Finding the position and orientation of the links of a manipulator required for attaining a certain position and orientation of the end-effector is just not sufficient. We also want to know the corresponding variation of velocity of the links and joints with the motion of the end-effector.

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### Velocity Analysis

In addition to finding the end-effector position for given joint parameters and vice versa in forward and inverse kinematics, the kinematics of manipulators also includes the analysis of manipulators in motion. Not only the final position of the links and joints to attain the desired position of the end-effector, but also the velocity, and its variation, of the links and joints of the manipulators while attaining the final position is important for analysis.

We discussed joint space and Cartesian space in the previous article. To find the final position of an end-effector for given joint parameters is like moving from joint space to Cartesian space, and obtaining joint parameters for a given final position of the end-effector is like expressing the Cartesian space in joint space.

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### The Jacobian Matrix

For the velocity analysis of the motion of manipulators a matrix is defined which represents the mapping of velocities from joint space to Cartesian space. Such matrix is called the jacobian of the manipulator. The Jacobian matrix is characteristic to a particular configuration of the manipulator. As the configuration of the manipulator changes while in motion, the mapping of velocities changes accordingly.

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### Singularities

During the motion of the manipulator, the links of the manipulator acquire different configurations. With these changing configurations, the description of the Jacobian changes accordingly. While moving from one configuration to another, sometimes the manipulator reaches a point beyond which it cannot go. These points describe the workspace of the manipulator. And there are some points which cannot be attained by the manipulator due to structural constraints.

At all such points the Jacobian of the manipulator is not invertible and these points are called Singularities of the Jacobian. Singularities of the manipulator can tell what the limits of workspace of the manipulator are. The points of singularities should be taken into consideration while designing and programming a manipulator for any specific task. The knowledge and understanding of singularities is very helpful for designers and users of manipulators.

The Jacobian of a manipulator is also used for mapping of forces and torques. When the force and moment at the end-effector are given and the set of joint torques required to generate that are to be found then also the Jacobian of the manipulator is very useful.

The previous two articles have presented a brief overview of the kinematics of manipulators. The kinematics of manipulators contains position, orientation, and velocity analysis of manipulators. Now we move on to Dynamics, the study of forces causing motion.