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MIT2_017JF09_p27

# MIT2_017JF09_p27 - 27 WALKING ROBOT CONSTRAINTS 27 87...

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27 WALKING ROBOT CONSTRAINTS 87 27 Walking Robot Constraints The ”Big Pig” project, which involves a pig-like quadruped robot, wants to understand how to control the device when it operates on a slippery surface. Viewed from above, each foot is at a location described by the vector r i , comprising the x (positive forward) and y (positive to the left) coordinates relative to the center of mass; the vertical direction is z . Clearly static stability is possible only if the centroid is contained within one of the triangles created by the planted foot locations. Hence, slow walking can be achieved by moving the centroid into different triangle sets, and moving the feet in turn. Also in every such configuration, the roll and pitch moments have to balance to zero. Things get harder during actual locomotion, because we do not want slippage on any of the feet. This occurs when force in the horizontal plane exceeds a critical fraction of the vertical (normal) force: when F i,x μF i,z , where μ is the Coulomb friction coeﬃcient. A similar constraint would apply of course in the y -direction. Here we take on a simplified, three-leg problem that lays out some very practical issues in large and slowly moving walking robots. Consider the statically stable condition r = [[1.1, 0.8];[1.2,-0.7];[-1.5,-0.7]] meters, where each pair gives the x and y coordinates of three planted legs (forward left, forward right, back right). The weight of the Big Pig is 4600 Newtons, and the value of μ is 0.12. Along with the non-slip constraint for each foot in the x -direction, we have a mechanical design constraint that the x force magnitude on any foot cannot exceed 250 Newtons. Also, since a foot cannot ”pull” on the ground, each F i,z has to be positive. This is a total of nine inequality constraints.

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