From Legible to Inscrutable Trajectories: (Il)legible Motion Planning Accounting for Multiple Observers

In cooperative environments, such as in factories or assistive scenarios, it is important for a robot to communicate its intentions to observers, who could be either other humans or robots. A legible trajectory allows an observer to quickly and accurately predict an agent’s intention. In adversarial environments, such as in military operations or games, it is important for a robot to not communicate its intentions to observers. An illegible trajectory leads an observer to incorrectly predict the agent’s intention or delays when an observer is able to make a correct prediction about the agent’s intention. However, in some environments there are multiple observers, each of whom may be able to see only part of the environment, and each of whom may have different motives. In this work, we introduce the Mixed-Motive Limited-Observability Legible Motion Planning (MMLO-LMP) problem, which requires a motion planner to generate a trajectory that is legible to observers with positive motives and illegible to observers with negative motives while also considering the visibility limitations of each observer. We highlight multiple strategies an agent can take while still achieving the problem objective. We also present DUBIOUS, a trajectory optimizer that solves MMLO-LMP. Our results show that DUBIOUS can generate trajectories that balance legibility with the motives and limited visibility regions of the observers. Future work includes many variations of MMLO-LMP, including moving observers and observer teaming.

💡 Research Summary

The paper addresses a nuanced problem in robot motion planning where a robot must simultaneously convey its intent to cooperative observers while concealing it from adversarial ones, all under limited visibility constraints. The authors formalize this as the Mixed‑Motive Limited‑Observability Legible Motion Planning (MMLO‑LMP) problem. In MMLO‑LMP, the environment contains a set of candidate goals, a start location, and a true goal. Each observer is characterized by a visibility region (a subset of the environment that the observer can see) and a motive value ranging from –1 (purely adversarial) to +1 (purely cooperative). Positive‑motive observers should find the robot’s trajectory legible—i.e., they should be able to infer the true goal quickly and confidently—whereas negative‑motive observers should find the trajectory illegible, either by being misled toward a decoy goal or by maintaining ambiguity among several goals.

Legibility is quantified using a Bayesian model that relates the probability of the true goal given a partial trajectory to the cost of that partial trajectory and the optimal cost from the current point to each goal. The authors adopt the same formulation as Dragan et al., approximating the likelihood with an exponential of the difference between actual and optimal costs. Illegibility is defined via two strategies: (1) a “decoy” strategy that maximizes the probability of a false goal, and (2) an “ambiguous” strategy that minimizes the average probability gap between the true goal and all other goals. The overall illegibility of a trajectory is the maximum of the two scores.

The MMLO‑LMP objective aggregates, over all observers, the weighted legibility of positive observers (weight = motive) and the weighted illegibility of negative observers (weight = |motive|). The robot must therefore maximize a sum of these terms while respecting the robot’s own efficiency cost (trajectory length, smoothness, etc.). The key difficulty lies in overlapping visibility regions where the robot must satisfy contradictory requirements at the same time.

To solve this, the authors propose DUBIOUS (Dual‑Based Limited‑Observability Unified trajectory Solver), a trajectory optimizer built on the STOMP (Stochastic Trajectory Optimization for Motion Planning) framework. Starting from a straight‑line baseline, DUBIOUS iteratively perturbs the trajectory. At each timestep, a cost function F(i) is computed that incorporates, for each observer whose visibility region contains the current point, the legibility or illegibility score scaled by the observer’s motive. Positive observers contribute a legibility term, negative observers contribute either the decoy or ambiguous illegibility term, depending on the chosen strategy. Because STOMP does not require differentiable costs, the method can handle the non‑smooth max‑operations inherent in the illegibility definition.

Experiments are conducted in 2‑D simulated environments with multiple static observers placed at various locations. The authors evaluate several configurations: (a) only cooperative observers, (b) only adversarial observers, (c) mixed motives with overlapping visibility, and (d) different placements of the adversarial observer (near the start vs. near the goal). Results show that DUBIOUS produces trajectories that remain near‑optimal in terms of path length and travel time while achieving high legibility scores for cooperative observers and high illegibility scores for adversarial ones. When the adversarial observer’s visibility overlaps the cooperative observer’s region, the optimizer finds a compromise path that briefly diverges toward the decoy goal before correcting toward the true goal, thereby satisfying both objectives. The decoy strategy is most effective when the adversarial observer watches the robot early in its motion; the ambiguous strategy works better when the adversary observes the robot near the goal.

The paper’s contributions are threefold: (1) a formal definition of MMLO‑LMP that integrates limited observability and mixed motives, (2) a unified cost formulation that can be optimized with a non‑gradient‑based planner, and (3) empirical evidence that the approach can balance competing informational goals in realistic settings. Limitations include the reliance on a fixed Bayesian observer model and the assumption of static observers; dynamic observers with moving fields of view are left for future work. The authors outline extensions such as modeling observer teams, incorporating learning‑based prediction of observer behavior, and real‑world user studies on human‑robot interaction.