Recursive higher-order constraints for linkages with lower kinematic pairs

The geometric constraints imposed on the bodies of a linkage determine its function and mobility. They are the basis for any computational analysis of the kinematics and dynamics of a linkage, which ultimately requires higher-order time derivatives of the constraints. A tailored formulation for linkages comprising lower kinematic pairs is the product of exponentials (POE) formula, where rigid body motions are represented as curves in the Lie group SE(3). The corresponding velocity constraints involve the instantaneous joint screws, and their derivatives involve the derivatives of these screws. It is known that partial derivatives of arbitrary order of the instantaneous joint screws are given explicitly and algebraically in terms of Lie brackets (i.e. screw products). This, however, leads to complex expressions that are very difficult to use in actual computations. In this paper a recursive formulation for the time derivatives of arbitrary order of the velocity constraints for lower-pair linkages is presented. This formulation applies to multi-loop linkages. To this end, the linkage topology is represented by a topological graph, and loop closure constraints are formulated for the (topologically independent) fundamental cycles. It is briefly discussed that this provides the basis for the higher-order kinematic analysis of linkages.