Flocking for Multiple Ellipsoidal Agents with Limited Communication Ranges

Abstract

This paper contributes a design of distributed controllers for flocking of mobile agents with an ellipsoidal shape and a limited communication range. A separation condition for ellipsoidal agents is first derived. Smooth step functions are then introduced. These functions and the separation condition between the ellipsoidal agents are embedded in novel pairwise potential functions to design flocking control algorithms. The proposed flocking design results in (1) smooth controllers despite of the agents’ limited communication ranges, (2) no collisions between any agents, (3) asymptotic convergence of each agent’s generalized velocity to a desired velocity, and (4) boundedness of the flock size, defined as the sum of all distances between the agents, by a constant.

1. Introduction

Flocking, referred to as a collective motion of a large number of self-propelled entities, has attracted a lot of attention of researchers in biology, physics, and computer science [1–4]. Engineering applications of flocking include search, rescue, coverage, surveillance, sensor networks, and cooperative transportation [5–14].

In 1987, Reynolds [2] introduced three rules of flocking: (1) separation: avoid collision with nearby flock-mates; (2) alignment: attempt to match velocity with nearby flock-mates; (3) cohesion: attempt to stay close to nearby flock-mates. Since then, there has been a number of modifications and extensions of the above three rules and additional rules to result in many algorithms to realize these rules. The graph theory and the Lyapunov direct method were used to solve consensus problems in [15–17]. Local artificial potentials between neighboring agents were used to deal with separation (collision avoidance) and cohesion problems in [12, 13, 18–23]. The leader-follower approach to a target tracking problem was used in [24, 25]. Other related work includes geometric formation optimization [26, 27], pattern formation [28], and task allocation [29]. In all the above cited references, the agents are considered as a single point, a circular disk, or a sphere.

In practice, many agents have a nonspherical, especially long and narrow, shape. If these agents are fitted to spheres, there is a problem of the large conservative volume. To illustrate this problem, we look at an example of fitting a cylindrical agent with a radius of 𝑟_𝑐 and a length of 2𝑙_𝑐 to an ellipsoid with semiaxes of 𝑎, 𝑏, and 𝑐, and a sphere with a radius of 𝑟_𝑠 as shown Figure 1. By shrinking the space along the direction of the major axis of the ellipsoid, we can find 𝑎=√2𝑙_𝑐, 𝑏=𝑐=√2𝑟_𝑐, and 𝑟_𝑠=𝑟²_𝑐+𝑙²_𝑐. Therefore, the conservative volume, 𝑉_con, defined as the difference between the volumes enclosed by the sphere and the ellipsoid, is given by 𝑉_con=(4𝜋𝑟³_𝑐/3)[(𝑙²_𝑐/𝑟²_𝑐+1)𝑙²_𝑐/𝑟²_𝑐+1−2√2(𝑙_𝑐/𝑟_𝑐)]. This means that the conservative volume is always nonnegative and is proportional to cubic of the half-length 𝑙_𝑐 over the radius 𝑟_𝑐 of an agent.

Figure 1 

Fitting an cylindrical agent to a sphere and an ellipsoid.

A spherical approximation of the shape of long and narrow agents can adversely affect performance of a flocking algorithm. An example is the case where it is a must to flock a group of long and narrow agents through a long and narrow passageway. In some cases, a spherical approximation of the agents's shape can result in failure of a flocking algorithm. As an illustration, we consider two cylindrical agents with a length of 2𝑙₁ and 2𝑙₂, and a radius of 𝑟₁ and 𝑟₂. Assuming that 𝑟₁ and 𝑟₂ are much less than 𝑙₁ and 𝑙₂, respectively, that is, the two agents have a long and narrow shape. We now require these two agents to flock in a way that they do not collide with each other and the distance 𝑑₁₂ between them is such that (𝑟₁+𝑟₂)+𝜖₁₂<𝑑₁₂<(𝑙₁+𝑙₂)−𝜖₁₂ with 𝜖₁₂ being a feasible positive constant. Clearly, a spherical approximation of the agents' shape is not applicable in this case for a flocking algorithm. On the other hand, an ellipsoidal approximation can be applicable. In addition, an ellipsoidal approximation of the agents' shape for collision avoidance between the agents in a flocking algorithm covers a spherical approximation of the agents' shape by setting the semiaxes of the ellipsoid equal, but not vice versa. The above discussion indicates that it is much more efficient to use an ellipsoidal approximation of the agents with a long and narrow shape for collision avoidance in designing flocking algorithms.

Despite of the above advantages of an ellipsoidal approximation of the agents' shape, flocking for ellipsoidal agents has not been addressed in the literature except for a recent paper [30] on coordination control of multiple ellipsoidal agents. This is partially due to difficulties in determining a separation condition between two ellipsoids. There have been two main methods to determine a separation condition between ellipsoids. The first method in [31, 32] consists of determining the intersection of the ellipsoids with the plane containing the line joining their centers and rotating the plane. The distance of closest approach [33] of the two ellipses formed by the intersection is a periodic function of the plane orientation, of which the maximum value corresponds to the closest distance between the two ellipsoids. The second method [34] is based on the discriminant of their characteristic polynomial. Both methods are too complicated for an application in flock control. If these methods are applied for collision avoidance, the condition, for which the minimum distance between two disks or the discriminant of their characteristic polynomial is positive, is extremely complicated to be embedded in a proper potential function for designing a flocking algorithm. In [35], a design of distributed controllers for flocking of mobile agents with an elliptical shape and with limited communication ranges was addressed. As it will be seen later, the design of a flocking algorithm in two dimensional space (i.e., elliptical agents are considered) is much harder than that in three-dimensional space (i.e., ellipsoidal agents are considered).

The aforementioned observations motivate contributions of this paper on a design of flocking algorithms for mobile agents with an ellipsoidal shape and limited communication ranges. The paper's contributions include (1) a new condition for separation between two ellipsoids, see Section 2.1; (2) smooth step functions; (3) a new pairwise potential function for two ellipsoidal agents, see Section 4.1.1; (4) a derivation of flocking algorithms based on the pairwise potential functions, see Section 4.4.

2. Preliminaries

2.1. Separation Condition Between Two Ellipsoids

This section presents a condition for separation of two ellipsoids applicable for collision avoidance in the flock control design later. As such, we consider two ellipsoids 𝑖 and 𝑗 shown in Figure 2. In this figure, 𝑂𝑋𝑌𝑍 is the earth-fixed frame, 𝑂_𝑖𝑋_𝑖𝑌_𝑖𝑍_𝑖 is the body-fixed frame attached to ellipsoid 𝑖, 𝐪_𝑖=[𝑥_𝑖𝑦_𝑖𝑧_𝑖]^𝑇 denotes the position of the center 𝑂_𝑖, and 𝜼_𝑖=[𝜙_𝑖𝜃_𝑖𝜓_𝑖]^𝑇 denotes the orientation (roll, pitch and yaw angles) of the ellipsoid 𝑖. Moreover, (𝑎_𝑖,𝑏_𝑖,𝑐_𝑖) denote the semiaxes of the ellipsoids 𝑖. These notations are similar to the ellipsoid 𝑗.

Figure 2 

Two ellipsoids and their coordinates.

Lemma 1. Consider two ellipsoids 𝑖 and 𝑗, which have semi-axes of (𝑎_𝑖,𝑏_𝑖,𝑐_𝑖) and (𝑎_𝑗,𝑏_𝑗,𝑐_𝑗), and orientation vectors 𝜼_𝑗=[𝜙_𝑖𝜃_𝑖𝜓_𝑖]^𝑇 and 𝜼_𝑗=[𝜙_𝑗𝜃_𝑗𝜓_𝑗]^𝑇, and are centered at 𝐪_𝑖=[𝑥_𝑖𝑦_𝑖𝑧_𝑖]^𝑇 and 𝐪_𝑗=[𝑥_𝑗𝑦_𝑗𝑧_𝑗]^𝑇, respectively, see Figure 2. Define the transformed distance Δ_𝑖𝑗 between the ellipsoids 𝑖 and 𝑗 as Δ_𝑖𝑗=𝐐_𝑖𝑗𝐪_𝑖𝑗²+𝐐_𝑗𝑖𝐪_𝑗𝑖²2−1, (1) where 𝐐_𝑖𝑗=𝐈_3×3−𝐈_3×3+𝜅_𝑖𝑗𝐓_𝑗⁻¹,𝐪_𝑖𝑗=𝐏_𝑖𝐪_𝑖𝑗, (2) with 𝐈_3×3 being a 3×3 identity matrix, and 𝐪_𝑖𝑗=𝐪_𝑖−𝐪_𝑗,𝐏_𝑖=−𝐀_𝑖⁻¹𝐑⁻¹𝜼_𝑖,𝐀_𝑖=diag𝑎_𝑖,𝑏_𝑖,𝑐_𝑖. (3) The matrix 𝐑(•) represents the three-dimensional rotational matrix with respect to the vector •. The matrix 𝐓_𝑗 is given by 𝐓_𝑗=𝑇_𝑗11𝑇_𝑗12𝑇_𝑗13𝑇_𝑗12𝑇_𝑗22𝑇_𝑗23𝑇_𝑗13𝑇_𝑗23𝑇_𝑗33, (4) where 𝑇_𝑗11=𝑎²₁₁+𝑎²₂₁+𝑎²₃₁,𝑇_𝑗12=𝑎₁₁𝑎₁₂+𝑎₂₁𝑎₂₂+𝑎₃₁𝑎₃₂,𝑇_𝑗22=𝑎²₁₂+𝑎²₂₂+𝑎²₃₂,𝑇_𝑗13=𝑎₁₁𝑎₁₃+𝑎₂₁𝑎₂₃+𝑎₃₁𝑎₃₃,𝑇_𝑗33=𝑎²₁₃+𝑎²₂₃+𝑎²₃₃,𝑇_𝑗23=𝑎₁₂𝑎₁₃+𝑎₂₂𝑎₂₃+𝑎₃₂𝑎₃₃, (5) with 𝑎_𝑚𝑛 for 𝑚=1,2,3 and 𝑛=1,2,3 being the element (𝑚,𝑛) of the matrix (𝐀_𝑖⁻¹𝐑(𝜼_𝑖𝑗)𝐀_𝑗)⁻¹ with 𝜼_𝑖𝑗=𝜼_𝑖−𝜼_𝑗,𝐀_𝑗=diag𝑎_𝑗,𝑏_𝑗,𝑐_𝑗. (6) The variable 𝜅_𝑖𝑗 is the largest root (the right most root) of the following equation: 𝐹_𝑖𝑗𝜅_𝑖𝑗∶=𝐪^𝑇_𝑖𝑗𝐈_3×3+𝜅_𝑖𝑗𝐓_𝑗^−𝑇𝐓_𝑗𝐈_3×3+𝜅_𝑖𝑗𝐓_𝑗⁻¹𝐪_𝑖𝑗−1=0, (7) where (𝐈_3×3+𝜅_𝑖𝑗𝐓_𝑗)^−𝑇 denotes the transpose of (𝐈_3×3+𝜅_𝑖𝑗𝐓_𝑗)⁻¹.
The vector 𝐪_𝑗𝑖 and the matrix 𝐐_𝑗𝑖 are defined accordingly. The two ellipsoids are externally separated, that is, the ellipsoids are outside of each other and do not contact with each other like Figure 2, if Δ_𝑖𝑗>0. (8)

Remark 2. The transformed distance Δ_𝑖𝑗 is symmetric, that is, Δ_𝑖𝑗=Δ_𝑗𝑖. Moreover, Δ_𝑖𝑗 is a smooth function of 𝐪_𝑖𝑗, 𝜼_𝑖, 𝜼_𝑗, and 𝜼_𝑖𝑗. Alternatively, Δ_𝑖𝑗 is a smooth function of 𝐪_𝑖𝑗, 𝐪_𝑗𝑖, and 𝜼_𝑖𝑗.

Proof. see Appendix A.

2.2. Smooth Step Function

This section 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 potential function to avoid discontinuities in the control law due to the agents' communication limitation ranges in solving the collision avoidance problem.

Definition 3. A scalar function ℎ(𝑥,𝑎,𝑏,𝑐) is said to be a smooth step function if it possesses the following properties: (1)ℎ()(],(2)ℎ()[),(3)()(),(4)ℎ()(5)ℎ𝑥,𝑎,𝑏,𝑐=0,∀𝑥∈−∞,𝑎𝑥,𝑎,𝑏,𝑐=1,∀𝑥∈𝑏,∞0<ℎ𝑥,𝑎,𝑏,𝑐<1,∀𝑥∈𝑎,𝑏𝑥,𝑎,𝑏,𝑐issmooth,_′()(),(6)ℎ𝑥,𝑎,𝑏,𝑐>0,∀𝑥∈𝑎,𝑏_′′(𝑥,𝑎,𝑏,𝑐)=0at𝑥=𝑥^∗∈(𝑎,𝑏), (9) where ℎ^′(𝑥,𝑎,𝑏,𝑐)=𝜕ℎ(𝑥,𝑎,𝑏,𝑐)/𝜕𝑥, ℎ^′′(𝑥,𝑎,𝑏,𝑐)=𝜕²ℎ(𝑥,𝑎,𝑏,𝑐)/𝜕𝑥², 𝑎 and 𝑏 are constants such that 𝑎<𝑏, and 𝑐 is a positive constant.

Lemma 4. Let the scalar function ℎ(𝑥,𝑎,𝑏,𝑐) be defined as ℎ()=𝑓(𝜏)𝑥,𝑎,𝑏,𝑐𝑓(𝜏)+𝑐𝑓(1−𝜏)with𝜏=𝑥−𝑎,𝑏−𝑎 (10) where 𝑓(𝜏)=0if𝜏≤0,𝑓(𝜏)=𝑒^−1/𝜏if𝜏>0, (11) with 𝑎 and 𝑏 being constants such that 𝑎<𝑏, and 𝑐 being a positive constant. Then the function ℎ(𝑥,𝑎,𝑏,𝑐) is a smooth step function.

Proof (see [30]). An alternative “symmetric” (i.e., 𝑐=1) smooth step function is available in [14] but it requires a numerical integration. The introduction of the positive constant 𝑐 in the smooth step function in Lemma 4 is to shift the location at which ℎ^′(𝑥,𝑎,𝑏,𝑐) attains its extremum value. An illustration of a smooth step function (𝑎=0,𝑏=3,𝑐=2) is given in Figure 3.

Figure 3 

A smooth step function and its first and second derivatives.

2.3. Barbalat-Like Lemma

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

Lemma 5. Assume that a nonnegative scalar differentiable function 𝑓(𝑡) satisfies the following conditions:(1𝑑)𝑓(𝑡𝑑𝑡)≤𝑘₁𝑓(𝑡)(2),∀𝑡≥0,_∞0𝑓(𝑡)𝑑𝑡≤𝑘₂, (12) where 𝑘₁ and 𝑘₂ are positive constants, then lim_𝑡→∞𝑓(𝑡)=0.

Proof (see [7]). Lemma 5 differs from Barbalat's lemma found in [36]. While Barbalat's lemma assumes that 𝑓(𝑡) is uniformly continuous, Lemma 5 assumes that |(𝑑/𝑑𝑡)𝑓(𝑡)| is bounded by 𝑘₁𝑓(𝑡). Lemma 5 is useful in proving convergence of 𝑓(𝑡) when it is difficult to prove uniform continuity of 𝑓(𝑡).

3. Problem Statement

3.1. Agent Dynamics

As mentioned before, this paper mainly focuses on difficulties caused by the ellipsoidal shape of the agents in designing flocking algorithms, we therefore assume that each ellipsoidal agent 𝑖 has the following dynamics: 𝐪̇_𝑖=𝐮_𝑖,𝜼̇_𝑖=𝝎_𝑖,𝑖∈ℕ, (13) where ℕ is the set of all agents in the group, 𝐮_𝑖=[𝑢_𝑥𝑖𝑢_𝑦𝑖𝑢_𝑧𝑖]^𝑇 and 𝝎_𝑖=[𝜔_𝜙𝑖𝜔_𝜃𝑖𝜔_𝜓𝑖]^𝑇 are the control input vectors of the agent 𝑖. It is recalled that 𝐪_𝑖=[𝑥_𝑖𝑦_𝑖𝑧_𝑖]^𝑇 with (𝑥_𝑖,𝑦_𝑖,𝑧_𝑖) being the position coordinates of the center of the agent 𝑖 and 𝜼_𝑖=[𝜙_𝑖𝜃_𝑖𝜓_𝑖]^𝑇 with (𝜙_𝑖,𝜃_𝑖,𝜓_𝑖) being the orientation angles of the agent 𝑖, see Figure 2. For agents with higher order dynamics, the backstepping technique [37] can be used because we will design the control input vectors 𝐮_𝑖 and 𝝎_𝑖 such that they are smooth.

3.2. Flock Control Objective

In order to design a flocking algorithm for a group of ellipsoidal agents, there is a need to specify a common goal for the group, some communication between the agents, and initial position and orientation of the agents. We therefore impose the following assumption on the flocking rendezvous trajectory, communication, and initial conditions between the agents.

Assumption 1. (1) The agents 𝑖 and 𝑗 have spherical communication spaces, which are centered at the points 𝑂_𝑖 and 𝑂_𝑗, and have radii of 𝑅_𝑖 and 𝑅_𝑗. The radii 𝑅_𝑖 and 𝑅_𝑗 are sufficiently large in the sense that Δ^𝑚_𝑖𝑗𝑅>0, (14) where Δ^𝑚_𝑖𝑗𝑅 is the greatest lower bound of Δ_𝑖𝑗 when the agents 𝑖 and 𝑗 are within their communication ranges, that is, Δ^𝑚_𝑖𝑗𝑅=infΔ_𝑖𝑗𝜼s.t.^𝑖𝑗∈ℝ³,𝐪_𝑖𝑗=min𝑅_𝑖,𝑅_𝑗, (15) for all (𝑖,𝑗)∈ℕ and 𝑗≠𝑖.
(2) The agents 𝑖 and 𝑗 can exchange their trajectories, (𝐪_𝑖,𝜼_𝑖) and (𝐪_𝑗,𝜼_𝑗) if these agents are in their communication spaces.
(3) At the initial time 𝑡₀≥0, all the agents in the group are sufficiently far away from each other in the sense that the following condition holds Δ_𝑖𝑗𝑡₀>0, (16) where Δ_𝑖𝑗(𝑡₀) is given in (1) evaluated at (𝐪_𝑖=𝐪_𝑖(𝑡₀),𝜼_𝑖=𝜼_𝑖(𝑡₀)) and (𝐪_𝑗=𝐪_𝑗(𝑡₀),𝜼_𝑗=𝜼_𝑗(𝑡₀)), and we have abused the notation of Δ_𝑖𝑗(𝐪_𝑖𝑗(𝑡₀),𝜼_𝑖(𝑡₀),𝜼_𝑗(𝑡₀)) as Δ_𝑖𝑗(𝑡₀) for simplicity of presentation.
(4) The flocking rendezvous position and orientation trajectory, 𝐪_𝑜𝑑=[𝑥_𝑜𝑑,𝑦_𝑜𝑑,𝑧_𝑜𝑑]^𝑇 and 𝜼_𝑜𝑑=[𝜙_𝑜𝑑,𝜃_𝑜𝑑,𝜓_𝑜𝑑]^𝑇, for the flock to follow has bounded derivatives 𝐪̇_𝑜𝑑 and 𝜼̇_𝑜𝑑, and is available for all the agents.

Remark 6. (1) In item (1), the condition (14) holds if there exists a positive constant 𝜚_𝑖 such that 𝑅_𝑖≥𝜚_𝑖+sup(𝑎_𝑖+𝑎_𝑗,𝑎_𝑖+𝑏_𝑗,𝑎_𝑖+𝑐_𝑗,𝑏_𝑖+𝑎_𝑗,𝑏_𝑖+𝑏_𝑗,𝑏_𝑖+𝑐_𝑗,𝑐_𝑖+𝑎_𝑗,𝑐_𝑖+𝑏_𝑗,𝑐_𝑖+𝑐_𝑗), for all (𝑖,𝑗)∈ℕ and 𝑗≠𝑖.
(2) Items (1) and (2) in Assumption 1 specify the way each agent communicates with other agents in the group within its communication range. In other words, the agents in the group are connected if they are inside their communication ranges.
(3) Item (3) in Assumption 1 implies from Lemma 1 that at the initial time 𝑡₀ there is no collision between any agents in the group.
(4) Items (1), (2), and (3) in Assumption 1 do not guarantee overall connectivity among all the agents in the group in general. Under item (4) in Assumption 1, we do not require overall connectivity among all the agents to design a flocking algorithm.
(5) Item (4) in Assumption 1 means that all the agents are aware of the flocking rendezvous trajectory. This item together with items (1) and (2) in Assumption 1 were also required in [13] to design a nonfragmentation flocking algorithm for point agents based on an attractive/repulsive potential field. In [13] Olfati-Saber also showed that under items (1) and (2) if all the agents are not aware of the flocking rendezvous trajectory, the flocking algorithm works only for a very restricted set of initial states and a small number of agents. In [21], a further analysis of the flocking algorithm proposed in [13] was carried out for the case where only several agents in a group are aware of the flocking rendezvous trajectory. It was shown in [21] that the agents, which are not aware of the flocking rendezvous trajectory and stay disconnected from the other agents in the group sufficiently long, would stay disconnected from the agents, which are aware of the flocking rendezvous trajectory, that is, the flocking algorithm leads to fragmentation. This represents a real situation. For example, the chance for a bird that has gone away from its flock a sufficiently long time coming back to the flock is very low. Hence, this paper imposes item (4) in Assumption 1 to design a flocking algorithm for elliptical agents. It is possible to use the analysis technique in [21] to analyze the flocking algorithm to be proposed later in this paper for ellipsoidal agents when only several agents are aware of the flocking rendezvous trajectory. This is because the flocking algorithm to be designed in this paper is also based on an attractive/repulsive potential field. However, since this paper focuses on solving difficulties due to an ellipsoidal shape of the agents for collision avoidance in a flocking algorithm, all the agents are assumed to be aware of the flocking rendezvous trajectory.

Flock Control Objective. Under Assumption 1, for each agent 𝑖 design the smooth control input vectors 𝐮_𝑖 and 𝝎_𝑖 to achieve a desired flocking including: (1) no collisions between any agents; (2) asymptotic convergence of each agent's generalized velocity to a desired velocity; (3) boundedness of the flock size 𝐹_𝐿(𝑡)=∑^𝑁_𝑗=𝑖+1∑^𝑖−1_𝑗=1Δ_𝑖𝑗(𝑡) by a constant when the time 𝑡 tends to infinity.

4. Flock Control Design

4.1. Potential Function

4.1.1. Pairwise Potential Function

This section defines and constructs pairwise potential functions that will be used in a Lyapunov function for the flock control design.

Definition 7. Let 𝜑_𝑖𝑗 be a scalar function of the transformed distance Δ_𝑖𝑗 given in (1) between the ellipsoidal agents 𝑖 and 𝑗. The function 𝜑_𝑖𝑗 is said to be a pairwise potential function if it has the following properties: (1)𝜑_𝑖𝑗=𝑘_𝑖𝑗,𝜑^′_𝑖𝑗=0,𝜑^′′_𝑖𝑗=0,∀Δ_𝑖𝑗∈Δ^𝑀_𝑖𝑗𝑅,(2)𝜑,∞_𝑖𝑗>0,∀Δ_𝑖𝑗∈0,Δ^𝑀_𝑖𝑗𝑅,(3)lim_{Δ𝑖𝑗→0}𝜑_𝑖𝑗=∞,lim_{Δ𝑖𝑗→0}𝜑^′_𝑖𝑗(4)𝜑=−∞,_𝑖𝑗issmooth,∀Δ_𝑖𝑗∈(),(5)𝜑0,∞_𝑖𝑗hasauniqueminimumvalueatΔ_𝑖𝑗=Δ_𝑖𝑗𝑑,∀Δ_𝑖𝑗∈0,Δ^𝑀_𝑖𝑗𝑅, (17) where 𝜑^′_𝑖𝑗=𝜕𝜑_𝑖𝑗/𝜕Δ_𝑖𝑗, 𝜑^′′_𝑖𝑗=𝜕²𝜑_𝑖𝑗/𝜕Δ²_𝑖𝑗, and 𝑘_𝑖𝑗 is a positive constant. The positive constant Δ_𝑖𝑗𝑑 is referred to as the desired transformed distance between the agents 𝑖 and 𝑗, and satisfies the condition 0<Δ_𝑖𝑗𝑑<Δ^𝑚_𝑖𝑗𝑅, (18) with Δ^𝑚_𝑖𝑗𝑅 defined in (15). The constant Δ^𝑀_𝑖𝑗𝑅 is the least upper bound of the transformed distance Δ_𝑖𝑗 when the agents 𝑖 and 𝑗 are within their communication ranges, that is, Δ^𝑀_𝑖𝑗𝑅=supΔ_𝑖𝑗𝜼s.t.^𝑖𝑗∈ℝ³,𝐪_𝑖𝑗=min𝑅_𝑖,𝑅_𝑗. (19)

Remark 8. Property (1) implies that the function 𝜑_𝑖𝑗 is constant when the agents 𝑖 and 𝑗 are outside of their communication ranges. Property (2) implies that the function 𝜑_𝑖𝑗 is positive definite when the agents 𝑖 and 𝑗 are inside of their communication ranges. By Lemma 1, Property (3) means that the function 𝜑_𝑖𝑗 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 [36] for continuous systems instead of techniques for switched and discontinuous systems found in [38] to handle the collision avoidance problem under the agents' limited communication ranges. Property (5) makes it effective to use a gradient-based method for the flock control design.

Lemma 9. Let the scalar function 𝜑_𝑖𝑗 be defined as𝜑_𝑖𝑗=Δ_𝑖𝑗−Δ_𝑖𝑗𝑑²1−ℎΔ_𝑖𝑗,𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑗Δ²_𝑖𝑗+𝑘_𝑖𝑗ℎΔ_𝑖𝑗,𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑗, (20) where the positive constants 𝑎_𝑖𝑗 and 𝑏_𝑖𝑗 satisfy the condition 𝑎_𝑖𝑗=Δ_𝑖𝑗𝑑,𝑎_𝑖𝑗<𝑏_𝑖𝑗<Δ^𝑀_𝑖𝑗𝑅, (21) and 𝑐_𝑖𝑗 is a positive constant. The positive constant 𝑘_𝑖𝑗 is chosen such that 𝜑^′_𝑖𝑗>0,∀Δ_𝑖𝑗∈Δ_𝑖𝑗𝑑,Δ^𝑀_𝑖𝑗𝑅, (22) where 𝜑^′_𝑖𝑗=2Δ_𝑖𝑗−Δ_𝑖𝑗𝑑1−ℎ(•)Δ²_𝑖𝑗−Δ_𝑖𝑗−Δ_𝑖𝑗𝑑²ℎ′(•)+2(1−ℎ(•))Δ³_𝑖𝑗+𝑘_𝑖𝑗ℎ′(•). (23) The function ℎ(•) with • stood for (Δ_𝑖𝑗,𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑗) is a smooth step function defined in Definition 3.
Then the function 𝜑_𝑖𝑗 is a pairwise potential function.

Proof (see Appendix B). A pairwise potential function 𝜑_𝑖𝑗 with Δ_𝑖𝑗𝑑=5, Δ^𝑀_𝑖𝑗𝑅=10, 𝑎_𝑖𝑗=Δ_𝑖𝑗𝑑, 𝑏_𝑖𝑗=0.9Δ^𝑀_𝑖𝑗𝑅, and 𝑐_𝑖𝑗=1 is plotted in Figure 4.

Figure 4 

A pairwise potential function.

4.1.2. Potential Function

Having constructed the pairwise potential function 𝜑_𝑖𝑗 for the agents 𝑖 and 𝑗, the potential function 𝜑 for all the agents in the group is the sum of all the pairwise potential functions, that is, 𝜑=^𝑁−1𝑁_{𝑖=1𝑗=𝑖+1}𝜑_𝑖𝑗, (24) where 𝜑_𝑖𝑗 is given in (20).

4.2. Derivative of Potential Function

To prepare for the flock control design later, we calculating the derivative of 𝜑 by differentiating both sides of (24) to obtain 𝜑̇=^𝑁−1𝑁_{𝑖=1𝑗=𝑖+1}𝜑^′_𝑖𝑗Δ̇_𝑖𝑗. (25) It is noted that Δ_𝑖𝑗 is a smooth function of 𝐪_𝑖𝑗, 𝜼_𝑖, 𝜼_𝑗, and 𝜼_𝑖𝑗, see Remark 2. However, there is a difficulty in determining an explicit dependence of Δ_𝑖𝑗 on 𝐪_𝑖𝑗, 𝜼_𝑖, 𝜼_𝑗, and 𝜼_𝑖𝑗 via the matrix 𝐐_𝑖𝑗, see (1) and (2), because 𝜅_𝑖𝑗 cannot be solved explicitly. To avoid this difficulty, we treat Δ_𝑖𝑗 as a smooth function of 𝜅_𝑖𝑗, 𝐪_𝑖𝑗, 𝐪_𝑗𝑖, and 𝜼_𝑖𝑗.

We first calculate the first time derivative of 𝜅_𝑖𝑗. From (7), we have 𝜅̇_𝑖𝑗=−𝜕𝐹_𝑖𝑗𝜕𝜅_𝑖𝑗⁻¹𝜕𝐹_𝑖𝑗𝜕𝐪_𝑖𝑗^𝑇𝐪̇‌_𝑖𝑗+𝜕𝐹_𝑖𝑗𝜕𝜼_𝑖𝑗^𝑇𝜼̇_𝑖𝑗. (26) It is noted that 𝜕𝐹_𝑖𝑗/𝜕𝜅_𝑖𝑗 is always nonzero, see A.10. Hence, the first time derivative of Δ_𝑖𝑗 is Δ̇_𝑖𝑗=𝐆_𝑖𝑗𝐪̇‌_𝑖𝑗+𝐆_𝑗𝑖𝐪̇‌_𝑗𝑖+𝐇_𝑖𝑗𝜼̇_𝑖𝑗+𝐇_𝑗𝑖𝜼̇_𝑗𝑖, (27) where 𝐆_𝑖𝑗=𝜕Δ_𝑖𝑗𝜕𝐪_𝑖𝑗−𝜕Δ_𝑖𝑗𝜕𝜅_𝑖𝑗𝜕𝐹_𝑖𝑗𝜕𝜅_𝑖𝑗⁻¹𝜕𝐹_𝑖𝑗𝜕𝐪_𝑖𝑗^𝑇,𝐆_𝑗𝑖=𝜕Δ_𝑖𝑗𝜕𝐪_𝑗𝑖−𝜕Δ_𝑖𝑗𝜕𝜅_𝑗𝑖𝜕𝐹_𝑗𝑖𝜕𝜅_𝑗𝑖⁻¹𝜕𝐹_𝑗𝑖𝜕𝐪_𝑗𝑖^𝑇,𝐇_𝑖𝑗=𝜕Δ_𝑖𝑗𝜕𝜼_𝑖𝑗−𝜕Δ_𝑖𝑗𝜕𝜅_𝑖𝑗𝜕𝐹_𝑖𝑗𝜕𝜅_𝑖𝑗⁻¹𝜕𝐹_𝑖𝑗𝜕𝜼_𝑖𝑗^𝑇,𝐇_𝑗𝑖=𝜕Δ_𝑖𝑗𝜕𝜼_𝑗𝑖−𝜕Δ_𝑖𝑗𝜕𝜅_𝑗𝑖𝜕𝐹_𝑗𝑖𝜕𝜅_𝑗𝑖⁻¹𝜕𝐹_𝑗𝑖𝜕𝜼_𝑗𝑖^𝑇. (28) From definition of 𝐪_𝑖𝑗 in (2), we have 𝐪̇‌_𝑖𝑗=𝐏_𝑖𝐪̇_𝑖𝑗+𝐏̇_𝑖𝐪_𝑖𝑗,=𝐏_𝑖𝐪̇_𝑖𝑗+𝐒_𝑖𝐪_𝑖𝑗+𝐒_𝑗𝐪_𝑗−𝐪_𝑜𝑑−𝐒_𝑗𝐪_𝑗−𝐪_𝑜𝑑, (29) where 𝐒_𝑖=𝐏_𝑖⁻¹𝐏̇_𝑖 and 𝐒_𝑗=𝐏_𝑗⁻¹𝐏̇_𝑗, and we have added and subtracted 𝐒_𝑗(𝐪_𝑗−𝐪_𝑜𝑑) to the first line of (29) to result in the second line. Now substituting 𝐪_𝑖𝑗=[𝐪_𝑖−𝐪_𝑜𝑑]−[𝐪_𝑗−𝐪_𝑜𝑑] into (29) gives 𝐪̇‌_𝑖𝑗=𝐏_𝑖𝐪̇_𝑖−𝐪̇_𝑜𝑑+𝐒_𝑖𝜼̇_𝑖−𝐪̇_𝑗−𝐪̇_𝑜𝑑+𝐒_𝑗𝜼̇_𝑗−𝐏_𝑖𝐒_𝑖𝑗𝜼̇_𝑖−𝐒_𝑗𝜼̇_𝑗, (30) where the matrices 𝐒_𝑖, 𝐒_𝑗, and 𝐒_𝑖𝑗 are defined by 𝐒_𝑖𝐪_𝑖−𝐪_𝑜𝑑=𝐒_𝑖𝜼̇_𝑖,𝐒_𝑗𝐪_𝑗−𝐪_𝑜𝑑=𝐒_𝑗𝜼̇_𝑗,𝐒_𝑖𝐪_𝑗−𝐪_𝑜𝑑=𝐒_𝑖𝑗𝜼̇_𝑖. (31) Similarly, we have 𝐪̇‌_𝑗𝑖=𝐏_𝑗𝐪̇_𝑗−𝐪̇_𝑜𝑑+𝐒_𝑗𝜼̇_𝑗−𝐪̇_𝑖−𝐪̇_𝑜𝑑+𝐒_𝑖𝜼̇_𝑖−𝐏_𝑗𝐒_𝑗𝑖𝜼̇_𝑗−𝐒_𝑖𝜼̇_𝑖. (32)

Substituting (30) and (32) into (27) results in the following:Δ̇_𝑖𝑗=Γ_𝑖𝑗𝐪̇_𝑖−𝐪̇_𝑜𝑑+𝐒_𝑖𝜼̇_𝑖+Λ_𝑖𝑗𝜼̇_𝑖+Γ_𝑗𝑖𝐪̇_𝑗−𝐪̇_𝑜𝑑+𝐒_𝑗𝜼̇_𝑗+Λ_𝑗𝑖𝜼̇_𝑗, (33) where Γ_𝑖𝑗=𝐆_𝑖𝑗𝐏_𝑖−𝐆_𝑗𝑖𝐏_𝑗,Γ_𝑗𝑖=𝐆_𝑗𝑖𝐏_𝑗−𝐆_𝑖𝑗𝐏_𝑖,Λ_𝑖𝑗=−𝐆_𝑖𝑗𝐏_𝑖𝐒_𝑖𝑗+𝐆_𝑗𝑖𝐏_𝑗𝐒_𝑖+𝐇_𝑖𝑗−𝐇_𝑗𝑖,Λ_𝑗𝑖=−𝐆_𝑗𝑖𝐏_𝑗𝐒_𝑗𝑖+𝐆_𝑖𝑗𝐏_𝑖𝐒_𝑗+𝐇_𝑗𝑖−𝐇_𝑖𝑗. (34)

Substituting (33) into (25) gives 𝜑̇=^𝑁_{𝑖=1𝑗≠𝑖}𝜑^′_𝑖𝑗Γ_𝑖𝑗𝐪̇_𝑖−𝐪̇_𝑜𝑑+𝐒_𝑖𝜼̇_𝑖+_𝑗≠𝑖𝜑^′_𝑖𝑗Λ_𝑖𝑗𝜼̇_𝑖. (35)

Remark 10. The transformed distance Δ_𝑖𝑗 depends not only on the relative distance vector 𝐪_𝑖𝑗 and the relative orientation vector 𝜼_𝑖𝑗 but also on the individual orientation vectors 𝜼_𝑖 and 𝜼_𝑗 of the ellipsoidal agents 𝑖 and 𝑗. This dependence creates a difficulty in designing flocking control algorithms using a gradient-based approach since it is hard to write the derivative of the potential function 𝜑 as a summation of the product of each individual agent's potential gradient and its state derivative. To overcome this difficulty, we have carefully calculated the derivative of the transformed distance by adding and subtracting the term 𝐒_𝑗(𝐪_𝑗−𝐪_𝑜𝑑) to the first line of (29). As a result, the expression of 𝜑̇ has been obtained in a feasible form, see (35), for the flocking control design later.
The expression of 𝜑̇ in (35) deserves some discussion. If we define 𝜑=∑^𝑁_𝑖=1𝜑_𝑖, where 𝜑_𝑖 is the potential function for each agent 𝑖. We then have 𝜑̇_𝑖=(∑_𝑗≠𝑖𝜑^′_𝑖𝑗Γ_𝑖𝑗)(𝐪̇_𝑖−𝐪̇_𝑜𝑑+𝐒_𝑖𝜼̇_𝑖)+(∑_𝑗≠𝑖𝜑^′_𝑖𝑗Λ_𝑖𝑗)𝜼̇_𝑖, where the term (∑_𝑗≠𝑖𝜑^′_𝑖𝑗Γ_𝑖𝑗) can be regarded as the gradient of 𝜑_𝑖 with respect to the relative distance, which is rotated by the orientation vector 𝜼_𝑖, from the agent 𝑖 to the rendezvous trajectory 𝐪_𝑜𝑑. The term (∑_𝑗≠𝑖𝜑^′_𝑖𝑗Λ_𝑖𝑗) can be considered as the gradient of 𝜑_𝑖 with respect to 𝜼_𝑖.

4.3. Lyapunov Function

Since the derivative of the potential function 𝜑 is the summation of the term (∑_𝑗≠𝑖𝜑^′_𝑖𝑗Γ_𝑖𝑗)(𝐪̇_𝑖−𝐪̇_𝑜𝑑+𝐒_𝑖𝜼̇_𝑖) instead of (∑_𝑗≠𝑖𝜑^′_𝑖𝑗Γ_𝑖𝑗)(𝐪̇_𝑖−𝐪̇_𝑜𝑑), and the term (∑_𝑗≠𝑖𝜑^′_𝑖𝑗Λ_𝑖𝑗)𝜼̇_𝑖 instead of (∑_𝑗≠𝑖𝜑^′_𝑖𝑗Λ_𝑖𝑗)(𝜼̇_𝑖−𝜼̇_𝑜𝑑), it is not possible to use a Lyapunov function candidate as a summation of the potential function 𝜑 and the square of all errors 𝐪_𝑖−𝐪_𝑜𝑑 and 𝜼_𝑖−𝜼_𝑜𝑑 for the flock control design.

To overcome the aforementioned impossibilities, we will construct a Lyapunov function candidate as a sum of the potential function 𝜑 in (24) and the square of errors 𝐏_𝑖(𝐪_𝑖−𝐪_𝑜𝑑) and 𝜼_𝑖−𝜼_𝑖𝑑. The vector 𝜼_𝑖𝑑 is considered as the virtual rendezvous orientation vector to be designed such that lim_𝑡→∞𝜼_𝑖𝑑(𝑡)=𝜼_𝑜𝑑(𝑡). As such, the Lyapunov function candidate for the flock control design in the next section is constructed as follows: 1𝑉=𝜑+2^𝑁_𝑖=1𝑐₁𝐏_𝑖𝐪_𝑖−𝐪_𝑜𝑑²+𝑐₂𝜼_𝑖−𝜼_𝑖𝑑², (36) where 𝑐₁ and 𝑐₂ are positive constants. Differentiating both sides of (36) along the solutions of (35) and recalling from (13) that 𝐪̇_𝑖=𝐮_𝑖 and 𝜼̇_𝑖=𝝎_𝑖 results in the folowing: 𝑉̇=^𝑁_𝑖=1Ω^𝑇_𝑖𝐮_𝑖−𝐪̇_𝑜𝑑+𝐒_𝑖𝝎_𝑖+Ξ^𝑇_𝑖𝝎_𝑖−𝜼̇_𝑖𝑑+_𝑗≠𝑖𝜑^′_𝑖𝑗Λ_𝑖𝑗𝜼̇_𝑖𝑑, (37) where we added and subtracted ∑^𝑁_𝑖=1(∑_𝑗≠𝑖𝜑^′_𝑖𝑗Λ_𝑖𝑗)𝜼̇_𝑖𝑑 to the right hand side of the equation (35) before substituting this equation into the derivative of 𝑉. In (37), we have defined Ω_𝑖=_𝑗≠𝑖𝜑^′_𝑖𝑗Γ_𝑖𝑗+𝑐₁𝐏_𝑖𝐪_𝑖−𝐪_𝑜𝑑^𝑇𝐏_𝑖^𝑇,Ξ_𝑖=_𝑗≠𝑖𝜑^′_𝑖𝑗Λ_𝑖𝑗+𝑐₂𝜼_𝑖−𝜼_𝑖𝑑^𝑇𝑇. (38)

4.4. Control Law

We first deal with the terms inside the square bracket in the right hand side of (37). As such, to avoid a large control effort when an agent in the group is close to the agent 𝑖 due to Property (3) of the function 𝜑_𝑖𝑗, see (17), for collision avoidance, we design a control law for 𝐮_𝑖 and 𝝎_𝑖 as follows: 𝐮_𝑖=−𝑘₁𝐖Ω_𝑖+𝐪̇_𝑜𝑑−𝐒_𝑖−𝑘₂𝐖Ξ_𝑖+𝜼̇_𝑖𝑑,𝝎_𝑖=−𝑘₂𝐖Ξ_𝑖+𝜼̇_𝑖𝑑, (39) where 𝑘₁ and 𝑘₂ are positive constants. The vector 𝐖(𝝌) denotes a vector of bounded and differentiable functions of elements of 𝝌 in the sense that 𝐖(𝝌)=[𝑤(𝜒₁)…,𝑤(𝜒_𝑙),…,𝑤(𝜒_𝑛)]^𝑇 with 𝜒_𝑙 the 𝑙^𝑡ℎ element of 𝝌, that is, 𝝌=[𝜒₁…,𝜒_𝑙,…,𝜒_𝑛]^𝑇. The function 𝑤(𝜒) is a scalar, differentiable and bounded function, and satisfies (1)𝑤𝜒≤𝑀₁,(2)𝑤𝜒𝜒(3)𝑤𝜒,𝑤𝜒(𝜔(4𝑤𝜒=0if𝜒=0,𝜒𝑤>0if𝜒≠0,−𝜒=−𝑤𝜒−𝜔−𝑤)≥0,)𝜒≤𝑀₂,𝜕𝑤𝜒𝜕𝜒≤𝑀₃,𝜕𝑤𝜒𝜕𝜒_𝜒=0=1, (40) for all 𝜒∈ℝ,𝜔∈ℝ, where 𝑀₁,𝑀₂,𝑀₃ are positive constants. Some functions that satisfy the above properties are arctan(𝜒) and tanh(𝜒).

We now deal with the term (∑_𝑗≠𝑖𝜑^′_𝑖𝑗Λ_𝑖𝑗)𝜼̇_𝑖𝑑 in the right hand side of (37). This term seems to be troublesome because 𝜼̇_𝑖𝑑 is nonzero in general since we are considering a time-varying rendezvous trajectory. To get around this problem, we will design an update law 𝜼̇_𝑖𝑑 such that _𝑗≠𝑖𝜑^′_𝑖𝑗Λ_𝑖𝑗𝜼̇_𝑖𝑑=0, (41) holds for all time. Moreover, it is desired to have the virtual rendezvous orientation vector 𝜼_𝑖𝑑 asymptotically approached the desired rendezvous orientation vector 𝜼_𝑜𝑑. As such, we utilize the smooth step function to design an update law 𝜼̇_𝑖𝑑 as follows: 𝜼̇_𝑖𝑑=_𝑗≠𝑖ℎΔ_𝑖𝑗,𝑎_𝑖𝑗𝑑,𝑏_𝑖𝑗𝑑,𝑐_𝑖𝑗𝑑−𝑘_𝑑𝜼_𝑖𝑑−𝜼_𝑜𝑑+𝜼̇_𝑜𝑑, (42) where 𝑘_𝑑 is a positive constant, and 𝜼_𝑖𝑑(𝑡₀)=𝜼_𝑜𝑑(𝑡₀). The function ℎ(Δ_𝑖𝑗,𝑎_𝑖𝑗𝑑,𝑏_𝑖𝑗𝑑,𝑐_𝑖𝑗𝑑) is a smooth step function with the constants 𝑎_𝑖𝑗𝑑, 𝑏_𝑖𝑗𝑑, and 𝑐_𝑖𝑗𝑑 chosen as 𝑎_𝑖𝑗𝑑=𝑏_𝑖𝑗,𝑎_𝑖𝑗𝑑<𝑏_𝑖𝑗𝑑≤Δ^𝑀_𝑖𝑗𝑅,𝑐_𝑖𝑗𝑑>0, (43) where 𝑏_𝑖𝑗 is chosen as in (21), and Δ^𝑀_𝑖𝑗𝑅 is given in (19). Using properties of the smooth step function, the choice of the constants 𝑎_𝑖𝑗, 𝑏_𝑖𝑗, 𝑎_𝑖𝑗𝑑, and 𝑏_𝑖𝑗𝑑 in (21) and (43) results in ℎ^′(Δ_𝑖𝑗,𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑗)ℎ(Δ_𝑖𝑗,𝑎_𝑖𝑗𝑑,𝑏_𝑖𝑗𝑑,𝑐_𝑖𝑗𝑑)=0 and (1−ℎ(Δ_𝑖𝑗,𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑗))ℎ(Δ_𝑖𝑗,𝑎_𝑖𝑗𝑑,𝑏_𝑖𝑗𝑑,𝑐_𝑖𝑗𝑑)=0. These equalities imply that (41) holds as long as Δ_𝑖𝑗>0, which is to be guaranteed by our control design. Moreover, the choice of the constants 𝑎_𝑖𝑗𝑑 and 𝑏_𝑖𝑗𝑑 in (43) ensures that the function ℎ(Δ_𝑖𝑗,𝑎_𝑖𝑗𝑑,𝑏_𝑖𝑗𝑑,𝑐_𝑖𝑗𝑑) approaches 1 whenever Δ_𝑖𝑗 approaches a value larger than 𝑏_𝑖𝑗𝑑. The inequality Δ_𝑖𝑗>0 and the limit lim_𝑡→∞(∑_𝑗≠𝑖𝜑^′_𝑖𝑗(𝑡)Λ_𝑖𝑗(𝑡))=0 will be guaranteed by our designed control input vectors 𝐮_𝑖 and 𝝎_𝑖 in (39). This will be shown in the proof of the main result.

Remark 11. (1) The control vectors 𝐮_𝑖 and 𝝎_𝑖 in (39) of the agent 𝑖 are smooth and depend on only its own state and the rendezvous trajectory, and the states of other agents 𝑗 in the communication range of the agent 𝑖 due to Property (1) of the pairwise potential function 𝜑_𝑖𝑗 in (17).
(2) The update law 𝜼̇_𝑖𝑑 in (42) ensures that when the collision avoidance is active, the virtual rendezvous orientation vector 𝜼_𝑖𝑑 is not updated. This implies that the control vectors 𝐮_𝑖 and 𝝎_𝑖 give priority to the collision avoidance mission or the rendezvous orientation tracking mission whenever which mission is more important.
Substituting the control vectors 𝐮_𝑖 and 𝝎_𝑖 in (39) and the update law 𝜼̇_𝑖𝑑 in (42) into (37) results in 𝑉̇=−^𝑁_𝑖=1𝜗_𝑖, (44) where 𝜗_𝑖=𝑘₁Ω^𝑇_𝑖𝐖Ω_𝑖+𝑘₂Ξ^𝑇_𝑖𝐖Ξ_𝑖. (45) On the other hand, substituting the control vectors 𝐮_𝑖 and 𝝎_𝑖 in (39) and the update law 𝜼̇_𝑖𝑑 in (42) into (13) results in the closed-loop system: 𝐪̇_𝑖=−𝑘₁𝐖Ω_𝑖+𝐪̇_𝑜𝑑−𝐒_𝑖−𝑘₂𝐖Ξ_𝑖+𝜼̇_𝑖𝑑,𝜼̇_𝑖=−𝑘₂𝐖Ξ_𝑖+𝜼̇_𝑖𝑑, (46) for all 𝑖∈ℕ. We now present the main result of our paper in the following theorem.

Theorem 12. Under Assumption 1, the smooth control vectors 𝐮_𝑖 and 𝝎_𝑖 in (39) and the update law 𝜼̇_𝑖𝑑 in (42) for the agent 𝑖 solve the flocking control objective. In particular, the following results hold
(1) There are no collisions between any agents and the closed-loop system (46) is forward complete;
(2) The relative distance between each agent 𝑖 and the flocking rendezvous trajectory 𝐪_𝑜𝑑 is bounded, that is, ‖𝐪_𝑖(𝑡)−𝐪_𝑜𝑑(𝑡)‖≤𝐴₀ for all 𝑡≥𝑡₀≥0 with 𝐴₀ a constant depending on the initial conditions;
(3) The generalized velocity of each agent 𝑖 asymptotically tends to the generalized flocking rendezvous velocity, that is, lim_𝑡→∞𝐪̇_𝑖−𝐪̇_𝑜𝑑+𝐒_𝑖𝜼̇_𝑖𝑑=0, (47) where 𝜼̇_𝑖𝑑 is given in (42).
(4) The flock size 𝐹_𝐿(𝑡)=∑^𝑁_𝑗=𝑖+1∑^𝑖−1_𝑗=1Δ_𝑖𝑗(𝑡) is bounded by a positive constant as time tends to infinity, that is, lim_𝑡→∞𝐹_𝐿(𝑡)=𝐹_𝐿𝑐, (48) with 𝐹_𝐿𝑐≤𝑐₀ where 𝑐₀ is a positive constant.

Proof. See Appendix C.

5. Simulation Results

In this section, we provide a numerical simulation to illustrate the effectiveness of the proposed flocking control design stated in Theorem 12. We use 𝑁=6 ellipsoidal agents with the geometric parameters as 𝑎_𝑖=3 and 𝑏_𝑖=𝑐_𝑖=1 for all 𝑖=1,…,𝑁. The initial position and orientation of these agents are chosen as follows: 𝐪_𝑖(0)=15[cos(2𝜋(𝑖−1)/𝑁)sin(2𝜋(𝑖−1)/𝑁)0]^𝑇 and 𝜼_𝑖(0)=[000]^𝑇. All the agents have the same communication range of 𝑅_𝑖=25. The control design parameters are chosen as follows: 𝑘_𝑖𝑗=10, Δ_𝑖𝑗𝑑=5, 𝑎_𝑖𝑗=Δ_𝑖𝑗𝑑, 𝑏_𝑖𝑗=1.5𝑎_𝑖𝑗, 𝑐₁=1, 𝑐₂=1, 𝑘₁=20, 𝑘₂=20, 𝑎_𝑖𝑗𝑑=𝑏_𝑖𝑗, 𝑏_𝑖𝑗𝑑=1.2𝑎_𝑖𝑗𝑑, 𝑘_𝑑=1. The function 𝑤(⋅) is chosen as arctan(⋅). The flocking rendezvous trajectory is chosen as 𝐪̇_𝑜𝑑=[111]^𝑇 with 𝐪_𝑜𝑑(0)=[0,0,0]^𝑇. This choice implies that the flocking rendezvous trajectory is a angled straight line. A calculation shows that the above initial conditions and the above choice of control design parameters satisfy all the conditions (14), (16), (18), (21), (22).

Simulation results are plotted in Figures 5 and 6. In Figure 5, several snapshots of the position and orientation of all agents are plotted. The representative Δ^∗_𝑖𝑗=(Π_{𝑗∈ℕ𝑗≠𝑖}Δ_𝑖𝑗)^1/5 is plotted in the first subfigure of Figure 6. The control inputs 𝐮=[𝐮₁,…,𝐮_𝑖,…,𝐮_𝑁]^𝑇 and 𝝎=[𝝎₁,…,𝝎_𝑖,…,𝝎_𝑁]^𝑇 are plotted in the second and third subfigures of Figure 6. It is clearly seen from Figures 5 and 6 that there is no collision between any agents as indicated by Δ^∗_𝑖𝑗>0 for all 𝑖∈ℕ. Moreover, all the agents manage to track the generalized flocking rendezvous trajectory 𝐪_𝑜𝑑. The mismatched velocity can be seen from the second and third subfigures of Figure 6.

Figure 5 

Snapshots of the agents' position and orientation.

(a)

(b)

(c)

Figure 6 

Representative Δ∗𝑖𝑗 and control inputs.

6. Conclusions

Flocking of 𝑁 mobile agents with an ellipsoidal shape and limited communication ranges was studied. The flock control design is based on a separation condition for ellipsoidal agents, smooth step functions, and novel pairwise potential functions. The proposed flock algorithms achieved desired flocking behaviors including smooth controllers despite of agents' limited communication ranges, no collisions between any agents, asymptotic convergence of each agent's velocity to a desired velocity, and boundedness of the flock size by a constant. The keys to success of our proposed flocking algorithm include the symmetric transformed distance Δ_𝑖𝑗 between two ellipsoids, smooth cut-off pairwise potential function 𝜑_𝑖𝑗, and a careful derivation of the derivative of the potential function 𝜑. An extension of the proposed flock control design and those controllers designed for single underactuated underwater vehicles in [39] to provide a flock control system for multiple underactuated underwater vehicles is under consideration.

Appendix

A. Proof of Lemma 1

From Figure 2, the boundaries of the ellipsoids 𝑖 and 𝑗 (equations of the points 𝑤_𝑖𝑏 and 𝑤_𝑗𝑏) coordinated in the 𝑂_𝑏𝑖𝑋_𝑏𝑖𝑌_𝑏𝑖𝑍_𝑏𝑖 frame attached to the ellipsoid 𝑖 can be described by 𝐸_𝑖∶𝐪^𝑇_𝑖𝑏𝐀_𝑖⁻²𝐪_𝑖𝑏𝐸=1,_𝑗∶𝐪_𝑗𝑏=−𝐑⁻¹𝜼_𝑖𝐪_𝑖𝑗+𝐑⁻¹𝜼_𝑖𝑗𝐀_𝑗𝝔_𝑗, (A.1) where 𝐀_𝑖, 𝐀_𝑗, 𝐪_𝑖𝑗, and 𝜼_𝑖𝑗 are defined in (3) and (6) and 𝝔_𝑗=[cos(𝛼_𝑗)cos(𝛽_𝑗)cos(𝛼_𝑗)sin(𝛽_𝑗)sin(𝛼_𝑗)]^𝑇 with 𝛼_𝑗∈[−𝜋/2,𝜋/2] and 𝛽_𝑗∈[−𝜋,𝜋] are auxiliary angles and 𝐪_𝑖𝑏 and 𝐪_𝑗𝑏 are vectors denoting position of the points 𝑤_𝑖𝑏 and 𝑤_𝑗𝑏, respectively. The ideas to prove Lemma 1 consists of two steps: (1) transforming the ellipsoids 𝑖 and 𝑗 to a unit sphere and an ellipsoid; (2) calculating the distance between the transformed sphere and the transformed ellipsoid.

A.1. Transformation

We transform the ellipsoids 𝑖 and 𝑗 to a unit sphere and an ellipsoid by the following coordinate transformation: 𝐪_𝑖𝑏=𝐀_𝑖⁻¹𝐪_𝑖𝑏+𝐑⁻¹𝜼_𝑖𝐪_𝑖𝑗,𝐪_𝑗𝑏=𝐀_𝑖⁻¹𝐪_𝑗𝑏+𝐑⁻¹𝜼_𝑖𝐪_𝑖𝑗. (A.2) With the above coordinate transformation, the ellipsoids (A.1) are transformed to a unit sphere and an ellipsoid as follows: 𝐸_𝑖∶𝐪_𝑖𝑏+𝐪_𝑖𝑗^𝑇𝐪_𝑖𝑏+𝐪_𝑖𝑗=1,𝐸_𝑗∶𝐪_𝑗𝑏=𝐀_𝑖⁻¹𝐑𝜼_𝑖𝑗𝐀_𝑗𝝔_𝑗. (A.3) Now, the ellipsoid 𝐸_𝑖 has become the unit sphere 𝐸_𝑖 centered at the point 𝑂_𝑢𝑠𝑖 whose coordinates are described by the first equation in (A.3). The ellipsoid 𝐸_𝑗 has become another ellipsoid 𝐸_𝑗 centered at the origin of the 𝑂_𝑏𝑖𝑋_𝑏𝑖𝑌_𝑏𝑖𝑍_𝑏𝑖 frame, that is, the point 𝑂_𝑏𝑖.

For convenience of calculating the distance between the unit sphere 𝐸_𝑖 and the ellipsoid 𝐸_𝑗, we will rewrite the ellipsoid 𝐸_𝑗 in an implicit form instead of parametric form given in the second equation of (A.3). By squaring both sides of each row of 𝐪_𝑗𝑏=𝐀_𝑖⁻¹𝐑(𝜼_𝑖𝑗)𝐀_𝑗𝝔_𝑗 then adding the results together, we have 𝐪^𝑇_𝑗𝑏𝐓_𝑗𝐪_𝑗𝑏=1, where 𝐓_𝑗 is defined in (4). Hence, the unit sphere and the ellipsoid defined in (A.3) can be rewritten as 𝐸_𝑖∶𝐪_𝑖𝑏+𝐪_𝑖𝑗^𝑇𝐪_𝑖𝑏+𝐪_𝑖𝑗=1,𝐸_𝑗∶𝐪^𝑇_𝑗𝑏𝐓_𝑗𝐪_𝑗𝑏=1. (A.4)

A.2. Distance Δ_𝑖𝑗

We now calculate the distance from the center of the unit sphere 𝐸_𝑖 described by the first equation in (A.4), that is, from the point 𝑂_𝑢𝑠𝑖 to the ellipsoid 𝐸_𝑗 described by the second equation in (A.4). A necessary condition for a point 𝐪_𝑗𝑏 to be the closest point to the point 𝑂_𝑢𝑠𝑖 is that 𝐪_𝑖𝑗−𝐪_𝑗𝑏 is perpendicular to the tangent plane to the ellipsoid 𝐸_𝑗 at 𝐪_𝑗𝑏. Since the surface gradient 𝜕𝑓(𝐪_𝑗𝑏)/𝜕𝐪_𝑗𝑏 with 𝑓(𝐪_𝑗𝑏)∶=(1/2)(𝐪^𝑇_𝑗𝑏𝐓_𝑗𝐪_𝑗𝑏−1) is normal to the ellipsoid's surface, the algebraic condition for the closest point 𝐪_𝑖𝑏 is 𝐪_𝑖𝑗−𝐪_𝑗𝑏=𝜅_𝑖𝑗𝜕𝑓𝐪_𝑗𝑏𝜕𝐪_𝑗𝑏. (A.5) For the point 𝑂_𝑢𝑠𝑖 outside the ellipsoid 𝐸_𝑗, there is only one point on the ellipsoid whose normal points toward the point 𝑂_𝑢𝑠𝑖. However, there can be as many as five other points whose surface normals point directly away from 𝑂_𝑢𝑠𝑖. The point on the ellipsoid whose normal points toward the point 𝑂_𝑢𝑠𝑖 corresponds to the largest root 𝜅_𝑖𝑗 of (A.5). Moreover, 𝐪_𝑗𝑏 must satisfy the ellipsoid equation, that is, the second equation of (A.4).

From (A.5), we have 𝐪_𝑗𝑏=𝐈_3×3+𝜅_𝑖𝑗𝐓_𝑗⁻¹𝐪_𝑖𝑗, (A.6) which is substituted into the second equation in (A.4) results in (7).

Now the distance from the point 𝑂_𝑢𝑠𝑖 to the closest point 𝐪_𝑗𝑏 on the ellipsoid 𝐸_𝑗 described by the second equation in (A.4) is given by 𝑑_𝑖𝑗=𝐪_𝑖𝑗−𝐪_𝑗𝑏−1, (A.7) where 𝐪_𝑗𝑏 is the solution of (A.5) and the second equation of (A.4) with 𝜅_𝑖𝑗 being the largest root. Substituting 𝐪_𝑗𝑏 in (A.6) and 𝐪_𝑖𝑗 in (2) into (A.7) results in the following: 𝑑_𝑖𝑗=𝐐_𝑖𝑗𝐪_𝑖𝑗−1, (A.8) where 𝐐_𝑖𝑗 is given in (2).

It is noted that after 𝐪_𝑗𝑏 is found, we can determine the intersection point with coordinates 𝐪_𝑖𝑏 on the unit sphere 𝐸_𝑖 between the line from the point 𝑂_𝑢𝑠𝑖 to the point with coordinates 𝐪_𝑗𝑏 and the unit sphere 𝐸_𝑖. Once we obtain the coordinates 𝐪_𝑖𝑏 and 𝐪_𝑗𝑏 with respect to the closest distance between the unit sphere 𝐸_𝑖 and the transformed ellipsoid 𝐸_𝑗, the corresponding coordinates 𝐪_𝑖𝑏 and 𝐪_𝑗𝑏 with respect to the shortest distance between the original ellipsoids 𝐸_𝑖 and 𝐸_𝑗 on the original ellipsoids 𝐸_𝑖 and 𝐸_𝑗 can be directly determined from (A.2). The actual distance between the original ellipsoids 𝐸_𝑖 and 𝐸_𝑗 is ‖𝐪_𝑖𝑏−𝐪_𝑗𝑏‖. For a flocking control application, the transformed distance 𝑑_𝑖𝑗 is sufficient because from the transformation (A.2) we can see that 𝑑_𝑖𝑗>0 implies that ‖𝐪_𝑖𝑏−𝐪_𝑗𝑏‖>0 and vice versa.

Similarly, we transform the two ellipsoids 𝑗 and 𝑖 to a unit sphere 𝐸_𝑗 and an ellipsoid 𝐸_𝑖. The distance from the center of the unit sphere 𝐸_𝑗, whose coordinates are 𝐪_𝑗𝑖, to the closest point 𝐪_𝑖𝑏 on the ellipsoid 𝐸_𝑖 (which is transformed from the ellipsoid 𝑖) is 𝑑_𝑗𝑖=𝐐_𝑗𝑖𝐪_𝑗𝑖−1. (A.9) Therefore, two ellipsoids 𝑖 and 𝑗 are separated if either 𝑑_𝑖𝑗>0 or 𝑑_𝑗𝑖>0. Moreover, 𝑑_𝑖𝑗>0 implies 𝑑_𝑗𝑖>0 and vice versa. Hence it is sufficient that the two ellipsoids are separated if the condition (8) holds. The reason why we use the distance Δ_𝑖𝑗 in (1) instead of Δ_𝑖𝑗=𝑑_𝑖𝑗 or Δ_𝑖𝑗=𝑑_𝑗𝑖 is to create a symmetrical Δ_𝑖𝑗, see Remark 2. This is crucial for the success of our flocking design.

A.3. Solution of (7)

We first show that (7) has a unique root on the domain of interest. Let us define 𝜅^𝐿_𝑖𝑗 be the largest root of the equation det(𝐈_3×3+𝜅_𝑖𝑗𝐓_𝑗)=0. This is a cubic equation and can be solved for its roots explicitly. We now observe that for any 𝐪_𝑖𝑗 such that ‖𝐪_𝑖𝑗‖>0, the following inequalities and limits hold 𝜕𝐹𝜅_𝑖𝑗𝜕𝜅_𝑖𝑗𝜕<0,²𝐹𝜅_𝑖𝑗𝜕𝜅²_𝑖𝑗>0,∀𝜅_𝑖𝑗∈𝜅^𝐿_𝑖𝑗,∞,lim_{𝜅𝑖𝑗→𝜅𝐿𝑖𝑗}𝐹_𝑖𝑗𝜅_𝑖𝑗=∞,lim_{𝜅𝑖𝑗→∞}𝐹_𝑖𝑗𝜅_𝑖𝑗=−1, (A.10) because the matrix 𝐓_𝑗 is symmetric and positive definite with its elements given in (5). Properties of 𝐹_𝑖𝑗(𝜅_𝑖𝑗) in (A.10) imply that the function 𝐹_𝑖𝑗(𝜅_𝑖𝑗) is strictly decreasing from ∞ to −1 on the domain 𝜅_𝑖𝑗∈(𝜅^𝐿_𝑖𝑗,∞). Therefore, (7) has a unique root on the domain of interest. Moreover, this root is also the largest root of (7).

Given an initial value 𝜅_𝑖𝑗(0)=𝜅^𝐿_𝑖𝑗+𝜖, where 𝜖 is a positive constant such that 𝐹_𝑖𝑗(𝜅^𝐿_𝑖𝑗+𝜖)>0, a numerical procedure using the Newton method to calculate the largest root 𝜅_𝑖𝑗 is given as follows [40]: 𝜅_𝑖𝑗(𝑛+1)=𝜅_𝑖𝑗(𝑛)−𝐹_𝑖𝑗𝜅_𝑖𝑗(𝑛)𝐹^′_𝑖𝑗𝜅_𝑖𝑗(𝑛), (A.11) where 𝐹^′_𝑖𝑗(𝜅_𝑖𝑗(𝑛))=𝜕𝐹_𝑖𝑗(𝜅_𝑖𝑗)/𝜕𝜅_𝑖𝑗|_{𝜅𝑖𝑗=𝜅𝑖𝑗(𝑛)} and 𝐹_𝑖𝑗(𝜅_𝑖𝑗(𝑛))=𝐹_𝑖𝑗(𝜅_𝑖𝑗)|_{𝜅𝑖𝑗=𝜅𝑖𝑗(𝑛)} with 𝐹_𝑖𝑗(𝜅_𝑖𝑗) given in (7). The algorithm (A.11) provides a quadratic convergence of 𝜅_𝑖𝑗(𝑛) to the largest root 𝜅_𝑖𝑗 of (7), since 𝜕𝐹(𝜅_𝑖𝑗)/𝜕𝜅_𝑖𝑗 is nonzero and 𝜕²𝐹(𝜅_𝑖𝑗)/𝜕𝜅²_𝑖𝑗 is bounded in the domain of interest, see Theorem 1.1 in [40] for a proof. Indeed, after the largest root 𝜅_𝑖𝑗 is found, 𝐪_𝑗𝑏 is obtained from (A.6). Proof of Lemma 1 is completed.

B. Proof of Lemma 9

We first show that there exists a positive constant 𝑘_𝑖𝑗 such that the condition (22) holds. To do so, we partition the interval (Δ_𝑖𝑗𝑑,Δ^𝑀_𝑖𝑗𝑅) into two intervals (Δ_𝑖𝑗𝑑,Δ^∗_𝑖𝑗] and (Δ^∗_𝑖𝑗,Δ^𝑀_𝑖𝑗𝑅) with Δ^∗_𝑖𝑗 being such that Δ_𝑖𝑗−Δ_𝑖𝑗𝑑−Δ_𝑖𝑗−Δ_𝑖𝑗𝑑²Δ_{𝑖𝑗∀Δ𝑖𝑗∈(Δ𝑖𝑗𝑑,Δ∗𝑖𝑗]}>0. (B.1) It is trivial to show that there exists Δ^∗_𝑖𝑗∈(Δ_𝑖𝑗𝑑,Δ^𝑀_𝑖𝑗𝑅) such that (B.1) holds.

For all Δ_𝑖𝑗∈(Δ_𝑖𝑗𝑑,Δ^∗_𝑖𝑗], we rewrite (23) as 𝜑^′_𝑖𝑗(•)=21−ℎΔ²_𝑖𝑗Δ_𝑖𝑗−Δ_𝑖𝑗𝑑−Δ_𝑖𝑗−Δ_𝑖𝑗𝑑²Δ_𝑖𝑗+ℎ^′(•)𝑘_𝑖𝑗−Δ_𝑖𝑗−Δ_𝑖𝑗𝑑²Δ²_𝑖𝑗. (B.2) Since (1−ℎ(•))/Δ²_𝑖𝑗>0 and ℎ^′(•)>0 for all Δ_𝑖𝑗∈(Δ_𝑖𝑗𝑑,Δ^∗_𝑖𝑗], we choose the constant 𝑘_𝑖𝑗 such that 𝑘_𝑖𝑗Δ≥sup^𝑖𝑗−Δ_𝑖𝑗𝑑²Δ²_𝑖𝑗,∀Δ_𝑖𝑗∈Δ_𝑖𝑗𝑑,Δ^∗_𝑖𝑗. (B.3) The above choice of 𝑘_𝑖𝑗 makes the condition (22) hold for all Δ_𝑖𝑗∈(Δ_𝑖𝑗𝑑,Δ^∗_𝑖𝑗].

For all Δ_𝑖𝑗∈(Δ^∗_𝑖𝑗,Δ^𝑀_𝑖𝑗𝑅), we rewrite (23) as𝜑^′_𝑖𝑗=2Δ_𝑖𝑗−Δ_𝑖𝑗𝑑1−ℎ(•)Δ²_𝑖𝑗+ℎ^′(•)𝑘_𝑖𝑗−Δ_𝑖𝑗−Δ_𝑖𝑗𝑑²Δ²_𝑖𝑗2((•1+1−ℎ))ℎ^′(•). (B.4)

Since (Δ_𝑖𝑗−Δ_𝑖𝑗𝑑)((1−ℎ(•))/Δ²_𝑖𝑗)>0 and ℎ^′(•)>0 for all Δ_𝑖𝑗∈(Δ^∗_𝑖𝑗,Δ^𝑀_𝑖𝑗𝑅), we can choose the constant 𝑘_𝑖𝑗 such that 𝑘_𝑖𝑗Δ≥sup^𝑖𝑗−Δ_𝑖𝑗𝑑²Δ²_𝑖𝑗2((•1+1−ℎ))ℎ^′(•),∀Δ_𝑖𝑗∈Δ_𝑖𝑗∈Δ^∗_𝑖𝑗,Δ^𝑀_𝑖𝑗𝑅. (B.5)

The above choice of 𝑘_𝑖𝑗 makes condition (22) hold for all Δ_𝑖𝑗∈(Δ^∗_𝑖𝑗,Δ^𝑀_𝑖𝑗𝑅). Since 𝑎_𝑖𝑗 and 𝑏_𝑖𝑗 satisfy condition (21), and Δ^∗_𝑖𝑗∈(Δ_𝑖𝑗𝑑,Δ^𝑀_𝑖𝑗𝑅), we have (1−ℎ(•))/(ℎ^′(•)) is bounded by some constant for all Δ_𝑖𝑗∈(Δ^∗_𝑖𝑗,Δ^𝑀_𝑖𝑗𝑅). This implies that the value of 𝑘_𝑖𝑗 satisfying (B.5) is finite.

Therefore, the positive constant 𝑘_𝑖𝑗 that satisfies both conditions (B.3) and (B.5) makes condition (22) holds for all Δ_𝑖𝑗∈(Δ_𝑖𝑗𝑑,Δ^𝑀_𝑖𝑗𝑅).

To prove Lemma 9, we show that the pairwise function 𝜑_𝑖𝑗 defined in (20) satisfies all properties listed in (17). Proof of Properties (1)–(4) is trivial using properties of the smooth function ℎ_𝑖𝑗(Δ_𝑖𝑗,𝑎_𝑖𝑗,𝑏_𝑖𝑗) listed in (9). We focus on proving Property (5).

Since the function 𝜑_𝑖𝑗 is smooth, Property (5) holds if 𝜑^′_𝑖𝑗<0,∀Δ_𝑖𝑗∈0,Δ_𝑖𝑗𝑑,𝜑^′_𝑖𝑗=0,ifΔ_𝑖𝑗=Δ_𝑖𝑗𝑑,𝜑^′_𝑖𝑗>0,∀Δ_𝑖𝑗∈Δ_𝑖𝑗𝑑,Δ^𝑀_𝑖𝑗𝑅,𝜑^′_𝑖𝑗=0,∀Δ_𝑖𝑗∈Δ^𝑀_𝑖𝑗𝑅,∞. (B.6) The above conditions mean that the smooth function 𝜑_𝑖𝑗 is decreasing for all Δ_𝑖𝑗∈(0,Δ_𝑖𝑗𝑑), is increasing for all Δ_𝑖𝑗∈(Δ_𝑖𝑗𝑑,Δ^𝑀_𝑖𝑗𝑅), and is constant for all Δ_𝑖𝑗∈[Δ^𝑀_𝑖𝑗𝑅,∞).

For all Δ_𝑖𝑗∈(0,Δ_𝑖𝑗𝑑), we have from (23) that 𝜑^′_𝑖𝑗=2Δ_𝑖𝑗𝑑Δ³_𝑖𝑗Δ_𝑖𝑗−Δ_𝑖𝑗𝑑, (B.7) where we have used ℎ(•)=0 and ℎ^′(•)=0 for all Δ_𝑖𝑗∈(0,Δ_𝑖𝑗𝑑) because 𝑎_𝑖𝑗 and 𝑏_𝑖𝑗 satisfy condition (21). Therefore 𝜑^′_𝑖𝑗<0,forallΔ_𝑖𝑗∈(0,Δ_𝑖𝑗𝑑). For Δ_𝑖𝑗=Δ_𝑖𝑗𝑑, clearly 𝜑^′_𝑖𝑗=0 since 𝑎_𝑖𝑗=Δ_𝑖𝑗𝑑, see (21). For all Δ_𝑖𝑗∈(Δ_𝑖𝑗𝑑,Δ^𝑀_𝑖𝑗𝑅), we have already proved that 𝜑^′_𝑖𝑗>0. For all Δ_𝑖𝑗∈[Δ^𝑀_𝑖𝑗𝑅,∞), we have ℎ(Δ_𝑖𝑗,𝑎_𝑖𝑗,𝑏_𝑖𝑗)=1 because 𝑎_𝑖𝑗 and 𝑏_𝑖𝑗 satisfy condition (21). Hence, from (20) we have 𝜑_𝑖𝑗=𝑘_𝑖𝑗, which implies that 𝜑^′_𝑖𝑗=0 for all Δ_𝑖𝑗∈[Δ^𝑀_𝑖𝑗𝑅,∞). Proof of Lemma 9 is completed.

C. Proof of Theorem 12

C.1. Proof of No Collisions and Complete Forwardness of the Closed Loop System

It is seen from (44) that 𝑉̇≤0. Integrating 𝑉̇≤0 from 𝑡₀ to 𝑡 and using the definition of 𝑉 in (36) with 𝜑 in (24) and 𝜑_𝑖𝑗 in (20) result in 𝑉(𝑡)≤𝑉(𝑡₀), where 𝑉(𝑡)=∑^𝑁−1_𝑖=1∑^𝑁_𝑗=𝑖+1𝜑_𝑖𝑗(𝑡)+(1/2)∑^𝑁_𝑖=1(𝑐₁‖𝐏_𝑖(𝐪_𝑖−𝐪_𝑜𝑑)‖²+𝑐₂‖𝜼_𝑖−𝜼_𝑖𝑑‖²), and 𝑉(𝑡₀) is 𝑉(𝑡) with 𝑡 replaced by 𝑡₀, for all 𝑡≥𝑡₀≥0. From the condition specified in item (4) of Assumption 1, and Property (3) of 𝜑_𝑖𝑗, we have 𝑉(𝑡₀) is bounded by a positive constant depending on the initial conditions. Boundedness of 𝑉(𝑡₀) implies that 𝑉(𝑡) must be also bounded. As a result, 𝜑_𝑖𝑗(Δ_𝑖𝑗(𝑡)) must be smaller than some positive constant depending on the initial conditions for all 𝑡≥𝑡₀≥0. From properties of 𝜑_𝑖𝑗, see (17), Δ_𝑖𝑗(𝑡), for all (𝑖,𝑗)∈ℕ and 𝑖≠𝑗, must be larger than 0 for all 𝑡≥𝑡₀≥0. This in turn implies from Lemma 1 that there are no collisions between any agents for all 𝑡≥𝑡₀≥0. Boundedness of 𝑉(𝑡) also implies that of 𝐏_𝑖(𝑡)(𝐪_𝑖(𝑡)−𝐪_𝑜𝑑(𝑡)) and 𝜼_𝑖(𝑡)−𝜼_𝑖𝑑(𝑡) for all 𝑡≥𝑡₀≥0. Since we have already proved that Δ_𝑖𝑗(𝑡)>0, the update law 𝜼̇_𝑖𝑑 in (42) implies that 𝜼_𝑖𝑑(𝑡)−𝜼_𝑜𝑑(𝑡) is also bounded. Therefore, the closed-loop system (46) is forward complete. Moreover, from 𝑉(𝑡)≤𝑉(𝑡₀) we have ‖𝐪_𝑖(𝑡)−𝐪_𝑜𝑑(𝑡)‖≤(2𝜆_max(𝐀_𝑖)/min(𝑐₁,𝑐₂))𝑉(𝑡₀)∶=𝐴₀ where 𝜆_max(𝐀_𝑖) denotes the maximum eigenvalue of 𝐀_𝑖. Hence we have proved the first two results listed in Theorem 12.

C.2. Mismatched Velocity Analysis

We first use Lemma 5 to find the equilibrium set, which the trajectories of the closed loop system (46) converge to. Integrating both sides of (44) yields ∫^∞₀∑^𝑁_𝑖=1𝜗_𝑖(𝑡)𝑑𝑡=𝜑(𝑡₀)−𝜑(∞)≤𝜑(𝑡₀), where 𝜗_𝑖 is given in (45). Indeed, the function ∑^𝑁_𝑖=1𝜗_𝑖(𝑡) is scalar, nonnegative, and differentiable. Now differentiating ∑^𝑁_𝑖=1𝜗_𝑖(𝑡) along the solutions of the closed loop system (46) and using the properties of the function 𝜑_𝑖𝑗 given in (17) and the function 𝑤(⋅) in (40) readily show that |𝑑∑^𝑁_𝑖=1𝜗_𝑖(𝑡)/𝑑𝑡|≤𝑀∑^𝑁_𝑖=1𝜗_𝑖(𝑡) with 𝑀 being a positive constant. Therefore, Lemma 5 results in lim_𝑡→∞∑^𝑁_𝑖=1𝜗_𝑖(𝑡)=0, which implies that lim_𝑡→∞𝜗_𝑖(𝑡)=0. Hence, from the expression of 𝜗_𝑖(𝑡) in (45) we have lim_𝑡→∞(Ω_𝑖(𝑡),Ξ_𝑖(𝑡))=0. Hence the position and orientation trajectory (𝐪_𝑖,𝜼_𝑖) of the agent 𝑖 `almost globally' converges to an (moving with (𝐪_𝑜𝑑,𝜼_𝑜𝑑) equilibrium set, 𝚼_𝐜, asymptotically. In the equilibrium set 𝚼_𝐜, we have Ω_𝑖(𝑡)=0 and Ξ_𝑖(𝑡)=0 for all 𝑖∈ℕ. The term “almost globally” refers to the fact that the agents start from a set, in which the condition (16) holds. Now substituting the limit lim_𝑡→∞(Ω_𝑖(𝑡),Ξ_𝑖(𝑡))=0 into the time limit both sides of the closed-loop system (46) to ∞ gives (47). Moreover, it is seen from the update law (42) that when the transformed distance Δ_𝑖𝑗 is larger than 𝑏_𝑖𝑗𝑑 (noting that Δ_𝑖𝑗𝑑<𝑏_𝑖𝑗𝑑, see (43) and (21)), we have lim_𝑡→∞(𝜼_𝑖𝑑(𝑡)−𝜼_𝑜𝑑(𝑡))=0 and lim_𝑡→∞(𝜼̇_𝑖𝑑(𝑡)−𝜼̇_𝑜𝑑(𝑡))=0.

C.3. Flock Size

Let (𝐪_𝑖𝑐,𝜼_𝑖𝑐), for all 𝑖∈ℕ, be the equilibrium state of the agent 𝑖, that is, (𝐪_𝑖𝑐,𝜼_𝑖𝑐)∈𝚼_𝐜 where 𝚼_𝐜 is the equilibrium set as defined above. In the set 𝚼_𝐜, we have Ω_𝑖(𝑡)=0 and Ξ_𝑖(𝑡)=0 for all 𝑖∈ℕ. Hence, from the expression of Ω_𝑖 and Ξ_𝑖 in (38), we have _𝑗≠𝑖𝜑^′_𝑖𝑗𝑐Γ_𝑖𝑗𝑐+𝑐₁𝐏_𝑖𝑐𝐪_𝑖𝑐−𝐪_𝑜𝑑^𝑇𝐏_𝑖𝑐=0,_𝑗≠𝑖𝜑^′_𝑖𝑗𝑐Λ_𝑖𝑗𝑐+𝑐₂𝜼_𝑖𝑐−𝜼_𝑖𝑑^𝑇=0, (C.1) where Γ_𝑖𝑗𝑐, Λ_𝑖𝑗𝑐, and 𝐏_𝑖𝑐 are Γ_𝑖𝑗, Λ_𝑖𝑗, and 𝐏_𝑖 with (𝐪_𝑖,𝜼_𝑖) replaced by (𝐪_𝑖𝑐,𝜼_𝑖𝑐) for all 𝑖∈ℕ. From (C.1), we have 𝐪_𝑖𝑗𝑐=𝐏^𝑇_𝑖𝑐𝐏_𝑖𝑐⁻¹𝑐_1𝑙≠𝑖𝜑^′_𝑖𝑙𝑐Γ^𝑇_𝑖𝑙𝑐−𝐏^𝑇_𝑗𝑐𝐏_𝑗𝑐⁻¹𝑐_1𝑘≠𝑗𝜑^′_𝑗𝑘𝑐Γ^𝑇_𝑗𝑘𝑐,𝜼_𝑖𝑗𝑐=𝜼_𝑖𝑗𝑑+1𝑐_2𝑙≠𝑖𝜑^′_𝑖𝑙𝑐Λ_𝑖𝑙𝑐−1𝑐_2𝑘≠𝑖𝜑^′_𝑗𝑘𝑐Λ_𝑗𝑘𝑐, (C.2) where 𝐪_𝑖𝑗𝑐=𝐪_𝑖𝑐−𝐪_𝑗𝑐 for all (𝑖,𝑗)∈ℕ,𝑗≠𝑖.

We make the following observations. All the terms Γ_𝑖𝑗𝑐 and Λ_𝑖𝑗𝑐 are vector functions of 𝐪_𝑖𝑗𝑐, and bounded functions (sin and cos) of elements of 𝜼_𝑖𝑗𝑐, 𝜼_𝑖𝑐, and 𝜼_𝑗𝑐 only for all (𝑖,𝑗)∈ℕ,𝑗≠𝑗, see (28) and (34) for the expression of Γ_𝑖𝑗 and Λ_𝑖𝑗 with a note that (𝐪_𝑖,𝜼_𝑖 are replaced by (𝐪_𝑖𝑐,𝜼_𝑖𝑐 for all 𝑖∈ℕ. Moreover, from (42), we can see that lim_𝑡→∞𝜼_𝑖𝑗𝑑(𝑡) tends to a value depending on Γ_𝑖𝑗𝑐 and Λ_𝑖𝑗𝑐 only.

The above observations imply from (C.2) that 𝐪_𝑖𝑗𝑐 and 𝜼_𝑖𝑗𝑐 must be bounded and depend on bounded functions (sin and cos) of elements of 𝜼_𝑖𝑐 only with 𝑖=1,…,𝑁. On the other hand, the flock size 𝐹_𝐿=∑^𝑁_𝑗=𝑖+1∑^𝑖−1_𝑗=1Δ_𝑖𝑗 is a function which depends on 𝐪_𝑖𝑗, and bounded functions (sin and cos) of elements of 𝜼_𝑖𝑗𝑐, 𝜼_𝑖𝑐, and 𝜼_𝑗𝑐 only for all (𝑖,𝑗)∈ℕ,𝑗≠𝑗. Therefore in the equilibrium set 𝚼_𝐜, the flock size, 𝐹_𝐿𝑐 being 𝐹_𝐿 with (𝐪_𝑖,𝜼_𝑖) replaced by (𝐪_𝑖𝑐,𝜼_𝑖𝑐), is a function that depends bounded functions (sin and cos) of elements of 𝜼_𝑖𝑐 only with 𝑖=1,…,𝑁. Moreover, the flock size is bounded whenever its arguments 𝐪_𝑖𝑗 and bounded functions (sin and cos) of elements of 𝜼_𝑖𝑗𝑐, 𝜼_𝑖𝑐, and 𝜼_𝑗𝑐 are bounded.

Hence, in the equilibrium set 𝚼_𝐜, the flock size is bounded by a function of bounded functions (sin and cos) of elements of 𝜼_𝑖𝑗𝑐, 𝜼_𝑖𝑐, and 𝜼_𝑗𝑐. Since functions sin and cos are bounded by −1 and 1, there exist a positive constant 𝑐₀ such that 𝐹_𝑐≤𝑐₀. This completes proof of Theorem 12.