Hybrid VORRT-COLREGs Path Planner

• The paper presents a hybrid algorithm that fuses RRT’s global search with VO’s local collision avoidance, enforcing COLREGs for compliant ASV trajectories.
• It employs a two-stage RRT sampling strategy and joint forward simulation to validate trajectory feasibility in dynamic maritime environments.
• Experimental results confirm COLREGs-compliant paths with consistent minimum CPA values and rapid replan times across varying traffic densities.

The Hybrid VORRT-COLREGs algorithm is a global path planning methodology that synthesizes the velocity obstacles (VO) framework and rapidly-exploring random trees (RRT) to generate physically feasible and COLREGs-compliant trajectories for autonomous surface vessels (ASVs) subjected to dynamic multi-agent environments. This approach systematically integrates the global search capability of RRT with the local collision avoidance guarantees of VO and further encodes constraints drawn from the International Regulations for Preventing Collisions at Sea (COLREGs), specifically addressing power-driven vessel rule sets (Rules 13–17). The methodology deploys a joint forward simulation to propagate all vessels and ensures trajectory validity and compliance throughout the iterative tree expansion process (Dubey et al., 2021).

1. Problem Formulation and State Space Representation

The planning scenario is formalized by defining each vessel $i$ with a state vector

$$s_i = (P_i, v_i, \psi_i) \in \mathbb{R}^2 \times \mathbb{R} \times S^1,$$

where $P_i = (x_i, y_i)$ denotes the planar position, $v_i \geq 0$ is the speed, and $\psi_i$ is the heading. The motion of each vessel within a step $\Delta t$ is governed by

$$P_i^+ = P_i + v_i \begin{bmatrix}\cos\psi_i \ \sin\psi_i\end{bmatrix} \Delta t$$

under the assumption of constant velocity and heading over each planning step. Each other vessel is abstracted as a dynamic circular obstacle with a radius $R_j$ typically set equal to its length. A minimum separation threshold $d_\text{th}$, interpreted as the Minimum Threshold Distance (MTD) at Closest Point of Approach (CPA), is strictly enforced. No explicit acceleration or turning constraints are included in the planning process; waypoint-to-waypoint heading changes ($\Delta\psi$) are smoothed in post-processing (Dubey et al., 2021).

2. RRT-Based Global Path Exploration

The algorithm grows a tree $T$ from the current ownship state $s_a$ in the 2D spatial domain. The RRT algorithm’s sampling strategy is staged to balance exploration and exploitation:

• Stage 1: For the first 200 iterations, sampling is biased entirely ($p_1=1.0$) towards the goal position $P_g$.
• Stage 2: In the subsequent 200 iterations ($p_1=0$), uniform spatial sampling resumes.

For each sampled point $P_s$, the algorithm selects the nearest tree node

$n = \arg\min_{n} \|P_n - P_s\|$

and extends by a fixed step $d' = 100$ m, computing

$$P_\text{new} = P_n + d' \cdot \frac{P_s - P_n}{\|P_s - P_n\|}.$$

The Joint Forward Simulator (JFS) propagates all vessel positions over $\Delta t = d'/v_a$. Prior to tree expansion, each candidate extension is validated via VO and CPA checks, as well as additional COLREGs conditions. If compliance is verified, the new node is inserted; otherwise, it is discarded and sampling continues (Dubey et al., 2021).

3. Velocity Obstacles and Local Collision Avoidance

The VO formalism is applied for every candidate extension:

• Relative position and velocity between ownship ($A$) and another vessel ($B$) are

$\Delta P = P_b - P_a, \quad \Delta V = V_a - V_b.$

• Minkowski summing: Both vessels are represented as disks with radii $r_a$ and $r_b$; the effective collision radius is $R = r_a + r_b$.
• Collision cone: The opening half-angle $\alpha$ of the collision cone is

$$\alpha = \arcsin\left( \frac{R}{\|\Delta P\|} \right).$$

The collision cone is thus

$$CC_{ab} = \{ \Delta V : \theta(\Delta V, \Delta P) \leq \alpha \},$$

where $\theta(\Delta V, \Delta P)$ is the angle between $\Delta V$ and $\Delta P$.

• VO region:

$$VO_b^a(P_a, P_b, V_b) = \left\{ V_a \in \mathbb{R}^2 \mid \angle((V_a - V_b), P_b - P_a) \leq \arcsin \frac{r_a + r_b}{\|P_b - P_a\|} \right\}.$$

• CPA criteria:

$$t_\mathrm{cpa} = \begin{cases} \frac{(P_a - P_b) \cdot (V_a - V_b)}{\|V_a - V_b\|^2} & \|V_a - V_b\| > 0 \ 0 & \text{otherwise} \end{cases}$$

$$d_\mathrm{cpa} = \|P_a + t_\mathrm{cpa} V_a - (P_b + t_\mathrm{cpa} V_b)\|$$

A velocity $V_a$ is considered unsafe if either $\Delta V$ is in the collision cone, or $d_\mathrm{cpa} < d_\mathrm{th}$ (Dubey et al., 2021).

4. Joint Forward Simulation and Algorithm Integration

The planner integrates global search and local safety verification at each node extension:

1. Propose $P_\text{new}$ from current node.
2. Simulate all vessels forward by $\Delta t = d'/v_a$ using the JFS.
3. For every other vessel, perform VO test and CPA distance check; if any violation occurs, reject the extension.
4. If all tests are passed, update the tree with the new node and corresponding state.

Although cost minimization (e.g., $c(n) = c(\text{parent}) + \|P_\text{new} - P_n\|$) is possible, no explicit optimization beyond RRT sampling bias is used in the original formulation (Dubey et al., 2021).

5. Maritime Collision Regulations (COLREGs) Encoding

Rules 13–17 pertinent to power-driven vessels are explicitly embedded:

• Encounter type (overtaking, head-on, crossing) is classified using relative bearing ($\beta$) and heading differences ($\Delta\psi$).
• Logical maneuvering constraints:
  • Rule 16 (give-way): Require starboard turn of at least $\delta_{\text{min}}$ (e.g., $10^\circ$) in give-way scenarios.
  • Rule 17 (stand-on): Enforce unchanged course and speed if ownship is stand-on.
• The safe VO region is dynamically altered to enforce COLREGs; for example, in a crossing give-way situation, only candidate headings indicating a starboard turn are permitted (Dubey et al., 2021).

6. Summary of Parameter Values and Operational Settings

Parameter values selected for the planning context include:

• Step length $d' = 100$ m.
• Stage 1 RRT bias $= 1.0$ for 200 iterations, Stage 2 bias $= 0$ for 200 more.
• Maximum iterations per replan: 400.
• CPA threshold $d_\text{th} = 200$ m for open seas, reduced to 100 m in Traffic Separation Schemes.
• Minimum heading change $\delta_{\text{min}} = 10^\circ$.
• Vessel radii $r_i$ equal to the vessel’s length (conservative assumption) (Dubey et al., 2021).

7. Experiments, Results, and Evaluation

Scenarios evaluated include two-vessel interactions (crossing, overtaking, head-on), multi-vessel scenarios (up to four vessels with simultaneous complex encounters), and dense environments such as traffic separation schemes (up to 15 vessels in a 1 nautical mile channel):

• For two-vessel scenarios, the planner consistently produces path lengths $1.2$–$1.5\times$ the straight-line distance, replans within $0.05$–$0.1$ s, no collisions, and CPA $\geq 200$ m.
• Four-vessel situations achieve safe path generation in fewer than 350 nodes, with minimum CPA $\geq 300$ m.
• In high-density channels, the planner achieves COLREGs-compliant trajectory crossing, minimum CPA $\approx 135$ m, and replan times below $0.2$ s. VORRT-COLREGs consistently maintains physically feasible and COLREGs-compliant paths, with dynamic replanning as vessel states evolve under a constant-velocity assumption (Dubey et al., 2021).

Scenario	Min CPA (m)	Path Length Ratio	Replan Time (s)
Two-vessel (cross/over/head)	$\geq$ 200	1.2–1.5	0.05–0.1
Four-vessel	$\geq$ 300	–	–
Traffic Separation Scheme	$\approx$135	–	$<$0.2

A plausible implication is that the coupling of RRT exploratory capacity with VO local safety filtering is effective for both sparse and congested environments, and the proposed framework is adaptable through parameter tuning for a range of maritime navigation contexts.

• "VORRT-COLREGs: A Hybrid Velocity Obstacles and RRT Based COLREGs-Compliant Path Planner for Autonomous Surface Vessels" (Dubey et al., 2021)