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Research Papers

Fast Equilibrium Test and Force Distribution for Multicontact Robotic Systems

[+] Author and Article Information
Yu Zheng

Control and Mechatronics Laboratory, National University of Singapore, Block EA #04-06, 9 Engineering Drive 1, Singapore 117576, Singaporeyuzheng001@gmail.com

Chee-Meng Chew

Control and Mechatronics Laboratory, National University of Singapore, Block EA #04-06, 9 Engineering Drive 1, Singapore 117576, Singaporechewcm@nus.edu.sg

J. Mechanisms Robotics 2(2), 021001 (Mar 24, 2010) (11 pages) doi:10.1115/1.4001089 History: Received September 24, 2008; Revised October 06, 2009; Published March 24, 2010; Online March 24, 2010

In the research of multicontact robotic systems, the equilibrium test and contact force distribution are two fundamental problems, which need to determine the existence of feasible contact forces subject to the friction constraint, and their optimal values for counterbalancing the other wrenches applied on the system and maintaining the system in equilibrium. All the wrenches, except those generated by the contact forces, can be treated as a whole, called the external wrench. The external wrench is time-varying in a dynamic system and both problems usually must be solved in real time. This paper presents an efficient procedure for solving the two problems. Using the linearized friction model, the resultant wrenches that can be produced by all contacts constitute a polyhedral convex cone in six-dimensional wrench space. Given an external wrench, the procedure computes the minimum distance between the wrench cone and the required equilibrating wrench, which is equal but opposite to the external wrench. The zero distance implies that the equilibrating wrench lies in the wrench cone, and that the external wrench can be resisted by contacts. Then, a set of linearly independent wrench vectors in the wrench cone are also determined, such that the equilibrating wrench can be written as their positive combination. This procedure always terminates in finite iterations and runs very fast, even in six-dimensional wrench space. Based on it, two contact force distribution methods are provided. One combines the procedure with the linear programming technique, yielding optimal contact forces with linear time complexity. The other directly utilizes the procedure without the aid of any general optimization technique, yielding suboptimal contact forces with nearly constant time complexity. Effective strategies are suggested to ensure the solution continuity.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

Multicontact robotic systems. Multiple contacts arise not only in the robotic systems having many (a) legs or (b) fingers, but also (c) making line or face contacts with the environment, which normally can be treated as two or more point contacts on the boundaries of the contact areas. The equilibrium of the six-legged robot in (a), the object in (b), or the bipedal robot in (c) is our concern; thus, the contact normal ni is chosen as in this figure.

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Figure 2

Illustration of the computations in six-dimensional wrench space, which are involved in (a) Step 2 and (b) Step 4 of Procedure 1. The point 0 denotes the origin of the wrench space. The convex cone and subspace indicated in (a) are generated by w1,w2,…,wl. The vector rl is orthogonal to the subspace spanned by w1,w2,…,wl, such that −wext−rl is the orthogonal projection of −wext on the subspace. If condition (C1) in Proposition 2 is satisfied, −wext−rl is also contained in the convex cone of w1,w2,…,wl.

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Figure 3

Illustration of the cause of discontinuity and the way to eliminate it. The squares and dots denote wij and −wext, respectively. The small triangles denote wσ, which, together with the curve, indicate the trajectory of wσ moving along with the variation in −wext. The dashed triangles and large squares depict the outputs of Procedure 1. (a) The CFD solution is not continuous because of the disordered outputs. (b) The adjoining outputs ensure the continuity of the CFD solution.

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Figure 4

Illustration of Procedure 1 directly producing a suboptimal solution of CFD

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Figure 5

Iterative values of rlTrl in running Procedure 1 with respect to wexta (circles) and wextb (squares)

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Figure 6

CPU times for running Procedure 1 with respect to n

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Figure 7

Contact forces computed by Method 1. (a)–(d) Contact forces at the four contacts. The solid curves depict the normal force components, while the dashed and dash-dot curves depict two tangential force components. (e) Sum of normal force components. (f) Maximum of normal force components.

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Figure 8

Contact forces computed by Method 2

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Figure 9

Contact forces computed by Method 1 without using any prior information

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Figure 10

Contact forces computed by Method 2 without using any prior information

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Figure 11

Average CPU times and total number of iterations with respect to n. (a) CPU time for solving the LP problem at a sampling instant. (b) CPU times for running Procedure 1 in Method 1 (solid line) and Method 2 (dashed line) at a sampling instant. (c) Numbers of iterations required by Procedure 1 in Method 1 (solid line) and Method 2 (dashed line) for all 201 sampling instants.

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