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This definition is one of convinience since it allows periodic locomotion such as

periodically.

walking or running to be discussed in terms limit cycles.

(a)

(b)

Figure 3.1 - Passive Limit Cycle Stability
(a) Passively stable
(b) Passively unstable

Limit cycles may be stableor unstable. A stable limit cycle is one in which slight perturbations to

the state-space trajectory are driven back into the limit cycle as indicated in Figure 3.1(a).

An

unstable limit cycle is one in which slight perturbations to the trajectory result in the system

deviating further from the limit cycle as shown in Figure 3.1 (b). We will call limit cycles that do

not require explicit control forces to maintain them passivelimit cycles.

Note that this definition

does not preclude a system with active components (motors etc.) from exhibiting passive limit

cycles. A motorized or windup toy is an example of such a system.

We wish to attain similar stable limit cycles with passively unstable bipedal systems by applying

suitable control forces to periodically drive the system back into an activelimit cycle.

We define

an activelimit cycle as one that requires corrective control forces to be applied to the system for

the explicit purpose of maintaining the cyclic trajectory. Figure 3.2 illustrates this idea.