25

Figure 3.2 - An Active Limit Cycle

3. 2

Control Formulation

The problem of choosing appropriate control perturbations to drive the entire stateof a system to a

desired value is a difficult one.

Assuming a solution does exist, the number of parameters to be

determined is large for all but very simple systems (a few DOFs or less).

Non-linearities in a

system mean that over the course of a full cycle even small perturbations of certain state variables

can cause large changes in final state and/or result in almost no cycle at all.

For example, a small

change in the roll angle of the ankle in a walk might cause the next foot to miss the ground

completely.

The essence of our control approach is to begin with a passively unstable system, discretize it into

individual cycles and stabilize each cycle in turn.

Each cycle is stabilized by applying control

perturbations which drive its final state to a suitable state from which to begin the next cycle.

The

motivation for using a discrete version of the system is that the discrete dynamics are much

simpler to model than the continuous system and therefore, simpler to control, as we shall see

shortly.