Using Symbolic Technology to Derive Inverse Kinematic Solutions for Actuator Control Development

In multibody mechanics, the motion analysis for a platform (the kinematics problem) can be classified into two cases: the forward kinematics and the inverse kinematics problems. For the forward kinematics problem, the trajectory of a point on a mechanism (for example, the end effector of a robot arm or the center of a platform support by a parallel link manipulator) is computed as a function of the joint motions. In the inverse kinematics case, the problem is reversed: the goal is to compute the joint motions necessary to achieve a prescribed end effector trajectory. In general, given the mechanism geometry, it is quite straightforward to solve the forward kinematics problem both numerically and symbolically. In contrast, solving for the inverse kinematic problem typically involves solving a nonlinear system or equation with trigonometric functions. Issues such as singularity, multiple solutions (as in the case of “elbow up” and “elbow down” configurations for a robot arm), and no solution (as in the case in which the specified trajectory goes beyond the workspace of the mechanism) can often come up, further complicating the solution process. The complexity in the inverse kinematics problem is compounded even more for parallel link manipulators. Because of the complexity involved, the inverse kinematics problem is often solved numerically through iterations, and is computationally expensive. With a numeric approach, however, information about the motion of the mechanism is often lost. In this paper, we will describe how to obtain a symbolic solution to the inverse kinematics problem for two real problems using tools available in MapleSimTM. The first, a 2 degrees-of-freedom (DOF) tracking radar gimbal is used to show the principal steps in a relatively simple mechanism. These principles are then demonstrated with a much more complex mechanism: a Stewart-Gough hydraulic platform. Furthermore, we will show how having access to the symbolic Jacobian of the constraint equations allows us to inspect and exploit the underlying matrix structure, which leads to a simplified solution process for obtaining the symbolic solution. An advantage of having a symbolic solution to the inverse kinematics problem is the possibility of codegenerating the symbolic solution so that it can be embedded in real-time hardware-in-the-loop (HIL) applications. This approach can be contrasted with a purely numerical approach where the iterative solution process makes it difficult to use in real-time applications.