The Orbiting Dubins Traveling Salesman Problem: planning inspection tours for a minehunting AUV

Abstract

The Orbiting Dubins Traveling Salesman Problem (ODTSP) is to plan a minimum-time tour for a Dubins vehicle model to inspect a set of targets in the plane by orbiting each target along a circular arc. This problem arises in underwater minehunting, where targets are mine-like objects on the sea bottom that are inspected by a sonar-equipped underwater vehicle. Each orbit subtends a prescribed angle so that the target’s acoustic response is measured from a variety of target-sensor relative geometries to aid in classifying it. ODTSP tours consist of circular-arc orbits joined by Dubins paths, and the optimization problem is to partition the set of targets into orbits and determine the position, radius, direction, and vehicle entry angle of each. Algorithms are presented for the restricted case, where each orbit inspects a single target (only), and the general case, where orbits inspect multiple targets. The approach is facilitated by analytical conditions that identify admissible clusters of targets as cliques of a disk graph. The ODTSP is extended to consider path planning in the presence of a steady uniform current.

Appendices

A Performance bounds for the ODTSP

The following Corollaries establishes performance bounds for the restricted ODTSP.

Corollary 2

(Worst case performance) Suppose that $||\mathbf{v}_i - \mathbf{v}_j || \ge 2 s_{\mathrm{min}}$ for all $\mathbf{v}_i, \mathbf{v}_j \in V_t$. Let $J_{\mathrm{NC}}^*$ be the cost of the optimal ODTSP tour with no clustering and a fixed orbit radius $s_{\mathrm{min}}$, i.e., $n_o = n_t$, and $\rho _i = s_{\mathrm{min}}$ for all $i \in {\mathcal {I}}$. Let $J_{\mathrm{NC}}$ be the cost of a tour obtained by either SymAngle or NoCluster. The cost $J_{\mathrm{NC}}$ is within a factor $J_{\mathrm{NC}} \le \lambda J_{\mathrm{NC}}^*$ of the optimal solution, where $\lambda$ is given in (38).

Proof

First, refer to Theorem 3.1 in (Savla et al. 2008) which establishes that the length of a Dubins path joining any two states $\mathbf{q}_i =(x_i,y_i,\psi _i)$ and $\mathbf{q}_j = (x_j,y_j,\psi _j)$ is bounded from above by

$$\begin{aligned} D_p(\mathbf{q}_i,\mathbf{q}_j) \le d_{ij} + \kappa \pi R \;, \end{aligned}$$

(33)

where $\kappa \in [2.657, 2.658]$ is a constant. For the ODTSP, if $||\mathbf{v}_i - \mathbf{v}_j || \ge 2 s_{\mathrm{min}}$, the cost of the inter-orbit path $D_{p}(\mathbf{q}_i, \mathbf{q}_{j})$ with $\mathbf{q}_i\in Q(\mathbf{v}_i, s_{\mathrm{min}})$ and $\mathbf{q}_{j} \in Q(\mathbf{v}_{j}, s_{\mathrm{min}})$ is bounded by

$$\begin{aligned} ||\mathbf{v}_i - \mathbf{v}_{j} || - 2 s_{\mathrm{min}} \le D_{p}(\mathbf{q}_i, \mathbf{q}_{j}) \le ||\mathbf{v}_i - \mathbf{v}_{j} || + 2 s_{\mathrm{min}} + \kappa \pi R \;. \end{aligned}$$

(34)

Now proceed in parallel to Theorem 3.1 in (Isaacs et al. 2011). Apply the inequality (34) to each pair of targets $\mathbf{v}_i, \mathbf{v}_j \in V_t$ to obtain the least lower bound

$$\begin{aligned} \lambda _{\mathrm{LB}} \triangleq \min _{\mathbf{v}_i, \mathbf{v}_j \in V_t} || \mathbf{v}_i - \mathbf{v}_j || - 2 s_{\mathrm{min}} \;, \end{aligned}$$

(35)

and the greatest upper bound

$$\begin{aligned} \lambda _{\mathrm{UB}} \triangleq \max _{\mathbf{v}_i, \mathbf{v}_j \in V_t} || \mathbf{v}_i - \mathbf{v}_j || + 2 s_{\mathrm{min}} + \kappa \pi R \;. \end{aligned}$$

(36)

Using (7) and (8), the ratio of tour costs is

$$\begin{aligned} \frac{J_{\mathrm{NC}}}{J_{\mathrm{NC}}^*}&= \frac{\sum _{i=1}^{n_t-1} D_p({\mathcal {T}}_i, {\mathcal {T}}_{i+1}) + D_p({\mathcal {T}}_{n_t}, {\mathcal {T}}_1) + n_t s_{\mathrm{min}} \Theta }{\sum _{i=1}^{n_t-1} D_p({\mathcal {T}}_i^*, {\mathcal {T}}_{i+1}^*) + D_p({\mathcal {T}}_{n_t}^*, {\mathcal {T}}_1^*) + n_t s_{\mathrm{min}} \Theta } \;, \end{aligned}$$

(37)

where ${\mathcal {T}}_i^*$ is the optimal set of orbit parameters, and ${\mathcal {T}}_i$ is the set of orbit parameters obtained with the SymAngle or NoCluster algorithm. For all $i \in \{1,\ldots ,n_t-1\}$ the bounds (35) and (36) imply that $\lambda _{\mathrm{LB}} \le D_p({\mathcal {T}}_i, {\mathcal {T}}_{i+1}) \le \lambda _{\mathrm{UB}}$. Similarly, the bounds imply $\lambda _{\mathrm{LB}} \le D_p({\mathcal {T}}^*_{n_t}, {\mathcal {T}}^*_{1}) \le \lambda _{\mathrm{UB}}$. Substitute these inequalities into (37) to obtain

$$\begin{aligned} \frac{J_{\mathrm{NC}}}{J_{\mathrm{NC}}^*}&\le \frac{\lambda _{\mathrm{UB}} + s_{\mathrm{min}} \Theta }{\lambda _{\mathrm{LB}} + s_{\mathrm{min}} \Theta } \triangleq \lambda \; . \end{aligned}$$

(38)

$\square$

Corollary 3

(Bounds on optimal tour cost) Suppose that $||\mathbf{v}_i - \mathbf{v}_j || \ge 2s_{\mathrm{min}}$ for all $\mathbf{v}_i, \mathbf{v}_j \in V_t$. The optimal cost, $J_{\mathrm{NC}}^*$, of the ODTSP tour without clustering is bounded by

$$\begin{aligned} L_{\mathrm{ETSP}} - \eta ^- \le J_{\mathrm{NC}}^* \le L_{\mathrm{ETSP}} + \eta ^+ \;, \end{aligned}$$

where $\eta ^- \triangleq s_{\mathrm{min}}n_t(2 - \Theta )$, $\eta ^+ \triangleq n_t(s_{\mathrm{min}}(2 + \Theta ) + \kappa \pi R )$, and $L_{\mathrm{ETSP}}$ is the length of the optimal ETSP tour through $V_t$.

Proof

Let $\gamma _{\mathrm{ETSP}}= \{{(\mathbf{v}}_{i_1},\mathbf{v}_{i_2}), \ldots , (\mathbf{v}_{i_{n_t}},\mathbf{v}_{i_{1}})\}$ be the optimal ETSP tour through the points $V_t$ with length

$$\begin{aligned} L_{\mathrm{ETSP}} = \sum _{j=1}^{n_t -1} || \mathbf{v}_{i_j} - \mathbf{v}_{i_{j+1}} || ~+~ || \mathbf{v}_{i_{n_t}} - \mathbf{v}_{i_{1}} || \;. \end{aligned}$$

(39)

Let $J_{\mathrm{NC}}$ be the cost of a ODTSP tour with orbit parameters $\{{\mathcal {T}}_1, \ldots , {\mathcal {T}}_{n_t} \}$ that passes through the targets in the order specified by $\gamma _{\mathrm{ETSP}}$. Using (7) and (8),

$$\begin{aligned} J_{\mathrm{NC}}&= \sum _{i=1}^{n_t-1} D_p({\mathcal {T}}_i, {\mathcal {T}}_{i+1}) + D_p({\mathcal {T}}_{n_t}, {\mathcal {T}}_1) + n_t s_{\mathrm{min}} \Theta \;. \end{aligned}$$

(40)

Applying the upper bound in (34) to each $D_p$ term in (40) gives

$$\begin{aligned} J_{\mathrm{NC}}&\ge \sum _{j=1}^{n_t -1} || \mathbf{v}_{i_j} - \mathbf{v}_{i_{j+1}} || ~+~ || \mathbf{v}_{i_{n_t}} - \mathbf{v}_{i_{1}} || \nonumber \\&+ n_t[s_{\mathrm{min}}(2+\Theta ) + \kappa \pi R )] \;. \end{aligned}$$

(41)

Substituting (39) into (41) yields $J_{\mathrm{NC}}^* \le J_{\mathrm{NC}} \le L_{\mathrm{ETSP}} + \eta ^+$.

To determine the lower bound, suppose that an optimal ODTSP tour passes through a sequence of points whose tour length is $L_{\mathrm{NC}}$ (with $L_{\mathrm{ETSP}} \le L_{\mathrm{NC}}$ by definition of an ETSP). Applying a similar procedure establishes $L_{\mathrm{ETSP}} - \eta ^- \le L_{\mathrm{NC}} - \eta ^- \le J_{\mathrm{NC}} \;.$ $\square$

Time complexity of the ODTSP

The time complexity¹ of the ODTSP algorithms is driven by the tour computation and, for clustering algorithms only, clique enumeration. The minimum enclosing circle computation is linear in the number of targets $O(n_t)$ (Megiddo 1983). Other procedures, such as evaluating the Dubins path cost matrix or sorting cliques, do not drive the complexity. The complexity is summarized in Table 1, with the number of targets $n_t$ and number of entry angles $n_e$ as inputs.

Table 1 Worst-case time complexity of the ODTSP

The complexity of the restricted ODTSP algorithms is determined as follows. The LKH solver in SymAngle has numerically established complexity $O(n^{2.2})$ (Helsgaun 2000). Since the ETSP in SymAngle has $n_t$ nodes, SymAngle is $O(n_t^{2.2})$. An asymmetric GTSP can be transformed into an asymmetric TSP with the same number of nodes (Behzad and Modarres 2002), and further into a TSP with twice as many nodes (Jonker and Volgenant 1983). Since NoCluster GTSP contains $n_e n_t$ nodes, it is $O([n_en_t]^{2.2})$.

The complexity of the general ODTSP algorithms is established similarily. A GTSP with intersecting nodesets can be transformed into an instance of the GTSP with non-intersecting nodesets by creating $|F_v|$ additional replica nodes, for each node $v \in V$, where $F_v$ is the membership set indicating which nodesets v belongs to (Lien et al. 1993). ExhaustCluster contains $n_e n_c$ nodes grouped into at most $n_c$ intersecting nodesets, where $n_c$ is the number of candidate orbits. The worst-case number of candidate orbits $n_c = 2^{n_t}$ occurs when each element of the powerset of ${\mathcal {I}}$ is an admissible orbit. Thus, this GTSP has at most $2^{n_t}n_e$ nodes and $2^{n_t}n_e n_t$ replica nodes. The corresponding TSP is $O([2^{n_t}n_e(1+n_t)]^{2.2})$. Determining admissible orbits requires enumerate all cliques of a disk graph with $n_t$ nodes, which is also computationally expensive. The following steps yield all cliques: enumerate all maximal cliques $O(3^{n/3})$ (Tomita et al. 2006), compute the powerset for each maximal clique $O(2^n)$, remove duplicate cliques from the resulting list O(n). Therefore, enumerating all cliques is no worse than $O( 2^{n_t} n_t )$, and the complexity of ExhaustCluster is dominated by the TSP. Clearly, ExhaustCluster is not practical for a large number of targets or entry angles, but it provides a useful benchmark for comparing the tour cost of MaxCluster. The GTSP in MaxCluster has the same complexity as NoCluster, $O([n_en_t]^{2.2})$, since the number of orbits is no greater than $n_t$. Enumerating all maximal cliques is $O(3^{n_t/3})$ (Tomita et al. 2006), thus MaxCluster is $O(\mathrm{max}\{[n_t n_e]^{2.2},3^{n_t/3}\})$. The analysis indicates that the algorithms SymAngle, NoCluster, MaxCluster, and ExhaustCluster increase in complexity and confirms the trend observed in the Monte Carlo simulation results.