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Friction Modeling and Experimental Identification of a Mitsubishi PA10-6CE Robot Manipulator

Permanent Link: http://ufdc.ufl.edu/UFE0045506/00001

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Title: Friction Modeling and Experimental Identification of a Mitsubishi PA10-6CE Robot Manipulator
Physical Description: 1 online resource (50 p.)
Language: english
Creator: Chen, Cheng
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: dynamics -- flexibility -- friction -- hyperbolic -- identification -- mocap -- model -- pa10-6ce -- robot
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The purpose of this paper is to evaluate the effects of extra loads, different velocity ranges and different friction models on friction parameter identification of a PA10-6CE robot manipulator. A reformulated rigid-link flexible-joint dynamic model including the friction function is applied to identify the friction parameters of a robot with harmonic drive joint transmissions. Motion capture is used to measure the link position as an additional experimental tool. Finite Fourier series and optimization methods are used in trajectory design and friction parameter fitting to provide a well conditioned computational system. A Coulomb plus viscous model, a four-parameter Stribeck model and a continuous hyperbolic model are implemented as friction models, and parameter estimation robustness is compared. The reliability, repeatability and applicability of each model are analyzed based upon the experimental data.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Cheng Chen.
Thesis: Thesis (M.S.)--University of Florida, 2013.
Local: Adviser: Banks, Scott Arthur.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045506:00001

Permanent Link: http://ufdc.ufl.edu/UFE0045506/00001

Material Information

Title: Friction Modeling and Experimental Identification of a Mitsubishi PA10-6CE Robot Manipulator
Physical Description: 1 online resource (50 p.)
Language: english
Creator: Chen, Cheng
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: dynamics -- flexibility -- friction -- hyperbolic -- identification -- mocap -- model -- pa10-6ce -- robot
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The purpose of this paper is to evaluate the effects of extra loads, different velocity ranges and different friction models on friction parameter identification of a PA10-6CE robot manipulator. A reformulated rigid-link flexible-joint dynamic model including the friction function is applied to identify the friction parameters of a robot with harmonic drive joint transmissions. Motion capture is used to measure the link position as an additional experimental tool. Finite Fourier series and optimization methods are used in trajectory design and friction parameter fitting to provide a well conditioned computational system. A Coulomb plus viscous model, a four-parameter Stribeck model and a continuous hyperbolic model are implemented as friction models, and parameter estimation robustness is compared. The reliability, repeatability and applicability of each model are analyzed based upon the experimental data.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Cheng Chen.
Thesis: Thesis (M.S.)--University of Florida, 2013.
Local: Adviser: Banks, Scott Arthur.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045506:00001


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1 FRICTION MODELING AND EXPERIMENTAL IDENTIFICATION OF A MITSUBISHI PA10 6CE ROBOT MANIPULATOR By CHENG CHEN A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2013

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2 2013 Cheng Chen

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3 To my parents for all their support to my graduate study and their encouragement which always inspires me to pursue academic go als

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4 ACKNOWLEDGMENTS I am foremost grateful to my advisor, Dr. Banks, who provided me with the opportunity to work in the Orthopaedic Biomechanics Laboratory, helped a lot with my research and taught me about how to live an academic life. I would lik e to thank Dr. Fregly for being my committee member and for many valuable suggestions. I also would like to thank my lab mates, Ira and Tim, for their great help in robotics and in how to become a qualified graduate student, and to thank Dr. Conrad for his arrangement in my experiment.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 6 LIST OF FIGURES ................................ ................................ ................................ .......... 7 ABSTRA CT ................................ ................................ ................................ ..................... 8 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ ...... 9 2 MODELING ................................ ................................ ................................ ............. 12 Rigid Link Flexible Joi nt Model ................................ ................................ ............... 12 Frictional Models ................................ ................................ ................................ ..... 13 3 METHODS ................................ ................................ ................................ .............. 16 Finite Fourier Series and Trajectory Design ................................ ............................ 16 Estimation of Link Position ................................ ................................ ...................... 18 4 EXPERIMENT ................................ ................................ ................................ ........ 22 5 RESULT AND DISCUSSION ................................ ................................ .................. 25 Extra End Effector Load ................................ ................................ .......................... 26 Velocity Ranges ................................ ................................ ................................ ...... 26 Different Friction Models ................................ ................................ ......................... 27 6 CONCLUSIONS ................................ ................................ ................................ ..... 33 APPENDIX A FRICTION PARAMETER IDENTIFICATION RESULTS ................................ ......... 35 B ROOT MEAN SQUARE ERROR OF THREE FRICTION MODELS ....................... 46 LIST OF REFERENCES ................................ ................................ ............................... 48 BIOGRAPH ICAL SKETCH ................................ ................................ ............................ 50

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6 LIST OF TABLES Table page 5 1 Friction parameter identification results of Coulomb plus viscous and Stribeck model for joint 5. ................................ ................................ ................................ 29 5 2 Friction parameter identification results of hyperbolic model for joint 5. ............. 29 5 3 Root mean square error of three different friction models for joint 4 and joint 5. ................................ ................................ ................................ ........................ 30 A 1 Friction parameter identification results of Coulomb plus viscous and Stribeck model for joint 1. ................................ ................................ ................................ 36 A 2 Friction parameter identification results of hyperbolic model for joint 1. ............. 37 A 3 Friction parameter identification resul ts of Coulomb plus viscous and Stribeck model for joint 2. ................................ ................................ ................................ 38 A 4 Friction parameter identification results of hyperbolic model for joint 2. ............. 39 A 5 Friction parameter identification results of Coulomb plus viscous and Stribeck model for joint 3. ................................ ................................ ................................ 40 A 6 Friction parameter identification results of hyperbolic model for joint 3. ............. 41 A 7 Friction parameter identification results of Coulomb plus viscous and Stribeck model for joint 4. ................................ ................................ ................................ 42 A 8 Friction parameter identification results of hyperbolic model for joint 4. ............. 43 A 9 Friction parameter identification results of Coulomb plus viscous and Stribeck model for joint 6. ................................ ................................ ................................ 44 A 10 Friction parameter identification results of hyperbolic model for joint 6. ............. 45 B 1 Root mean square error of three different friction models for joint 1 and joint 2. ................................ ................................ ................................ ........................ 4 6 B 2 Root mean square error of three different friction models for joint 3 and joint 6. ................................ ................................ ................................ ........................ 47

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7 LIST OF FIGURES Figure page 3 1 The designed, measured, and approximated trajectories for Link 3. .................. 17 3 2 The joint position, velocity and a cceleration from finite Fourier approximations for Link 3. ................................ ................................ .................. 18 3 3 Motion capture markers fixed on the surface of PA10 6CE. ............................... 21 4 1 Configuration of the robot in the experiment. ................................ ...................... 24 5 1 Experimental data and fitted Coulomb plus viscous friction model for three different velocity ranges for joint 5. ................................ ................................ ..... 31 5 2 Experimental data compared to three friction models in high velocity range for joint 5. ................................ ................................ ................................ ............ 32

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8 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science FRICTION MODELING AND EXPERIMENTAL IDENTIFICATION OF A MITSUBISHI PA10 6CE ROBOT MANIPULATOR By Cheng Chen May 2013 Chair: Scott Banks Major: Mechanical Engine ering The purpose of this paper is to evaluate the effects of extra loads, different velocity ranges and different friction models on friction parameter identification of a PA10 6CE robot manipulator. A reformulated rigid link flexible joint dynamic model including the friction function is applied to identify the friction parameters of a robot with harmonic drive joint transmissions. Motion capture is used to measure the link position as an additional experimental tool. Finite Fourier series and optimizatio n methods are used in trajectory design and friction parameter fitting to provide a well conditioned computational system. A Coulomb plus viscous model, a four parameter Stribeck model and a continuous hyperbolic model are implemented as friction models, a nd parameter estimation robustness is compared. The reliability, repeatability and applicability of each model are analyzed based upon the experimental data.

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9 CHAPTER 1 INTRODUCTION Dynamic modeling with accurate experimental parameter identification is essential to improv e the performance of robot control and realistic simulation. Velocity dependent f riction is regarded as one of the most complicated terms in robot dynamic modeling since it can var y dramatically over different velocity ranges. Although precise and reliable friction identification experiments for rigid joint robots have been well reported ( Khosla, 1985 and Johnson 1992 ), methods and results for robots with flexible joints, such as the Mitsubishi PA10 6CE with harmonic drive transmissions are less well established or generally used Based on considerations of tribology and related physics, Armstrong et al. (1991) survey ed friction compensation control strateg ies for robotics The c onventional Coulomb plus viscous friction model was compa r ed to a Stribeck friction model that includ ed a negative viscous period before Coulomb friction. Armstrong et al. developed a seven parameter integrated friction model with Stribeck effects based upon their review of previous work Several related studies have been conducted to identify friction properties in mechanisms and to verify the Stribeck phenomenon with various models. Tuttle (1992) implemented a cubic polynomial to approximate friction in harmonic drives and found a linear relationship between fr iction and velocity. Taghirad (1997) investigated friction in harmonic drives and described a Stribeck friction formulation using simulation and experimental results Canudas et al. (1995) created a bristle based dynamic friction model containing the Strib eck effect, hysteresis and spring like properties This model was subsequently adopted and modified by Swevers (2000). In order to fulfill the requirements of high performance controllers, a continuously

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10 differentiable friction model was proposed by Makkar et al. (2005) with six hyperbolic terms. Sim ultaneously exhibiting Coulomb viscous, static and Stribeck effects, was verified in a numerical simulation. F riction in harmonic drives mounted in robot manipulator s is not solely a function of joint velocity, so a practical model must account for other factors including assembly, load, and bearing preload. Th us, friction modeling and robot parameter identification require different approaches since removal of the harmonic drive from a robot is impractical and change s friction characteristics of real working condition s Kennedy et al. (2003) determined the friction velocity relationship for each joint in a PA 10 robot by moving single joint s at a constant velocity and recording the mean torque re quired to maintain this motion. A four parameter friction model was implemented to express Stribeck Coulomb and viscous properties. Bompos et al. (2007) carried out similar constant velocity experiment s with a six parameter friction model. In their experi ments, the link position was measured from the resolver in each joint. Lightcap et al. (2007) us ed motion capture and optimized trajectories to identif y all dynamic parameters of a PA10 6CE robot one joint at a time, using Coulomb and viscous terms for fr iction. One objective of this paper is to compare the consistency robustness and reliability of three different friction models for a range of experimental conditions. Kennedy et al. (2003) made a comparison between viscous, cubic and Stribeck friction models at a single joint velocity of 0.4 rad/s. Bompos and Lightcap identified the friction parameters under condition s where the maximum joint velocity was about half the velocity limit for that joint. However, there is both practical and theoretical inte rest to study robot dynamics, particularly friction, over the entire velocity range

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11 Trajectory optimization for system identification is another significant technical issue for appropriate experiment design and robustness of model parameters Swevers et a l. (1996) designed periodic trajectories by minimizing the condition number of a regress or matrix which reduced th e effects of measurement noise and improved parameter identification accuracy. Finite Fourier series were used to parameterize trajector ies an d the Fourier coefficients were treated as design variables in the optimization. Calafiore et al. (2001) developed an optimal trajectory design method which minimized the logarithmic determinant of the Fisher information matrix and was carried out on a two link planar manipulator. Since small position residue was achieved in their experiments, finite Fourier series were also implemented to calculate joint velocity and acceleration. This paper applies a rigid link flexible joint robot dynamic parameter ide ntification method to evaluate the effect s from external loads, velocity ranges and different friction model s on parameter identification Section 2 introduces three different types of friction models and reformulates the dynamic equation of a one link rob ot manipulator. Section 3 describes the computational methods for trajectory design and joint angle calculation from motion capture data. Section 4 discusses the experiment al process. In section 5 and 6, the experiment data is analyzed and discussed, and c onclusions are drawn from the comparison between each friction model.

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12 CHAPTER 2 MODELING Rigid Link Flexible Joint Model The motor position and the link position of the PA 10 robot arm should be parameterized separately in order to exhibit the joint fle xibility of the robot manipulator. Then the dynamics of an n link rigid link flexible joint (RLFJ) can be modeled as (Spong, 1987): ( 2 1 ) where and express the joint position, velocity and acceleration of the link and motor angle, respectively, and represent the inertia matrix, the centripetal Coriolis matrix, gravitational and frictional effects of the link dynamics respectively, denote the joint stiffness, motor viscous friction and motor inertia which are all constant, diagonal, positive definite matrices. To simplify the calculation of the dynamic equation, the gear ratio is included in the expression of the motor position. The dynamic model of a one link RLFJ robot can be expressed as: ( 2 2) where and are measurable parameters in the experiment representing the join t position, velocity and acceleration of the link and motor angle respectively, denot e the link inertia and mass respectively, are the motor inertia and viscous friction which can be obtained from the

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13 manufact urer or previous experiment s ( Lightcap, 200 7 ) and represents the frictional term to be modeled. As weak robot joint elasticity (i.e. stiff joints) leads to poorly scaled equations for computation, the stiffness term is substituted from Equation 2 2 Therefore, the one link dynamic equation can be re written as: ( 2 3) However, this scenario still requires very high experimental accuracy and computational sensi tivity to separate the link inertia and viscous coefficients and those of the motor since the deflections are small. Previously measured motor inertia and viscous friction parameters can be used in the calculation to enhance the robustness of the optimizat ion and the dynamic equation can be reformulated as: ( 2 4) Now the friction al gravitational and inertial term s in a one link model of the robot manipulator can be identified with a better conditioned numerical system. L ink positions are measur ed using a motion capture system and linear or nonlinear optimization is used to solve for parameters in an overdetermined system of experimental data Frictional Models Coulomb + Viscous model Velocity d ependent Coulomb friction and viscous friction with simple mathematical expressions are two basic factors in frictional models. This simple model can be written as: ( 2 5) Accordingly, the friction force is linearly dependent on ve locity with a constant Coulomb value based on the direction of motion.

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14 Substituting E quation 2 5 in to E quation 2 4 we have the regressor matrix: Least squares methods are applied to solve this model and this matrix can be implemented to optimize the manipulator trajectory in the experiment to im prove computational robustness. Four parameter Stribeck model A four parameter Stribeck model can be used to capture the negative viscous friction phenomenon at low velocities : ( 2 6) where and denote the Coulomb and viscous friction coefficient s respectively and represents the Stribeck effect of negative viscous friction at low velocities Nonlinear optimization methods in MATLAB (Mathworks, N atick, MA) are utilized to fit overdetermined data sets to this model. Hyperbolic model A continuously differentiable friction model for high performance controller s can be expressed as the combination of hyperbolic terms (Makkar, 2005). With six parameters, the hyperbolic friction model can present comprehensive velocity dependent frictional characteristics based upon a nonlinear symmetric form. Friction force is ex pressed as: ( 2 7) where the Coulomb and viscous dissipation friction are denoted by the terms and respectively, the static coefficient of friction is assumed to be approximated by and the Stribeck effect at low velocity is represented by

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15 Similar to the four parameter Stribeck friction model, t he six design variables in this hyperbolic model are also fitted by nonlinear optimization with an initial guess from the Coulomb + Viscous model.

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16 CHAPTER 3 METHODS Finite Fourier Series and Trajectory Design It has been shown that a well designed traject ory can sufficiently excite the dynamics and frictional properties of a rigid link flexible joint robot. Finite Fourier series are commonly used to approximate periodic trajectories with all the harmonics as design variables in the optimization. Therefore, the optimization criteria can be selected to minimize the condition number of the regressor matrix (Swevers, 1996). The purpose of using finite Fourier series approximation s for trajectory optimization is to enhance the computational robustness and improv e the reliability and accuracy of velocity and acceleration calculation s Thus, the nonlinear optimization with linear constraint s can be expressed as: ( 3 1 ) where the link position constitut ing the regressor matrix is approximated by the motor positio n (a rigid joint approximation), and the position and velocity limits are provided by the robot manufact urer Based on finite Fourier series, the link position, velocity and acceleration can be given as: ( 3 2 )

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17 w here represent s the fundamental frequency chosen as the natural frequency of the robot, and and denot e amplitudes of harmonics as design variables Different velocity ranges are used to assess friction model parameters from different cond itions, thus peak velocities corresponding to 100%, 50% and 25% of the manufacturer specified maximum joint velocity are used for separate tests. Zero initial and final velocity and acceleration conditions are design constraint s to ensure safe and realizab le trajectories Figure 3 1 The designed, measured, and approximated trajectories for Link 3. If the control system is properly designed, the robot arm will exhibit relatively good trajectory tracking behavior, and the finit e Fourier series approximation can also be applied to fit the experimental data to the link and motor positions. Figure 3 1 shows the optimization result of the designed trajectory and the experimental results for joint and link 3. The data indicate the fe ed forward controller used in the experiment has

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18 small tracking errors for well designed trajectories, and that finite Fourier series can provide excellent approximations with average errors less than 0.05 degree. The motor velocity and acceleration can th en be calculated according to the finite Fourier approximation shown in Figure 3 2. Similarly, the link velocity and acceleration can also be computed through this method. Figure 3 2. The joint position, velocity and acceleration from finite Fourier a pproximation s for Link 3. Estimation of Link Position Although the PA10 6CE robot manipulator is assembled with two encoders on both sides of the harmonic drive in each joint, we only have access to the data from the motor side encoder Therefore, the l ink position needs to be measured by the motion capture system due to joint flexibility. Figure 3 3 shows that motion capture markers are attached on the surface of the robot.

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19 To calculate the link position, we first have to know the relative position of the robot to the camera system, in other words, the transformation matrix from the cameras to the fixed base of the robot The unit vector of an axis of rotation and the coordinate of one point on that axis can be computed by rotating a single joint around a fixed axis (Halvorsen, 2003). The relative position of the robot to the camera system can be determined by individually rotating joints one and two and calculating the intersection of these two axes of rotation in the camera reference system. In a rigid link robot arm, the coordinates of each reflective marker evaluated in its corresponding link coordinate s ystem is constant in every data frame. Implementing coordinate transformations, we have: ( 3 3 ) where is the position of marker measured in the coordinate system fixed in link in frame is the position of that marker measured i n the motion capture system in frame and denotes the transformation matrix from the coordinate system fixed in link to the cameras in frame With all the constant geometric parameters of the robot from the manufact urer the only un known and time varying parameters in the transformation matrix are the link angles. Furthermore, the left hand side of the E quation 3 3 is constant in every frame since marker is fixed on that link. To approximate the relative position of each marker corresponding to the coordinate system of its fixed link, a static pose is applied. In that frame, the robot can be obtained directly from the joint encoder. Then th e relative position of the marker in a static pose can be calculated as:

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20 ( 3 4 ) After the static position of each marker corresponding to its fixed link is computed, a nonlinear optimization can be built to obtain the link position. The cost function of link in frame is defined as: ( 3 5 ) where is calculated by using the transformation matrix from the motion capture system to link and the measured position of marker in the camera s in frame In this case, markers are utilized to construct an overdetermined optimization and the motor position is chosen as the initial guess. This method can also be used t o calculate the link position in the situation where multiple link s of a robot are simultaneously moving As discussed above, the only design variable of the optimization i s the joint angle and the computation is carried out in a single frame. However, more design variables such as the geometric parameters can be applied and more frames can be simultaneously used in the optimization of this method.

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21 Figure 3 3. Motio n capture markers fixed on the surface of PA10 6CE (Photo courtesy of author)

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22 CHAPTER 4 EXPERIMENT Figure 4 1 shows the setup of the experiment where a PA10 4CE robot manipulator was mounted on a floor stand and placed in the center of the viewing volum e of a twelve camera motion capture system (Motion Analysis Corporation, Santa Rosa, CA). Sixteen passive reflective markers were attached on the exterior surface of the robot links. The motion capture system was calibrated and the 3D residual measurement error was recorded as 0.845 mm (S.D. 0.21 mm). A real time computer, CompactRio (National Instruments Corporation, Austin, TX), was used to control the robot and collect measurement data. Synchronized data with time stamps from the robot and the motion cap ture system were streamed to a host computer at a frequency of 200Hz. The experiment was conducted in two groups of trials. A 33.075 N load was mounted on the robot end effector for one group of trials and the second group of trials had no additional end effector load. The optimized single joint trajectories were followed for each individual robot joint and repeated for each group of trials. Electro magnetic brakes were used for inactive links. Three different velocity boundaries were designed for each joi nt with the highest maximum velocity in one trajectory being the velocity limit of that joint and the lowest velocity boundary being 10% of the limit. Each trajectory was repeated five times to establish repeatability. During the experiment, the motor pos ition was measured by the joint encoders with a resolution of 0.011 degree for the angular position. The motor torque was computed from the input motor current and motor torque constant. The link position was calculated offline by using the marker position data from the motion capture system.

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23 Finite Fourier series were used to fit the link position to a continuous function in order to synchronize the link and motor position. Friction models were fit to the experimental data using linear or nonlinear optimiz ation. The sample mean value and a coefficient of variation (CV) for each friction parameter were calculated as experiment results.

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24 Figure 4 1. Configuration of the robot in the experiment (Photo courtesy of author)

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25 CHAPTER 5 RESULT AND DISCUSSION The velocity dependent friction parameters for three friction models are re p or t ed in Table 5 1. The root mean square (RMS) error in the estimation of the friction torque and its percentage of the maximum motor torque for joint s 4 and 5 are shown in Table 5 2. The results of the remaining joints are reported in the A ppendix. Most CV values for Coulomb friction terms, and RMS errors in the Coulomb plus viscous model, were below 10% and indicate consisten t data was obtained from repeated trials The percentag es of the RMS error proportional to the motor torque of that joint, were less than 9 % of maximum motor torque in that trial. In terms of finite Fourier approximation, the computational error was less than 0.05 degree for motor angles and 0.2 degree for li nk positions. Both experimental and computational sources of error were produced in the identification process. With the calibration error from the motion capture system, the optimization for estimating the link position had a residual error of marker pos ition of approximate 0.9 mm in each joint. The measured input torque computed from the current did not include the variability of motor constant and the efficiency of gear transmission in different assembly and working conditions. The inaccuracy of the geo metric parameters of the robot from the manufacturer and the motor inertia and viscous parameters from previous studies (Lightcap, 2007 and Lightcap, 2008) contributed additional uncertainties to the optimization. In terms of friction, the sensitivity of j oint angle measurement caused difficulty in modeling static friction, and position dependent friction (Tuttle, 1992) was not included in the models. In addition, single link

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26 trajectories reduced the accuracy of the experiment since joints locked by electro magnetic brakes may still deflect slightly under load. Based on the experimental results, several factors influencing friction parameter estimation are identified. Extra End Effector Load Extra end effector load has been claimed to influence friction p arameters (Armstrong, 1991) and to better excite robot dynamics for better parameter estimation (Lightcap, 2008). By comparing the experimental results of the loaded (33.1 N) and unloaded groups, the following conclusions can be drawn: There was no obvious change in either Coulomb or viscous friction terms for any of the three friction models with extra end effector load. The extra load group estimates showed more consistent parameters with smaller CV values and better model fits with smaller RMS errors, e specially for joints 4, 5 and 6. Velocity Ranges For each joint, three different trajectories were designed with velocity boundaries ranging from 100% to about 20% of the maximum joint velocity Friction models were fit based upon the data from each traj ectory, and several conclusions can be drawn from the data shown in Table 5 1 and the Appendix. The Coulomb friction coefficient for all three friction models was not significantly affected by experiment velocity ranges. This indicates that Coulomb fricti on remains constant and does not change measurably as the velocity magnitude is varied. As the velocity boundary increased, the viscous term decreased. Figure 5 1 shows an example of the effect of experimental velocity range on friction parameters for join t 5. The CV for each model parameter was higher for low velocity conditions since friction torques are smaller, the signal to noise ratio is smaller, and the sensitivity for parameter estimation is reduced.

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27 RMS errors reached their lowest values for medi um velocity trajectories in some joints. This may be due to other mechanical factors, such as soft wind up and friction hysteresis at low velocities, and model applicability across the whole robot velocity range. Different Friction Models It is essentia l to verify the reliability and accuracy of different friction models in the PA10 robot. According to the experimental data and analysis the hyperbolic friction model with six parameters obtained the lowest RMS errors among the three models. It is obvious to conclude that greater degrees of freedom in the friction model contributes to lower model fitting error. On the other hand, the Coulomb plus viscous friction model exhibited the best parameter repeatability, with the smallest CV in all the joints. An e xample of a comparison among three friction models is shown in Figure 5 2. The Coulomb plus viscous friction model captures the basic frictional properties of the PA10 robot. The two most significant friction parameters were conveniently fitted with good convergence during a number of trials. However, relatively large RMS residual errors remain, and the negative viscous friction phenomenon at low velocities cannot be captured by this simple model. A four term Stribeck model was utilized to enhance the a bility to capture low velocity friction properties, which involved adding two terms, and to the Coulomb plus viscous friction model. Smaller RMS errors were achieved, and negative viscous friction at low joint velocities was approximated. However, due to the similarity with the Coulomb plus viscous friction model, the four parameter Stribeck model could converge to two entirely different forms: The first form had very close to 0, and the model degenerated to the Coulomb plus viscous model (joint 3). In the second form (joint 5 ), was approximately 0 with a small and the Coulomb friction was expressed by a

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28 negative Therefore, very large CV values for fitting and resulted without extremely high measurement sensitivity and properly designed optimization parameter boundaries. Furthermore, Stribeck effect can only be represented by a positive and a small value for normal mechanisms. In the six parameter hyperbolic model an d which denote the Coulomb and viscous friction coefficients, had the best repeatability during the repeated trials. The values of the remaining design parameters varied dramatically across trials even though the negative viscous friction phenomeno n was clearly shown. This indicated that non unique combinations of and can lead to similar overall values for the expression of the Stribeck effect using this model. The static friction term, claimed to be represented as could not be properly modeled since high enough experimental accuracy could not be obtained.

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29 Table 5 1 Friction parameter identification results of Coulomb plus viscous and Stribeck model for joint 5 Joint Extra load ( N) Maximum velocity (rad/s) Statistic Coulomb + viscous model Four parameter Stribeck model 5 33.1 2.43 mean 4.13 1.17 <0.01 1.01 4.63 <0.01 CV(%) 3.10 2.73 n/a 2.99 2.88 n/a 1.58 mean 3.88 1.56 <0.01 1.38 4.16 <0.01 CV(%) 0.57 1.05 n/a 1.17 0.55 n/a 0.80 mean 3.62 2.57 0.79 1.76 3.56 <0.01 CV(%) 1.32 2.49 200 .00 20.27 54.66 n/a 0 2.42 mean 3.90 0.84 <0.01 0.67 4.41 <0.01 CV(%) 2.67 4.46 n/a 3.45 0.88 n/a 1.58 mean 3.89 1.37 0.53 1.14 3.70 0.01 CV(%) 1.63 1.64 80.39 2.05 10.28 18.58 0.79 mean 3.62 2.66 0.80 1.83 3.58 0.0 1 CV(%) 2.70 12.36 200 .00 40.28 57.15 50.22 Table 5 2. Friction parameter identification results of hyperbolic model for joint 5 Joint Extra load (N) Maximum velocity (rad/s) Statistic Six parameter hyperbolic model 5 33.1 2.43 mean 5.02 161.4 0 5.37 5.01 4.18 0.81 CV(%) 7.11 9.31 11.83 2.47 6.54 4.88 1.58 mean 6.49 105.85 30.43 4.44 19.37 1.22 CV(%) 2.09 2.46 1.43 0.44 1.26 1.30 0.80 mean 27.63 88.87 22. 16 4.56 18.88 1.53 CV(%) 163.2 0 17.07 91.72 3.95 80.26 14.07 0 2.42 mean 4.70 157.77 4.82 4.87 3.77 0.42 CV(%) 0.67 16.19 5.21 2.70 5.33 10.34 1.58 mean 106.1 0 52.5 0 49.12 4.57 20.29 0.93 CV(%) 26.07 4.22 2.15 1.70 3.03 1.06 0.79 mea n 84.21 72.12 49.87 4.40 36.66 1.85 CV(%) 51.53 26.82 40.87 6.94 35.88 33.80

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30 Table 5 3 Root mean square error of three different friction models for joint 4 and joint 5 Joint Extra load (N) Maximum velocity (rad/s) RMS error Coulomb + viscous model Stribeck model Hyperbolic model 4 33.1 2.00 mean(Nm) 0.27 0.19 0.19 CV(%) 3.57 5.69 5.78 percentage(%) 5.48 3.89 3.80 1.51 mean(Nm) 0.18 0.17 0.17 CV(%) 4.83 4.89 4.91 percentage(%) 3.90 3.73 3.67 0.81 mean( Nm) 0.17 0.12 0.12 CV(%) 12.34 25.11 27.08 percentage(%) 3.96 2.89 2.80 0 2.00 mean(Nm) 0.32 0.25 0.24 CV(%) 6.18 10.97 11.65 percentage(%) 5.70 4.44 4.36 1.51 mean(Nm) 0.23 0.20 0.19 CV(%) 0.41 0.51 0.52 percentage(%) 4 .41 3.88 3.78 0.79 mean(Nm) 0.35 0.26 0.25 CV(%) 51.68 82.61 85.56 percentage(%) 6.37 4.78 4.66 5 33.1 2.43 mean(Nm) 0.83 0.61 0.58 CV(%) 8.72 12.88 12.25 percentage(%) 7.16 5.22 2.01 1.58 mean(Nm) 0.58 0.47 0.45 C V(%) 1.81 1.68 1.77 percentage(%) 4.69 3.79 3.63 0.80 mean(Nm) 0.66 0.53 0.49 CV(%) 3.55 9.63 3.04 percentage(%) 7.27 5.85 5.41 0 2.42 mean(Nm) 0.80 0.57 0.54 CV(%) 3.55 9.63 3.04 percentage(%) 7.27 5.85 5.41 1.58 mean(Nm ) 0.64 0.55 0.53 CV(%) 1.82 4.49 5.41 percentage(%) 6.66 5.72 5.53 0.79 mean(Nm) 0.67 0.53 0.50 CV(%) 7.40 22.45 16.03 percentage(%) 8.38 6.63 6.24

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31 Figure 5 1. Experimental data and fitted Coulomb plus viscous friction model for t hree different velocity ranges for joint 5.

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32 Figure 5 2. Experimental data compared to three friction models in high velocity range for joint 5.

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33 CHAPTER 6 CONCLUSIONS In this paper we have presented our approach for identifying friction parameters in three different models through experiments with a RLFJ robot manipulator. We first established the dynamic model of the robot including the friction function which took the joint flexibility of harmonic drives into account. A motion capture system was used to measure the link position as an additional tool of the experiment. Linear and nonlinear optimization methods were applied to solve for friction parameters of each model and comparison was made among different friction models bas ed on the experimental data. The reliability and repeatability of each friction model were analyzed at different velocity ranges. Compared to the dynamic model of rigid joint robot, the model involving joint flexibility is more accurate and practical to d escribe the robot arm installed with harmonic drives. With the development of precise cameras and real time data transmission, motion capture systems can contribute significantly in research of robotics and human robot interaction. The experimental method discussed in this paper has the advantage over the constant velocity method (Kennedy, 2003) since it includes gravitational term s in the dynamic model which reduce inaccuracy when installing the robot arm in typical standing configurations Different fric tion models have their corresponding applicable velocity ranges. For example, the four parameter Stribeck model can capture the negative viscous friction in low velocity, but tends to degenerate to Coulomb plus viscous model in relatively high velocity. Th erefore, it is crucial to use the friction model in control loops based on the working velocity to achieve high accuracy.

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34 Further efforts will be focused on two main areas. One is to improve the accuracy of dynamic identification and design multi link tra jectories for more practical working conditions. The other is to apply the identified friction model to the control strategy (Dixon, 2000) and to achieve better model based tracking performance.

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35 APPENDIX A FRICTION PARAMETER IDENTIFICATION RESULTS

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36 Tabl e A 1 Friction parameter identification results of Coulomb plus visco us and Stribeck model for joint 1. Joint Extra load (N) Maximum velocity (rad/s) Statistic Coulomb + viscous model Four parameter Stribeck model 1 33.1 1.00 mean 9.86 5.47 0.04 4.24 11.06 0.01 CV(%) 0.51 4.55 200.00 4.46 1.31 7.92 0.50 mean 8.44 12.52 <0.01 7.72 10.91 0.01 CV(%) 1.77 0.84 n/a 1.88 1.53 2.39 0.34 mean 8.87 15.30 1.02 8.12 10.37 0.01 CV(%) 1.15 1.72 200 .00 13.74 23.51 16.94 0 1.00 mean 9.53 5.62 <0.01 4.37 10.81 0.01 CV(%) 1.01 2.70 n/a 3.27 1.44 4.18 0.50 mean 8.68 12.09 <0.01 7.37 11.09 0.01 CV(%) 2.66 1.70 n/a 3.36 2.05 1.77 0.34 mean 10.05 19.31 <0.01 10.12 13.27 0.01 CV(%) 9 .14 9.06 n/a 9.70 9.77 1.21

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37 Table A 2. Friction parameter identification results of hyperbolic model for joint 1. Joint Extra load (N) Maximum velocity (rad/s) Statistic Six parameter hyperbolic model 1 33.1 1.00 mean 13.62 95.24 14.38 11.30 10.62 3.84 CV(%) 2.71 8.81 5.09 0.83 6.67 5.99 0.50 mean 261.4 0 62.75 59.48 10.61 39.75 8.42 CV(%) 13.06 3.34 2.68 1.59 3.82 1.77 0.34 mean 157.7 0 76.34 70.69 10.99 61.71 9.55 CV(%) 19.53 1.78 2.48 1.25 6.04 3.87 0 1.00 mean 14.01 86.94 16.13 10.92 11.98 4.12 CV(%) 3.54 4.27 9.34 1.71 9.26 3.65 0.50 mean 250.1 0 59.11 55.62 10.87 34.55 7.90 CV(%) 19.61 3.08 2.85 2.22 3.94 2.35 0.34 mean 199.5 0 75.39 70.26 12.56 60.95 12.47 C V(%) 25.09 0.93 0.57 9.97 4.93 9.17

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38 Table A 3 Friction parameter identification results of Coulomb plus viscous and Stribeck model for joint 2. Joint Extra load (N) Maximum velocity (rad/s) Statistic Coulomb + viscous model Four parameter Stribeck mo del 2 33.1 1.01 mean 12.44 26.96 2.00 23.82 14.39 0.00 CV(%) 14.87 3.25 200.00 0.53 22.22 66.15 0.51 mean 12.69 39.07 12.76 83.87 57.16 1.00 CV(%) 2.16 1.91 2.15 8.34 15.52 0.00 0.26 mean 4.41 79.81 4.45 435.29 401.9 0 1.00 CV(%) 44.02 8.56 32.55 16.50 18.57 0.00 0 1.01 mean 13.39 24.01 2.40 20.77 15.01 <0.01 CV(%) 20.87 2.82 200.00 2.22 20.14 n/a 0.51 mean 12.35 32.81 12.40 60.05 34.74 1.00 CV(%) 0.65 1.10 0.61 6.73 13.79 0.00 0.26 mean 5.42 65.92 5.25 323.05 290.2 0 1.00 CV(%) 3.57 2.09 3.72 4.79 5.91 0.00

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39 Table A 4. Friction parameter identification results of hyperbolic model for joint 2. Joint Extra load (N) Maximum velocity (rad/s) Statistic Six parameter hyperbolic model 2 33.1 1.01 mean 14.41 1015. 8 0 16.29 16.49 15.54 23.63 CV(%) 2.39 77.99 37.37 3.12 29.58 5.12 0.51 mean 14.51 231.73 3.11 1.04 17.76 60.90 CV(%) 2.96 3.60 5.35 157.19 104.8 0 4.56 0 .26 mean 86.93 51.73 12.50 0.21 9.85 99.37 CV(%) 170.6 0 25.87 48.55 112.83 77.80 0.89 0 1.01 mean 15.50 825.62 8.85 16.12 9.23 21.62 CV(%) 6.13 69.49 49.46 29.84 36.66 17.12 0.51 mean 14.26 224.90 2.62 <0.01 95.36 54.58 CV(%) 0.28 2.62 1 .85 n/a 116.7 0 0.90 0.26 mean 12.37 59.79 8.67 0.01 7.18 89.69 CV(%) 1.25 3.30 1.59 89.81 88.63 1.46

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40 Table A 5 Friction parameter identification results of Coulomb plus viscous and Stribeck model for joint 3. Joint Extra load (N) Maximum velocity (rad/s) Statistic Coulomb + viscous model Four parameter Stribeck model 3 33.1 2.03 mean 9.18 7.35 9.18 7.35 <0.01 1.00 CV(%) 0.90 4.49 0.90 4.49 n/a 0.00 0.51 mean 10.04 11.19 10. 05 16.60 6.81 1.00 CV(%) 1.76 6.15 1.74 15.78 48.00 0.00 0.27 mean 6.90 21.94 6.90 21.94 <0.01 1.00 CV(%) 2.72 3.13 2.72 3.13 n/a 0.11 0 2.03 mean 8.34 5.20 8.34 5.20 <0.01 1.00 CV(%) 1.58 8.53 1.58 8.53 n/a 0.00 0.51 mean 10.16 9.62 10.16 9.62 <0.01 1.00 CV(%) 2.76 9.34 2.76 9.35 n/a 0.10 0.25 mean 7.26 24.31 7.26 24.31 <0.01 1.00 CV(%) 2.46 7.23 2.46 7.23 n/a 0.00

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41 Table A 6. Friction parameter identification results of hyperbolic model for joint 3. Joint Extra load (N) Maximum velocity (rad/s) Statistic Six parameter hyperbolic model 3 33.1 2.03 mean 10.98 54.95 5.94 12.96 3.88 4.80 CV(%) 1.30 6.54 1.93 2.50 3.62 4.97 0.51 mean 8.16 1459.6 0 33.59 10.31 48.74 10.97 CV(%) 3.15 19.98 4.81 2.17 3.71 5.95 0.27 mean 9.26 148.54 5.15 2.37 10.27 36.95 CV(%) 2.80 2.36 4.10 8.93 6.35 1.86 0 2.03 mean 10.22 53.01 6.06 11.27 4.16 3.24 CV(%) 1.37 4.30 3.27 3.27 3.37 10.10 0.51 mean 8.61 1177.7 0 41.29 10.58 52.83 9.06 CV(%) 2.47 4.66 4.54 3.23 3.36 8.80 0.25 mean 9.80 140.90 5.62 3.25 11.53 38.35 CV(%) 1.55 1.06 1.20 11.52 2.00 7.23

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42 Table A 7 Friction parameter identification results of Coulomb plus viscous and Stribeck model for joint 4. Joint Extra load (N) Maximum velocity (rad/s) Statistic Coulomb + viscous model Four parameter Stribeck model 4 33.1 2.00 mean 1.37 0.12 <0.01 0.06 1.50 <0.01 CV(%) 1.16 17.80 n/a 32.22 1.12 n/a 1.51 mean 1.47 0.32 <0.01 0.30 1.50 <0.01 CV(%) 1.25 31.59 n/a 32.96 1.30 n/a 0.81 mean 1.62 1.31 0.01 1.03 1.86 0.01 CV(%) 3.99 11.88 200.00 17.09 4.45 14.49 0 2.00 mean 1.51 0.36 <0.01 0.29 1.66 0.01 CV(%) 1.23 2.51 n/a 3.96 0.14 30.51 1.51 mean 1.55 0.61 <0.01 0.57 1.63 <0.01 CV(%) 5.33 16.43 n/a 19.14 1.85 n/a 0.79 mean 1.50 2.37 <0.01 1.90 1.93 0.01 CV(%) 12.68 22.15 n/a 32.74 14.10 26.52

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43 Table A 8. Friction parameter identification results of hyperbolic mod el for joint 4. Joint Extra load (N) Maximum velocity (rad/s) Statistic Six parameter hyperbolic model 4 33.1 2.00 mean 30.97 118.96 111.57 1.50 61.41 0.06 CV(%) 5.77 11.73 11.39 1.15 10.36 31.63 1.51 mean 1.79 208.84 20.71 1.54 16.45 0.27 CV(%) 8.77 11.15 50.20 2.94 50.25 31.29 0.81 mean 38.73 49.75 34.09 1.92 20.21 0.96 CV(%) 78.29 34.91 12.84 5.22 8.95 20.35 0 2.00 mean 41.32 78.31 73.47 1.68 38.09 0.29 CV(%) 37.47 21.08 19 .14 0.37 12.40 3.64 1.51 mean 1.67 553.90 8.45 1.81 6.75 0.43 CV(%) 1.65 33.17 40.67 7.25 38.86 28.96 0.79 mean 25.49 103.93 22.12 1.82 13.86 2.01 CV(%) 116.7 0 45.04 99.31 42.79 61.67 61.57

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44 Table A 9 Friction parameter identification re sults of Coulomb plus viscous and Stribeck model for joint 6. Joint Extra load (N) Maximum velocity (rad/s) Statistic Coulomb + viscous model Four parameter Stribeck model 6 33.1 2.01 mean 1.18 0.30 <0.01 0.25 1.30 0.01 CV(%) 0.85 2.69 n/a 3.74 1.10 5.94 1.51 mean 1.13 0.42 <0.01 0.36 1.24 <0.01 CV(%) 1.25 9.04 n/a 9.89 1.24 n/a 0.80 mean 1.07 0.82 <0.01 0.55 1.32 0.01 CV(%) 4.27 11.14 n/a 19.19 4.31 7.35 0 2.00 mean 1.20 0.30 <0.0 1 0.26 1.30 0.01 CV(%) 0.92 2.58 n/a 3.32 1.35 6.22 1.50 mean 1.14 0.42 0.04 0.38 1.18 <0.01 CV(%) 2.86 8.66 200.00 7.38 6.80 n/a 0.80 mean 1.05 1.02 0.19 1.20 0.49 0.20 CV(%) 7.18 28.71 200.00 95.42 332.6 0 195.1 0

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45 Table A 10. Frict ion parameter identification results of hyperbolic model for joint 6. Joint Extra load (N) Maximum velocity (rad/s) Statistic Six parameter hyperbolic model 6 33.1 2.01 mean 1.55 75.54 11.45 1.21 11.5 6 0.30 CV(%) 2.27 6.68 13.85 1.54 13.13 6.09 1.51 mean 1.35 428.76 23.91 1.23 24.38 0.37 CV(%) 2.93 18.07 34.97 3.08 38.30 3.30 0.80 mean 20.06 62.45 58.12 1.30 41.43 0.57 CV(%) 20.29 6.72 6.31 4.76 17.15 19.52 0 2.00 mean 1.42 118.81 10.94 1.23 11.31 0.30 CV(%) 1.20 5.81 13.81 1.14 13.86 3.17 1.50 mean 1.25 687.14 22.64 1.24 23.95 0.36 CV(%) 2.30 32.53 56.82 3.99 62.04 7.00 0.80 mean 13.50 100.24 66.39 1.03 52.86 1.06 CV(%) 48.94 22.58 48.62 50.04 50.13 77.03

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46 APPE NDIX B ROOT MEAN SQUARE ERROR OF THREE FRICTION MODELS Table B 1. Root mean square error of three different friction models for joint 1 and joint 2. Joint Extra load (N) Maximum velocity (rad/s) RMS error Coulomb + viscous model Stribeck model Hyperbolic model 1 33.1 1.00 mean(Nm) 1.72 1.28 1.24 CV(%) 5.00 5.16 5.47 percentage(%) 6.39 4.75 4.62 0.50 mean(Nm) 1.86 1.45 1.38 CV(%) 1.37 2.33 2.45 percentage(%) 9.04 7.02 6.71 0.34 mean(Nm) 1.92 1.56 1.46 CV(%) 1.32 4.56 1.83 percentage(%) 10.49 8.52 7.96 0 1.00 mean(Nm) 1.77 1.33 1.30 CV(%) 4.20 4.02 4.17 percentage(%) 6.59 4.93 4.82 0.50 mean(Nm) 1.87 1.49 1.44 CV(%) 1.87 1.76 1.96 percentage(%) 9.01 7.21 6.94 0.34 mean(Nm) 2.27 1.82 1.73 CV(%) 9.79 8.77 8.68 percentage(%) 10.78 8.61 8.22 2 33.1 1.01 mean(Nm) 3.73 2.57 2.57 CV(%) 15.97 8.15 4.67 percentage(%) 2.88 1.99 1.99 0.51 mean(Nm) 3.01 2.99 2.37 CV(%) 3.52 3.56 3.84 percentage(%) 3. 59 3.58 2.83 0.26 mean(Nm) 3.11 3.02 2.79 CV(%) 6.76 5.56 9.91 percentage(%) 5.95 5.79 5.34 0 1.01 mean(Nm) 3.70 2.43 2.46 CV(%) 12.57 5.02 8.55 percentage(%) 3.56 2.34 2.36 0.51 mean(Nm) 2.93 2.92 2.28 CV(%) 1.40 1.36 0 .39 percentage(%) 4.29 4.28 3.34 0.26 mean(Nm) 2.86 2.82 2.51 CV(%) 2.06 2.05 2.04 percentage(%) 6.53 6.44 5.73

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47 Table B 2. Root mean square error of three different friction models for joint 3 and joint 6. Joint Extra load (N) Maximum velocity (rad/s) RMS error Coulomb + viscous model Stribeck model Hyperbolic model 3 33.1 2.03 mean(Nm) 3.10 3.10 2.67 CV(%) 3.04 3.04 3.32 percentage(%) 3.25 3.25 2.80 0.51 mean(Nm) 1.30 1.30 1.11 CV(%) 1.83 1.84 2.79 p ercentage(%) 2.77 2.77 2.35 0.27 mean(Nm) 2.03 2.03 1.68 CV(%) 2.78 2.78 2.83 percentage(%) 7.71 7.71 6.37 0 2.03 mean(Nm) 2.51 2.51 2.08 CV(%) 5.40 5.40 6.91 percentage(%) 3.96 3.96 3.27 0.51 mean(Nm) 1.12 1.12 0.97 CV( %) 4.52 4.52 6.09 percentage(%) 3.23 3.23 2.80 0.25 mean(Nm) 2.24 2.24 1.84 CV(%) 0.95 0.95 1.31 percentage(%) 9.77 9.77 8.02 6 33.1 2.01 mean(Nm) 0.25 0.20 0.19 CV(%) 4.10 3.74 3.89 percentage(%) 6.45 5.09 4.88 1. 51 mean(Nm) 0.16 0.12 0.12 CV(%) 7.76 12.05 12.04 percentage(%) 4.66 3.47 3.38 0.80 mean(Nm) 0.18 0.13 0.12 CV(%) 4.28 10.97 11.58 percentage(%) 6.41 4.57 4.43 0 2.00 mean(Nm) 0.21 0.17 0.16 CV(%) 3.82 3.07 3.22 percentag e(%) 5.38 4.31 4.19 1.50 mean(Nm) 0.14 0.11 0.11 CV(%) 17.75 18.18 18.06 percentage(%) 4.10 3.17 3.12 0.80 mean(Nm) 0.20 0.16 0.15 CV(%) 30.14 49.17 47.23 percentage(%) 6.49 5.20 4.97

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48 LIST OF REFERENCES 1. Proc. of 24th IEEE Conference on Decision and Control pp. 1754 1760, 1985. 2. compensation in precise, position controlled mech IEEE Trans. Ind. Applicat., vol. 28, pp. 1392 1398, Nov./Dec. 1992. 3. B. Armstrong Helouvry, Control of machines with friction Kluwer Academic Publishers, 1991. 4. transm 5. Centre for Intelligent Machines, McGill University. Tech. Rep. CIM TR 97 02, 1997. 6. C. Canudas de Wit, IEEE Transactions Automatic Control, vol. 40, pp. 419 425, Mar. 1995. 7. J. Swevers, F. Al Friction Model Structure with Improved Presliding Behavior for Accurate Friction IEEE Transactions Automatic Control, vol. 45, pp. 675 686, Apr. 2000. 8. differentiable friction model for c Proc. IEEE/ASME Int. Conf. Adv. Intell. Mechatronics Monterey, CA, 2005, pp. 600 605. 9. transmission in the Mitsubishi PA Proc. IEEE/ RSJ Int. Conf. Intell. Robots Syst., 2003, pp. 3331 3336. 10. N. Bompos, P. Artemiadis, A. Oikonomopoulos, and K. Kyriakopoulos Advanced intelligent mechatronics, 2007 IEEE/ASME international conference on, September 2007, pp. 1 6.

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49 11. PA10 6CE Proc. IEEE/RSJ Int. Conf. Intell. Robot. Syst ., San Diego, CA, 2007, pp. 3860 3865. 12. Mechanical Systems and Signal Processing vol. 10, no. 5, pp. 561 577, 1996. 13. Journal of Robotic Systems vol. 18, no. 2, pp. 55 68, 2001. 14. Journal of Dyn. Syst., Meas. and Cont., vol. 109, pp. 310 319, 1987. 15. Journal of Biomechanics vol. 36, pp. 999 1008, 2003. 16. accuracy of the PA10 IEEE Trans. on Robotics submitted for publication, Jan. 2007. 17. W.E. Dixon, E. Zergerog lu, D.M. Dawson, and M.W. Hannan, "Global adaptive partial state feedback tracking control of rigid link flexible joint robots ", Robotica vol. 18, no. 3, pp. 325 336, 2000.

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50 BIOGRAPHICAL SKETCH Cheng Chen received a Bachelor of Science degree in Mechani cal Engineering from Shanghai Jiaotong University, China. Then he entered University of Florida with research interest about dynamics and control. He decided to pursue his Master of Science degree at UF under the direction of Dr. Banks.