Shortest Trajectory of a Dubins Vehicle with a Controllable Laser

Abstract

We formulate a novel planar motion planning problem for a Dubins-Laser system that consists of a Dubins vehicle with an attached controllable laser. The vehicle moves with unit speed and the laser, having a finite range, can rotate in a clockwise or anti-clockwise direction with a bounded angular rate. From an arbitrary initial position and orientation, the objective is to steer the system so that a given static target is within the range of the laser and the laser is oriented at it in minimum time. We characterize multiple properties of the optimal trajectory and establish that the optimal trajectory for the Dubins-laser system is one out of a total of 16 candidates. Finally, we provide numerical insights that illustrate the properties characterized in this work.

Figures (14)

• Figure 1: Problem Description. The red dot depicts the static target located at the origin and the blue dot represents the Dubins vehicle. The laser is depicted by the yellow (longer) arrow and the blue (shorter) arrow depicts the orientation of the Dubins vehicle.
• Figure 2: Illustration for proof of Theorem \ref{['thm:singular_omega']}. The red dot denotes the target and the gray dashed circle denotes $\partial\mathcal{C}$. $|\psi(t_l^*)-\psi(t_f)| > \pi$ and $\text{sgn}(\psi(t_l^*)-\psi(t_f))=-1$. Thus, $\omega^*=\omega_M$.
• Figure 3: Illustration of trajectories $\mathcal{T}(t)$ and $\mathcal{T}'(t)$ for proof of Lemma \ref{['lem:same_sign']}. The red dot denotes the target and the gray dashed circle denotes $\partial \mathcal{C}$. The blue dot denotes the Dubins vehicle. Since the pose trajectory is same in both $\mathcal{T}(t)$ and $\mathcal{T}'(t)$, it is denoted by a single blue curve. The laser turns anti-clockwise (depicted in yellow color) in $\mathcal{T}(t)$ and clockwise (depicted in brown color) in $\mathcal{T}'(t)$. $\Delta\psi'>\Delta \psi$ holds.
• Figure 4: An illustration of $RS$ type trajectory when $t_l^*>0$ with the laser turning clockwise.
• Figure 5: An illustration of $RL$ type trajectory when $t_l^*=0$.

Theorems & Definitions (46)

• Theorem III.1
• proof
• Corollary 1
• proof
• Lemma 1
• proof
• Theorem III.2
• proof
• Lemma 2
• proof