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PAGE 1 1 THE DESIGN OF A SINGLE DEGREE OF FREEDOM OPENLOOP SPATIAL MECHANISM THAT INCORPORATES GEARED CONNECTIONS By JOSEPH M. BARI 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 2009 PAGE 2 2 2009 Joseph M. Bari PAGE 3 3 To my mom, for inspiring me to investigate the world, and to my dad, for providing me the foundation to do so PAGE 4 4 ACKNOWLEDGEMENTS I would like to acknowledge the support of my committee Dr. Scott Banks, Dr. Carl Crane, Dr. David Dooner and Dr. John Schueller. Dr. Dooner provided invaluable day to day support as well as the wonderful Ingear software that made this work possible. Dr. Crane has provided a vast amount of insight and dedication to this project. I would also like to thank the members of the Center for Intelligent Machines and Robotics for their friendship and knowledge. Finally, I would like to thank m y family for their support and love as well as the foundation that has enabled me to reach this point. Thank you to my mom, Gayle Bari, brothers Shane Bari and Adam Reinardy, sister Kaela Reinardy and my amazing girlfriend Carly Knoell. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGEMENTS ............................................................................................... 4 LIST OF TABLES ............................................................................................................ 6 LIST OF FIGURES .......................................................................................................... 7 ABSTRACT ..................................................................................................................... 8 CHAPTER 1 INTRODUCTION .................................................................................................... 10 1.1 Motivation ....................................................................................................... 10 1.2 Related Work ................................................................................................. 11 2 PROBLEM STATEM ENT ........................................................................................ 13 3 OPTIMIZATION ...................................................................................................... 15 4 EXACT PATH SPECIFICATION ............................................................................. 17 4.1 Pa th Creation ................................................................................................. 17 4.2 Scoring Function ............................................................................................ 20 4.2.1 Imaginary Solutions ............................................................................. 22 4.2.2 Singularities ......................................................................................... 23 4.2.3 Limitations ........................................................................................... 23 4.3 Numerical Example ........................................................................................ 23 5 PRECISON POINT SYNTHESIS ............................................................................ 30 5.1 Path Creation ................................................................................................. 30 5.2 Scoring Function ............................................................................................ 31 5.3 Numerical Example ........................................................................................ 33 6 CONCLUSION ........................................................................................................ 40 LIST OF REFERENCES ............................................................................................... 41 BIOGRAPHICAL SKETCH ............................................................................................ 43 PAGE 6 6 LIST OF TABLES Table page 2 1 Design p arameters ............................................................................................. 14 4 1 Spline parameters .............................................................................................. 25 4 2 Dimens ional bounds ........................................................................................... 25 4 3 Initial population range for genetic algorithm optimization .................................. 25 4 4 Genetic algorithm results .................................................................................... 26 4 5 Quasinewton results .......................................................................................... 26 5 1 Dimensional bounds ........................................................................................... 37 5 2 Initial population range for genetic algorithm optimization .................................. 37 5 3 Genetic algorithm results .................................................................................... 37 5 4 Quasinewton results .......................................................................................... 37 PAGE 7 7 LIST OF FIGURES Figure page 2 1 Example of openloop geared manipulator ......................................................... 14 4 1 Example desired path. ........................................................................................ 19 4 2 Example translational splines ............................................................................. 20 4 3 Example rotational splines .................................................................................. 20 4 4 Flow chart of design algorithm ............................................................................ 26 4 5 Three dimensional desired path (cm). ................................................................ 27 4 6 X/Y/Z coordinates of tool point for spline ......................................................... 27 4 7 Roll/pitch/yaw angles for spline ....................................................................... 28 4 8 Angular positions of joints ................................................................................... 28 4 9 Instantaneous gear ratios ................................................................................... 29 4 10 Input/output plots ................................................................................................ 29 5 1 Joint angles for precision point synthesis numerical example ............................ 38 5 2 Path traced by end effector ................................................................................ 38 5 3 Kinematic velocities for numerical example ........................................................ 39 5 4 Gear pair produced by Ingear software .............................................................. 39 PAGE 8 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 THE DESIGN OF A SINGLE DEGREE OF FREEDOM OPENLOOP SPATIAL MECHANISM THAT INCORPORATES GEARED CONNECTIONS By Joseph M. Bari December 2009 Chair: Carl D. Crane, III Major: Mechanical Engineering A means has been discovered to apply gear pairing to create a one degree of freedom openloop spatial mechanism. A specially chosen geometry consisting of three pairs of parallel joint axes is constricted by five sets of gears, three of which are planar, allowing for a reconfigurable mechanism that is suited for repetitive tasks. Previous work has examined threedimensional rigid body guidance in closedloop geared mechanisms, but has not come to a solution for the openloop case. Two design methods are introduced for the synthesis of mechanism parameters. Gear pairs are designed based upon a desired position and orientation path. Numerical optimization is performed to obtain physically realizable gear profiles. Noncircular gear centrodes must be c ontinuous and smooth. A gear pair must spin in a single direction and with consistent relative turning direction between joined gears These constraints eliminate nonrealizable or nonoptimal gears in favor of simple, more easily produced profiles. In the first design algorithm v ariable parameters include link lengths, joint offsets and twist angles. A second algorithm is developed that also determines the mechanism parameters and gear profiles that move the end effector through specified positions and orientations PAGE 9 9 without concern for the path that is followed between poses. Again, gear centrodes must be continuous, smooth and mono directional Path freedom allows greater flexibility in the design space of the mechanism. Similar to the rigid path des ign algorithm, variable parameters are mechanism link lengths, joint offsets and twist angles. The position and orientation of the base of the manipulator are also considered as variable parameters. PAGE 10 10 CHAPTER 1 INTRODUCTION 1.1 Motivation Robotic manipulators are utilized for a wide variety of tasks including surgical assistance, weapons disposal and hazardous waste removal. The most common use for manipulators is in an industrial setting. In this setting, motion is often repetitious. Currently, industrial robots achieve this repetitive motion with a series of controllers consisting of electronics and actuators. These systems require overhead in resources and engineering time. The development of a single degree of freedom manipulator would allow the electronics and actuators to be replaced by mechanical gearing. This would reduce the cost and increase repeated precision as t he mechanism is not su bject to electronic glitches or pneumatic failure. A novel mechanism ha s been developed that allows gear pairs to mechanically couple the variable joint angles in an openloop manipulator comprised of six revolute joints (6R) and six serial links. The general 6R manipulator has six degrees of freedom; this is constricted to a single degree of freedom by the gears. Non circular gearing is utilized to achieve predefined position and orientation requirements of the manipulators end effector For physical realization purposes, gear centrodes must be continuous and smooth. In addition, gear pairs must move in a single relative direction. These constraints lead to the design of a mechanism that is capable of moving through a series of ordered desired posit ions and orientations. PAGE 11 11 1.2 Related Work Non circular gearing has been studied notably by Dooner and Seireg (1995) who developed a unified geometric theory for the design and manufacturing of gears. The method described is useful for a variety of gearing needs including general noncircular gears. Danieli and Mundo (2004) also developed a method for the design of noncircular gears with constant pressure angle teeth. Manipulators with mechanically bound joints have been studied previously in both openloop planar and closedloop cases. Pang and Krovi (2000) studied both design and number synthes is. They attempted to achieve a desired tool path by the design of mechanism link lengths. However, they did not constrain the number of serial links. They used Fourier techniques to determine the optimal amount of links needed to trace a specified path. The mechanism is limited to movement on a single plane. Krovi, et al. (2002) developed an openloop planar single degree of freedom serial chain that was coupled using both gear trains and belt dri ves. Their mechanism consisted of three revolute joints with parallel joint axes. This allowed coupling between all three joints, but limited the workable area of the manipulator to a single plane perpendicular to the joint axes. Rodriguez et al. (2006) investigated the use of a bound openloop serial mechanism to create a single degree of freedom anthropomorphic finger. The mechanism consists of three revolute joints bound with tendons similar to those found in a human hand. The tendoncoupling of joints allowed for accurate and humanlike grasping by a set of mechanical fingers. Molder, et al. (2009b) investigated noncircular gears for coupling joints in a five degree of freedom closedloop manipulator with broad applications. Dooner (2001) studied the use of noncircular gears in the design of gear trains with a specific torque PAGE 12 12 curve or highly nonlinear Input Output relationship. Mckinley et al. (2007) used screw theory techniques to solve the reverse kinematic analysis for the special case 6R mechanism with three sets of parallel joint axes using the unified method defined by Duffy and Rooney (1975) The solution leads to sixteen separate solution sets for a given desired position and orientation. Mckinley (2008) worked further on designing noncircular gears for a closedloop serial manipulator made up of two sets of the special 6R geometry. Harshe (2009) advanced the design of the closedloop mechanism with the development of a scoring function for design synthesis optimization Dou and Ting (1996) defined criterion for serial chain design that is robust against the branching errors inherent to a mechanism with a higher order of reverse analys is solutions. PAGE 13 13 CHAPTER 2 PROBLEM STATEMENT Given an ordered set of desired positions and orientations, a mechanism is to be designed that is capable of achieving each successive pose in order. Variable mechanism parameters include three joint offs ets, three link lengths and two twist angles. All other offsets, link lengths and twist angles are zero by virtue of the special 6R geometry. The special geometry is pictured in Figure 21, and design parameters are displayed in Table 21. Joint offset vectors and are parallel, as are and and and Interior and exterior concentric cylindrical links are utilized, allowing interior links to spin freely inside of exterior links. The exterior links are rigidly attached to the nonplanar gears while the interior links are rigidly attached to the planar gear pairs. Five pairs of noncircular gears are utilized to couple adjacent joints This reduces the mobility of the mechanism to one, producing a mechanism with a single input necessitating external actuation. In order to design a mechanism with gears that are physically realizable, gear profiles are optimized to be as close to circular as possible. Also, gears must be without abrupt jumps or direction changes. Gears in mesh must move in a single direction, and the input gear motion must also be mono directional PAGE 14 14 S S S S a 2 a 4 a 1 a 3 a 5 S S 2 3 4 5 6 1 Table 21. Design p arameters Figure 21. Example of o penl oop g eared m anipulator Offset d istance (cm) Link l engths (cm) Twist angles (degrees) 2 12 12 = 0 3 = 0 23 = 0 23 4 34 34 = 0 5 = 0 45 = 0 45 6 56 56 = 0 PAGE 15 15 CHAPTER 3 OPTIMIZATION Dimensional optimization is performed to minimize a scoring function. The precise scoring function used in this research is discussed in Chapter 4. Nine parameters are optimized including three link lengths ( 12, 34, 56) three joint offsets ( 2, 4, 6) and two twist angles ( 23, 34) The reverse kinematic analysis produces sixteen possible solutions, or branches. The branch number is also chosen as a parameter as a different branch may provide drastically different circularity scores. First, a genetic algorithm optimization is performed with the use of the Matlab ga function. A genetic algorithm is chosen to avoid the many local minima present in the highly nonlinear scoring function. The algorithm creates a binary string representative of a vec tor of parameters and creates an initial random population of parameter values within a specified range. Next, bits in the binary parametric representations of the initial population are swapped with other parameter sets to create the next generation of solutions. This process is iterated over successive generations, with solutions with lower scores having a higher probability of surviving to the next generation. Periodically, a random mutation is performed by switching bit values of a solution. The initial population of the genetic algorithm is designated with a high level of diversity in order to cover the entire design space. This plays a large role in the success of the genetic algorithm. Because the genetic algorithm is seeded by a random initial population, solutions are not consistent. For this reason, the optimization is run multiple times and the best solution selected. There is no guarantee that the score returned by the genetic algorithm is the absolute minimum, only a high probability that the solution set is close PAGE 16 16 to the optimal solution. For this reason, an active set quasiNewton minimizing optimization is performed with the initial value returned by the genetic algorithm. The minimizing optimization is performed with the Matlab fminco n function. The approximate Hessian matrix is computed, and the search follows the path that alters the parametric vector in order to decrease the scoring function the most. The Hessian matrix is composed of the approximated gradient field of the scoring function with respect to the parametric set and is defined in Equation 3 1. ( ) = 2 1 2 2 1 2 2 1 9 2 2 1 2 2 2 2 2 9 2 9 1 2 9 2 2 9 2 (3 1 ) The quasiNewton algorithm finds the minimum value of the scoring function within close proximity to the parametric solution returned by the genetic algorithm. PAGE 17 17 CHAPTER 4 EXACT PATH SPECIFICATION 4.1 Path Creation A desired path is defined by a series of ordered precision poses, that is, the exact position and orientation requirements for the end effector These poses are described by a position vector the position of the tool point in the fixed coordinate system and a 3 3 rotation matrix describing the relative orientation of the coordinate system attached to the sixth link and the fixed coordinate system, 6 The matrix 6 is defined in E quation 41. 6 = [ 67 6 676 ] (4 1 ) The rotation matrix and location vector are combined into a 4 4 transformation matrix described by Crane and Duffy (1998) as Equation 42 6 = 6 6 0 0 0 1 (4 2 ) w here: 6 = (6 ) (6 )6 67 (6 )6 (4 3 ) The desired Cartesian coordinates of the origin of the coordinate system attached to the sixth link are easily obtained from the fourth column of the transformation matrix. Crane and Duffy as Equation 44. 6 = 111213212223313233 (4 4 ) PAGE 18 18 w here: 11= cos ( ) cos ( ) 12= cos ( ) sin ( ) 13= sin () 21= cos ( ) sin ( ) + sin ( ) sin ( ) cos ( ) 22= cos ( ) cos ( ) + sin ( ) sin ( ) cos ( ) 23= cos ( ) sin ( ) 31= sin ( ) sin ( ) cos ( ) sin ( ) cos ( ) 32= sin ( ) sin ( ) cos ( ) sin ( ) cos ( ) 33= cos ( ) cos ( ) 6 from Equation 45. = asin (r11) = atan 23 ( ) 33 ( ) (4 5 ) = atan 12cos ( ) 11cos ( ) The relative distance between precision points is calculated as a combination of translational and angular displacement. To compensate for unitary disagreement, a scalar mapping metric is chosen to compare the relative change in angular displace ment with the relative change in Cartesian displacement The path is parameterized with a variable such that when = 0 roll, pitch and yaw angles calculated in Equation 45 are with respect to the first pose and when = 1 rotation PAGE 19 19 2 4 6 8 10 14 16 18 20 22 6 8 10 12 14 16 18 20 22 x yz angles are calculated with respect to the final pose The mapping metric affects what value will have a t each of the intermediate poses. A continuous path is then created by creating splines between precision points. These splines are numerically approximated cubic fits of the data that is created by small linear approximations. Because of the nature of the fit, there is no explicit functional definition. Six separate splines are created; three each for translation and orientation. These splines can then be recombined to form a threedimensional path which containing all precision points. A numerical example is shown in Figures 41, 42 and 43. In Figure 41, the desired path is shown along with the orientation at each pose. Figures 42 and 43 respectively show the values for translational and rotational components of each transformation at values for computed using a weighting factor of 5= 0 .6 Values computed with the weighting factor are = { 0 0 .2957 0 .5130, 0 .7826, 1 } Figure 41. Example d esired p ath. PAGE 20 20 0 0.2 0.4 0.6 0.8 1 10 5 0 5 10 15 tcm x y z 0 0.2 0.4 0.6 0.8 1 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 tradians roll pitch yaw Figure 42. Example t ranslational splines Figure 43. Example r otational splines 4.2 Scoring Function A r everse kinematic analysis is performed at a series of one hundred evenly spaced points along the desired end effector path. This corresponds to = 0 .1 for the parameterized splines. This value provides balance between path resolution and computational time. At each point, a solution set is obtained that contains sixteen sets of joint angles that satisfy the specific position and orientation requirements. In general, PAGE 21 21 multiple sets of joint angles within a solution set will have nonzero imaginary values. These values indicate that the solution is in a configuration that is not physically attainable. This issue is addressed in Section 4.2.2. An individual solution is chosen at the first point along the path. This defines the solution branch to be eva luated. For the given branch, a list of sequential joint angle values is composed. It is necessary to ensure that a single branch is used for an entire list of joint angles. If a set of joint angles does not belong to the branch of the preceding set, a discontinuity is experienced. The list of joint angles is stored in a 100 6 array such as in Equation 46. = 11 12 13 14 15 16 21 22 23 24 25 26 1001 1002 1003 1004 1005 1006 (4 6 ) A numerical derivative with respect to the length of the parameterized path ( ) is taken of each column as in Equation 47. The numerical derivatives is found as a set of finite differences between joint angl e values along the parameterized path = = 1 6 (4 7 ) For a gear pair, the instantaneous gear ratio can be defined as Equation 48. = + 1 = 1 5 (4 8 ) A given set of gear profiles is scored as to how circular they are. A general circular gear has a constant gear ratio as in Equation 49. = 1 (4 9 ) PAGE 22 22 The circularity of a given gear can therefore be evaluated by summing the magnitude of the differences in instantaneous gear ratio at adjacent points along the length of the path in Equation 410. =  + 1  = 1 5100 = 1 (4 10) The circularity score of each gear is summed with equal weighting to produce a scoring function for optimization in Equation 411. = 1 + 2 + 3 + 4 + 5 (4 11) 4.2.1 Imaginary Solutions Non reachable solutions characterized by imaginary joint angles must be excluded by a scoring function. However, complete exclusion creates sections of the design space with no gradient field, causing poor optimization performance. Therefore, sets of imaginary joint angles are not excluded, but are heavily penalized. The real and imaginary portions of the instantaneous gear ratio are split, and the imaginary part is scaled by a weight ing factor ( ) in order to drive down the magnitude of imaginary elements in Equation 4 12. = { } + { } (4 12) Setting the weighting factor many orders of magnitude greater than ensures a steep gradient, quickly excluding nonreachable solutions. Making this substitution, the scoring function becomes Equation 413. = 1 + 2 + 3 + 4 + 5 (4 13) PAGE 23 23 4.2.2 Singularities For gear pair a mathematical singularity in the gear ratio occurs when = 0 which causes to approach infinity This represents the case when one gear changes direction relative to the other. This is not physically viable, as these gears cannot be readily fabrica ted. The singularity causes sudden spikes in the value of the instantaneous gear ratio. These spikes heavily weigh the scoring function ( ), providing a robust solution to designs that result in these singularities. Path resolution is chosen as to detect this singularity. The = 0 solution is not found explicitly. Rather, the intermediate poses immediately preceding and following the direction change get close to = 0 which results in large increases in the scoring function. 4.2.3 Limitation s This design algorithm provides a method for attaining noncircular gears to drive the mechanism through a desired path. However, due to the rigid method of path creation, many paths yield gear pairs that are not physically realizable. A design algorithm that solves this issue is presented in C hapter 5. Although not applicable to every desired path, this algorithm provides a straight forward method of design and scoring for the openloop geared spatial mechanism. 4.3 Numerical Example A desired path is defined by six transformation matrices. 1 = 0 .2424 0 .5198 0 .8192 2 .8388 0 .9063 0 .4226 0 10.3486 0 .3462 0 .7424 0 .5736 6 .3245 0 0 0 1 2 = 0 .1794 0 .6525 0 .7362 6 .9302 9 .208 0 .1520 0 .3591 6 .8499 0 .3462 0 .7424 0 .5736 6 .6232 0 0 0 1 PAGE 24 24 3 = 0 .5649 0 .6531 0 .5043 9 .3051 0 .7490 0 .1494 0 .6455 1 .7943 0 .3462 0 .7424 0 .5736 6 .9446 0 0 0 1 4 = 0 .8361 0 .5215 0 .1703 9 .3935 0 .4255 0 .4206 0 .8013 3 .9446 0 .3462 0 .7424 0 .5736 7 .2861 0 0 0 1 5 = 0 .9380 0 .2843 0 .1982 7 .0530 0 .0159 0 .6066 0 .7948 9 .2959 0 .3462 0 .7424 0 .5736 7 .6451 0 0 0 1 6 = 0 .9176 0 .1406 0 .3719 5 .0519 0 .1955 0 .6550 0 .7299 11.4881 0 .3462 0 .7424 0 .5736 7 .8303 0 0 0 1 A flow chart of the design algorithm is shown in Figure 44. Each transformation matrix can be broken into three translation and three rotational components which is expanded by a spline fit. A mapping metric is used such that five degrees equals 0.6 centimeters. The splines are defined by a variable t which varies from 0 to1. When = 0 the splines evaluate to the first end effector pose along the path. When = 1 the splines evaluate to the last on pose the path. The spline values are presented in Table 4 1 The complete path is shown in Figure 45 In the graphs below (Figure 46 Fig ure 4 7 ), open circles represent the location of precision points. Dimensional optimization is performed with this desired path. Optimization is bounded as in Table 42 Diversity is controlled by the initial range of the initial population, which is set as in Table 43 The genetic algorithm optimization is run ten times, and the best score is obtained with the parameters listed in T able 44 QuasiNewton optimization is then performed using the bounds from Table 4 2 The genetic algor ithm optimization results from Table 44 are used as the initial guess. PAGE 25 25 The results of the QuasiNewton optimization are summarized in Table 45 The QuasiNewton optimization provides a dimensional solution resulting in continuous, smooth noncircular gears. Angular position values for each joint are displayed in Figure 4 8 for each point at which the reverse kinematic analysis is performed. The instantaneous gear ratios ( : 1 ) are displayed in Figure 4 9 for each reverse analysis point. Input/Output plots for individual gear pairs are displayed in Figure 4 10. Table 41. Spline p arameters Pose Parameter Roll (rad) Pitch (rad) Yaw (rad) X (cm) Y (cm) Z (cm) 1 0 0.0000 0.9559 1.1345 2.8388 10.3486 6.3245 2 0.2150 0.5594 0.8275 1.8392 6.9302 6.8499 6.6232 3 0.4351 0.8443 0.5286 2.2840 9.3051 1.7943 6.9446 4 0.6591 0.9495 0.1711 2.5839 9.3935 3.9446 7.2861 5 0.8857 0.9457 0.1995 2.8473 7.0530 9.2959 7.6451 6 1 0.9407 0.3810 2.9896 5.0619 11.4881 7.8303 Table 42 Dimensional b ounds Offset d istance (cm) Link l engths (cm) Twist angles (degrees) 2 = [ 0 1 20 ] 12 = [ 0.1, 2 0] 12 = 0 3 = 0 23 = 0 23 = [ 0 1 280 ] 4 = [ 0 1 20 ] 34 = [ 0 1 20 ] 34 = 0 5 = 0 45 = 0 45 = [ 0 1 280 ] 6 = [ 0 1 20 ] 56 = [ 0.1, 2 0] 56 = 0 Table 43 Initial p opulation r ange for g enetic a lgorithm o ptimization Offset distance (cm) Link l engths (cm) Twist angles (degrees) 2 = [ 3 8 ] 12 = [ 1, 6] 12 = 0 3 = 0 23 = 0 23 = [ 70 120 ] 4 = [ 4 13 ] 34 = [ 1 6 ] 34 = 0 5 = 0 45 = 0 45 = [ 70 120 ] 6 = [ 6 11 ] 56 = [ 2, 7] 56 = 0 PAGE 26 26 Interpolate path betwen precision points to acheive entire path Define bounds on design parmeters and initial population of design candidates Perform genetic algorithm optimization Seed quais newton optimization algorithm with results from genetic optimization For final optimized design, find total gear profile with polynomial fit Use Ingear software to synthesize gear pairsTable 44 Genetic a lgorithm r esults Offset d istance (cm) Link l engths (cm) Twist angles (degrees) S 2 = 7 2729 a 12 = 9 9995 12 = 0 S 3 = 0 a 23 = 0 23 = 130 9404 S 4 = 13 8577 a 34 = 8 7209 34 = 0 S 5 = 0 a 45 = 0 45 = 118 9175 S 6 = 14 1003 a 56 = 9 9978 56 = 0 branch = 4 score = 1 9586 Table 45 Quasin ewton r esults Offset d istance (cm) Link l engths (cm) Twist angles (degrees) S 2 = 0 1731 a 12 = 18 9506 12 = 0 S 3 = 0 a 23 = 0 23 = 136 8717 S 4 = 8 1420 a 34 = 15 8695 34 = 0 S 5 = 0 a 45 = 0 45 = 118 2651 S 6 = 14 2196 a 56 = 19 4163 56 = 0 branch = 4 score = 1 2238 Figure 44. Flow chart of d esign a lgorithm PAGE 27 27 20 10 0 10 10 0 10 5 10 15 x yz 0 0.2 0.4 0.6 0.8 1 15 10 5 0 5 10 15 tcm x y z Figure 45 Three d imensional d esired p ath (cm) Figure 46 X/Y/Z coordinates of t ool p oint for spline PAGE 28 28 0 0.2 0.4 0.6 0.8 1 3 2.5 2 1.5 1 0.5 0 0.5 tradians roll pitch yaw Figure 47. Roll/ p itch/ yaw a ngles for spline Figure 48 Angular p ositions of j oints PAGE 29 29 Figure 49 Instantaneous g ear r atios Figure 410 Input/ o utput p lots PAGE 30 30 CHAPTER 5 PRECISON POINT SYNTH ESIS 5.1 Path Creation In the precision point design synthesis, a path is again defined by a set of desired positions and orientations of the end effector named precision points For precision point synthesis, roll, pitch and yaw angles as well as X, Y and Z coordinates of the base with respect to the path is included in the parameter set. Equations 42 to 4 5 describe how these angles are converted to an initial transformation matrix from the first pose to the fixed reference frame. Subsequent poses are defined by a series of translations and rotations from the first pose. In this manner, altering the first transformation matrix moves the entire path with respect to the fixed reference frame. This allows for the optimization of the position and orientation of the base as well as the link lengths, joint offsets and twist angles. Reverse analysis is performed on each precision point defined by a transformation matrix for a given candidate set of optimization parameters. The reverse kinematic analysis produces sixteen sets of joint angles for the given set of mechanism parameters. A solution branch is selected and an ordered list of angles is compiled for each joint angle ( 1, 2, 3, 4, 5, 6). For each joint angle list, a relative angular displacement is computed. This angular displacement is normalized as the basis for a spline fit used to create the joint angles for each point along the path between precision points. The forward kinematic analysis is performed with the computed joint angles and the candidate mech anism parameters to create the transformation matrix at each point. PAGE 31 31 This path creation algorithm allows a greater degree of design flexibility when compared to the algorithm described in Chapter 4. The path between precision points is defined by the candidate set of parameters and is not constricted beforehand. Since the path is not defined outside of the precision points, there exist far fewer limiting points on the path, increasing the likelihood of a physically viable solution. 5.2 Scoring Function With the precision point design algorithm, the joint angle array defined by Equation 4 6 is given by the splines produced by the joint angles at precision points. This eliminates the need to perfor m reverse analysis at each point along the path. This reduces the computational time by greater than an order of magnitude when compared to the exact path specification algorithm. A comparative test showed a decrease in optimization time from over sevent een hours to just over twenty three minutes for a largescale optimization problem with five specified positions and orientations The flexibility in the path necessitates more constraints on the scoring function. Gear motion direction changes and intragear pair relative motion direction changes must be explicitly penalized. The number of times that a given gear pair () changes directions is computed by counting the number of times that the instantaneous kinematic velocity of the output gear with respect to the input gear changes from positive to negative or negative to positive. The number of times that the gear pairs change direction relative to each other () is computed by counting the number of times that the plot of the output angle changes directions relative to path parameter These counts are weighted (, ) and summed as in Equation 51. = + = 1 ,5 (5 1 ) PAGE 32 32 The circularity of a given gear pair is scored by summing the deviance of the instantaneous gear ratio defined in Equations 47 and 48 from the gear ratio of 1 over the entire path as in Equation 52. This circularity score differs slightly from that used in Chapter 4. Both scores effectively grade the circularity of the gear pair, but the earlier circularity equation has a floating reference point from which the deviance is calculated. Equation 52 uses the gear ratio 1 as the absolute reference point. This eliminates the possibility of a gradual inc rease of the gear ratio without penalty. This change also penalizes gear pairs with a net ratio that is not 1. For non circular gears, the net ratio is chosen at 1 to ensure that every input revolution results in a single output revolution. =  1  100 = 1 = 1 5 (5 2 ) Again, the scoring function must compensate for imaginary solutions as in Equation 53. The imaginary portion is weighted heavily to quickly drive the scoring function out of imaginary regions. = + { } (5 3 ) The total circularity score of the candidate design is found by summing the circularity scores for individual gear pairs with a weighting factor as Equation 54. = (1+ 2+ 3+ 4+ 5) ( 5 4 ) In order to ensure gear continuity, the kinematic acceleration of the gear pair is computed and penalized. The angular acceleration of each gear is computed by Equation 55. = = 1 6 (5 5 ) PAGE 33 33 The kinematic acceleration of the gear pair is then defined by Equation 56. The complete kinematic acceleration score is created by summing the acceleration scores of the gear pairs with a weighting factor as in Equation 57. = + 1 = 1 5 (5 6 ) = ( 1+ 2+ 3+ 4+ 5) (5 7 ) The total score for a candidate design is created by summing all of the scoring components and their weights in Equation 58. = + + ( 5 8 ) This scoring function provides robustness against imaginary solutions and singularit ies. This is largely a function of the circularity score. The other aspects of the scoring function serve to guide the optimization towards a more physically attainable solution. 5.3 Numerical Example A desired path is defined by the following translations and roll, pitch and yaw angles relative to the candidate first pose 1 2 is produced by translating the 1 along the vector 4 .4048 3 .7189 4 .5084 cm with = 0 .5608 = 0 .1874 = 0 .5327 radians. 2 3 is produced by translating the 1 2 along the vector 10.0589 3 .3143 4 .0983 cm with = 1 .1386 = 0 .4502 = 3 .1335 radians. 3 4 is produced by translating the 2 3 along the vector 21.3910 1 .6621 12.4279 cm with PAGE 34 34 = 1 .9983 = 0 .0843 = 2 .7088 radians. 4 5 is produced by translating the 3 4 along the vector 0 .4276 22.5314 9 .4250 cm with = 0 .9918, = 0 .1168 = 0 .3558 radians. 5 6 is produced by translating the 5 4 along the vector 2 1 .3244 3 .9853 2 .3050 cm with = 0 .7752 = 0 .7008 = 0 .0588 radians. The genetic algorithm optimization is run 2 0 times with the desired path. Repeated optimizations are necessitated by the random nature of the genetic algorithm. A different set of initial candidate solutions, or even a chance parametric mutation, can cause the algorithm to fail. Bounds are set on the design space as defined in Table 51. The genetic algorithm is seeded within the initial population defined in Tabl e 5 2. The initial population is specified in order to lead the search algorithm towards a solution with desirable features such as low link length to joint offset ratio and twist angles close to 90 degrees. Because the genetic algorithm does no directed searching, there is no reason to believe that the candidate parameters result in an absolute minimum of the scoring function. Therefore, the quasiNewton algorithm is utilized to ensure the best solution possible. The results of the genetic algorithm opt imization highlighted in Table 53 serve as the starting point for this search. The results of the quasiNewton search are presented in Table 54. Joint angle plots are displayed in Figure 51. The joint angles at precision points are marked by open cir cles. The position and orientation of the first pose result in the initial transformation matrix about which all other poses are based. PAGE 35 35 The transformation matrix of pose is defined as The transformation matrices for the desired path are listed below. 1 = 0 .9216 0 .0676 0 .3822 20.0000 0 .3772 0 .0758 0 .9230 10.3754 0 .0913 0 .9948 0 .0444 14.6198 0 0 0 1 2 = 0 .9471 0 .3206 0 .0127 15.5952 0 .1311 0 .3507 0 .9273 14.0943 0 .2929 0 .8799 0 .3742 19.1282 0 0 0 1 3 = 0 .8517 0 .2744 0 .4465 5 .5363 0 .3277 0 .9437 0 .0451 17.4086 0 .4090 0 .1848 0 .8937 23.2265 0 0 0 1 4 = 0 .9053 0 .4230 0 .0392 15.8356 0 .1579 0 .4208 0 .8933 15.7465 0 .3943 0 .8025 0 .4478 10.7986 0 0 0 1 5 = 0 .9788 0 .0891 0 .1846 16.2632 0 .0806 0 .9953 0 .0530 6 .7849 0 .1885 0 .0370 0 .9814 1 .3736 0 0 0 1 6 = 0 .8654 0 .4377 0 .2439 5 .0612 0 .1254 0 .6605 0 .7403 2 .7996 0 .4851 0 .6101 0 .6265 0 .9314 0 0 0 1 This path is displayed in Figure 52 The path initially rises and maintains a steady orientation path. At the top of the path, the orientation of the end effector begins to rapidly rotate and the positi on drops vertically and in one horizontal direction. At the end of the path, the end effector maintains elevation and moves largely in the horizontal plane. This arbitrary path has a wide variety of uses including precision welding of highly nonlinear s urfaces and product pick and place. PAGE 36 36 The kinematic velocity of each gear pair is displayed in Figure 53 The instantaneous kinematic velocity is the same as the gear ratio (: 1 ) as in Figure 45. The k inematic velocities are extended to form the kinematic velocity plot for an entire noncircular gear pair with net ratio of 1:1. The nonactive portion of the gear is specified in the velocity domain by Dooner (2001) and is formed by fitting the polynomial with respect to input angle in Equation 59 subject to the constraints defined in Equation 5 10. The position, angular velocity and angular acceleration of the output with respect to the input at the last pose are denoted by and respectively. The angular velocity and angular accel eration at the first pose are specified as and The end points of the polynomial are defined as 1 and 2. ( ) = 4 4+ 3 3+ 2 2+ 1+ 0 ( 5 9 ) 1= 1 = 2= (5 10) 2 = ( ) + (2 0) = 2 In Figure 53, the solid line represents the active portion of the gear while the dashed line is the polynomial resulting from Equations 59 and 510. The kinematic velocity data for the complete gear is input into the INGEAR software. The software provides a robust method for the development of generalized gears. It is particularly potent for the creation of noncircular gears. Screw theory methods are used to create physical gears for a desired trajectory and parameter set. PAGE 37 37 Stereolithography files ar e generated and used to create prototype gears. An example of the gears produced for gear pair 2 is shown below in Figure 54. The gears have a shaft angle dictated by 23. The input gear is on the right and the gears are shown in mesh at pose 1. Table 5 1. Dimensional b ounds Offset d istance (cm) Link l engths (cm) Twist angles (degrees) Position of b ase (cm) Orientation of b ase (degrees) 2 = [ 2 20 ] 12 = [ 2,20 ] 12 = 0 X = [ 20,20 ] Roll = [ 0 359 ] 3 = 0 23 = 0 23 = [ 0 359 ] Y = [ 20,20 ] Pitch = [0,359 ] 4 = [ 2 20 ] 34 = [ 2 20 ] 34 = 0 Z = [ 20,20 ] Yaw = [0,359 ] 5 = 0 45 = 0 45 = [ 0 359 ] 6 = [ 2 20 ] 56 = [ 2,20 ] 56 = 0 Table 52. Initial p opulation r ange for g enetic a lgorithm o ptimization Offset d istance (cm) Link l engths (cm) Twist angles (degrees) Position of b ase (cm) Orientation of b ase (degrees) 2 = [ 9 12 ] 12 = [ 3,6 ] 12 = 0 X = [ 18 20 ] Roll = [90,180 ] 3 = 0 23 = 0 23 = [ 80 110 ] Y = [9,11 ] Pitch = [ 0,90 ] 4 = [ 9 12 ] 34 = [ 3 6 ] 34 = 0 Z = [14,16 ] Yaw = [ 0,90 ] 5 = 0 45 = 0 45 = [ 80 110 ] 6 = [ 9 12 ] 56 = [ 3,6 ] 56 = 0 Table 53. Genetic a lgorithm r esults Offset d istance (cm) Link l engths (cm) Twist angles (degrees) Position of b ase (cm) Orientation of b ase (degrees) 2 = 8 5948 12 = 4 3935 12 = 0 X = 19.7491 Roll = 265.9640 3 = 0 23 = 0 23 = 88 3247 Y = 10.4973 Pitch = 25.1538 4 = 9 5908 34 = 4 5661 34 = 0 Z = 13.8915 Yaw = 346.5936 5 = 0 45 = 0 45 = 91 8139 6 = 11 6491 56 = 4 5379 56 = 0 branch = 16 score = 233 5577 Table 54. Quasin ewton r esults Offset d istance (cm) Link l engths (cm) Twist angles (degrees) Position of b ase (cm) Orientation of b ase (degrees) S 2 = 9 1246 a 12 = 3 3398 12 = 0 X = 20 Roll = 267.2457 S 3 = 0 a 23 = 0 23 = 88 3247 Y = 10.3753 Pitch = 22.4679 S 4 = 9 8750 a 34 = 4 1289 34 = 0 Z = 14.6198 Yaw = 355.8072 S 5 = 0 a 45 = 0 45 = 91 8139 S 6 = 10 9394 a 56 = 4 8051 56 = 0 branch = 16 score = 109 4138 PAGE 38 38 25 20 15 10 5 0 5 10 15 20 25 20 0 20 40 5 0 5 10 15 20 25 30 y x z 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 Joint Angles for Phi 1 Position Along PathRadians 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 Joint Angles for Theta 2 Position Along PathRadians 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 Joint Angles for Theta 3 Position Along PathRadians 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 Joint Angles for Theta 4 Position Along PathRadians 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 Joint Angles for Theta 5 Position Along PathRadians 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 Joint Angles for Theta 6 Position Along PathRadians Figure 51. Joint a ngles for p recision p oint synthesis n umerical e xample Figure 52. Path t raced by e nd effector PAGE 39 39 0 1 2 3 4 5 6 0.5 1 1.5 Kinematic Velocity for Pair 1 Input Angle (rad)Kinematic Velocity 0 1 2 3 4 5 6 0.5 1 1.5 Kinematic Velocity for Pair 2 Input Angle (rad) 0 1 2 3 4 5 6 0.5 1 1.5 Kinematic Velocity for Pair 3 Position Along PathInput Angle (rad) 0 1 2 3 4 5 6 0.5 1 1.5 Kinematic Velocity for Pair 4 Input Angle (rad) 0 1 2 3 4 5 6 0.5 1 1.5 Kinematic Velocity for Pair 5 Input Angle (rad)Kinematic Velocity Figure 53. Kinematic velocit ies for n umerical e xample Figure 54. Gear p air p roduced by Ingear software PAGE 40 40 CHAPTER 6 CONCLUSION Two algorithms are presented for the design of a single degree of freedom openloop spatial manipulator with a special geometry that allows for the use of geared connections to control end effector movement. The first algorithm utilizes an exact desired path for the end effector Gears are scored on their degree of circularity. A candidate design including joint offset, link length and twist angle parameters is optimized to create circular gears. This algorithm succeeds, but is limited in use due to th e high degree of nonlinearity in the system and strict path adherence. A second algorithm is presented that generates a path that is flexible for a candidate design. The path is defined by an ordered series of desired positions and orientations. The desired poses are termed precision points. The end effector motion between the precision points is not constrained, allowing a greater ability to provide realistic noncircular gears for any generalized path. The precision point algorithm necessitates that the gears of a candidate design must be scored not only on circularity but also on absolute and relative motion directionality. Physically viable gears are designed and created using Ingear. Future work for this project includes dynamic force analysis including the identification of reaction loads and link balancing. These are necessary next steps to the broad realization of single degree of freedom openloop mechanisms in industry. Further path definition including velocity constraints on precision points is another area of exploration for this work. A driving mechanism to link a constant torque motor to the first link must also be developed. A closedloop single degree of freedom manipulator is also under investigation. PAGE 41 41 LIST OF REFERENCES Crane, C., and Duffy, J. (1998). Kinematic Analysis of Robot Manipulators Cambridge University Press. Danieli, G. A., and Mundo, D. (2004). "New Developments in Variable Radius Gears Using Constant Pressure Angle Teeth." Mechanism and Machine Theory 40, 203217. Dooner, D., and Seireg, A. (1995). The Kinematic Geometry of Gearing; A Concurrent Engineering Approach John Wiley, New York. Dooner, D. B. ( 2001). "Function generation utilizing an eight link mechanism and optimized noncircula r gear elements with application to automotive steering." Proceedings of the Insititution of Mechanical Engineers 847857. Dou, X., and Ting, K. L. (1996). "Branch Identification of Geared Five Bar Chains." Journal of Mechanical Design, 118, 384389. Duff y, J., and Rooney, J. (1975). "A foundation for a unified theory of analysis of spatial mechanisms." ASME J. Eng. Ind. 97B(4), 11591164. Harshe, M. (2009). "Design of One Degree of Freedom Closed Loop Spatial Chains Using NonCircular Gears," University of Florida, Gainesville, FL. Krovi, V., Ananthasuresh, G. K., and Kumar, V. (2002). "Kinematic and kinetostatic synthesis of planar coupled serial chain mechanisms." Journal of Mechanical Design, Transactions of the ASME 124(2), 301312. McKinley, J. R., Crane, C., and Dooner, D. ( 2007). "Reverse Kinematic Analysis of the Spatial Six Axis Robotic Manipulator with Consecutive Joint Axes Parallel." ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Confer ence Las Vegas, NV. McKinley, J. R., Crane, C., and Dooner, D. ( 2008). "Threedimensional rigid body guidance using gear connections in a robotic manipulator with parallel consectuive axes." ASME Design Engineering Technical Conferences New York, NY. Modler, K. H., Lovasz, E. C., Bar, G. F., Neumann, R., Perju, D., Perne r, M., and Margineanu, D. (2009 ). "General method for the synthesis of geared linkages with noncircular gears." Mechanism and Machine Theory 44(4), 726738. Pang, Y. W., and Krovi, V. ( 2000). "Kinematic Synthesis of Coupled Serial Chain Mechanisms for Planar Path Following Tasks Using Fourier Methods." ASME 2000 Design and Engineering Technical Conferences and Computers and Information in Engineering Conference Baltimore, Maryland. PAGE 42 42 Ro driguez, N. E. N., Carbone, G., and Ceccarelli, M. (2006). "Optimal design of driving mechanism in a 1DOF anthropomorphic finger." Mechanism and Machine Theory 41(8), 897911. PAGE 43 43 BIOGRAPHICAL SKETCH Joe Bari was born in Apple Valley, MN. He received B.A. degrees in p hysics and m athematics at St. Olaf College in Northfield, MN in May, 2008. In August, 2008 Joe joined the Center for Intelligent Machines and Robotics at the University of Florida as a Research assistant. He graduated in December 2009 with a M.S. in m echanical e ngineering. His research focuses on the design of spatial manipulators. 