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Abstract

This paper presents a cooperative strategy to achieve evenly spaced circular formation of a group of unmanned aerial vehicles. The strategy is claimed to be fail-safe because the circular formation remains unaltered even if one or more unmanned aerial vehicles fail. We can also add more unmanned aerial vehicles without altering the formation. The control law uses only bearing angle information. We can pre-specify the centre of the circular formation of the unmanned aerial vehicles. The unmanned aerial vehicles will maintain the desired distance from the centre with bearing angle measurement only. We assume that the unmanned aerial vehicles are identical and can measure the bearing angle of all the other unmanned aerial vehicles. The strategy is analysed using unicycle kinematic and verified using six-degree-of-freedom model of the unmanned aerial vehicles. Extensive simulations are carried out for noisy measurement, presence of wind and moving target.

Keywords

Balance circular formation encirclement fail-safe strategy bearing only measurement target tracking 6-DOF unmanned aerial vehicle model

Introduction

A team of multiple, relatively small vehicles working in cooperation, offers many advantages as compared to single vehicle missions. There are many military and civil applications such as surveillance, security systems, space and underwater exploration where such systems are used. There are various advantages of multi-vehicle systems such as better reliability, scalability, efficiency, operational capability and adaptability. More benefits can be derived from such systems if the vehicles are cooperating. In the recent past, there has been a lot of research in designing control strategies to make multiple entities work together to achieve a common goal. The control strategy, often referred to as cooperative control strategy, is decentralised and establishes coordination between the entities. The research topics in the area of cooperative control include consensus, formation control, distributed task assignment and distributed estimation (Cao et al.¹ and references therein).

There exists vast literature on formation control for a group of vehicles modelled by single integrator or double integrator dynamics^2–5 and by unicycles.^6–18 Unicycle model has close resemblance with kinematics of non-holonomic vehicles such as unmanned aerial vehicles (UAVs) and ground mobile robots. For analysing performance of guidance laws designed for the vehicles working in cooperation, it is prudent to reduce the complexity of vehicle dynamics and consider simplified model. Unicycle model is a very simple model. A controller designed using this model can be implemented with some modifications on a differential drive ground vehicle or a six-degree-of-freedom (6-DOF) model of aircraft with altitude hold. We present the existing work that achieves a balanced circular formation with the unicycle vehicles encircling a common centre. Paley and colleagues^6,7 have designed control laws for the stabilisation of multiple vehicles to a balanced circular formation, where each vehicle needs relative position and relative velocity information of neighbouring vehicles. Klein and Morgansen⁸ and Guo et al.⁹ have proposed control law for encircling a moving target with multiple vehicles assuming full information about the target as well as all other agents is based on the analysis of a planar two-vehicle formation control discussed in Justh and Krishnaprasad.¹⁰ The authors have proved a global convergence result for the two-vehicle formation and have characterized equilibria for the n-vehicle problem. In Marshall et al.¹¹ and Daingade and Sinha,¹² cyclic pursuit-based strategies are proposed to get a balanced circular formation. At equilibrium, it is shown that the agents settle along a circle with rigid polygonal formation. Controlling both orientation and speed, law designed in Lin et al.¹⁵ and Zheng et al.¹⁶ achieves circular formation. Mohamed and Maggiore¹⁷ have designed control law for making group of vehicles to converge to a circle of a pre-specified radius with desired separations and ordering. George et al.¹⁸ have addressed a problem of achieving collective circular motion of heterogeneous unicycle-type vehicles moving with different velocities. The control strategies discussed so far are based either on absolute or relative position or the measurement of velocity of own and/or neighbours. The absolute position information can be obtained using global positioning system (GPS) or by using some localisation algorithm. Communication network is required to communicate the position information to the neighbouring agents. The algorithms, which rely on relative position information, make use of sensors to measure range and bearing angle. Vision sensors can measure the bearing angle efficiently, but the range can also be estimated from the successive images. The advantage of vision-based formation control is that no explicit communication is required. Vision-based formation is addressed in the literature.^19–23 A vision-based control law proposed in Moshtagh et al.¹⁹ for achieving circular formation is based on bearing angle information, optical flow and estimation of time of collision. The agents finally converge to a circular formation but the point about which formation converges is not specified a priori. In applications such as target tracking or landmark surveillance, it is necessary to get the formations about a specific point (target). Ma and Hovakimyan²⁴ have proposed a vision-based target tracking strategy for tracking a ground vehicle using multiple UAVs. The authors have designed tracking control and coordination control separately. For tracking, both bearing angle and range measurements were used whereas for coordination the bearing angle information was enough.

This paper addresses the problem of encircling a point by multiple autonomous vehicles. Only bearing angle measurement is available and based on this information, we propose a simple control strategy that ensures a balanced circular formation of the vehicles about a point. The point may or may not be pre-specified. If pre-specified, then it can be considered as the target point about which the formation needs to be achieved. The vehicles are not informed about the desired radius of the circle since they do not have the instantaneous range information to correct it. The controller gains are pre-computed to achieve the desired radius. The absolute or relative positions of the vehicles or their neighbours or the point of interest (target point) are not measured or estimated. Similarly, the control law does not require the information of the absolute or relative velocities of the vehicles. The bearing angle can be measured using sensors onboard each vehicle, which can be an omni-directional vision sensor. Therefore, communication between the vehicles is not required. The strategy requires the bearing information of all the other UAVs and the target point. The UAVs are assumed to be identical and we do not need to identify the UAV to implement the control strategy. At equilibrium, the vehicles get evenly spaced around the desired circle. One of the merits of the proposed control law is that we can add or remove vehicles from the formation without affecting the radius of the circle while the vehicles adjust to maintain uniform spacing. This prompted us to state our strategy as fail-safe. The strategy is fail-safe in the sense that addition or removal of vehicles does not lead to failure of mission. However, the strategy does not ensure collision avoidance between the agents when new agents are added. We validate our strategy using the 6-DOF model of UAVs and extensive simulations are presented to illustrate the performance of the strategy under different scenarios such as measurement noise, presence of wind and moving target.

The paper is organised as follows. The system model with the proposed control law is presented in ‘Problem Formulation’ section. ‘Analysis with Unicycle Kinematics’ section discusses about possible equilibrium formations and stability analysis of these formations considering unicycle model of the vehicles. ‘Implementation in 6-DOF UAV Model’ section gives the realistic MAV model and the details of control implementations on the autopilot. The last but one section discusses the simulation results, and the final section summarizes and concludes the paper.

Problem formulation

Consider a set of n vehicles modelled as unicycle. The kinematics of ith vehicle is

{x \cdot}_{i} = V \cos (θ_{i}), {y \cdot}_{i} = V \sin (θ_{i}), {θ \cdot}_{i} = u_{i}

(1)

where

P_{i} = [x_{i}, y_{i}] T

represents the position, θ_i represents the heading angle and v_i represents the linear speed of the agent i. The motivation for unicycle model is that it can represent the kinematics of a point mass model of a UAV at a constant altitude or a point mass model of a wheeled robot on a plane. We use the generic term ‘agent’ to represent the aerial or ground vehicle. The linear speed of an agent is assumed to be a constant and the motion of the agent is controlled using lateral acceleration

u_{i}

which gets reflected in the angular velocity of the agent. In this paper, the objective is to get a circular formation about a stationary point with n agents. Consider this point to be P_o. Figure 1 shows positions of vehicle i and one of its neighbour j in a reference frame where the target point is at the origin. We call this as a target centric reference frame. Let P_i and P_j represent the position of agents i and j, respectively. The variables in Figure 1 are as follows:

$r_{io}$ – Distance between ith agent and the target,

$r_{ij}$ – Distance between ith agent and jth agent,

$β_{i}$ – angle made by the vector $r_{io}$ with respect to reference,

$β_{i}^{j}$ – angular separation between agents i and j taken with respect to the target,

$φ_{io}$ – bearing angle of agent i with respect to the target,

$φ_{ij}$ – bearing angle of agent i with respect to agent j.

Figure 1.

Positions of the vehicles in a target centric frame.

We assume that agent i can measure $φ_{io}$ and $φ_{ij} \forall j \neq i$ . It computes the average bearing angle with respect to the others agents $φ_{ia} = \frac{1}{n - 1} \sum_{j = 1 j \neq i}^{n} φ_{ij}$ . To incorporate the target information, we use a parameter $ρ \in (0, 1)$ , which decides the priority given to target information over neighbours information. Let ρ be defined as the pursuit gain. So, the resultant bearing angle of agent i is calculated as

φ_{i} = (1 - ρ) φ_{io} + ρ φ_{ia}

(2)

Then, the control input to the ith agent is defined as

u_{i} = k φ_{i}

(3)

where

k_{i} > 0

is the controller gain. We assume that

φ_{i} \in [0, 2 π)

for all time,

t \geq 0

. This condition ensures that the agents always rotate in counter clockwise direction. Note that equation (3) is a simple control law that can be easily computed. Let

β_{i}^{j} = β_{j} - β_{i}

and

β_{i}^{*} = \frac{1}{n} \sum_{j = 1}^{n} β_{i}^{j}

(4)

We assume that the agents are identical, that is, $v_{i} = V$ and $k_{i} = k$ for all i. Let us define the states of the n agent system as $x_{i (1)} = r_{io}, x_{i (2)} = β_{i}^{*}$ and $x_{i (3)} = θ_{i} - β_{i}$ for all i. Then, the kinematics (1) can be re-written in the target centric reference frame as

{x \cdot}_{i (1)} = V \cos (x_{i (3)})

(5a)

{x \cdot}_{i (2)} = \frac{1}{n} \sum_{j = 1}^{n} (\frac{V \sin (x_{j (3)})}{x_{j (1)}} - \frac{V \sin (x_{i (3)})}{x_{i (1)}})

(5b)

{x \cdot}_{i (3)} = k φ_{i} - \frac{V \sin (x_{i (3)})}{x_{i (1)}}

(5c)

Remark 1

In actual vehicles, there will be bounds on the angular speed $ω_{\max}$ , that is, $ω_{i} \leq ω_{\max}, \forall i$ . We can take this into account by imposing an upper bound on the value of k as, $k \leq k_{\max} where k_{\max} = \frac{ω_{\max}}{2 π}$ .

Analysis with unicycle kinematics

With the system as defined as in equation (5), we study its asymptotic behaviour under the control law (3).

Theorem 1

A group of agents with kinematics (5) and control (3) encircles a target point at a radius of $\frac{2 V}{k π}$ while being always uniformly distributed irrespective of the number of agents.

Proof

To prove this, we have to study the behaviour of the agents at equilibrium, that is, when ${x \cdot}_{i (st)} = 0$ for $i = 1, \dots, n and st = 1, 2, 3$ . In this proof, we assumed the right-hand side of equation (5) is zero. Then, at equilibrium, $x_{i (1)} = r_{io}$ is a constant for all i which implies that the agents move on circular paths with the target at the centre. Using equation (5a), $x_{i (3)} = (2 m + 1) \frac{π}{2}$ where $m = 0, \pm 1, \pm 2, \dots$ and using equation (5c), $k_{i} φ_{i} = \frac{V \sin (x_{i (3)})}{x_{i (1)}} = \pm \frac{V}{r_{io}}$ , but $k_{i} > 0, V > 0, r_{io} \geq 0 and φ_{i} \in [0, 2 π)$ , so

k_{i} φ_{i} = \frac{V}{r_{io}}

(6)

and therefore,

m = 0, \pm 2, \pm 4, \dots

Assuming

θ_{i} \in [0, 2 π) and β_{i} \in [0, 2 π)

, we get

x_{i (3)} \in (- 2 π, 2 π)

which implies

m = 0

. So, for all i

θ_{i} - β_{i} = \frac{π}{2}

(7)

Now, using equation (5b), $\frac{1}{r_{io}} = \frac{1}{n} \sum_{j = 1}^{n} \frac{1}{r_{jo}}$ . On rearranging, we get $\frac{(n - 1)}{r_{io}} - \sum_{j = 1, j \neq i}^{n} \frac{1}{r_{jo}} = 0$ for all i. Writing it in matrix form

[(n - 1) - 1 - 1 \dots - 1 - 1 (n - 1) - 1 \dots - 1 - 1 - 1 (n - 1) \dots - 1 : : : : : - 1 - 1 - 1 \dots (n - 1)] [1 / r_{1 o} 1 / r_{2 o} 1 / r_{3 o} : 1 / r_{no}] = 0

(8)

Solving this system of equations,²⁵ we get an unique solution $[\frac{1}{r_{1 o}} \frac{1}{r_{2 o}} \dots \frac{1}{r_{no}}] T = \frac{1}{R} 1_{n}$ for some $R \in ℝ$ . Therefore, $r_{io} = R$ for all i. Then, from equation (6)

φ_{i} = \frac{V}{kR}

and from equations (1) and (3),

{θ \cdot}_{i} = \frac{V}{R}

for all i, which implies the angular speed is constant and same for all agents. Thus, the agents encircle the target at a radius R and they maintain a fixed relative position with respect to each other.

Next, we show that the agents are uniformly distributed. From Figure 1 and equation (7)

φ_{io} = π - (θ_{i} - β_{i}) = \frac{π}{2}

(9)

for all i. Consider

Δ P_{i} P_{o} P_{j}

in Figure 2. Since

P_{i} P_{o} = P_{j} P_{o} = R, ∠ P_{o} P_{i} P_{j} = ∠ P_{o} P_{j} P_{i} = b

(say). Then,

b = π / 2 - φ_{ij}

and

β_{i}^{j} = π - 2 b

which implies

φ_{ij} = \frac{β_{i}^{j}}{2}

(10)

Figure 2.

Formation of agent i and agent j at equilibrium.

From equation (8), $φ_{i} = φ_{k}$ for all $i, k$ , then using equation (2) and (10), $\sum_{j = 1, j \neq i}^{n} β_{i}^{j} = \sum_{j = 1, j \neq k}^{n} β_{k}^{j}$ . Without loss of generality, we can assume that at equilibrium the agents are indexed sequentially in a counter clockwise order. For $k = i + 1, β_{i}^{i + 1} + β_{i}^{i + 2} + \dots + β_{i}^{i + n - 1} = β_{i + 1}^{i + 2} + β_{i + 1}^{i + 3} + \dots + β_{i + 1}^{i}$ where the indices are taken modulo n. On simplification, we get $(n - 1) β_{i}^{i + 1} = β_{i + 1}^{i}$ . Since the agents are on a circle, $β_{i}^{j} = 2 π - β_{j}^{i}$ , so $β_{i}^{i + 1} = \frac{2 π}{n}$ for all i, which implies that the agents are uniformly distributed around target. From equation (4), $β_{i}^{*} = \frac{n - 1}{n} π$ for all i. Then, from equations (2), (9) and (10)

φ_{i} = (1 - ρ) \frac{π}{2} + ρ \frac{π}{2} = \frac{π}{2}

(11)

and from equations (8) and (11)

R = \frac{2 V}{k π}

(12)

Thus, the agents will be uniformly placed around the target on a circle of radius given by equation (12), independent of the number of agents n. ▪

The equilibrium states of the system are given as

x_{i (1)} = \frac{2 V}{k π}

(13a)

x_{i (2)} = \frac{n - 1}{n} π

(13b)

x_{i (3)} = \frac{π}{2}

(13c)

and the inter agent distance

R_{aa}

R_{aa} = 2 R \sin (\frac{π}{n})

(14)

Remark 2

Since R is independent of n, agents can be added or removed from the group without affecting the encircling radius. The agents will uniformly distribute themselves around the target when the number of agents are changed. This emphasises the fail-safe property of the proposed strategy. In fact, the radius of the cycle depends on the linear speed V and controller gain k only. Pursuit gain ρ also does not play any role.

Next, we study the stability of the equilibrium formation of agents using linearisation.

Theorem 2

The equilibrium formation of $n \geq 2$ agents with kinematics (5) and control law (3) is locally asymptotically stable.

Proof

In order to study the local stability of the system (5), we linearise it about the equilibrium point (13). For this, we need to find a relation between $β_{i}^{*} and φ_{i}$ . Using trigonometry (refer Figure 1), we can express $φ_{ij}$ as a function of $β_{i}^{j}$ as

φ_{ij} = π - (θ_{i} - β_{i}) - \sin - 1 \frac{r_{jo} \sin (β_{i}^{j})}{\sqrt{r_{io}^{2} + r_{jo}^{2} - 2 r_{io} r_{jo} \cos (β_{i}^{j})}}

The relation between $β_{i}^{*} and β_{i}^{j}$ is $β_{i}^{j} = β_{i}^{*} - β_{j}^{*} + \frac{(j - i)}{n} 2 π$ which is derived in Appendix 1. Now, we can linearise the system (5) about the equilibrium point to get a linear model of the system as

\land \cdot X = A \land X \land

(15)

where

X \land = [{x \land}_{1}, {x \land}_{2}, \dots, {x \land}_{n}] T, {x \land}_{i} = [x_{i (1)}, x_{i (2)}, x_{i (3)}] T and A \land

is a circulant matrix given by

A \land = circ [A_{1} A_{2} \dots A_{n}]

with

where

2 \leq i \leq n

. We can determine the eigenvalues of block circulant matrix

A \land

(as given in Davis²⁶) by block diagonalising the system matrix using Fourier matrix

F_{n}

. The Fourier matrix

F_{n}

is given by

F_{n} = \frac{1}{\sqrt{n}} [111111 ς ς 2 \dots ς n - 1 : : : : : 1 ς n - 1 ς 2 (n - 1) \dots ς (n - 1) (n - 1)]

where

ς = e j \frac{2 π}{n} and j = \sqrt{- 1}

. The block circulant matrix

A \land

can be diagonalised as,

A \land = (F_{n} \otimes I_{3}) * D (F_{n} \otimes I_{3})

where

(⋆)

indicates conjugate transpose. The diagonal matrix D is given by,

D = diag (D_{1}, D_{2}, \dots, D_{n})

, where

D_{i} = \sum_{i = 1}^{n} ς i - 1 A_{i}

. Substituting the value of

A_{i}

we get

where

D_{i_{(31)}} = \frac{k ρ}{2 R (n - 1)} \sum_{p = 1}^{n - 1} \cot (\frac{π p}{n}) (1 - ς i - 1 - ς 2 (i - 1) - \dots - ς (n - 1) (i - 1)) + \frac{V}{R 2}

D_{i_{(32)}} = \frac{k ρ}{2 (n - 1)} ((n - 1) - ς i - 1 - \dots - ς (n - 1) (i - 1))

The eigenvalues of $A \land$ are same as eigenvalues of $D_{i}, i = 1, \dots, n$ . So we can comment about the stability of n vehicle system, by observing the eigenvalues of each block $D_{i}$ . After substituting the value of R from equation (12), each $D_{i}$ can be factorised as $D_{i} = kT {D ~}_{i} T - 1$ , where $T = diag [\frac{1}{k}, \frac{1}{V}, \frac{1}{V}]$ and

{D ~}_{i} = [00 - 1 \frac{π 2}{4 n} ((n - 1) - ς i - 1 - \dots - ς (n - 1) (i - 1)) 00 {D ~}_{i_{31}} {D ~}_{i_{32}} - 1]

where

{D ~}_{i_{(31)}} = \frac{ρ}{4 (n - 1)} \sum_{p = 1}^{n - 1} \cot (\frac{π p}{n}) (1 - ς i - 1 - ς 2 (i - 1) - \dots - ς (n - 1) (i - 1)) + \frac{π 2}{4}

{D ~}_{i_{(32)}} = \frac{ρ}{2 (n - 1)} ((n - 1) - ς i - 1 - \dots - ς (n - 1) (i - 1))

It can be observed that the stability of equilibrium formation is independent of $k and V$ , as long as $k > 0$ . In ${D ~}_{i_{31}}$ , $\cot (\frac{π p}{n}) = - \cot (π - \frac{π p}{n}) since \frac{π p}{n} < π$ . So $\sum_{p = 1}^{n - 1} \cot (\frac{π p}{n}) = 0$ . Then, the value of ${D ~}_{i_{31}}$ reduces to $\frac{π 2}{4}$ . The value of $ς i - 1 + ς 2 (i - 1) + \dots + ς (n - 1) (i - 1) = n - 1 when i = 1 and ς i - 1 + ς 2 (i - 1) + \dots + ς (n - 1) (i - 1) = - 1 for 2 \leq i \leq n$ . Therefore, the characteristic equation of block ${D ~}_{1}$ , is $s 3 + s 2 + (\frac{π 2}{4}) s = 0$ . The eigenvalues corresponding to block ${D ~}_{1} are s = 0 and s = - \frac{1}{2} \pm \frac{1}{2} \sqrt{1 - π 2}$ . As the system of n agents is constrained to evolve along manifold $M$ as defined in Marshall et al.,¹¹ following similar procedure, we can disregard the zero eigenvalue of matrix ${D ~}_{1}$ and hence that of $A \land$ . The characteristic equation corresponding to rest of the blocks is $s 3 + s 2 + (\frac{π}{2}) 2 (s + \frac{ρ n}{2 (n - 1)}) = 0$ . Applying Routh–Hurwitz criterion, we see that the eigenvalues will be in the left half of s – plane if $1 - \frac{ρ n}{2 (n - 1)} > 0$ , which implies $\frac{2}{(2 - ρ)} < n .$ Since $0 < ρ < 1$ , this condition is always true for $n \geq 2$ and for cooperative missions n is always more than one. Hence proved.▪

Implementation in 6-DOF UAV model

To validate the design, the proposed control strategy is tested in high-fidelity simulation. Each vehicle is simulated with full 6-DOF dynamical model with aerodynamic parameters that match the small fixed-wing miniature autonomous vehicles (MAV). Figure 3 shows formation geometry in three-dimensional space. The direction of force due to gravity is along the Z axis. The autopilot code is emulated to match actual flight conditions. The flight model is taken from Krishnan et al.²⁷ and the wind tunnel data were obtained from National Aerospace Laboratories, Bangalore. The aerodynamic equations are given below. The variables with subscript a refer to parameters of aerial vehicle.

{x \cdot}_{a} = [u_{a} \cos θ_{a} + (v_{a} \sin φ_{a} + w_{a} \cos φ_{a}) \sin θ_{a}] \cos ψ_{a} - (v_{a} \cos φ_{a} - w_{a} \sin φ_{a}) \sin ψ_{a}

{y \cdot}_{a} = [u_{a} \cos θ_{a} + (v_{a} \sin φ_{a} + w_{a} \cos φ_{a}) \sin θ_{a}] \sin ψ_{a} + (v_{a} \cos φ_{a} - w_{a} \sin φ_{a}) \cos ψ_{a}

{z \cdot}_{a} = - u_{a} \sin θ_{a} + (v_{a} \sin φ_{a} + w_{a} \cos φ_{a}) \cos θ_{a}

θ_{a} \cdot = q_{a} \cos φ_{a} - r_{a} \sin φ

φ_{a} \cdot = p_{a} + (q_{a} \sin φ_{a} + r_{a} \cos φ_{a}) \tan θ_{a}

ψ_{a} \cdot = \frac{(q_{a} \sin φ_{a} + r_{a} \cos φ_{a})}{\cos θ_{a}}

u_{a} . = r_{a} v_{a} - q_{a} w_{a} + \frac{1}{m_{a}} f_{x_{a}}

v_{a} . = p_{a} w_{a} - r_{a} u_{a} + \frac{1}{m_{a}} f_{y_{a}}

w_{a} . = q_{a} u_{a} - p_{a} v_{a} + \frac{1}{m_{a}} f_{z_{a}}

p_{a} . = Γ_{1} p_{a} q_{a} - Γ_{2} q_{a} r_{a} + Γ_{3} l_{a} + Γ_{4} n_{a}

q_{a} . = Γ_{5} p_{a} r_{a} - Γ_{4} (p_{a}^{2} - r_{a}^{2}) + Γ_{5} m_{a}

r_{a} . = Γ_{6} p_{a} q_{a} - Γ_{1} q_{a} r_{a} + Γ_{4} l_{a} + Γ_{7} n_{a}

where

[x_{a}, y_{a}, z_{a}]

represent position of MAV,

[u_{a}, v_{a}, w_{a}]

represent velocity components of MAV in body frame,

[φ_{a}, θ_{a}, ψ_{a}]

represent Euler angles,

[p_{a}, q_{a}, r_{a}]

represent body axis angular rates and

[l_{a}, m_{a}, n_{a}]

are roll, pitch and yaw moments, respectively. The Runge–Kutta fourth-order method has been used to solve the system of equations with the time step of 2 ms.

Figure 3.

Formation geometry in 3D.

Each vehicle has three basic controls that are implemented on the on board computer to regulate heading, speed and altitude²⁷ (see Figure 4). Lateral and longitudinal motions of the aircraft and their corresponding control loops are assumed to be decoupled for simplicity of implementation and gain tuning, and for small pitch and roll angles, these assumptions are valid. The MAVs that are simulated do not use a rudder for yaw control, and thus it is not considered in the control loop. During the flight, the altitude and airspeed are held constant. In the speed control loop, commanded speed ( $V_{a_{c}}$ ) is held constant. This is achieved by generating a suitable command for throttle control as shown in Figure 4(c). The commanded altitude ( $h_{a_{c}}$ ) is held constant to simulate planar condition. To avoid possibility of collisions each MAV is commanded to fly at different altitudes. The altitude control loop generates appropriate commands for elevator defection of the MAV (Figure 4(b)).

Figure 4.

MAV autopilot control loops. (a) Heading control, (b) Altitude control and (c) Speed control.

Lateral autopilot command is generated from the desired heading angle. The proposed control strategy (3) is implemented in heading loop. The desired heading angle or heading command $χ_{a_{c}}$ for the next cycle is calculated as $χ_{a_{c}} = χ_{a_{m}} + H_{K} φ where χ_{a_{m}}$ is measured values of the current heading angle and φ is the bearing angle as defined in equation (2). We relate $H_{K}$ to the controller gain k in equation (3) by $H_{K} = Vk / g where V$ is the speed of the vehicle and g is acceleration due to gravity. From flight mechanics, the heading rate of MAV can be calculated as

{χ \cdot}_{a} = - p_{a} \sin θ_{a} + q_{a} \cos θ_{a} \sin φ_{a} + r_{a} \cos φ_{a} \sin θ_{a}

(18)

It is assumed that the bearing angle information is available at discrete instances (1 s) and the autopilot control loops run at a frequency of 20 ms. The bearing angle command of MAV is updated at every 1 s using the bearing angle information of the other MAVs and the target. The error $χ_{a_{e}}$ between the command heading $χ_{a_{c}}$ and measured heading $χ_{a_{m}}$ is computed as $χ_{a_{e}} = χ_{a_{c}} - χ_{a_{m}}$ . Using this heading error $χ_{a_{e}}$ and the heading rate ${χ \cdot}_{a}$ from equation (18), the roll angle command $φ_{a_{c}}$ for the inner loop roll controller is generated as

φ_{a_{c}} = H_{Kp} χ_{a_{e}} - H_{Kd} χ_{a} \cdot

where

H_{Kp}

is the proportional gain and

H_{Kd}

is the derivative gain. The roll command is tracked by the inner PID control loop, and it generates appropriate actuation command for aileron defection as shown in Figure 4(a).

In the next section, we present and compare the simulation results obtained with unicycle and 6-DOF model of MAVs.

Simulation results

We simulate the trajectories of the autonomous vehicles under different conditions to verify the effectiveness of the proposed control law. The control law is applied to both the unicycle agents and 6-DOF dynamical model of MAVs and the results are compared. For the MAV, the flight model is programmed in MATLAB and Runge–Kutta fourth-order method is used to carry out the simulation. Since the measurements are available at discrete intervals, we implement it in discrete time steps. It is assumed that the bearing angle is measured at every 1 s and control algorithm runs at every 2 ms.

We considered a group of seven agents moving with linear speed of 15 m/s, controller gain $k = 0.1 and ρ = 0.5$ . The target point is the origin $(0, 0)$ . The agents start from random initial positions with random heading direction. Figure 5(a) and (b) shows the trajectories of agents having unicycle kinematics and 6-DOF dynamic model, respectively. The trajectories in both the cases are very similar, and we can see that the agents are able to encircle the target point continuously while being uniform distribution around it.

Figure 5.

Trajectories of seven vehicles: ⋄ represents initial position and ⋆ represents final position of the agents.

Next, the simulations were carried out for different values of controller gain k and pursuit gain ρ. Table 1 shows the target to agent and inter-agent distances. It can be observed that the calculated values of these distances match exactly with those obtained by simulating unicycle kinematics. For the 6-DOF dynamical model, these distances are very close to their analytical values and the difference between them can be attributed to the discrete time computation carried out in the simulations. Also, it can be observed that the radius of the circle is a function of the controller gain k and is independent of the value of ρ for both the unicycle and 6-DOF model of the agents. This shows that the proposed strategy when implemented on the MAVs will produce results as predicted in the analysis.

Table 1.

Comparison between unicycle and 6-DOF model.

			$R (m)$				$R_{aa} (m)$
V (m/s)	k	ρ	Analytical	Unicycle	6-DOF	% Error	Analytical	Unicycle	6-DOF
$15$	0.1	0.2	95.493	95.493	100.187	4.91	82.866	82.866	86.239
$15$	0.1	0.5	95.493	95.493	100.106	4.83	82.866	82.866	86.052
$15$	0.2	0.2	47.746	47.746	49.638	3.96	41.433	41.433	42.758
$15$	0.2	0.5	47.746	47.746	49.557	3.79	41.433	41.433	42.661

Effect of addition/removal of agents

To study the robustness of the proposed strategy against failure of agents, the simulation was initiated with seven agents. After 300 s, one of the agent was removed and the simulation proceeded with the remaining agents. After another 300 s, we introduced a new agent from a random position. Figure 6 shows the distances between the agents and the target (R) and the inter agent distance ( $R_{aa}$ ) for unicycle and 6-DOF model of the vehicle. From the figures, it can be observed that the agents converge to the same R even if agents are removed or added. However, the inter agent distances vary with the number of agents since the agents get uniformly distributed around the target. This proves the robustness of the formation in case of failure of agents.

Figure 6.

Simulation with 7 agents for 0 ≤ t ≤ 300, with six agents for 300 < t ≤ 600 and with seven agents for 600 < t ≤ 900. (a) Agent to target distances using (Unicycle model), (b) Inter agent distances (Unicycle model), (c) Agent to target distances (6-DOF model) and (d) Inter agent distances (6-DOF model).

Effect of ρ on the time of convergence to formation

Next, we study the effect of the pursuit gain ρ on the time of convergence to equilibrium formation. The value of ρ was varied from $0.1 to 0.9$ . Considering the unicycle model of the agents, Figure 7 shows the settling time of inter-agent distances and target to agent distances for three different initial conditions. The inter-agent distances settle faster with the increase in ρ whereas the time taken by the agents to reach the desired radius about the target increases with ρ. This is because when ρ is small, priority is given for target capturing whereas when ρ is large, priority is given for uniform formation.

Figure 7.

Effect of ρ for three different initial conditions. (a) Settling time of inter-agent distance and (b) Settling time of target to agent distance.

Effect of measurement noise

To study the effect of measurement noise, we added white noise of magnitude

\pm 5 \circ

to the bearing angle. Table 2 shows comparison of simulation results with and without noise in the measurement. Simulations were carried out for six unicycle agents for different values of linear speed, controller gain and ρ. It is observed from the table that the error in the radius of the circle is less than

1 %

. This shows the robustness of the strategy in the presence of noisy measurement.

Table 2.

Simulation with different magnitude of measurement noise.

			Without noise	Noise magnitude $\pm 5 \circ$		Noise magnitude $\pm 10 \circ$		Noise magnitude $\pm 15 \circ$
V (m/s)	k	ρ	$R (m)$	$R (m)$	$% \| \| Error \| \|$	$R (m)$	$% \| \| Error \| \|$	$R (m)$	$% \| \| Error \| \|$
$10$	0.1	0.2	63.66	63.65	0.024	63.52	0.345	64.16	0.284
$10$	0.1	0.5	63.66	63.51	0.156	63.82	0.853	63.48	0.218
$10$	0.2	0.2	31.83	31.83	0.022	31.84	0.659	31.60	0.271
$10$	0.2	0.5	31.83	31.78	0.167	31.71	0.630	31.98	0.612
$15$	0.1	0.2	95.49	95.18	0.324	96.25	0.486	94.45	0.005
$15$	0.1	0.5	95.49	95.90	0.428	94.99	0.388	95.99	0.11
$15$	0.2	0.2	47.75	47.84	0.196	47.97	0.152	47.47	0.158
$15$	0.2	0.5	47.75	47.83	0.175	47.88	0.136	47.74	0.750

Effect of wind

The proposed strategy is tested in presence of constant wind along X direction. We assumed five agents having 6-DOF model and considered different wind speeds. With V = 15 m/s,

k = 0.1

and

ρ = 0.1

, the simulation results are tabulated in Table 3 and Figure 8 shows two specific cases. The target to agent distance R is calculated as the average distance of the agents to the target since they no longer move on a fixed circle. It is observed that in the presence of wind, formation is no longer uniform and there is shift in the centre of formation. Ideally in no wind case, centroid of formation should coincide target location. The deviation of the centre of the formation from the target point is shown by line inside the formation (Figure 8). This difference increases with the increase in the wind speed. The agents are able to enclose the target even in the presence of wind but with non-uniform distribution.

Table 3.

Simulation with different wind speed.

V_wind (m/s)	R (analytical)	R (simulation)	Distance between target point and centre of formation
0	95.49	95.86	0.13
1	95.49	101.25	21.72
2	95.49	99.87	44.98
3	95.49	103.66	67.94
4	95.49	104.77	94.04

Figure 8.

Trajectories of five agents in presence of wind: ⋄ represents initial position and ⋆ represents final position of the agents. (a) Wind speed = 2 m/sec and (b) Wind speed = 4 m/sec.

When the target is moving

We considered a target moving on a circular path and five agents to encircle it. Considering 6-DOF model for the agents and V = 15 m/s,

k = 0.2, ρ = 0.1

, the results are presented in Table 4. Trajectories of the vehicles for two different linear velocities of the target are shown in Figure 9. It is observed that the centre of the formation lies behind the instantaneous target position and the distance between formation centre and the target is larger for higher values of the target speed. However, the target remains enclosed by the agents.

Table 4.

Simulation results with moving target.

Target velocity (m/s)	Radius	Distance between target and centre of formation
$0$	47.95	0.05
$1$	41.26	22.73
$2$	55.36	42.01
$3$	61.98	57.83

Figure 9.

Trajectories of five agents for a moving target: ⋄ represents initial position and ⋆ represents final position of the agents. (a) Target speed = 1m/sec and (b) Target speed= 3m/sec.

Conclusions

In this paper, we proposed a fail-safe strategy for a group of identical autonomous vehicles to achieve a balanced circular formation about a point which may or may not be pre-specified. When the point is not pre-specified, the pursuit gain ρ is assumed to be one. At equilibrium, the vehicles get uniformly distributed on the circle whose radius depends on the linear speed and controller gain of the vehicles. Since the number of vehicles does not influence the radius of the circle, the formation remains unchanged even if one or more vehicles fail. This justifies the control strategy to be called fail-safe. In fact, we can add or remove vehicles from the formation and still achieve uniform formation at a desired radius as long as there are at least two vehicles. The control law required only the bearing angles information of the other vehicles and the target point (if specified). No measurement or estimation of position or velocity is necessary. The control law can be easily computed and is found to work well even if the measurements are noisy. The effect of the pursuit gain ρ on the performance of the control law is observed. Extensive simulations are carried out for different parameter values. The effect of wind is presented. The simulations are also presented for the case when the target is slowly moving on a curved path. All the simulations are carried out using 6-DOF model of UAVs and are found to produce satisfactory results that are closed to analytical solutions obtained using unicycle kinematics. It remains to verify the performance of the control strategy when implemented in actual hardware. Future work includes mathematical analysis of the system in presence of wind, noisy measurements or moving target.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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Fail-safe encircling of multiple unmanned aerial vehicles with bearing only measurements

Abstract

Keywords

Introduction

Problem formulation

Remark 1

Analysis with unicycle kinematics

Theorem 1

Proof

Remark 2

Theorem 2

Proof

Implementation in 6-DOF UAV model

Simulation results

Effect of addition/removal of agents

Effect of ρ on the time of convergence to formation

Effect of measurement noise

Effect of wind

When the target is moving

Conclusions

Footnotes

Declaration of conflicting interests

Funding

Appendix 1

References