Data-Grounded Trajectory Synthesis Framework

• Data-grounded trajectory synthesis is a formal method that integrates logical constraints and data inputs to generate feasible, operation-aware action sequences.
• It leverages SMT solving combined with discrete waypoint planning and 3D Bézier curve smoothing to ensure physical, temporal, and resource constraints are met.
• The framework supports scalable multi-agent mission planning by addressing resiliency, data freshness, and budget constraints for robust real-world deployments.

A data-grounded trajectory synthesis framework is an integrated methodology for generating feasible, constraint-compliant, and operation-aware action sequences for autonomous or multi-agent systems, where the synthesis process is explicitly tied to data-driven constraints, real-world operational objectives, and, in many cases, logical or mathematical formalizations. In these frameworks, trajectory generation is “grounded” in specific requirements arising from the physical system, the environment, and the end-task—rather than being purely abstract or synthetic—ensuring both practical viability and adaptation to adversarial, stochastic, or dynamically changing conditions.

1. Formal Model-Based Trajectory Synthesis

Data-grounded trajectory synthesis frameworks rely on formal, logic-based models that encode the operational environment, agent (e.g., UAV) capabilities, resource budgets, and a diverse set of task-oriented objectives. In the Synth4UAV framework (Jakaria et al., 2020), the aerial space is mathematically modeled as a set of 3D waypoints, partitioned into “data points” (sensor sites to be visited), “forbidden zones,” and neutral locations. The UAV’s possible actions (moves between waypoints) are represented via Boolean variables: $M^{(\hat{p},p)}_u$ indicates whether UAV $u$ moves from $\hat{p}$ to $p$, and $V^p_u$ records visitation status. These logic variables, along with temporal and cost variables (e.g., $T^p_u$ for time to reach $p$), provide a formal basis for encoding spatiotemporal and operational constraints.

The synthesis problem is then defined as a constraint satisfaction problem typically solved using Satisfiability Modulo Theories (SMT). Synth4UAV concretely employs Z3 to solve for SAT assignments of these variables, producing routes (full control trajectories) for each UAV that meet mission requirements.

2. Constraint Encoding: Physical, Operational, and Security Objectives

The framework systematically encodes a comprehensive set of constraints informed by system data and mission needs:

• Physical Flight Constraints: Each move is subject to constraints on turn angle, climb angle, and connectivity (i.e., whether a physically valid transition exists), as in

$$M^{(\hat{p}, p)}_u \to (V^{\hat{p}}_u \wedge L^{(\hat{p})}_p \wedge (\delta^{(p',\hat{p},p)}_\alpha \leq \theta^{t_{th}}) \wedge (\delta^{(p',\hat{p},p)}_\beta \leq \theta^{c_{th}}))$$

• Resource (Fuel, Time) Budgets: Motion costs are a function of Euclidean distance, turn angle, climb angle, and UAV fuel efficiency:

$$f^{(\hat{p}, p)}_u = \frac{ d^{(\hat{p}, p)} + k_1 \cdot |\alpha^{(\hat{p},p)}_u| + k_2 \cdot \beta^{(\hat{p},p)}_u }{r_u}$$

The accumulated fuel or time per trajectory is summed and constrained not to exceed the allotted budget.

• Coverage and Freshness Objectives: Coverage is enforced by requiring that a fraction $v_{th}$ of relevant sensors are visited:

$$\frac{\sum_{u, p \in \mathbb{S}} V^p_u}{|\mathbb{S}|} \geq v_{th}$$

Data freshness is implemented by restricting the time difference between multiple visits:

$$\hat{V}^p_{u,u'} : (V^p_u \wedge V^p_{u'} \wedge (T^p_u \leq T^p_{u'}) \wedge (T^p_{u'} - T^p_u \leq t^p_{th}))$$

• Resiliency to Failures/Compromises: To tolerate up to $r$ lost agents, each critical point $p$ is required to be visited by $r + 1$ UAVs:

$$\forall p \in \mathbb{S}, R^p_c \rightarrow \left(\sum_u V^p_u \geq r + 1\right)$$

By casting all requirements formally, the solution to the SMT instance yields trajectories that satisfy all encoded properties.

3. Trajectory Planning and Satisfiability

Trajectory synthesis is not merely a search for geometric paths but involves the global solution of logical and arithmetic constraints:

• State-space is explored with respect to all constraints: Rather than greedily assigning moves, the solver computes assignments to all variables such that the entire set of operational constraints is met (or, if infeasible, declares UNSAT). For each UAV, the variables $(V^p_u, M^{(\hat{p}, p)}_u)$ together define a feasible, complete, and non-colliding path through the 3D space.
• Incorporation of smoothing and simulation: Synth4UAV augments the discrete, waypoint-based solution with 3D Bézier curve smoothing, enhancing the physical plausibility of the synthesized trajectories. The Unity 3D-based simulator allows for visualization and verification of both synthesized paths and temporal schedules.

A strong feature is scalability: Although constraint complexity naturally grows with the number of waypoints, UAVs, and resiliency requirements, the SMT approach is shown to scale to practical instance sizes (e.g., tens of waypoints and UAVs).

4. Evaluation and Systemic Trade-offs

Evaluation in Synth4UAV exemplifies a rigorous, data-driven approach:

• Case Study: A scenario with 30 waypoints, 15 sensors, and 5 UAVs is used to validate algorithmic soundness. The synthesized routes collectively meet >80% data coverage and strict resiliency (e.g., all sensors are visited by $k=3$ UAVs within time windows to guarantee both coverage and cross-verification).
• Parametric Analysis: Graphs demonstrate how data coverage and budget are interrelated; as more UAVs are added or resiliency is relaxed, the per-UAV cost of achieving full coverage drops.
• Scalability: Synthesis time is empirically correlated to problem size and constraint tightness. Higher resiliency (increased $k$) yields more UNSAT instances, revealing “hard” operational regions and illuminating trade-offs in mission planning.

This evaluation strategy provides both proof of concept and operational insight for real-world deployments.

5. Mathematical Formulation Summary

The key mathematical backbone includes movement feasibility, link constraints, resource-aware cost functions, and logical expressions for resilience and data freshness. Table 1 (see below) consolidates pivotal constraints:

Name	Mathematical Expression	Function
Move constraint	$$M^{(\hat{p},p)}_u \rightarrow (V^{\hat{p}}_u \wedge L^{(\hat{p})}_p)$$	Ensure move is allowed
Link feasibility	$$L^{(\hat{p},p)} \rightarrow (D^{(\hat{p},p)} \wedge (\delta_\alpha \leq \theta^{t_{th}}) \wedge (\delta_\beta \leq \theta^{c_{th}}))$$	Physical path existence
Fuel consumption	$$f^{(\hat{p},p)}_u = \frac{d^{(\hat{p},p)} + k_1|\alpha^{(\hat{p},p)}_u| + k_2\beta^{(\hat{p},p)}_u}{r_u}$$	Resource limiting
Time update	$$T^p_u = T^{(\hat{p})}_u + t^{(\hat{p},p)}_u + H^{(\hat{p})}_u$$	Cumulative timing
Resiliency	$$\forall p \in \mathbb{S}, R^p_c \to (\sum_u V^p_u \geq r + 1)$$	Tolerance to failures
Data freshness	$$\forall p \in \mathbb{S}, R^p_t \to (\vee_u \sum_{u' \neq u} \hat{V}^p_{u,u'} \geq k+1)$$	Temporal cross-verification

These expressions, formalized into an SMT instance, underpin every synthesized trajectory.

6. Practical Implications and Deployment

The data-grounded trajectory synthesis approach embodied by Synth4UAV directly addresses the needs of collaborative UAV data collection operations:

• Robustness: By grounding trajectory synthesis in logical (and thus verifiable) constraints, the approach guarantees operational properties, e.g., resiliency against agent loss and data collusion/dilution.
• Scalability and Realism: SMT encoding supports the generation of routes for multiple UAVs with practical requirements (fuel, time, maneuvering) and supports late-stage smoothing for continuous execution.
• Flexible Reconfiguration: Parametric constraints (coverage, resiliency) are not hardcoded; operators can adjust them, and the SMT solver either synthesizes a solution or reports infeasibility, informing real-world planning.
• Operational Awareness: The approach, by design, combines spatial awareness (waypoints, forbidden zones), agent limits (maneuverability, endurance), and mission-level priorities (coverage, data freshness), serving as a practical blueprint for field deployment in remote sensing and surveillance missions.

This synthesis was derived from the formal, technical account given in (Jakaria et al., 2020), and all statistics, models, and behaviors are taken verbatim from the described Synth4UAV system.

7. Connections to Broader Research Context

The formal, data-grounded trajectory synthesis paradigm represents a critical methodological advance over heuristic-based or purely geometric planning approaches. By leveraging logical modeling, formal verification, and parametric constraint satisfaction, these frameworks are adaptable to a wide array of multi-agent robotics systems, especially where resilient, data-valid, and dynamically feasible path planning is required. Synth4UAV sets a template for similar approaches in multi-UAV coordination, resilient mission management, and secure data gathering, informing both theoretical and applied research across robotics, formal methods, and autonomous system design.