The present invention generally relates to the herding of avians using unmanned aerial vehicles (UAVs, or colloquially “Drones”), and namely systems and methods for modifying flight path while maintaining flock integrity.
Unmanned Aerial Vehicles (UAVs), also referred to as “Drones” are autonomous robotic vehicles capable of flight. One of the most common UAV designs is the quadcopter, a four-rotor device capable of immediate movement in each of the three axes. However, there are many fixed-wing UAV designs as well.
Sharing the skies, many avian species travel in flocks. Flocks can be made up of any number of birds, from two to tens of thousands, depending on the species. Mixed flocks can also form from birds of two or more species. Flock movements, such as their flight paths and formations, are highly complex and dependent upon many factors.
Systems and methods for autonomously herding birds in accordance with embodiments of the invention are illustrated. One embodiment includes an autonomous bird flock herding system, including at least one bird location sensor, an unmanned aerial vehicle; and a flock herding control system, including a processor, and a memory, the memory containing a flock herding application, where the flock herding application directs the processor to obtain bird position data from the at least one bird location sensor, where the bird position data describes the location of birds in a flock of birds, determine if the flock of birds will enter a protected zone, generate a set of waypoints using a flock dynamics model, instruct the unmanned aerial vehicle to navigate to at least one waypoint in the set of waypoints such that the flock of birds will, in response to the presence of the unnamed aerial vehicle at the at least one waypoint, change trajectory away from the protected zone.
In another embodiment, in response to the presence of the unmanned aerial vehicle, the flock maintains integrity.
In a further embodiment, the protected zone is a cylinder.
In still another embodiment, the protected zone is over an airport.
In a still further embodiment, the at least one bird location sensor is an avian radar.
In yet another embodiment the system includes at least one environmental sensor, and the flock herding application further directs the processor to obtain environment data describing environmental conditions proximal to the protected zone from the at least one environmental sensor, and generate the set of waypoints using the environment data.
In a yet further embodiment, the environment data describes the position of airplanes.
In another additional embodiment, the environment data describes wind speed.
In a further additional embodiment, the flock herding application further directs the processor to obtain updated bird position data, and update the set of waypoints based on the updated bird position data.
In another embodiment again, the system includes a second unmanned aerial vehicle, and wherein the flock herding application further directs the processor to generate a second set of waypoints using the flock dynamics model, and instruct the second unmanned aerial vehicle to navigate to a given at least one waypoint in the second set of waypoints such that the flock of birds will, in response to the presence of the second unmanned aerial vehicle at the given at least one waypoint, change trajectory away from the protected zone.
In still yet another embodiment, in response to the presence of the unmanned aerial vehicle, the flock maintains integrity.
In a still yet further embodiment, the protected zone is a cylinder.
In still another additional embodiment, the protected zone is over an airport.
In a still further additional embodiment, the at least one bird location sensor is an avian radar.
In still another embodiment again, the method further includes obtaining, using the flock herding control system, environment data from at least one environmental sensor describing environmental conditions proximal to the protected zone from the at least one environmental sensor, and generating the set of waypoints using the environment data.
In a still further embodiment again, the environment data describes the position of airplanes.
In yet another additional embodiment, the environment data describes wind speed.
In a yet further additional embodiment, the method further includes obtaining, using the flock herding control system, updated bird position data, and updating, using the flock herding control system, the set of waypoints based on the updated bird position data.
In yet another embodiment again, the method further includes generating, using the flock herding control system, a second set of waypoints using the flock dynamics model, and instructing, using the flock herding control system, a second unmanned aerial vehicle to navigate to a given at least one waypoint in the second set of waypoints such that the flock of birds will, in response to the presence of the second unmanned aerial vehicle at the given at least one waypoint, change trajectory away from the protected zone.
Additional embodiments and features are set forth in part in the description that follows, and in part will become apparent to those skilled in the art upon examination of the specification or may be learned by the practice of the invention. A further understanding of the nature and advantages of the present invention may be realized by reference to the remaining portions of the specification and the drawings, which forms a part of this disclosure.
The description and claims will be more fully understood with reference to the following figures and data graphs, which are presented as exemplary embodiments of the invention and should not be construed as a complete recitation of the scope of the invention.
Turning now to the drawings, systems and methods for avian flight path modification using UAVs are illustrated. Flocks of birds are a common sight, and their dynamic movements are highly choreographed. The nature of these movements both in formation and flight path are complex and take place at high speeds. Unfortunately, there are many situations where the presence of birds are highly problematic to human activities and infrastructure. For example, in the case of US Airways Flight 1549, colloquially referred to as the “Miracle on the Hudson,” the airplane on ascent after takeoff struck a flock of geese and was forced to ditch into the Hudson river. While keeping the skies clear over a small area has been traditionally handled by trained birds of prey, for example the famous Harris's Hawks of Wimbledon, this solution is not easily scaleable nor feasible in many situations.
Drones, however, as opposed to biological solutions, are mass-producible, highly controllable, and can be stored long-term in many different conditions with considerably less maintenance. While there have been some attempts to utilize drones to scare away birds from airports (for example RoBird of the Netherlands), these systems are often remote controlled and designed to scare birds away. Both of these factors can cause significant issues, as both the ineffectiveness of human piloting and unrefined scare tactics can lead to flocks of birds to split up, which simply causes more problems in the form of multiple sub-flocks which must now be controlled. In turn, this requires additional UAVs or other interdiction methods (e.g. air cannons, lasers, biological solutions, etc.) which escalate cost and increase the likelihood of strikes.
In contrast to the conventional “scare-tactics” which can split flocks, systems and methods described herein can mitigate these issues by utilizing autonomous drones to manipulate the movements of flocks of birds to guide them outside of a predefined geographic region without disrupting the flock unit. This process is referred to as “herding,” because the flock is left intact. In many embodiments, multiple drones acting in accordance with methods described herein can work in tandem to herd a single flock, or to herd multiple flocks at once. Autonomous drones with appropriate sensing platforms can be pre-loaded with control software and deployed to safely remove birds in any number of scenarios not limited to airports. However, for ease of description, airports will be discussed in greater detail, and one of ordinary skill in the art would understand that the systems and methods described herein could easily be deployed in alternative scenarios and locations.
In order to maintain flock integrity, in many embodiments, herding processes include real-time modeling of a flock and the expected response of the flock to external pursuers. In many embodiments, a number of waypoints are calculated for the drone to maneuver to in relation to the flock. The model of the flock dynamics can be used to generate waypoints that avoid flock fragmentation and/or reduce the amount of physical space that the flock occupies (e.g. promote tighter flying formations). In numerous embodiments, multiple drones can be utilized, each with their own set of waypoints collectively calculated to herd the flock. Flock herding systems are discussed in more detail below.
Flock herding systems are systems which prevent flocks of birds from entering a pre-defined geographical region, or a “protected zone.” In many embodiments, a protected zone includes the airspace above a particular region. As mentioned above, protected zones can be any number of different regions, and are not defined by any particular geographic feature. Drones can be utilized either alone or in tandem to keep birds out of the protected zone. In many embodiments, drones have sensors to enable them to at least detect birds and/or other drones. However, in numerous embodiments, external sensors and/or sensor arrays are utilized to track bird movements.
Turning now to
Flock herding system 100 further includes a bird location sensor 130. In many embodiments, bird location sensors 130 are avian radars. However, bird location sensors can be any sensor platform capable of reporting the location of birds in a protected zone in at least near real-time. In numerous embodiments, flock herding systems include additional sensors for tracking other environmental objects or conditions (collectively “environmental sensors”). For example, if an airport is in a protected zone, additional environmental data can be produced regarding the position of nearby aircraft. Further, environmental conditions such as wind speed, humidity, temperature, cloud cover, time, date, and any other environmental information can be captured by sensors and provided to flock herding control systems for use in waypoint generation and/or other control functions.
Flock herding system 100 additionally includes an interface device 140. Interface devices are any device capable of enabling a user to communicate with a flock herding control system or a drone. In many embodiments, interface devices are personal computers, but can be any number of digital communication devices such as, but not limited to, smart phones, smart TVs, tablet computers, and/or any other digital communication device as appropriate to the requirements of specific applications of embodiments of the invention. Components of the flock herding system 100 communicate via a network 150. In many embodiments, the network is a composite network made of multiple different interconnected networks which can be implemented using the same or different communications protocols. For example, networks can be wired, wireless, or a combination thereof. For example, in some embodiments, flock herding control systems communicate with bird location sensors on a particular network, and the flock herding control system provides data to the drones via a separate network. One of ordinary skill in the art would appreciate that any different number of system architectures and/or communications architectures can be utilized as appropriate to the requirements of specific applications of embodiments of the invention, including, but not limited to, those that utilize different environmental sensors and/or different numbers of drones.
Turning now to
Flock herding control system 200 further includes a memory 230. Memories can be implemented using volatile memory, non-volatile memory, or a combination of both. Memory 230 contains a flock herding application 232. Flock herding applications instruct the processor to generate waypoints for drones to navigate to in order to herd flocks away from protected zones. In many embodiments, memory 230 further contains bird position data 234 and/or environment data 236. As noted above, bird position data describes the 3D coordinate location of birds in range of the protected zone, and environment data describes various environmental conditions.
In many embodiments, flock herding control systems are implemented on standard computing platforms. However, flock herding control systems can be integrated into drones so long as the drone platform has sufficient computing capabilities. As one can appreciate, any number of different implementations of system architectures and flock herding control systems can be utilized to generate waypoints that enable drones to successfully herd flocks away from a protected zone as appropriate to the requirements of specific applications of embodiments of the invention. Processes for generating waypoints are discussed below.
As noted above, a key problem with conventional bird dispersal techniques is the fracturing of flocks into individual birds and/or subflocks. This creates significant inefficiencies as the fractured units are dispersed over a wider area and then must each individually be dealt with. In contrast, processes described herein can generate waypoints that, which when navigated to by a drone, will herd a flock away from a protected zone without fracturing the flock. This leads can lead to an increase in efficiency at least by requiring fewer drones, quicker dispersal times, and more efficient use of resources generally.
By way of example, a high-level conceptual illustration of the effect of a flock herding process in accordance with an embodiment is illustrated in
Turning now to
To accomplish successful herding, the “m-waypoint herding algorithm” has been developed. At a high level, a drone interacts with the intruding flock by positioning itself sequentially at a periodically refreshed set of m points. The inherent stability of the flocking dynamics are leveraged to prevent fragmentation, and the m points can be selected to maximize the deflection of the flock's flight path.
Turning now to
In order to generate the waypoints, an understanding of flock dynamics can be modeled from which the basis of the m-waypoint herding algorithm can be formulated. Consider a flock of n (usually >>1) birds. The position and the velocity of the ith bird are denoted by xi∈3 and vi ∈3, respectively. The vector between the ith and jth birds is denoted by rij=xj−xi. The subscript ‘p’ is used for the pursuer, so that xp denotes its position vector, and rpi=xi−xp is the vector from the pursuer to the ith bird. Given a vector r, the unit vector along r is denoted by {circumflex over (r)}.
Let Rcom denote the communication range for interaction between two birds. The neighborhood set of the ith bird is defined using the standard Euclidean distance as,
i
={j≤n:|∥x
i
−x
j
∥≤R
com}. (1)
and it is assumed that two birds interact only if their distance is less than or equal to Rcom. The set of birds (ν) thus forms a graph G=(ν, ε) where
ν={1,2, . . . ,n},
ε={(i,j)∈ν×ν|j∈⊆i}, card(ε)=p (2)
Note that if =i for all i, the graph is undirected. Otherwise, it is directed. The incidence matrix of the graph is denoted as Bc∈n×p. Recall that the (i,j)thb entry of Bc is 1 if the jth edge is incoming at i, −1 if outgoing at i, and 0 otherwise. An undirected edge is treated as two separate directed edges in this formulation of Bc. Δ(ν) denotes the diagonal matrix formed by the elements of the tuple ν. Finally, the centroid and the radius of the flock are defined as
The dynamics of an individual (ith) bird are described by a nonlinear, second-order differential equation based on Reynolds' rules and augmented with an evasive response:
where Rsafe denotes the steady-state distance between two adjacent members of the flock; vd denotes the pre-selected, steady velocity of each member of the flock; and H(rpi)=H(xi−xp) denotes the evasive response to the pursuer. It is assumed that vd lies in the horizontal plane.
Since the out-turning or 3-D behavior can have adverse implications for the stability of the flock, Ragg can be treated as a lower bound for the permissible distance between the pursuer and the flock. H(⋅) can be modeled as follows:
and a restriction that ∥rpi∥ Ragg+ϵ for all i, where ϵ>0 is an arbitrarily-chosen constant can be imposed.
As noted above, the protected zone (PZ) is the spatial volume that needs to be kept free of birds. For illustrative purposes, in this example, PZ is assumed to be cylindrical with radius RPZ. However, in many embodiments, the PZ can be other shapes. For modeling purposes, it can be assumed that birds fly along a constant, pre-selected heading, {circumflex over (v)}d, when not disturbed by a pursuer. This is in fact true, for instance, for flocks flying to migration grounds considerably far from PZ. It is clear that the pursuer only needs to move the flock to a point from which its flight path (along {circumflex over (v)}d) would no longer cross PZ. This target point is denoted as xdiv(t), noting that its exact position may change with time as described presently.
The global coordinate frame is defined with its origin placed at the centre of PZ, and its axes pointing north, east and up, respectively. Let h(t) denote the instantaneous altitude of the flock. First, two vertical lines (in the global frame) can be constructed which serve as loci for all possible target points. Recall that vd in (4) is the predetermined nominal flock velocity which satisfies {circumflex over (k)}Tvd=0 where {circumflex over (k)} is the unit vector pointing upwards in the global frame. Unit vector {circumflex over (v)}d⊥ is defined which satisfies vdt{circumflex over (v)}d⊥={circumflex over (k)}T{circumflex over (v)}d⊥=0. The two candidate loci are then given by
In order to avoid fracturing the flock, conditions for flock stability can be modeled. These conditions can be used to select the gains and the underlying graph in (4) for analytical investigation when sufficient empirical data is not available to determine these parameters. The resulting robustness of the flock implies that the pursuer is free to approach the flock from any direction. As long as it maintains a certain minimum distance between itself and the flock, the flock will not undergo fragmentation.
To model snapshot dynamics, steady states, and linearization, two column vectors x, v∈3n, formed by permuting the components of xi and vi, respectively, ∀i:
x=[x11, . . . ,xn1,x12, . . . ,xn2,x13, . . . ,xn3]T
are defined, and v is constructed in a similar manner. There exists a permutation matrix, denoted by P3n∈3n×3n, such that
Ignoring the pursuer-dependent terms, the flocking dynamics (4) can be written as
{dot over (x)}=v
{dot over (v)}=−k
a(I3⊗DcBcT)v+kg(vd⊗1n−v)−ks(I3⊗DcQ(x)BcT)x (6)
where Dc(i,j)=−1 if the jth edge is incoming at i, and Dc(i,j)=0 otherwise. Here, I3 the 3×3 identity matrix, while ⊗ denotes the Kronecker product. Dc can be replaced with Bc/2 if the graph is undirected. The matrix Q(x)∈p×p is a diagonal matrix satisfying Q(j,j)=1−Rsafe3/∥ej∥3.
In order to analyze the stability of the flock, a change of coordinates can be made: {tilde over (x)}i(t)=xi(t)−vdt, {tilde over (v)}i(t)=vi(t)−vd. Since Q(x)=Q({tilde over (x)}), flocking dynamics can be written as
{tilde over ({dot over (x)})}={tilde over (v)}
{tilde over ({dot over (v)})}=−ks(I3⊗DcQ({tilde over (x)})BcT){tilde over (x)}−ka(I3ÐDcBcT){tilde over (v)}−kg{tilde over (v)} (7)
This is a nonlinear, autonomous system whose equilibrium solutions are given by [{tilde over (x)}0, 1n⊗03], where {tilde over (x)}0 satisfies (I3⊗DcQ({tilde over (x)}0)BcT){tilde over (x)}0=0.
Given an equilibrium configuration {tilde over (x)}0, the family of equilibrium solutions obtained by rigid-body translation and rotation the flock are defined by
S({tilde over (x)}0)={{tilde over (x)}|{tilde over (x)}=b⊗1n+αP3n(In⊗J)P3n−1{tilde over (x)}0} (8)
where b∈3, J∈ so (3), and α∈.
Let δ· denotes a small perturbation in the corresponding variable, and ei0 is the steady-state ith edge vector. Then, the linearization of (7) about an equilibrium point is given by
These equations can be written compactly as
where the Jacobian G is given by
and
In order to determine the spectral properties of A and G:
Lemma 1. The null space of A defined in (10) is a superset of eigenvectors which correspond to a rigid body translation and rotation of the flock; i.e.,
(A)⊇≙span([1,0,0]T⊗1n[0,1,0]T⊗1n,
[0,0,1]T⊗1n,P3n(In⊗J1)P3n−1{tilde over (x)}0,P3n(In⊗J2)P3n−1{tilde over (x)}0,
P
3n(In⊗J3)P3n−1{tilde over (x)}0)
where J1, J2, J3 form a basis for so (3).
Proof: It is evident from (10) that (A) is a superset of the vectors δx which satisfy (In⊗BcT)δx=0 or e0iTδei=0 ∀i={1, . . . , p}. The former condition implies that δx=k1[1, 0, 0]T⊗1n+k2[0, 1, 0]T⊗1n+k3[0, 0, 1]T⊗1n, where k1, k2, and k3 are arbitrary constants. The latter condition is satisfied by δxi=Σj=13k′jJjxi, for all constants k′j.
Assumption 1. The matrix A has exactly six eigenvalues which are equal to 0. Therefore, u(A)=(A).
Assumption 1 is satisfied by connected undirected graphs and by strongly connected and balanced digraphs.
Assumption 2. The Jacobian C has the following properties:
To perform the stability analysis, the center manifold theorem can be used to show that trajectories which start away from S({tilde over (x)}0), at least locally, converge to S({tilde over (x)}0) at an exponential rate. The proof proceeds in two steps. First, it is shown that there exists a center manifold when Assumption 2 is satisfied. It is noted that the set S({tilde over (x)}0) is also a center manifold. In the second step, the property that center manifolds are arbitrarily close to each other can be used to complete the proof.
Lemma 2. Suppose that the flocking dynamics satisfy Assumption 2. Then, for every steady state {tilde over (x)}0, there exists an invariant manifold εc ≙εc({tilde over (x)}0) such that any trajectory starting around εc converges at an exponential rate to εc.
Proof: From Assumption 2, it is known that the Jacobian G, evaluated at {tilde over (x)}0, has six zero eigenvalues and the other eigenvalues are negative. Thus, it follows that from the center manifold theorem that there exists a six-dimensional center manifold εc such that trajectories starting outside EC converge at an exponential rate to εc.
Theorem 1. The set S({tilde over (x)}0) is exponentially stable.
Proof: It is noted that S({tilde over (x)}0) is itself a center manifold of the dynamics. From Lemma 2, it is known that trajectories starting in a neighborhood of x0 converge at an exponential rate to the center manifold εc({tilde over (x)}0). Recall that any two center manifolds about an equilibrium point {tilde over (x)}0 differ by transcendentally small terms, and every equilibrium point in the vicinity of {tilde over (x)}0 lies on every center manifold εc({tilde over (x)}0). Thus, a trajectory starting in a neighborhood of {tilde over (x)}0 converges exponentially fast to S({tilde over (x)}0).
The discussion above shows that S({tilde over (x)}0) is exponentially stable, and therefore, robust to small perturbations. In the context of herding, this implies that, for every direction in relation to the flock (or its center), there exists a nontrivial set of pursuer positions for which the underlying graph of the flock is preserved. Below are analytical formulae for estimating a set of permissible pursuer positions based purely on local topological considerations.
In addition to stability, for the purpose of herding, it is useful to have the property of time-scale separation between the synchronization of the flocking dynamics to S({tilde over (x)}0) on the one hand, and the translational dynamics of the center of gravity of the flock (whose time constant, as shown in above, is 1/kg) on the other. This can be achieved by a suitable choice of ks and ka for a given graph topology.
The objective of this section is to derive approximations for the approach distance between a pursuer and the flock, based purely on the local interaction between the pursuer and the birds in a flock. This estimate supersedes the distance Ragg in
The influence exerted by the pursuer can cause one of two effects in extreme circumstances. In one case, the pursuer can push a bird to within an unacceptable distance of its neighbor. This scenario is likely in dense flocks, as shown in
Consider the situation wherein a bird on the boundary of the flock (labeled ‘1’) is engaged by the pursuer. The bird tries to move away radially from the pursuer, and into the flock. In the worst case, the bird's evasive path points in the direction of a neighboring bird (‘2’) and there is force from other neighboring birds pulling ‘1’ away from ‘2.’ This is depicted in accordance with an embodiment of the invention in the dense flock of
A conservative estimate for the minimum approach distance between the pursuer and the bird ‘1’ can be found from the local neighborhood of
In sparse flocks, it is commonplace to find a linear topology, involving two or three agents forming an angle or a straight line with no other common neighbors. Consider a single edge, as shown in
Below, example estimates for ∥rp1∥min are used as part of the motion planning strategy for the pursuer.
Definition 1. Define a set Xp of permissible pursuer positions in relation to the flock as:
where Rmin is the maximum of Ragg and either (11) or (12) depending on the topology of the flock.
The above established conditions for stability and robustness of the flock, and derived approximations for the approach distance between the flock and the pursuer. In this section, the problem of herding flocks without fracturing them is solved using waypoints generated by modeling the ensemble behavior of the flock. A version of the m-waypoint herding algorithm in accordance with an embodiment of the invention is illustrated in
The coordinates of the flock's centroid and its velocity are given by
Using (4) and (5), obtain
where denotes the subset of the flock that lies within Rfear of the pursuer; the pursuer's velocity, up, is a control variable for the herding algorithm. The term f(x,v) in zero when the graph is undirected (since 1nTBc=0) or if it has synchronized to a steady state configuration. This would occur naturally when the flock synchronizes on a faster time scale than the dynamics of its CG. Since the goal is for the flock to point in the direction of the herding target point, it suffices to ensure that its velocity normal to an axis pointing towards xdiv is driven to zero. Define
r
dc(t)≙xdiv(t)−xcg(t),q≙rdc×vcg (15)
Recall that {circumflex over (k)}Txdiv(t)={circumflex over (k)}Txcg(t)=h(t) (the altitude of the flock). Since xdiv(t) lies on a fixed vertical line for all t, then {circumflex over (k)}×{dot over (x)}div(t)=0. Since {dot over (r)}dc(t)={dot over (x)}div(t)−vcg(t), the following expression is obtained for the dynamics of q:
{dot over (q)}=r
dc
×{dot over (v)}
cg
+{dot over (x)}
div
×v
cg
=−kgq+kg(rdc×vd)+rdc×Fp(xp)+rdc×f(x,v)+{dot over (x)}div×vcg
Next, define qk ≙{circumflex over (k)}Tq. Note that {circumflex over (k)}T({dot over (x)}div×vcg)=vcgT({circumflex over (k)}×{dot over (x)}div)=0. Thus, the dynamics of qk are given by
{dot over (q)}
k
+k
g
q
k
={circumflex over (k)}
T(rdc×(kgvd+f(x,v))+rdc×Fp(xp)) (16)
Denote the amount of deviation that can be produced by placing the pursuer at xp using the right hand-side of (16):
ρ(xp)≙{circumflex over (k)}T(rdc×(kgvd+f(x,v))+rdc×Fp(xp)) (17)
If ρ(xp)=0 for all t, then it is possible to deduce from (16) that qk→0 and the CG of the flock moves in a straight line (represented as a solid line in
A solution to the aforementioned control problem is to compute the trajectory of xp(t) which maximizes ρ(xp), and get the pursuer to track it. It is known that a flock tends to deform into a concave shape locally under persistent pressure from a pursuer, which is known to dent the effectiveness of the pursuit. Furthermore, over-stressing one or more birds continuously over an extended period of time carries the risk of the distressed birds attempting aggressive evasive maneuvers and fragmenting the flock. To avoid these problems, a sub-optimal approach can be employed wherein the flock is engaged through different waypoints in a given timefranme.
The waypoints are chosen by sampling the set Xp from Definition 1. Sampling is preferred to statically defined waypoints because it allows the algorithm to be agnostic to the exact geometry of the flock. This is useful when the flock is not necessarily best represented as a convex shape; for instance, star-shaped flocks and flocks with a curvilinear geometry. The set of m waypoints can be formally identified as follows.
Definition 2. The set Xpm is defined by construction. The waypoint selection algorithm, illustrated in accordance with an embodiment of the invention in
Even for a convex shaped flock, the set Xpm need not be connected. This enables the use of random sampling for construction.
The motion planner solves for the pursuer's velocity up(t) by commanding one of two motions, as follows.
In order to promote successful herding, a condition is derived in the form of the minimum allowable distance from the protected zone at which the herding must begin. If the herding commences beyond this distance, it will succeed with an elevated likelihood. Assume that f(x, v)=0; i.e., the underlying graph is undirected or the flock has synchronized to a steady state.
To begin, ignore the dynamics of the pursuer, and recall that the control objective is to maximize ρ(xp)≥0. Consider the conservative case ρ(xp)=0 for all t during the course of herding. This case is limiting in that the distance at which the herding needs to be commenced can be lesser when larger values of ρ(xp) are attainable.
In order to achieve ρ(xp)=0, set xp=xp* where Fp(xp*)=−kg({circumflex over (k)}T({circumflex over (r)}dc×vd))({circumflex over (k)}×{circumflex over (r)}dc). Since {circumflex over (r)}dc=rdc/∥rdc∥, and vd lie in the horizontal plane, it follows that ∥Fp∥=kg∥{circumflex over (r)}dc×vd∥. Let ρF denote the upper bound on ∥Fp∥, which is known. It is clear then that the flock should be made to satisfy ∥{circumflex over (r)}dc×vd∥<ρF/kg ∀t, which, in turn, means that the flock must be kept outside the cone shown in
where Rdiv was defined above.
While the solution xp* appears to solve the problem in principle, there is the possibility that the trajectory xp* may not be feasible. Furthermore, the motion planning algorithm adopted here requires that the pursuer engage with an entire “front” of the flock rather than specific individual birds. In particular, it means that the engagement between the pursuer and the flock takes place in pulses. Processes described herein can be refactored in light of this pulsed interaction.
Let τe denote the duration of any given engagement between the pursuer and the flock, and let τf denote the time taken by the bird to fly between two waypoints. During the time τf, the flock receives no external stimulus and its velocity tends to align to vd. Assuming that an engagement begins at time t0, the dynamics of the flock in the time interval [t0, t0+τe+τf] can be described via the switching dynamics
where vd∥=vd+Fp/kg (see
The terms ∥vd∥∥ and θ can be be estimated empirically since the precise interaction between the flock and the pursuer is highly case-specific. The magnitude of vd∥ depends on whether the flock is driven from the front or from the rear. Since the intent is for the net motion to be perpendicular vd, it is reasonable to assume that the average force also acts perpendicular to vd. Thus,
Consequently,
Theorem 2. Given the approximate values of ∥vd∥∥ and θ, the refined value of Dp,min is given by
Proof Let ns denote the number of engagement pulses. From
Furthermore, during each engagement phase, the flock translates horizontally through a distance ∥vd∥∥ τe cos θ. During the non-engagement phase, the horizontally translation is at moat ∥vd∥τf. Thus, the minimum horizontal translation is given by
Setting cos θ=∥vd∥/∥vd∥∥ yields Dp,min=Rdiv cot θ (1+τf/τe). Notice that (18) is recovered if when τf=0 and θ=θmax. This completes the proof. Using the above, flock flight dynamics can be modeled and effective waypoints can be generated.
Although the present invention has been described in certain specific aspects, many additional modifications and variations would be apparent to those skilled in the art. It is therefore to be understood that the present invention can be practiced otherwise than specifically described without departing from the scope and spirit of the present invention. Thus, embodiments of the present invention should be considered in all respects as illustrative and not restrictive. Accordingly, the scope of the invention should be determined not by the embodiments illustrated, but by the appended claims and their equivalents.
The present invention claims priority to U.S. Provisional Patent Application Ser. No. 62/703,759 entitled “Robotic Herding of a Flock of Birds using an Unmanned Aerial Vehicle” to Chung et al., filed Jul. 26, 2018, the disclosure of which is herein incorporated by reference in its entirety.
This invention was made with government support under Grant No(s). IIS1253758 and IIS1664186 awarded by the National Science Foundation. The government has certain rights in the invention.
Number | Date | Country | |
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62703759 | Jul 2018 | US |