Persistent Monitoring by Multiple Unmanned Aerial Vehicles Using Bernstein Polynomials

Abstract

A framework for monitoring a target modeled as Dubins car using multiple UAVs is proposed. The UAVs are subject to minimum and maximum speed, maximum angular rate constraints, as well as inter-vehicle safety requirements and no-fly-zones. The problem is formulated as a continuous time nonlinear optimal control problem. This problem is first simplified by using a sequential approach, which significantly reduces its complexity. Then, by defining the desired trajectories to be tracked by the UAVs as Bernstein polynomials, it is transcribed into a nonlinear optimization problem. It is shown through numerical simulations that the present approach is computationally efficient, and thus it is well suited for trajectory planning/re-planning to monitor a target of unknown speed, heading direction and unexpected detours. Moreover, the proposed method guarantees satisfaction of feasibility and safety constraints for the whole planning time period, rather than only at discrete time points.

Appendix

An nth order Bernstein polynomial, $\mathbf {p}_n(t)$, is defined as

$$\begin{aligned} \mathbf {p}_n(t) = \sum _{k=0}^n{{{\bar{\mathbf{p}}}}_{k,n}b_{k, n}(t)}, \quad b_{k, n}(t) = \left( {\begin{array}{c}n\\ k\end{array}}\right) \frac{(t-t_0)^k(t_f-t)^{n-k}}{(t_f-t_0)^n} \, , \end{aligned}$$

(24)

$t \in [t_0, t_f]$, where ${{\bar{\mathbf{p}}}}_{k,n} \in \mathbb {R}^{N}$ is the kth control point, $b_{k, n}(t)$, $k = 0, \dots , n$, is the Bernstein basis, and $\left( {\begin{array}{c}n\\ k\end{array}}\right)$ is the binomial coefficient. Bernstein polynomials (BPs) can be used to describe 2D (or 3D) spatial curves. In this case, BPs are often referred to as Bézier curves.

Furthermore, an nth order rational Bernstein polynomial, $r_n(t)$ is defined as

$$\begin{aligned} r_n(t) = \frac{\sum _{k=0}^n{{\bar{p}}_{k,n} {\bar{w}}_{k,n} b_{k, n}(t)}}{\sum _{k=0}^n{{\bar{w}}_{k,n} b_{k, n}(t)}}, \quad t \in [t_0, t_f], \end{aligned}$$

(25)

where ${\bar{w}}_{k,n} \in \mathbb {R}$, $k = 0, \dots , n$ are referred to as weights.

Next, we provide a review of relevant properties of (rational) BPs, which are used in this paper. For an extensive review on BPs, the reader is referred to [9, 17, 19].

Property A.1

End Point Values

The first and last control points of a (rational) Bernstein polynomial are its endpoints, i.e., $\mathbf {p}_n(t_0) = {{\bar{\mathbf{p}}}}_{0,n} \quad \text {and} \quad \mathbf {p}_n(t_f) = {{\bar{\mathbf{p}}}}_{n, n}.$

Property A.2

Integration

The definite integral of a Bernstein polynomial is computed as

$$\begin{aligned} \int _{t_0}^{t_f}{\mathbf {p}}_n(t)dt = w \sum _{k = 0}^{n} {{\bar{\mathbf{p}}}}_{k,n} \, , \qquad w = \frac{t_f-t_0}{n+1} \, . \end{aligned}$$

(26)

Property A.3

Differentiation

The derivative of a Bernstein polynomial is an $(n-1)$th order Bernstein polynomial with vector of control points ${{\bar{\mathbf{p}}}}^\prime _{n-1} = [{{\bar{\mathbf{p}}}}^\prime _{0,n-1} , \ldots , {{\bar{\mathbf{p}}}}^\prime _{n-1,n-1}]$ given by ${{\bar{\mathbf{p}}}}_{n-1}^\prime = {{\bar{\mathbf{p}}}}_n \mathbf {D}_{n-1},$ where $\mathbf {D}_{n-1}$ is the $\mathbb {R}^{(n-1) \times n}$ differentiation matrix (see [19]).

Property A.4

Degree Elevation

Any Bernstein polynomial of degree n can be expressed as a Bernstein polynomial of degree m, $m > n$. The vector of control points of the degree elevated Bernstein polynomial, namely ${{\bar{\mathbf{p}}}}_m = [{\bar{p}}_{0,m}, \dots , {\bar{p}}_{m, m}]$, can be calculated as ${{\bar{\mathbf{p}}}}_{m} = {{\bar{\mathbf{p}}}}_{n} \mathbf {E}_{n}^{m},$ where $\mathbf {E}_{n}^{m} = \{e_{j,k}\} \in \mathbb {R}^{(n+1) \times (m+1)}$ is the degree elevation matrix with elements given by $e_{i, i+j} = \frac{\left( {\begin{array}{c}m-n\\ j\end{array}}\right) \left( {\begin{array}{c}n\\ i\end{array}}\right) }{\left( {\begin{array}{c}m\\ i+j\end{array}}\right) },$ where $i = 0, \dots , n$ and $j = 0, \dots , m - n$, all other values in the matrix are zero, and ${{\bar{\mathbf{p}}}}_n = [{\bar{p}}_{0, n}, \dots , {\bar{p}}_{n, n}]$ is the vector of control points of the curve being elevated (see [23]).

Property A.5

Arithmetic Operations

The sum (difference) of two polynomials of the same order can be performed by simply adding (subtracting) their control points.

Let $f_m(t)$ and $g_n(t)$ be two 1-dimensional BPs of degree m and n, respectively, with control points ${\bar{a}}_{0,m}, \dots , {\bar{a}}_{m,m}$ and ${\bar{b}}_{0,n}, \dots , {\bar{b}}_{n,n}$. The product $h_{m+n}(t)=f_m(t)g_n(t)$ is a Bernstein polynomial of degree $(m+n)$ with control points ${\bar{p}}_{k, m+n}, k \in \{0, \dots , m+n\}$ given by

$$\begin{aligned} {\bar{p}}_{k,m+n} = \sum _{j=\max (0,k-n)}^{\min (m,k)} \frac{\left( {\begin{array}{c}m\\ j\end{array}}\right) \left( {\begin{array}{c}n\\ k-j\end{array}}\right) }{\left( {\begin{array}{c}m+n\\ k\end{array}}\right) } {\bar{a}}_{j, m} {\bar{b}}_{k-j, n}. \end{aligned}$$

(27)

The ratio between two 1-dimensional BPs, $f_n(t)$ and $g_n(t)$, with control points ${\bar{a}}_{0,n}, \dots , {\bar{a}}_{n,n}$ and ${\bar{b}}_{0,n}, \dots , {\bar{b}}_{n,n}$, i.e., $r_n(t)=f_n(t)/g_n(t)$, can be expressed as a rational Bernstein polynomial as defined in (25), with control points and weights ${{\bar{p}}}_{i,n} = \frac{{{\bar{a}}}_{i,n}}{{{\bar{b}}}_{i,n}}, \quad {{\bar{w}}}_{i,n} = {{\bar{b}}}_{i,n},$ respectively.