A Systematic Approach for the Jacobian Analysis of Parallel Manipulators with Two End-Effectors

Parallel manipulators with two end-effectors (PM2Es) enable the design of gripping robots with high dynamic performance. The gripping action is enabled by internal, relative degrees of freedom (DoFs) between the two end-effectors. Many standard methods for the analysis and control of parallel manipulators rely on a Jacobian, where a complete Jacobian analysis includes constraint relations. These constraint relations have not been consistently included in previous analyses of PM2Es, while they are specifically relevant for PM2Es because constraints play an important role in the static force analysis of a PM2E. This is because wrenches applied by the actuators can be transferred to the end-effectors through internal constraints, an effect which is not captured by kinematic relations alone. This paper presents a systematic approach to perform the Jacobian analysis of PM2Es, which is based on screw theory, and that takes all constraint relations into account. The approach is applied to a PM2E with three legs and one internal closed-loop chain. An example mechanism was built to experimentally validate the resulting Jacobian analysis using a static force analysis.