Motion planning via optimal control theory

摘要

We use optimal control theory to solve motion planning problems. <P xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">The development of geometric control theory using differential geometric methods to deal with control problems on arbitrary manifolds (in particular Lie groups and their symmetric spaces), has provided methods which are particularly useful for motion planning problems in areas such as robotics, spacecraft attitude control and the control of underwater vehicles as the kinematics involved in such problems can always be represented as a dynamical system on a Lie group of admissible motions. To find suitable control laws, we formally treat the velocity variables as controls, introduce time-dependent weighting functions, introduce penalty terms in the cost functional, and formulate the problem as a fixed-endtime problem but derive duration estimates which ensure that the control algorithm can be performed during the chosen time interval in the presence of actuator constraints.</P>