Currently, there are three different approaches to the inverse kinematics problem, each of which has some disadvantages. The algebraic methodology suffers from the fact that the mathematical trees hide the wood of the solution, making it hard for the user to select the proper solution from the theoretically possible ones. On the other hand, the iterative approach is computationally more demanding and cannot guarantee convergence under all circumstances. Finally, the geometric methodology can only handle relatively simple manipulators. The problem of calculating the inverse kinematics can be restated as: ''given a target position of the end effector of a manipulator in a Cartesian coordinate system, calculate for all joints all possible positions''. From these positions all possible sets of joint angles which place the end effector in its target position can be easily calculated. The new ''intersection'' method is based on intersection of joint-axis ranges R(i,j): all possible positions of joint-axis(j) relative to any other joint-axis(i). This joint-axis(i) can be the base, the end effector or any joint axis between them. One can regard R(i,j) as the set of possible positions of a reduced manipulator which has only (j - i - 1) joints. Note that one works both in the forward and in the backward direction of the original manipulator.The method is intuitively clear, easy to understand and is capable of finding all feasible solutions. It is applicable to fairly complex manipulators such as the PUMA, as will be shown.
