Spatiotemporal Tube-Based Safety-Certificate for Autonomous Navigation of Articulated Vehicles

Notation and Preliminaries

The symbols $\mathbb{R},\mathbb{R}_{0}^{+},\mathbb{R}^{+},\mathbb{Z}^{+},\mathbb{Z}_{0}^{+},\mathbb{S}^{1}$, and $\mathbb{S}^{2}_{++}$ denote sets of reals, non-negative reals, positive reals, positive integers, non-negative integers, unit circle, and $2\times 2$ symmetric positive definite matrices, respectively.

The symbol $u^{\perp}$ represents an orthogonal component of the vector $u$ expressed as $u^{\perp}=R(90^{0})u$, $u\in\mathbb{R}^{2}$ and $R(90^{0})$ is the rotation matrix corresponding to a $90^{0}$ counter-clockwise rotation.

Problem Formulation

Autonomous navigation of kinematically-coupled interconnected systems is a complex task and reactive approaches cannot guarantee safety. In this study, we consider synthesizing a confinement zone in the form of an STT for the prime mover and subsequently for $N$ trailers using kinematic coupling, such that the complete articulated vehicle remains within the given route at all times, as shown in Figure 1 .

The problem considered in this paper includes the prime mover towing $N$ trailers, each assumed to have one axle.

PATH PLANNING

The path planning problem here now becomes finding the STET of the prime mover such that all trailers remain confined to the route $\mathcal{R}(t)$. We use the kinematic geometry as illustrated in Figure 1 to find the feasible tubes for the towed trailers. Each towed trailer can sway by an angle $\delta_{i}(t)\in[-\theta_{i},\theta_{i}]$ relative to its leading vehicle hinge.

Safe path plan certificate for a route

In this section, we arrive at a safety certificate for an articulated vehicle in a particular configuration on a given route.

For a multi-trailer configuration consisting of a prime mover with each body represented by a STET, we deduce an algorithm, as stated in Algorithm 1 , that adjusts the center and shape matrix of the prime mover’s ellipsoid based on violations by the last trailer, and plans the center and shape matrix of the prime mover such that all follower trailers are safe.

Simulations

We prove the effectiveness of our planning approach using a truck as a prime mover towing multiple trailers. To test our planning method, we consider multiple scenarios, as listed in Table I , configured using different parameters, viz.

, the number of subsystems $N+1$, hitch length $L$, sway limit $\pm\theta$, trailer dimension $\mathcal{D}(length\times width)$, bounds $h_{\Omega}$ and $h_{\Xi}$, and the minimum route half width $w_{min}$.

Sources and Demonstrations

Figures are reproduced from Faruqui et al. See the full paper for experimental details and the project page, when the paper links one, for demonstrations and videos.