A surface integral algorithm for the motion planning of nonholonomic mechanical systems
Anderson, David P.
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The number of coordinates needed to completely describe the configuration of a holonomic mechanical system is equal to the number of degrees of freedom possessed by that system. In contrast, nonholonomic systems always require more coordinates for their description than there are degrees of freedom due to the nonintegrable nature of the governing velocity constraints. The task of nonholonomic motion planning applied to a given system is to develop trajectories of the independent coordinate variables such that the system is driven to some desired point in its configuration space. An algorithm for constructing these trajectories is presented. In this algorithm, the independent variable are first converged to their desired values. The dependent variables are subsequently converged using closed trajectories of the independent variables. The requisite closed trajectories are planned using Stoke's Theorem which converts the problem of finding a closed path in the space of the independent variables to that of finding a surface area in that same space such that the dependent variable converge to their desired values as the independent variables traverse along the boundary of the surface area. The use of Stoke's Theorem simplifies the motion planning process and also answers important questions pertaining to the system. The salient features of the algorithm are apparent in the two examples discussed: a planar space robot and a disk rolling without slipping on a flat surface.
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