Sensor-based Trajectory Deformation: Application to Reactive Navigation of Nonholonomic Robots - Visual Servoing via Advanced Numerical Methods - page 305

configuration space of the robot, we naturally model a trajectory deformation pro-
cess as a mapping of two real variables s and can

be considered as time (or more generally as an increasing function of time), while s
is the abscissa along each trajectory.
The chapter is organized as follows. Section 17.2 defines trajectory deformation
as an infinite-dimensional dynamic control system the state of which is a trajec-
tory. In Section 17.3, we describe an iterative algorithm controlling the deformation
process to make an optimization criterion decrease. In Section 17.4, the trajectory
deformation algorithm is applied to mobile robot Hilare 2 towing a trailer. In Section
17.5, the method is extended to perform docking for nonholonomic robots.

τ into the configuration space. τ

17.2 Nonholonomic Trajectory Deformation as a Dynamic

Control System

A trajectory for a robotic system is usually represented by a mapping from an inter-
val of into the configuration space of the system. In this section, we introduce the

notion of trajectory deformation as a mapping from an interval of

R

into the set of
trajectories. Equivalently, a trajectory deformation is a mapping from two intervals
into the configuration space as explained later in this section.

R

17.2.1 Admissible Trajectories

<

A nonholonomic system of dimension n is characterized by a set of k n vector

n is the configuration of the system. For
each configuration q , the set admissible velocities of the system is the set of linear
combinations of the X i ( q ). A trajectory q ( s ) is a smooth curve in the configuration
space defined over an interval [0

,..., ∈ C R

fields X 1 ( q ) X k ( q ),where q =

S ]. A trajectory is said to be admissible if and
only if there exists a k-dimensional smooth vector valued mapping u =( u 1

,

,..., u k )

defined over [0 , S ] and such that:

k
i =1 u i ( s ) X i ( q ( s ))

S ] q ( s )=

∀∈ ,

[0 (17.1)

where from now on, denotes the derivative with respect to s .

17.2.2 Admissible Trajectory Deformation

2

We call trajectory deformation a mapping from a subset [0 , S ] × [0 , ∞ ) of R to the

configuration space of the system:

( s , τ ) → q ( s , τ ) .