IEEE Robotics & Automation Magazine - September 2011 - 58
used in the framework. Given the
decomposition of the workspace, the
elements of the decomposition correHigh-Level Search Layer
Low-Level Search Layer
spond to the states of the abstraction.
Specification
Automaton Aφ
LTL Formula φ
Monitor
The transition relations between abDiscrete
stract states are determined using the
Robot Model Η
Abstraction Μ
adjacency information among the eleand
ments of decomposition. Every system
LTL Semantics
state is mapped to an abstract state by
first projecting it onto the workspace
Discrete Search over
and then using the workspace decomExploration Information
Aφ × Μ
Sampling-Based
position to map it to the set D. Further
Tree Search
High-Level Guide
details on the construction of the
discrete abstraction are presented in
Solution
Synergy Layer
the "Construction and Exploration of
Discrete Abstraction" section.
Figure 3. Synergistic multilayered approach for motion planning with LTL specifications.
Given a specification / defined
over a set of proposition P, the
formulas that contain only the Next (X), Until (U), and high-level layer searches the product A/ 3 M for promisFinally (F) operators, when written in positive normal ing high-level guides. To construct such guides, ideas and
form (i.e., the negation operator : occurs only in front of tools from model checking and graph search algorithms
atomic propositions; see [64] for more details). As an are used [31]. The idea of using the product A/ 3 M for
example, the specification described in Figure 1, "In the discrete planning has also been used before [22], [25],
future, visit the region where p1 is true, and then visit a [63]. An important difference in the proposed approach
region where p2 or p3 is true," can be written as the LTL is that the vertices and edges in the graph representation
formula / ¼ F(p1 ^ F(p2 _ p3 )). Given a syntactically of A/ 3 M are assigned weights. These are used to
cosafe LTL formula /, it has been shown that a nondeter- synergistically convey the low-level exploration informaministic finite automaton (NFA) A/ can be constructed tion to the high-level layer. Every high-level state
(with at most exponential blowup) that describes all the (z, d) 2 A/ 3 M is assigned a feasibility estimate q(z, d).
finite good prefixes satisfying / [64]. For the LTL formula Here, z is a state of A/ and d is a state of abstraction. An
described previously, the corresponding NFA is shown in edge connecting a pair of high-level states ((zi , di ), (zj , dj ))
Figure 4. It has been recently shown that using a mini- in A/ 3 M is assigned the weight (q(zi , di ) Á q(zj , dj ))À1 .
mized deterministic finite automaton (DFA) for an NFA The edge weight estimates the feasibility of transition
can offer significant computational speedups for model between the corresponding high-level states (zi , di ) and
checking and falsification of temporal specifications for (zj , dj ) while respecting the dynamics of the robot. Further
hybrid systems [27], [66]. In view of this result, a mini- details on the real-valued function q are discussed as part
of the synergy layer. A high-level guide is computed as
mized DFA is used in the proposed approach.
the shortest path from an initial state of A/ 3 M to the set
of accepting states using Dijkstra's algorithm. To account
High-Level Search Layer
This is the layer that handles most of the discrete nature of for the fact that the weights are an estimate of feasibility,
the problem. Given a robot model H, a discrete abstraction the high-level search layer also computes a random path
M of the system is constructed. The abstraction M contains a occasionally, which need not be the shortest one.
set D of states and the transition relations between the
abstract states. The abstraction techniques proposed in [28] Low-Level Search Layer
and [29] are based on the decomposition of the workspace. A high-level guide f constructed by the high-level search
However, alternate abstraction techniques that use additional layer may or may not be feasible for the robot. This is
information about the system (e.g., dynamics) can also be incrementally checked at the low-level search layer by
exploring the state space S of the robot. The suggested
high-level guide f is used to bias the search so that the
1
resulting exploration of S improves the chances of finding
1
1
a feasible trajectory satisfying the LTL specification /. The
1
p2 || p3
p1
low-level exploration is done by repeatedly selecting a
Init
2
(p1 & p2) || (p1 & p3)
high-level state from the high-level plan f. The probability
of selecting a high-level state (z, d) is proportional to its
feasibility estimate q(z, d) (described as part of the synergy
Figure 4. NFA A/ describing all the finite good prefixes for the
syntactically cosafe LTL formula / ¼ F ðp1 ^ Fðp2 _ p3 ÞÞ.
layer). Let V(z, d) denote the set of tree vertices that belong
Problem Definition
58
*
IEEE ROBOTICS & AUTOMATION MAGAZINE
*
SEPTEMBER 2011
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