Optimization and Systems Theory, Royal Institute of
Technology (KTH), Stockholm, Sweden
ABSTRACT:
This report investigates how Optimal Control and above all, the
Pontryagin Maximum Principle (PMP), can be used to find algorithms for
steering a mobile platform between two pre-specified points in the state
space.
First the problem of finding time-optimal paths for Dubins' as well as
Reeds-Shepp's car models is considered. They turn out to be the
concatenation of circular arcs of maximum curvature and straight line
segments, all tangentially connected (i.e. essentially bang-bang
solutions). For each of these car models, a sufficient family of paths
$\mathcal{F}$, is presented. In addition, an algorithm for synthesizing
the optimal path, i.e. to pinpoint the global optimum inside
$\mathcal{F}$, is provided. The nature of the presented algorithm is
geometrical, which makes it highly suitable for numerical computations.
We then consider the problem of generating smoother, more flexible and
pliable paths. This helps us to reduce the impairment of the steering
device, but also raises the robustness with respect to any possible
uncertainties. Despite some rewarding simulation results, the presented
concept turns out to suffer from severe numerical instability
properties. Upon a more careful investigation, it turns out that the
problem at hand is singular. Finally, in an effort to reduce the
numerical difficulties, an alternative approach, viz. the Method of
Perturbation is adopted. Taking the synthesized time-optimal paths as
the starting point, the idea is to study how a small change in the
design parameter influences the generated paths. Unfortunately, this
approach turns out to be numerically divergent as well.