Foot forces in walking machines and finger forces in multi-fingered grippers are usually indeterminate due to the multi-closed-chain geometry of the systems. While many methods were proposed to solve the force distribution of such systems, a method called stiffness matrix method  was developed based on the concept that the force must satisfy the equations of material deformations. However, only leg compliances were considered in the stiffness matrix method. In this paper, the stiffness matrix method is generalized to include all the major system compliances, i.e., those of legs (fingers), actuators and terrain (object to be grasped). Based on the developed generalized stiffness matrix method, an example of foot force analysis of a quadruped is presented to demonstrate the effects of different system compliances on the foot forces.
A Generalized Stiffness Matrix Method for Force Distribution of Robotic Systems with Indeterminancy
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Gao, X., Song, S., and Zheng, C. Q. (September 1, 1993). "A Generalized Stiffness Matrix Method for Force Distribution of Robotic Systems with Indeterminancy." ASME. J. Mech. Des. September 1993; 115(3): 585–591. https://doi.org/10.1115/1.2919230
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