Trajectory Planning and Control for Planar Robots with Passive Last Joint

We present a method for trajectory planning and control of planar robots with a passive rotational last joint. These underactuated mechanical systems, which are subject to nonholonomic second-order constraints, are shown to be fully linearized and input-output decoupled by means of a nonlinear dynamic feedback. This objective is achieved in a unified framework, both in the presence or absence of gravity. The linearizing output is the position of the center of percussion of the last link. Based on this result, one can plan smooth trajectories joining in finite time any initial and desired final state of the robot; in particular, transfers between inverted equilibria and swing-up maneuvers under gravity are easily obtained. We also address the problem of avoiding the singularity induced by the dynamic linearization procedure through a careful choice of output trajectories. A byproduct of the proposed method is the straightforward design of exponentially stable tracking controllers for the generated trajectories. Simulation results are reported for a 3R robot moving in a horizontal and vertical plane. Possible extensions of the approach and its relationships with the differential flatness technique are briefly discussed.

[1]  Kevin M. Lynch,et al.  Dynamic Nonprehensile Manipulation: Controllability, Planning, and Experiments , 1999, Int. J. Robotics Res..

[2]  Mark W. Spong.  Bipedal Locomotion, Robot Gymnastics, and and Robot Air Hockey: A Rapprochement , 1999 .

[3]  S.K. Agrawal,et al.  Optimal planning of an under-actuated planar body using higher-order method , 1998, Proceedings. 1998 IEEE International Conference on Robotics and Automation (Cat. No.98CH36146).

[4]  Giuseppe Oriolo,et al.  Motion planning and trajectory control of an underactuated three-link robot via dynamic feedback linearization , 2000, Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065).

[5]  A. Isidori.  Nonlinear Control Systems , 1985 .

[6]  Susumu Tachi,et al.  Position control of manipulator with passive joints using dynamic coupling , 1991, IEEE Trans. Robotics Autom..

[7]  Naoji Shiroma,et al.  Time-scaling control of an underactuated manipulator , 1998, Proceedings. 1998 IEEE International Conference on Robotics and Automation (Cat. No.98CH36146).

[8]  Jun Nakanishi,et al.  A brachiating robot controller , 2000, IEEE Trans. Robotics Autom..

[9]  Giuseppe Oriolo,et al.  Stabilization of a PR planar underactuated robot , 2001, Proceedings 2001 ICRA. IEEE International Conference on Robotics and Automation (Cat. No.01CH37164).

[10]  Keigo Kobayashi,et al.  Design of a desirable trajectory and convergent control for 3-DOF manipulator with a nonholonomic constraint , 2000, Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065).

[11]  M. Fliess,et al.  Flatness and defect of non-linear systems: introductory theory and examples , 1995 .

[12]  Richard M. Murray,et al.  Real Time Trajectory Generation for Differentially Flat Systems , 1996 .

[13]  Georges Bastin,et al.  Control of Nonholonomic Wheeled Mobile Robots by State Feedback Linearization , 1995, Int. J. Robotics Res..

[14]  J. Lévine,et al.  On dynamic feedback linearization , 1989 .

[15]  Russell H. Taylor,et al.  Constrained Cartesian motion control for teleoperated surgical robots , 1996, IEEE Trans. Robotics Autom..

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

[17]  Naoji Shiroma,et al.  Nonholonomic Motion Planning for Coupled Planar Rigid Bodies with Passive Revolute Joints , 2002, Int. J. Robotics Res..

[18]  Giuseppe Oriolo,et al.  Stabilization of the Acrobot via iterative state steering , 1998, Proceedings. 1998 IEEE International Conference on Robotics and Automation (Cat. No.98CH36146).

[19]  Mark W. Spong,et al.  Underactuated mechanical systems , 1998 .

[20]  M. Rathinam,et al.  Configuration flatness of Lagrangian systems underactuated by one control , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[21]  Antonio Bicchi,et al.  Hands for dexterous manipulation and robust grasping: a difficult road toward simplicity , 2000, IEEE Trans. Robotics Autom..

[22]  Giuseppe Oriolo,et al.  Motion planning under gravity for underactuated three-link robots , 2000, Proceedings. 2000 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2000) (Cat. No.00CH37113).

[23]  Steven Dubowsky,et al.  The Kinematics and Dynamics of Space Manipulators: The Virtual Manipulator Approach , 1990, Int. J. Robotics Res..

[24]  Keigo Kobayashi,et al.  Design of a Desirable Trajectory and Convergent Control for 3-D.O.F Manipulator with a Nonholonomic Constraint , 2000 .

[25]  R. Mattone,et al.  Control problems in underactuated manipulators , 2001, 2001 IEEE/ASME International Conference on Advanced Intelligent Mechatronics. Proceedings (Cat. No.01TH8556).

[26]  Mark W. Spong,et al.  The Pendubot: a mechatronic system for control research and education , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[27]  Richard M. Murray,et al.  Configuration Controllability of Simple Mechanical Control Systems , 1997, SIAM Rev..

[28]  Yoshihiko Nakamura,et al.  Nonlinear behavior and control of a nonholonomic free-joint manipulator , 1997, IEEE Trans. Robotics Autom..

[29]  Kevin M. Lynch,et al.  Kinematic controllability for decoupled trajectory planning in underactuated mechanical systems , 2001, IEEE Trans. Robotics Autom..

[30]  Alessandro De Luca,et al.  Stabilization of an underactuated planar 2R manipulator , 2000 .

[31]  Giuseppe Oriolo,et al.  Control of mechanical systems with second-order nonholonomic constraints: underactuated manipulators , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[32]  Jean-Paul Laumond,et al.  Robot Motion Planning and Control , 1998 .

[33]  Mark W. Spong,et al.  The swing up control problem for the Acrobot , 1995 .

[34]  R. W. Brockett,et al.  Asymptotic stability and feedback stabilization , 1982 .

[35]  Alessandro De Luca,et al.  A general algorithm for dynamic feedback linearization of robots with elastic joints , 1998, Proceedings. 1998 IEEE International Conference on Robotics and Automation (Cat. No.98CH36146).

[36]  J. Descusse,et al.  Decoupling with Dynamic Compensation for Strong Invertible Affine Non Linear Systems , 1985 .

[37]  R. Murray,et al.  Flat Systems , 1997 .

[38]  Naoji Shiroma,et al.  Nonholonomic control of a three-DOF planar underactuated manipulator , 1998, IEEE Trans. Robotics Autom..