A VARIATIONAL DERIVATION OF EQUATIONS OF MOTION WITH CONTACT CONSTRAINTS USING SE(3)

摘要

For rigid-body systems subjected to non-holonomic constraints, a streamlined method is presented to derive a minimum number of analytical equations of motion. To illustrate the method, a rolling disk problem is considered. In kinematics, an orthonormal coordinate system is attached to the center of mass together with additional coordinate systems introduced to define the connection path. For each coordinate system, a moving frame is defined by explicitly writing the coordinate vector basis and the position vector of the origin, whereby the attitude of the coordinate vector basis and the coordinates of the origin are compactly stored in a 4 × 4 frame connection matrix of the special Euclidean group, SE(3). Contact velocity constraints are transformed to pfaffians to obtain the associated variational constraints. In kinetics, the principle of virtual work is employed. The desired equations of motion are obtained by expressing the translational and angular velocities at the center of mass as the linear functions of the generalized velocities with the coefficients stored in [B]-matrix, and reducing it to [B~*]-matrix after incorporating the contact constraints. The method can be easily extended to multi-body systems with both holonomic and non-holonomic constraints.