Appendix 16.2: Derivation of the Least Squares Solution for Control Moves

The objective function is

Expanding the objective function, we find

Unconstrained System

The minimization with respect to the control move vector, when there are no constraints, is

and solving for Du_f we find

Equation 16.23

Constrained System

The objective function for a constrained system is no different than the unconstrained

but this is often written in the following form

and since

the objective function in the form of most quadratic programs is

and

The constraints

are shown in Section 16.4.