Dynamics and contouring control of a 3-DoF parallel kinematics machine

In machining applications, instead of tracking error (the difference between the actual position and the desired position), contouring error (the minimum distance from the actual position to the desired trajectory) characterizes product quality. In this paper, we propose a generalized moving task coordinate frames based contouring control for parallel kinematics machines, whose dynamics is in general coupled and strongly nonlinear. The Orthopod, a 3 degree-of-freedom purely translational parallel kinematics machine, is introduced as a control plant. The Lagrange-D’Alembert formulation is used to model the system dynamics. The developed dynamic model in Cartesian space is transformed and parametrized by tangential error, normal error, and binormal error in moving task coordinate frames. The contouring error is then approximated by the normal error and the binormal error, which is the projection of tracking error to the normal plane at the desired position. By employing the structural properties of the transformed dynamics, a special feedback linearization, the computed torque control is applied. It leads to a stabilization problem for a second-order linear time-invariant system. Coulumb plus viscous friction model is used to compensate friction effects. Friction parameters are identified by least-squares approach. For comparison purpose, the tracking error based computed torque control is also carried out. Experiments demonstrate that the proposed control scheme not only leads to improved contouring accuracy, but also produces smaller and smoother control input torques, which may contribute to smaller vibration.