INTERCEPTING OR SURROUNDING A GROUP OF HOSTILE UAVS WITH A NET

Abstract

The present disclosure addresses the problem of UAVs pursuing a swarm of target UAVs. The target UAVs are flying together as a flock that are initially modeled as a circle having a time-varying radius or an arbitrarily-shaped swarm that may change in size. Guidance of the pursuing UAVs is developed based on a collision cone framework, wherein the pursuing UAVs cooperatively steer the velocity vector of any point in their convex hull, to intercept the target. Also, the problem of capturing a swarm of intruder UAVs using a net manipulated by a team of defense UAVs is disclosed. The intruder UAV swarm may be stationary, in motion, and even maneuver. Collision cones in 3-dimensional space are used to determine the strategy used by the net carrying UAVs to maneuver or manipulate the net in space in order to capture the intruders.

Claims

1. A method of surrounding target unmanned aerial vehicles (UAVs), comprising: modeling the target UAVs as a circle having a radius R; determining accelerations of pursuing UAVs based on a collision cone framework; and using the accelerations determined by the collision cone framework to cooperatively steer the pursuing UAVs and subsequently surround the target.

2. The method of claim 1, further comprising enclosing the target UAVs within the convex hull of the pursuing UAVs.

3. The method of claim 1, further comprising: carrying, by the pursuing UAVs, a ropeline or a suspended net in an open-chain configuration; and orienting an open end of the ropeline or suspended net toward the target UAVs, wherein surrounding the target UAVs further includes the pursuing UAVs approaching the target UAVs at an appropriate orientation that corresponds with the open end of the ropeline or suspended net.

4. The method of claim 3, wherein the ropeline or suspended net is an open chain, the method further comprising: surrounding the target UAVs by rotating a formation of the pursuing UAVs to approach the target UAVs from a direction of an open end of the open chain.

5. The method of claim 1, wherein the size of the swarm of target UAVs varies with time, the method further comprising: adjusting inter-vehicle distances of the pursuing UAVs in correspondence with the size of the target UAV swarm; and enclosing the target UAVs within the convex hull of the pursuing UAVs.

6. The method of claim 1, further comprising cooperatively driving the velocity vector of any point that lies within the convex hull of the pursuing UAVs into the collision cone of the target UAVs to intercept the target UAVs.

7. The method of claim 1, wherein the target UAVs are moving in a two-dimensional plane.

8. The method of claim 1, further comprising observing the target UAVs to determine risk associated with the target UAVs.

9. The method of claim 1, wherein the collision cone framework uses collision cones to represent a collection of velocity vectors of the pursuing UAVs which leads to a collision with the target UAVs; and using the collision cones to specify accelerations for the pursuing UAVs to enable them to cooperatively steer the velocity vector of any chosen point that lies in the convex hull of the pursuing UAVs into the collision cone to the target to facilitate surrounding the target UAVs.

10. The method of claim 1, further comprising generalizing the circle as having a time-varying radius or as having an arbitrary shape that changes in size.

11. A method of capturing a swarm of target unmanned aerial vehicles (UAVs) by pursuing UAVs carrying a net, comprising: modeling the swarm of target UAVs as being enclosed in a sphere; determining, by the pursuing UAVs, a trajectory to intercept the sphere using a collision cone in 3-dimensional space; maneuvering the heading angle of the net into the collision cone by coordinated guidance of the pursuing UAVs; and maintaining the heading angle to stay inside the collision cone to intercept the intruder UAV swarm modeled within the sphere.

12. The method of claim 11, further comprising: modeling the swarm of target UAVs as a flock enclosed by a sphere; and determining the capturing as an interception between the net and the sphere.

13. The method of claim 11, further comprising engaging the target UAVs by the pursuing UAVs in two phases, wherein during the first phase, the pursuing UAVs move the net close to the swarm of target UAVs, and wherein during the second phase, the pursuing UAVs rotate the net to achieve a desired net angle.

14. The method of claim 13, further comprising rotating the net to orient the net such that the normal to the net is approximately parallel to the velocity vector of the intruder swarm, or any other desired angle.

15. The method of claim 13, further comprising altering the velocity headings and/or speed of the pursuing UAVs to rotate the net.

16. The method of claim 13, further comprising commencing the second phase when a time-to-go is less than a threshold time determined as a function of the resolution of cameras mounted on the pursuing UAVs.

17. The method of claim 13, further comprising commencing the second phase when a time-to-go is less than a threshold distance to the target UAVs.

18. The method of claim 11, further comprising: determining if the size of the swarm is larger than the net, and if so, intercepting a subset of the swarm of target UAVs by choosing an interception point on the net that lies close to the centroid of the net.

19. The method of claim 18, further comprising intercepting the subset of the target UAVs that lie within a sphere whose radius is less than or equal to the radius of the in-circle of the net.

20. The method of claim 11, further comprising intercepting elongated swarms which are modeled as being enclosed in a spheroid.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0006] The components in the drawings are not necessarily to scale relative to each other. Like reference numerals designate corresponding parts throughout the several views.

[0007] FIGS. 1A and 1B illustrate time snapshots of engagement between 5 pursuing UAVs carrying a ropeline, and a target swarm B;

[0008] FIG. 2 illustrates an engagement geometry between a point and a circle;

[0009] FIG. 3A illustrates a representation of the cone in relative velocity space;

[0010] FIG. 3B illustrates a representation of the cone in physical space;

[0011] FIG. 4 illustrates a schematic of 5 UAVs A.sub.1, . . . , A.sub.5 pursuing an intruder UAV swarm;

[0012] FIG. 5 illustrates a schematic of 5 UAVs pursuing a target UAV swarm, while carrying a ropeline A.sub.1 A.sub.2 A.sub.3 A.sub.4;

[0013] FIG. 6. illustrates a schematic of a ropeline between UAVs A.sub.i and A.sub.j;

[0014] FIG. 7 illustrates a schematic of 5 UAVs A.sub.1, . . . , A.sub.5 pursuing a target UAV swarm, while carrying a stretchable ropeline A.sub.1A.sub.2A.sub.3A.sub.4A.sub.5;

[0015] FIG. 8 illustrates an engagement geometry between a point and an arbitrarily shaped object;

[0016] FIG. 9 illustrates a schematic of 5 UAVs carrying a ropeline A.sub.1 A.sub.2 A.sub.3 A.sub.4 A.sub.5, pursuing a target UAV swarm of arbitrary shape;

[0017] FIG. 10 illustrates a schematic of three UAVs A.sub.1, A.sub.2, A.sub.3 carrying a net to intercept a swarm of UAVs;

[0018] FIG. 11 illustrates a schematic representing various angles used herein;

[0019] FIG. 12A illustrates an interceptor-target engagement in 2-D;

[0020] FIG. 12B illustrates net-intruder swarm engagement;

[0021] FIG. 13 illustrates partitioning of the 3-D Space based on the signs of y and V.sub.r;

[0022] FIG. 14 illustrates a schematic of CC.sub.XB and CC.sub.NB;

[0023] FIG. 15A illustrates a collision cone in the azimuth, elevation angle space between the net and the sphere;

[0024] FIG. 15B illustrates a surface plot of y(λ.sub.1,λ.sub.2);

[0025] FIG. 16A illustrates a choice of interception point X on the net;

[0026] FIG. 16B illustrates a choice of interception point X on the sphere;

[0027] FIG. 16C illustrates an Intercepting a subset of intruder swarm;

[0028] FIG. 17A shows the trajectory of the point object for several intercept points on the circle;

[0029] FIG. 17B shows the difference between the heading angles of the point object and the circle;

[0030] FIG. 17C shows that the acceleration profiles also vary for these different interception points;

[0031] FIG. 18 illustrates the upper bounds on |U.sub.1|, |U.sub.2|, |U.sub.3|;

[0032] FIG. 19 illustrates a schematic of two UAVs A.sub.1 and A.sub.2 carrying a ropeline;

[0033] FIGS. 20A-20B illustrate a change in orientation of the net and distance between A.sub.1 and A.sub.2 in the case of α.sub.A1≠α.sub.A2≠0 and α.sub.A1=0, α.sub.A2≠0, respectively;

[0034] FIGS. 21A-21B illustrates contour plots of ΔL, (b) Δθ.sub.L, on the (α.sub.A2, ΔT) plane, for a given value of α.sub.A1;

[0035] FIG. 22 illustrates a schematic of net connected to UAVs with cables; and

[0036] FIG. 23 illustrates a schematic of three UAVs carrying a net to intercept a swarm of UAVs enclosed in a virtual spheroid.

[0037] The APPENDIX, comprising 25 pages and incorporated herein by reference as part of the disclosure of this application, shows simulation results based on the principles disclosed herein.

DETAILED DESCRIPTION

[0038] Cooperative Pursuit Guidance to Surround Intruder Swarms Using Collision Cones

[0039] With reference to FIGS. 1A and 1B, three example classes of problems are considered. In the first problem, capture is said to occur when the n pursuing UAVs have succeeded in enclosing the target within their convex hull. In the second problem, the pursuing UAVs are assumed to be carrying a ropeline (or a suspended net). This ropeline (or suspended net) is considered to be an open chain, and the n UAVs need to surround the intruder UAV, while rotating their formation appropriately such that they approach the intruder from the direction of the open end of the chain. This is schematically represented in FIGS. 1A and 1B, which show five pursuing UAVs A.sub.1, A.sub.2, A.sub.3, A.sub.4, A.sub.5 carrying a ropeline, whose open end is A.sub.1A.sub.5. The pursuing UAVs need to rotate their formation appropriately such that they eventually approach the target swarm B from the direction corresponding to A.sub.1A.sub.5 and enclose the swarm. In both of the above problems, the size of the intruder swarm is assumed to be constant. In the third problem, the intruder swarm is assumed to spread out or contract and thus the size of the target may change with time. Therefore, in this case, the UAVs need to perform the additional task of adjusting their inter-vehicle distances (in correspondence with the changing size of the target) so that they successfully surround the intruder swarm at the time of capture. In all three problems, we consider that the intruder swarm may either move with constant velocity or may maneuver. The target swarm is initially modeled as a circle (of possibly time-varying radius), and this is subsequently generalized to arbitrarily-shaped swarms.

[0040] To address the three problems, a collision cone approach is disclosed herein. Collision cones represent a collection of velocity vectors of an object which leads to collision with another moving object. Guidance laws to avoid collision are then designed to pull the current velocity vector of the object outside the collision cone. As described herein, collision cones can be used to develop analytical guidance laws for cooperative pursuit being performed by a group of UAVs. Differently from the collision avoidance problem, these guidance laws specify accelerations, which when applied by the pursuing UAVs, enable them to cooperatively drive the velocity vector of any chosen point (that lies in the convex hull of the pursuing UAVs) into the collision cone to the target, thereby facilitating subsequent capture of the target.

[0041] Collision Cone Preliminaries

[0042] The present disclosure details how the concepts from collision cones can be used to design the cooperative pursuit guidance laws. FIG. 2 illustrates a point object A and a circle B of radius R, moving with speeds V.sub.A and V.sub.B, and heading angles α and β, respectively. Let P represent the center of the circle. Let r represent the distance AP, and θ represent the bearing angle of AP. Then, V.sub.θ represents the component of the relative velocity (of B with respect to A) normal to AP, while V.sub.r represents the component along AP. These are defined as follows:

V.sub.θ=r{dot over (θ)}=V.sub.B sin(β-θ)—V.sub.A sin(α−θ) V.sub.r={dot over (r)}=V.sub.B cos(β-θ)−V.sub.A cos(α−θ)  (1)

[0043] When A and B move with constant velocities, the miss-distance r.sub.m (which is the predicted distance between the point A and the center P of the circle at the instant of closest approach), is given by the following equation:

[00001]$\begin{array}{cc}{r}_m^2=\frac{r^2{V}_{\theta }^2}{\left(V_{\theta }^2+V_r^2\right)}& \left(2\right)\end{array}$

[0044] and the time at which the two objects are at their point of closest approach is given by:

[00002]$\begin{array}{cc}{t}_m=\frac{-{\mathrm{rV}}_r}{V_{\theta }^2+V_r^2}& \left(3\right)\end{array}$

[0045] When V.sub.r<0, the above expression leads to a positive value of t.sub.m. When V.sub.r>0, it leads to a negative value of t.sub.m, and this is to be interpreted as the time of closest approach if the trajectories of both the objects are projected backwards in time. Since the objects are moving with constant velocities, projection backwards implies that the trajectories of the objects before time t, are also straight lines.

[0046] One can see that if the miss distance r.sub.m is less than R, then the point object A is on a collision course with the circle B. Based on the above, one may define a miss-distance function y as follows:

[00003]$\begin{array}{cc}y=\frac{r^2{V}_{\theta }^2}{V_{\theta }^2+V_r^2}-R^2& \left(4\right)\end{array}$

[0047] If the two objects continue to move with constant velocities for all future time, then the predicted miss distance r.sub.m is equal to the actual miss distance, and therefore, the conditions y<0 and V.sub.r<0 are both necessary and sufficient conditions for one object to intercept the other object. When the two objects move with varying velocities, then the condition y<0, V.sub.r≤0 for all future time is a sufficient condition for interception.

[0048] The condition y<0, V.sub.r<0 can be represented by a cone in the relative velocity space, as follows:

R=(V.sub.θ,V.sub.r):y<0∩V.sub.r<0  (5)

[0049] This cone R is schematically depicted in FIG. 3A. If the instantaneous value of (V.sub.θ, V.sub.r) lies inside this cone, then the two objects are on a collision course with one another. When the speeds V.sub.A and V.sub.B of the objects are constant, the cone can equivalently be represented in physical space by the set of heading angles of A that lead to collision, that is,

R.sub.α=α:y<0∩V.sub.r<0  (6)

[0050] The cone R.sub.α in physical space is shown in FIG. 3B. Cones R and R.sub.α are different representations of the collision cone.

[0051] Cooperative Pursuit Guidance Laws

[0052] Below is described details cooperative pursuit by a group of UAVs. Consider n UAVs A.sub.1, . . . , A.sub.n that are pursuing an intruder UAV swarm. The UAVs are all moving on a plane. We assume that the vehicles in the intruder swarm need to stay reasonably close together. The reason for this could be, for example, that the intruders seek to perform a coordinated attack on a protected area, or because the intruder UAVs are flying in a leader-follower configuration and the followers need to be close to the leader. Because of their need to stay close together, the intruder UAVs are moving as a flock. We furthermore assume that this flock of intruder UAVs lies within a circle of radius R. A scenario comprising five UAVs performing a cooperative pursuit of an intruder swarm bounded within a circle is illustrated in FIG. 4.

[0053] Engagement Geometry Kinematics

[0054] Let {right arrow over (r)}.sub.1, . . . , {right arrow over (r)}.sub.n represent the position vectors of the pursuing UAVs, and V.sub.1, V.sub.n represent their respective velocity vectors. Let V.sub.B represent the velocity vector of the circle encompassing the intruder UAV swarm. Consider a virtual point X that lies in the convex hull of A.sub.1, . . . , A.sub.n, defined by:

{right arrow over (X)}=Σ.sub.1=1.sup.nλ.sub.i{right arrow over (r)}.sub.i,Σ.sub.i=1.sup.nλ.sub.i=1,λ.sub.i>0,i=1, . . . ,n  (7)

[0055] An algorithm to compute (λ.sub.1, . . . , λ.sub.n) is discussed in Section D. The chosen (λ.sub.1, . . . , λ.sub.n) combination is then used to compute the velocity vector {right arrow over (V)}.sub.X of point X as follows:

{right arrow over (V)}.sub.X=λ.sub.1{right arrow over (V)}.sub.1+ . . . +λ.sub.n{right arrow over (V)}.sub.n  (8)

[0056] Let V.sub.1, . . . , V.sub.n represent the speeds of A.sub.1, . . . , A.sub.n, and let the angles of the associated velocity vectors (with respect to a reference line) be α.sub.1, . . . , α.sub.n, respectively. Then,

[00004]$\begin{array}{cc}{\stackrel{.fwdarw.}{V}}_X=\left[\begin{array}{c}{\lambda }_1{V}_1\cos {\alpha }_1+.Math.+{\lambda }_n{V}_n\cos {\alpha }_n\\ {\lambda }_1{V}_1\sin {\alpha }_1+.Math.+{\lambda }_n{V}_n\sin {\alpha }_n\end{array}\right]& \left(9\right)\end{array}$

[0057] The magnitude of {right arrow over (V)}.sub.X is then given by:

V.sub.X=(λ.sub.1V.sub.1 cos α.sub.1+ . . . +λ.sub.nV.sub.n cos α.sub.n).sup.2+(λ.sub.1V.sub.1 sin α.sub.1+ . . . +λ.sub.nV.sub.n sin α.sub.n).sup.2  (10)

[0058] The angle made by {right arrow over (V)}.sub.X with respect to the reference line is denoted by α.sub.X and is given by:

[00005]$\begin{array}{cc}{\alpha }_X={\tan }^{-1}\left(\frac{{\lambda }_1{V}_1\sin {\alpha }_1+.Math.+{\lambda }_n{V}_n\sin {\alpha }_n}{{\lambda }_1{V}_1\cos {\alpha }_1+.Math.+{\lambda }_n{V}_n\cos {\alpha }_n}\right)& \left(11\right)\end{array}$

[0059] Let V.sub.B represent the speed of B, and let β denote the angle made by {right arrow over (V)}.sub.B with respect to the horizontal. Defining r.sub.X=XP, and θ.sub.X as the angular bearing of the line XP, the relative velocity components of B with respect to X are given by:

V.sub.θ,X=V.sub.B sin(β−θ.sub.X)−V.sub.X Sin(α.sub.X−θ.sub.X) V.sub.r,X=V.sub.B cos(β−θ.sub.X)−V.sub.X cos(α.sub.X−θ.sub.X)  (12)

[0060] Let α.sub.1, . . . , α.sub.n represent the magnitudes of the applied accelerations of A.sub.1, . . . A.sub.n, and these are applied at angles δ.sub.i, . . . , δ.sub.n, respectively, with these angles measured with respect to a reference line. Then, the nonlinear state equations governing the kinematics between the point X and center P of the circle, represented in a polar coordinate frame, are as follows:

[00006]$\begin{array}{cc}{\stackrel{.}{r}}_X=V_{r,X}& \left(13\right)\end{array}$$\begin{array}{cc}{\dot{\theta }}_X=\frac{V_{\theta ,X}}{r_X}& \left(14\right)\end{array}$$\begin{array}{cc}{\stackrel{.}{V}}_{\theta ,X}=-\frac{V_{\theta X}{V}_{r,X}}{r_X}-{\dot{V}}_X\sin \left({\alpha }_X-{\theta }_X\right)-{\dot{\alpha }}_X{V}_X\cos \left({\alpha }_X-{\theta }_X\right)& \left(15\right)\end{array}$$\begin{array}{cc}{\stackrel{.}{V}}_{r,X}=\frac{V_{\theta ,X}^2}{r_X}-{\stackrel{.}{V}}_X\cos \left({\alpha }_X-{\theta }_X\right)+{\stackrel{.}{\alpha }}_X{V}_X\sin \left({\alpha }_X-{\theta }_X\right)& \left(16\right)\end{array}$$\begin{array}{cc}{\dot{V}}_X=\frac{1}{V_X}\left[\left(-{\lambda }_X\sin {\alpha }_1+{\lambda }_Y\cos {\alpha }_1\right){\lambda }_1{V}_1{\dot{\alpha }}_1+.Math.+\left(-{\lambda }_X\sin {\alpha }_n+{\lambda }_Y\cos {\alpha }_n\right){\lambda }_n{V}_n{\dot{\alpha }}_n+\left({\lambda }_X\cos {\alpha }_1+{\lambda }_Y\sin {\alpha }_1\right){\lambda }_1{\dot{V}}_1+.Math.+\left({\lambda }_X\cos {\alpha }_n+{\lambda }_Y\sin {\alpha }_n\right){\lambda }_n{\dot{V}}_n\right]& \left(17\right)\end{array}$$$\begin{gathered} \begin{array}{cc}{\dot{\alpha }}_X=\frac{1}{{\lambda }_X^2+{\lambda }_Y^2}\text{⁠}\left[\text{⁠}\left({\lambda }_X\cos {\alpha }_1+{\lambda }_Y\sin {\alpha }_1\right){\lambda }_1{V}_1{\dot{\alpha }}_1+.Math.+\left({\lambda }_X\cos {\alpha }_n+{\lambda }_Y\sin {\alpha }_n\right){\lambda }_n{V}_n{\dot{\alpha }}_n \\ +\left({\lambda }_X\sin {\alpha }_1-{\lambda }_Y\cos {\alpha }_1\right){\lambda }_1{\dot{V}}_1+.Math.+\left({\lambda }_X\sin {\alpha }_n-{\lambda }_Y\cos {\alpha }_n\right){\lambda }_n{\dot{V}}_n\right]& \left(18\right)\end{array} \end{gathered}$$$\begin{array}{cc}\begin{array}{cc}{\dot{\alpha }}_i=\frac{a_i\sin \left({\delta }_i-{\alpha }_i\right)}{V_i},,& i=1,.Math.,n\end{array}& \left(19\right)\end{array}$$\begin{array}{cc}\begin{array}{cc}{\stackrel{.}{V}}_i=a_i\cos \left({\delta }_i-{\alpha }_i\right),& i=1,.Math.,n\end{array}& \left(20\right)\end{array}$

[0061] Note that Eqs. 17 and 18 above are obtained by differentiating Eqs. 10 and 11, respectively, with respect to time. In these equations, λ.sub.X and λ.sub.Y are defined as:

λ.sub.X=λ.sub.1V.sub.1 cos α.sub.1+ . . . +λ.sub.nV.sub.n cos α.sub.n  (21)

λ.sub.Y=).sub.1V.sub.1 sin α.sub.1+ . . . +λ.sub.nV.sub.n sin α.sub.n  (22)

[0062] Eqs 19 and 20 represent the lateral and longitudinal acceleration components, respectively, of the ith UAV. Eqs 13-20 thus govern the kinematics of the relative velocity between any point X residing in the convex hull of the pursuing UAVs A.sub.1, . . . , A.sub.n, and the center of the target circle B, when B is non-maneuvering, that is, B moves with constant velocity. The guidance problem is addressed in this section is to determine suitable accelerations of the pursuing UAVs so that they cooperatively steer the velocity vector of X, such that X intercepts the target circle.

[0063] Cooperative Pursuit Laws for Non-Maneuvering Target

[0064] Cooperative pursuit laws for the scenario when the target moves with constant velocity, i.e., it is non-maneuvering are determined below. From Eq 4, a collision cone function y.sub.X can be defined as follows:

[00007]$\begin{array}{cc}{y}_X=\frac{r_X^2{V}_{\theta ,X}^2}{V_{\theta ,X}^2+V_{r,X}^2}-R^2& \left(23\right)\end{array}$

[0065] The condition y.sub.X<0, V.sub.r,X<0 indicates that the current velocity vector of X, (that is, {right arrow over (V)}.sub.X) is such that X is on a path to intercept with the target circle B. If the initial conditions of the engagement are such that y.sub.X<0, V.sub.r,X<0 is not satisfied (See FIG. 4 for an illustration), then the pursuing UAVs need to apply appropriate accelerations so that the velocity vector of X is driven into the y.sub.X<0,V.sub.r,X<0 region. Toward this end, we employ a Lyapunov function technique. Lyapunov functions can be used to design controls for systems with nonlinear dynamics, and may provide global asymptotic stability properties. A scalar positive definite Lyapunov function Z may be defined as follows:

Z=1/2(y.sub.X−w).sup.2  (24)

[0066] where, w is a reference value, which is chosen to satisfy −R.sup.2<w<0. Note that driving this Lyapunov function to zero is equivalent to driving the velocity vector of X into the collision cone to the target. Then, the derivative of the Lyapunov function (Eq. 24) is Ż=(y.sub.X-w){dot over (y)}.sub.X. where {dot over (y)}.sub.X is as follows:

[00008]$\begin{array}{cc}\frac{d{y}_X}{dt}=\frac{\partial y_X}{\partial r_X}{\stackrel{.}{r}}_X+\frac{\partial y_X}{\partial V_{r,X}}{\dot{V}}_{r,X}+\frac{\partial y_X}{\partial V_{\theta ,X}}{\dot{V}}_{\theta ,X}& \left(25\right)\end{array}$

[0067] Evaluating the above along the system trajectories given by Eqs. 13-20, and rearranging terms in the resulting equation leads to the following:

[00009]$\begin{array}{cc}\begin{array}{c}{\stackrel{.}{y}}_X=\frac{\partial y_{X_{}}}{\partial V_{r,X}}\left[-{\stackrel{.}{V}}_X\cos \left({\alpha }_X-{\theta }_X\right)+{\stackrel{.}{\alpha }}_X{V}_X\sin \left({\alpha }_X-{\theta }_X\right)\right]+\frac{\partial y_X}{\partial V_{\theta ,X}}\left[-{\stackrel{.}{V}}_X\sin \left({\alpha }_X-{\theta }_X\right)-{\stackrel{.}{\alpha }}_X{V}_X\cos \left({\alpha }_X-{\theta }_X\right)\right]\\ =-{\stackrel{.}{V}}_X=\left[\frac{\partial y_x}{\partial V_{r,X}}\cos \left({\alpha }_X-{\theta }_X\right)+\frac{\partial y_X}{\partial V_{\theta ,X}}\sin \left({\alpha }_X-{\theta }_X\right)\right]+{\stackrel{.}{\alpha }}_X{V}_X[\frac{\partial y_X}{\partial V_{r,x}}\sin \left({\alpha }_X-{\theta }_X\right)-\frac{\partial y_X}{\partial V_{\theta ,X}}\cos ({\alpha }_X-{\theta }_{X)]}\end{array}& \left(26\right)\end{array}$

[0068] The control inputs may be defined as u.sub.α,i and u.sub.V,i, where u.sub.α,i≡{dot over (α)}.sub.i and u.sub.V,i≡{dot over (V)}.sub.i, represent the acceleration components governing the heading and speed change, respectively, of UAV A.sub.i. Here, u.sub.V,i represents the longitudinal acceleration and the lateral acceleration is computed from u.sub.α,i V.sub.i These control inputs may be determined so that they will force the Lyapunov function Z to follow the dynamics Ż=−KZ. Choosing the constant K to satisfy K>0 will render Ż to be negative definite, and Z to be globally exponentially stable. By substituting {dot over (V)}.sub.X from Eq. 17 and {dot over (α)}.sub.X from Eq. 18 into Eq. 26, it is evident that we can enforce Z to follow the dynamics Ż=−KZ, if the accelerations of the pursuing UAVs satisfy the equation:

λ.sub.1(N.sub.1αu.sub.α,1+N.sub.1vu.sub.V,1)+ . . . +λ.sub.n(N.sub.nau.sub.α,n+N.sub.nvu.sub.V,n)=−K(y.sub.X−w)  (27)

[0069] where, N.sub.iα and N.sub.iv are as follows:

[00010]$\begin{array}{cc}{N}_{i\alpha }=\frac{V_i}{V_X}\left[\frac{\partial y_X}{\partial V_{r,X}}\cos \left({\alpha }_X-{\theta }_X\right)+\frac{\partial y_X}{\partial V_{\theta ,X}}\sin \left({\alpha }_X-{\theta }_X\right)\right]\left[{\lambda }_X\sin {\alpha }_i-{\lambda }_Y\cos {\alpha }_i\right]+\frac{V_X{V}_i}{{\lambda }_X^2+{\lambda }_Y^2}\left[\frac{\partial y_X}{\partial V_{r,X}}\sin \left({\alpha }_X-{\theta }_X\right)-\frac{\partial y_X}{\partial V_{\theta ,X}}\cos \left({\alpha }_X-{\theta }_X\right)\right]\left[{\lambda }_X\cos {\alpha }_i+{\lambda }_Y\sin {\alpha }_i\right]& \left(28\right)\end{array}$$\begin{array}{cc}{N}_{i\nu }=\frac{-1}{V_X}\left[\frac{\partial y_X}{\partial V_{r,X}}\cos \left({\alpha }_X-{\theta }_X\right)+\frac{\partial y_X}{\partial V_{\theta ,X}}\sin \left({\alpha }_X-{\theta }_X\right)\right]\left[{\lambda }_X\cos {\alpha }_i-{\lambda }_Y\sin {\alpha }_i\right]+\frac{V_X}{{\lambda }_X^2+{\lambda }_Y^2}\left[\frac{\partial y_X}{\partial V_{r,X}}\sin \left({\alpha }_X-{\theta }_X\right)-\frac{\partial y_X}{\partial V_{\theta ,X}}\cos \left({\alpha }_X-{\theta }_X\right)\right]\left[{\lambda }_X\sin {\alpha }_i+{\lambda }_Y\cos {\alpha }_i\right]& \left(29\right)\end{array}$

[0070] It is evident that Eq. 27 represents a single equation in 2n unknowns (u.sub.α,1, . . . , u.sub.α,n) and (u.sub.V,1, . . . , u.sub.V,n), which are the heading angle rates and longitudinal accelerations, respectively, of the pursuing UAVs. There are thus multiple combinations of the lateral and longitudinal accelerations of the pursuing UAVs that can satisfy. If the pursuing UAVs are capable of only changing their heading angles (that is, they always move with constant speed), then in order for X to intercept B, the equation governing the heading angle rates of the pursuing UAVs is obtained by setting u.sub.V1, . . . , u.sub.Vn in Eq. 27 to be identically zero. This leads to the following equation:

λ.sub.1N.sub.1αu.sub.α,1+ . . . +λ.sub.nN.sub.nau.sub.α,n=K(y.sub.X−w)  (30)

[0071] In Eq. 27, K is a constant. If the initial conditions are such that V.sub.r,X<0, then we choose K>0 to ensure that the Lyapunov function Z decays to zero. Note that since we are enforcing Ż=−KZ, therefore Z will follow the dynamics Z(t)=Z(0)e.sup.−Kt, where Z(0) represents the initial value of the Lyapunov function. The decay in Z is thus exponential in nature. We now choose K such that Z decays to E (where E is a small number) within a time that is shorter than t.sub.m,X, which (from Eq. 3) is:

[00011]$\begin{array}{cc}{t}_{m,X}=\frac{-r_X{V}_{r,X}}{V_{\theta ,X}^2+V_{r,X}^2}& \left(31\right)\end{array}$

[0072] From the expression for Z(t), it can be seen that the requisite K that will achieve this is given by:

[00012]$\begin{array}{cc}K>\frac{1}{t_{m,X}}\ln \left[\frac{Z\left(0\right)}{ϵ}\right]& \left(32\right)\end{array}$

[0073] Thus, for a constant velocity target, when the initial conditions are such that V.sub.r,X<0, then if the control inputs of the pursuing UAVs satisfy Eq. 27, and K>0 is chosen as above, then X will intercept B.

[0074] The required sensor data is the target velocity, range and bearing to the target from each of the individual UAVs, as well as the radius of the target UAV circle. The radius R of the target UAV circle can be extracted from image data obtained from camera(s) mounted on the pursuing UAVs. Then, knowing R, the reference point w is chosen so as to satisfy −R.sup.2<w<0, in order to ensure interception.

[0075] To compute K from Eq. 32, Z(0) is needed and t.sub.m,X. Z(0) is computed from Eq. 24, and requires the use of y.sub.X(0). y.sub.X(0) is computed from Eq. 23, while t.sub.m,X is computed from Eq. 31, and these require the values of r.sub.X(0), V.sub.θ,X(0), V.sub.r,X(0). These are computed using Eqs. 7-12, in conjunction with the chosen values of (λ.sub.1, . . . ,λ.sub.n).

[0076] When the initial conditions are such that V.sub.r,X>0, then the velocity vector {right arrow over (V)}.sub.X first needs to be steered into the V.sub.r,X<0 region (See FIG. 3). This can be achieved by using Eq. 27 with an initial value of K<0. This will drive {right arrow over (V)}.sub.X into the V.sub.r,X<0 region, after which we subsequently switch the sign of K to a value K>0.

[0077] Cooperative Pursuit Laws that Account for Target's Acceleration

[0078] In this section, the developed guidance law Eq. 27 is modified so that it explicitly accounts for the target's lateral acceleration a.sub.B. It is assumed that the target swarm maneuvers are such that its acceleration vector acts normal to its velocity vector, that is, the target may change its direction but otherwise moves with constant speed. When B has an acceleration of magnitude a.sub.B, then the time derivatives of the relative velocity components and the heading angle of B are as follows:

[00013]$\begin{array}{cc}{\dot{V}}_{\theta ,X}=-\frac{V_{\theta ,X}{V}_{r,X}}{r,X}-{\dot{V}}_X\sin \left({\alpha }_X-{\theta }_X\right)-{\dot{\alpha }}_X{V}_X\cos \left({\alpha }_X-{\theta }_X\right)+a_B\cos \left(\beta -\theta \right)& \left(33\right)\end{array}$$\begin{array}{cc}{\dot{V}}_{r,X}=\frac{V_{\theta ,X}^2}{r_X}-\dot{V_X}\cos \left({\alpha }_X-{\theta }_X\right)+{\dot{\alpha }}_X{V}_X\sin \left({\alpha }_X-{\theta }_X\right)-a_B\sin \left(\beta -\theta \right)& \left(34\right)\end{array}$$\begin{array}{cc}\dot{\beta }=a_B/V_B& \left(35\right)\end{array}$

[0079] The kinematics of the engagement between then pursuing UAVs and the maneuvering target are given by Eqs. 13, 14, 17-20 and 33-35.

[0080] The quantity y.sub.X is, in general, a function that provides the predicted miss-distance. When all the pursuing UAVs A.sub.1, . . . , A.sub.n as well as the target B are not maneuvering, then y.sub.X is a constant in time, and is a function of the actual miss-distance. When B is maneuvering, the quantity y.sub.X represents a predicted miss-distance function and varies with time. At any time t.sub.1, y.sub.X(t.sub.1) represents a prediction of what the miss distance function would be if the target were to move with constant velocity for all future time t>t.sub.1. Eventually, this predicted miss-distance converges to the actual miss distance.

[0081] If we drive the Lyapunov function Z to zero, then we are essentially forcing this predicted miss-distance to the reference value w, at each instant in time. Since this predicted miss-distance only eventually converges to the actual miss-distance, therefore, for a maneuvering target, driving Z to zero represents a sufficient condition for X to intercept B.

[0082] Differentiating the Lyapunov function Eq. 24 along the trajectories of the system defined by Eqs. 13, 14, 17-20 and 33-35, ensures that Eq. 24 follows the dynamics Ż=−KZ, if the control inputs of the pursuing UAVs follow the equation:

[00014]$\begin{array}{cc}{\lambda }_1\left(N_{1\alpha }{u}_{\alpha ,1}+N_{1v}{u}_{V,1}\right)+.Math.+{\lambda }_n\left(N_{n\alpha }{u}_{\alpha ,n}+N_{nv}{u}_{V,n}\right)=-K\left(y_X-w\right)+a_B\left[\frac{\partial y_X}{\partial V_{\theta ,X}}\cos \left(\beta -{\theta }_X\right)-\frac{\partial y_X}{\partial V_{r,X}}\sin \left(\beta -{\theta }_X\right)\right]& \left(36\right)\end{array}$

[0083] In Eq. 36, if K is large enough to ensure that Z is driven to zero in time less than t.sub.m,X, then X is on a path to intercept B.

[0084] Assume that at t=0, all the pursuing UAVs are moving in a formation with identical velocities. Refer Eqs. 27 and 36. Each of these represent a single equation with 2n unknowns, and in each of them, one of the solutions corresponds to the case of equal lateral accelerations and equal longitudinal accelerations of the n pursuing UAVs. Employing this solution will cause the pursuing UAVs to move towards the target with no change in either their formation shape, or the formation orientation. On the other hand, when Eqs 27 and 26 are employed with unequal accelerations, then the formation shape and orientation will change as it approaches the target. In Section 4, we show how to obtain the particular solution of Eqs. 27 and 36 that will ensure that the formation approaches the target with an orientation change, while otherwise remaining rigid. In Section 5, we show how to obtain the particular solution of Eqs. 27 and 36 that will ensure that the formation performs an orientation change, as well as a change in the lengths of the formation edges.

[0085] Computation of λ.sub.1, . . . , λ.sub.n

[0086] The computation of the quantities (λ.sub.1, . . . , λ.sub.n) in Eq. 7 will now be introduced. From Caratheodory's Theorem, when n>3, then the representation of {right arrow over (X)} may not be unique in the sense that there can be more than one (λ.sub.1, . . . , λ.sub.n) tuple that corresponds to the same point {right arrow over (X)}. So in our case, we first choose the coordinates of point X at time t=0, and then take any (λ.sub.1, . . . , λ.sub.n) tuple that will correspond to that point X. Note that if we choose λ.sub.i=0 for any i, then that UAV does not participate in the guidance laws Eqs. 27 and 36, as a consequence of which the acceleration of that UAV cannot be computed from Eqs. 27 and 36. To ensure that this does not occur, we necessarily require that λ.sub.i>0, ∀i.

[0087] The chosen point X lies at the centroid of the convex hull, we then chose

[00015]${\lambda }_1=.Math.={\lambda }_n=\frac{1}{n}.$

There can be scenarios when it would be better to choose X to be a point different from the centroid of the convex hull.

[0088] The scenarios requiring unequal values of λ.sub.1 can be understood from the notion of the Chebyshev center. Here, a definition that the Chebyshev center is the center of the largest radius circle that can be inscribed within a polygon is adopted. If the Chebyshev center is not located at the centroid of the polygon formed by the pursuing UAVs, then unequal values of λ.sub.i will be required.

[0089] An algorithm to compute λ.sub.1, . . . , λ.sub.n is as follows. For a given {right arrow over (X)}, find λ.sub.1, . . . , λ.sub.n:

[00016]$\mathrm{minimize}\underset{i=1}{\overset{n}{.Math.}}{\left({\lambda }_i-\frac{1}{n}\right)}^2$

[0090] subject to constraints:

λ.sub.i>0,Σ.sub.i=1.sup.nλ.sub.i=1,Σ.sub.i=1.sup.nλ.sub.iA.sub.i=X  (37)

[0091] Guidance Laws for Pursuing UAVs with a Ropeline (or Suspended Net)

[0092] In a scenario where the pursuing UAVs are carrying a ropeline (or suspended net) with which they wish to surround the target. The rope essentially forms an open chain as depicted in FIG. 5 (where the solid edges of the polygon represent the rope). In such a scenario, the UAVs may need to reorient the formation appropriately so that the formation approaches the target from the direction of the open end of the chain. More precisely, if ĥ represents the unit normal to the line joining the pair of UAVs that are carrying the open end of the chain, we would like the UAVs to apply the requisite accelerations that will enable them to move towards the target, while simultaneously rotating the formation so as to orient ĥ appropriately. In FIG. 5, the ropeline is formed by UAVs A.sub.1A.sub.2A.sub.3A.sub.4A.sub.5, while UAVs A.sub.1 and A.sub.5 carry the open ends of the rope. Therefore, the normal of interest in this figure is the normal to the line joining UAVs A.sub.1 and A.sub.5.

[0093] Assuming that the sizes of the target circle and the polygon A.sub.1 . . . A.sub.n formed by the pursuing UAVs are such that (i) the open edge of the polygon is of length greater than the diameter of the circle, and (ii) the perimeter of the polygon is greater than the circumference of the circle B.

[0094] The problem addressed in this section is to develop guidance laws governing the longitudinal and lateral accelerations of then pursuing UAVs with which, the pursuing UAVs can simultaneously achieve the following: [0095] Objective a) Drive the velocity vector of any chosen point X in their convex hull into the collision cone to the target swarm B, [0096] Objective b) Rotate their formation appropriately so that they approach the target swarm along a direction that corresponds with the open end of the chain, and [0097] Objective c) Maintain the distances between the UAVs to be constant, that is, the formation to be rigid.

[0098] Objective a: To meet this objective, we determine the equation governing the longitudinal and lateral accelerations of the pursuing UAVs that will drive y.sub.X to a reference value w<0, since y.sub.X<0 corresponds to the physical scenario of the velocity vector of X being inside the collision cone to the target. For a maneuvering target, this objective is met if the control inputs of the pursuing UAVs satisfy Eq. 36. If the target is non-maneuvering, then Eq. 36 reduces to Eq. 27.

[0099] Objective b: In order to meet this objective, Eq. 36 is combined with additional conditions that will enable the formation to be oriented appropriately. Toward determining these additional conditions, we look at the kinematics of each of the lines A.sub.iA.sub.j. The states governing each line A.sub.iA.sub.j are represented by the quantities r.sub.ij, θ.sub.ij, V.sub.θ,ij and V.sub.r,ij. These quantities are schematically shown in FIG. 6. The governing kinematic equations are as follows:

[00017]$\begin{array}{cc}\left[\begin{array}{c}{\stackrel{.}{r}}_{\mathrm{ij}}\\ {\dot{\theta }}_{\mathrm{ij}}\\ {\dot{V}}_{\theta ,\mathrm{ij}}\\ {\dot{V}}_{r,\mathrm{ij}}\end{array}\right]=\left[\begin{array}{c}{V}_{r,\mathrm{ij}}\\ V_{\theta ,\mathrm{ij}}/r_{\mathrm{ij}}\\ \left(-V_{\theta ,\mathrm{ij}}{V}_{r,\mathrm{ij}}\right)/r_{\mathrm{ij}}\\ \left(V_{\theta ,\mathrm{ij}}^2\right)/r_{\mathrm{ij}}\end{array}\right]-\left[\begin{array}{c}0\\ 0\\ \sin \left({\delta }_i-{\theta }_{\mathrm{ij}}\right)\\ \cos \left({\delta }_i-{\theta }_{\mathrm{ij}}\right)\end{array}\right]a_i+\left[\begin{array}{c}0\\ 0\\ \sin \left({\delta }_j-{\theta }_{\mathrm{ij}}\right)\\ \cos \left({\delta }_j-{\theta }_{\mathrm{ij}}\right)\end{array}\right]a_j& \left(38\right)\end{array}$

[0100] where, δ.sub.i and δ.sub.i represent the directions of the acceleration vectors of a.sub.i and a.sub.j, respectively. From Eq. 38, it is evident that when {right arrow over (a)}.sub.i={right arrow over (a)}.sub.j then the relative velocity component of A.sub.j with respect to A.sub.i is zero, that is, V.sub.r,ij=0, V.sub.θ,ij=0 and r.sub.ij is constant.

[0101] Now consider the two UAVs carrying the open ends of the rope. Without loss of generality, assume that these two UAVs are A.sub.1 and A.sub.n. Then, θ.sub.1n represents the angle made by the line A.sub.1A.sub.n. Let θ.sub.1n,d represent the desired orientation of this line. We define an error quantity e.sub.θ,1n=θ.sub.1n−θ.sub.1n,d, Differentiating e.sub.θ,1n twice, we get the following:

[00018]$\begin{array}{cc}{\stackrel{.Math.}{e}}_{\theta ,1n}=\frac{r_{1n}{\stackrel{.}{V}}_{\theta ,1n}-V_{\theta ,1n}{V}_{r,1n}}{r_{1n}^2}-{\stackrel{.Math.}{\theta }}_{1n,d}& \left(39\right)\end{array}$

[0102] The quantity e.sub.θ,1n follows the dynamics ë.sub.θ,1n=−K.sub.1ė.sub.θ,1n−K.sub.2 e.sub.θ,1n, (where, K.sub.1>0 and K.sub.2>0 are constants). Such a choice ensures that e.sub.θ,1n follows stable dynamics and decays to zero. After substituting terms from Eq. 38, it is required that the accelerations of A.sub.1 and A.sub.n satisfy the following equation:

—a.sub.1 sin(δ.sub.1−θ.sub.1n)+a.sub.n sin(δ.sub.n−θ.sub.1n)=V.sub.θ,1n(2V.sub.r,1n/r.sub.1n−K.sub.1)−r.sub.1nK.sub.2(θ.sub.1n−θ.sub.1n,d)+r.sub.1nK.sub.1{dot over (θ)}.sub.1,nd  (40)

[0103] The above equation can then be written in terms of the control inputs u.sub.α,1, u.sub.V,1, u.sub.α,n, u.sub.V,n as follows:

−u.sub.α,1V.sub.1 cos(α.sub.1−θ.sub.1n)−u.sub.V,1 sin(α.sub.1−θ.sub.1n)+u.sub.α,nV.sub.n cos(α.sub.n−θ.sub.1n)+u.sub.V,n sin(α.sub.n−θ.sub.1n)=V.sub.θ,1n(2V.sub.r,1n/r.sub.1n−K.sub.1)−r.sub.1nK.sub.2(θ.sub.1n−θ.sub.1n,d)+r.sub.1nK.sub.1{dot over (θ)}.sub.1,nd  (41)

[0104] If the control inputs are applied such that (36) and (41) are satisfied, then the pursuing UAVs will be able to surround the target with the rope at the appropriate orientation. The reference value θ.sub.1n,d would be equal (or close) to

[00019]${\alpha }_X+\frac{\pi }{2}.$

[0105] Objective c: This requirement dictates that the distances between the UAVs need to remain constant throughout the engagement. a n-sided polygon can be triangulated into n−2 distinct triangles. As an example, the pentagon A.sub.1A.sub.2A.sub.3A.sub.4A.sub.5 in FIG. 6 can be triangulated into three triangles A.sub.5A.sub.1A.sub.2, A.sub.5A.sub.2A.sub.3 and A.sub.5A.sub.3A.sub.4. It can be ensured that the pentagon will remain rigid if each of these three triangles remain rigid, or in other words, each of the seven sides A.sub.1A.sub.2, A.sub.2A.sub.3, A.sub.3A.sub.4, A.sub.4A.sub.5, A.sub.5A.sub.1, A.sub.5A.sub.2 and A.sub.5A.sub.3 remain of constant length. A general n-sided polygon will remain rigid, if 2n−3 inter UAV-distances (between UAV pairs chosen based on a triangulation of the polygon) remain constant. Let E represent the set of these 2n−3 inter UAV-distances.

[0106] The distance between a UAV pair (A.sub.i,A.sub.j) can be kept constant if the accelerations of UAVs A.sub.i and A.sub.j are such that {dot over (V)}.sub.r,ij=0 (with the assumption that the initial conditions are such that V.sub.r,ij=0). From Eq. 38, {dot over (V)}.sub.r,ij=0 will hold, if the following condition is satisfied:

−a.sub.i cos(δ.sub.i−θ.sub.ij)+a.sub.j cos(δ.sub.j−θ.sub.ij)=V.sub.θ,ij.sup.2/r.sub.ij,∀(i,j)∈E  (42)

[0107] The above equation can then be written in terms of the control inputs u.sub.α,i, u.sub.V,i, u.sub.α,j, u.sub.V,j as follows:

u.sub.V,i cos(α.sub.i−θ.sub.ij)−u.sub.α,iV.sub.i sin(α.sub.i−θ.sub.ij)−u.sub.V,j cos(α.sub.j−θ.sub.ij)+u.sub.α,jV.sub.j sin(α.sub.j−θ.sub.ij)=V.sub.θ,ij.sup.2/r.sub.ij,∀(i,j)∈E  (43)

[0108] Eqs 27, 41 and 43 can now be combined and written in the following matrix form AU=Y:

[00020]$\begin{array}{cc}\left[\begin{array}{cccccccccc}{\lambda }_1{N}_{\alpha ,1}& {\lambda }_1{N}_{V,1}& {\lambda }_2{N}_{\alpha ,2}& {\lambda }_2{N}_{V,2}& .Math.& .Math.& {\lambda }_{n-1}{N}_{\alpha ,n-1}& {\lambda }_{n-1}{N}_{V,n-1}& {\lambda }_n{N}_{\alpha ,n}& {\lambda }_n{N}_{V,n}\\ -V_1{s}_1& c_1& V_2{s}_2& -c_2& 0& 0& 0& 0& 0& 0\\ .Math.& .Math.& .Math.& .Math.& .Math.& .Math.& .Math.& .Math.& .Math.& .Math.\\ 0& 0& 0& 0& 0& 0& -V_{n-1}{s}_{n-1}& c_{n-1}& V_n{s}_n& -c_n\\ -V_1{c}_1& -s_1& 0& 0& 0& 0& 0& 0& V_n{c}_n& s_n\end{array}\right]& \left(44\right)\end{array}$$\left[\begin{array}{c}{u}_{\alpha ,1}\\ u_{V,1}\\ u_{\alpha ,2}\\ u_{V,2}\\ .Math.\\ u_{\alpha ,n-1}\\ u_{V,n-1}\\ u_{\alpha ,n}\\ u_{V,n}\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}-K\left(y_X-w\right)+a_B\left[\frac{\partial y_X}{\partial V_{\theta ,X}}\cos \left(\beta -{\theta }_X\right)-\frac{\partial y_X}{\partial V_{r,X}}\sin \left(\beta -{\theta }_X\right)\right]\\ V_{\theta ,12}^2/r_{12}\end{array}\\ .Math.\end{array}\\ V_{\theta ,n-1,n}^2/r_{n-1,n}\end{array}\\ V_{\theta ,1n}\left(2V_{r,1n}/r_{1n}-K_1\right)-r_{1n}{K}_2\left({\theta }_{1n}-{\theta }_{1n,d}\right)+r_{1n}{K}_1{\stackrel{.}{\theta }}_{1,\mathrm{nd}}\end{array}\right]$

[0109] where, we have used the shorthand notation that for every pair of adjacent UAVs i and j, s.sub.i≡sin (α.sub.i−θ.sub.ij), s.sub.j=sin(α.sub.j−θ.sub.ij) and c.sub.i=cos(α.sub.i−θ.sub.ij), c.sub.j=cos(α.sub.j−θ.sub.ij).

[0110] Eq 44 represents a total of 2n−1 equations with 2n unknowns. The 2n−1 equations comprise one equation that drives the collision cone function y.sub.X, 2n−3 equations that ensure that the inter-UAV distances are such that the formation remains rigid, and one equation that drives the orientation of the open end A.sub.1A.sub.n of the chain. The 2n unknowns correspond to the control inputs (longitudinal acceleration and heading angle rate) of each of the n UAVs. This thus represents an under-determined system of equations, with there being one more unknown than the number of equations. Solution of this linear system of equations leads to control inputs of the pursuing UAVs with which objectives (a)-(c) are simultaneously satisfied.

[0111] In general, an undetermined system of equations (that are consistent), have non-unique solutions. For the current system (Eq. 44), from the given sizes of the A and Y matrices, it can be determined that the general solution will lie along a line in the R.sup.2n control (U) space. A particular solution U.sub.1 can be found as U.sub.1=A.sup.+Y, where A.sup.+ is the Moore-Penrose pseudoinverse of A. The general solution is then obtained as U=U.sub.1+cN, where N is the null space of A, and c is a scalar.

[0112] it can sometimes happen that the accelerations lead to a trajectory that puts one of the n pursuing UAVs on a collision course with the target. The occurrence of such a collision can be predicted by defining functions y.sub.1, . . . , y.sub.n representing the miss distance functions of each of n pursuing UAVs to the target circle, similar to Eq. 4, along with the corresponding miss-times t.sub.m1, . . . , t.sub.mn, similar to Eq. 3. If it is determined that y.sub.1<0,V.sub.r,i<0,t.sub.mi<t.sub.m,X holds for some UAV A.sub.i, this means that UAV A.sub.i will collide with the target before X intercepts the target. This collision can be averted by rotating the formation appropriately, and this can be achieved by modifying the reference value θ.sub.1n,d used in Eq. 44. Collisions between the rope and the target circle can be detected and avoided by following a similar process.

[0113] Guidance Laws for Time-Varying Target Size

[0114] Below is described a scenario where the size of the target changes with time. This could happen, for example, when the vehicles in the intruder swarm spread apart during their flight. At the same time however, we have assumed at the outset that their mission requires them to remain as a flock (given that they are performing a coordinated attack or flying in a leader-follower configuration), and they cannot stray too far apart from one another. We incorporate the effect of such a size change by making the radius of the circle that bounds the target, to be a function of time, that is, R is now R(t). In this case, the pursuers not only need to surround the target, but they may also need to increase their inter-vehicle distances appropriately so that they can enclose the target within their convex hull.

[0115] It is assumed that the pursuing UAVs have the ability to vary the length of the rope to some extent. Such a length change can be achieved by means of a spinning reel deployment mechanism (similar to the mechanism used in a fishing rod) that is carried by each UAV. Then, in response to an increase in the size of the target, if the distance between a pair of UAVs needs to increase, those UAVs can extend the rope between them till it is of desired length. Similarly, if the distance between a pair of UAVs needs to decrease, those UAVs reel in the rope between them. since each UAV can carry only a finite amount of cable, the permissible change in the length of the rope is upper bounded. This is schematically depicted in FIG. 7.

[0116] The problem addressed in this section is to develop guidance laws governing the longitudinal and lateral accelerations of then pursuing UAVs with which, the pursuing UAVs can simultaneously achieve the following: [0117] Objective a) Drive the velocity vector of any point X in their convex hull into the collision cone to the target swarm B, [0118] Objective b) Rotate their formation appropriately so that they approach the target swarm with the open end of the chain at a desired angle, and [0119] Objective c) Adjust the edges of their formation to desired values (and these desired values depend on the size of the target).

[0120] It is preferable to determine a radius R(t) of the smallest circle that bounds the intruder swarm. The desired values of the edges of the formation then need to be such that the perimeter of the polygon is larger than the circumference of this computed circle.

[0121] Objective a: This objective can be met if Eq. 27 is modified to account for the changing radius of the circle. Toward this end, Eq. 4 takes the following form:

[00021]$\begin{array}{cc}{y}_X=\frac{r_X^2{V}_{\theta ,X}^2}{V_{\theta ,X}^2+V_{r,X}^2}-{R\left(t\right)}^2& \left(45\right)\end{array}$

[0122] The time derivative of y.sub.X is:

[00022]$\begin{array}{cc}\frac{{\mathrm{dy}}_X}{\mathrm{dt}}=\frac{\partial y_X}{\partial r_X}{\stackrel{.}{r}}_X+\frac{\partial y_X}{\partial V_{r,X}}{\dot{V}}_{r,X}+\frac{\partial y_X}{\partial V_{\theta ,X}}{\dot{V}}_{\theta ,X}+\frac{\partial y_X}{\partial R}\dot{R}& \left(46\right)\end{array}$

[0123] Note that

[00023]$\frac{\partial y_X}{\partial R}=-2R\left(t\right).$

We differentiate the Lyapunov function Eq. 24 along the trajectories of the system defined by Eq. 13-20. After doing so, it is seen that that we can ensure Eq. 24 follows the dynamics Ż=−KZ, if the control inputs of the pursuing UAVs satisfy the following equation, which is the counterpart of Eq. 27:

λ.sub.1(N.sub.1αu.sub.α,1+N.sub.1vu.sub.V,1)+ . . . +λn(N.sub.nαu.sub.α,n+N.sub.nvu.sub.V,n)=−K(y.sub.X-w)+2R{dot over (R)}  (47)

[0124] When the lateral acceleration a.sub.B of the maneuvering swarm is explicitly incorporated in the guidance law, we get the following equation that will ensure that Ż=−KZ continues to be satisfied even when the target maneuvers:

[00024]$\begin{array}{cc}{\lambda }_1\left(N_{1\alpha }{u}_{\alpha ,1}+N_{1v}{u}_{V,1}\right)+.Math.+{\lambda }_n\left(N_{n\alpha }{u}_{\alpha ,n}+N_{nv}{u}_{V,n}\right)=-K\left(y_X-w\right)+2R\dot{R}+a_B\left[\frac{\partial y_X}{\partial V_{\theta ,X}}\cos \left(\beta -{\theta }_X\right)-\frac{\partial y_X}{\partial V_{r,X}}\sin \left(\beta -{\theta }_X\right)\right]& \left(48\right)\end{array}$

[0125] Objective b: This is met if the accelerations of A.sub.1 and A.sub.n satisfy Eq. 41.

[0126] Objective c: Consider UAVs A.sub.i, A.sub.j∈E that have an inter-UAV distance of r.sub.ij(t). Let r.sub.ij,d represent the desired inter-UAV distance between this pair of UAVs. We define an error quantity e.sub.r,ij=r.sub.ij−r.sub.ij,d. Differentiating e.sub.r,ij twice, we get the following:

ë.sub.r,ij={dot over (V)}.sub.r,ij−{umlaut over (r)}.sub.ij,d  (49)

[0127] We enforce that the quantity e.sub.r,ij follows the dynamics ë.sub.r,ij=−K.sub.3ė.sub.r,ij−K.sub.4e.sub.r,ij, (where, K.sub.3>0 and K.sub.4>0 are constants). This choice of dynamics will ensure that e.sub.r,ij decays to zero. Then, substituting terms from Eq. 38, we eventually arrive at the following equation governing the accelerations of UAVs A.sub.i and A.sub.j:

−a.sub.i cos(δ.sub.i−θ.sub.ij)+a.sub.j cos(δ−θ.sub.ij)=—V.sub.θ,ij.sup.2/r.sub.ij−K.sub.3(V.sub.r,ij−{dot over (r)}.sub.ij,d)−K.sub.4(r.sub.ij−r.sub.ij,d),∀(i,j)∈E  (50)

[0128] The above equation can then be written in terms of the control inputs u.sub.α,i, u.sub.V,i, u.sub.α,j, u.sub.V,j as follows:

u.sub.α,iV.sub.i sin(α.sub.i−θ.sub.ij)−u.sub.V,i cos(α.sub.i−θ.sub.ij)−u.sub.α,jV.sub.j sin(α.sub.j−θ.sub.ij)+u.sub.V,j cos(α.sub.j−θ.sub.ij)=−V.sub.θ,ij.sup.2/r.sub.ij−K.sub.3(V.sub.r,ij−r.sub.ij,d)−K.sub.4(r.sub.ij−r.sub.ij,d),∀(i,j)∈E  (51)

[0129] Eqs. 47, 51 and 43 can now be combined and written in the following matrix form AU=B:

[00025]$\begin{array}{cc}\left[\begin{array}{cccccccccc}{\lambda }_1{N}_{\alpha ,1}& {\lambda }_1{N}_{V,1}& {\lambda }_2{N}_{\alpha ,2}& {\lambda }_2{N}_{V,2}& .Math.& .Math.& {\lambda }_{n-1}{N}_{\alpha ,n-1}& {\lambda }_{n-1}{N}_{V,n-1}& {\lambda }_n{N}_{\alpha ,n}& {\lambda }_n{N}_{V,n}\\ V_1{s}_1& -c_1& -V_2{s}_2& c_2& 0& 0& 0& 0& 0& 0\\ .Math.& .Math.& .Math.& .Math.& .Math.& .Math.& .Math.& .Math.& .Math.& .Math.\\ 0& 0& 0& 0& 0& 0& -V_{n-1}{s}_{n-1}& c_{n-1}& V_n{s}_n& -c_n\\ -V_1{c}_1& -s_1& 0& 0& 0& 0& 0& 0& V_n{c}_n& s_n\end{array}\right]& \left(52\right)\end{array}$$\left[\begin{array}{c}{u}_{\alpha ,1}\\ u_{V,1}\\ u_{\alpha ,2}\\ u_{V,2}\\ .Math.\\ u_{\alpha ,n-1}\\ u_{V,n-1}\\ u_{\alpha ,n}\\ u_{V,n}\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}-K\left(y_X-w\right)+2R\stackrel{.}{R}+a_B\left[\frac{\partial y_X}{\partial V_{\theta ,X}}\cos \left(\beta -{\theta }_X\right)-\frac{\partial y_X}{\partial V_{r,X}}\sin \left(\beta -{\theta }_X\right)\right]\\ -V_{\theta ,12}^2/r_{12}-K_3\left(V_{r,12}-{\stackrel{.}{r}}_{12,d}\right)-K_4\left(r_{12}-r_{12,d}\right)\end{array}\\ .Math.\end{array}\\ -V_{\theta ,n-1,n}^2/r_{n-1,n}-K_3\left(V_{r,n-1,n}-{\stackrel{.}{r}}_{n-1,n,d}\right)-K_4\left(r_{n,n-1}-r_{n,n-1,d}\right)\end{array}\\ V_{\theta ,1n}\left(2V_{r,1n}/r_{1n}-K_1\right)-r_{1n}{K}_2\left({\theta }_{1n}-{\theta }_{1n,d}\right)+r_{1n}{K}_1{\stackrel{.}{\theta }}_{1,\mathrm{nd}}\end{array}\right]$

[0130] By solving the above matrix equation for the 2n unknowns u.sub.α,i, u.sub.V,i, i=1, . . . , n, we can determine the control inputs of the n pursuing UAVs that will simultaneously satisfy Objectives (a)-(c).

[0131] Guidance Laws for Arbitrarily-Shaped Intruder Swarms

[0132] The guidance laws developed in the preceding sections assumed that the shape of the intruder swarm is bounded within a circle. In cases where the intruder swarm has a somewhat elongated shape, then the circular approximation can represent an over-approximation to the shape of the swarm, and this can be therefore inadequate. In this section, we show how the guidance laws can be modified when the shape of the intruder swarm is arbitrary.

[0133] By way of background, the following is a result on the collision cone between a point and an arbitrarily shaped object with reference to FIG. 8 point object A is on a collision course with an arbitrarily shaped object B if the condition y<0, V.sub.r,b<0 is satisfied, where y and V.sub.r,b are defined as follows:

[00026]$\begin{array}{cc}y=\frac{{\left[V_{\theta }\cos \left({\theta }_b-\theta \right)-V_r\sin \left({\theta }_b-\theta \right)\right]}^2}{V_{\theta }^2+V_r^2}-{\sin }^2\left(\frac{\psi }{2}\right)& \left(53\right)\end{array}$$\begin{array}{cc}{V}_{r,b}=V_r\cos \left({\theta }_b-\theta \right)+V_{\theta }\sin \left({\theta }_b-\theta \right)& \left(54\right)\end{array}$

[0134] In the above equations, ψ represents the angle subtended at A by the tangent lines AT.sub.1 and AT.sub.2 to B. T.sub.1AT.sub.2 represents the conical hull of B, relative to A, and ψ thus represents the angle of this conical hull. When there exist more than two tangent lines to B, then AT.sub.1 and AT.sub.2 represent the pair of tangents that are such that B is completely contained within the sector T.sub.1AT.sub.2. Note that since B is moving, the conical hull is distinct from the collision cone, which is an entity computed in the relative velocity space. The quantity θ.sub.b represents the angle made by the angular bisector of T.sub.1AT.sub.2 with the horizontal. V.sub.θ and V.sub.r are the relative velocity components that are respectively normal to, and along, a line from A that passes through a reference point in B. Note that when the arbitrary shape is replaced by a circle, (and the reference point P is the center of this circle), then

[00027]${\theta }_b=\theta ,\sin \left(\frac{\psi }{2}\right)=\frac{R}{r}.$

Substituting these in Eqs. 53 and 54. these reduce to Eq. 4, and V.sub.r, respectively.

[0135] Now consider the engagement geometry between n pursuing UAVs carrying a stretchable ropeline and an arbitrarily shaped swarm, as shown in FIG. 9. We use Eq. 53 to define a function y.sub.X between a point X in the convex hull of the pursuing UAVs and B. In this equation, ψ.sub.X represents the angle subtended by B at the point X. The time derivative of y.sub.X is as follows:

[00028]$\begin{array}{cc}\begin{array}{c}\frac{\partial y_X}{\partial dt}=\frac{\partial y_X}{\partial {\theta }_X}{\dot{\theta }}_X+\frac{\partial y_X}{\partial {\theta }_b}{\dot{\theta }}_b+\frac{\partial y_X}{\partial V_{r,X}}{\dot{V}}_{r,X}+\frac{\partial y_X}{\partial V_{\theta ,X}}{\dot{V}}_{\theta ,X}+\frac{\partial y_X}{\partial {\psi }_X}{\dot{\psi }}_X\\ =-{\dot{V}}_X\left[\frac{\partial y_X}{\partial V_{r,X}}\cos \left({\alpha }_X-{\theta }_X\right)+\frac{\partial y_X}{\partial V_{\theta ,X}}\sin \left({\alpha }_X-{\theta }_X\right)\right]+\\ {\dot{\alpha }}_X{V}_X\left[\frac{\partial y_X}{\partial V_{r,X}}\sin \left({\alpha }_X-{\theta }_X\right)-\frac{\partial y_X}{\partial V_{\theta ,X}}\cos \left({\alpha }_X-{\theta }_X\right)\right]+\\ \frac{\partial y_X}{\partial {\theta }_b}{\dot{\theta }}_b+\frac{\partial y_X}{\partial {\psi }_X}{\stackrel{.}{\psi }}_X\end{array}& \left(55\right)\end{array}$

[0136] Substituting V.sub.X from Eq. 17 and {dot over (α)}.sub.X from Eq. 18 in the above equation, it is evident that we can enforce the Lyapunov function Z to follow the dynamics Ż=−KZ, if the quantities u.sub.α,1, u.sub.V,1, . . . , u.sub.α,n, u.sub.V,n of the pursuing UAVs satisfy the equation:

[00029]$\begin{array}{cc}{\lambda }_1\left(N_{1\alpha }{u}_{\alpha ,1}+N_{1v}{u}_{V,1}\right)+.Math.+{\lambda }_n\left(N_{n\alpha }{u}_{\alpha ,n}+N_{nv}{u}_{V,n}\right)=-K\left(y_X-w\right)-\frac{\partial y_X}{\partial {\theta }_b}{\dot{\theta }}_b-\frac{\partial y_X}{\partial {\psi }_X}{\dot{\psi }}_X& \left(56\right)\end{array}$

[0137] where, N.sub.iα, i=1, . . . , n is as in Eq. 28, and N.sub.iv, i=1, . . . , n is as in Eq. 29.

[0138] Eq 56 thus represents a generalization of Eq. 27 to the general case of arbitrarily shaped swarms. This equation can be combined with Eq. 41 and Eq. 43 to reorient the formation and lead to a matrix equation along the lines of Eq. 44. Similarly, Eq. 56 can be combined with Eqs. 41 and 51 to reorient the formation as well as change the inter-UAV distances and lead to a matrix equation along the lines of Eq. 44. In the latter case, the reference lengths r.sub.ij,d of the formation edges need to be such that the perimeter of the polygon formed by the pursuing UAVs is greater than the perimeter of the swarm.

[0139] Thus, as described herein, the present disclosure teaches solutions to the problem of n UAVs needing to pursue, and subsequently surround, a swarm of hostile UAVs (flying as a flock) that has intruded into a protected airspace. The development of analytical cooperative pursuit guidance laws that will meet the objective of the pursuing UAVs, is addressed. These cooperative pursuit guidance laws are developed based on a collision cone framework. Using this framework, analytical cooperative pursuit guidance laws that enable the n UAVs to enclose the intruder swarm within their convex hull are developed, for scenarios where the intruder swarm is enclosed in a bounding circle, and the circle may change size with time. In subsequent variants of this problem, the pursuing UAVs are assumed to be carrying a dragnet in an open-chain configuration, and the collision cone-based guidance laws are further developed to enable the UAVs to re-orient their formation appropriately (so that they approach the target from the open end of this chain), and also adjust their inter-vehicle distances appropriately (so that they are able to surround the target even as the target increases in size). These guidance laws are then generalized to the case when the intruder swarm is arbitrarily shaped. Simulations demonstrate the efficacy of the cooperative guidance laws.

[0140] Collision Cone-Based Net Capture of a Swarm of UAVs

[0141] Below addresses the problem of capturing a swarm of intruder UAVs, using a net manipulated by a team of defense UAVs. The intruder UAV swarm may be stationary, in motion, and even maneuver. The concept of collision cones in 3-dimensional space is used to determine the strategy used by the net carrying UAVs to maneuver or manipulate the net in space in order to capture the intruders. The manipulation of the net involves guiding the net to pursue the intruders and orienting it in space appropriately so as to maximize the effectiveness of capture. The net manipulation strategy is derived from the concept of collision cones defined in a relative velocity framework, and analytical expressions of nonlinear guidance laws are obtained. Simulations are presented to demonstrate the efficacy of these guidance laws.

[0142] Nomenclature

[0143] The following nomenclature is used hereinbelow.

TABLE-US-00001 ϕ Azimuth angle of position vector θ Elevation angle of position vector ψ Azimuth angle of velocity vector γ Elevation angle of velocity vector β Azimuth angle of acceleration vector α Elevation angle of acceleration vector a Acceleration magnitude V.sub.r, V.sub.θ, V  Relative velocity components V speed R Radius of sphere indicates data missing or illegible when filed

[0144] Introduction

[0145] The problem of a team of UAVs capturing a swarm of intruder UAVs with a net is disclosed hereinbelow. This class of net capture applications, in which a net carried by one or more UAVs, is used to capture a UAV intruding into a guarded airspace, has received considerable recent attention from researchers working on drone technologies due to the obvious need to protect the airspace from a proliferation of unauthorized drones. A related (but distinct) problem is that of net recovery, in which a UAV approaching a landing site (which could be an unprepared runway or a ship deck), is made to land on a net. Most net recovery problems involve a guidance algorithm that guides the UAV to a particular point on the net, with the net being fixed on the ground or on a stationary or moving platform. These methods may be termed as passive when the net carrying platform is either stationary or mobile, but does not maneuver to aid the landing operation, and it is the landing UAV that uses a guidance algorithm to modulate its trajectory in order to land on the recovery net. On the other hand, the net capture algorithms are active methods in which the net-carrying platform maneuvers itself to capture the intruder UAVs, which may themselves be maneuvering. Some of the net recovery methods can also be termed as active in the sense that the net is maneuvered in such a way as to bring it in the path of the landing UAV, thereby aiding the landing operation.

[0146] Given the rapid pace of advance in UAV technologies, it appears that UAV swarms will soon become common in many applications and may sometimes pose a threat for other vehicles in their airspace. In the present disclosure, we consider an interception problem in which three UAVs carrying a net actively attempt to trap a swarm of intruder UAVs. The capturing UAVs are assumed to be capable of flying at speeds higher than the intruder UAVs which, in turn, may maneuver in space. The problem is to decide how the UAVs need to manipulate, as well as orient the net in order to ensure that the net intercepts the intruder UAVs. In the guidance literature, interception problems have been addressed in the context of a missile intercepting an intruder aircraft. However, the problem there is formulated as an interceptor (modeled as a point), trying to intercept a target (which is also a point). In fact much of the guidance laws in the literature (PN being one of them) are based on the idea of guiding a point to intercept a target which may or not maneuver. In the present disclosure, we do not consider the problem as that of guiding a point but rather that of guiding multiple points (UAVs) together. Such a coordinated guidance is required in order to ensure that the net-carrying UAVs achieve the simultaneous objectives of target interception as well as appropriate net orientation.

[0147] To solve this problem, we use the notion of collision cones which are primarily used for collision detection and avoidance. The collision cone is a cone of velocity vectors of an arbitrarily shaped object in motion which lead to collision with another arbitrarily shaped object moving in space. Initially, the collision cones were defined for objects moving on a plane and then extended to 3-D space. The concept was later extended to define what is known as a safe passage cone which was used to maneuver vehicles through narrow orifices.

[0148] Apart from the novelty of using the collision cone approach for the net capture problem, there are several other differences that the present disclosure has with similar problems addressed in the literature. We consider a swarm of intruder UAVs flying as a flock. We do not consider any restriction on their trajectory, which may be a straight line or a curved trajectory. We model the intruder UAV flock as being enclosed inside a sphere, and formulate the problem as one of interception between the net and the sphere, with the net at a desired orientation. The kinematics-based guidance laws are developed in a relative velocity framework, and thus the kinematic effect of the wind acting on the vehicles are implicitly accounted for in the analysis. As far as dynamic effects are concerned, we assume that the drones are robust enough to take care of the disturbances caused by the wind and the net.

[0149] Problem Formulation and Solution Approach

[0150] With reference to FIGS. 10 and 11, consider three UAVs carrying a net, with the objective of capturing a swarm of intruder UAVs. The three net carrying UAVs are denoted as A.sub.1, A.sub.2, and A.sub.3, and these UAVs form a triangle in 3-D space. B represents a virtual sphere of radius R that notionally encloses the swarm of intruder UAVs. Let the speed of each of the three net-carrying UAVs be denoted by V.sub.A, and the directions of their velocity vectors by (ψ.sub.A1, γ.sub.A1), (ψ.sub.A2, γ.sub.A2), (ψ.sub.A3, γ.sub.A3). Here, (ψ, γ) represents an azimuth-elevation angle pair, as schematically depicted in FIG. 11. Let the speed of the sphere be V.sub.B and the direction of its velocity vector be (ψ.sub.Bγ.sub.B).

[0151] Net carrying UAVs: In the present disclosure, we assume that the net-carrying UAVs are of a fixed-wing type. There are several scenarios where fixed-wing UAVs have advantages over quadcopters for this application: Fixed-wing UAVs generally have speeds that are higher than quadrotors [fwing], have longer flight times and longer ranges, as well as the ability to carry heavier payloads. Also, it is often desirable that the intruder is intercepted by the net-carrying UAVs some safe distance away from the protected area, and for such operations the longer flight range of fixed-wing UAVs offers benefits.

[0152] Since fixed wing UAVs always need to fly at speeds above a certain threshold in order to generate lift, therefore the permissible speed variation for such UAVs is somewhat small. When the net carrying UAVs move with equal velocities, the formation is rigid. However, with equal velocities, these UAVs cannot rotate or orient the net in space. In the present disclosure, we assume that while the speeds of the net carrying UAVs are all equal (and constant in time), the velocity headings of these UAVs can be different from one another. This difference in velocity headings can be used to rotate the net whenever required.

[0153] if the UAVs are very light and/or the net is very heavy, then the UAVs may not be robust to disturbances caused by the net. We assume that the UAVs and the net are so chosen that the drone controller is robust to these disturbances.

[0154] Intruder Swarm: We assume that the vehicles in the intruder swarm need to stay reasonably close to each other. This could be either because the swarm is in a leader-follower configuration and the followers need to stay close to the leader, or because the swarm has an objective to carry out a coordinated attack on a specific point in a protected area, and in order to increase the effectiveness of this attack, they need to stay close together.

[0155] We also assume that while the adversaries are trying to perform such a focused attack on a target, they do not have the intelligence to simultaneously detect that they themselves are being attacked. This thus belongs to a class of scenarios wherein the adversaries have been given a predefined trajectory (possibly computed off-line) to their target, and as they fly towards their target, they do not have the sensors to detect a threat. Therefore, the adversaries do not have the ability to take reactive, evasive action to avoid being intercepted by the net, or the ability to re-compute their trajectory to the target, based on sensor data. This is close to the technology level of present day drones. Smart drones working in swarms and capable of detecting an attacker and taking evasive maneuvers is still a futuristic concept. We also assume that the target of the adversaries, as well as their intended trajectory to the target, is unknown to the net-carrying UAVs.

[0156] The intruder swarm need not remain confined to a sphere of constant radius, and the geometry of the swarm can indeed be arbitrary. Our approach then can be used to intercept as many vehicles as possible. Algorithms like the circumcenter algorithm [circumcenter] can be employed to determine the center and radius of the smallest sphere that will bound a chosen subset of vehicles in the intruder swarm, and the net-carrying UAVs will then attempt to intercept this sphere.

[0157] Guidance Objectives: We define a point X on the net as a weighted centroid of the three UAVs. The guidance objective is to maneuver the net so as to close the distance between X and B with the additional requirement that the net approaches B at a certain angle, where the angle may be defined with respect to the current direction of motion of B. This is similar to the impact angle requirement in the interceptor guidance literature, but with the important difference that the net itself needs to maintain a pre-specified angle with respect to the motion of the point X. The impact angle and the net angle are illustrated in FIGS. 12A-12B.

[0158] Phases of engagement: The engagement comprises two phases. During the initial phase of the engagement, the objective is primarily to get the net sufficiently close to the intruder swarm. Subsequently, when the time-to-go becomes smaller than a threshold, we begin to rotate the net, in order to achieve a desired net angle. The reasons for which we rotate the net only during the latter phase of the engagement are the following: (i) The purpose of net rotation is to orient the net such that it's normal is roughly parallel to the velocity vector of the intruder swarm. If the intruder swarm performs continual maneuvers, then doing net orientations early (in response to these maneuvers) can lead to unnecessary wastage of control effort by the net-carrying UAVs. (ii) The interceptor seeker range is also relevant here, since the UAV sensors tracking the intruder swarm may be less effective at larger distances and more effective at closer distances. So it makes sense to first guide the UAVs to get close to the intruder UAV and then do the orientation maneuver when its sensors are able to track the intruder trajectory better.

[0159] The threshold value of the time-to-go at which to start rotating the net, will be a function of the resolution of the cameras mounted on the UAVs. This threshold value can be set based on the appropriate distance to the target, at which the camera's resolution is adequate for switching to the net rotation phase.

[0160] When we desire a point object moving with constant speed to intercept another point object moving with constant velocity, there is typically a single (at most, two) heading angle(s) of the former that will lead to collision with the latter. On the other hand, when we desire a finite-sized object (such as a net) to intercept with another finite-sized object (such as a sphere), there is a range (or more precisely, a cone) of heading angles of the net, with which such an interception can be achieved. We refer to this cone as the collision cone. By maneuvering the heading angle of the net into the collision cone, and ensuring that the heading angle stays inside this cone, it can be ensured that the net intercepts the intruder UAV swarm. The UAVs use the accelerations a.sub.Ai, i=1, 2, 3, to orient the net and carry it along a trajectory that leads to interception of the intruder UAV swarm.

[0161] Let {right arrow over (r)}.sub.A1, {right arrow over (r)}.sub.A2, {right arrow over (r)}.sub.A3 represent the position vectors of A.sub.1, A.sub.2, and A.sub.3, respectively (with respect to some inertial reference frame). Then, any point X on the plane A.sub.1A.sub.2A.sub.3 can be represented as a convex combination of these three vectors as follows:

{right arrow over (r)}.sub.X=λ.sub.1{right arrow over (r)}.sub.A1+λ.sub.2{right arrow over (r)}.sub.A2+λ.sub.3{right arrow over (r)}.sub.A3,λ.sub.1+λ.sub.2+λ.sub.3=1,λ.sub.1,λ.sub.2,λ.sub.3≥0  (1-1)

[0162] The weighted centroid X of the triangle A.sub.1A.sub.2A.sub.3, has a velocity {right arrow over (V)}.sub.X:

i. {right arrow over (V)}.sub.X=λ.sub.1{right arrow over (V)}.sub.A1+λ.sub.2{right arrow over (V)}.sub.A2+λ.sub.3{right arrow over (V)}.sub.A3  (1-2)

[0163] where, the components of {right arrow over (V)}.sub.X are as follows:

[00030]$\begin{array}{cc}\mathrm{ii}.& \\ {\stackrel{.fwdarw.}{V}}_X=V_A\left[\begin{array}{c}\begin{array}{c}{.Math.}_{i=1}^3{\lambda }_i\cos {\gamma }_{Ai}\cos {\psi }_{Ai}\\ {.Math.}_{i=1}^3{\lambda }_i\cos {\gamma }_{Ai}\sin {\psi }_{Ai}\end{array}\\ {.Math.}_{i=1}^3{\lambda }_i\sin {\gamma }_{Ai}\end{array}\right]& \left(1-3\right)\end{array}$

[0164] From (1-3), the magnitude of {right arrow over (V)}.sub.X is as follows:

[00031]$\begin{array}{cc}\mathrm{VX}={\mathrm{VA}\left[{.Math.}_{i=1}^3{\lambda }_i^2+2{\lambda }_1{\lambda }_2\sin {\gamma }_{A1}\sin {\gamma }_{A2}+2{\lambda }_1{\lambda }_3\sin {\gamma }_{A1}\sin {\gamma }_{A3}+2{\lambda }_2{\lambda }_3\sin {\gamma }_{A2}\sin {\gamma }_{A3}+2{\lambda }_1{\lambda }_3\cos {\gamma }_{A1}\cos {\gamma }_{A3}\cos \left({\psi }_{A3}-{\psi }_{A1}\right)+2{\lambda }_2{\lambda }_3\cos {\gamma }_{A2}\cos {\gamma }_{A3}\cos \left({\psi }_{A2}-{\psi }_{A3}\right)+2{\lambda }_1{\lambda }_2\cos {\gamma }_{A1}\cos {\gamma }_{A2}\cos \left({\psi }_{A1}-{\psi }_{A2}\right)\right]}^{\frac{1}{2}}& \left(1-4\right)\end{array}$

[0165] The azimuth and elevation angles of {right arrow over (V)}.sub.X are denoted by ψ.sub.X and γ.sub.X, respectively, and are given by:

[00032]$\begin{array}{cc}{\psi }_X={\tan }^{-1}\left[\frac{{\Sigma }_{i=1}^3{\lambda }_i\cos {\gamma }_{Ai}\sin {\psi }_{Ai}}{{\Sigma }_{i=1}^3{\lambda }_i\cos {\gamma }_{Ai}\cos {\psi }_{Ai}}\right]& \left(1-5\right)\end{array}$${\gamma }_X={\sin }^{-1}\left[\frac{V_A{\Sigma }_{i=1}^3{\lambda }_i\sin {\gamma }_{Ai}}{V_X}\right]$

[0166] In the remainder of the paper, for ease of notation, we will drop the subscript X in the terms {right arrow over (V)}.sub.X, V.sub.X, ψ.sub.X and γ.sub.X, except when necessary. Therefore, in the sequel, the terms {right arrow over (V)}, V, ψ and γ are all to be interpreted as quantities associated with the interception point X.

[0167] The orientation of the net is defined by the normal to the plane A.sub.1A.sub.2A.sub.3, denoted by {circumflex over (n)}. Then, defining {right arrow over (r)}.sub.21={right arrow over (r)}.sub.A2−{right arrow over (r)}.sub.A1 and {right arrow over (r)}.sub.31={right arrow over (r)}.sub.A3−{right arrow over (r)}.sub.A1, we can write {circumflex over (n)} as the normal to the plane that contains these vectors, as follows:

[00033]$\begin{array}{cc}\hat{n}=\frac{{\stackrel{.fwdarw.}{r}}_{21}\times {\stackrel{.fwdarw.}{r}}_{31}}{r_{21}{r}_{31}}& \left(1-6\right)\end{array}$

[0168] Let F represent the center of the sphere enclosing the swarm of intruder UAVs moving with speed V.sub.B, at an azimuth-elevation angle pair (ψ.sub.B, γ.sub.B). Consider the line XF. Defining r=XF, and (ϕ, θ) as the azimuth-bearing angle pair of XF, the relative velocity components of F with respect to X are:

V.sub.ϕ=V.sub.B cos γ.sub.B sin(ψ.sub.B−ϕ)−V sin γ sin(ψ−ϕ)  (1-7)

V.sub.θ=V.sub.B{−cos γ.sub.B sin θ cos(ψ.sub.B−ϕ)+sin γ.sub.B cos θ}−V{−cos γ sin θ cos(ψ−ϕ)+sin γ cos θ}  (1-8)

V.sub.r=V.sub.B{cos γ.sub.B cos θ cos(ψ.sub.B−ϕ)+sin γ.sub.B sin θ}−V{cos γ cos θ(ψ−ϕ)+sin γ sin θ}  (1-9)

[0169] where, V.sub.r is the relative velocity component along XF, and V.sub.ϕ and V.sub.θ represent relative velocity components orthogonal to XF. These quantities are defined as follows:

{dot over (r)}=V.sub.r,ϕ=V.sub.ϕ/(r cos θ),{dot over (θ)}=V.sub.θ/r  (1-10)

[0170] The derivatives of these relative velocity components are:

{dot over (V)}ϕ=−ϕ(V.sub.r cos θ−V.sub.θ sin θ)−{dot over (V)} cos γ sin(ψ−ϕ)+V sin γ{dot over (γ)}sin(ψ−ϕ)−V cos γψ cos(ψ−ϕ)  (1-11)

{dot over (V)}.sub.θ=−{dot over (θ)}V.sub.r−ϕV.sub.ϕ sin θ−{dot over (V)}[cos γ sin θ cos(ψ−ϕ)+sin γ cos θ]−V[−{dot over (γ)}sin γ sin θ cos(ψ−ϕ)−cos γ sin θ{dot over (ψ)}sin(ψ−ϕ)+{dot over (γ)}cos γ cos θ]  (1-12)

{dot over (V)}.sub.r={dot over (θ)}V.sub.θ+{dot over (ϕ)}V.sub.ϕ cos θ−{dot over (V)}[cos γ cos θ cos(ψ−ϕ)+sin γ sin θ]−V[−{dot over (γ)}sin γ cos θ cos(ψ−ϕ)−cos γ cos θ{dot over (ψ)}sin(ψ−ϕ)+{dot over (γ)}cos γ sin θ]  (1-13)

[0171] Eqs (1-11)-(1-13) are valid with the assumption that B moves with a constant velocity, that is, {dot over (V)}.sub.B=0, {dot over (ψ)}.sub.B=0, {dot over (γ)}.sub.B=0. The guidance laws for capture are initially designed in Section 3.C with this constant velocity assumption. We subsequently relax this assumption in Section 3.F. From (1-11)-(1-13), it is evident that the derivatives of the relative velocity components vary with the quantities {dot over (V)}, {dot over (γ)}, {dot over (ψ)}, which govern the acceleration of the weighted centroid X of the net. These derivatives are as follows.

[00034]$\begin{array}{cc}\stackrel{.}{\gamma }=\frac{V_A\left({\Sigma }_{i=1}^3{\lambda }_i{\stackrel{.}{\gamma }}_i\cos {\gamma }_i\right)}{V\cos \gamma }-\frac{\stackrel{.}{V}{V}_A\left({\Sigma }_{i=1}^3{\lambda }_i\sin {\gamma }_i\right)}{V^2\cos \gamma }& \left(1-14\right)\end{array}$$\begin{array}{cc}\dot{\psi }=\frac{{\cos }^2\psi }{{\left({.Math.}_{i=1}^3{\lambda }_i\cos {\gamma }_i\cos {\psi }_i\right)}^2}\left[\left(\underset{i=1}{\overset{3}{.Math.}}{\lambda }_i\cos {\gamma }_i\cos {\psi }_i\right)\left(-\underset{i=1}{\overset{3}{.Math.}}{\lambda }_i{\dot{\gamma }}_i\sin {\gamma }_i\sin {\psi }_i+\underset{i=1}{\overset{3}{.Math.}}{\lambda }_i{\dot{\psi }}_i\cos {\gamma }_i\cos {\psi }_i\right)+\left({.Math.}_{i=1}^3{\lambda }_i\cos {\gamma }_i\sin {\psi }_i\right)\left({.Math.}_{i=1}^3{\lambda }_i{\dot{\gamma }}_i\sin {\gamma }_i\cos {\psi }_i+{.Math.}_{i=1}^3{\lambda }_i{\dot{\psi }}_i\cos {\gamma }_i\sin {\psi }_i\right)\right]& \left(1-15\right)\end{array}$$\begin{array}{cc}\dot{V}=\frac{V_A^2}{V}\left[\underset{i=1}{\overset{3}{.Math.}}{\lambda }_i\sin {\gamma }_i\left(\underset{\underset{j\ne i}{j=1}}{\overset{3}{.Math.}}{\lambda }_j{\dot{\gamma }}_j\cos {\gamma }_j\right)-\underset{i=1}{\overset{3}{.Math.}}{\lambda }_i{\dot{\gamma }}_i\sin {\gamma }_i\left\{\underset{\underset{j\ne i}{j=1}}{\overset{3}{.Math.}}{\lambda }_j\cos {\gamma }_j\cos \left({\psi }_j-{\psi }_i\right)\right\}+{.Math.}_{i=1}^3{\lambda }_i{\dot{\psi }}_i\cos {\gamma }_i\left\{{.Math.}_{\underset{j\ne 1}{j=1}}^3{\lambda }_j\cos {\gamma }_j\sin \left({\psi }_j-{\psi }_i\right)\right\}\right]& \left(1-16\right)\end{array}$

[0172] In (1-14)-(1-16), for the sake of brevity, we have used ψ.sub.i and γ.sub.i for ψ.sub.Ai and γ.sub.Ai, respectively. It is seen from the above that {dot over (V)}, {dot over (γ)}, and {dot over (ψ)} depend linearly on {dot over (γ)}Ai and {dot over (ψ)}.sub.Ai, which are the velocity heading angle rates for A.sub.1, A.sub.2 and A.sub.3. They evolve as follows:

{dot over (γ)}.sub.Ai=a.sub.Ai[cos α.sub.Ai sin γ.sub.Ai cos(β.sub.Ai−ψ.sub.Ai)−cos γ.sub.Ai sin α.sub.Ai]/V.sub.A,  (1-17)

{dot over (ψ)}.sub.Ai=−a.sub.Ai cos α.sub.Ai sin(β.sub.Ai−ψ.sub.Ai)/(V.sub.A cos γ.sub.Ai)  (1-18)

[0173] where, (β.sub.Ai, α.sub.Ai) are the azimuth-elevation angle pair of the acceleration vector {right arrow over (a)}.sub.Ai. Eqs (1-10)-(1-18) thus represent the kinematic state equations for the engagement between the net and the threat UAV sphere B. Additionally, after substitution of (1-18) in (1-14)-(1-16), the equations corresponding to (1-14), (1-15) and (1-16) are seen to be linear functions of a.sub.A1, a.sub.A2, a.sub.A3 and can be written in the following form:

[00035]$\begin{array}{cc}\stackrel{.}{\gamma }=\left[\begin{array}{cc}\begin{array}{cc}{G}_1& G_2\end{array}& G_3\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{a}_{A1}\\ a_{A2}\end{array}\\ a_{A3}\end{array}\right],& \left(1-19\right)\end{array}$$\stackrel{.}{\psi }=\left[\begin{array}{cc}\begin{array}{cc}{H}_1& H_2\end{array}& H_3\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{a}_{A1}\\ a_{A2}\end{array}\\ a_{A3}\end{array}\right],$$\stackrel{.}{V}=\left[\begin{array}{cc}\begin{array}{cc}{J}_1& J_2\end{array}& J_3\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{a}_{A1}\\ a_{A2}\end{array}\\ a_{A3}\end{array}\right]$

[0174] where, the expressions for G.sub.i,H.sub.i,J.sub.i,i=1,2,3 are lengthy (and therefore not presented here), but can be readily inferred from (1-14)-(1-16).

[0175] Guidance Laws for Net Capture

[0176] In the following we will show how the concepts from collision cones can be used to design the net capture guidance law.

[0177] Point Object and a Sphere Moving in 3-D

[0178] When a point object and a sphere of radius R are moving with constant velocities, the miss-distance r.sub.m (which is the predicted distance between the point and the center of the sphere at the instant of closest approach), is given by the following equation:

r.sub.m.sup.2=r.sup.2(V.sub.ϕ.sup.2+V.sub.θ.sup.2)/(V.sub.ϕ.sup.2+V.sub.r.sup.2+V.sub.θ.sup.2)  (1-20)

[0179] and the time of closest approach is given by:

t.sub.m=(−rV.sub.r)/(V.sub.ϕ.sup.2+V.sub.r.sup.2+V.sub.θ.sup.2)  (1-21)

[0180] When V.sub.r<0, the above expression leads to a positive value of t.sub.m. When V.sub.r>0, it leads to a negative value of t.sub.m, which can be interpreted as the time of closest approach if the trajectories of both the objects are projected backwards in time.

[0181] If the miss distance r.sub.m is less than R, then the point object is on a collision course with the sphere. Based on this, one may define a miss-distance function y as follows [ref2]:

[00036]$\begin{array}{cc}y=\frac{r^2\left(V_{\varphi }^2+V_{\theta }^2\right)}{V_{\varphi }^2+V_r^2+V_{\theta }^2}-R^2& \left(1-22\right)\end{array}$

[0182] If the two objects move with constant velocities, then the predicted miss distance is equal to the actual miss distance, and therefore, the condition y<0 and V.sub.r<0 are both necessary and sufficient conditions for one object to intercept the other object. When the two objects move with varying velocities, then the condition y<0,V.sub.r<0 for all future time is a sufficient condition for interception.

[0183] In the 3-D physical space, we can construct a surface defined by the set of heading angles of {right arrow over (V)} that satisfy y=0, that is, (ψ,γ):y=0. This surface is shown in FIG. 13, and is a double cone, with vertex at X. Also, the set of heading angles of {right arrow over (V)} that satisfy V.sub.r=0, that is, (ψ,γ):V.sub.r=0 forms a plane, which partitions the above double cone into two single cones as shown in FIG. 13. We can accordingly partition the physical 3-D space into four sectors as follows: (i) y<0,V.sub.r<0, (ii) y≥0,V.sub.r≤0, (iii) y>0,V.sub.r>0, (iv) y≤0,V.sub.r>0. The collision cone between X and B is the sector defined by (i) and represents the set of heading angles of V for which the point X is on a collision course with B.

[0184] Collision Conditions Between the Net A.sub.1A.sub.2A.sub.3 and the Sphere B

[0185] Eq (1-22) can now be used to represent the miss distance from any point on the net A.sub.1A.sub.2A.sub.3 to the sphere, as follows. As stated in (1-1), any point X on the net can be written as a convex combination of {right arrow over (r)}.sub.A1, {right arrow over (r)}.sub.A2 and {right arrow over (r)}.sub.A3. Therefore, the vector r from any point X on the net to the center of the sphere B is:

{right arrow over (r)}={right arrow over (r)}.sub.B−(λ.sub.1{right arrow over (r)}.sub.A1+λ.sub.2{right arrow over (r)}.sub.A2+λ.sub.3{right arrow over (r)}.sub.A3)  (1-23)

[0186] where, {right arrow over (r)}.sub.B is the position vector of the center of B. Similarly, the relative velocity vector of the center of the sphere with respect to X on the net is given by:

{right arrow over (V)}.sub.REL={right arrow over (V)}.sub.B−(λ.sub.1{right arrow over ({dot over (r)})}.sub.A1+λ.sub.2{right arrow over ({dot over (r)})}.sub.A2+λ.sub.3{right arrow over ({dot over (r)})}.sub.A3)  (1-24)

[0187] Then, {right arrow over (V)}.sub.REL can be resolved into components V.sub.r, V.sub.ϕ, and V.sub.θ, where V.sub.r=<{right arrow over (V)}.sub.REL,{circumflex over (r)}> is the relative velocity component along the unit vector {circumflex over (r)} that corresponds to (1-23), while V.sub.ϕ, and V.sub.θ are the two mutually orthogonal components of V.sub.REL that are also orthogonal to {circumflex over (r)}.

[0188] We can then state the following. The net is on a collision course with B if there exists a (λ.sub.1,λ.sub.2,λ.sub.3) with Σ.sub.i=1.sup.3λ.sub.i=1,λ.sub.i≥0, such that:

[00037]$\begin{array}{cc}y\equiv \frac{r^2\left(V_{\varphi }^2+V_{\theta }^2\right)}{V_{\varphi }^2+V_r^2+V_{\theta }^2}-R^2<0,V_r<0& \left(1-25\right)\end{array}$

[0189] is satisfied. Since each point on the net is parametrized by (λ.sub.1,λ.sub.2,λ.sub.3), we can use (1-25) to write the miss distance function from each point on the net to the sphere as y(λ.sub.1,λ.sub.2,λ.sub.3). Along similar lines, we can also write V.sub.r associated with each point on the net as a function V.sub.r(λ.sub.1,λ.sub.2,λ.sub.3). The collision cone between the net A.sub.1A.sub.2A.sub.3 and the sphere B can be defined in the (ψ,γ) space as:

CC.sub.NB={(ψ,γ):y(λ.sub.1,λ.sub.2,λ.sub.3)<0 and V.sub.r(λ.sub.1,λ.sub.2,λ.sub.3)<0,∀λ.sub.i≥0,Σ.sub.i=1.sup.3λ.sub.i=1}  (1-26)

[0190] FIG. 14 demonstrates the distinction between the collision cone of the point X to B (referred to as CC.sub.XB in the figure) and the collision cone of the net to B. while the vertex of the cone CC.sub.XB is necessarily to be placed at X, the vertex of the cone CC.sub.NB can be placed anywhere on the net. If the actual velocity vector lies along the boundary of CC.sub.XB, it corresponds to the physical scenario of X grazing the surface of B. If the actual velocity vector lies along the boundary of CC.sub.NB, it corresponds to the physical scenario of a point on the perimeter of the net grazing the surface of B.

[0191] As an illustrative example, assume that the three UAVs A.sub.1, A.sub.2, A.sub.3 are at a distance 10 m apart from each other and they are moving with a speed of 5 m/sec. The intruder swarm is moving with a speed of 2.3 m/sec and a heading angle given by ψ=45 deg, γ=20 deg. The initial positions of the net-carrying UAVs are (0,0,8.7), (−5,0,0) and (5,0,0), respectively, while that of the center of the intruder sphere is (30,0,4.3). Assuming identical heading angles of the UAVs A.sub.1, A.sub.2, A.sub.3, the collision cone CC.sub.NB from the net to the intruder sphere (computed from (1-26)), is depicted on a (ψ,γ) plane as shown in FIG. 15A. Consider a heading angle inside this collision cone, say, ψ.sub.A1=ψ.sub.A2=ψ.sub.A3=200 deg, γ.sub.A1=γ.sub.A2=γ.sub.A3=171 deg. For this particular heading angle, FIG. 15B shows the surface plot of y(λ.sub.1,λ.sub.2) (with λ.sub.3 determined from λ.sub.3=1−(λ.sub.1+λ.sub.2)). The surface of V.sub.r is nearly flat, and negative for all (λ.sub.1,λ.sub.2), while the surface of y is positive for some (λ.sub.1,λ.sub.2) and negative for other (λ.sub.1,λ.sub.2). The portion of the net for which y(λ.sub.1,λ.sub.2)<0 is currently on a collision course with the intruder swarm.

[0192] FIG. 15A gives the range of (ψ,γ) that represents the collision cone between the entire net (not just the center of the net) and the intruder sphere. From this (ψ,γ) range, some considerations that govern the choice of the intercept point on the net and the intercept point on the sphere, are given below:

[0193] Intercept point on the net: The interception point on the net needs to be chosen such that the net hits the entire intruder swarm (or at least most of it). While there will be many engagement scenarios where the ideal interception point on the net is located close to its center, there can also be other scenarios where this is not necessarily the case. For example, consider the scenario when the net triangle is isosceles (with its two equal sides being significantly longer than the third side), and the intruder swarm is arranged in a somewhat linear configuration, with the length of this configuration being greater than the longest side of the net. If we now choose the center of the net as the intercept point, then the entire swarm cannot be intercepted by the net. It would be better to choose the intercept point closer to the longer side of the net, so as to intercept as many intruder UAVs as possible. This is shown in FIG. 16A.

[0194] Intercept point on the sphere: In many engagements, the ideal interception point on the sphere is located close to the center of the sphere. However, there can also be scenarios where the radius of the bounding sphere on the intruder swarm is larger than the size of the net, and moreover, the intruders are not uniformly distributed inside this bounding sphere. In such cases, we choose an intercept point on the intruder sphere that lies in the denser part of the swarm (where the denser part basically contains more vehicles). This will enable the interception of more vehicles in the swarm. This is shown in FIG. 16B.

[0195] Different intercept points on the sphere also lead to different impact angles. This is illustrated in FIGS. 17A-17C for a 2-D scenario involving an engagement between a point object and a moving circle. FIG. 17A shows the trajectory of the point object for several intercept points on the circle. FIG. 17B shows the difference between the heading angles of the point object and the circle, from which it is clearly evident that the impact angle varies with the intercept point. FIG. 17C shows that the acceleration profiles also vary for these different interception points. Choosing the best combination of interception point on the net and interception point on the sphere is an interesting problem in its own right, however it is beyond the scope of the present disclosure. The guidance schemes are developed in the present disclosure to work for any arbitrary choice of intercept point on the net and intercept point on the sphere.

[0196] Intercepting a subset of the intruder swarm: As mentioned earlier, if the size of the swarm is larger than the net, we then attempt to intercept a subset of the swarm. If we choose an interception point on the net that lies close to the centroid of the net, then the net will be able to intercept those vehicles in the swarm that lie within a sphere whose radius is less than (or equal to) the radius of the in-circle of the net. If there are members of the intruder swarm outside this sphere, then those intruders can potentially escape â€″ however our approach ensures that we have intercepted at least some of the intruding vehicles. For a given geometry of the intruder swarm, the larger the size of the net, the larger the radius of the sphere of intruders that can be intercepted. Refer to FIG. 16C for an illustration.

[0197] Acceleration Magnitude for Capture

[0198] In this section, we determine guidance laws for the three net carrying UAVs so as to enable them to intercept the swarm of threat UAVs enclosed in a sphere. The guidance laws are initially designed with an assumption that B moves with a constant velocity, that is, {dot over (V)}.sub.B=0, {dot over (ψ)}B=0, {dot over (γ)}B=0. In a subsequent section, we will relax this assumption. Define a function Z as follows:

Z=(1/2)(y−w).sup.2  (1-27)

[0199] where, w≤0 is a specified reference value, which may be time-varying. having an arbitrary w<0 provides us the flexibility to choose different interception points on the sphere. By determining a.sub.Ai, i=1,2,3 which will make Ż negative definite, interception of the swarm of threat UAVs B can be guaranteed.

[0200] Theorem 1: Let UAV A.sub.1 have an acceleration a.sub.Ai whose magnitude is given by:

N.sub.1a.sub.A1+N.sub.2a.sub.A2+N.sub.3a.sub.A3=−K(y−w)  (1-28)

[0201] where, N.sub.i, i=1,2,3, is given in (1-34). Then, the error function Z in (1-27) is globally asymptotically stable almost everywhere.

[0202] Proof: Ż=(y−w)({dot over (y)}−{dot over (w)}), where the time derivative of y(t) is as follows:

[00038]$\begin{array}{cc}\frac{dy}{dt}=\frac{\partial y}{\partial r}\dot{r}+\frac{\partial y}{\partial V_{\varphi }}{\dot{V}}_{\varphi }+\frac{\partial y}{\partial V_{\theta }}{\dot{V}}_{\theta }+\frac{\partial y}{\partial V_r}{\dot{V}}_r& \left(1-29\right)\end{array}$

[0203] Evaluating (1-29) along the system trajectories defined by (1-10)-(1-18), we get:

[00039]$\begin{array}{cc}\frac{dy}{dt}=\stackrel{.}{V}\left[-\frac{\partial y}{\partial V_{\varphi }}\cos \gamma \sin \left(\psi -\varphi \right)-\frac{\partial y}{\partial V_{\theta }}\left\{\cos \gamma \sin \theta \cos \left(\psi -\varphi \right)+\sin \mathrm{\gamma cos\theta }\right\}-\frac{\partial y}{\partial V_r}\left\{\cos \gamma \cos \theta \cos \left(\psi -\varphi \right)+\sin \gamma \sin \theta \right\}\right]+\dot{\gamma }V\left[\frac{\partial y}{\partial V_{\varphi }}\sin \gamma \sin \left(\psi -\varphi \right)-\frac{\partial y}{\partial V_{\theta }}\left(-\sin \mathrm{\gamma sin\theta cos}\left(\psi -\varphi \right)+\cos \gamma \cos \theta \right)-\frac{\partial y}{\partial V_r}\left(-\sin \mathrm{\gamma cos\theta cos}\left(\psi -\varphi \right)+\cos \mathrm{\gamma sin\theta }\right)\right]+\dot{\psi }V\cos \gamma \left[-\frac{\partial y}{\partial V_{\varphi }}\cos \left(\psi -\varphi \right)+\frac{\partial y}{\partial V_{\theta }}\sin \theta \sin \left(\psi -\varphi \right)+\frac{\partial y}{\partial V_r}\cos \mathrm{\theta sin}\left(\psi -\varphi \right)\right]& \left(1-30\right)\end{array}$

[0204] which can be written in a compact form as follows:

[00040]$\begin{array}{cc}\frac{dy}{dt}=\dot{V}{Y}_1+\dot{\gamma }{\mathrm{VY}}_2+\dot{\psi }V\cos \gamma Y_3& \left(1-31\right)\end{array}$

[0205] where, Y.sub.1, Y.sub.2 and Y.sub.3 are the expressions within the square brackets in (1-30), and the partial derivatives of y are as follows:

[00041]$\begin{array}{cc}\frac{\partial y}{\partial V_{\varphi }}=\frac{2r^2{V}_{\varphi }{V}_r^2}{{\left(V_{\varphi }^2+V_{\theta }^2+V_r^2\right)}^2},& \left(1-32\right)\end{array}$$\frac{\partial y}{\partial V_{\theta }}=\frac{2r^2{V}_{\theta }{V}_r^2}{{\left(V_{\varphi }^2+V_{\theta }^2+V_r^2\right)}^2},$$\frac{\partial y}{\partial V_r}=\frac{2r^2\left(V_{\varphi }^2+V_{\theta }^2\right)V_r}{{\left(V_{\varphi }^2+V_{\theta }^2+V_r^2\right)}^2}$

[0206] Substituting for {dot over (γ)}, {dot over (ψ)} and {dot over (V)} from (1-19), we can rewrite (1-31) in the form:

[00042]$\begin{array}{cc}\frac{\mathrm{dy}}{\mathrm{dt}}=\left[\begin{array}{cc}\begin{array}{cc}{Y}_1{G}_1+Y_2{H}_{1}V+Y_3{J}_{1}V\cos \gamma & Y_1{G}_2+Y_2{H}_{2}V+Y_3{J}_{2}V\cos \gamma \end{array}& Y_1{G}_3+Y_2{H}_{3}V+Y_3{J}_{3}V\cos \gamma \end{array}\right]\left[\begin{array}{c}\begin{array}{c}{a}_{A1}\\ a_{A2}\end{array}\\ a_{A3}\end{array}\right]& \left(1-33\right)\end{array}$