Optimization and Systems Theory, Royal Institute of
Technology (KTH), Stockholm, Sweden
Johan Hamberg
Dept. of Autonomous Systems, Swedish Defence Research
Agency (FOI), Stockholm, Sweden
ABSTRACT:
It is well known that the sufficient family of time-optimal paths for
both Dubins' as well as Reeds-Shepp's car models consist of the
concatenation of circular arcs with maximum curvature and straight line
segments, all tangentially connected. These time-optimal solutions
suffer from some drawbacks. Their discontinuous curvature profile,
together with the wear and impairment on the control equipment that the
bang-bang solutions induce, calls for ``smoother'' and more supple
reference paths to follow. Avoiding the bang-bang solutions also raises
the robustness with respect to any possible uncertainties.
In this paper, our main tool for generating these ``nearly
time-optimal'', but nevertheless continuous-curvature paths, is to use
the Pontryagin Maximum Principle (PMP) and make an appropriate and
cunning choice of the Lagrangian function. Despite some rewarding
simulation results, this concept turns out to be numerically
divergent at some instances. Upon a more careful investigation, it
can be concluded that the problem at hand is nearly singular. This is
seen by applying the PMP to Dubins' car and studying the corresponding
two point boundary value problem, which turn out to be singular.
Realizing this, we are able to contradict the widespread belief that all
the information about the motion of a mobile platform lies in the
initial values of the auxiliary variables associated with the PMP.