9.1.   INTRODUCTION

A large class of industrial and military control problems consist of planning
and following a trajectory in the presence of noise and uncertainty. Examples
include unmanned airplanes and submarines for surveillance and combat,
mobile robots in factories and on the surface of Mars, and medical robots
performing inspection and manipulation tasks inside the human body under
the control of a surgeon. All of these systems are highly nonlinear and
demand accurate performance.

To control such systems, we make use of the notion of two-degree-of
freedom controller design. This is a standard technique in linear control
theory that separates a controller into a feedforward compensator and a feed-
back compensator. The feedforward compensator generates the nominal input
required to track a given reference trajectory. The feedback compensator
corrects for errors between the desired and actual trajectories. This is shown
schematically in Figure 9.1.

In a nonlinear setting, two-degree-of-freedom controller design decouples
the trajectory generation and asymptotic tracking problems. Given a desired
output trajectory, we first construct a state space trajectory x_d and a nominal
input u_d that satisfy the equations of motion. The error system can then be
written as a time-varying control system in terms of the error, e = x - x_d.
Under the assumption that that tracking error remains small, we can
linearize this time-varying system about e = 0 and stabilize the e = 0 state.
A more detailed description of this approach, including references to some of
the related literature, is given in reference 1.

In optimization-based control, we use the two-degree-of-freedom paradigm
with an optimal control computation for generating the feasible trajectory. In
addition, we take the extra step of updating the generated trajectory based
on the current state of the system. This additional feedback path is denoted
by a dashed line in Figure 9.1 and allows the use of so-called receding horizon
control techniques: a (optimal) feasible trajectory is computed from the
current position to the desired position over a finite-time T horizon, used for
a short period of time δ < T, and then recomputed based on the new
position.