Trajectory Generation and Control

Nonholonomic motion planning: Steering using sinusoids
RM Murray and SS Sastry  (IEEE Transactions on Automatic Control, 38 (5), 1993)
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In this paper, we investigate methods for steering systems with nonholonomic constraints between arbitrary configurations. Early work by Brockett derives the optimal controls for a set of canonical systems in which the tangent space to the configuration manifold is spanned by the input vector fields and their first order Lie brackets. Using Brocket's result as motivation, we derive suboptimal trajectories for systems which are not in canonical form and consider systems in which it takes more than one level of bracketing to achieve controllability. These trajectories use sinusoids at integrally related frequencies to achieve motion at a given bracketing level. We define a class of systems which can be steered using sinusoids (chained systems) and give conditions under which a class of two-input systems can be converted into this form.

Real Time Feedback Control for Nonholonomic Mobile Robots With Obstacles
Stephen R. Lindemann, Islam I. Hussein, Steven M. LaValle  (CDC '06)
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We introduce a method for constructing smooth feedback laws for a nonholonomic robot in a 2-dimensional polygonal workspace. First, we compute a smooth feedback law in the workspace without taking the nonholonomic constraints into account. We then give a general technique for using this to construct a new smooth feedback law over the entire 3- dimensional configuration space (consisting of position and orientation). The trajectories of the resulting feedback law will be smooth and will stabilize the position of the robot in the plane, neglecting the orientation. Our method is suitable for real time implementation and can be applied to dynamic environments.