User:Tt37129/Uncontrolled manifold

The Uncontrolled Manifold Hypothesis is a concept relating to the degrees of freedom problem in motor coordination. The degree of freedom problem arises when a controller- for example, the nervous system of an organism- has more degrees of freedom available for control than are required to perform a task. This leads to a multitude of solutions to a task, from which the controller must "choose" which is best. The uncontrolled manifold (UCM) hypothesis states that within this high dimensional space, a low dimensional space (manifold) of task variables can be identified which are relevant to performance of the task. Motion orthogonal to this manifold is left uncontrolled.

=Uncontrolled Manifold Hypothesis=

The uncontrolled manifold hypothesis states that in the higher dimensional space of the degrees of freedom available to the controller, there is a lower dimensional space of task variables which do not affect task performance. For example, in a sit to stand task, while there are degrees of freedom associated with every joint in the body, the subspace of joint movement which affects center of mass motion (controlled manifold) is more tightly controlled than the subspace of movement which affects hand motion (uncontrolled manifold) since center of mass motion is more relevant to the task at hand.

Example Task
As an example, consider a three link arm tasked with drawing a circle. The three joints- shoulder, elbow and wrist- represent three degrees of freedom about which rotation can occur. However, since the circle is two dimensional, the task is under-constrained. The equations $ x = l_1 cos(\theta_1) + l_2 cos(\theta_2) + l_3 cos(\theta_3) $ and $ y = l_1 sin(\theta_1) + l_2 sin(\theta_2) + l_3 sin(\theta_3) $  describe the end point of the arm $ (x,y) $  which are the 2 dimensional task variables (variables which must be controlled to perform a task)  in terms of the three dimensional state variables (degrees of freedom directly available to the controller) $ \theta_1 $, $ \theta_2 $ , and $ \theta_3 $  which are the angle each arm segment makes with the horizontal axis. Since only x and y need to be strictly controlled, any combination of changes in $ \theta_1 $, $ \theta_2 $ , and $ \theta_3 $ which leave $ (x,y) $  unchanged does not need to be controlled for. This forms the uncontrolled manifold.

Identifying the Uncontrolled Manifold
The uncontrolled manifold can be identified by variance along different axes in experimental data. For example, imagine a task involving applying a constant force with two fingers, F1 and F2. In the figure to the right, the dashed line represents the sum of forces F1 and F2. Variation along this axis will not affect task performance while variation perpendicular to it will. The uncontrolled manifold hypothesis would predict experimental data, represented by the ellipse, would show greater variation along the dashed line and smaller variation along the orthogonal axis.

The uncontrolled manifold can also be solved for directly by experimentally measuring the task variables and state variables for a given task. In experiments involving gripping or pressing with multiple or single fingers, Electromyography (EMG) data indicates force output from skeletal muscles which form the high dimensional state variables. Force sensors in the gripped object record the lower dimensional task variables. From this, the equation $F = A \cdot e$ can be written in which $F$  is the vector of the $n$  dimensional task variables and $e$  is the vector of the $m$  dimensional state variables. $A$ is then the $n \times m$  matrix which transforms the two vectors. Solving for $A$ can yield the UCM. This requires knowing what the state and task variables are and measuring them.

In a complicated task, the task and state variables may not be fully known. If assumptions can be made about what the task variables are, an uncontrolled manifold analysis can be used to test if those assumptions are valid. In a sit-to-stand task, one might assume some task variables could be center of mass control and hand control. The state variables can be approximated by the major body joints. A matrix $J$, called the Jacobian, can then be calculated, the elements of which indicate how task variables change as state variables change. By solving the equation $J \times \hat{e}=0$, the uncontrolled manifold can be identified by the vectors $\hat{e}$.