transformation 1 11 1 L G L Gp q T vect p (3.31) where 11 Lp is defined as the local marker pos ition in homogeneous coordinates, and x vect x f is an operator which eliminates the homo geneous coordinate from the last row of a position vector [73]. The partial derivative of the global marker pos ition with respect to the link position is defined as PAGE 59 59 1 11 1 L G L Gp q T vect q q p (3.32) 1 1 1 11 1 1 1 L N G L L G L Gp q q T vect p q q T vect q p (3.33) where T T T q T T T q q TL L G G L 1 3 4 2 3 2 1 2 0 1 0 21 1 1 The partial derivative of the global marker pos ition with respect to the kinematic parameter vector is defined as 1 11 0 0 1 L L G Gp q T q T vect p (3.34) 1 1 1 11 0 0 1 0 1 0 1 L L N G L L G Gp q T q T vect p q T q T vect p (3.35) The partial derivative of the global marker position with respect to the local marker position is defined as q R p p p q R p pG L L G L L G L L Go 1 1 1 1 1 11 1 1 1 (3.36) where 1G L R q is the rotational transformation between global and local coordinate systems. Observability in Nonlinear Systems The determination of observability in nonlinear systems is not as straightforward as the procedure for linear systems. The main differences in the observability of linear systems are 1) the observability matrix is constant, and 2) inde pendent of the control i nput. Nonlinear systems can exhibit singular inputs that rend er the statespace realization unobservable [74]. The statespace realization PAGE 60 60 ) ( ) (x h y x f x (3.37) is said to be locally observable at n ox if there exists a neighborhood n oU of xo in which every state ox x is distinguishable from xo. In other words, there must exist a unique mapping between each observation and respective st ate. For the nonlinear system described in (3.37), the condition for local observability in a neighborhood n oU containing the origin is n rank (3.38) where h L hn f 1, (3.39) and h Lf is the Lie derivative of the function h( x) with respect to the function f(x). The relationship in (3.38) reduces to the wellknown observability criterion given a linear system. The statespace realization is locally weakly observable (LWO) if the observability rank condition is satisfied at the origin. The observability matrix for th e nonlinear system described in (3.6) can be written as 313637 41434445 31313637 21222325 444141444343444425 31323334353637 111222232315 2122232425 31323334353637000000 000000 0000 000000 000 000 000 00 00 00 I I HHH I FFFF H FFFF FFFFFFFF FFF (3.40) PAGE 61 61 as described in Appendix A. This expression has M=8n+12m rows and N=4n+6+p+3m columns. For the case in which the number of columns is less than the number of rows, i.e. p + 6 < 4n + 9m, the rank of the matrix can be at most N, the number of states in the extended Kalman Filter. From inspection, the columns are not dependent upon each other. Therefore, the observability matrix has full rank. It may be possible for either 25F or 45F to equal zero, but not at the same time, and the matrices 41F or 43F are linearly dependent upon each other. Neither condition affects the rank of the observability matrix. For the case in which the number of columns is greater than the number of rows, higher order Lie derivatives are required to increase the number of rows in order to achieve an Nrank matrix. The observability matrix, without measurem ents of marker positions, reduces to the following expression: 41434445 21222325 444141444343444425 111222232315 21222324250000 0000 0000 0 0I I I FFFF FFFF FFFFFFFF FFF (3.41) This expression has M=8n rows and N=4n+p columns. As long as p < 4n, the number of columns will be less than the number of rows and the rank of the matrix can be at most N. From inspection, the columns are not dependent upon each other. As a result the observability rank condition is satisfied, a necessary and sufficient condition for observability in nonlinear systems. PAGE 62 62 RigidLink Flexib leJoint Control Theoretical Background The principal theory behind the Lyapunovbased stability analysis a nd controller design was first developed by Alek sandr Mikhailovich Lyapunov (18571918) at the turn of the 20th century. He wrote that a con tinuously differentiable and posit ivedefinite function with a negativedefinite derivative would converge, along with the states that co nstitute its value, to zero. It is a simple idea but an extremely power ful design tool. Unfortunate ly, its application for control system design would not become popular until more than 50 years after his death. It was in 1983 that Artstein [75] and Sontag [76] first introduced the idea of the control Lyapunov function. The idea was to design a cont rol input to ensure the negative definite derivative of a Lyapunov function comprised of the error signal. This method bridged the distance between descriptions of stab ility and constructive design tools. A breakthrough occurred a few years later with the technique called integrator backstepping. Lyapunovbased methods could now be applied to a much broader class of systems, those which are extended or cascaded syst ems. The first use of backstepping as a design tool occurred in [77]. Soon afterward, an adaptive backstepping method was developed by Kanellakopoulos et al. for nonlinear systems with unknown parameters [78]. Later, Freeman and Kokotovic developed robust backste pping to achieve global stabiliz ation despite the presence of unknown disturbances [79]. In the last decade, the research community has continued to advance the control of nonlinear systems, with update la w modifications and progress in non modelbased controllers, such as ne ural networks and fuzzy logic controllers. A brief understanding of Lyapunovbased stability is necessary for understanding the following controller design. PAGE 63 63 Stability Definition: The equilibrium point 0 x of ) (x f x is stable if, for each 0 there exists 0 such that 0 ) ( ) 0 ( t t x x unstable if it is not stable. asymptotically stable if it is stable and can be chosen such that 0 ) ( lim ) 0 ( t x xt. Lyapunov Stability Theorem: Let 0 x be the equilibrium point of ) (x f x and nD be a domain containing 0 x. Let D V : be a continuously differentiable function such that } 0 { in 0 ) ( and 0 ) 0 ( D x V V, and D x Vin 0 ) ( then, the equilibrium point0x is stable. Also, given } 0 { 0 ) ( D in x V then 0 x is asymptotically stable. Control Objective An adaptive rigidlink flexiblej oint controller is described below based on the work in [63], [61]. The controller has a re duced parameterization to d ecrease the total number of integration variables, and assumes that both motor positions and velocities are measured and available for feedback, and that link positions and velocities are estimated in the extended Kalman filter. These modifications significantly reduce the complexity of the controller and make its design unique. PAGE 64 64 The control objective is to design a global fullstate feedback tracking controller for the robot model described in (3.6). The link position tracking error nt e ) (is defined as q q ed (3.42) To facilitate the control design, define a filtered tracking error n Lt r ) ( as f Le e e r2 1 (3.43) where n ft e ) ( is an auxiliary filter variable and 0 ,2 1 The auxiliary filter variable, introduced to eliminate the dependence of the controller on higher order derivatives in the stability analysis, is defined as p e k ef 1, (3.44) where e k p e k k p2 1 1 1 3 2 1 and 0 ,3 1 k. Define the joint deflection nt ) ( as the difference between the motor and link position, q qm In order to account for constant pa rametric uncertainties in the robot model, define the parameter estimation error pt ) ( ~ as ) ( ) ( ~t t (3.45) where p contains constant unknow n model parameters and pt ) ( is an estimate of those parameters. Define two subsets of parameters, L, and M, representing unknown parameters in the link dynamics (e.g. li nk mass, inertia, and friction), and motor dynamics (e.g. motor inertia, friction, and joint s tiffness), respectively. De fine a third subset of parameters K, such that n KI diag K The following properties [80] aid in the subsequent development of the adaptive controller. PAGE 65 65 Property 1. The inertia matrix, () M q, is a symmetric, positivedefinite matrix that satisfies the following inequalities 22 12()TnmMqm. (3.46) where m1, and m2 are known positive constants. Property 2. A skew symmetric relationship exists between the timederivative of the inertia matrix, () M q and the centripetalCoriolis matrix, (,)mVqq This enables the useful relationship 1 ()(,)0 2Tn mMqVqq (3.47)Property 3. Diagonalization of nx and ny is commutative, such that ()()diagxydiagyx (3.48)Property 4. The cascaded diagonalization of nx and ny can be simplified as diagdiagxydiagxdiagy (3.49)Error System for Filtered Tracking Error Define the first backstepping error nt ) (as mq u where nu is an auxiliary control input. It is an alogous to the desired joint defl ection. Define the desired link regressor matrix p n d d d dR q q q Y) ( as the linear parameterization of the link dynamics ) ( ) ( ) )( ( ) )( (d d d d d m d d L dq F q G q q q V q q M Y (3.50) The openloop error dynamics for the filtered tracking error are expressed as f d Le q M e q M q q M q q M r q M ) ( ) ( ) ( ) ( ) (2 1 (3.51) Substituting the link dynamics in (3.7), and the following expressions for the error dynamics, PAGE 66 66 12 Lferee (3.52) and 123 f Lfekree (3.53) the filtered tracking error dyna mics can be rewritten as L L m L d Lr q M k u K Kq r q q V Y r q M) ( ) ( ) (2 1 (3.54) where K r q M e q M e q M Y q F q G e e q q q V q q ML f L d f d m d ) ( ) ( ) ( ) ( ) ( ) ( ) (1 2 2 2 1 3 2 2 1 2 1 (3.55) The function ) (t contains the undesired terms in the Lyapunov derivative. Using the mean value theorem and knowledge of system dynamics it is possible to place an upper bound on this expression as 2 22 011223121 MMMK fLwweer (3.56) where T fLweer K and M represent the maximum eigenvalues of the inertia and stiffness matrices, respectively, and 0 and 1 are positive bounding constants [60]. Partial Stability Analysis for Filtered Tracking Error Begin a partial stability analysis for the link position tracking errors and parametric uncertainties by defining a Ly apunov candidate function as 11 1121212()1212TTTTT f fLLLLLKKKVeKeeKerMqrK (3.57) Evaluate the time derivative, subs titute expressions for the error and filter dynamics in (3.52) (3.54), and utilize the skew symmetric relationship in Property 2 to yield K K T K L L T L L L d T L L T f L T f T f TK r q M k u K Kq Y r Kr e k Kr e Ke e Ke e V ~ ~ ~ ~ ) (1 1 2 1 1 3 1 1 (3.58) Design the first auxiliary control and parameter estimate update laws as PAGE 67 67 1fKdLuekediagYq (3.59) L T d L Lr Y (3.60) and L d L K KY r diag ) ( (3.61) Add and subtract d K T LY diag K r ,and substitute (3.59) (3.61) into (3.58) to yield T L L T L f T f Tr r q M k r Ke e Ke e V ) (2 1 3 1 1. (3.62) Using Property 1 to bound the inertial term, and given the upperbound for ) (t in (3.56), simplifies the derivative of the Lyapunov function into 2 222 11312011223 2 22 121M KKM f LLLfL MKM LLLVeekrwrwrer errr (3.63) The first three terms in the expression above ar e negative for all time, wh ile the remaining terms are positive for all time. Howe ver, using nonlinear damping [81] and designing the control gain 2 2 1 2 0 2 1 2 11M K n M r Mc k k k (3.64) where 2 2 2 2 1 2 3 2 2 1 c, one can write a much leaner form of the derivative. 2 2 222422 11311111 44444KK frLf nnnnnVeekrwwee kkkkk (3.65)Auxiliary Control and Backstepping Error System Begin with the backstepping error mq u and differentiate with respect to time to yield mq u (3.66) where L d K fY diag q e k e u 1 (3.67) PAGE 68 68 Differentiating the expression for the auxiliary control input and utiliz ing the diagonalization properties yields m Lq r 2 1 (3.68) where L d K d fY diag q e k e k 3 1 2 1 1 (3.69) and T d L d K L d KY Y diag Y diag k 2 2 1 2. (3.70) Insert the auxiliary control input u through addition and subtraction to yield u rL 2 1. (3.71) Design 2 2 1 nk k u to enable the following expressi on for the dynamics of the first backstepping error: 2 2 2 n Lk k r. (3.72) Define the second backstepping error as mq u Obtain the backstepping error dynamics by differentiating with re spect to time, and substituti ng the expression for the motor dynamics. u K q B u J Jm (3.73) In order to evaluate this expr ession, determine an expression for the timederivative of the auxiliary control input, such that 2 2 2 2 3 1 2 12 n n L d L d K L d K d fk k k Y Y diag Y diag q e k e k u. (3.74) This simplifies to Lr u4 3 where T d L d T d L d K n L d L d K n m n L d K d fY Y Y Y diag k Y diag Y diag k q k k Y diag q e k e k 2 4 2 2 1 2 2 2 3 2 2 1 3 2 2 1 1 3. (3.75) PAGE 69 69 and T d L d L d K n T d L d L d K n n T d L d K L d L d KY Y diag Y diag k Y Y diag Y diag k k k Y Y diag Y diag Y diag k k 2 4 2 2 2 2 2 3 1 2 1 4 (3.76) Define the linear parameterization of the motor dynamics as K q B r J Ym L M M 4 3, (3.77) and substitute this expression in to the error dynamics to yield u Y JM M (3.78) Design the control effort k Y uM M to yield the following expression for the second backstepping error dynamics: k Y JM M ~ (3.79)Partial Stability Analysis for Backstepping Error System Begin a partial stability analysis for the back stepping errors and para metric uncertainties by defining a second Lyapunov candidate function as 1 2121212TTT M MMVJ (3.80) Differentiate with respect to time and substitu te expressions for the two backstepping error dynamics to yield M M T M M M T n L Tk Y k k r V ~ ~ 1 2 2 2 2 (3.81) Design the parameter estimate update law T M M MY such that k k k r VT n L T 2 2 2 2. (3.82) The Lyapunov derivative can be upperbounded as L nr k k k V 2 2 2 2 2 2 2 (3.83) Using nonlinear damping [81], the expression can be si mplified into the following. PAGE 70 70 2 2 2 24 1L nr k k k V (3.84)Controller Stability Result Theorem 1. Given the control strategy describe d above, semiglobal asymptotic link position tracking is obta ined in the sense lim()0tet (3.85)provided that the controller and dam ping gains are selected to satisfy 2(0)2 4nx k (3.86) where T fLxeer (3.87) and 13min,,,,KK rkkk (3.88)Proof. Define the composite Lyapunov function 12()()()VtVtVt The upper bound on the timederivative can be written as follows: 2 2222 13 2 242221 4KK frL fL nVeekrkk wweer k (3.89) Simplify the expression for the upper bound into 2241 2 4nVxxx k (3.90) and rearrange terms to yiel d the following expression: 2211 24nnVxx kk (3.91) If the condition in (3.86) is satisfied, then the derivative can be further simplified into the form PAGE 71 71 2Vx, (3.92) where is some positive bounding constant. From (3.92) it can be seen that the Lya punov function is a radially unbounded, positivedefinite function for all 5()nzt and t, where ()fLLKMzteer (3.93) Since its timederivative is negative semidefinite, ()VzL and as a result ()ztL. Given that the actual parameters of the manipulator are bounde d, it can be shown that L K and ML. The control efforts ,,and uuu, and the timederivatives of the error signals, represented in (3.52) (3.54), (3.72), and (3.78), are also bounded for all time, based on the signals that define them. Therefore, () x tL () x tL and 2() x tL based on the preceding information and the structure of (3.92). Use Barbalats Lemma [82] to show that lim()0txt and as a result lim()0tet (3.94) i.e. semiglobal asymptotic stability for the proposed adap tive controller. ClosedLoop Stability Result The exponential convergence of the EKF has been demonstrated in [70]. Using (3.5), it is possible to approximate the error dynamics as with / 1 The closedloop controller and observer d ynamics when using the EKF state estimate are written as follows. ) ( x f x (3.95) ) ( x g (3.96) where T L fr e e x PAGE 72 72 The system represented in (3.95) and (3.96) is in standard singul arly perturbed form [83]. Asymptotic and exponentia l stability of controll er and observer dynamics, respectively, ensures asymptotic stability of the combined closedloop system. Simulation The EKF observer and corresponding RLFJ controller were implemented in MATLAB and Simulink (The MathWorks, Inc., Natick, MA). The equations of motion and associated differentiation of the six degreeoffreedom model of the PA106C E manipulator were determined with the analytic software progr am AutoLev (OnLine Dynamics, Inc., Sunnyvale, CA). The desired trajectory of the robot was based on the motion of the knee joint in gait. The position of a marker fixed to the lateral condyl e was recorded using a motion capture system (Motion Analysis, Corp., Santa Rosa, CA), and a fourterm Fourier seri es approximation of the trajectory was determined usi ng nonlinear optimization in MATLAB and transformed into joint space using an inverse kinematics algorithm. Five ma rkers were arbitrarily a ttached to the virtual PA106CE model, two on link 2, and three on link 6. Inertial, fr ictional, and stiffness parameters in the simulation model were determined from the dynamic identification experiment in Ch. 2. Random zeromean Gaussian noi se with covariances 0.1 mm2 ( = 0.30 mm) and 1x107 rad2 ( = 0.02 deg) were added to the measurement of each marker position and motor position, respectively. In order to limit the number of stat es in the extended Kalman filter, only mass and inertia properties for link six are included as unknown parameters. The remaining robot parameters, as well as initial conditions for unknown paramete rs, are corrupted by random Gaussian noise. The standard deviation is chos en such that 99.7% of chosen parameters fall within 10% of their nominal amount (% 10 3 ). The performance of the EKFRL FJ controller was assessed fr om rootmeansquare (RMS) and steadystate errors for the kne egait trajectory while carrying no payload and a 5 kg payload. PAGE 73 73 For comparison, the same trajectory and load ing conditions were ex ecuted using a desiredcompensation adaptationlaw (DCAL) controller [84]. The DCAL controller is an adaptive modelbased controller based on the backstep ping design method. It compensates for nonlinearities arising from Coriolis and centrif ugal effects, but does not incorporate joint flexibilities into its model. Results. The EKFRLFJ contro ller and DCAL controller were implemented with the set of gains in Tables IA and IB, respectively. As e xpected, the tracking errors were smaller for the EKFRLFJ controller in both loading cases, especia lly for joints 2, and 3, where there was a 71% and 83% reduction in steadyst ate tracking error. The signifi cant improvement in tracking performance for joints 2 and 3 was most likely a re sult of the increased defl ection in these joints due to a gravitational load (Tab les 31 and 32). There was not a significant difference between the maximum and RMS motor torques between cont rollers. The mean RMS and steadystate link position estimation errors amongst all joints were less than 0.1 deg for both the unloaded and loaded cases. The mass and inertia estimation er rors converged to less than 0.1 kg and 0.3 kg m2 in the unloaded case, and 0.1 kg and 0.2 kg m2 in the loaded case. The mean RMS and steadystate marker position estimation errors amongst a ll markers were equal to 1.3 mm and 1.7 mm, respectively. The link pos ition tracking errors and estimation e rrors are shown in Fig. 31 and 32, respectively. PAGE 74 74 0 20 40 60 80 100 6 4 2 0 2 4 Time (sec)Tracking Error (deg) Joint 1 Joint 2 Joint 3 Joint 4 Joint 5 Joint 6 Figure 31. Link position tracking er rors in simulation while loaded 0 20 40 60 80 100 0.4 0.2 0 0.2 0.4 0.6 Time (sec)Estimation Error (deg) Joint 1 Joint 2 Joint 3 Joint 4 Joint 5 Joint 6 Joint 1 Figure 32. Link position estimation errors in simulation while loaded PAGE 75 75 Table 31. Simulation results without payload Table 32. Simulation results with payload DCAL Controller Jnt. 1 2 3 4 5 6 RMS Torque (Nm) 21.5 68.2 14.5 3.4 1.9 1.4 Max Torque (Nm) 34.7 93.8 24.8 4.8 3.7 1.8 RMS Tracking Error (deg) 0.6 1.6 0.7 0.2 0.2 0.2 Max Tracking Error (deg) 3.4 7.1 1.8 0.7 0.5 0.3 SteadyState Tracking Error (deg) 0.6 1.4 1.0 0.3 0.2 0.2 EKFRLFJ Controller Jnt. 1 2 3 4 5 6 RMS Torque (Nm) 22.2 71.6 13.9 3.5 1.9 1.5 Max Torque (Nm) 38.1 97.3 25.7 4.8 3.7 1.8 RMS Tracking Error (deg) 0.4 0.4 0.2 0.2 0.1 0.2 Max Tracking Error (deg) 2.4 2.1 1.6 0.3 0.3 0.3 SteadyState Tracking Error (deg) 0.5 0.4 0.2 0.3 0.2 0.2 DCAL Controller Jnt. 1 2 3 4 5 6 RMS Torque (Nm) 21.8 97.7 22.8 3.3 1.8 1.4 Max Torque (Nm) 40.7 126.7 34.2 4.8 4.8 1.8 RMS Tracking Error (deg) 0.7 2.2 0.8 0.2 0.2 0.2 Max Tracking Error (deg) 3.9 10.9 3.9 0.6 0.7 0.3 SteadyState Tracking Error (deg) 0.6 1.5 0.8 0.3 0.4 0.2 EKFRLFJ Controller Jnt. 1 2 3 4 5 6 RMS Torque (Nm) 22.9 102.0 22.4 3.5 1.9 1.5 Max Torque (Nm) 49.1 128.5 43.9 4.8 4.8 1.8 RMS Tracking Error (deg) 0.3 0.8 0.3 0.2 0.1 0.1 Max Tracking Error (deg) 2.9 5.1 2.2 0.3 0.4 0.3 SteadyState Tracking Error (deg) 0.3 0.7 0.3 0.2 0.1 0.2 PAGE 76 76 Experiment A Mitsubishi PA106CE robot was mounted on a steel frame table with a height of 0.6 m. The table was attached to the fl oor with a set of four bolts a nd internallythreaded concrete anchors. The robot was positioned in the cente r of the viewing volume of an eightcamera passivemarker motion camera system (Motion An alysis, Corp., Santa Rosa, CA). The sampling frequency of the motion capture system was set to 250 Hz. Time delays due to image acquisition and processing are on the order of 3 to 4 frames or 12 16 ms for this setup. The residual measurement error, obtained th rough calibration of the motion cam era system, was on the order of 0.3 mm, with a standard de viation of 0.1 mm. Groups of two and three retroreflective markers were attached to link 2, and the ende ffector, respectively. R ealtime measurement and control of the robot and motion capture system was attained with a GNU/Linux kernel patched with RTAILXRT, and the help of th e Orocos and Orca software libraries [34], [35]. An endeffector apparatus was designed and constructe d to support additional load on the robot endeffector. The extra payload was added in order to excite robot inertial and stiffness dynamics. The experimental trajectory a nd loading conditions were identic al to those in simulation. Figure 33. Mitsubishi PA 106CE robot with markers PAGE 77 77 RealTime Implementation Issues The realtime implementation of an EKFRLFJ cont roller presents many obstacles, such as numerical integration errors, tim edelay artifacts, and measurem ent and synchronization errors. The most challenging obstacle is to achieve an accurate numerical inte gration given the limited step size and the computational burden in eval uating the partial derivatives of the dynamics. Several integration methods were evaluated on th e EKF algorithm, including an explicit Euler method, a modified predictorc orrector Euler method, a 4th order RungeKutta algorithm, and an AdamsBashforth AdamsMoulton method [85]. The host PC contai ned an Intel Pentium processor running at 3.2 GHz with 3 GB of RAM. The minimum step size required for stability, and the computation time required for one itera tion of the EKF were recorded for each method and are displayed in Table 33. Table 33. Comparison of nu merical integration methods All integration methods except the explicit Euler me thod fail to yield a stable estimate in the time required for one iterati on. Additionally, the difference between the expected execution time and the maximum step size is too small to safely account for computation time from other processes. Instability in the numerical integration arises b ecause the link dynamics and motor dynamics exist on two different time scales. Imp licit integration methods, which are successful for such numerical systems, require an unpred ictable amount of processing time in order to iterate through a rootf inding method. The increased computa tional time increases the step size, further increasing the co mputational time. In order to avoi d this computational problem, the dynamics were discretized and the state estim ate was obtained with a discretetime EKF. Explicit Euler Modified Euler 4th Order RungeKutta ABAM Execution time (ms) 0.8 1.5 2.9 variable Max. step size (ms) 1.1 1.4 1.6 1.6 PAGE 78 78 Given the continuoustime system and perturbation model ) (0 0 0u x f x (3.97) Gw x F x (3.98) one can define the discretetime model k k k k kw x x 1, (3.99) where t F ke (3.100) The solution for the predicted state estimate can be obtained from the state transition matrix as k k kx x 1. (3.101) However, since the partial derivatives of the dynami cs with respect to the states are known, the predicted state estimate is obtained with firstorder implicit Euler integration. d x f x xk kt t k k k 11. (3.102) such that t x f x xk k k 1 1. This expression can be re arranged into the form 01 1 1 t x f x x x gk k k k. (3.103) and solved using the NewtonRaphson me thod for rootfinding, such that n n n nx g x J x x1 1 (3.104) where t x F I x x g x Jk k k k 1 1 1 1 1. (3.105) The solution for the predicted error covariance matrix is obtained from the state transition matrix and can be written as T k k k T k k k kQ P P 1, (3.106) PAGE 79 79 where t GQG QT T k k k The updated state estimate and erro r covariance matrix are obtained from the discretetime EKF equations k k k k k kx H z K x x. (3.107) k k k kP H K I P. (3.108) and 1 k T k k k T k k kR H P H H P K. (3.109)Experimental Results Initially, the EKFRLFJ contro ller did not have improved tracking performance in joints 1, 4, and 6, the result of relatively high joint s tiffnesses coupled with low stiffness torques. However, there were improvements in the link position tracking errors for joint 2 and 3, stemming from the much larger signaltonoise ra tios for estimated joint deflections in heavilyloaded joints. In order to take advantage of th is observation, a mixed rigidjoint/flexiblejoint model was used for the EKFRLFJ controller. The joints which did not incur significant stiffness torques during the activity, i.e. jo ints 1, 4, and 6, were defined as rigid. The remaining joints were defined as flexible. The mixed model provided smalle r steadystate tracking errors for most joints in the PA106CE manipulator (Tables 34 and 35). RMS tracking errors decr eased for all joints while loaded, yet there was not a consistent reduction in RM S tracking errors while unloaded. RMS torques were similar between controllers in both loadi ng scenarios, and maximum torques were slightly greater in the EKFRLFJ contro ller in both loading scenarios. The link position tracking errors and estimated joint stiffness coefficients for the mixed controller are shown in Fig. 34 and Fig. 35, respectively. PAGE 80 80 0 50 100 150 15 10 5 0 5 10 Time (sec)Tracking Error (deg) Joint 1 Joint 2 Joint 3 Joint 4 Joint 5 Joint 6 Figure 34. Experimental link posit ion tracking errors while loaded 0 50 100 150 0 5000 10000 15000 Time (sec)Joint Stiffness (Nm/rad) Joint 2 Joint 3 Joint 5 Figure 35. Joint stiffness coefficients from experiment while loaded PAGE 81 81 Table 34. Experimental results without payload Table 35. Experimental results with payload DCAL Controller Jnt. 1 2 3 4 5 6 RMS Torque (Nm) 18.9 82.3 16.9 3.0 5.1 2.0 Max Torque (Nm) 39.3 117.2 34.5 4.4 11.6 2.8 RMS Tracking Error (deg) 1.3 1.0 0.7 1.0 0.8 0.7 Max Tracking Error (deg) 4.6 3.7 2.4 3.4 2.3 3.5 SteadyState Tracking Error (deg) 1.6 1.7 1.5 1.7 1.1 1.3 EKFRLFJ Controller Jnt. 1 2 3 4 5 6 RMS Torque (Nm) 19.3 80.9 18.3 3.1 5.1 2.0 Max Torque (Nm) 47.1 128.9 53.2 7.3 12.5 3.1 RMS Tracking Error (deg) 1.1 0.8 0.8 0.6 1.6 0.7 Max Tracking Error (deg) 5.6 2.5 3.1 5.0 4.4 4.3 SteadyState Tracking Error (deg) 1.3 1.3 1.4 1.5 2.1 1.7 DCAL Controller Jnt. 1 2 3 4 5 6 RMS Torque (Nm) 20.2 120.3 29.7 3.3 8.8 2.0 Max Torque (Nm) 39.2 161.9 53.6 5.6 14.3 2.8 RMS Tracking Error (deg) 1.4 1.2 0.9 2.2 3.3 0.7 Max Tracking Error (deg) 4.8 4.3 5.0 5.4 5.5 2.8 SteadyState Tracking Error (deg) 1.6 2.2 1.7 3.7 4.2 1.2 EKFRLFJ Controller Jnt. 1 2 3 4 5 6 RMS Torque (Nm) 20.2 118.4 31.5 3.3 8.5 2.1 Max Torque (Nm) 45.3 196.0 61.1 6.1 14.4 4.4 RMS Tracking Error (deg) 0.9 0.8 0.9 1.6 2.3 0.6 Max Tracking Error (deg) 6.0 2.9 3.1 4.6 8.4 2.9 SteadyState Tracking Error (deg) 1.2 1.2 1.8 2.4 3.4 1.4 PAGE 82 82 Conclusion The performance of the EKFRL FJ controller, compared to the DCAL controller, was significantly better in simulation. Mean steadystat e tracking errors decreased more than 50% for both unloaded and loaded trials. RMS torques were nearly identical between the two controllers, indicating almost equivalent energy expenditure. Performanc e improvements using the EKFRLFJ controller were not as universal in real time tracking experiments. Again, RMS torques were almost identical between controllers, but strong improvements in tracking performance only occurred while the endeffector was loaded Mean steadystate tracking errors decreased 22% while loaded, but increased 4% while unloa ded. Reduced performance while unloaded is expected, given that reduced load ing conditions lead to reduced ex citation of stiffness dynamics. Joints 2, and 5 had the greatest reduction in st eadystate tracking errors while loaded (> 50%), and also had the greatest stiffness torques relative to joint stiffnesses. Differences between simulation and experiment al results can be at tributed to unmodeled dynamics, limited actuation bandwidth, discreti zation errors, synchr onization errors, and measurement noise. Closer investig ation of these effects in simu lation revealed that parameter error in the EKF model ha d the greatest effect on simulation performance. Greater accuracy of identification methods, or inclusi on of these parameters as states in the EKF model, may improve performance. However, computational resources n eeded to include more parameters in a realtime experiment are beyond our current capabilities. This paper discusses a novel observercontroll er combination for accu rate estimation and control of a RLFJ manipulator The EKFRLFJ controller cont rasts previous work by not requiring direct measurements of link positions, cr itical for estimation and control of industrial manipulators with flexible joints, since these manipulators are often assembled without link position resolvers. The tracking performance of the Mitsubishi PA106CE improved with the PAGE 83 83 proposed EKF observer and RLFJ controller wh ile loaded. This observercontroller should provide an even greater improvement, and for any loading conditions, in manipulators with greater joint flexibility. PAGE 84 84 CHAPTER 4 MEASUREMENT ANALYSIS Introduction The paramount objective of the dynamic radiogr aphic imaging system is the measurement of skeletal kinematics. This goal is achieved with modelbased registra tion of bone or implant models to radiographic images. These methods require knowledge of the 3D geometry and calibration parameters, such as focal length and principal point. Fortunately, it has been shown that kinematic measurements obtained from registra tion are not strongly influenced by errors in calibration parameters [86]. However, it has been suggested that this measurement is degraded by motion blur, arising from relati ve motion between the xray sour ce, flatpanel detector, and object, but the magnitude of this effect is unknow n. This section discusses the uncertainties in the relative motion of the dynamic radiographi c imaging system, thei r propagation to motion blur, and their influence on image contrast and image registration. Literature Review Motion blur is an imaging artifact introduced by relative motion between the image sensor and object. During image capture, the elements of the image se nsor collect photons, in their respective bins, and transform them into electrical signals. The motion of the object relative to the image sensor will project these photons in a line or curve along the image sensor, thus blurring the resulting image. Th e amount of motion blur increa ses as the exposure time or relative motion increases, and is more pronounced at higher resolutions. A point spread function (PSF) is the result of a single poi nt of light projected on the image sensor. Motion blur can be arti ficially generated with accurate knowledge of the PSF, and is often added in the field of comput er graphics to generate the fee ling of motion. In addition, an image with motion blur can be corrected given accurate knowledge of the blurgenerating PSF. PAGE 85 85 The theoretical foundation behind correcting image blur given re lative motion is not a recent development. Motion blur is equivalent to convolution (or multiplicati on in the frequency domain) between the original im age and a PSF, written as ghfn (4.1) where f is the original image, h is the PSF, n is shot noise, and g is the resulting image. Thus, the simplest method of image correction, inverse filtering, amounts to the inverse discrete Fourier transform of the complex division 1g f h F F F. (4.2) The Weiner filter expands upon the inverse filter ing method by incorporating information about the signaltonoise rati o of the image, however, this in formation must be known a priori [87]. An improvement upon this method is the LucyRicha rdson algorithm. It yields the maximum likelihood estimate of the or iginal image give n unknown Poisson noise statistics [88]. For an insightful review of moti on deblurring methods, see [89]. Nonetheless, these methods are not often implemented on physical systems because motion information is scarcely available. In [90], motionbased deblurri ng methods are revisited given motion estimates based on a secondary image detector with low spa tialresolution and high temporalresolution. In the context of the dynami c radiographic imaging system, these additional costs are unnecessary given that we can estimate the motion of the xray source and flatpanel detector from joint resolver and motion capture measurements. However, residual motion blur, or that which is leftover after deconvolution, will remain desp ite a solid theo retical foundation PAGE 86 86 and reliable experimental da ta. This residual motion blur may impair image registration [91], [92], and further investigation is needed to establish the extent of this degradation. Much literature discusses the methods for regist ration of 3D implant or bone models to 2D radiographic images. The most co mmon objective or cost functions can be divided into three categories, featurebased, contourbased, and intensitybased. Using a contourbased metric, Banks and Hodge assessed the difference betw een the segmented image and the predicted contour with Fourier descriptors [93]. As an alternative, one can count the number of pixels that disagree between two binary images [94]. Mahfouz et al. defined a cost function having both an intensity matching score and a contour matching score in [95]. The intensity matching score and contour matching score are similar to a crossco rrelation between the two images, and the edgedetected images [96], in that order. These methods provide a foundation for estimating implant or skeletal kinematics from radiographic images. Uncertainty Analysis Imaging Geometry Motion blur is a direct result of relative motion between the im age sensor and the object. It can be characterized as the length in pixels in which a point of light is smeared across the image sensor during a fixed exposure time. The inplane translation and inplane rotation of the object relative to the image sensor will generate the principal motion blur. The outofplane translation and rotation of the object relative to the image sensor will increase or decrease the size of the object in the image plane and therefore generate some motion blur. Howe ver, it can be shown that this amount is negligible for typical pr ojection geometries. This section describes the imaging geometry of the robotic imaging sy stem and its connecti on with motion blur. The relationship between the size of the obj ect on the image sensor and the imaging geometry can be described as PAGE 87 87 obj imD L L L D2 1 1 (4.3) where Dim is the diameter of the object on the image sensor, Dobj is the diameter of the object in the physical world, and L1 and L2 represent the source to sensor distance and object to sensor distance, respectively (Fig 41). The change in size of the object on the image sensor will result from the following expression: obj imD L L L L L L L D2 1 2 1 2 2 1 (4.4) where 22object sensorLLvt is the new object to sensor distance. Using 10 cm/s and 10 ms as approximations for the outofplane motion of the object and the exposure time of the image sensor, the change in L2 can be written as 3 2 210 01 0 1 0 L L. (4.5) Therefore, the change in the size of the object in the image plane will be approximately 0.1% of the size of the image, resulting in 0 imD. This analysis demonstrates that outofplane translations will not generate motion blur in the dynamic radiographic imaging system. The analysis of motion blur is further comp licated because the motion of the xray source relative to the object and image sensor will also generate motion blur (Fig. 41). In order to facilitate the analysis, define an effective obj ect to image sensor velocity which generates the same motion blur as the relative motion betw een xray source and object, written as 2 1'objectsource object sensorL vv L (4.6) The effect of xray source motion on motion blur is small compared to the object and sensor motion, and it is in the o pposite direction of the na tural motion of the object. PAGE 88 88 The outofplane translation of the xray source relative to the object and image sensor will not generate a significant amount of motion blur. Using similar tria ngles, the size of the object on the image sensor can be written as obj imD L L L D2 1 1 (4.7) where Dim is the diameter of the object on the image sensor, Dobj is the diameter of the object in the physical world, and L1 and L2 represent the source to sensor distance and object to sensor distance, respectively (Fig 41). The change in size of the object on the image sensor can be written as obj imD L L L L L L L D2 1 2 1 2 1 1 (4.8) where 11 source objectLLvt It can be shown that the change in length is negligible when using an approximation for the outofplane motion and exposure time, 3 1 110 01 0 1 0 L L, (4.9) and therefore the change in the size of the object in the image is negligible,0 imD. Using similar arguments, it can be shown that the inp lane and outofplane ro tational motion of the xray source will not have an effect on motion blur. In conclusion, the inplane motion of the xray source will lead to an effective inplane object to sensor motion, resulti ng in the following combined or total inplane relative sensor motion: source object sensor object sensor objectv L L v v1 2 (4.10) On the other hand, only the inplane rotation of the image sensor relative to the object will lead to rotational image blur, such that PAGE 89 89 objectobject s ensorsensor (4.11) Figure 41. Imaging configurati on of robotic imaging system Covariance of Relative Motion An estimate for the relative motion between the target object and the endeffector of the robot is needed in order to simulate motion blur. It can be written as objectobjectobject s ensorsensorsensorvvv (4.12) where object s ensorv is the measured relative motion between the target object and endeffector, i.e. the tracking error, and object s ensorv is the measurement uncertainty. The covariance of the relative motion can be written as 22 ,,covobjectobjectobject vv s ensorsensorsensorve (4.13) where 2 covobjectobject v s ensorsensorev and 2 ,covobjectobject v s ensorsensorv source objectv 'sensor objectv L1 L2 PAGE 90 90 Endeffector tracking error The relative motion between the target objec t and the endeffector of the robot was measured in a series of tracking experiment s with the Mitsubishi PA 106CE. The robot was commanded to follow a path approximated from the location of the femoral condyle of the knee in gait (also used in the previous chapter). The same path was followed at three different fundamental frequencies in order to observe the relationship between the objects velocity and tracking error. The fastpaced trajectory, which re sembled the knee joint in a fast walk, exceeded the joint velocity limits of the PA106CE. This scenario might arise for some faster moving activities, and is worth considering in the analys is. The DCAL controller was used in place of the EKFRLFJ controller becau se it has a much simpler design and produced similar tracking results. In the slowpaced tracking experiment, th e DCAL controller produced the following tracking error covariance matrix: 2 2 2 ,58623404 23282917 40417817object v sensormm e s (4.14) In the mediumpaced tracking experiment, the DC AL controller yielded the following tracking error covariance matrices: 2 2 2 ,80199464 99653644 464441308object v sensormm e s (4.15) In the fastpaced tracking experiment, the DCAL controller yielded the following tracking error covariance matrices: PAGE 91 91 2 23 2 ,71.81.64.9 1.6176.32.410 4.92.42.2object v sensormm e s (4.16) The crosscovariance terms are small compared to the diagonal elements, and as a result, the covariance matrices are simplified as diagonal matr ices for the remaining analysis. The strongest correlation existed between tracking errors in the x and z directions (r2 = 0.2), however, outofplane motions are neglected in the motion blur simulation. Measurement uncertainty The relative motion between the target objec t and endeffector is a function of both motions, such that objectobjectsensor g lobal sensorglobalvvv (4.17) Therefore, the relative motion measurement uncerta inty is dependent upon the uncertainties in target object and endeffector ve locities. The law of propagation of uncertainty, which explains the relationship between the stat istics of the returning value of a function and the individual uncertainties which constitute its value, can be used to determine the measurement uncertainty. This leads to the following expression: 222 ,,objectobjectsensor v vv g lobal sensorglobal, (4.18) and 22 ,, objectobject vv s ourcesensor. Uncertainty in EndEffector Velocity Estimate. The law of propagation of uncertainty can also be used to determine the uncertainty in endeffector velocity. The joint measurement noise covariance matrix can be written as PAGE 92 92 ) ( ) ( ) ( ) (6 2 6 1 2 6 1 2 1 2 2q q q q q qq (4.19) This is equivalent to the diagonal matrix ) ( ), (6 2 1 2 2q q diagq because of the independence of angular position measurement noi se. Using the Jacobian relationship between joint space and task space velocities, q q J x ) (, the endeffector measurement noise covariance, 2 x can be expressed as T q xq J q J ) ( ) (2 2 (4.20) where J(q) is the manipulator Jacobian. The uncertain ties in joint positions are neglected since they are small in comparison to those in joint velocities, 2 2 q q The upper bound on the uncertainty in linear and angular velocities can be evaluated from 2 22 ,()sensorTq v globalJq (4.21) and 2 22 ,()sensorRq globalJq (4.22) where 6 3) ( q JT and 6 3) ( q JR are the translational and rotational parts of the manipulator Jacobian, respectivel y. The upper bound for the norm of 2 q was determined from the output of the extended Kalman filter during a simple tracking experiment. The robot was commanded to follow the gaitlike path as the estim ated error covariance matrix, returned from the extended Kalman filter, was recorded. The error covariance matrix did not have a strong dependence upon the selected trajecto ry. The upper bounds on the norms of ) (q JT and ) (q JR were determined from a nu merical simulation sampling 1e4 random configurations inside the workspace of the manipulator (Fig. 42, Fig. 43). As a result, it can be written that PAGE 93 93 2 22 24 22 ,11402.1610281sensor v globalmmradmm rad ss, (4.23) 2 2 2 2 24 22 ,degdeg 1.732.161057.32.1sensor globalrad rad ss (4.24)Uncertainty in Object Velocity Estimate. The uncertainty in the estimate of the object velocity was obtained from the results of the MATLAB simulation of an extended Kalman Filter for rigidbody pose estimation [97]. The measurement error cova riance matrices returned from the filter equaled 2 2 2 ,320.10.9 0.1320.6 0.90.634object v globalmm s (4.25) and 2 2 2 ,5.10.20.5 deg 0.24.40.4 0.50.45.2object globals (4.26) Although small in this example, the differences in angular velocity uncertainties between directions can be attributed to the location of markers in the local coordinate system. It is more difficult to estimate outofplane rotations wh en markers are placed in a single plane. Covariance of Relative Motion. The uncertainty in the relative motion estimate can be determined using (4.23) (4.26), and (4.18), such that 2 2 2 ,318object v sensormm s, (4.27) and 2 2 2 ,deg 7.3object sensors. (4.28) Using (4.15), (4.27), (4.28) and (4.13), the covariance of the relative motion between endeffector and object can be determined as PAGE 94 94 cov111668511623object sensorvdiag (4.29) and cov14.217.456.9object sensordiag (4.30) given the results from the mediumpaced tracking experiment. 0 200 400 600 800 1000 0 0.2 0.4 0.6 0.8 1 1.2 Sample numberMatrix norm Figure 42. Norm of translation part of manipulator Jacobian PAGE 95 95 0 200 400 600 800 1000 1 1.2 1.4 1.6 1.8 2 Sample numberMatrix norm Figure 43. Norm of rotational part of manipulator Jacobian Image Sharpness The spatial frequency response (SFR), or modul ation transfer function (MTF), of an image depicts relative image contrast for a range of spatial frequencies, and is the spatialdomain analog to the (timedomain) frequency response of a mechanical system. The sharpness, or contrast, of an image is most often assessed fr om its SFR. A slantededge is often used to determine the frequency response because pixels along the slanted edge can be subsampled into multiple bins. This procedure increases the effective resolution of the image and permits observation of frequencies greate r than the normal Nyquist frequenc y. The SFR is then obtained from the Fourier transform of the derivative of the oversampled edge. The spatialdomain behavior of motion blur is an alogous to a common lowpass filt er, attenuating high frequency content in the image, and can be assessed from th e SFR. This section discusses the adverse effect of motion blur on image contrast, and its im provement with image deconvolution, through simulations and experiments. PAGE 96 96 Theoretical Improvement The SFR of an image with a slantededge was determined in three different situations: 1) without motion blur, 2) with mo tion blur, and 3) with motion blur and image deconvolution. Motion blur was generated from the probability dist ribution of the expected relative motion, as determined in the preceding uncertainty analysis It is a result of bot h source to object motion and sensor to object motion, leading to the following propagation of uncertainties: 2 2 1covcovcovobjectobjectobject s ensorsensorsourceL vvv L (4.31) Using approximations for the covariance of the relative motion in (4.29), and the approximation 2 10.1 L L the total uncertainty in lin ear motion can be upperbounded as cov112769201639object sensorvdiag (4.32) The total uncertainty in angular motion is small compared to the uncertainty in linear motion and is neglected in the motion blur simulation. Image deconvolution was performed using th e damped LucyRichardson deconvolution algorithm [88]. The PSF, used in this algorithm, was de fined as the true relative motion plus the measurement uncertainty perturbation term in (4.27): 2 2 2 ,318object v sensormm s. (4.33) Each probability distribution was sampled at 200 positions, and the response of each sample was recorded. The SFR of the slantededge was obtai ned with the SlantedEdge Analysis Tool in MATLAB [98]. Image sharpness was evaluated from the area under the SFR curve from zero to the Nyquist frequency, such that PAGE 97 97 0Nfd F. (4.34) As expected, there was a significant improve ment in image contrast with the motionbased deblurring method, despite uncertainties in the precise endeffect or and tracking object velocities. The mean value increased from 0.43 in the set of blurred images to 0.69 in the set of blurred/corrected images, an increas e of more than 60% contrast. The 0 value from the static (undistorted) image was 1.04. The mean responses of the blurred and corr ected image sets, those which corresponds to their respective mean value, are shown in Fig. 44. The blue solid line represents the SFR of the original image, without mo tion blur. Although the mean values are much different between the two se ts of images, the ranges of cont rast are similar, extending from 0.14 1.04 and 0.23 1.34 in the blurred a nd blurred/corrected images, respectively. In a second simulation, motion blur was added in even increments to the slantededge image. The image contrast, as expected, decrea sed with an increase in relative object/sensor motion (Fig. 45). The thin red solid line and thick black solid line represent the probability density functions of the tracking e rror and its estimation error. It can be seen from Fig. 45 that tracking errors will lead to values in the range of 0.1 1. 0, while measurement uncertainties alone will lead to values in the range of 0.5 1.0. PAGE 98 98 0 0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1 1.2 Spatial Frequency (cycles/pixel)Spatial Frequency Response Undistorted Image Corrected Image Blurred Image Figure 44. Spatial frequency resp onse of motion blurred images 0.5 0 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Integral of Spatial Frequency Response 0.5 0 0.5 0 5 10 15 20 25 Relative Motion (m/s)Normalized Probability Density FunctionProbability Distribution of Measurement Uncertainty Probability Distribution of Tracking Error Figure 45. Relationship between re lative motion and image contrast PAGE 99 99 Experimental Methods The improvement in image contrast from motionbased deblurring methods was also evaluated in a dualrobot tracki ng experiment. For this experiment, two Mitsubishi PA106CE manipulators were positioned in the workspace of an elevencamera optical motion camera system (Motion Analysis Corp., Santa Rosa, CA). Each manipulator was positioned on a 0.6 m tall steel frame table, located 2.4 m from each other. A monochrome video camera (JAI Pulnix, San Jose, CA) and 25 x 20 cm board were attached to the endeffectors of the source (right) and sensor (left) robots (Fig. 46). The ISO 12233 test chart was prin ted and attached to the bo ard in order for the video camera to capture images of a slantededge dur ing the experiment. The sensor manipulator was commanded to follow a path resembling the path of the kneejoint cen ter during normal gait, while the source robot was commanded to track the endeffector of the sensor robot, such that the ISO test chart would remain in sight for the complete traject ory. Four tracking markers were attached to the board, while an EKF, developed by Halvorsen et al., was used to estimate the location of the target from this set of markers [97]. Four tracking markers were also attached to the surface of the source robot in order to implem ent a second EKF [Ch. 3]. A set of 20 images was captured along one period of the gaitlike trajectory using an asynchronous trigger and 10 ms exposure time. The position and orientation of the camera and the target object were recorded from the two EKF observers. The SFR of each image, evaluated from the slantededge, was determined before and after image deconvolut ion. The bestcase SFR was defined from the static, thus undistorted, image of the test chart. Image contrast was evaluated from the area und er the SFR curve from 0 2 cycles/pixel, defined as Residual motion blur was defined as the percent difference between in the undistorted image and in the corrected image. PAGE 100 100 Figure 46. Experimental setup of dualrobot imaging system Experimental Results As expected the spatial fr equency response of each image was improved using motion information acquired from the EKF. The mean value increased from 0. 11 in the blurred image group to 0.19 in the blurred/corrected image group, an increase of more than 70%. The 0 value from the static, undistorted image (0.33) serv ed as a gold standard for comparison. The value of each image relative to the static image, expre ssed as a percentage, is shown in Fig. 47. The greatest and smallest improvements in image cont rast were 418% (Image #6) and less than 1% (Image #17). The blurred and blurred/corrected imag es of the slantededge for these cases are shown in Fig. 48. The mean relative contrast was equal to 34.3% and 58 .2% in the blurred and blurred/corrected image groups, respectively. The mean residua l motion blur, expressed as a percentage, was equal to 41.8%. PAGE 101 101 0 5 10 15 20 0 20 40 60 80 100 120 140 Image NumberRelative Contrast (%) Blurred Images Blurred/Corrected Images Undistorted Image Figure 47. Relative contrast in bl urred and blurred/corrected images A B C D Figure 48. Image sharpening from LucyRic hardson algorithm A) Blurred image #6, B) Blurred/Corrected image #6 (bestcase), C) Blurred image #17, D) Blurred/Corrected image #17 (worstcase) PAGE 102 102 Image Registration It has been shown in the preceding section that motion blur will exist in the acquired images, yet it has not been seen how motion bl ur affects registration accuracy. This section discusses the impact of motion bl ur on image registration in the context of the robotic imaging platform, and the potential improve ments with image deconvolution. Methods Four different radiographic images were collect ed from four different studies, representing the range of contrast and resoluti on that might be expected in typi cal clinical studies. Image RU is an 800x800 pixel, underexposed radiograph of an uncemented total knee arthroplasty (TKA). Image RC is a 1024x1024, high contrast radiograph of a cemented TKA. Image FC is a 352x288, low contrast fluoroscopic image of a cemented TKA, and image FN is a 512x512, high contrast fluoroscopic image of a natural knee. Twenty randomly blurred images were genera ted from each of the four original images. Blur statistics were determined by sampling a distri bution of relative tracki ng errors obtained in a series of experiments with a Mitsubishi PA106C E robot during a gait tracking task, described in (4.29). Tracking errors resulted in standard devia tions of blur in the horiz ontal/vertical directions of 4.9/2.4, 6.0/2.9, 1.1/0.5, and 1.4/0.7 pi xels for the RU, RC, FC, and FN images, respectively, given a conserva tive exposure time of 10 ms. Four additional sets of 20 randomly blurred and subsequently deblurred, or deconvol ved, images were also generated from each original image. Each blurred image was d econvolved using the damped LucyRichardson algorithm in MATLAB (Mathworks, Natick, MA) and estimated measurement uncertainties, described in (4.27). The JointTrack opensource software program was used to register the implant or bone models to their respective images [99]. Registration was performed using simulated annealing PAGE 103 103 optimization algorithm and an edge plus grayscale cost function [95]. The mean and standard deviation of the image registrati on error were recorded for each image and used to compare the blurred and blurred/corrected sets of images. A pa ired ttest was used to determine if the RMS translational and rotational errors were greater in the blurred images compared to the blurred then corrected images 0.05. Results Motion blur generated from typi cal tracking errors measured in a robotic imaging system did not have a significant impact on image regist ration accuracy. Both mean anteriorposterior (AP) and superiorinferior (SI) RMS errors for the blurred and blurred/corrected image sets were less than 0.5 mm. The mean flexionextension (FE) RMS errors equaled 0.4 and 0.3 deg. for the blurred and blurred/corrected image sets, respec tively, and the mean varusvalgus (VV) and internalexternal (IE) RMS errors were less than 1 deg. Motion blur had th e greatest impact in mediallateral (ML) translation where mean RMS errors totaled 2.0 mm for both the blurred and blurred/corrected image sets. Kinematic errors for both image groups are summarized in Table 41. Mean RMS errors decreased in the blurred/corrected image group for five of the six degrees of freedom (Table 42). Image deblurring significa ntly reduced RMS registration errors only for flexionextension. (p < 0.05). PAGE 104 104 A B C D Figure 49. Undistorted ra diographic images A) RU, B) RC, C) FC, D) FN PAGE 105 105 Table 41. Kinematic errors for Blurred and Corrected image groups (RMS errors) RU RC FC FN Parameter B C B C B C B C AnteriorPosterior Translation (mm)0.32 0.34 0.77 0.44 0.66 0.58 0.22 0.17 SuperiorInferior Translation (mm) 0.12 0.18 0.44 0.51 0.59 0.33 0.63 0.28 MedialLateral Translation (mm) 1.87 2.61 2.58 2.07 2.66 1.71 0.96 1.95 FlexionExtension (deg) 0.26 0.18 0.77 0.60 0.19 0.18 0.42 0.22 VarusValgus (deg) 0.15 0.18 0.47 0.26 0.26 0.27 0.47 0.15 InternalExternal Rotation (deg) 0.47 0.46 1.23 0.66 1.08 0.38 0.39 0.32 Table 42. Kinematic differences between Blu rred and Corrected image groups (RMS errors) Parameter RU RC FC FN mean AnteriorPosterior Translation (mm)0.02 0.33 0.08 0.05 0.11 SuperiorInferior Translation (mm) 0.06 0.07 0.26 0.35 0.12 MedialLateral Translation (mm) 0.74 0.51 0.95 0.99 0.07 FlexionExtension (deg) 0.08 0.17 0.01 0.20 0.12* VarusValgus (deg) 0.03 0.21 0.01 0.32 0.12 InternalExternal Rotation (deg) 0.01 0.57 0.70 0.06 0.34 statistically greater than zero (p < 0.05) Conclusion Motion blur, representative of a dynamic robotic imaging system, has a harmful effect on image contrast, as shown in simulations and ex periments. However, contrast can be improved with motionbased deblurring methods. Increases in the value of more than 60% and 70% were observed once the images were deconvolved usi ng the estimated motion, in simulations and experiments. On the other hand, motion blur, representative of a d ynamic robotic imaging system, did not significantly degrade kine matic measurements based upon modelimage registration. Inplane transla tional and rotational errors we re less than 0.5 mm and 1.0, respectively. These errors are within bounds fo r research and clinically useful kinematic PAGE 106 106 measurements. Errors in the ML direction exceeded those in ot her directions because of low registration sensitivity to outofplane translations. Mean RMS erro rs decreased in blur corrected images, but only differences in fl exionextension errors were signi ficantly improved (p < 0.05). Typical absolute errors with this cost f unction and optimization for unblurred images have been reported as 0.1 mm and 0.4 [95]. Additive errors due to expected image blur would result in net RMS errors of 0.6mm and 1.4, wh ich are sufficiently accurate for many research and clinical assessments. Shorter exposure times can directly reduce measurement degradation if motion blur is problematic. The limited impact of motion blur on image registration can be attributed to several factors. First, various registration cost functions are robust to imaging artifacts, such as motion blur, since they do not solely depend on edge info rmation. Second, the simulated annealing (SA) method used in this study is a global optimi zation method, reducing the chance of spurious solutions from local minima. Lastly, user supe rvision of modelimage registration safeguards against poor or unrealistic so lutions, and reduces the error covariance between images. As an alternative to motionbased deblurring, one might choose to blur rendered images before comparison with acquired images, in or der to improve their registration. Blurring velocities could also provide a missing bridge be tween the kinematics of a single frame and the entire sequence of motion. Modeling joint kine matics as a continuous process would enforce continuous kinematic estimates and likely improve results. Further investigated is warranted to determine the accuracy of this method. Radiographic imaging systems provide superior joint kinematic measurement accuracy compared to optical motion capture systems with skin affixed markers. By definition, DRI systems will be subject to motion blur and poten tially significantly degradation of skeletal PAGE 107 107 kinematic measurements. This investigate demonstrates that knee kinematics from modelimage registration are not significantly degraded by moti on blur for subject and robot velocities typical of normal gait. PAGE 108 108 CHAPTER 5 CONCLUSION The dynamic radiographic imaging system desc ribed in these pages will be one of the first to acquire natural dynamic joint motion. Its results will likely improve the current standards for treatment and diagnosis of musculoskeleta l conditions. With a rising need for improved healthcare worldwide, especially for the aging po pulation, this imaging system may develop into a great success. The potential is large, but there are challenges concerning the measurement and control of such as system. The tracking performance of th e robotic imaging system is hindered from nonlinearities arising from its revolute geomet ry, friction and joint flexibili ties. In order to account these modeling intricacies, novel calibration and id entification methods were developed. Their estimates initialized ad aptive modelbased control methods and enabled the PA106CE to meet stringent tracking requirements. The taskspace RMS and steadystate tracking errors of the robotic imaging system were 1.9 cm and 2.3 cm, respectively, allowing th e joint to remain in sight of the flatpanel detector for the duration of the activity. The measurement performance of the robotic imaging system is hampered by motion blur arising from the dynamic nature of the system. Bu t, these artifacts can be reduced with accurate tracking and motionbased deblurri ng methods. RMS tracking errors were less than 9.0 cm/s, providing a manageable amount of motion blur, and the corrected images showed more than 60% improvement in contrast compared to bl urred images. Furthermore, it was shown that accurate registration is possible given motion bl urred images. RMS inplane translational and rotations errors in motion blurred imag es were less than 0.5 mm and 1.0 deg. PAGE 109 109 The results in this dissertation indicate th at a dynamic radiographic imaging system can acquire clinically useful kinematic measuremen ts. The dualrobot imaging system can provide acceptable tracking errors despite no nlinear flexiblejoint dynamics, and images acquired in such a system can provide accurate motion informa tion despite motion blur. Mobile and flexible radiographic imaging systems are the future of medical imaging, and further development is critical to the field of orthopedics and beyond. PAGE 110 110 APPENDIX A OBSERVABILITY MATRIX As shown in Chapter 3, the state dynamics and partial derivatives can be written as 0 mq q u x f x (A.1) and 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) (45 44 43 41 25 23 22 21F F F F I F F F F I t F x f (A.2) where , ,1q F q G q q q V q q K q Mm m (A.3) and q q K q B u Jm m 1. (A.4) The observation h(x) is based on measurements of motor positions and velocities, and marker positions. The measurement dynamics and partial derivatives can be written as T T G T m T mp q q x h (A.5) and 37 36 310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) (H H H I I t H x h (A.6) The first Lie derivative of function h(x) with respect to f(x) is as follows: q H q q q H H H I I h Lm m f 31 37 36 310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (A.7) PAGE 111 111 where T T G M T Gq q p q q p q H 1 31, (A.8) and N L N G L L G L Gq p q q T vect q p q q T vect q q p 1 1 1 11 1 1 1 1. (A.9) The gradient of the first Lie derivative with respect to the states can be written as 37 36 31 31 45 44 43 410 0 0 0 0 0 0 0 0 0 0 0H F F F F I h Lf. (A.10) where q q H 31 31, q H 31 36, and q p HL 31 37. The second Lie derivative of function h(x) with respect to f(x) is as follows: 0 0 0 0 0 0 0 0 0 0 0 0 037 36 31 31 45 44 43 41 2 m fq q H F F F F I h L (A.11) 36 31 31 44 43 41 2H q F q F q F h Lm f (A.12) The gradient of the second Lie derivative w ith respect to the states can be written as 37 36 35 34 33 32 31 25 44 44 43 43 44 41 41 44 25 23 22 21 20 0 0 0 0 F F F F F F F F F F F F h Lf. (A.13) where 45 44 44 43 41 25F F F q F q Fm (A.14) PAGE 112 112 41 36 36 21 31 31 31 31F q F H q H q q (A.15) 42 36 31 22 31 31 31 32F H F H q q H (A.16) 43 36 23 31 33F F H (A.17) 44 36 34F (A.18) 45 36 25 31 35F F H (A.19) 36 31 31 36H q (A.20) L L Lp p H q p 36 31 31 37. (A.21) The third Lie derivative of function h(x) with respect to f(x) is as follows: 0 0 0 0 0 037 36 35 34 33 32 31 25 44 44 43 43 44 41 41 44 25 23 22 21 3 m fq q F F F F F F F F F F F F h L (A.22) 34 33 32 31 44 44 43 43 44 41 41 44 23 22 21 3 m m m fq q F F F q F F F q F F q F F q F h L (A.23) The gradient of the third Lie derivative with respect to the states can be written as 111222232315 3 2122232425 3132333435363700 00fFFF Lh (A.24) where PAGE 113 113 2122 112221FF qFF qq (A.25) 2122 12212222FF qFFF qq (A.26) 23 2122 152225 mF FF qFFq (A.27) 21412143444441FFFFFF (A.28) 2244414122FFFF (A.29) 23412343444443 F FFFFF (A.30) 24444343444444FFFFFF (A.31) 43 44414144 25414441254344 43 44 44434444452mF FFFF FFqFFFFq F F FFFFF (A.32) 31323334 3132213441mqFqF qqqq (A.33) 31323334 32313222mqFq qqqq (A.34) 313233 3332233443m mmmqFqF qqq (A.35) 313233 34333444 m mmmqqF qqq (A.36) 31323334 3532253445 mqFqF (A.37) 31323334 36 mqq (A.38) PAGE 114 114 31323334 37 m LLLLqq pppp (A.39) This leads to the observability matrix 313637 41434445 31313637 21222325 444141444343444425 31323334353637 111222232315 2122232425 31323334353637000000 000000 0000 000000 000 000 000 00 00 00I I HHH I FFFF H FFFF FFFFFFFF FFF (A.40) where the elements are described in (A .14) (A.21) and (A.25) (A.39). PAGE 115 115 APPENDIX B OBSERVER DESIGN FOR TWOLINK ROBOT The equations of motion for a tw olink manipulator with joint fl exibility can be written as 0 ) ( ) ( ) ( ) ( ) ( m mq q K q F q G q q q V q q M (B.1) u q q K q B q Jm m m ) ( (B.2) where 2 2 3 2 2 3 2 2 3 12 ) (p c p p c p p c p p q M, (B.3) 0 ) (1 2 3 2 1 2 3 2 2 3q s p q q s p q s p q q Vm (B.4) 12 5 12 5 1 4) (s p s p s p q G, (B.5) 2 11 2 10 9 1 8 1 7 6tanh tanh ) (q p q p p q p q p p q F (B.6) 13 120 0p p J, (B.7) 15 140 0p p B, (B.8) and 17 160 0p p K. (B.9) The coefficients in the inertia matrix, centrip etalCoriolis matrix, a nd gravitational component are based on the mass and inertia properties of the each link, such that 222 111122212 p ImlImll (B.10) 2 2222 p Iml (B.11) PAGE 116 116 32212 p mll (B.12) 411212 p mglmgl (B.13) and 522 p mgl (B.14)The State Model The partial derivative of the robot dynam ics with respect to the states is 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) (45 44 43 41 25 23 22 21F F F F I F F F F I t F x f (B.15)Solution for F21: Given the definition for the inertia matrix, the centripetalCoriolis acceleration matrix, and the gravitational and fr ictional components of the link dynamics, the so lution to the partial differential equations contained in F21 can be written as follows: q q M q q M q q M 2 1, (B.16) where 0 01q q M and 1 2 3 2 2 3 1 2 3 22q s p q s p q s p q q M such that 1 2 3 2 2 3 1 2 30 2 0q s p q s p q s p q q M (B.17) Furthermore, q q V q q V q q Vm m m 2 1, (B.18) where 0 01q q Vm and 2 1 2 3 2 2 1 2 3 2 1 2 3 2q c p q q q c p q q c p q q Vm such that PAGE 117 117 2 1 2 3 2 2 1 2 3 2 1 2 30 0q c p q q q c p q q c p q q Vm (B.19) 12 5 12 5 12 5 12 5 1 4c p c p c p c p c p q G, (B.20) The inertia matrix is positive definite for all robot configurations, and therefore, the inverse of the inertia matr ix is nonsingular. 2 3 1 2 3 2 2 3 2 2 2 2 2 3 2 2 2 1 12 1c p p c p p c p p p c p p p p M, (B.21) Therefore, K q G q q V q q M M t Fm 1 21) (, (B.22) 17 16 12 5 12 5 12 5 12 5 1 4 2 1 2 3 2 2 1 2 3 2 1 2 3 1 2 3 2 2 3 1 2 3 2 3 1 2 3 2 2 3 2 2 2 2 2 3 2 2 2 1 210 0 0 0 0 2 0 2 1 ) ( p p c p c p c p c p c p q c p q q q c p q q c p q s p q s p q s p c p p c p p c p p p c p p p p t F (B.23)Solution for F22: The partial derivatives necessary for the solution to F22 can be written as 0 2 2 21 2 3 2 1 2 3 2 2 3q s p q q s p q s p V q q Vm m (B.24) and 11 2 10 2 10 9 8 1 7 2 7 6sech 0 0 sechp q p p p p q p p p q F (B.25) Therefore, q F V q q V M t Fm m 1 22) (, (B.26) PAGE 118 118 11 2 10 2 10 9 8 1 7 2 7 6 1 2 3 2 1 2 3 2 2 3 2 3 1 2 3 2 2 3 2 2 2 2 2 3 2 2 2 1 22sech 0 0 sech 0 2 2 2 2 1 ) ( p q p p p p q p p p q s p q q s p q s p c p p c p p c p p p c p p p p t F (B.27)Solution for F23: The solution to F23 can be written as K M t F1 23) ( (B.28) 17 16 2 3 1 2 3 2 2 3 2 2 2 2 2 3 2 2 2 1 230 0 2 1 ) ( p p c p p c p p c p p p c p p p p t F, (B.29)Solution for F25: The partial derivatives necessary for the solution to F25 can be written as q M q M q M q MP 2 1, (B.30) where 01 1q q M 2 1 2 2q q q q M and 1 2 2 2 1 2 32 q c q c q c q M such that 8 1 1 2 2 1 8 1 2 2 1 2 2 10 0 0 2 q c q q q c q c q q q M (B.31) 8 1 2 1 2 2 1 8 1 2 2 1 2 2 1 2 2 10 0 0 0 q s q q q s q q s q Vm (B.32) 6 1 12 3 1 6 1 12 1 3 10 0 0 0 0 s s s G, (B.33) 2 2 2 10 2 9 2 10 5 1 1 1 1 7 2 6 1 7 5 1sech tanh 0 0 0 0 0 0 0 sech tanh 0 q q q p p q p q q q p p q p F (B.34) PAGE 119 119 2 2 15 1 1 1 15 10 0 0 0m m mq q q q q q K, (B.35) Therefore, m mq q K F G q V q M M t F 1 25) ( (B.36) 2 2 15 1 1 1 15 1 2 2 2 10 2 9 2 10 5 1 1 1 1 7 2 6 1 7 5 1 6 1 12 3 1 6 1 12 1 3 1 8 1 2 1 2 2 1 8 1 2 2 1 2 2 1 2 2 1 8 1 1 2 2 1 8 1 2 2 1 2 2 1 2 3 1 2 3 2 2 3 2 2 2 2 2 3 2 2 2 1 250 0 0 0 sech tanh 0 0 0 0 0 0 0 sech tanh 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 1 ) (m mq q q q q q q p p q p q q q p p q p s s s q s q q q s q q s q c q q q c q c q q c p p c p p c p p p c p p p p t F (B.37)Solution for F41: The motor inertia matrix is constant, di agonal, and positive definite. The inverse is defined as 12 13 13 12 10 0 1 p p p p J. (B.38) The solution for F41 can be written as K J t F1 41) ( (B.39) 17 12 16 13 13 12 17 16 12 13 13 12 410 0 1 0 0 0 0 1 ) ( p p p p p p p p p p p p t F, (B.40)Solution for F43: PAGE 120 120 The solution for F43 can be written as K J t F1 43) ( (B.41) 17 12 16 13 13 12 17 16 12 13 13 12 430 0 1 0 0 0 0 1 ) ( p p p p p p p p p p p p t F, (B.42)Solution for F44: The solution for F44 can be written as B J t F1 44) ( (B.43) 15 12 14 13 13 12 15 14 12 13 13 12 440 0 1 0 0 0 0 1 ) ( p p p p p p p p p p p p t F, (B.44)Solution for F45: The partial derivatives necessary for the solution to F45 can be written as 4 1 2 11 1 4 1 1 11 10 0 0 0 0 0m m mq q q J (B.45) 2 1 2 13 1 2 1 1 13 10 0 0 0 0 0m m mq q q B (B.46) 1 1 15 1 1 1 15 10 0 0 0 q q q q q q Km m m, (B.47) Therefore, q q K q B q J J t Fm m m 1 45) ( (B.48) 2 2 12 2 12 2 12 11 1 1 1 13 1 13 1 13 11 1 13 12 450 0 0 0 0 0 0 0 1 ) ( q q p q p q p q q p q p q p p p t Fm m m m m m (B.49) The Measurement Model PAGE 121 121 The partial derivative of the robot dynam ics with respect to the states is 37 36 310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) ( H H H I I t H x h. (B.50)Solution for H31: Define the transformation betw een the global coordinate syst em and the base coordinate system of the robot as an xyz Euler angl e sequence of rotations, represented as 3 2 1 and a translation 6 5 4 such that 1 0 0 06 2 1 3 1 3 2 1 3 1 3 2 1 5 2 1 3 1 3 2 1 3 1 3 2 1 4 2 3 2 3 2 0 c c c s s s c s s c s c c s c c s s s s c c s s s s c c c TG. (B.51) The transformation from one link coordinate system to another is a function of the link position. 1 0 0 0 02 12 12 12 2 12 2 2 12 12 12 2 12 2 12 2 2 1 2S c c s c s s S s s c c c s a s c T (B.52) where 2 2sin q s 2 2cos q c 12 12sin s, 12 12cos c, and a and S represent the constant link lengths, twist angles, and joint offsets, respectively [16]. The partial derivative of the transformation between link coordinate systems, with respect to the link positions, is defined as follows: 0 0 0 0 0 0 0 0 0 012 2 12 2 12 2 12 2 2 2 2 1 2s s s c c s c c c s q T (B.53) The solution for H31 can be written as PAGE 122 122 M ML M G L M G L G L Gp q T T T vect p T q T T vect p q T T T vect p T q T T vect t H2 1 2 0 1 0 1 2 1 0 1 0 1 2 1 2 0 1 0 1 1 2 1 0 1 0 311 1) ( (B.54)Solution for H36: The partial derivatives of the global to base transformation matrix, with respect to the kinematic parameters, are defined as 0 0 0 0 0 0 0 0 0 02 1 3 1 3 2 1 3 1 3 2 1 2 1 3 1 3 2 1 3 1 3 2 1 1 0 c s c c s s s s c c s s c c c s s s c s s c s c TG (B.55) 0 0 0 0 0 0 02 1 3 2 1 3 2 1 2 1 3 2 1 3 2 1 2 3 2 3 2 2 0 s c s c c c c c s s s c s c c s c s s c s TG (B.56) 0 0 0 0 0 0 0 0 0 03 1 3 2 1 3 1 3 2 1 3 1 3 2 1 3 1 3 2 1 3 2 3 2 3 0 s s c s c c s s s c s c c s s c c s s s c c s c TG (B.57) 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 04 0TG (B.58) 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 05 0TG (B.59) PAGE 123 123 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 06 0TG (B.60) The solution to H36 can be written as M M M ML M L N G L M L G L L N G L L Gp q T q T vect p q T q T vect p q T q T vect p q T q T vect t H0 0 0 1 0 1 0 0 1 0 1 0 361 1 1 1) ( (B.61)Solution for H37: The solution to H37 is dependent upon rotation from the global to local link coordinate systems. 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Soderstr om, V. Stokes, and H. Lans hammar, Using an extended Kalman filter for rigid pose estimation, J. Biomech. Eng. vol. 127, pp. 475, 2005. [98] P. D. Burns, Slantededge MTF for digital camera and scanner analysis, Proc. PICS Conf. IS&T pp. 135138, 2000. PAGE 131 131 [99] S. Mu, J.D. Yamokoski, and S.A. Banks, JointTrack: An opensource program for measuring skeletal kinematics using modelim age registration. [Online]. Available: https://sourceforge.ne t/projects/jointtrack/ PAGE 132 BIOGRAPHICAL SKETCH Chris A. Lightcap was born in Philadelphia, Pennsylvania. He graduated from Lehigh University in 2004 with a B.S. in mechanical en gineering, and received the Cornelius Award for his outstanding merit as an underg raduate student. He moved to Gainesville, Florida in 2004 to begin research in robotics and medi cal imaging at the University of Florida. He received his M.S. in mechanical engineering and b ecame a Ph.D. candidate in the fi eld of mechanical engineering in 2006. His research interests include orthopaed ic biomechanics, medical imaging, and robotassisted surgery. 