IEEE Robotics & Automation Magazine - September 2011 - 48

If Algorithm 1 terminates, the fixed point is equal to the
^ ^q ) (see [38] for details). We next
^ ^q , . . . , C
tuple of sets (C
1
M
show how to calculate the steps of this algorithm for the
hybrid automaton of Figure 2.
Application Scenario
^ such that
Referring to Figure 2, we have the system H
^
Q ¼ f^q1 , ^q2 , ^q3 g with ^q1 ¼ fA, Bg, ^q2 ¼ fAg, and
^q3 ¼ fBg. As a consequence, Algorithm 1 leads to
2

3
Pre(^q1 , S2 [ S3 [ Bad)
5
Pre(^q2 , Bad)
G(S) ¼ 4
Pre(^q3 , Bad)

control map p^(^qi ,x) that maintains the state x outside
^ ^q for the application, is given by
Pre(^qi ,Bad), which is equal to C
i
8
>
>
<

if x 2 Pre(^qi , Bad)uL \ @Pre(^qi , Bad)uH
uH
uL
if x 2 Pre(^qi , Bad)uH \ @Pre(^qi , Bad)uL
:
fuH , uL g if x 2 @Pre(^qi , Bad)uH \ @Pre(^qi , Bad)uL
>
>
:
U
otherwise:
Since we have that Pre(^qi , Bad)  Pre(^q1 , Bad) for
i 2 f2, 3g, when the mode switches from ^q1 to ^q2 or from
^q1 to ^q3 , the continuous state x being outside Pre(^q1 , Bad)
implies that it is also outside Pre(^q2 , Bad) and Pre(^q3 , Bad).
Therefore, the above feedback map guarantees that the
state never enters the capture set.

so that
2

3
Pre(^q1 , Bad)
S1 ¼ 4 Pre(^q2 , Bad) 5
Pre(^q3 , Bad)
and
2

3
Pre(^q1 , Pre(^q2 , Bad) [ Pre(^q3 , Bad) [ Bad)
5:
Pre(^q2 , Bad)
S2 ¼ 4
Pre(^q3 , Bad)
The first component of this expression means that, when
the system starts in mode ^q1 , the trajectory can enter Bad
by flowing in ^q1 or by first transitioning to ^q2 or ^q3 and
then by flowing in either of these modes. By the properties of the Pre operator (refer to [37] and [38]), since
^q2 , ^q3  ^q1 , it can be shown that Pre(^q1 , Pre(^q2 , Bad) [
Pre(^q3 , Bad) [ Bad) ¼ Pre(^q1 , Bad) so that Algorithm
1 terminates at the second step. Therefore, we have
^ ^q ¼ Pre(^q2 , Bad), and C
^ ^q ¼
^ ^q ¼ Pre(^q1 , Bad), C
that C
1
2
3
Pre(^q3 , Bad).
Computational Tools
The sets Pre(^q, Bad) can be computed by linear complexity
algorithms. This is because for every mode estimate ^q the
continuous dynamics is the parallel composition of two
order-preserving systems, and the bad set is convex [13],
[20]. Specifically, for the application example, define the
restricted Pre operators for i 2 f1, 2, 3g Pre(^qi ,Bad)uL :¼
fx 2 X j9d, t ! 0 s:t: some /^x (t,(^qi , x), uL , d,) 2 Badg and
Pre(^qi , Bad)uH :¼ fx 2 X j 9 d, t ! 0 s:t: some /^x (t, (^qi , x),
uH , d, ) 2 Badg. Then, we have that (refer to [20])
Pre(^qi ,Bad) ¼ Pre(^qi ,Bad)uL \ Pre(^qi ,Bad)uH fori 2 f1,2, 3g.
Each of the sets Pre(^qi ,Bad)uL and Pre(^qi ,Bad)uH can be
computed by linear complexity discrete time algorithms
(see the "Experimental Setup" section).
For each mode ^qi for i 2 f1, 2, 3g, a safe control map
p^(^qi , x) acts in such a way to maintain the state outside the cur^ ^q . This results in a map
rent mode-dependent capture set C
^ ^q when x
p^(^qi , x) that makes the vector field point outside set C
i
^
is on the boundary of C^qi . One can show (refer to [20]) that a
48

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IEEE ROBOTICS & AUTOMATION MAGAZINE

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SEPTEMBER 2011

Experimental Setup
The two-vehicle conflict scenario of Figure 1 was
implemented in an in-scale multivehicle lab. The laboratory is equipped with an overhead camera-based positioning system, a control station, a human-driver interface, the
roundabout system, and six scaled vehicles (https://wikis.mit.edu/confluence/display/DelVecchioLab).
A car chassis (length 0.375 m, width 0.185 m, and
wheelbase 0.257 m) is used as the hardware platform for
the scaled vehicle. The vehicles are equipped with an
onboard computer (Mini ITX) and a motion controller.
The longitudinal dynamics is dynamically similar to that
of a high-mobility multipurpose wheeled vehicle (HMMWV)
[40]. One of the scaled vehicles is configured to be an
autonomous vehicle that can follow a predefined path
and control its throttle/brake input while another acts as
a human-driven vehicle that can be driven by a human
driver using a human-driver interface. The human-
driver interface comprises a steering wheel and two pedals for throttle and brake commands (see Figure 3). The
hardware used is a Logitech MOMO force feedback racing wheel and pedal set. The hardware is connected to the
control station via a Universal Serial Bus (USB) cable,
and the input command from the hardware is transmitted
to the vehicle via the wireless connection.
Figure 3 shows the roundabout system. There are
two circular paths that share a common section on a
6 m 3 6 m arena. The human-driven vehicle follows
the outer path while the autonomous vehicle follows the
inner path. Both vehicles travel in an anticlockwise
direction. A collision is possible at the intersection
when both vehicles are in the area shaded red (Figure 3)
at the same time. This area corresponds to the set
f(p1 , p2 ) j (p1 , p2 ) 2 ½L1 , U1  3 ½L2 , U2 g. The maximum
vehicle speed is 1,100 mm/s, and the minimum speed is
350 mm/s. A software module on all the vehicles maintains
the speed between the specified bounds. When the two
vehicles are simultaneously present in the shared path
(between points Pt1 and Pt2 ), another software module prevents rear-end collision by appropriately accelerating or
decelerating the autonomous vehicle when the two vehicles
https://wikis http://www.mit.edu/confluence/display/DelVecchioLab
Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - September 2011
