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Bounded Coordination Control of Second-order Dynamic Agents | OMICS International
ISSN: 2168-9695
Advances in Robotics & Automation
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Bounded Coordination Control of Second-order Dynamic Agents

Khac Duc Do*

Department of Mechanical Engineering, Curtin University, Perth, WA 6845, Australia

*Corresponding Author:
Khac Duc Do
Department of Mechanical Engineering
Curtin University
Perth, WA 6845, Australia
E-mail: [email protected]

Received July 04, 2013; Accepted July 18, 2013; Published July 22, 2013

Citation: Do KD (2013) Bounded Coordination Control of Second-order Dynamic Agents. Adv Robot Autom 2:105. doi:10.4172/advances-robotics-automation.1000105

Copyright: © 2013 Do KD. This is an open-access article distributed under theterms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

This paper presents a constructive design of distributed and bounded coordination controllers that force mobile agents with second-order dynamics to track desired trajectories and to avoid collision between them. The control design is based on the new bounded control design technique for second-order systems, and new pairwise collision avoidance functions. The pair wise collision functions are functions of both the relative position and velocity of the agents instead of only the relative position as in the literature. Desired features of the proposed control design include: 1) Boundedness of the control inputs by a predefined bound despite collision avoidance between the agents considered, 2) No collision between any agents, 3) Asymptotical stability of desired equilibrium set, and 4) Instability of all other undesired critical sets of the closed loop system. The proposed control design is then applied to design a coordination control system for a group of vertical take-off and landing (VTOL) aircraft.

Keywords

Second-order agents; Bounded coordination control; Collision avoidance; Potential functions; VTOL aircraft

Introduction

Coordination control of multiple agents has received a lot of attention from researchers in the control community due to its various applications to search, rescue, coverage, surveillance, reconnaissance and cooperative transportation. Therefore, a number of approaches have been available for coordination control of networked agents. Here, three common approaches are briefly mentioned. The leaderfollower approach (e.g., [1-4] uses several agents as leaders and others as followers. This approach is easy to understxand and ensures coordination maintenance if the leaders are disturbed. However, the desired coordination shape cannot be maintained if followers are perturbed unless a feedback is implemented [5]. The behavioral approach (e.g., [6,7]), where each agent locally reacts to actions of its neighbors, is suitable for decentralized control but is difficult in control design and stability analysis since group behavior cannot explicitly be defined. The virtual structure approach (e.g., [8-17]) treats all agents as a single entity. The virtual structure approach is amenable to mathematical analysis but has difficulties in controlling critical points, especially when collision avoidance between the agents is a must. The control coordination design in this paper belongs to the virtual structure approach. In the literature, cooperative control of multiple agents with second-order dynamics was also addressed, see for example [12,18-30]. However, the problem of bounded control has not been addressed for the case where collision avoidance between the agents must be considered.

Let us discuss the reasons why it is difficult to design a bounded coordination controller for multiple agents with second-order dynamics when collision avoidance between the agents must be considered using the virtual structure approach. In this approach, the control design is usually based on a nontrivial potential function and the direct Lyapunov method. The potential function essentially consists of two parts. The first part is usually referred to as the goal part. This part is designed such that it puts penalty on the tracking errors of the agents, and is equal to zero when all the agents perfectly track their desired trajectory. The second part is usually referred to as the collision avoidance part, which is chosen such that it is equal to infinity whenever any two agents come in contact, i.e., a collision occurs, and attains the minimum value when the agents perfectly track their desired trajectories. Since the collision avoidance part of the potential function is designed to be equal to infinity when there is a collision between any two agents, the control signals become extremely large when any two agents are close to each other. To overcome this drawback, a method was proposed in [10] to design bounded controllers for the agents with the first-order dynamics. However, the method in [10] cannot be extended to the agents with second-order dynamics by applying the back stepping technique in [31]. To see where the problem lies, let us consider a group of N agents with the dynamics qi= ui , i ∈ N, where q1 and u1 denote the position and control input vectors of the agent i, respectively, and N is the set containing all the agents in the group. Next, let us denote the collision avoidance part of the potential function be β, which is equal to infinity when there is a collision between any two agents. Then the function β must be a summation or a product of all pairwise collision avoidance functions βij of the agents i and j, i.e., equation or equation. Moreover, the function βij must be a positive definite function of equation , where qij = qi-qj, i.e., the relative position vector between the agents i and j, As a result, the control ui must be a function of equation. Since βij is a positive definite function of equation , βij tends to infinity faster than βij when equation approaches zero, i.e., when a collision between the agents i and j tends to occur. Consequently, it is not possible to extend the bounded control design in [10] using the back stepping technique to design a bounded coordination controller for the agents with second-order dynamics. Using the same analysis as above shows that it is not possible (or at least not clear how) to extend the bounded controllers for connectivity preservation of a network of the agents with the first order dynamics in [32] and [33] to design a bounded controller for the agents with second-order dynamics.

Motivated by the above discussion and the fact that many mechanical systems in practice are of second-order dynamics and require bounded control inputs with a predefined bound, this paper contributes a new method to design bounded coordination controllers for mobile agents with second-order dynamics. The method is based on a new technique to design a bounded controller for second-order systems, and a construction of new pairwise collision avoidance functions, which are functions of both the relative position and velocity of the agents. The proposed bounded coordination controllers guarantee no collision between any agents, asymptotical stability of desired equilibrium set, and instability of all other undesired critical sets of the closed loop system. The applicability of the proposed method is illustrated through an example of designing a coordination control system for a group of VTOL aircraft.

The rest of the paper is organized as follows. In the next section, the control objective is stated. Section 3 gives essential preliminary results including saturation functions, a technique for designing bounded controllers for a second-order system, a non-zero convergent lemma for a differential inequality, smooth step functions, pair wise collision avoidance functions, and Barbalat-like lemma. Proofs of preliminary results are given in [34,10] and Appendices A and B. These results are to be used in the control design in Section 4 and stability analysis in Appendix C. An application of the proposed coordination control design to a group of VTOL aircraft is presented in Section 5. Stability analysis of coordination control of VTOL aircraft is briefly given in Appendix D. Finally, conclusions are given in Section 6.

Problem Statement

Agent dynamics

We consider a group of mobile agents, of which each agent has the following dynamics

equation     (1)

Where qi € Rn and equation are the state and control input of the agent i. We assume that n>1 and N>1.

Coordination control objective

In order to design a coordination control system, it is necessary to specify a common goal for the group and initial positions and velocities of the agents. We therefore impose the following assumption on the reference trajectories and initial conditions between the agents.

Assumption 4.1:

i) The reference position vector qid (t) for the agent i to track satisfies the condition:

equation      (2)

for all (i, j) € N, j ≠ i and t ≥ t0 where qijd =qid(t)-qjd(t),ε1ijd is a strictly positive constant, ε2od and ε3od are nonnegative constants, and qod(t) is referred to as the common reference trajectory.

ii) Let us define

equation     (3)

for all (i, j) € N, j ≠ i, with N the set of all the agents, k is a symmetric positive definite matrix, In is an n×n dimensional identity matrix, and Δ(x) with x € Rn is an n×n dimensional diagonal and nonnegative definite matrix, whose elements are bounded if x is bounded. In addition, the matrix Δ(x) possesses the following properties:

equation    (4)

Where L is a diagonal positive definite matrix, ? is a nonnegative constant, and A(x1,x2) is a diagonal and nonnegative definite matrix for all (x1,x2) € R2n. There are many matrices that satisfy the above properties. An example is Δ(x)= diag (x1 2,..., x1 2,...xn 2 ) where x1,.... xi,...,xn are elements of x, i.e., x=[x1,..., xi,...xn]T. At the initial time t0 ≥ 0, each agent starts at a different location and all the agents do not approach each other at high relative velocities. Specifically, there exist strictly positive constants ?ij0 and χij0 such that for all (i, j) € N, with i ≠ j, with, where N is the set of all the agents, the following conditions hold:

equation        (5)

iii) The agents i and j can communicate with each other, i.e., the agent i can measure the states qj and pj of the agent j, and the agent j can measure the states qi and pi of the agent i, if the following condition holds:

equation            (6)

Where χij M is a strictly positive constant.

Remark 4.1:

i) Assumption 4.1.1 specifies feasible reference trajectory qid (t) for the agent i in the group to track since it has to satisfy the condition (2). A desired coordination shape can be specified by the reference trajectories qid (t) with i € N. Let us consider the virtual structure approach in [8-10] to generate the reference trajectories qid (t) the agent i to track. First, a virtual structure consisting of N vertices is designed as a desired coordination shape. Second, we let the center of the virtual structure move along the common reference trajectory qod (t). Third, as the virtual structure moves, its vertex i generates the reference trajectory qid (t) for the agent i to track. Specifically, the reference trajectory qid (t) can be generated as qid (t) = qod (t) + li where li is a constant vector.

Moreover, the second equation in (2) implies that all the agents have the same desired velocity and acceleration. Since this paper focuses on designing bounded coordination controllers for the agents with second-order dynamics, we do not consider the case where the shape of the virtual structure is time-varying, i.e., the vectors li are time-varying, to avoid complication of the presentation. When li are time-varying, the technique proposed in [35] can be used in conjunction with the coordination control design in this paper.

ii) If at the initial time t0 the agents approached each other at high relative velocities, a bounded controller would not be able to prevent the agents from colliding with each other. Therefore, it is reasonable to impose Assumption 1.2 for the design of bounded coordination controllers, which guarantee collision avoidance between the agents.

iii) Assumption 4.1.3 implies that we need to design a distributed coordination control system since the condition (6) specifies that when χij > χijM the agents and do not communicate with each other. The constant equation can be considered as the generalized communication range between the agents and . This generalized communication range is different from usual communication ranges used in the literature, e.g. [12,10,19] in the sense that the generalized communication range relates to both the relative position and the relative velocity between the agents while the usual communication range relates only to the relative position. Dependence of equation on the relative velocity between the agents is necessary because we are to design bounded controllers.

Coordination control objective 4.1: Under Assumption 2.1, for each agent i design the bounded and distributed control ui such that the position vector qi of the agent i tracks its reference position vector qid while avoiding collision with all other agents in the group. Specifically, we will design ui such that

equation          (7)

For all (i, j) € N, i ≠ j and, t ≥ t0 ≥ 0 where δ and ?ij are strictly positive constants. The constant δ is strictly larger than ε3od, which is defined in (2). Moreover, the control ui should depend only on its own states and the states of other agents j if the condition (6) between the agent i and the agents j holds.

Preliminaries

This section presents saturation functions, a technique for designing bounded controllers for a second-order system, and nonzero convergent lemma for a differential inequality, smooth step functions, pairwise collision avoidance functions, and Barbalat-like lemma. These preliminary results will be used in the control design and stability analysis later.

Saturation functions

This subsection defines saturation functions that will be used in the control design later.

Definition 5.1: The function σ(x) is said to be a smooth saturation function if it possesses the following properties:

equation      (8)

for all (x, y) €R2. Some functions satisfying the above properties include σ(x) = than (x) and equation . For the vector x=[x1,...,xi,...,xn]T, we use the notation σx=[σ(x1),..,σ(xi)...,σ(xn)]T to denote the smooth saturation function vector of x.

Bounded control design for second-order systems

This subsection presents the idea of a bounded control design technique through a simple example. This technique will be used to design bounded cooperative controllers later. As such, consider the following second-order system:

equation         (9)

where x1 and x2 are the states, and is the control input. Let us address a control problem of designing the control input u to asymptotically stabilize (9) at the origin for any initial values(x1(t0),x2(t0))€R2 at the initial time t0 ≥ 0 such that | u(t) | ≤ ε for all t ≥ t0. A solution to the above control problem is given in the following lemma.

Lemma 5.1: Let the positive constants K and C be chosen such that 0.5k + c ≤ ε, and let σ (•) be a smooth saturation function of • defined in Definition 3.1. The bounded control law

equation        (10)

Globally asymptotically stabilizes the system (9) at the origin.

Proof: Refer to Appendix A.

Remark 5.1: There are other methods (e.g., [36,37]) to design bounded control laws for a chain of integrators inspired by the work in [38]. However, these methods are not suitable for designing a bounded coordination controller for multiple agents in this paper.

Non-zero convergent lemma

This subsection presents a non-zero convergent result for a first order system. This result will be used to construct pairwise collision avoidance functions in Subsection 3.5.

Lemma 5.2: Assume the continuous vector x (t) € Rn and its derivative x, which is also continuous, satisfy the following conditions:

equation    (11)

where t0 ≥ 0 is the initial time, B is a symmetric positive definite matrix, In is an n×n dimensional identity matrix, Q(t) is a diagonal and nonnegative definite matrix whose elements are bounded for all, t ≥ t0 ≥ 0, a0 and a are strictly positive constants. Then

equation   (12)

where λM(B) is the maximum eigenvalue of the matrix .

Proof: Refer to Appendix B.

Smooth step function

This subsection gives a definition of the smooth step function followed by a construction of this function. The smooth step function is to be embedded in a pairwise collision avoidance function to avoid discontinuities in the control law in solving the collision avoidance problem.

Definition 5.2: A scalar function h(x, a, b) is said to be a smooth step function if it possesses the following properties

equation       (13)

where equation , and a and b are constants such that a<b.

Lemma 5.3: Let the scalar function h(x,a,b) be defined as

equation       (14)

where

equation       (15)

with a and b being constants such that a<b. Then the function (x,a,b) is a smooth step function.

Proof: See [34]. An alternative smooth step function is available in [39] but it requires a numerical integration.

Pair wise collision avoidance functions

This subsection defines and constructs pair wise collision avoidance functions. In constructing these functions, we utilize Lemma 3.1, Lemma 3.2, Definition 3.2, and Lemma 3.3. The pair wise collision avoidance functions will be embedded in a potential function for the coordination control design in the next section.

Definition 5.3: Let βij with (i, j) € N and i ≠ j be a scalar function of χij, which is defined in (3). The function βij is said to be a pair wise collision avoidance function if it possesses the following properties:

equation         (16)

where equation and χij* is a strictly positive constant satisfying the following condition:

equation                          (17)

where equation , and the constant χijM is defined in (6).

Remark 5.2: Property 1) implies that the function βij is zero when the agents i and j are at their desired locations or are sufficiently faraway from each other since the constant χij* satisfies the condition (17). Property 2) implies that the function βij is positive definite when the agents i and j are sufficiently close to each other. Property 3) means that the function βij is equal to infinity when a collision between the agents i and j occurs. Property 4) allows us to use control design and stability analysis methods found in [40] for continuous systems instead of techniques for switched and discontinuous systems found in [41] to handle the collision avoidance problem.

Using the smooth step function given in Definition 3.2, we can find many functions that satisfy all the properties listed in (16). An example is

equation                        (18)

where lij is a positive constant, and the positive constants aij and bij satisfy the following condition:

equation                                          (19)

The function h(χij,aij,bij) is a smooth step function defined in Definition 3.2. It can be directly verified that the function βij given in (18) possesses all the properties listed in (16). In the rest of the paper, the function βij defined in (18) will be used.

Barbalat-like lemma

The following Barbalat-like lemma is to be used in the stability analysis of the closed loop system.

Lemma 5.4: Assume that a nonnegative scalar differentiable function f(t) satisfies the following conditions

equation          (20)

where k1 and k2 are positive constants, then lim t → ∞ f ( t ) 0 = .

Proof: See [10]. Lemma 3.4 differs from Barbalat’s lemma found in [40]. While Barbalat’s lemma assumes that f(t) is uniformly continuous, Lemma 3.4 assumes that equation is bounded by k1f(t). Lemma 4 is useful in proving convergence of f(t) when it is difficult to prove uniform continuity of f(t).

Coordination Control Design

To design the control ui for the agent i that achieves Coordination Control Objective 1, we will construct a potential function ?. This potential function puts penalty on all the tracking errors for all the agents in the group, and includes all the pairwise collision avoidance functions βij for all i ≠ j. As such, the potential function ? is chosen as follows:

equation        (21)

where the matrices K and Δ(•) , and the pairwise collision avoidance function βij are given in Definition 5.3, and is the set of all the agents except for the agent i. Differentiating both sides of (21) along the solutions of (1) and using the first property of Δ(•) in (4), we have

equation      (22)

Where

equation            (23)

It is noted that in deriving (22), we have used pij=pie-pje since equation,see (2). Now using Property 3) of Δ(•) in (4), we can write the term equation in (22) as follows:

equation      (24)

Where we have again used pij=pie-pje. Substituting (24) into (22) gives

equation      (25)

Since βij ≤ 0 for all χij ∈ R , see Property 2 of βij in (16), and A(pie,pje) is a diagonal and nonnegative definite matrix, see Property 3) of Δ(•) in (4), we have the following inequality

equation      (26)

For all qij € Rn, pij € Rn, pie € Rn, and pje € Rn. Substituting (26) and equation into (25) results in

equation        (27)

which suggests that we choose the control ui as follows:

equation     (28)

Where C is a positive definite matrix.

Remark 6.1:

1. Substituting Ωi defined in (23) into (28) results in

equation      (29)

The elements of ui are explained as follows. The term equation is to make the control ui bounded with a predefined bound since σ (•) and equation are bounded with a predefined bound. The term equation , which is a part of the argument of σ (•) and the term −2 ie Kp referred to as the attractive force plays the role of forcing the agent to track its desired trajectory. The term equation , which is a part of the argument of σ (•) , referred to as the repulsive force, takes care of collision avoidance for the agent i with the other agents. Moreover, the control ui of the agent i given in (28) depends only on its own state and the states of other neighbor agents j if these agents are communicated with the agent i, i.e., when the condition (6) holds. This is because if the condition (6) does not satisfy, we have βij=0 due to Property 1) of βij listed in (16) with the constants aij and bij of the function βij chosen according to (19), and χij* satisfied the condition (17).

2. Since Δ(pie) satisfies Property 2) in (4) and Δ(pie) is a nonnegative definite matrix, L is a positive definite matrix, a calculation shows that the control ui is bounded by

equation     (30)

completeness of the closed loop system (32) is to be proved in Appendix C. In (30), (•) M a is the maximum eigenvalue of •, and ? and ε3od are defined in (4) and (2), respectively. Therefore, the upper-bound δ of the control ui can be specified as

equation    (31)

It is seen from (31) that the upper-bound δ of the control ui can be predetermined as long as the pre-specified value of δ is strictly larger than ε3od, which is the suprenum value of the reference trajectory acceleration, equation . This is because when δ is strictly larger than ε3od we can always appropriately choose the matrices and , and the constant ?, which comes from a proper choice of the matrix Δ(•) , see (4).

Substituting the control ui given in (28) into (1) gives the closed loop system

equation    (32)

On the other hand, substituting the control ui into (27) results in

equation    (33)

Where

equation    (34)

The coordination control design has been completed. We summarize the main results in the following theorem.

Theorem 6.1: Under Assumption 4.1, the smooth bounded control vector ui given in (28) with a predetermined bound δ defined in (31) for the agent i solves the coordination control objective. In particular, no collisions between any agents can occur for all t ≥ t0 ≥ 0 the closed loop system (32) is forward complete, and the trajectory qi (t) of the agent i asymptotically track its reference trajectory qid (t), for all i € N.

Proof: Refer to Appendix C.

Coordination Control of VTOL Aircraft

In this section, we present an application of the proposed coordination control design in the previous section to design a coordination control system for a group of VTOL aircraft.

Related work

Control of a single VTOL aircraft has been considered by many researchers since it is under actuated and non minimum phase. An approximate input-output linearization approach was used in [42-46] to develop controllers for stabilization and output tracking/regulation of a VTOL aircraft. In [47], by noting that the output at a fixed point with respect to the aircraft body (Huygens center of oscillation) can be used, an interesting approach was introduced to design a local output tracking controller. Simple approaches were developed in [48,49] to provide global controllers for the stabilization and tracking control of a VTOL aircraft. A dynamic high-gain approach was used in [50] to design a controller to force the VTOL aircraft to globally practically track a reference trajectory.

Since the VTOL aircraft are underactuated it is necessary to address the bounded control problem for the agents with secondorder dynamics in order to design a coordination control system for a group of VTOL aircraft, see discussion in Subsection 7.4. Due to the mentioned difficulties, only few results on cooperative control of multiple aircraft are available. In [51], direct coordination (using inertial measurements) and nearest-neighbor coordination (using relative measurements) were addressed. In [52], a sliding mode formation controller was proposed using the leader-follower approach. In [53,54], several formation controllers were designed to force a group of the quadrotor aircraft to track a desired reference linear velocity and to maintain a desired formation. In the above works, collision avoidance between the aircraft is not considered. The above review motivates an inclusion of this section to present an application of the coordination control design for agents with second-order dynamics to a group of VTOL aircraft.

Mathematical model and coordinate transformations

A scaled mathematical model of the ith VTOL aircraft in the group can be described [43]:

equation      (35)

where xi, zi and θi denote position of the aircraft center of mass and roll angle of the ith aircraft, respectively, equation , and τi are the vertical control force and rotational moment, g>0 is the gravitational acceleration, and equation is the constant coupling between the roll moment and the lateral force, see Figure 1.

advances-robotics-automation-votl-aircraft

Figure 1: The VTOL aircraft.

Since the VTOL dynamics (35) is non-minimum phase if its position (xi,zi) is controlled directly, we introduce the following coordinate transformations [49].

equation     (36)

With the coordinate transformations (36), we can write (35) as

equation         (37)

where

equation                    (38)

If we consider equation as the new position of the aircraft to be controlled, it is seen that (37) is of a triangular form. It is noted equation coincides with the aircraft’s center of oscillation [47]. Let us define

equation       (39)

With the definition of q1, p1 and ui in (39), we can write (37) as

equation     (40)

This is a convenient form that will be used for the coordination control design in Subsection 5.4.

Coordination control objective

Roughly speaking, we will consider a group of N aircraft with a coordination control objective of designing the controls equation and τi for the aircraft such that its position equation tracks its reference trajectory equation and avoids collision with all other aircraft in the group. We will state the coordination control objective precisely after the following assumption stated.

Assumption 7.1:

1. Assumption 2.1 holds. In addition, the reference trajectories qid satisfy the following conditions:

equation   (41)

for all t ≥ t0 ≥ 0 and i € N, where ε4od and ε5od are nonnegative constants.

2. ε3od ≤ g -μ , where ε3od is specified in (2), and μ is a strictly positive constant.

Assumption 7.1 covers both the stabilization and trajectory tracking of the VTOL aircraft and implies that the aircraft is not allowed to land faster than it freely falls under the gravitational force.

Coordination Control Objective 7.1: Under Assumption 7.1, for each aircraft i design the smooth control inputs equationand τi such that the position vector qi of the aircraft i tracks its reference position vector qid while avoiding collision with all other agents in the group. Specifically, we will design equationand τi such that

 

equation    (42)

for all (i, j)∈N , i ≠ j and t ≥ t0 ≥ 0 , where ?ij is a positive constant.

Coordination control design

It is seen that (40) consists of two subsystems, namely the subsystem describes linear motion of the aircraft’s center of oscillation and the subsystem gives rotational motion of the aircraft. These two subsystems are connected in a lower triangular structure. The subsystem is of the form (1) with n=2. This structure suggests that we should use the back stepping technique [31] to design the controls equationand τi .

Let us discuss the reason why we need to apply the smooth bounded coordination control design proposed in the previous section to the VTOL aircraft described by (40). Assume that we have already designed the smooth vector ui and that θi is a control. We will need to solve the third equation of (39) for equation and θi. As such, we write the third equation of (39) as

equation   (43)

where u1i and u2i are the first and second elements of ui, i.e., equation Clearly, we need a bounded ui with equation to obtain a bounded equation and a bounded equation . Motivated by the aforementioned discussion, we will proceed the coordination control design for the group of VTOL aircraft in two stages.

Stage 1: In this stage, we consider the subsystem A defined in (40) with ui , i.e., equation and θi, as a control vector to achieve asymptotical tracking of qi (t)-qid (t) and to guarantee no collision between any aircraft. As such, define

equation     (44)

where αθi is a virtual control of θi, and αui is a virtual control of uidefined by

equation      (45)

With the third equation of (39), the second equation of (44) and (45), we can write uie as

equation   (46)

The virtual control vector αui is designed in the same way as the one in (28), i.e.,

equation   (47)

where the matrices In ,L, C, and K, Δ(pie),and the constant ? are defined in the previous section. It is noted that equation , see (30). Therefore, to make it possible to solve (45) for αθi, we impose the following condition on the matrices L , K and C, and the constant ?:

equation  (48)

There always exist the matrices L ,K and C, and the constant ? such that the condition (48) under Item 2) of

Assumption 7.1: Solving (45) gives

equation      (49)

where αu1i and αu2i are the first and second elements of αui, i.e., equation . It is noted that αθi in (49) is well defined because equation .

Remark 7.1:

1. The virtual control αθi is a smooth function of q1,..,qN, p1,..,pN, equation and equation because equation and equation , see Assumption 2.1.1.

2. The virtual control αui depends only on its own state and the states of other neighbor aircraft j if these aircraft are communicated with the aircraft , i.e., when the condition (6) holds, see Remark 4.1.2 for more details. Therefore, the control equation of the aircraft i depends only on its own state and the states of other aircraft j if these aircraft are communicated with the aircraft i.

Due to ui=uieui, see (44), the derivative of the function ? defined in (21) is

equation     (50)

where, ϑi, Ωi and Δ(pie), are defined in (34), (23), and Assumption 1.1, respectively.

Stage 2: In this stage, we will design the control τi to asymptotically stabilize uie at the origin by using the back stepping technique [31]. We first note that the subsystem B defined in (40) can be written as

equation       (51)

Now, we define

equation       (52)

where αwi is a virtual control of ωi. We now design the virtual control αwi to asymptotically stabilize uie at the origin by considering the following Lyapunov function candidate

equation    (53)

where θie is defined in (44). Differentiating both sides of (53) along the solutions of (50), (52), and the first equation of (51) gives

equation   (54)

Where

equation    (55)

It is noted that the term equation is well defined because equation and equation, which are smooth for all ie θie ∈R . The equation (54) suggests that we choose the virtual control αwi as follows

equation  (56)

where k1 is a positive constant.

Remark 7.2: The virtual control αwi is a smooth function of q1,..,qN, p1,..,pN,q1d,..,qNd, equation, equation, and θi. Moreover, due to Remark 5.1 on αθi, the virtual control αwi of the aircraft i depends only on its own state and the states of other neighbor aircraft j if these aircraft are communicated with the aircraft i.

Substituting (56) into (54) results in

equation  (57)

We finally design the control τi to asymptotically stabilize ωie at theorigin by considering the following Lyapunov function candidate

equation   (58)

whose derivative along the solutions of (57), (52), and the last equation of (51) satisfies

equation  (59)

where

equation (60)

The equation (59) suggests that we design the control τi as

equation  (61)

where K2 is a positive constant. Due to Remark 7.2 on the virtual control αwi, the control τi of the aircraft i depends only on its own state and the states of other neighbor aircraft j if these aircraft are communicated with the aircraft i, i.e., when the condition (6) holds. Substituting (61) into (59) gives

equation    (62)

The above control design results in the following closed loop system:

equation      (63)

We summarize the results of the proposed coordination control design for VTOL aircraft in the following theorem:

Theorem 7.1: Under Assumption 5.1, the smooth controls equation with equation given in (49) and τi given in (61) for the aircraft i solve the coordination control objective 1 as long as the control design matrices K and C, and the constant ?, which comes from a proper choice of the matrix Δ(•) , see (4), are chosen such that the condition (48) holds. In particular, no collisions between any aircraft can occur for all t ≥ t0 ≥ 0 the closed loop system (63) is forward complete, and the trajectory qi (t) of the aircraft i asymptotically tracks its reference trajectory qid (t), for all i∈N.

Proof: Refer to Appendix D.

Simulation results

In this subsection simulates the coordination control design proposed in the previous subsection on a group 6 aircraft. All the aircraft have the same coupling constant 0 1 i ξi = 0.1 ,i=1,.....,6. The communication constant χijM is specified as χijM= 40. The initial conditions of the aircraft are chosen as follows:

equation      (64)

where R0=10. This choice of the initial conditions means that the aircraft are uniformly distributed on a circle centered at the origin with a radius of 10. The reference trajectories are chosen as follows:

equation   (65)

where Rd = 5. This choice of the reference trajectories mean that the aircraft are first to be uniformly placed on a circle centered at the origin with a radius of 5 but their positions are opposite to their initial positions, are then to track sinusoidal signals. The purpose of choosing the above reference trajectories is to illustrate both collision avoidance and reference trajectory tracking properties of the proposed coordination control. Indeed, with the above choices of the initial conditions and reference trajectories all the aircraft need to cross the origin, i.e., the center of the aforementioned circles. This is an effective illustration of the collision avoidance capacity of the proposed coordination controller.

The matrix Δ(pie), is chosen as equation where xie and zie are the first and the second elements of pie, i.e., equation. With this choice of Δ(pie), a simple calculation shows that the matrices L and A(x1,x2) and the constant ? in (4) are L=diag(2,2), equation, for all equation and equation, and ?=1. Moreover, the reference trajectories specified in (65) ensure that the condition (2) holds with ε1ijd=5, ε2od=0.5, and ε30d=0.05. From the condition (48), we choose the control design matrices as C=I2 and K=I2. From the initial condition (64) and the chosen K, it is directly verified that ?ij0 and χij0 in (5) are ?ij0 = 10 andχij0 = ?2ij0 . The constants aij and bij are chosen as aij=4 and bij=16. The other control design parameters are chosen as σ (•) = tanh(•) , lij=102 for all (i, j) €b N, i ≠ j,K1=2, and K2=4. A calculation shows that the condition (48) satisfies with this choice of the control parameters and the above choice of the reference trajectories. that the proposed coordination controller forces all the aircraft to perform collision avoidance and trajectory tracking very well, see also Figure 3a for a plot of the scaled product of all relative distance equation, which is always larger than zero, Figure 3b for convergence of the tracking error equation of the aircraft 1 to zero. Figure 2b and Figure 2d plot the trajectories zi and xi of all the aircraft versus time, and Figure 2c plots θi of all the aircraft versus time. It is seen that θi(t) is bounded for all i € N and t ≥ 0Finally, the controls F1 and τ1 of the aircraft 1 are plotted in Figure 3c and Figure 3d, respectively.

advances-robotics-automation-trajectories-aircraft

Figure 2: Trajectories of the aircraft.

advances-robotics-automation-scaled-value-relative-distances

Figure 3: Scaled value of relative distances, tracking errors, and control inputs.

Conclusions

constructive method has been presented to design distributed controllers for coordination control of a group of N mobile agents. The most desired feature of the proposed coordination control design is that the controllers are bounded with a pre-specified bound. The keys to the control design include the new bounded control design technique for second-order systems, and new pair wise collision avoidance functions of both the relative position and velocity of the agents. An extension of the proposed coordination control design to agents with higher order dynamics is under consideration.

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