Torque Task

Finally, it may also be desirable to specify torque tasks for purposes of regularization, among other possibilities. As with wrench tasks, torque tasks, can be written simply as,

(1) $\torque = \torque^{\text{des}} \tp$

To formulate the control objective function, $\ftor$ , the square norm of the task error is written,

(2) $\ftor = \left\| \torque - \torque^{\text{des}} \right\|_{2}^{2} \tc$

which can be reformulated in terms of $\optvar$ as,

(3) $\ftor = \left\| \bmat{\0 & S^{\top} & \0 } \optvar - \torque^{\text{des}} \right\|_{2}^{2} \tp$

Once again regrouping terms,

(4) $\Etor &= \bmat{\0 & S^{\top} & \0 }$

(5) $\fvector &= \torque^{\text{des}} \tc$

the least-squares form of the torque task is written,

(6) $\ftor = \left\| \Etor \optvar - \fvector \right\|_{2}^{2} \tp$