The target points for the end effectors are specified as per the task in world coordinate frame. But to make the end effectors move to the specific points the actuators attached to each joints have to be provided with the values of their respective movements producing the final effect as desired. This means that the control variables are the values provided to the actuators, hence, the coordinates of the target points have to be converted to the control coordinates.
The global coordinates and the control coordinates are represented in the matrix form. Control Coordinates are obtained from Global Coordinates through various transformations of the matrices. For a particular transformation of a matrix it is multiplied with a transformation matrix to obtain the required transformation. All these calculations are incorporated in a computer program and implemented through micro controllers and processors such that it takes the global coordinates and supply the actuators with the required values of the control coordinates.