A Characterization of Acceleration Capability. Alan Bowling and Oussama Khatib. Robotics Laboratory. Stanford University. Stanford, CA, USA 94305

Transcription

1 The Motion Isotropy Hypersurface: A Characterization of Acceleration Capability Alan Bowling and Oussama Khatib Robotics Laboratory Computer Science Department Stanford Uniersity Stanford, CA, USA 9405 Abstract The study of acceleration capability is concerned with the responsieness of a manipulator to controller commands In this paper we present a general model for the analysis of end-eector linear and angular accelerations that accounts for the elocity eects The separate treatment of linear and angular motion directly addresses the inhomogeneities of end-eector motions, aoiding the use of indeterminate scaling factors The elocity eects considered are the Coriolis and centrifugal forces, as well as the relationships associated with actuator's speed-torque performance cures This study results in a characterization referred to as the \motion isotropy hypersurface" which describes the relationships between isotropic end-eector linear and angular elocities and accelerations The utility of this surface and its associated information is demonstrated in a design application inoling the PUMA 560 manipulator Introduction In recent years, arious models hae been pro- _ posed for the analysis of manipulator performance The aim of these models _ is to proide a characterization of the responsieness of of a manipulator in the execution of a task Our approach to addressing this problem is to an- Figure : PUMA alyze the relationships between the end-eector linear and angular accelerations and elocities, _, _,, and ; Figure Note that these are directional quantities When analyzing mechanisms intended for general tasks it is necessary to consider its capabilities in all directions, and therefore these quantities are explored in terms of their isotropic magnitudes In earlier work, both the kinematic and dynamic aspects of this problem hae been considered The kinematic studies focused on the analysis of the Jacobian matrix Proposed kinematic measures include the Jacobian condition number [0, 6], manipulability measure [8, 0], and norm of the Jacobian rate of change [5] In dynamic analyses, the proposed measures and characterizations include the generalized inertia ellipsoid [], the end-eector mass matrix [6], dynamic manipulability ellipsoid [9, ], dynamic conditioning index [], acceleration set theory [8, 9], isotropic acceleration radius [5], acceleration hyperparallelipiped [7], and other measures discussed in [,, 4, ] Thomas, Yuan-Chow, and Tesar [7] come the closest to the treatment presented here Howeer, a common diculty encountered in these studies is the treatment of the inhomogeneities between linear and angular motions This diculty is often addressed by the introduction of scaling factors Our earlier studies of end-eector motion [, 4] resulted in a characterization of end-eector acceleration capability referred to as the motion isotropy hypersurface In this paper, we extend these analyses with a comprehensie treatment of linear and angular elocities This treatment includes the eects of the actuator's speed-torque performance cure, which describes the reduction of torque capacity due to rotor elocity This study of isotropic properties proides a \worst case" characterization of achieable elocities and accelerations These worst-case motions are described by

2 the motion isotropy hypersurface and the information obtained from its deelopment This article presents the deelopment of the hypersurface and its associated information, for non-redundant manipulators The utility of this analysis is demonstrated through a design application inoling the PUMA 560 manipulator Model In this section the equations describing the motion isotropy hypersurface for a non-redundant manipulator are deeloped A more extensie discussion is presented in [] The deelopment begins with the familiar joint space equations of motion A q + b + g =? = G : () where q, A, b, g,?,, and G, represent the n joint coordinates, joint space kinetic energy matrix, centrifugal and Coriolis forces, graity forces, joint torques and forces, actuator torques produced between the rotor and stator, and transformation between actuator and joint torques The goal is to transform equation () into a set of inequalities relating the actuator torque capacities to end-eector accelerations This is accomplished using the Jacobian, J, dened as # 4 = = J _q () where and are end-eector linear and angular elocities Just as joint torques are related to actuator torques, joint speeds are related to actuator, or rotor, speeds as follows _ = G T _q = G T J? # () where _ is the ector of n rotor speeds The bound on actuator torque capacity can be expressed as; Ideally boundi is a linear function of rotor elocity referred to as the speed-torque cure, Figure Howeer, using this linear function complicates the elocity analysis, so the speed-torque cure is approximated by a second order function without a linear term, Figure This j i j boundi ( _ i ; : : :): (4) _i bound i Figure : Speed-Torque Cure choice is made because the actual speed-torque cure may be nonlinear The nal model has the form boundi = 0 i? iii _ i (5) where 0 i is usually the peak torque and iii is a scalar used to approximate the slope of the speedtorque cure It is expressed as an element of a third order tensor where all elements other than the iii th are zero Equation (4) can be expressed in terms of an upper and lower bound using equation (5)? 0 i? b(x) iii _ i i 0 i? b(x) iii _ i(6) where b(x) equals positie or negatie one depending on the boundary of interest b(x) = if x = upper? if x = lower : (7) These bounds are normalized and expressed in terms of end-eector elocities using equation () to obtain? N ( + b(x)t ) (8) where is a ector of ones and t = 6 4 # T T# # T Tn# 7 5 = N 6 4 # T J?T T? G G J # # T J?T T G n G J? # 7 5 (9) where N is a diagonal matrix where N ii = 0i Omitting the details, equations (), (), and (9) can be used to obtain where lower E _ + E _ + h upper (0) h = [ E E ] = E = N G? AJ? 4 _qt B _q _q T Bn _q 4 # T M # # T Ma# 4 # T H # 5 = b? AJ? _ J _q 5 = N G? 4 # T J?T B J? # 5 = 4 # T M # # T J?T BnJ? # 5 + b(x)t # T Ha # # T Ma#? upper = N 0? G? g lower =?N? 0 + G? g : () The separation of linear and angular characteristics in equation (0) is motiated by the need to analyze these properties separately 5

3 Isotropy Equations Geometric and algebraic approaches are used to analyze equation (0) In the geometric approach each term is represented by a geometric object and the relationship between them is explored [, 4] Insights obtained from the geometric approach facilitate the deelopment of the algebraic approach which is more useful for analyzing elocities The isotropic acceleration is determined by expanding the ellipsoid until it is inscribed within the bounds, Figure Note that only the ectors associated with the tangency points, referred to as tangency ectors, need to be examined The ellipsoids seldom align with the coordinate axes, making it dicult to realistically illustrate the geometric approach Howeer, although unrealistic, these gures still represent alid concepts A similar process is followed to obtain the torque ellipsoid representing angular accelerations E _ T (E E T ) + = k _k : (5) k _k Figure : Ellipse E _ k _k All possible sums of the ectors and are n considered by mapping the center of one ellipsoid onto eery point on the surface of the other, Figure 4 Corresponding tangency ectors from each ellipsoid are added to obtain the resultant Figure 5: Tangency which touches a particular bound, Figure 5 Vectors The resultant ector touching the boundary in Figure 5 must satisfy _ n ( + ) = upper (6) k _k _ where n is normal to the boundary plane as shown Figure 5 If the ectors and are dened as, E Figure 4: Ellipsoids Added In the geometric approach, the torque bounds are interpreted as an n-dimensional hypercube, the square in Figure, whose center is shifted away from the origin by the graity eect, ie N G? g in equation () Now consider the linear acceleration term in equation (0), E _ Motions are interpreted as isotropic quantities represented by spheres _ T _ = k _k : () This sphere is transformed by E into an ellipsoid in torque space using the following relationship _ = E + () where E + is the left inerse of E Equations () and () are used to obtain the ellipsoid T (E E T )+ = k _k : (4) T (E E T )+ = T (E E T )+ = (7) (the dashed ellipse in Figure ), and can be expressed in terms of these ectors in order to obtain n? k _k + k _k = upper : (8) Equation (8) represents a line with respect to k _k and k _k n relations similar to equation (8) are obtained for each boundary plane The isotropic accelerations are determined by the most limiting relationship(s) from the set Determining the line(s) which represent the most limiting relationships corresponds to k _k k _k Figure 6: Isotropy Cure nding the innermost enelope formed around the origin when the n lines are oerlayed in the same space

4 This cure is conex and piecewise linear, shown in Figure 6 A discontinuity in the cure occurs when the limiting boundary changes The geometric approach does not work well for the elocity dependent terms, because they cannot be separated into linear and angular subspaces In addition, the torque surface corresponding to the isotropic elocity surfaces is dicult to determine because the elocity dependent terms are second order Howeer, equation (8) shows that only the component of the tangency ector in the direction of its associated boundary is important These components, n and n, can be expressed succinctly as A [ k _k k _k ] T = T (9) where A is a n matrix containing the aforementioned components and T contains the elements of upper and lower The goal is to determine the corresponding components from the elocity dependent terms Doing this requires a more algebraic approach This approach will rst be presented in terms of acceleration since this facilitates consideration of elocities Of the ectors comprising an ellipsoid, the tangency ector has the largest component in the direction of its associated boundary For accelerations, the components of the ectors comprising the ellipsoid in the direction of the i th actuator are determined by the i th row of the E matrix, equation () The largest/smallest alue of this component, which is the tangency ector component, can be found as follows: minimize=maximize i = [ E i E in ] subject to _ T _ = k _k _ T _ = k _k : (0) This system is maximized, for the upper bound, or minimized, for the lower bound, with respect to _ and _ yielding a solution expressed in terms of k _k and k _k Applying the method of Lagrange multipliers to this system yields an easily solable eigenalue problem where the elements of A can be dened as u A j?; = A j; = + ut X p k= u u X A j?; = A j; = + t n k=p+ E jk E jk () where j = (; : : :; n), has dimension p, and has dimension r = n? p When considering elocities, the component of the tangency ector in the direction of i th actuator torque boundary is determined by the i th quadratic term in the h ector; minimize=maximize i = # T H i# subject to _ T _ = k _k _ T _ = k _k : () Recall that when considering the upper bound, H i = M i + b(upper)ti (and similarly for the lower bound) For this system, it is ery dicult to nd a closed form analytical solution, in terms of k _k and k _k Howeer, a useful partial solution can be found, which will be discussed momentarily This solution can be included in equation (9) as C, A [ k _k k _k ] T + C ( kk; kk ) = T: () where C ( kk; kk ) is a second order function of kk and kk These relations are referred to as the motion isotropy equations With the addition of the new terms each relation in equation () represents a hypersurface in four-space, as opposed to the lines represented by equation (9) Similarly to Figure 6, the motion isotropy hypersurface is determined by the innermost enelope formed by these hyperplanes Howeer, a four-dimensional hypersurface is dicult to display, so it must be sectioned by a hyperplane in order to show its characteristics The sectioning hyperplane is usually chosen as kk = 0, kk = 0, or kk = kk (note that the rst two sections are special cases of the third where = 0 and = ) This hyperplane is chosen because the useful partial solution for the elocity terms can be obtained with this added constraint on the optimization problem of equation () Also, since this hyperplane contains the origin, the resulting section gies a better picture of the olume encompassed by the hypersurface A description of the full surface can be obtained from the set of surfaces where is aried from 0 to The partial solution for the elocity dependent terms can be obtained as follows Since the minimization and maximization problems are similar only the former will be discussed First note that H i can be made positie semi-denite thereby ensuring the existence of a maximum alue Doing this inoles multiplying the constraints by the smallest eigenalue of H i, and adding them to i, i = # T shift Hi # + min? kk + kk (4) where min is a real number since H i is symmetric A change of ariables is performed, kk = kk (5)

5 # = [ kk kk ] T (6) where is a positie scalar and and are unit ectors, yielding = kk : (7) Using equation (7), the problem expressed in equations () and (4) is transformed into i = kk T shift H i T = T = : + min? + (8) Soling this problem only requires the maximization of the rst term in parenthesis with respect to the unit ectors and, since kk is only a multiplicatie factor and the second term can be considered constant Still, an analytical solution to this problem is ery dicult to nd (it does not correspond to any eigenectors of the matrix inoled except in special cases) Howeer, assigning a alue to yields a problem solable using publicly aailable search algorithms The solution has the form imax = c kk : (9) These components are added into equation (9) as the ector C (kk; kk) in the motion isotropy equations The elements of C can be dened similarly to the elements of A in equation (), although the expressions are more complex 4 Motion Isotropy Hypersurface The motion isotropy hypersurface describes the bounds on isotropic accelerations and therefore any motions lying beneath it are possible in eery direction An example of the isotropy surface where kk = 0 is shown in Figure 7a for the PUMA 560 at the conguration shown in Figure 7b Howeer, the actuators used here are not the original ones shipped with this manipulator First note that the cure in the k _k-k _k plane is conex, recall Figure 6 Also note that, by equation (9), any section obtained using a hyperplane of the form kk = kk is also conex Howeer, it is dicult to know whether the entire hypersurface is conex The hypersurface is physically meaningful, describing the worst case combinations of linear and angular accelerations and elocities which saturate the actuators The actuator which saturates rst is termed the limiting actuator In the example, the motor actuating the rst joint of PUMA 560 is the limiting actuator, indicated by the numeric label on the surface of Figure 7a The resultant tangency ectors for the boundary corresponding to the limiting actuator can be used to determine the combination of motion which saturated that actuator, displayed in Figure 7b The magnitudes of these worst case motions are described by the hypersurface Consider the point marked by the 4 in Figure 7a This point represents the isotropic linear acceleration achieable from rest if the end-eector moes without changing its orientation, _ = = = 0 At 4 the actuator of the rst joint saturated attempting to proide a linear acceleration of 5:m=s in the direction denoted by _ in Figure 7b The cure in the k _k-k _k plane describes the combinations of isotropic linear and angular acceleration, at zero elocity = = 0, which saturate the rst actuator The intercept along k _k axis, marked by, denotes the isotropic angular acceleration achieable from rest if the end-eector rotates without translating the operational point, 48:4rad=s Coming out of the k _k-k _k plane, the bounds on acceleration capability when accelerating from a particular elocity state are displayed End-eector elocities can limit performance because the manipulator must oercome Coriolis/centrifugal forces while the torque capacities decline in accordance with the speedtorque cure The shape of the surface shows that the magnitude of acceleration attainable in eery direction decreases as the isotropic linear elocity increases The point marked by in Figure 7a marks the magnitude of linear elocity, :4m=s, where the manipulator cannot proide any arbitrary accelerations because all of the aailable torque from the limiting actuator is used to compensate for the Coriolis, centrifugal, and graity forces The motion isotropy hypersurface and its contained information can be applied to the manipulator design problems; specically the actuator selection problem This inoles determining the actuators which will proide a desired leel of performance In this process the motion isotropy hypersurface is used to ealuate a manipulator's current performance The limiting actuator then indicates which actuator torque to alter in order to improe performance This information is used within a heuristic algorithm for actuator selection, howeer the algorithm will not be discussed here An example of its use is presented to illustrate the utility of the motion isotropy hypersurface and its limiting actuator information For example, consider increasing the k _k intercept

6 _ k _krad=s k _krad=s _ 50 5 kkm=s k _km=s kkm=s k _km=s (a) (b) Figure 7: Isotropy Surface (c) of Figure 7a to g, 9:8m=s The requisite actuators are shown in Table and the resulting isotropy surface is shown in Figure 7c The elocity information for these actuators is gien in Table, where iii = 0 i = _ i max Actuator Peak Torque 0 i Nm Joint Original New Table : Actuator Selection Joint iNm 0: 0:4 0:65 0:86 : :60 _imax rad=s Table : Actuator Maximum Velocity Note that improed motion isotropy hypersurface section is comprised of two surface patches which means the limiting actuator changes between the rst and third actuators (the label indicating third actuator is omitted from Figure 7c) As expected, it is necessary to increase the rst actuator It is also necessary to increase the second actuator in order to achiee the desired performance, howeer the heuristic knows to do this based on the limiting actuator information obtained at dierent stages in the procedure Since the hypersurface is four-dimensional it is necessary to deelop some measures which summarize its characteristics These measures also reduce the necessity for actually generating the hypersurface, by summarizing the features discussed aboe Howeer, due to space limitations, they will not be presented here A complete description of them can be found in [] and will be presented in a later article 5 Conclusion The motion isotropy hypersurface presented in this paper proides a characterization of the end-eector isotropic acceleration capabilities at a gien conguration By addressing the inhomogeneity problem, this approach allows full analyses of linear and angular accelerations and elocities This procedure also yields insights into the limiting actuators, as motor saturation determines the extent of isotropic motions In addition, the analysis proides information about the directions of acceleration and elocity in which an actuator will saturate The motion isotropy hypersurface also proides a physically meaningful worst-case analysis of the end-eector motions These models hae been extended to redundant manipulators [4] The analysis of end-eector force capability can also be included in these models [] 6 Acknowledgments The nancial support of NASA/JPL, Boeing, Hitachi Construction Machinery, and NSF, grant IRI- 9007, are gratefully acknowledged Also many thanks to Professor Walter Murray References [] Haruhiko Asada A geometrical representation of manipulator dynamics and its application to arm design Journal of Dynamic Systems, Measurement, and Control, 05():{5, September 98 [] Alan Bowling Analysis of Robotic Manipulator Dynamic Performance: Acceleration and Force Capabilities PhD thesis, Stanford Uniersity, June 998

7 [] Alan Bowling and Oussama Khatib Analysis of the acceleration characteristics of manipulators In A Morecki, G Bianchi, and K Jaworek, editors, Proceedings Tenth CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators (Romansy 0), pages 59{64 Springer- Verlag, September 995 Gdansk, Poland [4] Alan Bowling and Oussama Khatib Analysis of the acceleration characteristics of non-redundant manipulators In Proceedings IEEE/RSJ International Conference on Intelligent Robots and Systems, olume, pages {8 IEEE Computer Society Press, August 995 Pittsburgh, Pennsylania [5] Timothy J Graettinger and Bruce H Krogh The acceleration radius: A global performance measure for robotic manipulators IEEE Journal of Robotics and Automation, 4(), February 988 [6] Oussama Khatib Inertial properties in robotic manipulation: An object-leel framework The International Journal of Robotics Research, ():9{6, February 995 [7] Oussama Khatib and Joel Burdick Optimization of dynamics in manipulator design: The operational space formulation The International Journal of Robotics and Automation, ():90{98, 987 [8] Yong-Yil Kim and Subhas Desa The denition, determination, and characterization of acceleration sets for spatial manipulators In The st Biennial Mechanisms Conference: Flexible Mechanism, Dynamics, and Robot Trajectories, September 990 Chicago, Illinois [9] Yong-Yil Kim and Subhas Desa The denition, determination, and characterization of acceleration sets for planar manipulators In The st Biennial Mechanisms Conference: Flexible Mechanism, Dynamics, and Robot Trajectories, September 990 Chicago, Illinois [0] Charles A Klein and Bruce E Blaho Dexterity measures for the design and control of kinematically redundant manipulators The International Journal of Robotics Research, 6():7{8, 987 [] Kazuhiro Kosuge and Katsuhisa Furuta Kinematic and dynamic analysis of robot arm In Proceedings IEEE International Conference on Robotics and Automation, olume, 985 [] Ou Ma and Jorge Angeles The concept of dynamic isotropy and its applications to inerse kinematics and trajectory planning In Proceedings IEEE International Conference on Robotics and Automation, olume, 990 [] Ricardo Matone and Bernard Roth Designing manipulators for both kinematic and dynamic isotropy In A Morecki, G Bianchi, and C Rzymkowski, editors, Proceedings Tenth CISM- IFToMM Symposium on Theory and Practice of Robots and Manipulators (Romansy ), 997 [4] Ricardo Matone and Bernard Roth The eects of actuation schemes on the kinematic performance of manipulators Journal of Mechanical Design, Transactions ASME, 9, 997 [5] R V Mayorga, B Ressa, and A K C Wong A dexterity measure for robot manipulators In Proceedings IEEE International Conference on Robotics and Automation, olume, 990 [6] J Kenneth Salisbury and John J Craig Articulated hands: Force control and kinematic issues The International Journal of Robotics Research, ():4{7, 98 [7] M Thomas, H C Yuan-Chou, and D Tesar Optimal actuator sizing for robotic manipulators based on local dynamic criteria Transactions of the ASME Journal of Mechanisms, Transmissions, and Automation in Design, 07():6{69, 985 [8] Masaru Uchiyama, Kunitoshi Shimizu, and Kyojiro Hakomori Performance ealuation of manipulators using the jacobian and its application to trajectory planning In Robotics Research, The Second International Symposium, 985 [9] Tsuneo Yoshikawa Dynamic manipulability of robot manipulators In Proceedings 985 IEEE International Conference on Robotics and Automation, pages 0{08, 985 St Louis, Missouri [0] Tsuneo Yoshikawa Manipulability of robotic mechanisms The International Journal of Robotics Research, 4(), 985 [] Tsuneo Yoshikawa Translational and rotational manipulability of robotic manipulators In Proceedings of the 99 International Conference Industrial Electronics, Control, and Instrumentation, October 99 [] Zhanfang Zhao, Zhen Wu, Jilian Lu, Weihai Chen, and Guanghua Zong Dynamic dexterity of redundant manipulators In Proceedings IEEE International Conference on Robotics and Automation, olume, 995 7

8 Alan Bowling Robotics Laboratory Department of Computer Science Stanford Uniersity MS: Gates 8 Stanford, CA, USA 9405 Tel: (650) Fax: (650) bowling@roboticsstanfordedu Professor Oussama Khatib Robotics Laboratory Department of Computer Science Stanford Uniersity MS: Gates 44 Stanford, CA, USA 9405 Tel: (650)7-975 Fax: (650) ok@csstanfordedu Abstract The study of acceleration capability is concerned with the responsieness of a manipulator to controller commands In this paper we present a general model for the analysis of end-eector linear and angular accelerations that accounts for the elocity eects The separate treatment of linear and angular motion directly addresses the inhomogeneities of end-eector motions, aoiding the use of indeterminate scaling factors The elocity eects considered are the Coriolis and centrifugal forces, as well as the relationships associated with actuator's speed-torque performance cures This study results in a characterization referred to as the \motion isotropy hypersurface" which describes the relationships between isotropic end-eector linear and angular elocities and accelerations The utility of this surface and its associated information is demonstrated in a design application inoling the PUMA 560 manipulator Contribution The contribution of this paper is in the area of performance analysis The characterization of acceleration capability presented here has mainly been applied to the manipulator design problem Howeer, control related applications will be explored in the near future 8