A Lower Bound on Observability for Target Tracking with Range Sensors and its Application to Sensor Assignment

State estimation is a fundamental problem in sensorics and finds many applications such as localization, mapping, and target tracking [1, 2]. The estimator performance can be improved by exploiting the observability of the underlying system [3, 4, 5, 6]. We study selecting sensors to improve the observability in tracking a potentially mobile target.

Observability is a basic concept in control theory and has been widely applied in robotics. Observability for range-only beacon sensors, in particular, has been closely studied for underwater navigation. Gadre and Stilwell [3] analyzed the local and global observability [7] for the localization of an Autonomous Underwater Vehicle by an acoustic beacon. The problems of single vehicle localization and multi-vehicle relative localization are studied in [5] using an observability criterion introduced in [8]. In these works, it is the sensors that are moving. Consequently, the sensors know their control vectors and can thus, compute the observability matrix and its measures. In target tracking with fixed or mobile sensors, however, the control inputs for the targets are unknown. In recent work, Williams and Sukhatme [6] studied a multi-sensor localization and target tracking problem where they showed how to leverage graph rigidity to improve the observability for sensor team localization and robust target tracking.

While others have studied similar problems in the past [6, 5], we focus uniquely on the case when the control inputs for the target are not known to the sensors. Consequently, we cannot compute the observability matrix of the resulting system. Our first contribution is to present a novel lower bound on the observability for the case of unknown target motion tracked by range-only sensors. Specifically, we show how to lower bound the condition number [8] of the partially known observability matrix using only the known part (Section III).

We then study two sensor assignment problems. In the first problem, the goal is to assign unique pairs of sensors to a target. Our second contribution is a greedy assignment algorithm for this problem. We prove that the greedy algorithm achieves a $1/3$–approximation of the optimal solution. We show that the greedy algorithm performs much better than $1/3$ in practice (Section V-C). Since the optimal solution cannot be computed, in order to compare the greedy approach we also consider a relaxed version of the assignment problem which gives an upper bound for the optimal.

We then study a general assignment problem where a set of sensors are to be assigned to a specific target. There are no restrictions on the number of targets assigned to a specific sensor. Instead, we let the algorithm decide the optimal configuration of sensor teams assigned to each target. If the weight function is submodular and monotone¹¹1We use “monotone” and “monotone increasing” interchangeably., a greedy algorithm gives a $1/2$–approximation [9]. However, our third contribution is to prove that the lower bound of the inverse of the condition number is neither submodular nor monotone. Instead, we use other observability measures such as the trace, log determinant and rank of symmetric observability matrix, and the trace of inverse symmetric observability matrix and show them to be submodular and monotone. We evaluate this algorithm through simulations where we find that sensors are assigned to targets almost uniformly (Section V-D).

We consider a scenario where there are $N$ sensors and $L$ targets in the environment. Our goal is to assign sensors to track the target. We use the notation $\sigma(l)$ to represent the set of sensors assigned to target $t_{l}$. Similarly, let $\sigma^{-1}(i)$ give the set of targets assigned to sensor $s_{i}$. We also use $\sigma_{i}(l)$ to give the $i^{th}$ sensor assigned to $t_{l}$. We order the assigned sensors by using their IDs such that $\sigma_{1}(l)<\sigma_{2}(l)<\sigma_{3}(l)<\ldots$. Let $\omega(s_{i},s_{j},t_{l})$, and $\omega(S_{l},t_{l})$ be some measure of the observability of tracking $t_{l}$ with $s_{i}$ and $s_{j}$, and with a set of sensor(s) $S_{l}$, respectively.

We study the following sensor assignment problems. We start with the problem of assigning pairs of sensors to each target.²²2Theorems 2 and 3 show that at least two sensors are necessary when $\omega(\cdot)$ is the inverse condition number of the observability matrix.

We then study the general version of the problem where each target can be tracked by more than two sensors. That is, the sensors form teams of varying sizes to track individual targets. Sensors within a team can share measurements so as to better track the targets. We constrain each sensor to be assigned to only one target, that is, communicate with only one team. This is motivated by scenarios where sensing multiple targets can be time consuming (as is the case with radio sensors [10]) or communicating multiple measurements can be time and energy consuming.

We present a $1/3$–approximation to solve Problem 1 for any $\omega(\cdot)$. Problem 2 is the more general assignment problem which is difficult to solve for arbitrary $\omega(\cdot)$. However, for the specific class of submodular functions, there exists a $1/2$–approximation by a greedy algorithm [9]. Submodularity captures the notion of diminishing returns, i.e., the marginal gain of assigning an additional sensor diminishes as more and more sensors are assigned to track the same target.

A typical measure of observability is the condition number of the observability matrix. When the target is moving, the condition number of the observability matrix cannot be computed, since the control input for the target is unknown to the sensors. We find a lower bound on the inverse of the condition number of the observability matrix. We use this lower bound as our measure, i.e., $\omega(\cdot)$, to find the assignments in Problem 1. However, we show that this measure is not submodular (in fact, not even monotone increasing). Instead, we can use a number of other measures (e.g., the trace of symmetric observability matrix, the trace of inverse symmetric observability matrix, log determinant and rank of the symmetric observability matrix) which we know to be submodular and monotone increasing (Theorem 8). We start by how to bound the inverse of condition number.

Consider a mobile target whose position is denoted by $o$. Suppose there are $N$ stationary sensors that can measure the distance³³3We use the square of the distance/range for mathematical convenience. to the target. We have:

where $o:=[o_{x},o_{y}]^{T}$ gives the 2D position of the target, and $u_{o}:=[u_{ox},u_{oy}]^{T}$ defines its control input, which is unknown. We assume an upper bound on the control input, given by $u_{o,\max}=\max\|u_{o}\|_{2}$. $z_{i}$ defines the range-only measurement from each sensor $s_{i}$ whose position is given by $p_{i}=[p_{ix},p_{iy}]^{T}$. For simplicity, we also assume that the target does not collide with any sensor, i.e., $\|p_{i}-o\|_{2}\neq 0$ and no two sensors are deployed at the same position.

We analyze the weak local observability matrix, $O(o,u_{o})$, of this multi-sensor target tracking system. We show how to lower bound the inverse of the condition number of $O(o,u_{o})$, given by $C^{-1}(O(o,u_{o}))$, independent of $u_{o}$. We also show that the lower bound, $\underline{C}^{-1}(O_{i}(o,u_{o}))$, is tight.

We compute the local nonlinear observability matrix [7, 6] for this system (Equation 3) as,

This equation can be rewritten as,

The state of the target $o$ is weakly locally observable if the local nonlinear observability matrix has full column rank [7]. However, the rank test for the observability of the system is a binary condition which does not tell the degree of the observability or how good the observability is. The condition number [8], defined as the ratio of the largest singular value to the smallest, can be used to measure this degree of unobservability. A larger condition number suggests worse observability. We use the inverse of condition number given as,

Note that, $C^{-1}\in[0,1]$. $C^{-1}=0$ means $O(o,u_{o})$ is singular and $C^{-1}=1$ means $O(o,u_{o})$ is well conditioned. A larger $C^{-1}$ means better observability (see more details in [5]).

In the local nonlinear observability matrix $O(o,u_{o})$, $u_{o}$ is unknown and not controllable by the sensor. On the other hand, $o-p_{i}$, depends on the relative state between each sensor $s_{i}$ and target $o$ and is known to the sensor (assuming an estimate of the target’s position is known). The system can control $o-p_{i}$ either by moving the sensors or assigning new sensors to track the target.

We present the full proof for this and all other results in the in the appendix.

We wish to improve the worst case, i.e., the lower bound of $C^{-1}(O(o,u_{o}))$, by optimizing the sensor-target relative state, which can be controlled by the sensor. For example, if the sensors are mobile, they can move so as to improve the lower bound. If only a subset of sensors are active at a time, we can choose the appropriate subset to improve the lower bound. In the following, we will show that at least two sensors are required to improve the lower bound.

When the number of sensors, $N\geq 2$, we have a positive result that shows that the sensors can improve the lower bound on the condition number of optimizing their positions.

Consider the special case when $N=2$. We use $i$ and $j$ to denote the two sensors tasked with tracking the target. The local observability matrix $O_{i,j}(o,u_{o})$ for the $i-j-o$ system is given by,

For ease of notation, we represent the relative sensor-target position and sensor-target orientation in polar coordinates with the target at the center:

where $d_{io}$, $d_{jo}$, $\theta_{i}$ and $\theta_{j}$ indicate the relative distances and orientations between sensors $i,j$ and target $o$ as shown in Figure 1.

We plot $\underline{C}^{-1}(O_{i,j}(o,u_{o}))$ as a function of $\alpha\in[0,5]$ and angle $\theta_{ji}\in[-\pi,\pi]$ with $u_{o,\max}$ and $d_{io}$ selected as $1~m/s$ and $1~m$, respectively (Figure 2). Note that $\underline{C}^{-1}(O_{i,j}(o,u_{o}))$ reaches its maximum when $\alpha=1$ and $\theta_{ji}=\pm\frac{\pi}{2}$. That is, both sensors are at the same distance from the target and are perpendicular with respect to the target. On the other hand, when $\theta_{ji}=0$, $\theta_{ji}=\pi$, $\alpha=0$ or $\alpha\to\infty$, $\underline{C}^{-1}(O_{i,j}(o,u_{o}))$ reaches zero. We summarize the results in the following theorem.

Next, we solve two assignment problems for selecting sensor(s) to track target(s).

So far, we have assumed that we know the true position, $o(k)$, of the target at time $k$. In practice, we only have an estimate, $\hat{o}(k)$, for $o$ along with its covariance $\Sigma(k)$. The estimate is obtained by fusing the past measurements obtained by the sensors using, for example, an Extended Kalman Filter (EKF).

Since the sensors know the probability distribution of target’s state with mean $\hat{o}$, and covariance $\Sigma$, they can calculate the relative distances, $d_{io}$, by using the Mahalanobis distance [11],

We illustrate the performance of the pairing and assignment strategies for sensor selection using observability measure as the performance criterion. We first consider the two sensors case for tracking a moving target, and then focus on the multi-sensor multi-target assignment problems. The video⁴⁴4https://youtu.be/Kt0yeYXyrKQ and the code⁵⁵5https://github.com/lovetuliper/observability-based-sensor-assignment.git of our simulations are available online.

In this paper, we solved sensor assignment problems to improve the observability for target tracking. We derived the lower bound on the inverse of the condition number of the observability matrix for a system with a mobile target and $N$ stationary sensors. The lower bound considers only the known part of the observability matrix — the sensor-target relative position and an upper bound on the target’s speed. We showed how this lower bound can be employed for sensor selection. We considered two sensor assignment problems for which we presented constant-factor approximation algorithm. Our immediate work is focused on assigning sensors to cover an area instead of tracking a group of targets. Another avenue is designing an efficient set covering strategy based on observability measures which are submodular and monotone.