# Cartesian Acceleration Task

Probably the most important, if not most prevalent, task is to move a link on the robot from one pose to another. Typically it is the end-effector(s) which are of interest.
These tasks, which are generally expressed as desired positions or orientations, are converted to **acceleration tasks**, through means of task servoing. More details on task servoing are provided in Task Servoing.
Once given a desired operational-space acceleration for a link, , an acceleration task consists in finding the joint-space values which produce ,

(1)

where and are the link Jacobian and its derivative. For the control objective, one simply rewrites the task as an error which must be minimized,

(2)

Using the squared -norm produces a quadratic error term, which defines the objective function to be minimized. The objective function is then rewritten in terms of the optimization variable, ,

(3)

In (3) the term represents a matrix of zeros. Regrouping terms as,

(4)

(5)

allows (3) to be written in the classical least-squares form as,

(6)

The dependencies of and have been removed for brevity.