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RESEARCH PAPERS

Visualization of the Contact Force Solution Space for Multi-Limbed Robots

[+] Author and Article Information
Dennis W. Hong

Mechanical Engineering Department, Virginia Tech, Blacksburg, VA 24061dhong@vt.edu

Raymond J. Cipra

School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, IN 47907-2088cipra@ecn.purdue.edu

J. Mech. Des 128(1), 295-302 (Jun 24, 2005) (8 pages) doi:10.1115/1.2118732 History: Received June 28, 2004; Revised June 24, 2005

Abstract

A new analytical method for determining, describing, and visualizing the solution space for the contact force distribution of multi-limbed robots with three feet in contact with the environment in three-dimensional space is presented. The foot contact forces are first resolved into strategically defined foot contact force components to decouple them, and then the static equilibrium equations are applied. Using the friction cone equation at each foot contact point, the problem is then transformed into a geometrical one. Using geometric properties of the friction cones and by simple manipulation of their conic sections, the entire solution space which satisfies the static equilibrium and friction constraints at each contact point can be found. Two representation schemes, the “force space graph” and the “solution volume representation,” are developed for describing and visualizing the solution space which gives an intuitive visual map of how well the solution space is formed for the given conditions of the system.

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Figures

Figure 2

The force system

Figure 3

The example system

Figure 4

Geometric interpretation of finding the range of FC1γ and FC1α

Figure 5

Examples of conic sections

Figure 6

Three friction cones sliced by their force planes

Figure 7

Projection of the cross section regions onto the foot contact plane

Figure 8

Transformation of the projected cross section regions

Figure 9

The force space graph representation

Figure 1

The coordinate system

Figure 10

Valid/invalid solution on the force space graph. (a) A valid solution; (b) An invalid solution (slip at C1).

Figure 11

The solution volume representation. (a) The intersection of EC1, EC2, and EC3; (b) The solution space as a volume.

Errata

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