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The continuous dynamic system which we eventually wish to control can be modelled by the

following system state equation:

x(t+ Dt)=V(x(t),U).

(3.1)

where xis the system state and Uis a set of applied control forces defined over [t, t+[!]t].

Vis a highly complex function which involves the integration of the forward dynamics of the

animated model over time and includes the effect of both internal and external applied forces such

as gravity, ground collisions, and muscular control forces.

Instead of working directly with this

complex continuous system, we assume that a strictly cyclic motion is desired and discretize Eq.

3.1 into individual motion cycles to obtain

x_i+1=g(x_i,U_i).

(3.2)

Here, the subscript idenotes the cycle number.

U_iis the set of time-varying control forces

applied over the ith cycle. The function gis a special case of Vin which the sample times are not

necessarily regular¹, depending on the definition of a motion cycle.

For example, the end of a

motion cycle could be defined as the time of a particular transition in a state machine.

We further

assume that a user-supplied open loop controller, U^nom, produces a near-cyclic motion when

applied to the system being controlled.

To drive the final motion into a cycle, additional control

forces are required. We denote these forces as DU_i^*, which are the control perturbations required

to drive each cycle of the nominal motion, toward a limit cycle. The discrete system then becomes

x_i+1=g₍x_i,U^nom+ DU_i^*₎.

(3.3)

where DU_i^*are still to be determined. Figure 3.3 illustrates this discrete dynamical system.

1_{Note that strictly speaking, g is a different function for each cycle since the size of the interval over which it is}

defined may vary from one cycle to the next.