Motion Recovery of Parallel Manipulators using Generalized Inverse and Partitioned Jacobian Matrix
Keywords:
parallel manipulators, motion recovery, generalized inverse, Penrose equationsAbstract
In this paper, the motion recovery of parallel manipulators due to joint failure is addressed. The failed joints have either zero velocities, locked joints, or nonzero velocities which are different from the desired velocities. For the Jacobian matrix of the failed leg, its generalized inverse, which satisfies the first, second, and fourth equations among four equations of Penrose, is utilized to characterize the minimum norm of the correctional velocity vector related to the healthy joints. The platform twist is fully recovered if the Jacobian matrix of the failed leg is of full rowrank or if the platform twist is in the range space of the Jacobian matrix of the failed leg. When the platform twist is in the orthogonal complement of the range space of the Jacobian matrix of the failed leg, or when the Jacobian matrix of the failed leg is not of full row-rank, the partial recovery of the platform twist is achieved using the partitioned Jacobian matrix, while the correctional joint velocity after the failure is minimized.
References
NOTASH, Leila, HUANG, Li, On the Design of Fault Tolerant Parallel Manipulators, Mech. Mach. Theory, 38, 1, pp. 85-101, 2003.
JING, Zhao, CHENG, Fang, On the Joint Velocity Jump during Fault Tolerant Operations for Manipulators with Multiple Degrees of Redundancy, Mech. Mach. Theory, 44, 6, pp. 1201–1210, 2009.
NOTASH, Leila, Motion Recovery after Joint Failure in Parallel Manipulators, Spec. ed. Trans. Can. Soc. Mech. Eng., 35, 4, pp. 559-571, 2011.
NOTASH, Leila, On the Twist Recovery Methodologies after Failure, Latest Adv. in Rob. Kin. (editors: J. Lenarcic and M. Husty), Springer, 2012, pp. 11-18.
ABDI, Hamid, NAHAVANDI, Saeid, FRAYMAN, Yakov, MACIEJEWSKI, Anthony A., Optimal Mapping of Joint Faults into Healthy Joint Velocity Space for Fault Tolerant Redundant Manipulators, Robotica, 30, 4, pp. 635-648, 2012.
TING, Yung, TOSUNOGLU, Sabri, FREEMAN Robert, Torque Redistribution and Time Regulation Methods for Actuator Saturation Avoidance of Fault-tolerant Parallel Robots, Rob. Syst., 12, 12, pp. 807-820, 1995.
NOTASH, Leila, A Methodology for Actuator Failure Recovery in Parallel Manipulators, Mech. Mach. Theory, 46, 4, pp. 454-465, 2011.
NAKAMURA, Y., Advanced Robotics: Redundancy and Optimization, Addison Wesley, 1991.
BEN-ISRAEL, A., GREVILLE, T.N.E., Generalized Inverses: Theory and Applications, 2nd edition, Springer, New York, 2003.
BJERHAMMAR, Arne, A Generalized Matrix Algebra, Trans. Roy. Inst. Tech., 124, Stockholm, 1958.
URQUHART, N. Scott, Computation of Generalized Inverse Matrices which Satisfy Specified Conditions, Soc. Indus. App. Math., 10, 2, pp. 216-218, 1968.
PENROSE, Roger, TODD, John Arthur, On Best Approximate Solutions of Linear Matrix Equations, Math. Proc. of the Cambridge Philo. Soc., 52, 1, pp. 17-19, 1956.
Published
Issue
Section
Copyright (c) 2020 The Romanian Journal of Technical Sciences. Applied Mechanics.
This work is licensed under a Creative Commons Attribution 4.0 International License.