In recent years, Signal Temporal Logic (STL) has gained traction as a practical and expressive means of encoding control objectives for robotic and cyber-physical systems. The state-of-the-art in STL trajectory synthesis is to formulate the problem as a Mixed Integer Linear Program (MILP). The MILP approach is sound and complete for bounded specifications, but such strong correctness guarantees come at the price of exponential complexity in the number of predicates and the time bound of the specification. In this work, we propose an alternative synthesis paradigm that relies on Bayesian optimization rather than mixed integer programming. This relaxes the completeness guarantee to probabilistic completeness, but is significantly more efficient: our approach scales polynomially in the STL time-bound and linearly in the number of predicates. We prove that our approach is sound and probabilistically complete, and demonstrate its scalability with a nontrivial example.
Another promising approach is to use Satisfiability Modulo Theories (SMT) to find a feasible solution, though not necessarily the optimal one @cite_12 @cite_13 . This approach is intuitively attractive in the context of robotics, where satisfying the specification may be more desirable than finding a perfectly optimal trajectory. Early results for Linear Temporal Logic (LTL) specifications indicate good potential on a variety of interesting problems. SMT is a generalization of the NP-complete Boolean satisfiability checking problem, however, and avoiding the associated worst-case exponential complexity may be nontrivial.
@cite_13 @cite_12
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Satisfiability Modulo Theories (SMT) problem is a decision problem for logical first order formulas with respect to combinations of background theories such as: arithmetic, bit-vectors, arrays, and uninterpreted functions. Z3 is a new and efficient SMT Solver freely available from Microsoft Research. It is used in various software verification and analysis applications. We present an efficient algorithm for multi-robot motion planning from linear temporal logic (LTL) specifications. We assume that the dynamics of each robot can be described by a discrete-time, linear system together with constraints on the control inputs and state variables. Given an LTL formula specifying the multi-robot mission, our goal is to construct a set of collision-free trajectories for all robots, and the associated control strategies, to satisfy We show that the motion planning problem can be formulated as the feasibility problem for a formula p over Boolean and convex constraints, respectively capturing the LTL specification and the robot dynamics. We then adopt a satisfiability modulo convex (SMC) programming approach that exploits a monotonicity property of p to decompose the problem into smaller subproblems. Simulation results show that our algorithm is more than one order of magnitude faster than state-of-the-art sampling-based techniques for high-dimensional state spaces while supporting complex missions. The problem of deciding the satisfiability of a quantifier-free formula with respect to a background theory, also known as Satisfiability Modulo Theories (SMT), is gaining increasing relevance in verification: representation capabilities beyond propositional logic allow for a natural modeling of real-world problems (e.g., pipeline and RTL circuits verification, proof obligations in software systems). In this paper, we focus on the case where the background theory is the combination T1∪T2 of two simpler theories. Many SMT procedures combine a boolean model enumeration with a decision procedure for T1∪T2, where conjunctions of literals can be decided by an integration schema such as Nelson-Oppen, via a structured exchange of interface formulae (e.g., equalities in the case of convex theories, disjunctions of equalities otherwise). We propose a new approach for SMT(T1∪T2), called Delayed Theory Combination, which does not require a decision procedure for T1∪T2, but only individual decision procedures for T1 and T2, which are directly integrated into the boolean model enumerator. This approach is much simpler and natural, allows each of the solvers to be implemented and optimized without taking into account the others, and it nicely encompasses the case of non-convex theories. We show the effectiveness of the approach by a thorough experimental comparison. We consider partially observable Markov decision processes (POMDPs), that are a standard framework for robotics applications to model uncertainties present in the real world, with temporal logic specifications. All temporal logic specifications in linear-time temporal logic (LTL) can be expressed as parity objectives. We study the qualitative analysis problem for POMDPs with parity objectives that asks whether there is a controller (policy) to ensure that the objective holds with probability 1 (almost-surely). While the qualitative analysis of POMDPs with parity objectives is undecidable, recent results show that when restricted to finite-memory policies the problem is EXPTIME-complete. While the problem is intractable in theory, we present a practical approach to solve the qualitative analysis problem. We designed several heuristics to deal with the exponential complexity, and have used our implementation on a number of well-known POMDP examples for robotics applications. Our results provide the first practical approach to solve the qualitative analysis of robot motion planning with LTL properties in the presence of uncertainty.