IEEE Robotics & Automation Magazine - September 2011 - 39

Results
_ trajectory of three example
Figure 6 shows the (/, /)
experimental validations through the designed capture sets
for the backflip maneuver. As the figure illustrates, the trajectories are contained within the capture sets, and the
switches between the maneuvers are contained within each
of their goal regions. Video of the backflip maneuver being
successfully performed by the STARMAC vehicle can be
viewed at http://hybrid.eecs.berkeley.edu/aerobatics.html.
Synthesis of Robust Motion Control Policies
Another example is the application of reachability tools to the
synthesis of motion control policies that are robust to
bounded disturbances [10]. In this case, we consider a problem where the task specification is to generate a state feedback
policy that steers the robot into a set of desired target configurations R & Rn using a finite set of high-level maneuvers Q
while avoiding a set of unsafe configurations A & Rn , subject
to worst-case bounds on run-time uncertainties (e.g., model
inaccuracies, actuator noise, and environment disturbances).
For practical implementation, the feedback policy selects
maneuvers based on sampled measurements of the system
state obtained at T time units apart.
This can be viewed as another special case of the hybrid
control problem described in the "Reachability: Hybrid
System Reachability and Control" section, where the target
and unsafe sets are the product of the sets R and A in each
mode i 2 Q, while the timing specifications are given by
s0 ¼ s1 ¼ Á Á Á ¼ skÀ1 ¼ T. Under the assumption that a
switch can be taken from mode i to any other mode j 2
SQ, the
computation of Pre(S) in each mode is simplified to i2Q Si ,
where Si is the component of S in mode i. Thus, the reach-
avoid set over a time interval ½0, kT is identical for each mode
of the system and can be obtained using Algorithm 2.

It should be noted that the various uncertainties are
accounted for in the computation of the set RAi (Sj , A, T),
which is carried out according to the system dynamics (1),
subject to bounds on the disturbance parameter.
As described in the "Reachability: Hybrid System
Reachability and Control" section, the result of this reachability calculation also gives an explicit representation of
the feedback policy for choosing the switching controls.
Specifically, given a state measurement x(kT), the minimum number of time
steps needed to reach R *
while avoiding A can be
Reachable sets were used
computed by iterating
through the reach-avoid
to guarantee that the
sets and finding the smallest set Sj containing x(kT).
quadrotor would be able to
By definition of Sj , there
exists some maneuver i
safely complete the backflip
such that x(kT) 2 RAi
(SjÀ1 , A, T). Then, chooseven under worst-case
ing of i guarantees that
the state trajectory will
disturbances.
enter SjÀ1 in one sampling interval. In repeat- *
ing this procedure, the
system state is steered through successively smaller reach-
avoid sets, until the target set is attained.
For conceptual illustration, this approach is applied to
the problem of controlling STARMAC to some neighborhood of the origin in the two-dimensional (2-D) plane,
while satisfying hard velocity bounds, and subject to model
uncertainties and motor noise. Under a previously designed inner control loop, the position-velocity dynamics

2
0

R(R 1, τ1)

-2
⋅
φ

Figure 6. (Calculating these sets using the Level Set Toolbox
on a standard laptop computer, e.g., 2-GHz dual-core CPU
and 4-GB RAM, took at most a few minutes of computation
time.) In Figure 6, R(R3 , s3 ) is the set in the recovery mode,
R(R2 , s2 ) is the set in the drift mode, and R(R1 , s1 ) is the
capture set in the impulse mode. Using the procedures
described in the "Reachability: Hybrid System Reachability
and Control" section, a provably safe switching controller
can be designed for the backflip maneuver.

-4

R2
R(R 3, τ3)

R(R 2, τ2)
R1

-6

*
Algorithm 2. Computation of exact finite
horizon reach-avoid set.

-8
0

Require: R, A & Rn , T > 0

3
φ

1: S0 ( R
2: for j ¼ 0 to k À 1 do
S
3: Sjþ1 ( i2Q RAi (Sj , A, T) [ Sj
4: end for
5: return Sk .

Figure 6. Three experimental validations (solid, dash, and dashdot lines) of the backflip maneuver overlaid on the composite
reach sets plotted in the rotational state space. The maneuver
begins in RðR1 ; s1 Þ and ends in R3 . The transitions from the
impulse to drift mode are shown as black diamonds, and the
transitions from the drift to the recovery mode are indicated by
the black squares.

SEPTEMBER 2011

IEEE ROBOTICS & AUTOMATION MAGAZINE

http://hybrid.eecs.berkeley.edu

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - September 2011