Nonlinear Dyn DOI 10.1007/s11071-016-3303-2
ORIGINAL ARTICLE
A geometric extension design for spherical formation tracking control of second-order agents in unknown spatiotemporal flowfields Yang-Yang Chen · Zan-Zan Wang · Ya Zhang · Cheng-Lin Liu · Qin Wang
Received: 18 January 2016 / Accepted: 17 December 2016 © Springer Science+Business Media Dordrecht 2016
Abstract This article proposes a solution to the problem of directing multiple second-order agents sphere landing, orbit tracking and formation moving around a family of given concentric spheres in unknown spatiotemporal flowfields. The flowfields put stress on each agent’s velocity and acceleration at once, and the specification of each one is composed of three known base vectors and unknown responding coefficients. First, our pervious two-dimensional geometric extension design is extended to deal with the extension of surface in three-dimensional space. Then, two new adaptive estimators and the cooperative control law are constructed to accomplish the robust spherical formation tracking motion by using the tools of adaptive backstepping, geometric extension and consensus. The asymptotic stability of system is proved when the bidirectional Y.-Y. Chen (B) · Z.-Z. Wang · Y. Zhang School of Automation, Southeast University, Nanjing 210096, People’s Republic of China e-mail:
[email protected] Y.-Y. Chen · Z.-Z. Wang · Y. Zhang Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Southeast University, Nanjing 210096, China C.-L. Liu Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Institute of Automation, Jiangnan University, Wuxi 214122, China Q. Wang College of Information Engineering, Yangzhou University, Yangzhou 225127, China
communication topology is connected. The effectiveness of the analytical result is verified by numerical simulations. Keywords Spherical formation tracking control · Second-order dynamics · Geometric extension design · Flowfields · Adaptive estimation
1 Introduction Nowadays, cooperative exploration executed by a family of sensor-equipped vehicles has aroused a great attention in the world. Such a mobile sensor network can provide a cheap, robust platform for the synoptic data collection of spatiotemporal processes in air [1–3], sea [4–6], and space [7,8]. To provide persistent and consistent revisit data collections and utilize the ability of sensors, each vehicle should track its planned orbit and the family forms a desired formation along given orbits, which is called the formation tracking control problem. Previous results related to formation tracking control focus on flow-free motion models in 2D space. The leader-following strategy [9–11] has been widely used in 2D formation tracking control during the past twenty years since it was proposed. However, group robustness in this method is weak as no information flows from the followers to the leaders. Over the same period, another method called virtual structure is used for rigid formation tracking motions. Some details can be
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found in [12,13]. To overcome the limitations of leaderfollowing strategy and virtual structure, Ghabcheloo et al. [14,15] decouple orbit tracking and formation by designing an extra upgrade law for the curve parameters. However, the introduce of the extra upgrade law leads to the complexity of controller design and system analysis. With the launching of ocean exploration [6,16], a geometric solution has emerged on planar formation tracking control recently and received a lot of attention. The key idea of this approach is to extend the given curve to be a set of level curves of the orbit function. Orbit tracking is achieved through driving the orbit value (that is the value of the orbit function) to the desired value. Formation motion along orbits is accomplished by forcing relative arc-lengths to the reference values. Zhang and Leonard [17] solve the even distribution over the orbit by extending the given orbit along its normal vector. To avoid the difference between the given orbit and the extended orbits in geometric topology [17], a novel geometric extension approach called concentric compression is proposed to deal with the convex loops [18] and then a general kind of nonconvex, closed curves [19]. Actually, agents outside are often disrupted by some external flowfields (e.g., the ocean current and the atmospheric wind), and then, the sensory performance fails to be maintained by above flow-free control laws. Therefore, the control of formation tracking motion in flowfields is an ongoing challenge. When the flowfield is steady, uniform and known to each agent, a temporally balanced formation algorithm is given to control the constant-speed particles [20,21] at beginning. Then, some spatiotemporal formation tracking controllers are proposed to solve the bidirectional communication [22] and the directed communication [23]. As we all know, flowfields such as winds or currents in nature often vary with time and space, and thus, it is hard to supply each agent all the information of flowfield [6,24]. In [25], a constant-velocity wind is estimated by the traditional adaptive design method, and the balanced formation motion around a circle is achieved. A similar idea is used to solve the robust formation tracking control with a unknown rotating flowfield [26]. In [27], a distributed information-consensus filter is given to estimate the coefficients of the parameterized flowfield and incorporated into the formation control law for the exploration motion in the ocean. However, all above results are not out of the scope of 2D formation tracking control.
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In the event that ocean scales [4,5] drive the need for the formation surrounding motion of mobile sensor networks in 3D space and planet exploration [7,8] requires vehicles looking like concomitant satellites to formation surround the planet, any methodologies derived for two dimensions need to be extended. In addition, nature group behaviors (e.g., the shoal of fishes) usually clusters a center, like a ball, for the purpose of foraging and avoiding predators [6,28]. Therefore, it is a trend to consider the cooperative motion on spheres. Paley [20] gives a result about analyzing the steady motion around a fixed, small circle on a sphere by using the tool of Lie group. A similar idea is used in distributed tracking control of multiple rigid bodies [32]. Wu and Zhang [29] propose a curve tracking control when four agents move on the given surface in the desired formation; Zhu [30] shows the consensus equilibria of highorder Kuramoto model on the unit sphere. However, all above results limit in the self-organization motion on a given sphere, which does not involve the control of sphere landing and the coordinated orbit tracking on spheres. In fact, adaptive data collection, whether in ocean, planet or space, the (near) optimal sampling trajectories for sensor platforms are often designed as set of closed curves on the different concentric spheres. Therefore, it is urgently needed a systematic method to deal with the spherical formation tracking control problem that includes sphere landing and formation tracking orbits on the responding spheres. For the purpose of making the control law applicable to the complex external environment, how to fight against the effect of the unknown spatiotemporal flowfields should be involved simultaneously. In this paper, an integral solution is proposed to the spherical formation tracking control problem of second-order agents moving in unknown spatiotemporal flowfields. Although spherical formation tracking motion is the simplest cooperative control problem on surfaces, we must emphasize its essentiality since it is the first step to more complex formation tracking control systems in 3D space. The unknown spatiotemporal flowfields under consideration not only put stress on the agent’s velocity (e.g., wind or current [20–22,25–27]) but also act on the agent’s acceleration (e.g., the gravitational force [33–36]), which covers almost all forms of flowfield in the literature. The specification of flowfield is composed of three known base vectors and unknown responding coefficients. One contribution of this paper is that we apply
A geometric extension design
the geometric extension method to deal with spherical formation tracking control problem, which is the first attempt of using the geometric method to achieve cooperative multi-agent exploration on surfaces. Concretely, we introduce the tubular neighborhood theorem to prove that the given sphere can be concentric compressed to be a set of level spheres of a smooth function (we call it sphere function). Then, the sphere function which incorporated into the polar and azimuth angle function to propose the solution to spherical formation tracking control. The other contribution of this paper is that we bring the adaptive backstepping method into the design process for the purpose of giving two new adaptive estimations of the unknown flow coefficients that are required for the spherical formation tracking algorithm. It must be emphasized that such adaptive backstepping design for 3D spatiotemporal flowfields putting stress on both agent’s velocity and acceleration is different from the adaptive design for the 2D constant velocity or rotating flowfield who only acts on each agent’s velocity [25,26] and the coupling strength of each pair of agents for consensus [32]. Meanwhile, our proposed adaptive upgrade law for the flowfield acting on each agent’s velocity can avoid the assumption that the time period of consensus filter is greater than the consensus time of information matrix and measurement [27]. The design procedure follows two steps: (1) Regard each agent’s velocity projected onto the normal vector, the direction of spherical meridian and the direction of spherical parallel as three virtual control, and then, we separately design three virtual control for sphere landing, orbit tracking and formation maintenance by using the estimations of flowfields; (2) Adaptive backstepping design is used to construct the control inputs of each agent projected onto the normal vector, the direction of spherical meridian and the direction of spherical parallel with the adaptive estimators for the unknown flow coefficients. Since the control inputs along three directions are linearly independent, the whole control input for each agent can be calculated. The paper is organized as follows: Sect. 2 summarizes the model of second-order agents moving in the external flowfields and formulates the control problem based on the geometric extension and consensus. Section 3 gives a solution to the robust spherical formation tracking control problem based on the adaptive backstepping design. Simulation results are given in Sect. 4. Conclusion is proved in Sect. 5 .
2 Problem statement 2.1 Second-order dynamics in unknown spatiotemporal flowfields Consider a family of N second-order nonlinear agents moving in the spherical coordinate system We = O, ex , e y , ez , where O is the origin that locates at the center of each sphere, e j , j = x, y, z, are unit vectors, and each one satisfies
p˙ i = υi + f θi ( pi , t)θi υ˙ i = f δi ( pi , υi , t) δi + u i
(1)
y
y
where pi = [ pix , pi , piz ]T , υi = [υix , υi , υiz ]T and y u i = [u ix , u i , u iz ]T are its position, velocity and control input, respectively. The flowfield putting stress on the agent’s velocity is spatiotemporal and denoted by f pi = f θi ( pi , t) θi , where
f θi = f θTαi , f θTβi , f θTγ i
T
⎡
f θxαi ⎢ x = ⎣ f θβi f θxγ i
y
f θαi y f θβi y f θγ i
⎤ f θzαi f θzβi ⎥ ⎦ f θzγ i
is a matrix consisted of three linear independent base vectors of flowfield and each base vector is a C 1 smooth function of agent’s position and time, θi =
x y z T θi , θi , θi represents the coefficients corresponding to three base vectors, which is unknown to each agent. The flowfield acting on the agent’s acceleration is denoted by f si = f δi ( pi , υi , t) δi , where
f δi = f δTαi , f δTβi , f δTγ i
T
⎡
f δxαi ⎢ x = ⎣ f δβi f δxγ i
y
f δαi y f δβi y f δγ i
⎤ f δzαi f δzβi ⎥ ⎦ f δzγ i
is a known matrix consisted of three linear independent base vectors of flowfield, and each base a uniis
vector T y formly continuous function, δi = δix , δi , δiz is the unknown flow coefficients. Remark 1 The agent’s dynamics represented in an inertial reference frame can be translated into the spherical coordinate system as (1) by coordinate translation and rotation. Remark 2 The uniform, time-invariant flowfield f pi =
x y T θ ,θ discussed in [26] and the uniform rotating
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flowfield f pi = θ eit with a constant rotation rate and an unknown constant θ [21] can be regarded as a special case of the flowfield in this paper. In [27], the parameterized flowfield with a set of known basis vectors and unknown coefficients is also considered. Although the known basis vectors are greater than the number dimension of space, the redundancy base vectors also can be represented as the linear combinations of the linear independent basis vectors. Therefore, we only consider the flowfield composed of three linear independent base vectors and the corresponding coefficients. The flowfield f si (e.g., the gravitational force) acting on the agent’s acceleration is also considered in this paper, which is not involved in [21,26,27]. In addition, we add a subscript i to the flow coefficients, which implies the measurements of the flow coefficients can be different even if two agents locate at the same place and the same time. It is due to the distinctions between the sensors installed in the different agents and the various attitude, material of agents. 2.2 Geometric extension and cooperative design method Suppose each given orbit Ci associated with the ith 2 . Notice agent is a circle located at a spherical surface Si0 that practical applications usually present cooperative motion around concentric spheres [4–8,30], we assume that all the spheres are concentric and the center of Ci locates at z−axis in We for the simple description of the 2 surface, control problem. Since sphere Si0
isπ a πregular we can use the polar angle αi ∈ − 2 , 2 , the azimuth angle φi ∈ (−π, π] and the fixed radius ρi0 > 0 2 function F(αi , φi ) : to parameterize xSi0 as a smooth y 3 → R = pi (αi , θi ), pi (αi , θi ), piz (αi , θi ) . Sim2 , a family of spheres F ilar to [18,19], near Si0 iλsi is constructed through concentric compression, that is Fiλsi
αi , θi ; λsi = 1 + λsi Fi0 (αi , θi )
(2)
ez
(1 + c ) Fi 0 (αi , φi ) Fi 0 (αi , φi )
ρi 0
ex
O
Sio2 : λsi = 1 −
ey
pi
ρi 0
=0
Sic2 : λsi = c
Fig. 1 Sphere extension via concentric compression 2 (precisely a (C1) An open set i ⊂ R3 with i ⊂ Si0 2 tubular neighborhood of Si0 ). (C2) A smooth function λsi : i → (−εi , εi ) (precisely a sphere function) with ∇λsi = 0 for all points in i , where εi > 0. Furthermore, set can be the open expressed as i = pi ∈ R3 λsi ( pi ) < εi , Fic (αi , θi ; c) is a level sphere of λsi (·) with its value belonging to a constant c, that is, λsi ( pi ) = c if pi ∈ Fic , and the sphere value (the value of λsi (·)) associ2 is zero. ated to the given sphere Si0
From concentric compression (2), the expression of sphere function λsi in We can be written as λsi = 1 −
pix
2
y 2 2 + pi + piz ρi0
= 1−
pi , (3) ρi0
where i = pi ∈ R3 λsi ( pi ) < ρi0 (see Fig. 1).
where λsi ∈ R is the proportion of concentric compression. Since the direction of concentric compres2, sion is the same as the unit normal vector Ni to Si0 the following lemma is gained according to the tubular neighborhood theorem on page 114 in [38].
Remark 3 It must be emphasized that our proposed design method is also suitable to the non-concentric spheres. If the center of Ci does not locate at z−axis, one can use the coordinate transformation matrix to make the proposed control law suitable to the general situation.
2 is a regular compact oriLemma 1 Since sphere Si0 2 with the geometric extenentable surface, extending Si0 sion (2), there exists
From Lemma 1, the key idea of sphere landing is to drive the sphere value λsi ( pi (t)) converge to 0 asymptotically, at the same time, the trajectory of each agent to
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restrict in the set i . In this paper, each agent’s control input projected onto the normal vector Ni to the surface of sphere is designed to accomplish sphere landing. Due to the fact that each point (except the poles of sphere) on the given circular orbit Ci has the same polar angle, we use αi∗ ∈ − π2 , π2 to represent Ci . To drive 2 to track C , the polar an agent moving on sphere Si0 i angle αi associated with the position of the ith agent is required to reach αi∗ asymptotically. Each agent’s control input projected onto the direction of spherical meridian Bi is constructed for this purpose. When each agent moves around its given orbit on 2 , the control object is to keep the family moving Si0 in the desired formation with the given orbits adopted. Its implementation requires communication among the agents. Let G = {V , E } be a bidirectional communication topology, where V = {v1 , v2 , · · · , v N } denotes the set of N agents and E ⊆ V × V is a set of data links. An edge Ei j denotes that the information can be exchanged between the jth agent and the ith agent. Let Ni denote the neighbor set of the ith agent and we assume that Ni is time-invariant. Two matrices such as the adjacency matrix A = [ai j ] and the Lapalacian matrix L = [li j ] are used to represent the bidirectional graph. Note that a sphere except its poles can be decom2 except posed as a set of level circles. Each point on Si0 its poles can be regarded as a point located at a special level circle. In the event that the center rotation angle of each level circle equals to the azimuth angle φi . In this paper, the generalized azimuth angle ξi (φi ) is used to represent the formation on given orbits. For example, three agents moving around various circles on a set of concentric spheres while maintaining in-linear formation as shown in Fig. 2. To keep the desired formation, it is required that the generalized azimuth angle ξi = φi research to consensus. It is said that formation is accomplished if the generalized azimuth angle ξi (t) chosen as Assumption 1 for inter-agent formation reach to consensus. We design each agent’s control input projected onto the direction of spherical parallel Ti to accomplish formation. Since the line l p between the poles of each sphere lead to the undefined azimuth angle of each level circle, the trajectory of the ith agent should not pass l p . Assumption 1 Each generalized azimuth angle ξi (φi ) is defined a C 2 smooth function of the azimuth angle 2 ∂ξi ∂ξi φi such that ∂φ and ∂∂φξ2i are bounded and ∂φ never i i equal to 0 for all φi .
i
sz
C3
φ3
p3
C2
φ2
p2
φ1
p1
C1
sx
O
S102
S202
sy
S302
Fig. 2 Formation description on spheres
From the above discussion, we give the definition of the robust spherical formation tracking control problem in unknown spatiotemporal flowfields as follows. Definition 1 Robust spherical formation tracking control problem in unknown spatiotemporal flowfields is said to be solved if there exist a sphere landing control Ni ·u i , a orbit tracking control Bi ·u i and formation control law Ti · u i with the adaptive upgrade laws θ˙ˆ i , δ˙ˆi for {θi , δi } such that the following conditions: lim λsi ( pi (t)) = 0,
(4)
λs ( pi (t)) < εi , i
(5)
lim αi (t) − αi∗ = 0,
(6)
π , 2
(7)
t→∞
t→∞
|αi (t)| <
lim ξi (t) − ξ j (t) = 0; j ∈ Ni ,
(8)
t→∞
hold for any initial trajectory of each agent in π . ϒi = pi ∈ R3 λsi ( pi ) < ρi0 , |αi ( pi ) | < 2
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Define the polar angle αi as
Remark 4 From Definition 1, condition (4) forces each agent to land on its given sphere while condition (5) comes from the domain of sphere function. Condition (6) guarantees each agent to arrive at its given orbit, and the same time, condition (7) makes sure the definition of azimuth angle of each level circle. The formation is maintained when (8) is tenable.
cos αi =
3 Main results
Taking the time derivative of (13b) with (13a) yields
3.1 Cooperative control model
α˙ i = −
In next subsection, we first regard each agent’s velocity projected onto the normal vector, the direction of spherical meridian and the direction of spherical parallel (that is {υNi , υ Bi , υTi }) as three virtual control υˆ Ni , υˆ Bi , υˆ Ti . Then, three control parts such as {Ni · u i , Bi · u i , Ti · u i} are designed to make the real velocity components υ Ni , υBi , υTi converge to the virtual control υˆ Ni , υˆ Bi , υˆ Ti . Therefore, the dynamics of {λsi , υ Ni , αi , υ Bi , ξi , υTi } should be given. Along the trajectory of the ith moving agent in the set ϒi , the position error of sphere landing for each agent can be represented as λsi ( pi ) because the sphere value associated to the given sphere is zero. Thus, the dynamics of position error of sphere landing is given by 1 υ N + λsi θi , λ˙ si = ρi0 i
(9)
sin αi =
( pix )2 + ( pix )2 pi
,
(13a)
piz . pi
(13b)
1 υ B + αi θi pi i
(14)
where υ Bi = Bi · υi
(15)
is the agent’s velocity projected onto the direction of spherical meridian T 1 pix piz , piy piz , − px yi 2 , (16) pi px yi
Bi =
y T px yi = pix , pi , 0 , αi = − p1i Bi T f θi . Differentiating both sides of (15), one gets υ˙ Bi = αυ Bi + βυ Bi θi + γυ Bi δi + Bi · u i ,
(17)
where β
where υ Ni = Ni · υi
(10)
is the agent’s velocity projected onto the normal vector p ∇λsi =− i Ni = ∇λs pi i
(11)
1 2 and T to Si0 λsi = ρi0 Ni f θi . Differentiating both sides of (10), the direction error between the movement of agent and the tangent plane to the level sphere is
υ˙ Ni = αυN i + βυ N i θi + γυ N i δi + Ni · u i , β
β
(12) γ
where αυN i = υiT αNi , υ N i = υiT Ni , υ N i = NiT f δi ,
αNi
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=
−υi + NiT υi Ni , pi
β Ni
=
− f θi + NiT f θi Ni pi
.
αυ Bi = υiT αBi , βυ Bi = υiT Bi , γυ Bi = BiT f δi , T υ Tυ p p υ D i i x y 1i i i i αBi = − Bi , + px y pi px y 2 pi 2 i i pxTyi f θi piT f θi D1i f θi β − Bi = Bi , + px y pi px y 2 pi 2 i i ⎤ ⎡ z pi 0 pix y piz pi ⎦ . D1i = ⎣ 0 y x −2 pi −2 pi 0 Since the variation of the azimuth angle φi is only related to the movement projected onto spherical parallel, the dynamics of ξi can be written as ξ˙i =
1 ∂ξi ∂ξi φ˙ i = υT + ξi θi , ∂φi ∂φi ρi0 1 − λsi cos αi i (18)
A geometric extension design
where υTi = Ti · υi
(19)
is the velocity projected onto the direction of spherical parallel Ti =
1 px yi
and ξi =
y
pi , − pix , 0
T
∂ξi 1 ∂φi ρi0 1−λs cos αi i
(20) TiT f θi . Taking the time
for each agent, we replace θi and δi with the parameter estimates θˆi and δˆi , respectively. The estimate errors are defined as θ˜i = θi − θˆi , δ˜i = δi − δˆi .
υ˙ Ti =
+ βυT i θi
+ γυT i δi
+ Ti · u i ,
(21)
where β
αυT i = υiT αT i , βυT i = υiT T i , γυT i = TiT f δi , αT i = β
T i =
pxTy υi −Ti i 2 px y i T px y f θi −Ti i 2 px y i ⎡
0 D2i = ⎣ −1 0
1 0 0
D2i υi , + px y i
0 0⎦. 0
From above discussion, the equations of robust spherical formation tracking control system for one agent in unknown spatiotemporal flowfields are summarized as follows: 1 υ N + λi θi , (22a) λ˙ si = ρi0 i υ˙ Ni = αυN i + βυ N i θi + γυ N i δi + Ni · u i , 1 α˙ i = − υ B + αi θi , pi i
(22b)
υ˙ Bi = αυ Bi + βυ Bi θi + γυ Bi δi + Bi · u i , 1 ∂ξi ξ˙i = υT + ξi θi , ∂φi ρi0 1 − λsi cos αi i
(22d)
υ˙ Ti =
αυT i
+ βυT i θi
+ γυT i δi
+ Ti · u i .
VI =
n
(22c)
(22e) (22f)
3.2 Controller design based on adaptive backstepping In this subsection, we give a cooperative control strategy for the robust spherical formation tracking problem with the adaptive estimation of the unknown flowfields. For the robust spherical formation tracking control law
λi (λi ) +
i=1
+ +
D2i f θi , + px y i ⎤
(24)
Step 1 With {υ Ni , υ Bi , υTi } viewed as the virtual control {υˆ Ni , υˆ Ti , υˆ Ti } in Eqs. (22a), (22c) and (22e), we begin to the design based on the following Lyapunov function:
derivative of (19), one gains αυT i
(23)
n
αi (αi )
i=1
n n 2 kξ ai j ξ¯i − ξ¯ j 4 i=1 j=1 n
1 2γ1
θ˜ i 2 ,
(25)
i=1
t where ξ¯i = ξi − a0 0 ξ˙ ∗ (τ )dτ and ξ˙ ∗ (t) is C 1 smooth function of t, a0 = 0 means there is no a demanded consensus dynamic, otherwise a0 = 1, kξ is greater than 0, γ1 > 0 is the adaptation gain, λsi λsi is a C 2 smooth, nonnegative function on (−εi , εi ) and satisfies: (C3) ψλsi λsi → +∞ and ∇ψλsi → −∞ as λsi → −ρi0. (C4) ψλsi λsi → +∞ and ∇ψλsi → +∞ as λsi → ρi0 . (C5) There exists a C 1 smooth function ϕλsi λsi such that ϕλsi ∇λsi ≥ 0 and ϕλsi ∇λsi = 0 if and only if λsi = 0. αi (αi ) is a C 2 smooth, nonnegative function on Also π π − 2 , 2 and satisfies: (C6) αi (αi ) → +∞ and ∇αi → −∞ as αi → − π2 . (C7) αi (αi ) → +∞ and ∇αi → +∞ as αi → π2 . (C8) There exists a C 1 smooth function ϕαi (αi ) such that ϕαi ∇αi ≥ 0 and ϕαi ∇αi = 0 if and only if αi = αi∗ . There aremanyfunctions that all the above prop satisfy erties of λsi λsi , ϕλsi λsi . An example is
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ψλsi
λsi =
λsi λ∗s i
c1
1 1 − ρi0 − τ ρi0 + τ
+ c2 (ln(ρi0 + τ ) − ln(ρi0 − τ ))] dτ and ϕλsi λsi = λsi , where λ∗si = λsi ( pi (0)) ∈ i and c1 , c2 > 0. Similarly, we can choose ϕαi (αi ) = αi −αi∗ and αi (αi ) as 1 1 c1 π αi (αi ) = − π αi∗ 2 −τ 2 +τ π π +τ −ln −τ −c3 dτ + c2 ln 2 2
αi
where αi∗ = αi ( pi (0)) ∈ − π2 , π2 , c3 = −c1 1 1 − c2 (ln( π2 + αi∗ ) − ln( π2 − αi∗ )). − π +α π ∗ −α ∗ i
2
2
n
1 ∇λsi υ N + λsi θˆi + ∇αi ρi0 i i=1 i=1 n 1 ˆ × − υ B + αi θi + kξ pi i i=1 1 ∂ξi ∗ ˆ ˙ × υT + ξi θi − a0 ξ ∂φi ρi0 (1 − λi ) cos αi i n n 1 − θ˙ˆiT + αˆ θ˜i , × ai j ξ¯i − ξ¯ j + θi γ1
V˙ I =
j=1
i=1
(26) where αˆ = ∇λsi λsi +∇αi αi + kξ ξi θi
n
ai j ξ¯i − ξ¯ j .
j=1
(27) To make V˙ I ≤ 0, let the virtual control υˆ Ni , υˆ Bi and υˆ Ti as follows:
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(28) (29)
(30)
where the control gain k1 is greater than 0. Next, we introduce the error variables υ˜ Ni = υ Ni − υˆ Ni ,
(31)
υ˜ Bi = υ Bi − υˆ Bi ,
(32)
υ˜ Ti = υTi − υˆ Ti
(33)
i
In the function (25), the first term guides the ith agent to land on the given sphere and stay in i when it starts from ϒi . It vanishes when λsi = 0. The next term in (25) forces the ith agent to the given circle and never pass l p . It vanishes when αi − αi∗ = 0. The third term ensures the consensus of the generalized azimuth angle ξi (t). The last term guarantees the converge of θ˜i . Note that θ˙˜i = −θ˙ˆi , the derivative of VI is n
υˆ Ni = −ρi0 ϕλsi + λsi θˆi , υˆ Bi = pi ϕαi + αi θˆi ∂ξ −1 υˆ Ti = ρi0 (1 − λsi ) cos αi ∂φi × a0 ξ˙ ∗ − k1 ξ¯i − ξi θˆi ,
for the purpose of designing {Ni · u i , Bi · u i , Ti · u i }. In the next step, the error variables υ˜ Ni , υ˜ Bi , υ˜ Ti will be driven to zero. According to (28)–(33), V˙ I can be rewritten as V˙ I = −
n
∇λsi ϕλsi −
i=1 n
n
∇αi ϕαi − kξ k1 ξ¯ T L ξ¯
i=1
n 1 + ∇λsi υ˜ Ni + ∇ψαi ρi0 i=1 i=1 n 1 × − υ˜ B + kξ pi i i=1 n ∂ξi 1 × υ˜ Ti ai j ξ¯i − ξ¯ j ∂φi ρi0 1 − λsi cos αi j=1 n 1 − θ˙ˆiT + αθˆ θ˜i . + (34) i γ1 i=1
We postpone the choice of update for θˆi until the next step. The position error subsystem of spherical formation tracking becomes 1 υ˜ N + λsi θ˜ i , ρi0 i
(35a)
1 υ˜ B + αi θ˜ i , pi i
(35b)
λ˙ si = −ϕλsi + α˙ i = −ϕαi −
A geometric extension design
1 ∂ξi υ˜ T + ξi θ˜ i . ξ˙¯ i = −k1 ξ¯i + ∂φi ρi0 1 − λsi cos αi i (35c)
piT υi + f θi θˆi
− BiT f θi θ˙ˆ i + ϕαi p i 1 υ B + αi θˆi , − pi i
+ pi ∇ϕαi
Step 2 The derivative of υ˜ Ni is υ˙˜ Ni = αυN i + βυ N i θi + γυ N i δi + Ni · u i − υ˙ˆ Ni , (36)
β υˆ Bi
˙
T β = − f θi θˆi Bi − BiT
where υ˙ˆ Ni = α˙
υˆ Ni
β + ˙ θ˜i , υˆ Ni
(37)
T β αNi + Ni θˆi − ρi0 λsi θ˙ˆ i = − f θi θˆi υˆ Ni 1 − ρi0 ∇ϕλsi υ Ni + λsi θˆi − NiT ρi0 ∂ f θi ∂ f θi ˆi I1 υi + f θi θˆi + ˆ θ y θi I2 x ∂ pi ∂ pi ∂f ∂f θi ˆ θi ˆ ˆ ˆ θi I3 υi + f θi θi + θi , υi + f θi θi + ∂ piz ∂t
α˙
T β = − f θi θˆi Ni − ρi0 ∇ϕλsi λsi − NiT ! ∂f ∂f ∂ f θi ˆθi I1 f θi + θyi θˆi I2 f θi + θzi θˆi I3 f θi , ∂ pix ∂ pi ∂ pi
I1 = [1, 0, 0]T , I2 = [0, 1, 0]T and I3 = [0, 0, 1]T . Taking the derivative of υ˜ Bi , one gains υ˙˜ Bi = αυ Bi + βυ Bi θi + γυ Bi δi + Bi · u i − υ˙ˆ Bi , (38) where
υˆ Bi
υˆ Bi
β + ˙ θ˜i , υˆ Bi
(39)
T β αBi + Bi θˆi = − f θi θˆi
∂ f θi ∂ f θi ˆi I1 υi + f θi θˆi + ˆ θ y θi I2 x ∂ pi ∂ pi " ∂f ∂f θi ˆ θi ˆ ˆ ˆ θi I3 υi + f θi θi + θi υi + f θi θi + ∂ piz ∂t
− BiT
Meanwhile, the derivative of υ˜ Ti is υ˙˜ Ti = αυT i + βυT i θi + γυT i δi + Ti · u i − υ˙ˆ Ti , (40) where β υ˙ˆ Ti = α˙ + ˙ θ˜i , υˆ Ti
υˆ Ti
(41)
∂ξi −2 ∂ 2 ξi a0 ξ˙ ∗ − k1 ξ¯i − ξi θˆi TiT υi + f θi θˆi =− 2 υˆ Ti ∂φi ∂φi 1 ∂ξi −1 ρi0 a0 ξ˙ ∗ −k1 ξ¯i − ξi θˆi υ Ni +λi θˆi × − ∂φi ρi0 1 ∂ξi −1 cos αi +(1 − λsi ) sin αi − + υ Bi + αi θˆi pi ∂φi ∂ξ 1 i × ρi0 (1 − λi ) cos αi a0 ξ¨ ∗ − (k1 υT ∂φi ρi0 (1 − λi ) cos αi i ∂ f θi + ξi θˆi − a0 ξ˙ ∗ − ξi θˆi + TiT θˆi I1 υi + f θi θˆi ∂ pix ∂ fθ ∂ fθ ∂ f θi i ˆ i ˆ ˆ ˆ ˆ + I + f I + f υ + υ + θ θ θ θ θ θ θ 2 3 i i i i i i i y i i ∂t ∂ piz ∂ pi T β + f θi θˆi αT + T θˆi , i
υ˙ˆ Bi = α˙ α˙
" piT f θi ∂ f θi ˆ + + pi ∇ϕαi αi . z θi I3 f θi +ϕαi pi ∂ pi
α˙
β υˆ Ni
˙
∂ f θi ∂ f θi ˆ θˆi I1 f θi + y θi I2 f θi x ∂ pi ∂ pi
i
−2 2 ∂ ξi a0 ξ˙ ∗ − k1 ξ¯i − ξi θˆi TiT f θi 2 ∂φi −1 ∂ξi − ρi0 cos αi ξ˙ ∗ − k1 ξ¯i − ξi θˆi λsi + k1 ξi ∂φi + (1 − λsi ) sin αi a0 ξ˙ ∗ − k1 ξ¯i αi +k1 ξi ∂ f θi ∂f ∂f ˆi I1 f θ + θyi θˆi I2 f θ + θzi θˆi I3 f θ + TiT θ i i i ∂ pix ∂ pi ∂ pi T β + f θi θˆi T .
∂ξi β =− υˆ Ti ∂φi
˙
i
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To design the control {Ni · u i , Bi · u i , Ti · u i }, we consider the augmented Lyapunov function n n 1 1 2 VI I =VI + υ˜ Ni + υ˜ B2 i + υ˜ T2i , δ˜i 2 + 2γ2 2 i=1
where γ2 > 0. Note that δ˙˜i = −δ˙ˆi and differentiating (42) along the solution (35) and {υ˙˜ Ni , υ˙˜ Bi , υ˙˜ Ti }, one gets V˙ I I = −
∇λsi ϕλsi −
i=1
+ +
n i=1 n
Ni · u i = gin = −u Ni − k2 υ˜ Ni ,
(44)
Bi · u i = gib = −u Bi − k3 υ˜ Bi ,
(45)
Ti · u i = git = −u Ti − k4 υ˜ Ti
(46)
i=1
(42)
n
From (43), we set
n
with the adaptive upgrade law θ˙ˆiT = γ1 θˆi ,
(47)
δ˙ˆiT = γ2 δˆi .
(48)
∇αi ϕαi − kξ ξ¯ T L ξ¯
i=1
υ˜ Ni Ni · u i + u Ni
where the control gains k2 , k3 and k4 are greater than 0. Substituting the expressions (44)–(48) into (43), we obtain
υ˜ Bi Bi · u i + u Bi
i=1
+
n
υ˜ Ti Ti · u i + u Ti
V˙ I I = −
n
∇λsi ϕλsi −
i=1
i=1 n
1 − θ˙ˆiT + θˆi θ˜i + γ1 i=1 n 1 − δ˙ˆiT + δˆi δ˜i , + γ2
− k2
n
∇αi ϕαi − kξ k1 ξ¯ T L ξ¯
i=1
n i=1
υ˜ N2 i − k3
n
υ˜ B2 i − k4
i=1
n
υ˜ T2i ≤ 0.
i=1
(49) (43) Then, the error system of robust spherical formation tracking control becomes
i=1
where u N i =
αυN i
+ βυ N θˆi i
+ γυ N δˆi i
1 + ∇λsi − α˙ , υˆ Ni ρi0
u Bi = αυ B +βυ B θˆi +γυ B δˆi − i
i
i
1 ∇αi −α˙ , υˆ Bi pi
u Ti = αυT + βυT θˆi + γυT δˆi − α˙ i
i
n k5 ∂ξi ai j ξ¯i − ξ¯ j , ∂φi ρi0 1 − λsi cos αi j=1
+
β υˆ Ni
θˆi = αθˆ + βυ N + βυ B + βυT − υ˜ Ni ˙ i
i
i
β υˆ Bi
i
β υˆ Ti
− υ˜ Bi ˙ υ˜ Ti ˙ , δˆi = γυ N + γυ B + γυT . i
123
i
i
1 υ˜ N + λi θ˜ i , ρi0 i
(50a)
1 β θ˜i+γυ N δ˜i − υ˙˜ Ni = βυ N − ˙ ∇λsi −k2 υ˜ Ni , υ ˆ i i Ni ρi0 (50b) α˙ i = −ϕαi −
υˆ Ti
i
λ˙ si = −ϕλsi +
1 υ˜ B + αi θ˜ i , pi i
(50c)
1 ˙υ˜ B = β −β θ˜i +γυ B δ˜i + ∇αi −k3 υ˜ Bi , υ Bi i υˆ˙ Bi i pi (50d) 1 ∂ξi ξ˙¯ i = −k1 ξ¯i + υ˜ T + ξi θ˜ i . ∂φi ρi0 1 − λsi cos αi i (50e)
A geometric extension design
β θ˜i + γυT δ˜i − k4 υ˜ Ti υ˙˜ Ti = βυT − ˙ i
−
υˆ Ti
v1
v3
i
n k1 ∂ξi ai j ξ¯i − ξ¯ j , ∂φi ρi0 1 − λsi cos αi j=1
θ˙˜iT = −γ1 θˆi ,
(50f) (50g)
δ˙˜iT = −γ2 δˆi .
(50h)
Remark 5 From (27) to (47), one can obviously see that the adaptive upgrade law for θi uses the neighbor-toneighbor information such as ξ¯i − ξ¯ j , and thus, we call it the coordinated adaptive estimator, which is different from the estimation of the unknown uniform, timeinvariant/rotating flowfield based on the agent’s position [25,26] and the distributed information-consensus filter [27]. Although the adaptive estimator (47) becomes more complex compared with the estimator used in [25–27], it can accomplish the estimate of the spatiotemporal flowfield and avoid to setting the time period of consensus filter that is required to be greater than the consensus time of information matrix and measurement. From the precedence of adaptive backstepping design, we give the following theorem. Theorem 1 Consider a family of desired orbit on a set of given spheres. Assume that each given circle orbit Ci associated with the ith agent is a circle located at 2 and all the spheres are concena spherical surface Si0 2 , a set of level tric. Through concentric compressing Si0 spheres of the sphere function is given by Lemma 1. Also assume that each generalized azimuth angle satisfies Assumption 1. Suppose that the initial conditions of agents make the initial value of VI I given in (42) finite. When the bidirectional communication topology is connected, robust spherical formation tracking control problem in unknown spatiotemporal flowfields is solved via ⎡
⎤−1 ⎡ n ⎤ gi NiT T ⎣ ⎦ ⎣ u i = Bi gib ⎦ , T Ti git
(51)
with the adaptive control (47)–(48), where gin , gib and gib are given in (44)–(46), respectively.
v5
v2
v4
Fig. 3 Communication topology
Proof The set I I = {(λsi , υ˜˙ Ni , αi , υ˙˜ Bi , ξ¯i − ξ¯ j , υ˙˜ Ti , θ˙˜iT , δ˙˜iT ) |VI I c} such that VI I ≤ c, for c > 0 is closed by continuity. Because of |λsi | ≤ ρi0 , υ˙˜ Ni ≤ √ √ 4c 2c, |αi | ≤ π2 , υ˙˜ Bi ≤ 2c, ξ¯i − ξ¯ j ≤ kξ , √ √ √ ˙ ˜˙ T ˜˙ T υ˜ Ti ≤ 2c, θi ≤ 2c and δi ≤ 2c, the set I I is compact. Then, the closed-loop system (50) is Lipschitz continuous on the set I I , and a solution exists and is unique. Note that the value of VI I is time-independent and non-increasing, we conclude that if the initial value of VI I is finite, then the entire solution stays in I I which means VI I is finite for all time. This implies that (5) and (7) are tenable by (C3)–(C4) and (C6)–(C7). Applying the invariance theorem, it follows that as t → ∞, the trajectories of closed-loop system equations (50) will converge to the set inside the region E I I = ˙ ˙ T T ˙ ˙ ˙ ˙ ¯ ¯ ˜ ˜ VI I = 0 λsi , υ˜ Ni , αi , υ˜ Bi , ξi − ξ j , υ˜ Ti , θi , δi that is ∇λsi ϕλsi = 0,
(52a)
∇αi ϕαi = 0,
(52b)
ξ¯i − ξ¯ j = ξi − ξ j = 0,
(52c)
υ˙˜ Ni = υ˙˜ Bi = υ˙˜ Ti = 0.
(52d)
From (52a) to (C5), one can easily see that λsi → 0 as t → ∞. Similarly, αi → αi∗ as t → ∞ according to (52b) and (C8). Remark 6 The control input (51) has an unique solution due to the fact that Di = 1, where
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i
4.5
(f)
(a)
y
θ 1 estimation
4
y 2 y
θ estimation
3.5
θ yi estimation
Fig. 4 Formation motion along different orbits on the same sphere: a Plot of movements, b Plot of λsi , c Plot of αi , d Plot of ξi − ξ j , y e Plot of θˆix , f Plot of θˆi , g z Plot of θˆi , h Plot of δˆix , i y Plot of δˆ , j Plot of δˆ z
θ 3 estimation y 4
3
θ estimation
2.5
θ y estimation 5
2 1.5 1
i
0.5 0 0
(b)
(g)
0.5
5
10
t
15
20
25
4 3.5
λ
si
λ
s1
λs
−0.5
2
λs λ λ
−1.5 0
5
10
15
θ z estimation θ z estimation 2
θ z3 estimation
2
s4
1.5
s5
1
20
1
2.5
i
3
−1
3
θ z estimation
0
0.5 0
25
θ z estimation 4
θ z5 estimation
5
10
(c) 1.5
(h) 30
α
1
x δ i estimation
α
4
0.5
α
α
i
5
0
25
δ x2 estimation
20
α3
20
δ x1 estimation
25
α2
1
15
t
t
x
δ 3 estimation
15
δ x estimation 4 x
10
δ 5 estimation
5 0 −5
−0.5
−10 −15
−1 0
5
10
15
20
−20 0
25
5
10
3
ξ1−ξ2 2
2
ξ4−ξ5 ξ5−ξ1
j 0 −1
δ y estimation 2 y
δ 3 estimation δ y estimation 4 y 5
5
δ estimation
0
−5
5
10
15
20
−10 0
25
5
10
(j)
(e) 1.6 1.4
1 0.8
x
θ 1 estimation
0.6
x 2 x 3 x 4 x 5
θ estimation
0.4
θ estimation
0.2
θ estimation
0
θ estimation
−0.2 5
10
15
t
20
25
δ iz estimation
x
θ i estimation
20
25
20 15
1.2
123
15
t
t
−0.4 0
25
1
−2 −3 0
20
δ y estimation
10
i
i
15
3
ξ3−ξ4
1
ξ −ξ
(i)
ξ −ξ
δ y estimation
(d)
15
t
t
10 5 0
z
δ 1 estimation
−5
δ estimation
−10
δ z estimation
−15
δ 4 estimation
−20 0
z 2 3 z z
δ 5 estimation 5
10
15
t
20
25
A geometric extension design
⎤ NiT Di = ⎣ BiT ⎦ TT ⎡ i x p − i ⎢ pi z ⎢ pix pi =⎢ ⎢ px yi pi ⎣ py ⎡
px y i
i
p
y
pz
− pii y
p pz i i px y pi i px − p i xy
− pii 2 ( p x )2 + p y − pi pi x yi
0
i
⎤ ⎥ ⎥ ⎥. ⎥ ⎦
i
4 Simulation results In this section, we give a simulation case about five agents’ spherical formation tracking motions with unknown spatiotemporal flowfields. The bidirectional communication topology among agents is shown in Fig. 3. The 1st, 2nd and 3rd agents are required to fly on the same sphere with the fixed radius 10 while the radius of given spheres associated with the 4th and 5th agents are 6. The given circles associated with agents are denotes as α1 = π/6, α2 = 0, α3 = −π/6, α4 = −π/6, α5 = π/6. The desired formation is a pentagon, and thus, we choose the general center angles as ξi = φi and the consensus dynamics is ξ˙ ∗ (t) = 2. The external flowfields are ⎡ ⎤⎡ ⎤ 1 sin pix sin piz sin t y z ⎦ ⎣ ⎣ 2⎦, cos t f υi = f θi θi = cos pi sin pi y x 3 0 cos( pi + pi ) 0 ⎡ ⎤ ⎡ ⎤ sin pix 0 1 cos t y ⎣ ⎦⎣1⎦, f ai = f δi δi = 0 cos pi 0 y sin piz sin pi 0 1 where θi and δi are unknown. The control gains are selected as k1 = k2 = k3 = k4 = 10, kξ = γ1 = γ2 = 1. The movement of agents is shown in Fig. 4a. From this figure, we can see that agents finally arrive at the given sphere and move along the set of given orbits with the desired formation. The sphere landing error λsi tends to zero and is plotted in Fig. 4b. Figure 4c shows that αi converges to αi∗ . Figure 4d demonstrates ξi reaches consensus. Figure 4e–j show the dynamics of θˆi and δˆi . According to these pictures, robust spherical formation tracking control problem in unknown spatiotemporal flowfields can be solved via our proposed controller. 5 Conclusion In this paper, the geometric extension design is developed to deal with 3D surface extension and incorpo-
rated into adaptive backstepping method to achieve the estimations of the unknown flowfields and the robust spherical formation tracking control. In ongoing work, we will devote ourselves to the coordination problem of formation surrounding superellipsoid. Acknowledgements This work was supported in part by the National Natural Science Foundation of China Under Grants 61673106, 61473081, 61473138, 61503329, Natural Science Foundation of Jiangsu Province Under Grant BK20141341 and in part by a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
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