Motion Planning with Metric Temporal Logic Using Reachability Analysis and Hybrid Zonotopes

Metric temporal logic (MTL) provides a formal framework for defining time-dependent mission requirements on autonomous vehicles. However, optimizing control decisions subject to these constraints is often computationally expensive. This article presents a method that uses reachability analysis to implicitly express the set of states satisfying an MTL specification and then optimizes to find a motion plan. The hybrid zonotope set representation is used to efficiently and conveniently encode MTL specifications into reachable sets. A numerical benchmark highlights the proposed method’s computational advantages as compared to existing methods in the literature. Further numerical examples and an experimental application demonstrate the ability to address time-varying environments, region-dependent disturbances, and multi-agent coordination.

💡 Research Summary

The paper tackles the problem of generating motion plans for autonomous systems that must satisfy Metric Temporal Logic (MTL) specifications, a task that is notoriously computationally demanding when approached with conventional methods. Traditional approaches fall into three categories: (i) automaton‑based synthesis that abstracts the dynamics and searches a product graph, (ii) mixed‑integer programming (MIP) formulations that encode the temporal logic using binary variables and “big‑M” constraints, and (iii) set‑based verification using reachability analysis (e.g., Hamilton‑Jacobi or hybrid zonotope complements). While the automaton approach can be fast for simple specifications, it often yields infeasible or sub‑optimal trajectories because of coarse abstractions. MIP methods guarantee global optimality but suffer from an exponential blow‑up in binary variables and loose convex relaxations caused by big‑M, leading to long solve times. Reachability‑based verification can be exact but scales poorly with state dimension or requires expensive set operations.

The authors propose a novel framework that fuses forward reachability analysis with the hybrid zonotope (HZ) set representation to encode MTL specifications directly, thereby avoiding big‑M and dramatically reducing the number of binary variables. A hybrid zonotope extends constrained zonotopes by adding binary generators, enabling the exact representation of unions of polytopes (e.g., obstacles, goal regions, disturbance zones). Each atomic proposition π in the MTL formula is associated with a polytope Xπ; the HZ map introduces a binary variable ξb,π,k such that xk∈Xπ ⇔ ξb,π,k=1. This creates a one‑to‑one correspondence between region membership and binary decision variables, which can be directly incorporated into the system state vector.

The dynamics considered are discrete‑time linear time‑invariant (LTI): xk+1 = A xk + B uk + wk, where wk models spatially varying disturbances (e.g., wind, terrain slope). The disturbance is expressed as a linear map wk = W ξb,k, where W stacks the disturbance vectors associated with each region. By augmenting the state with the binary variables, the forward reachable set over a horizon N can be computed recursively using standard linear reachability tools, but the result is stored as a single high‑dimensional hybrid zonotope that already contains the binary region indicators.

MTL specifications are first translated into half‑space (H‑rep) polytopes. The key technical contribution is the derivation of closed‑form generalized intersection formulas (Proposition 1) that intersect the reachable HZ with the H‑rep representation of each temporal operator. The authors provide new, compact encodings for the “until” operator that require far fewer disjunctions than existing literature, cutting the number of binary variables roughly in half. The other operators (eventually, always, conjunction, negation) are expressed using simple linear constraints on the binary variables.

All constraints—system dynamics, state/input bounds, disturbance mapping, and MTL satisfaction—are linear (or linear with binary variables). Consequently, the motion‑planning problem becomes a mixed‑integer quadratic program (MIQP):

 min ∑k‖xk‖Q² + ∑k‖uk‖R² + ‖xN‖QN²
 subject to: dynamics, bounds, reachable‑MTL intersection, and initial condition.

Because the formulation avoids big‑M and uses a compact HZ representation, commercial solvers (e.g., Gurobi, CPLEX) solve the MIQP far more efficiently than baseline methods. The authors benchmark their approach on the classic “door‑key” problem, showing an order‑of‑magnitude reduction in binary variables (from dozens to a few) and a 5× speed‑up in solve time. Additional case studies demonstrate the method’s flexibility:

• Time‑varying environments – the map M′k is updated each step, allowing moving targets or dynamic obstacles.
• Region‑dependent disturbances – wind fields or terrain gradients are modeled via W ξb,k, and the planner automatically selects disturbance‑aware trajectories.
• Multi‑agent coordination – multiple robots’ states and binary region indicators are stacked into a joint HZ, enabling simultaneous planning with collision avoidance and shared temporal goals.

Experimental validation on a small ground robot platform confirms that the planner can generate feasible trajectories in a few seconds, suitable for near‑real‑time operation.

In summary, the paper’s contributions are:

1. A direct encoding of MTL specifications into hybrid zonotopes, eliminating the need for big‑M constraints.
2. A novel, low‑complexity formulation of the “until” operator that reduces binary variables and disjunctions.
3. Integration of spatially varying disturbances into the reachability‑based planning pipeline.
4. Comprehensive simulation and hardware experiments that demonstrate superior computational performance and scalability compared to state‑of‑the‑art automaton‑based and MIP‑based methods.

The work opens a practical pathway for deploying rich, time‑dependent logical requirements on high‑dimensional autonomous systems while maintaining tractable computation times.