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Research Papers

# A Jacobian-Based Algorithm for Planning Attitude Maneuvers Using Forward and Reverse Rotations

[+] Author and Article Information
Sung K. Koh

Department of Mechanical Engineering, Pohang University of Science and Technology, Pohang, 790-784, Republic of Koreashkoh@postech.ac.kr

Gregory S. Chirikjian

Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218gregc@jhu.edu

G. K. Ananthasuresh

Department of Mechanical Engineering, Indian Institute of Science, Bangalore, 560012, Indiasuresh@mecheng.iisc.ernet.in

J. Comput. Nonlinear Dynam 4(1), 011012 (Dec 12, 2008) (12 pages) doi:10.1115/1.3007903 History: Received June 18, 2007; Revised May 20, 2008; Published December 12, 2008

## Abstract

Algorithms for planning quasistatic attitude maneuvers based on the Jacobian of the forward kinematic mapping of fully-reversed (FR) sequences of rotations are proposed in this paper. An FR sequence of rotations is a series of finite rotations that consists of initial rotations about the axes of a body-fixed coordinate frame and subsequent rotations that undo these initial rotations. Unlike the Jacobian of conventional systems such as a robot manipulator, the Jacobian of the system manipulated through FR rotations is a null matrix at the identity, which leads to a total breakdown of the traditional Jacobian formulation. Therefore, the Jacobian algorithm is reformulated and implemented so as to synthesize an FR sequence for a desired rotational displacement. The Jacobian-based algorithm presented in this paper identifies particular six-rotation FR sequences that synthesize desired orientations. We developed the single-step and the multiple-step Jacobian methods to accomplish a given task using six-rotation FR sequences. The single-step Jacobian method identifies a specific FR sequence for a given desired orientation and the multiple-step Jacobian algorithm synthesizes physically feasible FR rotations on an optimal path. A comparison with existing algorithms verifies the fast convergence ability of the Jacobian-based algorithm. Unlike closed-form solutions to the inverse kinematics problem, the Jacobian-based algorithm determines the most efficient FR sequence that yields a desired rotational displacement through a simple and inexpensive numerical calculation. The procedure presented here is useful for those motion planning problems wherein the Jacobian is singular or null.

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## Figures

Figure 1

Fully-reversed 90deg rotations about the x-, y-, z-, −x-, −y-, and −z-axes of a body-fixed coordinate frame

Figure 2

Mechanisms for implementing FR rotations: (a) ETC pseudo-wheels attached to a rigid-body, (b) ETC pseudo-wheel, and (c) ETC actuator (2)

Figure 3

Translation of the configuration of a rigid body in SO(3). The initial orientation Ri and the final orientation Rf are translated to I and RiTRf, respectively, so that the rotation angle ϕ and rotation axis ŵ of log[RiTRf] can be evaluated by Eqs. 2,3.

Figure 4

Flow chart of the single-step Jacobian-based algorithm

Figure 5

The angle of rotation θ(t)=[θx,θy,θz] of (a) Rxyzx−y−z− for the desired orientation Rf=Rxyz(θx=−1.9422,θy=2.1605,θz=−2.0488) and (b) Ryzxz−y−x− for the desired orientation Rf=Ryzxz−y−x−(θx=π∕3,θy=π∕3,θz=π∕3)

Figure 6

Trajectories of θ in T3 converged from 50 random initial orientations. ◻ represents analytical solutions to the inverse kinematics problem in T3. ○ represents the angle of rotation corresponding to the initial orientations provided to the single-step Jacobian algorithm.

Figure 7

Trajectories of ξ=[log[R]]V converged into 100 random desired orientations. The random desired orientations are located at the origin, and the initial orientation Ri is represented by ○.

Figure 8

Flow chart of the multiple-step Jacobian-based algorithm

Figure 9

Trajectories of ξ converging into 100 random desired orientations. ○ represents an initial orientation Ri, and random desired orientations are located at the origin.

Figure 10

Distance error D of (a) the single-step and (b) the multiple-step Jacobian methods that are run for the desired orientation Ryzxz−y−x−(θx=π∕3,θy=π∕3,θz=π∕3)

Figure 11

(a) Rotational path length L of the single-step Jacobian, two-step Jacobian; (b) multiple-step Jacobian methods (Δϕ=0.0001rad) that are run for the desired orientation Ryzxz−y−x−(θx=π∕3,θy=π∕3,θz=π∕3)

Figure 12

Multiple-step Jacobian method: (a) trajectory of ξ, (b) distance error D, and (c) rotational path length L

Figure 13

Pair-wise method: (a) trajectory of ξ, (b) distance error D, and (c) rotational path length L

Figure 14

Single-sequence method: (a) trajectory of ξ, (b) distance error D, and (c) rotational path length L

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