solutions. We also added a control algorithm that is
perfectly integrated with the planning method.
The result is a Polynomial Curvature Sliding
control, PC-Sliding, a novel RT procedure for
planning and control that can be summarised as
follows. The steering commands are designed by
means of the polynomial curvature model applying a
two-point boundary value problem driven by the
differential posture (pose plus curvature). While
following the path the vehicle replans iteratively the
path with a repetition rate that must not necessarily
be deterministic. To the actual curvilinear coordinate
it is added a piece forward, than computed the
corresponding posture in the original planned path,
finally replanned the differential path steering the
vehicle from the actual posture to the one just
computed. The result is to force the vehicle to
correct for deviations while sliding over the desired
path. Those little pieces of correcting path have the
property of fast convergence, thanks also to an
optimised mathematical formulation, allowing a
Real Time implementation of the strategy.
Advantages of the proposed method are its
essentiality thanks to the use of the same strategy
both for planning and control. Controlling vehicles
in curvature assures compatibility with respect to
kinematics and partially to dynamics if the
maximum rate of curvature variation is taken into
consideration. The method doesn’t need chained
form transformations and therefore is suitable also
for systems that cannot be transformable like for
example non-zero hinged trailers vehicles (Lucibello
2001). Controls are searched over a set of admissible
trajectories resulting in corrections that are
compatible with kinematics and dynamics, thus
more robustness and accuracy in path following.
Resulting trajectories are convenient to manipulate
and execute in vehicle controllers and they can be
computed with a straightforward numerical
procedure in real-time. Disadvantages could be the
low degrees of freedom to plan obstacle-free path
(Baglivo 2005), but the method can readily be
integrated with Reactive Simulation methods (De
Cecco 2007), or the degree of the polynomial
representing curvature can be increased to cope
with those situations (Kelly 2002, Howard 2006).
Parametric trajectory representations limit
computation because they reduce the search space
for solutions but this at the cost of potentially
introducing suboptimality. The advantage is to
convert the optimal control formulation into an
equivalent nonlinear programming problem faster in
terms of computational load. Obviously the dynamic
model is not explicitly considered thus missing
effectiveness in terms of optimisation (Biral 2001).
By means of this iterative planning sliding
control strategy we obtain a time-varying control in
feedback which produces convergence to the desired
path, guaranteeing at the same time robustness. In
particular, small non-persistent perturbations are
rejected, while ultimate boundedness is achieved in
the presence of persistent perturbations.
Verification of stability convergence and
robustness can be achieved analytically or
statistically. The first way has the merit to synthesise
the results in a general and compact fashion. By
means of its analytic representation it is mostly easy
to isolate the influence parameters and quantify its
effect. The second way has the merit to cope easily
with complex models where interact different
effects. In the present paper we aim at verifying the
performances of the proposed method that embody a
planner, a controller and a sensor fusion strategy for
the vehicle pose estimation that takes into account
an iterative measurement system and an
environment referred one (De Cecco 2003, De
Cecco 2007). The fusion technique takes into
account also systematic effects. This last part
interacts with the control strategy injecting step
inputs of different entity at each fusion step. For the
above reasons we decided to take the second way of
verification employing a Monte Carlo method.
Generally research focuses on control or
measurement goal separately. Seldom the deep
interaction between them is taken into consideration.
In this work the whole system robustness is
investigated by simulation.
2 PATH PLANNING AND
CONTROL PROBLEMS
The main problem with Wheeled Mobile Robots
path planning and control tasks is well known and
it’s strictly related to the nonholonomic constraints
on velocity. These constraints limit possible
instantaneous movements of the robot and cause the
number of controlled states to be less than the
control inputs one. WMR state equations constitute
a non linear differential system that cannot be solved
in closed form in order to find the control inputs that
steer the robot from an initial to a goal posture
(position, heading and curvature). As a consequence
also the control task is non standard with respect to
the case of holonomic systems like the most part of
manipulators. A suitable control law for precise and
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