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3.DISCRETE LIMIT CYCLE CONTROL

This chapter describes our basic control approach and its application to the generation of bipedal

walking motions.

Section 3.1 begins by describing the notion of limit cycles, on which our

control formulation is based. Next, Section 3.2 presents the overall control strategy and develops

a discrete system model to be used with a number of user-specified control elements to stabilize

periodic open-loop motions.

Section 3.3 discusses the application of this control system to

bipedal walking. The underlying open-loop control, which serves as the basis for a desired gait,

is discussed in Section 3.4.

Sections 3.5 and 3.6 then go on to describe various possible

observed and controlled variables for walking.

Section 3.7 provides details on the application of

the control elements introduced in earlier sections to the generation of balanced walks.

Finally,

minor variations on the basic control which are of particular use in improving the aesthetics of the

human model's motion are described in Section 3.8. While the basic formulation is applied to

bipedal walking, it is not inherently tied to any particular model or motion and could potentially be

applied to the generation of limit cycles in other types of animated figures.

Some of the terms we

define in this and subsequent chapters have different meanings in the context of control system

theory.

While we make efforts to avoid such conflicts, we have chosen to sometimes give

preference to the colloquial usage of terms. Our relaxed definition of a limit cycle is one example

of such usage.

3. 1

Limit Cycles

One common characteristic of many non-linear dynamical systems is the existence of system-wide

limit cycles.

A limit cycle is a periodic, cyclic trajectory through the state-space of a system.

Strictly speaking, a limit cycle involves the full state of the system.

However, within the context

of this thesis we will use a relaxed definition in which only part of the full system state must cycle