Traversal Risk Graphs (TRG)

• Traversal Risk Graphs (TRG) are graph-based models that combine geometric connectivity and explicit traversal risk metrics to optimize safety and efficiency in navigation.
• They employ deterministic and stochastic risk formulations, including Bayesian updates and CVaR-based policy optimization, to guide path planning in uncertain terrains.
• Empirical results show that TRGs improve planning success, reduce path lengths, and enable scalable coordination in both single-agent and multi-agent navigation scenarios.

Traversal Risk Graphs (TRG) are graph-theoretic models for safe and efficient navigation in environments where traversability and risk exhibit spatial heterogeneity, stochasticity, and correlation. TRGs encode not only the geometric connectivity of a terrain but also explicit measures of traversal risk on edges, enabling deployment of principled path-planning algorithms that account for both cost-efficiency and safety. Recent research articulates distinct TRG constructions and optimization paradigms, including deterministic geometric risk weighting (Lee et al., 3 Jan 2025), correlated stochastic edge costs for risk-averse planetary mobility (Lamarre et al., 19 May 2025), and multi-agent coordination under edge-risk constraints (Zhou et al., 2024).

1. Formal Definitions and Core Element Structures

Geometric Risk-Weighted TRG

In unstructured environments, a TRG is modeled as an undirected graph $G = (V, E)$ with

• Nodes $v_i \in V$ corresponding to locally stable and reachable terrain regions (e.g., inscribed circles of radius $r_{\rm robot}$ around point $P_i \in \mathbb{R}^3$),
• Edges $e_{ij} \in E$ signifying candidate traversable transitions, each assigned a traversal risk weight $w_{ij}$.

Node validity is encoded by terrain stability and reachability criteria: $$s_i = \begin{cases} \mathrm{valid}, & g(c_i)=1\land r(E_i)=1 \ \mathrm{invalid}, & \text{otherwise} \end{cases}$$ where $g(c_i)$ measures local height deviation and $r(E_i)$ requires incident connectivity.

Edge construction mandates (i) sufficient sample density, (ii) satisfaction of height/slope constraints,

$$\frac{|P_i(z)-P_j(z)|}{\|P_i(x,y)-P_j(x,y)\|} < \tan^{-1}\left(\frac{h_{\max}}{r_{\rm robot}}\right),$$

and risk weights

$v_i \in V$0

where $v_i \in V$1 and $v_i \in V$2 are direction-dependent tip/slip risks parameterized by PCA eigenvectors of local elevation (Lee et al., 3 Jan 2025).

Stochastic and Correlated Risk TRG

For planetary exploration, TRGs generalize as directed graphs $v_i \in V$3 with edge partition $v_i \in V$4:

• $v_i \in V$5: deterministic edges, cost $v_i \in V$6;
• $v_i \in V$7: stochastic edges, two realizations $v_i \in V$8 (low-cost), $v_i \in V$9 (high-cost).

Edge risk is modeled via conditional probability functions: $r_{\rm robot}$0 where $r_{\rm robot}$1 tracks observed edge statuses, admitting cost correlation via shared latent traversability functions and Bayesian belief updates (Lamarre et al., 19 May 2025).

Multi-Agent Edge-Risk TRG

For team coordination, TRGs annotate edges with risk and support dynamics:

• Risky edge $r_{\rm robot}$2 costs $r_{\rm robot}$3; with synchronized support from a robot at a designated node, traversal cost reduces to $r_{\rm robot}$4 for the traversing robot, with support incurring $r_{\rm robot}$5 for the supporter.
• System state comprises robots' positions and scheduled coordination variables (Zhou et al., 2024).

2. Construction and Hierarchical Management

Wavefront Propagation

TRGs are constructed incrementally via controlled sampling ("wavefront propagation"):

• Initialize at the robot's current location.
• Expand candidate nodes within a radius $r_{\rm robot}$6, prune unstable regions, merge nearby samples, and add edges if slope/height/risk criteria are satisfied.
• At each expansion, only locally reachable, stable nodes and edges are added; terminal graph is sparse, reflecting real terrain constraints (Lee et al., 3 Jan 2025).

Hierarchical Management

To maintain scalability for real-time planning:

• Extract local subgraphs within sensing range $r_{\rm robot}$7 for each planning cycle.
• Detect "frontier nodes"—valid nodes with adjacent unexplored space—seed further wavefront expansion as needed.
• Merge expanded local graphs into the global TRG, maintaining feasible coverage while avoiding combinatorial explosion (Lee et al., 3 Jan 2025).

3. Traversal Risk Modeling and Information Representation

Risk within TRGs can be deterministic (geometric weighting) or stochastic (cost distribution), with advanced models accommodating inter-edge correlations:

• Geometric risk weights combine slope, elevation variance, and directionality (e.g., principal axes of fitted elevation ellipses).
• Stochastic cost realizations reflect uncertain traversability, with the risk function $r_{\rm robot}$8 determined by Bayesian integration over candidate feature-to-cost functions, updating beliefs on each new observation to capture correlation structures (e.g., edges sharing terrain features) (Lamarre et al., 19 May 2025).
• In multi-agent TRGs, risk includes both the base edge cost and the cost-reducing effect of synchronized support actions by other robots, requiring explicit coordination schedules (Zhou et al., 2024).

4. Optimization Paradigms

Single-Agent Path Planning

TRG-based planners seek paths $r_{\rm robot}$9 that optimize a weighted sum of distance and risk: $P_i \in \mathbb{R}^3$0 with node/edge validity constraints. A*-style search is used, with cost function

$P_i \in \mathbb{R}^3$1

and admissible heuristics (Lee et al., 3 Jan 2025).

Risk-Averse Policy Optimization (CVaR-CTP)

With stochastic/correlated risks, the policy $P_i \in \mathbb{R}^3$2 is optimized for conditional value-at-risk: $P_i \in \mathbb{R}^3$3 where CVaR quantifies expected cost in the worst $P_i \in \mathbb{R}^3$4-tail of outcomes. This requires non-trivial policy search due to time-inconsistency; AND–OR forward search trees are grown, with nodes augmented by running cost, and policy evaluation reduced to truncated expectation subproblems: $P_i \in \mathbb{R}^3$5 for all candidate thresholds $P_i \in \mathbb{R}^3$6. The CVaR-CTP-AO* algorithm incrementally expands the policy tree, propagating $P_i \in \mathbb{R}^3$7 bottom-up and selecting $P_i \in \mathbb{R}^3$8 minimizing $P_i \in \mathbb{R}^3$9 (Lamarre et al., 19 May 2025).

Team Coordination Optimization

Team TRG traversal is formulated as a constrained optimization: $e_{ij} \in E$0 subject to joint-move and coordination constraints between $e_{ij} \in E$1 robots. Methods include:

• Joint-State Graph (JSG) expansion (exponential in $e_{ij} \in E$2),
• Coordination-Exhaustive Search (CES) (optimal when support use is limited),
• Receding-Horizon Sub-Team Planning (RHOC-A*) (scalable, near-optimal) (Zhou et al., 2024).

5. Empirical Validation and Applications

Geometric TRGs in Unstructured Navigation

• In simulation over 50×50 m mountainous settings, TRG-planner achieved $e_{ij} \in E$3 planning and travel success for tasks up to 30 m, with 10–30% shorter paths and 2–5× faster planning times than vanilla A*, PRM*, T-Hybrid planners.
• Real-world experiments: TRG-planner reliably planned safe and distance-efficient paths over 245 m in challenging terrain, including identifying a unique safe ascent route on a 35° slope.
• At the ICRA 2023 Quadruped Robot Challenge, TRG-planner supported autonomous navigation through five difficult obstacle sections, enabling competitive first-place performance (Lee et al., 3 Jan 2025).

CVaR-CTP in Planetary Mobility

• Jezero instance: varying risk-aversion $e_{ij} \in E$4 altered policy complexity (mean cost $e_{ij} \in E$5 sols for $e_{ij} \in E$6, with 3% chance of $e_{ij} \in E$7-sol tail; more conservative plans at lower $e_{ij} \in E$8 values executed information-gathering detours and reduced risk exposure).
• Midway instance: for nine stochastic edges, risk-averse policies increasingly avoided probing risky edges, balancing mean cost and tail risk as $e_{ij} \in E$9 decreased.
• Policy CVaR convergence was monotonic; correlation-aware models naturally resulted in adaptive, information-seeking traverses (Lamarre et al., 19 May 2025).

Team Coordination on Graphs

• Across 45 random graphs and $w_{ij}$0 trials, JSG methods attained optimality for small teams but lacked scalability. CES remained near-optimal when support pair count was limited; RHOC-A* managed 85–95% optimality and practical runtimes for larger teams and graphs (Zhou et al., 2024).

6. Comparative Summary and Methodological Implications

Aspect	TRG (Lee et al., 3 Jan 2025)	CVaR-CTP TRG (Lamarre et al., 19 May 2025)	Team Coordination TRG (Zhou et al., 2024)
Edge Model	Geometric risk weights	Stochastic, correlated costs	Risky vs. supported cost
Optimization	Weighted A* search	AND–OR search, CVaR objective	Joint-state, coordination-based, RHOC-A*
Uncertainty Handling	Deterministic	Bayesian update, correlation	Explicit scheduling, NP-hard combinatorics
Application Domain	Unstructured terrain	Planetary routes, Martian maps	Multi-robot teams, risky traverses
Empirical Results	Fast, safe paths; robust	Policy adaptation, risk tails	Near-optimal coordination, scalability

TRGs support formal risk modeling, uncertainty propagation, and context-sensitive policy optimization. Their construction and use allow both single-agent and multi-agent systems to adapt navigation strategies to uncertain and adverse environments, with empirical advantages for safety and efficiency.

7. Ongoing Extensions and Research Directions

Current advancements include installing TRGs across diverse robot morphologies (wheeled, hybrid systems), integrating semantic terrain classification, and extending risk-aware path-planning frameworks for planetary, urban, and disaster domains (Lee et al., 3 Jan 2025). In multi-agent contexts, scalable algorithms balancing optimality and runtime remain an active area (Zhou et al., 2024). For stochastic risk TRGs, improving correlation modeling and adaptive exploration is ongoing (Lamarre et al., 19 May 2025). A plausible implication is that richer TRG models will facilitate safe autonomy in increasingly challenging environments, especially where traversability is ambiguous and catastrophic outcomes must be proactively avoided.