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!

What Can You Do With Uncontrolled Manifold Analysis?

There is generally more than one way to perform a task (the ‘bliss of motor abundance’) and so it’s possible for a movement to incur a little noise that doesn’t actually affect performance that much.
Uncontrolled manifold analysis (UCM) is a technique for analysing a high-dimensional movement data set with respect to the outcome or outcomes that count as successful behaviour in a task. It measures the variability in the data with respect to the outcome and decomposes it into variability that, if unchecked, would lead to an error and variability that still allows a successful movement.

In the analysis, variability that doesn’t stop successful behaviour lives on a manifold. This is the subspace of the values of the performance variable(s) that lead to success. When variability in one movement variables (e.g. a joint angle, or a force output) is offset by a compensation in one or more other variables that keeps you in that subspace, these variables are in a synergy and this means the variability does not have to be actively controlled. This subspace therefore becomes the uncontrolled manifold. Variability that takes you off the manifold takes you into a region of the parameter space that leads to failure, so it needs to be fixed. This is noise that needs control.

With practice, both kinds of variability tend to decrease. You produce particular versions of the movement more reliably (decreasing manifold variance, or V-UCM) and you get better at staying on the manifold (decreasing variance living in the subspace orthogonal to the UCM, or V-ORT). V-UCM decreases less, however (motor abundance) so the ratio between the two changes. Practice therefore makes you better at the movement, and better at allocating your control of the movement to the problematic variability. This helps address the degrees of freedom control problem.

My current interest is figuring out the details of this and related analyses in order to apply it to throwing. For this post, I will therefore review a paper using UCM on throwing and pull out the things I want to be able to do. All and any advice welcome!

Uncontrolled Manifold Analysis

Human movement is hard to study, because there are many ways to perform even simple tasks and given the opportunity, different people will take different routes. It becomes hard to talk sensibly about average performance, or typical performance, or even best performance. 

This fact - that the action system contains more elements than are needed to solve a given task - was first formalised by Bernstein as the degrees of freedom problem. Anything that can change state is a degree of freedom that can contribute to movement stability and if you have more than you need then there is immediately more than one way to perform a task. This means you have to select the best action, and even then there are always variations in the details of how you perform that action (Bernstein called this 'repetition without repetition'). From this perspective, selecting the right action means freezing out redundant degrees of freedom and working with just the ones you need.

A more recent way to think about the problem is as the bliss of motor abundance (Gelfand & Latash, 1998; Latash, 2012; see this recent post too). From this perspective, selecting the right action is about balancing the contributions of all the degrees of freedom so that the overall behaviour of the system produces the required outcome. Nothing is frozen out, but errors incurred by one degree of freedom are compensated for by changes in other degrees of freedom. If (and only if) this compensation happens, then you have a synergy in action. 

This analysis leads to a prediction and an analysis. It predicts that there are two kinds of movement variability - variability that pulls you away from your target state and variability that doesn't. The former is a problem that must be corrected by another element in the synergy compensating. Successful movement requires clamping down on this variability. The latter requires no correction, no control, and successful movements can still happen even if this variability is high. An analysis of movement then follows. You can decompose the variability of movement in the total state space of that movement into that which pulls you away from the target, and that which does not. Successful movement lives on a subspace of the total space of possible values of your degrees of freedom. If the ratio of the 'good' variability to the 'bad' variability is high, you are hanging out close to that subsapce and working to keep yourself there, although not working to keep yourself doing anything in particular. You have a system that is working to compensate for 'bad' variability while ignoring the rest; a synergy defined with respect to the task demands. 

This subspace is referred to as the uncontrolled manifold. It is uncontrolled because when the system is in this subspace of it's total state space, it does not work to correct any variability because that variability is not affecting the outcome. Control only kicks in when you come off the manifold.