Analysis of Indistinguishable Trajectories of a Nonholonomic Vehicle Subject to Range Measurements

Abstract

We propose a global constructibility analysis for a vehicle moving on a planar surface. Assuming that the vehicle follows a trajectory that can be uniquely identified by the sequence of control inputs and by some intermittent ranging measurements from known points in the environment, we can model the trajectory as a rigid body subject to rotation and translation in the plane. This way, the localisation problem can be reduced to finding the conditions for the existence of a unique roto-translation of the trajectory from a known reference frame to the world reference frame, given the collected measurements. As discussed in this paper, such conditions can be expressed in terms of the shape of the trajectory, of the layout of the ranging sensors, and of the numbers of measurements collected from each of them. The approach applies to a large class of kinematic models. Focusing on the special case of unicycle kinematics, we provide additional local constructibility results.

Figures (7)

• Figure 1: Example \ref{['ex:rotation']}. The same trajectory $\mathcal{T}$ rotated about the pivot anchor $B_1$. When $\mathcal{P}_3$, $\mathcal{P}_4$ and $B_1$ are collinear, we always have two roto-translations of $\mathcal{T}$ that are compliant with the measurements.
• Figure 2: Example \ref{['ex:translation']}. The same trajectory $\mathcal{T}$ translated by $2\Delta$ orthogonally to $\mathcal{S}_{0,1}$. When $\mathcal{S}_{0,1}$, $\mathcal{S}_{2,3}$ and $\mathcal{S}_{4,5}$ are parallel and have the same distance $\Delta$ from the anchor collecting the measurements, we always have two translations of $\mathcal{T}$ that are compliant with the measurements.
• Figure 3: Locus where the third measurement point $\mathcal{P}_2$ can lie after the first two measurements $\rho_0$ and $\rho_1$ (dashed lines) are collected from the first two anchors. The blue and red colours are associated with the two intersections between the aforementioned circles. The solid green circle represents the third measurement $\rho_2$ collected by $B_3$ in a $1+1+1$ setting. After $\rho_2$, the blue and the red trajectories are no more indistinguishable ($\mathcal{P}_2^{(b)}$ does not lie on the green circle), but there are still $6$ intersections of the locus with the green circle, and thus $\mathcal{T}$ is $\operatorname{Ind}({6})$.
• Figure 4: New reference frame showing the setting $2+1$. The blue and red lines represent the two trajectories $\mathcal{T}^{(a)}$, $\mathcal{T}^{(b)}$. Each of them has a circle centred on their last point $\mathcal{P}_2$, hence yielding an overall number of $4$ intersections (i.e. possible positions of $B_2$) with the circle centred on $B_1$ of radius $D$.
• Figure 5: Summarising picture subsuming the taxonomy derived in this paper as a function of the overall number of measurements and of their distribution among the different anchors. The number in brackets denotes the number of indistinguishable trajectories. The red part is referred to Theorem \ref{['thm:negative_result']}, while the green part is associated with the results obtained in Theorem \ref{['thm:positive result']}.

Theorems & Definitions (22)

• Remark 1
• Definition 1
• Definition 2
• Remark 2
• Lemma 1
• proof
• Remark 3
• Definition 3
• Theorem 1
• proof