This module solves for the joint angles given the desired position and orientation in Cartesian space. This is a more complex problem than forward kinematics. The complexity of this problem arises from the nature of the transformation equations, which are nonlinear. There are two issues in solving these equations: existence of solutions and multiple solutions. A solution can exist only if the given position and orientation lies within the workspace of the manipulator's end-effector. By workspace, we mean all points in space that can be reached by the manipulator's end-effector. On the other hand, the problem of multiple solutions forces the designer to set a criterion for choosing one solution, e.g., a good choice is the solution that minimizes the amount that each joint is required to move.
There are two methods for solving the inverse kinematics problem: closed form solutions and numerical solutions. Numerical solutions are much slower than closed form solutions, but, for some configurations it is too difficult to find a closed form solution. In our case, we will use closed form solutions, since our models are three link manipulators with easy closed form formulas.
A software package called SRAST (Symbolic Robot Arm Solution Tool) that symbolically solves the forward and inverse kinematics for n-degree of freedom manipulators has been developed by Herrera-Bendezu, Mu, and Cain [18]. The input to this package is the Denavit-Hartenberg parameters, and the output is the direct and inverse kinematics solutions. Another method of finding symbolic solutions for the inverse kinematics problem was proposed in [46]. Kelmar and Khosla proposed a method for automatic generation of forward and inverse kinematics for a reconfigurable manipulator system [23].