Page 459 - Automotive Engineering Powertrain Chassis System and Vehicle Body

CHAP TER 1 4. 2 Decisional architecture

9
curvature of the xy-curve traced by A. k is the inverse of small-time controllability of a differential system im-
the distance between C and A: k ¼ l 1tanf, where l w plies that the existence of an admissible collision-free
w
is the wheelbase of A. Since f is mechanically limited, path is equivalent to the existence of a collision-free path
jfj f max , the following constraint holds (bounded (Laumond et al., 1998). Using tools from differential
curvature constraint): geometric control theory, it proved possible to show the
small-time controllability of the RS car and therefore to
jkj k max ¼ l 1 tanf max (14.2.37) redemonstrate Laumond’s result (Barraquand and
w
Latombe, 1990).
According to Equation 14.2.36, q is always tangent to In summary, in spite of the presence of nonholonomic
the xy-curve traced by A and its derivative, i.e. the an- constraints, the RS car can reach any configuration within
_
gular velocity u, therefore satisfies q ¼ u ¼ vk. the connected component of the collision-free configu-
Selecting v and u as (coupled) control parameters, the ration space where it is located.
model of A can be described by the following differential Optimal paths for the RS car Once the small-time
system: controllability of the RS car has been established, it is

0 1 0 1 0 1 interesting to find out the shortest path between two
_ x cos q 0 configurations in the absence of obstacles. Reeds and
@ A ¼ @ sin q A v þ @ A u (14.2.38) Shepp (1990) used differential calculus tools to give
_ y
0
q _ 0 1 a first characterization of the shortest paths for the RS
car. Later, Boissonnat et al. (1991), and Sussmann and
with jvj v max ; u ¼ vK; and jkj k max . Because path Tang (1991) used optimal control theory to refine Reeds
planning is generally interested in computing shortest and Shepp’s result.
paths, it is furthermore assumed that jvj¼ 1 (thus the The optimal path for the RS car is made up of line
time and the arc length of a path are the same). The segments and circular arcs of radius 1=k max ; it is the
system (Equation 14.2.38) under the different control shortest among a set of 46 paths that belong to one of the
constraints defines the Reeds and Shepp car. nine following families:
Admissible paths for the RS car Let C denote the
2
þ þ
þ þ
configuration space of the RS car A:ChR ½0; 2pŠ.Let ðiÞ l l l or r r r
P denote a path for A, it is a continuous sequence of ðiiÞðiiiÞ AjAA or AAjA
configurations: PðtÞ¼ðxðtÞ; yðtÞ; qðtÞÞ. An admissible ðivÞ AAjAA
path must satisfy both constraints (Equations 14.2.36 ðvÞ AjAAjA (14.2.40)
and 14.2.37), it is a solution to the differential system ðviÞ AjASAjA
(Equation 14.2.38); it is such that: ðviiÞðviiÞ AjASA or ASAjA
8 Ð t ðixÞ ASA
> xðtÞ¼ xð0Þþ Ð 0 vðsÞcosqðsÞds
<
yðtÞ¼ yð0Þþ t vðsÞsinqðsÞds (14.2.39) where A (resp. S) denotes a circular arc (resp. line seg-
> Ð 0 ment). j denotes a change of direction of motion (a cusp
:
qðtÞ¼ qð0Þþ 0 t uðsÞds point). A may be replaced by r or l to specify a right
(clockwise) or left (counterclockwise) turn. A þ or
with jvðsÞj ¼ 1; uðsÞ¼ vðsÞkðsÞandjkðsÞj k max: superscript indicates a forward or backward motion.
Reachable configuration space for the RS car As Fig. 14.2-30 depicts two examples of optimal paths for
mentioned earlier, the first question raised by the RS car.
nonholonomy is to determine whether the presence of This result enables definition of what Laumond et al.
nonholonomic constraints reduces the set of configu- (1998) call a steering method for the RS car, i.e. an
rations that the RS car can reach. This question was algorithm that computes an admissible path between two
first answered by Laumond (1986). Through an ad hoc configurations in the absence of obstacles. Let Steer RS
geometric reasoning, Laumond established that the RS denote the steering method returning the optimal path
car could reach any configuration within the same for the RS car, i.e. the shortest path among the set de-
connected component of the collision-free configuration fined by Equation 14.2.40.
space. Collision-free path planning for the RS car The com-
In fact, it turns out that this question is directly plete path planning problem for the RS car must consider
related to the controllability of differential systems. The the constraints imposed both by the nonholonomic

9 A differential system is locally controllable if the set of configurations reachable from any configuration q by an admissible path contains
a neighbourhood of q. It is small-time controllable if the set of configurations reachable from q before a given time t contains a neighbourhood of
q for any t.

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