Nearly time-optimal paths for a ground
vehicle (revised)
David A. Anisi and Xiaoming Hu
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
enhances 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 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. This is thus a counterexample to 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.