Maps of Dynamics (MoDs) for Robot Planning

• Maps of Dynamics (MoDs) are spatio-temporal, queryable models that capture probabilistic human activity from historical trajectory data for anticipatory planning.
• MoDs are constructed by discretizing environments into grid cells over time, using kernel smoothing to estimate occupancy probabilities and inform path planning.
• Integrating MoDs into multi-robot task allocation yields measurable improvements, reducing mission time by up to 26% and waiting time by 53%, while enhancing collision robustness.

Maps of Dynamics (MoDs) are spatio-temporal, queryable models devised to capture, represent, and exploit the patterns of historical human motion within an environment. Unlike traditional static maps—focused solely on fixed, immutable obstacles—MoDs encode, at fine spatial and temporal granularity, the probability of human presence in each region of space, thus enabling robots to anticipate dynamic environmental conditions likely to impact their trajectories, timing, or collaborative task execution. In multi-robot task allocation (MRTA), MoDs allow for explicit integration of human-centric stochasticity, making possible a more efficient and collision-robust deployment of robotic fleets within shared human-robot spaces (Eskeri et al., 27 Aug 2025).

1. Conceptual Foundation and Distinction from Static Maps

MoDs are constructed from historical trajectory data, most commonly by discretizing the environment into a grid and recording, per cell and over regular time intervals, the aggregate or probabilistic occupancy by humans. This results in a time-indexed, queryable map where each grid cell encodes the likelihood of human activity across the day or week.

Comparison Table: MoDs vs. Static Maps

Feature	Static Map	Map of Dynamics (MoD)
Encodes dynamic agents	No	Yes (probabilistic presence)
Temporal resolution	None (static)	Time-indexed
Model update frequency	Infrequent/manual	Routinely, from data
Planning role	Collision avoidance	Anticipatory task/cost planning

Static maps are static and blind to temporal patterns; MoDs offer predictive insight based on learned crowd flow, allowing integrated anticipation of delays and congestion, a critical distinction in environments where human activity induces substantial non-deterministic effects on robot trajectories (Eskeri et al., 27 Aug 2025).

2. Methodological Integration in Multi-Robot Task Allocation

Cost Function Design with MoDs

MoDs are incorporated into MRTA via a stochastic cost function built upon robot–task path evaluations. For a robot $r_i$ assigned to task $t_j$, the path traversed is discretized into grid cells, and for each segment $k$ the cost is: $c_{ij} = \sum_k (w_0 \cdot d_k + w_1 \cdot \eta_k)$ where:

• $d_k$ is the Euclidean length of cell $k$.
• $\eta_k$ is a Bernoulli variable (expectation equals the probability of human presence in cell $k$ at the planned time, derived from MoDs).
• $w_0$, $w_1$ are learned weights (via Bayesian Optimization with GPR) that trade off path length and expected delay due to human encounter.

This cost is used in an auction-based or binary programming framework: $$\text{Minimize}\quad Z = \min_{x_{ij}} \max_i(c_{ij} \cdot x_{ij})$$ with assignment constraints: $$\sum_i x_{ij} = 1\quad\forall j;\quad \sum_j x_{ij} = 1\quad\forall i;\quad x_{ij} \in \{0,1\}$$ This MRTA formulation explicitly internalizes the stochastic disturbance of human presence along traversed paths, as informed by MoDs. Bayesian Optimization iteratively updates $w_0$ and $w_1$ to best match observed mission outcomes.

MoD Construction and Query

• Historical trajectories yield a discretized spatio-temporal tensor (e.g., 0.05m spatial granularity, 30min temporal bins).
• Presence estimates in each cell are smoothed, e.g., with a disk kernel, to capture realistic occupancy and motion uncertainty.
• During deployment, for any spatio-temporal query, the MoD instantly provides the current or predicted likelihood of encountering a pedestrian.

3. Experimental Validation and Quantitative Findings

The methodology was validated on real-world trajectory data (ATC dataset, Osaka). MoDs were precomputed for the environment and used in simulation with fleets of 5–15 robots, each operating in parallel with human activity modeled as ground truth precedence (pedestrians always have right-of-way).

Key Metrics and Outcomes:

• Mission completion time: reduced up to 26% vs. path-based (dynamics-agnostic) allocation and 19% vs. HA-Alloc baseline.
• Robot waiting time: reduced up to 53% vs. path-based method.
• Failure/deadlock rate: substantially reduced for all fleet sizes.
• Computational overhead: negligible compared to full mission time, even as path queries use the MoD at every allocation step.

These gains stem from the ability of MoDs to steer robots away from probable high-density (human) regions, effectively desaturating congested paths in both the planning and real-world execution phases.

4. Implications, Limitations, and Future Directions

Integration of MoDs enables proactive and adaptive multi-robot planning in non-stationary, human-populated spaces. By anticipating where and when congestion and delays occur, robots not only achieve higher throughput but also improve safety and minimize deadlock.

Current limitations acknowledged in the paper:

• Evaluation was limited to fleets of ≤15 robots, owing to the simulation framework.
• MoD usage is based primarily on single-cell occupancy probabilities; more granular models could consider velocity and directionality of human traffic, occupancy correlations, or group behaviors.
• Human encounter likelihood calculations increase computational complexity, though this effect is mitigated by the low frequency of execution (one task allocation per mission).

Directions for future research:

• Expanded scalability via advanced coordination algorithms.
• Richer semantic modeling (direction, velocity, interaction patterns) in MoDs.
• Real-time adaptive MoDs, where the model is continually updated with live sensing, for maximal robustness to changes in environment occupancy and flow.

5. Broader Significance and Applicability

The MoD paradigm marks a shift toward dynamics-aware MRTA, in which robots are furnished not only with geometric maps but also with actionable knowledge about the dynamics of space usage by other agents—in this context, humans. This is especially critical in intralogistics, autonomous delivery, service robotics, and search and rescue, where operational efficiency is highly sensitive to dynamic interactions (Eskeri et al., 27 Aug 2025).

By systematically incorporating MoDs into optimization, the field advances from naive, static planning toward integrated, data-driven anticipation of a complex, evolving environment. This approach enables deployment of larger, more effective, and more human-compatible robot teams in natural, populated spaces, raising both the ceiling for mission efficiency and the floor for operational safety.