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*Objective*: To perform inverse kinematics on a 5-axis robotic arm.

Just as in direct kinematics, we find the position and orientation of the tool of a robotic manipulator for a given vector of joint variables; the reverse procedure of determining the joint variables given a desired or known tool position and orientation is addressed in inverse kinematics.

The inverse kinematics problem is more important from a practical point of view because any manipulation task is naturally stated in terms of the required position and orientation of the tool.

Again when sensors for visual feedback such as cameras are used the information fedback is the position and orientation of the tool or the object to be manipulated and not the state of the joint variables.

In direct kinematics, the “Arm Equation” served as a solution for the direct kinematics problem. It was a kind of one to one mapping tool since any particular vector of joint-variables would yield one and only one end-effector or tool configuration.

In inverse kinematics, the solution would be a set or vector of joint-variables that will result in the given end-effector or tool configuration. But unlike the arm equation which behaves as a one to one mapping tool the solution to an inverse kinematics problem is not so straight forward. The obvious reason for this is, there can be more than one set of joint-variables that can ultimately result in the same end-effector configuration.

In addition to the existence of multiple solutions especially for robots with more than 6 axes called “kinematically redundant” robots, a solution may not exist at all in many cases. Therefore what is usually done in inverse kinematics is that each robot or each class of robots is analysed separately in order to determine its inverse kinematic parameters.

There are two approaches to the inverse kinematics problem:

· To find a closed form solution using algebra or geometry.

· To find a numerical solution by a successive approximation algorithm.

Although it is highly desirable to go for the first approach, it is not always possible to find a closed form solution for arbitrary robot architecture. However, most of the industrial robots do have a fairly known tool configuration and this paves the way for using the first approach.

The algebraic approach involves finding the joint angles through algebraic transformation and relations whereas the geometric approach involves finding the same through geometrical relations and heuristics by analyzing the structural geometry of the robot.

Since the finding of a closed form solution in joint angles is a tool-configuration dependent task, the methods for finding the same for some specific robot architectures are discussed in theory.