Geometric Modelling and Control of Flexible Serial Robot Links

The article presents the initial attempts to provide a geometric framework for modelling and control of flexible link robotic manipulators whose end-effector interacts with a compliant surface. The case study presented here restricts itself to planar flexible link manipulators. Ideas from Lie groups and Lie algebras have been employed to derive the forward and inverse kinematic maps, and their associated Jacobian. Finally we employ the geodesic equation to write down the comprehensive dynamical equations of the system. A geometrical interpretation of the Jacobian is presented and a velocity-level control law proposed earlier in literature is recast in the geometric language. Lastly we incorporate Christoffel symbols in the dynamic model of the flexible manipulator highlighting the underlying geometry. Using this framework, an existing velocity based control law is synthesized and simulated. Results demonstrate that the geometric framework yields a smoother and faster error convergence.