The Kinematics of Manipulators – Forward and Inverse

In general, kinematics is the study of motion without any mention of the forces causing it. When any object moves, the parameters attached to it are its position, velocity and acceleration. Velocity and acceleration are derivatives of the position. Sometimes even higher derivatives of position are required to be calculated for the study of jerkiness and other irregularities in the motion.

The structure of a manipulator is like a kinematic chain with the end-effector at the free end. It has rigid links connected by joints such that there can be relative motion. The joints can be rotary or revolute and sliding or prismatic. At the rotary joints the motion is angular and measured in terms of joint angles. At the prismatic joint there is translational motion and is specified in units of length. The angle and length between links is measure by the position sensors attached to the joints.

For a particular set of values of joint angles and distance between the links the end-effector will be at a particular position in space. The forward kinematic analysis is to find the position and orientation of the end-effector of the manipulator with respect to the base frame for the given set of joint parameters.

For describing the position and orientation of the links, we attach coordinate frames to each of them and then the position and orientation of these frames are used for specifying the links. This scheme is known as the “joint space description.” The other form for describing position and orientation can be by directly specifying them with reference to base frame. This representation is called the Cartesian description.

Forward kinematics is like defining the position of manipulator links in a base frame. This is done by sequential transformation of the reference frames attached to the links.

We are provided with the position and orientation of the end-effector of the manipulator, and the exercise is to find the values of joint angles and displacements with which the specified position and orientation of the end-effector can be attained. There can be one or more such set of values and even no such set of parameters for which the specified position and orientation of the end-effector can be attained.

The equations formulated for solving the inverse kinematic problem are nonlinear and it’s very difficult to obtain closed form solutions for that. There may be multiple solutions to the problem, and maybe any solution doesn’t exist at all.

The solutions of the inverse kinematic problem for manipulators are helpful to define the workspace of manipulators. If solutions exist for a point then it is in the workspace of the manipulator, and if no solution exists then the point is not in the workspace.