💻 computer science

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

This paper introduces the Mixed-Motive Limited-Observability Legible Motion Planning (MMLO-LMP) problem and the trajectory optimizer, which enable robots to generate motion plans that are simultaneously legible to cooperative observers and illegible to adversarial ones while accounting for their distinct motives and limited visibility.

Original authors: Ananya Yammanuru, Maria Lusardi, Nancy M. Amato, Katherine Driggs-Campbell

Published 2026-09-02

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Ananya Yammanuru, Maria Lusardi, Nancy M. Amato, Katherine Driggs-Campbell

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a robot moving through a busy factory floor or a hospital corridor. It is not just a machine following a set of instructions; it is a social actor. Every time it turns, stops, or speeds up, it is sending a silent message to the people and other machines around it. In the field of robotics, this silent communication is known as "legibility." A legible path is one that allows an observer to quickly and correctly guess where the robot is going. If a robot moves in a way that makes its destination obvious, it helps humans coordinate with it safely. But the rules change depending on who is watching. In a cooperative setting, like a warehouse where humans and robots work together, the robot should be as clear as possible. In a competitive or adversarial setting, such as a military operation or a game, the robot might need to hide its true destination to prevent an opponent from predicting its next move. This creates a complex puzzle: what happens when a robot must be clear to one person while remaining hidden from another, all while navigating around walls that block some people's view but not others?

Researchers at the University of Illinois at Urbana-Champaign have tackled this exact challenge by introducing a new problem they call Mixed-Motive Limited-Observability Legible Motion Planning. In their work, they describe a scenario where a robot must plan a single path that satisfies two conflicting demands. It needs to be easily understood by "positive" observers—those who are friendly or cooperative—while simultaneously being confusing or misleading to "negative" observers, who might be adversaries. Complicating matters further, these observers do not all see the same thing. Some might only see a small corner of the room, or their view might be blocked by obstacles. The researchers developed a new software tool, which they named the trajectory optimizer, to solve this problem. This tool acts as a trajectory optimizer, calculating the exact path a robot should take to balance these competing needs. Instead of trying to be legible to everyone or invisible to everyone, the trajectory optimizer figures out how to be legible to the right people and illegible to the wrong ones, based on what each person can actually see.

To understand how this works, consider the two main strategies the robot can use when dealing with a negative observer. The first is the "decoy" strategy. Here, the robot deliberately moves toward a fake destination, making the observer believe it is going there. Only at the last possible moment does it turn toward its true goal. This is effective when the robot wants to actively mislead an opponent. The second strategy is the "ambiguous" or avoidance approach. In this case, the robot tries to minimize the time it spends in the negative observer's line of sight, or it moves in a way that makes all possible destinations look equally likely, so the observer cannot guess the true goal at all. The researchers found that the best choice depends entirely on the layout of the room and the motives of the people watching. If a friendly observer and an unfriendly one are standing in the same spot, the robot has to weigh their importance. If the friendly observer is more important, the robot will prioritize being clear to them, even if it means the unfriendly one sees a bit more. If the unfriendly observer is more dangerous, the robot might choose to hide completely or use a decoy, even if it makes the path less clear to the friendly observer.

The team tested their system in various simulated environments to see how well the trajectory optimizer performed. In one scenario, a robot needed to be clear to a friendly observer in a green zone but hidden from an unfriendly one in a red zone. The software successfully generated paths that curved through the green zone to show its intentions clearly, while either slipping out of the red zone entirely or moving in a way that confused the observer in the red zone. In another test, the researchers placed the friendly and unfriendly observers in overlapping areas. They found that the robot could still find a path that worked for both, though it had to make trade-offs. For instance, if the unfriendly observer had a stronger "motive" or influence, the robot would take a more deceptive route, even if it meant the friendly observer had to wait a moment longer to understand the robot's goal. The results showed that the trajectory optimizer could generate paths that were significantly better than simple straight lines or paths that only tried to be efficient. It could produce trajectories that were highly legible to the right people and highly illegible to the wrong ones, adapting its behavior based on the specific constraints of the environment.

The researchers also explored what happens when the goals are in completely opposite directions. In these difficult cases, a straight line is usually the most efficient path, but it might be too easy to read. The trajectory optimizer showed that by using a decoy strategy, the robot could move toward a fake goal for a long time, convincing the observer it was going the wrong way, before suddenly pivoting to the true destination. This delayed the observer's correct guess until the very end of the path. However, the team noted that their current work relies on simulations where the environment is static and the observers do not move. They acknowledge that real-world applications would require the robot to handle moving people and changing views in real time. They also point out that their system assumes the robot knows exactly where the observers are and what they can see, which might not always be true in a chaotic real-world setting. Despite these limitations, the study provides a clear mathematical framework for a problem that has not been solved before: how to be clear to some and hidden from others at the same time.

Ultimately, this work moves beyond the idea that a robot should simply be efficient or simply be polite. It recognizes that in a world with multiple observers, a robot's path is a form of communication that can be tailored to different audiences. The trajectory optimizer system demonstrates that it is possible to calculate a path that serves multiple masters, balancing the need for transparency with the need for privacy. As robots become more common in our daily lives, from warehouses to hospitals, the ability to navigate these social and strategic complexities will become increasingly important. This research offers a new way to think about robot movement, not just as a physical act of getting from point A to point B, but as a strategic decision about who gets to know the robot's plans and who does not.

Technical Summary: From Legible to Inscrutable Trajectories

Problem Definition: Mixed-Motive Limited-Observability Legible Motion Planning (MMLO-LMP)
This paper addresses a gap in existing legible motion planning literature, which typically assumes observers have full visibility of a trajectory and share the agent's cooperative motives. The authors introduce the Mixed-Motive Limited-Observability Legible Motion Planning (MMLO-LMP) problem. This problem arises in environments where an agent must navigate toward a goal while interacting with multiple observers who possess:

Limited Observability: Observers may only see specific portions of the environment due to occlusions or sensor range, meaning they cannot view the entire trajectory from start to finish.
Mixed Motives: Observers may have "positive" motives (cooperative, wishing to understand the agent's intent) or "negative" motives (adversarial, wishing to conceal or misinterpret the agent's intent).

The objective of MMLO-LMP is to generate a single trajectory that is legible to positive observers (allowing them to quickly and accurately predict the true goal) while simultaneously being illegible to negative observers (delaying their correct prediction or leading them to predict a decoy goal), all while accounting for the specific visibility regions of each observer.

Methodology: The Solver
To solve MMLO-LMP, the authors propose a trajectory optimizer based on Stochastic Trajectory Optimization for Motion Planning (STOMP). Unlike gradient-based methods that require differentiable cost functions, the optimizer utilizes STOMP to handle non-differentiable costs arising from visibility constraints and discrete strategy choices.

The core of the methodology involves constructing a cost function $F$ that balances competing objectives:

Legibility for Positive Observers: The cost function maximizes the probability that a positive observer correctly identifies the true goal ( $G_R$ ) early in the trajectory, but only considers the portion of the trajectory visible within the observer's region ( $V_o$ ).
Illegibility for Negative Observers: The cost function minimizes the probability of correct identification for negative observers. The authors propose two distinct strategies for achieving this, controlled by a hyperparameter $\alpha_{neg}$ $α_{n e g}$ :
1. Decoy Strategy: The agent intentionally moves toward a decoy goal ( $G_D$ ) to maximize the probability that the observer predicts the decoy goal instead of the true goal. This mimics "simulation" (showing the false).
2. Ambiguous/Avoidance Strategy: The agent minimizes time spent in the negative observer's visibility region entirely or maintains high ambiguity among all goals. This mimics "dissimulation" (hiding the true) and effectively reduces the observer's information gain.

The final cost function normalizes these legibility and illegibility scores based on the motives ( $M_o \in [-1, 1]$ ) of the observers and the specific visibility regions they occupy at each timestep.

Key Contributions

Problem Formulation: The paper formally defines the MMLO-LMP problem, extending legible motion planning to scenarios with multiple observers, partial visibility, and conflicting objectives.
Algorithm Development: The introduction of the optimizer, an optimization-based solver capable of generating trajectories that satisfy complex, multi-objective constraints involving mixed-motive observers.
Strategic Analysis: The identification and implementation of distinct strategies (decoy vs. ambiguous/avoidance) for handling adversarial observers, demonstrating that the optimal strategy depends on the spatial relationship between the agent, the goals, and the observers' visibility regions.
Empirical Validation: Quantitative and qualitative analysis showing that the optimizer can generate trajectories that outperform standard efficient (straight-line) paths and single-objective baselines in terms of balancing legibility and privacy.

Results
Experiments were conducted in 2D environments with varying configurations of observers and goals:

Single Observer Scenarios: The optimizer successfully generated trajectories that increased legibility for positive observers by spending more time in their visibility regions compared to efficient straight-line paths. For negative observers, the optimizer successfully employed either the decoy strategy (moving toward a fake goal) or the ambiguous strategy (avoiding the observer's region entirely), significantly reducing the observer's ability to predict the true goal.
Multiple Observer Scenarios: In environments with overlapping visibility regions and opposing motives, the optimizer demonstrated the ability to weigh the trade-offs. For instance, when a positive observer's region was fully contained within a negative observer's region, the optimizer prioritized the stronger motive (e.g., a -1 motive observer dominated a +0.25 observer), resulting in trajectories that sacrificed some legibility for the positive observer to maintain illegibility for the negative one.
Opposing Goals: In scenarios where goals were in opposite directions, the decoy strategy proved highly effective, allowing the agent to mislead an adversarial observer for a significant portion of the trajectory before pivoting to the true goal.

Significance and Claims
The authors claim that their work bridges the gap between cooperative legibility and adversarial obfuscation in realistic settings where observers have limited and varying viewpoints. They emphasize that the combination of limited observability and mixed motives fundamentally changes how legibility can be optimized; a strategy that is optimal for a fully visible cooperative observer may fail or be counterproductive when an adversarial observer shares the same view.

The paper modestly notes that the optimizer currently relies on offline planning with static observer models and requires hyperparameter tuning. However, the results suggest that the proposed framework provides a flexible foundation for future work involving dynamic observers, teaming, and more complex real-world constraints like obstacles and task planning. The work establishes that agents can effectively navigate the tension between transparency and privacy by strategically manipulating their visibility and motion relative to specific observers.

Technical Summary: From Legible to Inscrutable Trajectories

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