Optimal Pursuit of Surveilling Agents Near a High Value Target

Abstract

We introduce a tracking evasion game comprising a single mobile pursuer, two mobile trackers and one static high value target. The trackers rely on individual measurements of the location of the target using, for instance, their individual distance to the target and are assumed to be slower than the pursuer. The pursuer seeks to minimize the square of the instantaneous distance to one of the trackers, while the trackers aim to jointly maximize a weighted combination of the determinant of the Fisher Information Matrix and the square of the distance between the pursuer and the tracker being pursued. This formulation models the objective of the trackers which is to maximize the information gathered about the target, while delaying capture. We show that the optimization problem for the trackers can be transformed into a Quadratically Constrained Quadratic Program. We then establish that the game admits a Nash equilibrium in the space of pure strategies and provide several numerical insights into the trajectories and the payoff of the mobile agents. Finally, we outline how this work can be generalized to the case of multiple trackers and multiple targets.

This work was supported by NSF Award ECCS-2030556.

7 Appendix

In this section, we provide the expression for the matrices $P,Q_j,M$ and L, respectively. For ease of notation, denote $a_i = \hat{X}_i^t$, $b_i = \hat{Y}_i^t$. Further, let ${\textbf {I}}_{n\times p}$ (resp. ${\textbf {0}}_{n\times p}$) denote the identity (resp. zero) matrix of dimension $n\times p$. Then,

$$\begin{aligned} P =\frac{1}{\sigma _{\nu }^2}\times \begin{bmatrix} {\textbf {0}}_{6\times 6} &{}~ {\textbf {0}}_{6\times 1} &{}~ {\textbf {0}}_{6\times 1} &{}~ {\textbf {0}}_{6\times 8} &{}~ {\textbf {0}}_{6\times 1}\\ {\textbf {0}}_{1\times 6} &{}~ \delta \sigma _{\nu }^2 &{}~ -\delta \sigma _{\nu }^2 &{}~ {\textbf {0}}_{1\times 8} &{}~ 0\\ {\textbf {0}}_{1\times 6} &{}~ -\delta \sigma _{\nu }^2 &{}~ \delta \sigma _{\nu }^2 &{}~ {\textbf {0}}_{1\times 8} &{}~ 0\\ {\textbf {0}}_{8\times 6} &{}~ {\textbf {0}}_{8\times 1} &{}~ {\textbf {0}}_{8\times 1} &{}~ {\textbf {0}}_{8\times 8} &{}~ {\textbf {0}}_{8\times 1}\\ {\textbf {0}}_{1\times 6} &{}~ 0 &{}~ 0 &{}~ {\textbf {0}}_{1\times 8} &{}~ 1 \end{bmatrix}, F = \begin{bmatrix} F_1 &{}~ {\textbf {0}}_{8\times 9}\\ {\textbf {0}}_{9\times 8} &{}~ F_2 \end{bmatrix}, \end{aligned}$$

where $F_1=$

$$\begin{aligned} \small \begin{bmatrix} b_2^2 &{} -a_2b_2 &{} -b_1b_2 &{} 2a_1b_2-a_2b_1 &{} -b_2 &{} b_2 &{} 0 &{} a_2b_1b_2-a_1b_2^2\\ -a_2b_2 &{} a_2^2 &{} 2a_2b_1-a_1b_2 &{} -a_1a_2 &{} a_2 &{} -a_2 &{} 0 &{} a_1a_2b_2-b_1a_2^2\\ -b_1b_2 &{} 2a_2b_1-a_1b_2 &{} b_1^2 &{} -a_1b_1 &{} b_1 &{} -b_1 &{} 0 &{} a_1b_1b_2-a_2b_1^2\\ 2a_1b_2-a_2b_1 &{} -a_1a_2 &{} -a_1b_1 &{} a_1^2 &{} -a_1 &{} a_1 &{} 0 &{} a_1a_2b_1-b_2a_1^2\\ -b_2 &{} a_2 &{} b_1 &{} -a_1 &{} 1 &{} -1 &{} 0 &{} 0\\ b_2 &{} -a_2 &{} -b_1 &{} a_1 &{} -1 &{} 1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ a_2b_1b_2-a_1b_2^2 &{} a_1a_2b_2-b_1a_2^2 &{} a_1b_1b_2-a_2b_1^2 &{} a_1a_2b_1-b_2a_1^2 &{} 0 &{} 0 &{} 0 &{} (a_1b_2-a_2b_1)^2 \end{bmatrix} \end{aligned}$$

and $F_2=$

$$\begin{aligned} \small \begin{bmatrix} -(a_2^2+b_2^2) &{} 0 &{} -2a_1a_2 &{} -2a_1b_2 &{} b_2 &{} 0 &{} a_2 &{} 0 &{} a_1(a_2^2+b_2^2)\\ 0 &{} -(a_2^2+b_2^2) &{} -2a_2b_1 &{} -2b_1b_2 &{} 0 &{} a_2 &{} 0 &{} b_2 &{} b_1(a_2^2+b_2^2)\\ -2a_1a_2 &{} -2 a_2b_1 &{} -(a_1^2+b_1^2) &{} 0 &{} 0 &{} b_1 &{} a_1 &{} 0 &{} a_2(a_1^2+b_1^2)\\ -2a_1b_2 &{} -2b_1b_2 &{} 0 &{} -(a_1^2+b_1^2) &{} a_1 &{} 0 &{} 0 &{}b_1 &{} b_2(a_1^2+b_1^2)\\ b_2 &{} 0 &{} 0 &{} a_1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} a_2 &{} b_1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0\\ a_2 &{} 0 &{} a_1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0\\ 0 &{} b_2 &{} 0 &{} b_1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0\\ a_1(a_2^2+b_2^2) &{} b_1(a_2^2+b_2^2) &{} a_2(a_1^2+b_1^2) &{} b_2(a_1^2+b_1^2) &{} 0 &{} 0 &{} 0 &{} 0 &{} -(a_1^2+b_1^2)(a_2^2+b_2^2) \end{bmatrix}. \end{aligned}$$

Moreover,

$$\begin{aligned} Q_1 = \begin{bmatrix} {\textbf {I}}_{2\times 2} &{}~ {\textbf {0}}_{2\times 6} &{}~ {\textbf {0}}_{2\times 9}\\ {\textbf {0}}_{5\times 4} &{}~ {\textbf {0}}_{5\times 4} &{}~ {\textbf {0}}_{5\times 9}\\ {\textbf {0}}_{1\times 7} &{}~ -\mu _1^2 &{}~ {\textbf {0}}_{1\times 9}\\ {\textbf {0}}_{9\times 4} &{}~ {\textbf {0}}_{9\times 4} &{}~ {\textbf {0}}_{9\times 9} \end{bmatrix}, Q_2 = \begin{bmatrix} {\textbf {0}}_{2\times 2} &{}~ {\textbf {0}}_{2\times 2} &{}~ {\textbf {0}}_{2\times 4} &{}~ {\textbf {0}}_{2\times 9}\\ {\textbf {0}}_{2\times 2} &{}~ {\textbf {I}}_{2\times 2} &{}~ {\textbf {0}}_{2\times 4} &{}~ {\textbf {0}}_{2\times 9}\\ {\textbf {0}}_{3\times 2} &{}~ {\textbf {0}}_{3\times 2} &{}~ {\textbf {0}}_{3\times 4} &{}~ {\textbf {0}}_{3\times 9}\\ {\textbf {0}}_{1\times 2} &{}~ {\textbf {0}}_{1\times 5} &{}~ -\mu _2^2 &{}~ {\textbf {0}}_{1\times 9}\\ {\textbf {0}}_{9\times 2} &{}~ {\textbf {0}}_{9\times 5} &{}~ {\textbf {0}}_{9\times 1} &{}~ {\textbf {0}}_{9\times 9} \end{bmatrix}, \end{aligned}$$

$$\begin{aligned} M_1 = \begin{bmatrix} {\textbf {I}}_{2\times 2} &{} {\textbf {0}}_{2\times 5} &{} [e_1^t-p^t] &{} {\textbf {0}}_{2\times 9}\\ {\textbf {0}}_{4\times 2} &{} {\textbf {0}}_{4\times 4} &{} {\textbf {0}}_{4\times 2} &{} {\textbf {0}}_{4\times 9}\\ {\textbf {0}}_{1\times 6} &{} -1 &{} 0 &{} {\textbf {0}}_{1\times 9}\\ [e_1^t-p^t]' &{} {\textbf {0}}_{1\times 5} &{} \Vert {e_1^t-p^t}\Vert ^2 &{} {\textbf {0}}_{1\times 9}\\ {\textbf {0}}_{9\times 2} &{} {\textbf {0}}_{9\times 2} &{} {\textbf {0}}_{9\times 2} &{} {\textbf {0}}_{9\times 11} \end{bmatrix},\\ M_0 = \begin{bmatrix} {\textbf {0}}_{2\times 2} &{} {\textbf {0}}_{2\times 2} &{} {\textbf {0}}_{2\times 2} &{} {\textbf {0}}_{2\times 2} &{} {\textbf {0}}_{2\times 9}\\ {\textbf {0}}_{2\times 2} &{} {\textbf {I}}_{2\times 2} &{} {\textbf {0}}_{2\times 3} &{} [e_2^t-p^t] &{} {\textbf {0}}_{2\times 9}\\ {\textbf {0}}_{2\times 2} &{} {\textbf {0}}_{2\times 2} &{} {\textbf {0}}_{2\times 2} &{} {\textbf {0}}_{2\times 2} &{} {\textbf {0}}_{2\times 9}\\ {\textbf {0}}_{1\times 2} &{} {\textbf {0}}_{1\times 4} &{} -1 &{} 0 &{} {\textbf {0}}_{1\times 9}\\ {\textbf {0}}_{1\times 2} &{} [e_2^t-p^t]' &{} {\textbf {0}}_{1\times 3} &{} \Vert {e_2^t-p^t}\Vert ^2 &{} {\textbf {0}}_{1\times 9}\\ {\textbf {0}}_{9\times 2} &{} {\textbf {0}}_{9\times 2} &{} {\textbf {0}}_{9\times 2} &{} {\textbf {0}}_{9\times 2} &{} {\textbf {0}}_{9\times 9} \end{bmatrix}. \end{aligned}$$

We now define the matrices $L_g \in \mathbb {R}^{17\times 17}, \forall g \in \{1,\dots ,10\}$. Let $L_g(k,l)$ denote an element at the $k^{th}$ row and the $l^{th}$ column of the matrix $L_g, g\in \{1,\dots ,10\}$. Then,

$$\begin{aligned} L_1(k,l) = {\left\{ \begin{array}{ll} 0.5, \text { if } k=1, l=4,\\ 0.5, \text { if } k=4, l=1,\\ -0.5, \text { if } k=5, l=8,\\ -0.5, \text { if } k=8, l=5,\\ 0 \text { otherwise} \end{array}\right. }, L_2(k,l) = {\left\{ \begin{array}{ll} 0.5, \text { if } k=2, l=3,\\ 0.5, \text { if } k=3, l=2,\\ -0.5, \text { if } k=6, l=8,\\ -0.5, \text { if } k=8, l=6,\\ 0 \text { otherwise} \end{array}\right. }, \end{aligned}$$

$$\begin{aligned} L_3(k,l) = {\left\{ \begin{array}{ll} 0.5, \text { if } k=1, l=17,\\ 0.5, \text { if } k=17, l=1,\\ -0.5, \text { if } k=9, l=8,\\ -0.5, \text { if } k=8, l=9,\\ 0 \text { otherwise} \end{array}\right. }, L_4(k,l) = {\left\{ \begin{array}{ll} 0.5, \text { if } k=2, l=17,\\ 0.5, \text { if } k=17, l=2,\\ -0.5, \text { if } k=10, l=8,\\ -0.5, \text { if } k=8, l=10,\\ 0 \text { otherwise} \end{array}\right. }, \end{aligned}$$

$$\begin{aligned} L_5(k,l) = {\left\{ \begin{array}{ll} 0.5, \text { if } k=3, l=17,\\ 0.5, \text { if } k=17, l=3,\\ -0.5, \text { if } k=11, l=8,\\ -0.5, \text { if } k=8, l=11,\\ 0 \text { otherwise} \end{array}\right. }, L_6(k,l) = {\left\{ \begin{array}{ll} 0.5, \text { if } k=4, l=17,\\ 0.5, \text { if } k=17, l=4,\\ -0.5, \text { if } k=12, l=8,\\ -0.5, \text { if } k=8, l=12,\\ 0 \text { otherwise} \end{array}\right. }, \end{aligned}$$

$$\begin{aligned} L_7(k,l) = {\left\{ \begin{array}{ll} 0.5, \text { if } k=5, l=17,\\ 0.5, \text { if } k=17, l=5,\\ -0.5, \text { if } k=13, l=8,\\ -0.5, \text { if } k=8, l=13,\\ 0 \text { otherwise} \end{array}\right. }, L_8(k,l) = {\left\{ \begin{array}{ll} 0.5, \text { if } k=6, l=17,\\ 0.5, \text { if } k=17, l=6,\\ -0.5, \text { if } k=14, l=8,\\ -0.5, \text { if } k=8, l=14,\\ 0 \text { otherwise} \end{array}\right. }, \end{aligned}$$

$$\begin{aligned} L_9(k,l) = {\left\{ \begin{array}{ll} 0.5, \text { if } k=3, l=9,\\ 0.5, \text { if } k=9, l=3,\\ -0.5, \text { if } k=15, l=8,\\ -0.5, \text { if } k=8, l=15,\\ 0 \text { otherwise} \end{array}\right. }, L_{10}(k,l) = {\left\{ \begin{array}{ll} 0.5, \text { if } k=4, l=10,\\ 0.5, \text { if } k=10, l=4,\\ -0.5, \text { if } k=16, l=8,\\ -0.5, \text { if } k=8, l=16,\\ 0 \text { otherwise} \end{array}\right. }, \end{aligned}$$