Gauss-Udwadia Control

More than a decade ago, Udwadia (1996) suggested that mechanical constraints in Nature are just feedback control laws and there suggested that Gauss' principle (Gauss, 1829) and its generalizations should be used in order to compute feedback control laws. This can be generalized in a way by representing the control task of mechanical or structural systems in the form

$A(q,\dot{q})\ddot{q} = b(q,\dot{q})$

and that a system with the dynamics equations

$M(q)\ddot{q} = F(q,\dot{q}) + u$

will fullfill the task perfectly while minimizing the squared motor command with respect to a metric, i.e., the cost function

$J(u)=u^T N u.$

is minimized. This yields the general optimal solution

$u = N^{-1/2} (AM^{-1}N^{-1/2})^{+}(b-AM^{-1/2}F),$

which can be applied to any structural in order to create control laws. Udwadia (2003) gives a thorough overview over this approach in a general way outlining most of the occuring difficulties.

Robot Control

We have applied this approach to robotics. While there is a lot of work in progress, let me outline the major points in this context:

Gauss-Udwadia's control laws can be brought over to robotics successfully.
Several major control laws in tracking, force and hybrid control from the literature can be derived from this single common building principle.
It can be applied to the control of non-holonomic, over- and underactuated systems. We are currently having two different projects in progress on these topics.

In the near future, we are moving towards Learning Gauss Control ? laws. These are necessary when the robot system can longer be modeled well.

Downloadable References

Carl Friedrich Gauss (1829), Ueber ein neues algemeines Grundgesetz der Mechanik (Engl.: On a novel, general fundamental law for mechanics), Journal fuer die reine und angewandte Mathematik (Journal of pure and applied mathematics), v. 4, p. 232-235. [German Original] [English Translation by Jan Peters]

Jan Peters, Michael Mistry, Firdaus Udwadia, Stefan Schaal (2005), A new methodology for robot controller design, ASME 5th International Conference on Multibody Systems, Nonlinear Dynamics, and Control (MSNDC). [pdf]

Jun Nakanishi, Rick Cory, Michael Mistry, Jan Peters, Stefan Schaal (2005), Comparative Experiments on Task Space Control with Redundancy Resolution, IEEE/RSJ Conference on Intelligent Robots and Systems (IROS). [pdf] Δ

Jan Peters, Michael Mistry, Firdaus Udwadia, Rick Cory, Jun Nakanishi and Stefan Schaal (2005), A Unifying Framework for the Control of Robotic Systems, IEEE/RSJ Conference on Intelligent Robots and Systems (IROS). [pdf] Δ