Robot path planning problem is formulated in tool-configuration space, and robot motion is controlled at the joint space

For the calculation of the Jacobian, we need to:
- Get the forward kinematics relationship, X= w(θ)
- Differentiate X wtr θ

One of the potential problems with solving for joint space velocity is the non-existence of inverse. The Jacobean may not be invertible for all the values of θ

At certain points in joint space, Jacobian loses its rank; i.e. there is a reduction in the number of independent rows and columns. The points at which the Jacobian loses rank are called Joint spaces singularities.

It is worthy of note that the Jacobian matrix J(q) is of full rank as long as q is not a joint space singularity.

The interior singularity is potentially troublesome. It is formed when two or more axes form a straight line. The effects of rotation about one axis may be cancelled due to a counteracting rotation about the other axis. Tool configuration may remain the same even though the robot moves on joint space

Jacobian can be calculated as the cross product of the joint axis vector and the position vector of the joint to end effector.