Minimum effort motions for open chain manipulators with task-dependent end-effector constraints

In this article we examine the solution of minimum effort optimal control problems for open-chain manipulators. A local solution to the optimal control problem is determined by a constrained parameter optimization over a set of B-spline basis functions. We demonstrate that the parameter optimization formulation of the problem is numerically ill-conditioned and that it is therefore essential to include analytic, or exact, gradients of the objective function and the constraints in order to guarantee a solution. A main contribution of our approach is an explicit expression for these gradients for general serial chains. Our formulation relies on the use of matrix exponentials for the manipulator kinematics, dynamics, and task constraints. Several examples are presented that demonstrate the power and flexibility of our approach.

[1]  E. Blum,et al.  The Mathematical Theory of Optimal Processes. , 1963 .

[2]  Bernard Roth,et al.  The Near-Minimum-Time Control Of Open-Loop Articulated Kinematic Chains , 1971 .

[3]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[4]  I. Faux,et al.  Computational Geometry for Design and Manufacture , 1979 .

[5]  David Q. Mayne,et al.  Combined phase I—phase II methods of feasible directions , 1979, Math. Program..

[6]  R. Hettich.  A comparison of some numerical methods for semi-infinite programming , 1979 .

[7]  Roger W. Brockett,et al.  Robotic manipulators and the product of exponentials formula , 1984 .

[8]  Kang G. Shin,et al.  Minimum-time control of robotic manipulators with geometric path constraints , 1985 .

[9]  Elmer G. Gilbert,et al.  Distance functions and their application to robot path planning in the presence of obstacles , 1985, IEEE J. Robotics Autom..

[10]  見城 尚志,et al.  Permanent-magnet and brushless DC motors , 1985 .

[11]  Lino Guzzella,et al.  Time-optimal motions of robots in assembly tasks , 1985, 1985 24th IEEE Conference on Decision and Control.

[12]  J. Bobrow,et al.  Time-Optimal Control of Robotic Manipulators Along Specified Paths , 1985 .

[13]  J. Murray,et al.  Linearization and Sensitivity Models of the Newton-Euler Dynamic Robot Model , 1986 .

[14]  Rainer Hettich,et al.  An implementation of a discretization method for semi-infinite programming , 1986, Math. Program..

[15]  Steven Dubowsky,et al.  Time optimal trajectory planning for robotic manipulators with obstacle avoidance: A CAD approach , 1986, Proceedings. 1986 IEEE International Conference on Robotics and Automation.

[16]  Rajnikant V. Patel,et al.  Recursive evaluation of linearized dynamic robot models , 1986, IEEE J. Robotics Autom..

[17]  John M. Hollerbach,et al.  Local versus global torque optimization of redundant manipulators , 1987, Proceedings. 1987 IEEE International Conference on Robotics and Automation.

[18]  James E. Bobrow,et al.  Optimal Robot Path Planning Using the Minimum-Time Criterion , 2022 .

[19]  Jorge Angeles,et al.  Trajectory planning in robotic continuous-path applications , 1988, IEEE J. Robotics Autom..

[20]  John Baillieul,et al.  Resolution of Kinematic Redundancy using Optimization Techniques , 1988, 1988 American Control Conference.

[21]  Zhaoyu Wang,et al.  Global versus Local Optimization in Redundancy Resolution of Robotic Manipulators , 1988, Int. J. Robotics Res..

[22]  A. Tits,et al.  A globally convergent algorithm with adaptively refined discretization for semi-infinite optimization problems arising in engineering design , 1989 .

[23]  Arthur E. Ryson,et al.  Efficient Algorithm for Time-Optimal Control of a Two-Link Manipulator , 1990 .

[24]  Rajnikant V. Patel,et al.  Dynamic analysis of robot manipulators - a Cartesian tensor approach , 1991, The Kluwer international series in engineering and computer science.

[25]  Frank Chongwoo Park,et al.  The optimal kinematic design of mechanisms , 1991 .

[26]  E. Polak,et al.  Unified steerable phase I-phase II method of feasible directions for semi-infinite optimization , 1991 .

[27]  Chih-Cheng Chen,et al.  A combined optimization method for solving the inverse kinematics problems of mechanical manipulators , 1991, IEEE Trans. Robotics Autom..

[28]  Michael C. Mulder,et al.  Behavior of a minimum-effort control algorithm for a multi-jointed robotic arm , 1992, Proceedings of the 1992 IEEE International Symposium on Intelligent Control.

[29]  E. Polak,et al.  Rate-preserving discretization strategies for semi-infinite programming and optimal control , 1992 .

[30]  Ping Lu.  Use of approximate gradients in trajectory optimization , 1992 .

[31]  John T. Wen,et al.  A global approach to path planning for redundant manipulators , 1993, [1993] Proceedings IEEE International Conference on Robotics and Automation.

[32]  Elmer G. Gilbert,et al.  Robot path planning with penetration growth distance , 1994, Proceedings of the 1994 IEEE International Conference on Robotics and Automation.

[33]  Fan-Tien Cheng,et al.  Torque optimization schemes for kinematically redundant manipulators , 1994, J. Field Robotics.

[34]  Vijay Kumar,et al.  Optimal trajectories and force distribution for cooperating arms , 1994, Proceedings of the 1994 IEEE International Conference on Robotics and Automation.

[35]  Roger W. Brockett,et al.  Kinematic Dexterity of Robotic Mechanisms , 1994, Int. J. Robotics Res..

[36]  Beno Benhabib,et al.  Optimal motion coordination of two robots - a polynomial parameterization approach to trajectory resolution , 1994, J. Field Robotics.

[37]  Frank Chongwoo Park,et al.  A recursive algorithm for robot dynamics using Lie groups , 1994, Proceedings of the 1994 IEEE International Conference on Robotics and Automation.

[38]  Richard M. Murray,et al.  A Mathematical Introduction to Robotic Manipulation , 1994 .

[39]  Sung-Woo Kim,et al.  Redundancy resolution of robot manipulators using optimal kinematic control , 1994, Proceedings of the 1994 IEEE International Conference on Robotics and Automation.

[40]  Chee Leong Teo,et al.  Relation between global velocity and local torque optimization of redundant manipulators , 1994, J. Field Robotics.

[41]  Zvi Shiller,et al.  Time-energy optimal control of articulated systems with geometric path constraints , 1994, Proceedings of the 1994 IEEE International Conference on Robotics and Automation.

[42]  Frank Chongwoo Park,et al.  A Lie Group Formulation of Robot Dynamics , 1995, Int. J. Robotics Res..

[43]  Frank Chongwoo Park,et al.  Geometric optimization algorithims for robot kinematic design , 1995, J. Field Robotics.

[44]  Shugen Ma.  A stabilized local torque optimization technique for redundant manipulators , 1995, Proceedings of 1995 IEEE International Conference on Robotics and Automation.

[45]  Yasmina Bestaoui,et al.  Actuator constraints in optimal motion planning of manipulators , 1995, Proceedings of 1995 IEEE International Conference on Robotics and Automation.

[46]  James E. Bobrow,et al.  Determination of minimum-effort motions for general open chains , 1995, Proceedings of 1995 IEEE International Conference on Robotics and Automation.

[47]  Yury Stepanenko,et al.  Iterative dynamic programming: an approach to minimum energy trajectory planning for robotic manipulators , 1996, Proceedings of IEEE International Conference on Robotics and Automation.

[48]  Hiroaki Ozaki,et al.  Optimal B-spline joint trajectory generation for collision-free movements of a manipulator under dynamic constraints , 1996, Proceedings of IEEE International Conference on Robotics and Automation.

[49]  Elmer G. Gilbert,et al.  Robot path planning with penetration growth distance , 1994, Proceedings of the 1994 IEEE International Conference on Robotics and Automation.