Geometric Control of the Chaplygin Sleigh and the Double Gimbal System

Abstract:The Chaplygin sleigh--a rigid body with attached blade moving in a plane--is an example of a non-holonomic mechanical system with symmetry and constraints. Allowing the sleigh to have a moving center of mass, we can steer the sleigh in any direction and through any point in the plane. The control algorithm uses the controlled dynamics to initiate short-time transitions between the natural/un-controlled dynamics. The double gimbal--a two angular degree of freedom system--is a typical component of larger multi-body systems like the relay mirror. The dynamics of the double gimbal can be be viewed on an appropriate torus where geodesics on this torus play a critical role in the control design. The control objective is optimal re-orientation and tracking of the double gimbal.