Page 94 - Dynamic Vision for Perception and Control of Motion

3 Subjects and Subject Classes
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Simple steering maneuvers: Applying a constant steering rate A (considered the
standard lateral control input and representing a good approximation to the behav-
ior of real vehicles) over a period T SR yields the final steering angle and path curva-
ture
a
O O A t ˜ , C ( O A t ˜ )/ a C A t ˜ / ;
0 0 0
(3.11)
O O A T ˜ , C C A T ˜ / . a
f 0 SR f 0 SR
Integrating Equation 3.10 with the top relation 3.11 for C yields the (idealistic!)
change in heading angle for constant speed V
³
d
˜
ǻȤ =(CV )dt V ˜ ³ [C 0 At / ]
t
a
˜
(3.12)
˜
˜
VC T A T 2 /(2 )].
˜
a
[
0 SR SR
The first term on the right-hand side is the heading change due to a constant steer-
ing angle (corresponding to C 0); a constant steering angle for the duration IJ thus
leads to a circular arc of radius 1/C 0 with a heading change of magnitude
' F VC ˜ ˜ . W (3.13a)
C 0
The second term (after the plus sign) in Equation 3.12 describes the contribution of
the ramp-part of the steering angle. For initial curvature C 0 = 0, there follows
2
t
t
' ramp F V ˜ ³ [ / ]dt a 0.5 V A / . (3.13b)
a
˜
˜
A
Turn behavior of road vehicles can be characterized by their minimal turn radius
(R min = 1/C max). For cars with axle distance “a” from 2 to 3.5 m, R may be as low
as 6 m, which according to Figure 3.10 and Equation 3.9 yields Ȝ max around 30°.
This means that the linear approximation for the equation in Figure 3.10 is no
longer valid. Also the bicycle model is only a poor approximation for this case.
The largest radius of all individual wheel tracks stems from the outer front wheel
R fout. For this radius, the relation to the radius of the center of the rear axle R r, the
width of the vehicle track b Tr and the axle distance are given at the lower left of
Figure 3.10. The smallest radius for the rear inner wheel is R r - b Tr/2. For a track
width of a typical car b Tr = 1.6 m, a = 2.6 m, and R fout = 6 m, the rear axle radius
for the bicycle model would be 4.6 m (and thus the wheel tracks would be 3.8 m
for the inner and 5.4 m for the outer rear wheel) while the radius for the inner front
wheel is also 4.6 m (by chance here equal to the center of the rear axle). This gives
a feeling for what to expect from standard cars in sharp turns. Note that there are
four distinct tracks for the wheels when making tight turns, e.g., for avoiding nega-
tive obstacles (ditches). For maneuvering with large steering angles, the linear ap-
proximation of Equation 3.9 for the bicycle model is definitely not sufficient!
Another property of curve steering is also very important and easily measurable
by linear accelerometers mounted on the vehicle body with the sensitive axis in the
direction of the rear axle (y-axis in vehicle coordinates). It measures centrifugal ac-
celerations a y which from mechanics are known to obey the law of physics:
a V 2 / R V ˜ 2 C . (3.14)
y
For a constant steering rate A over time t this yields with Equation 3.11 a con-
stantly changing curvature C, assuming no other effects due to dynamics, time de-
lays, bank angle or soft tires:
a y V 2 ( ˜ 0 O A ˜ ) t a . (3.15)
/