Optimal Feedback Control and Its Relation to Uncontrolled Manifold Analysis

Motor control theories must propose solutions to the degrees of freedom problem, which is the fact that the movement system has more ways to move than are ever required to perform a given task. This creates a problem for action selection (which of the many ways to do something do you choose?) and a problem for action control (how do you create stable, repeatable movements using such a high dimensional system?).

Different theories have different hypotheses about what the system explicitly controls or works to achieve, and what is left to emerge (i.e. happen reliably without explicitly being specified in the control architecture). They are typically about controlling trajectory features such as jerk. Are you working to make movements smooth, or does smoothness pop out as a side effect of controlling something else? This approach solves the degrees of freedom control problem by simply requiring the system to implement a specific trajectory that satisfies some constraint on that feature you are controlling (e.g. by minimising jerk; Flash & Hogan, 1985). They internally replace the solutions afforded by the environment with one desired trajectory. 

Todorov and Jordan (2002a, 2002b; thanks to Andrew Pruszynski for the tip!) propose that the system is not optimising performance, but the control architecture. This is kind of a cool way to frame the problem and it leads them to an analysis that is very similar in spirit to uncontrolled manifold analysis (UCM) and to the framework of motor abundance. In these papers, they apply the mathematics of stochastic optimal feedback control theory and propose that working to produce optimal control strategies is a general principle of motor control from which many common phenomena naturally emerge. They contrast this account (both theoretically and in simulations) to the more typical 'trajectory planning' models.

The reason this ends up in UCM territory is that it turns out, whenever it's possible, the optimal control strategy for solving motor coordination problems is a feedback control system in which control is deployed only as required. Specifically, you only work to control task-relevant variability, noise which is dragging you away from performing the task successfully. The net result is the UCM patterns; task-relevant variability (V-ORT) is clamped down by a feedback control process and task-irrelevant variability (V-UCM) is left alone. The solution to the degrees of freedom control problem is to simply deploy control strategically with respect to the task; no degrees of freedom must be 'frozen out' and the variability can be recruited at any point in the process if it suddenly becomes useful - you can be flexible.

What follows is me working through this paper and trying to figure out how exactly this relates to the conceptually similar UCM. If anyone knows the maths of these methods and can help with this, I would appreciate it!