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A Jacobian-based algorithm for planning the motion of an underactuated rigid body undergoing forward and reverse rotations1

Published online by Cambridge University Press:  28 August 2009

Sung K. Koh
Sung K. Koh*
Affiliation:
Department of Mechanical Engineering, Pohang University of Science & Technology, Pohang, Republic of Korea
*
*Corresponding author. E-mail: skkoh@postech.ac.kr
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Summary

A Jacobian-based algorithm that is useful for planning the motion of a floating rigid body operated using two input torques is addressed in this paper. The rigid body undergoes a four-rotation fully reversed (FR) sequence of rotations which consists of two initial rotations about the axes of a coordinate frame attached to the body and two subsequent rotations that undo the preceding rotations. Although a Jacobian-based algorithm has been useful in exploring the inverse kinematics of conventional robot manipulators, it is not apparent how a correct FR sequence for a desired orientation could be found because the Jacobian of FR sequences is singular as well as being a null matrix at the identity. To discover the FR sequences that can synthesize the desired orientation circumventing these difficulties, the Jacobian algorithm is reformulated and implemented from arbitrary orientations where the Jacobian is not singular. Due to the insufficient degrees-of-freedom of four-rotation FR sequences required to achieve all possible orientations, the rigid body cannot achieve certain orientations in the configuration space. To best approximate these infeasible orientations, the Jacobian-based algorithm is implemented in the sense of least squares. As some orientations can never be attained by a single four-rotation FR sequence, two different four-rotation FR sequences are exploited alternately to ensure the convergence of the proposed algorithm. Assuming the orientation is supposed to be manipulated using three input torques, the switching Jacobian algorithm proposed in this paper has significant practical importance in planning paths for aerospace and underwater vehicles which are maneuvered using only two input torques due to the failure of one of the torque-generation mechanisms.

Keywords

Underactuated systemAttitude controlFully reversed sequences of rotationsNoncommutativity of finite rigid rotationsJacobian algorithmMotion planning
Type
Article
Information
Robotica , Volume 28 , Issue 5 , September 2010 , pp. 747 - 757
Copyright
Copyright © Cambridge University Press 2009

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Footnotes

1

Some of the results contained in this paper were published in the proceedings of the 2008 ASME International Design Engineering Technical Conference, paper number DET2008-49413, and are reproduced here with the permission of ASME.

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