Acceleration Sets of Planar Manipulators. Part 1. Theory

This report develops a systematic approach for determining the acceleration capability and the acceleration properties of the end-effector of a planar two degree-of-freedom manipulator. The acceleration of the end-effector at a given configuration of the manipulator is a linear function of the actuator torques and a nonlinear quadratic function of the joint-velocities. By decomposing the functional relationships between the inputs actuator torques and joint velocities and the output acceleration of the end-effector into two fundamental mappings, a linear mapping between the actuator torque space and the acceleration space of the end-effector and a quadratic nonlinear mapping between the joint velocity space and the mapping acceleration space of the end- effector, and by deriving the properties of these two mappings, it is possible to determine the properties of all acceleration sets which are the images of the appropriate inputs sets under the two fundamental mappings. The determination of the properties of the quadratic mapping, a key feature of the present work, allows us to obtain analytic expressions relating important acceleration properties of the end-effector to all the manipulator parameters and input variables of interest.