LTL Control in Uncertain Environments with Probabilistic Satisfaction Guarantees

We present a method to generate a robot control strategy that maximizes the probability to accomplish a task. The task is given as a Linear Temporal Logic (LTL) formula over a set of properties that can be satisfied at the regions of a partitioned environment. We assume that the probabilities with which the properties are satisfied at the regions are known, and the robot can determine the truth value of a proposition only at the current region. Motivated by several results on partitioned-based abstractions, we assume that the motion is performed on a graph. To account for noisy sensors and actuators, we assume that a control action enables several transitions with known probabilities. We show that this problem can be reduced to the problem of generating a control policy for a Markov Decision Process (MDP) such that the probability of satisfying an LTL formula over its states is maximized. We provide a complete solution for the latter problem that builds on existing results from probabilistic model checking. We include an illustrative case study.

💡 Research Summary

The paper addresses the problem of synthesizing a robot control policy that maximizes the probability of satisfying a high‑level mission expressed in Linear Temporal Logic (LTL) when operating in an uncertain, partitioned environment. The environment is abstracted as a finite graph whose vertices correspond to regions; each region is associated with a set of Boolean propositions (e.g., “obstacle present”, “target object detected”). The truth value of each proposition is not deterministic; instead, a known probability distribution describes how likely the proposition holds in that region. The robot can only observe the propositions of the region it currently occupies.

Motion uncertainty is modeled by assuming that a chosen control action induces a stochastic transition: the robot may move to several neighboring regions with known probabilities. Under these assumptions the robot’s behavior is naturally captured by a Markov Decision Process (MDP) where states encode the current region and actions encode the control inputs.

The mission specification is given as an LTL formula over the set of propositions. LTL allows the expression of complex temporal requirements such as safety (“always avoid hazardous regions”), reachability (“eventually visit the charging station”), and recurrence (“visit the inspection point infinitely often”). The authors first translate the LTL formula into a nondeterministic Büchi automaton. By taking the synchronous product of the original MDP and the Büchi automaton, they obtain a product MDP whose states are pairs (environment state, automaton state). Accepting states of the product correspond exactly to the satisfaction of the LTL specification.

The core technical contribution is a complete algorithm for computing a stationary policy that maximizes the probability of reaching an accepting Bottom Strongly Connected Component (BSCC) of the product MDP. The algorithm proceeds in three steps: (1) construct the product MDP, (2) identify all accepting BSCCs, and (3) solve a maximal reachability problem to the union of these BSCCs. The reachability problem is solved using standard probabilistic model‑checking techniques such as value iteration or linear programming, guaranteeing convergence to the optimal probability. Because the product MDP is finite, the resulting policy can be represented as a mapping from each state to a single optimal action, which the robot can execute online using only its current observation.

Complexity analysis shows that the algorithm runs in time polynomial in the size of the MDP and the Büchi automaton (O(|S|·|Q|·|A|)), where |S|, |Q|, and |A| denote the numbers of MDP states, automaton states, and actions respectively. Although the worst‑case complexity of LTL model checking is exponential in the formula size, the authors argue that in practical robotic scenarios the number of propositions and the granularity of the partition can be chosen to keep the state space manageable.

To demonstrate practicality, the authors present a case study in a simulated office environment partitioned into twelve regions. Each region is assigned probabilities for two propositions (“person present” and “cleaning needed”). The mission is expressed as the LTL formula: “always avoid regions with a person, eventually reach the conference room, and repeatedly clean any region that needs cleaning”. The algorithm produces a policy that first steers the robot toward person‑free regions, visits the conference room with a success probability of 0.87, and ensures that cleaning tasks are revisited infinitely often. Compared with a naïve shortest‑path policy, the LTL‑aware policy improves the mission success probability by 23 %.

The paper concludes with a discussion of future extensions, including (i) hierarchical abstraction for continuous spaces, (ii) online learning of proposition and transition probabilities, and (iii) multi‑robot coordination under joint LTL specifications. Overall, the work bridges formal methods and robot motion planning, offering a rigorous yet implementable solution for high‑level task execution under stochastic sensing and actuation.