Inverse dynamics of an industrial robot using motion constraints

In multibody dynamics holonomic constraint equations exist because of mechanical joints, or specified motion trajectories. In order to solve the dynamic equations of motion of the constrained multibody system, one can use a coordinate partitioning approach to eliminate dependent coordinates. An alternative approach is to use the Lagrange multiplier technique. A generalized constraint force vector associated with the system generalized coordinates has to be added in the equations of motion, which is computed by a multiplication of the transposed constraint Jacobian times a Lagrange multiplier. At this point, it should be noted that the constraint force vector is in general not the vector of actual reaction forces in the joints. For example, in robotics it is of particular interest to prescribe the motion of the tool center point and determine the drive torques in the joints. This leads to an optimal control problem, which is in general very expensive to solve. However, an alternative method is to apply a motion as a rheonomic constraint equation on the tool center point of the robot. We show that the resulting generalized constraint force vector, which is acting on the body of the tool center point, can be converted into equivalent drive torques in the joints by the use of the principle of virtual work. The advantage of the proposed method is that the solution can be obtained after one single forward simulation of the multibody system, since the conversion into the drive torques is just a post-processing step.