IEEE Robotics & Automation Magazine - September 2011 - 52

Mode Estimator Implementation
We use a discrete time form of the estimator proposed in the
"Problem Solution" section. Since the driver decides to switch
the mode to brake or accelerate once the human-driven vehicle crosses DP , the mode estimator running on the autonomous vehicle uses the
* continuous state measureThe human-driver interface ments of the humandriven vehicle after it
P
comprises a steering wheel crosses D . The instance
n ¼ 0 corresponds to the
time step when the humanand two pedals for throttle
driven vehicle crosses this
decision point. We take
and brake commands.
N ¼ 20 and consider
* n > N: At the nth time
step after the humandriven vehicle crosses the human-decision point, the estimate
^ ¼ (1=n À 1)Rn a(k).
is calculated by using the formula: b(n)
k¼2
Hence, n time steps after the human-driven vehicle crosses
the decision point, y(n) is given by y(n) ¼ yA if
^ À b j > c d,
 y(n) ¼ yB if jb(n)
 and
^ À b j > c d,
jb(n)
B
B
A
A
y(n) ¼  otherwise.
Control Map Implementation
We introduce the following discretization of system H
given in (1) and (2) (employing forward Euler approximation) with step size dT > 0, i 2 f1, 2g, and index
j: pi ½j þ 1 ¼ pi ½j þ F1i (vi ½j, ai ½j) and vi ½j þ 1 ¼ F i (vi ½j,
ai ½j), where F1i ¼ dT vi ½j, F i (vi ½j, ai ½j) ¼ vi ½j þ dTc
(vi ½j, ai ½j), c(vi , ai ) :¼ ai if vi þ ai dT < vmax and vi þ
ai dT > vmin , c(vi , ai ) :¼ (vmax À vi )=dT if vi þ ai dT >
vmax , and c(vi , ai ) :¼ (vmin À vi )=dT if vi þ ai dT < vmin .
We define the notation for a sequence of constant
inputs ai for i 2 f1, 2g: F i, 0 (vi , ai ) :¼ vi and F i, kþ1
(vi , ai ) :¼ F i (F i, k (vi , ai ), ai ) with k 2 N. The value of
pi ½k starting from initial conditions (pi , vi ) can be calcuP
i  i, j
lated as pi ½k ¼ pi þ kÀ1
j¼0 F1 (F (vi , ai ), ai ): Since Bad ¼
½L1 ,U1  3 R 3 ½L2 ,U2  3 R, define for i 2 f1;2g the sequenP
i  i,j
k
ces Lk1 (v1 ,a1 ):¼L1 À kÀ1
1 ,a1 ):¼
j¼0 F1 (F (v1 ,a1 ),a1 ), U1 (vP
PkÀ1 i i,j
kÀ1 i
k
U1 À j¼0 F1 (F (v1 ,a1 ),a1 ), L2 (v2 ,max(a2 )):¼L2 À j¼0
F
PkÀ1 1i
i,j
k
(F (v2 ,max(a2 )),max(a2 )), U2 (v2 ,min(a2 )):¼U2 À j¼0 F1
i,j
(F (v2 ,min(a2 )),min(a2 )), where max(a2 )¼ bq þcq d and
min(a2 )¼bq Àcq d when ^q ¼q, while max(a2 )¼bA þcA d
and min(a2 )¼bB ÀcB d when ^q¼fA,Bg. Then, one can
È
show that Pre(^q, Bad)u ¼ x 2X j9k!0 s:t: Lk1 (v1 , a1 )p1 , since the sequences fLk1 gk!0 ,fU1k gk!0 ,
fLk2 gk!0 , and fU2k gk!0 are strictly decreasing [20]. Thus,
we only need to make a finite number of computations.
To implement the feedback map p^(^q, x) of the "Problem
Solution: Computational Tools" section, we need to track
when the continuous flow hits the boundary of the relevant
set Pre(.,.). In discrete time, we consider the continuous
state to be on the boundary of Pre(.,.) when it is outside it
while its prediction forward in time is inside it. To make
this procedure robust to both communication and actuator
delays, we consider ten forward predictions in time instead
of only one.
Experimental Results
The cumulative time for which the trials were conducted is
3,479 s, resulting in a total of 97 instances of collision
avoidance in which the autonomous vehicle applied control to avoid a collision. In doing so, the autonomous vehicle entered the capture set in three such instances and
resulted in a collision in one such instance, resulting in an
overall success rate of 96.9%. During the total duration of
the experiments, the mode was estimated as A (acceleration) 102 times, as B (braking) 45 times, and remained at
fA, Bg (acceleration or braking) nine times. These results
are presented in Table 1. All mode estimations are correct.
Figure 4 shows a collision-avoidance instance when the
human-driven vehicle mode was identified as A.
Discussion and Conclusions
In this article, we have illustrated the application of a formal
hybrid control approach to design semiautonomous multivehicle systems that are guaranteed to be safe. Our experimental results illustrate that, in a structured task, such as
driving, simple human-decision models can be effectively
learned and employed in a feedback control system that
enforces a safety specification. They also highlight how the
incorporation of these models in a safety control system
makes the control actions required for safety less conservative. In fact, by virtue of the mode estimate, the current
(mode-dependent) capture set to avoid guaranteeing safety
is considerably smaller than the capture set to be avoided
when the mode estimate is not available. This is essential for
the practical applicability of cooperative active safety systems. In our data set, the flow entered the capture set only
3% times. These failures are mainly due to communication
delays between the vehicles and the workstation. These
delays, when significant, cause the calculated capture set to
be different from the actual one and hence may cause to
enforce control too late. These delays, in future work,
should be formally accounted for in the models and in the
safety control algorithm.
More complex models of human decisions in the
proximity of an intersection and the incorporation of additional details, such as weather conditions and road geometry, offer the potential for reducing the conservatism of safe
control actions even further. Future work will also consider

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