M.P. Hennessey


Unicycle, rolling disk, sphere, Lie algebra, nonholonomic mechanicalsystem, kinematics, computer graphics


The kinematics, dynamics, and control of a unicycle (with yaw and roll inputs) moving without slip on a planar surface has been studied extensively in the geometric mechanics and nonholonomic literature. This paper considers the kinematic extension to the case of a unicycle moving on a nonplanar surface: specifically, a spherical surface and the associated visualization, including use of computer-aided design (CAD) software packages (e.g., SolidWorks r& MATLAB r ). Such a scenario allows one to move closer to an understanding of the motion of an actual multiwheeled vehicle traversing nonplanar surfaces that may be approximated as spherical surface patches as well the realistic portrayal of virtual vehicles in computer graphics environments, in addition to other applications, such as those that employ kinematic inversion (e.g., industrial web handling). Differential equations of motion are derived based on the nonholonomic no-slip constraint and spherical trigonometry from which the associated Lie algebra is revealed and a related example foliation formulation is illustrated that offers further generalization. For a fixed body yaw rate it is shown that a circular trajectory of a specific radius results—this can also form the basis of an instantaneous centre integration (ICI) algorithm. Numerical studies designed to exercise the equations of motion (using MATLAB r and SIMULINK r ) were performed based on combinations of several different types of input trajectories: constant, non-infinitesimal Lie bracket manoeuvres, sinusoidal, and that induced by a prescribed contact point trajectory. Finally, the resulting interesting motions were plotted in 2D and 3D and provide a source of qualitative corroboration of the equations in addition to computer-generated artwork.

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