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3.4 Behavioral Capabilities for Locomotion 85

magnitude A, but with opposite signs and one with zero output in between. The two
characteristic time durations are T SR for ± A and IJ for the central zero-output.
A·T SR yields the maximum steering angle Ȝ f (fixing the turn radius), with which
a circular arc of duration IJ is driven (see Table 3.2); the total maneuver time T DC
for a change in heading direction then is 2·T SR + W. The total angular change in
heading is the integral of curvature over the arc length and depends on the axle dis-
tance of the car (see Figure 3.10 for the idealized case of infinitely stiff tires).
Proper application of Equation 3.12 yields the (idealized) numerical values.
A special case is the 90° heading change for turning off onto a crossroad. If the
vehicle chosen drives at 27 km/h (V § 7.5 m/s, column 2 in Table 3.2) then T SR =
T 2 is § 5.6 seconds, and the limit of 2 m/s² for lateral acceleration is reached with
í1
ǻȜ f = 6.4° and ǻȤ f § 42.6°. The radius of curvature R is 28.1 m (C = 0.0356 m ,
Equation 3.9); this yields a turn rate C·V (Equation 3.10) of 15.3°/s. Steering back
to straight-ahead driving on the crossroad with the mirrored maneuver for the steer-
ing angle leaves almost no room for a circular arc with radius R f [W = (90 –
2·42.6)/15.3 § 0.3 s]; the total turn–off–duration then is § 11.2 s and the total dis-
tance traveled is about 84 m.
For tight turns on narrow roads, either the allowed lateral acceleration has to be
increased, or lower speed has to be selected. A minimal turn radius of 6 m driven at
V = 7 m/s yields an ideal turn rate V/R of about 67°/s and a (nominal) lateral accel-
eration V²/R of about 0.82 g (~ 8 m/s²); this is realizable only on dry ground with
good homogeneous friction coefficients at all wheels. Slight variations will lead to
slipping motion and uncontrollable behavior. For the selected convenient limit of
maximum lateral acceleration of 2 m/s² with the minimal turn radius possible (6
m), a speed of V § 3.5 m/s (§ 12.5 km/h or 7.9 mph)should be chosen. These ef-
fects have to be kept in mind when planning turns.
The type of control according to Figure 3.13 is often used at higher speeds with
smaller values for A and T SR (W close to 0) for heading corrections after some per-
turbation. Switching the sequence of the sign of A results in a heading change in
the opposite direction.
Lane change maneuvers: Combining two extended pulses of opposite sign with
proper control of magnitude and duration results in a “lane change maneuver” dis-
cussed above and displayed in Figure 3.16.
The numerical values and the temporal extensions of these segments for a lateral
translation of one lane width depend on the speed driven and the maximum lateral
acceleration level acceptable. The behavioral capability of lane changing may thus
be represented symbolically by a name and the parameters specifying this control
output (just a few numbers, as given in the legend of the figure). Together with the
initial and final boundary values of the state variables and maybe some extreme
values in between, this is sufficient for the (abstract) planning and decision level.
Only the processor directly controlling the actuator needs to know the details of
how the maneuver is realized. For very high speeds, maneuver times for the pulses
become very small [see T2–curve (solid) in Figure 3.11]. In these cases, tire stiff-
ness effects play an important role; there will be additional dynamic responses
which interact with vehicle dynamics. This will be discussed in Section 3.4.5.2.