Delft University of Technology
Neuro-Adaptive Cooperative Tracking Rendezvous of Nonholonomic Mobile Robots
Lu, Peifen; Baldi, Simone; Chen, Guanrong; Yu, Wenwu DOI
10.1109/TCSII.2020.2969373 Publication date
2020
Document Version
Accepted author manuscript Published in
IEEE Transactions on Circuits and Systems II: Express Briefs
Citation (APA)
Lu, P., Baldi, S., Chen, G., & Yu, W. (2020). Neuro-Adaptive Cooperative Tracking Rendezvous of
Nonholonomic Mobile Robots. IEEE Transactions on Circuits and Systems II: Express Briefs, 67(12), 3167-3171. https://doi.org/10.1109/TCSII.2020.2969373
Important note
To cite this publication, please use the final published version (if applicable). Please check the document version above.
Copyright
Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons. Takedown policy
Please contact us and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim.
This work is downloaded from Delft University of Technology.
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/338807246
Neuro-Adaptive Cooperative Tracking Rendezvous of Nonholonomic Mobile
Robots
Article in Circuits and Systems II: Express Briefs, IEEE Transactions on · January 2020
DOI: 10.1109/TCSII.2020.2969373 CITATIONS 0 READS 30 4 authors, including:
Some of the authors of this publication are also working on these related projects:
LOCAL4GLOBAL: “Systems-of-systems that act locally for optimizing globally”View project
ACMMS: “Adaptive Control with Multiple Models and Switching”View project Simone Baldi
Southeast University (China)
141PUBLICATIONS 1,472CITATIONS
SEE PROFILE
All content following this page was uploaded by Simone Baldi on 03 February 2020.
Neuro-Adaptive Cooperative Tracking
Rendezvous of Nonholonomic Mobile Robots
Peifen Lu, Simone Baldi, Senior Member, IEEE, Guanrong Chen, Fellow, IEEE,
Wenwu Yu, Senior Member, IEEE,
Abstract—This paper proposes a neuro-adaptive method for the unsolved problem of cooperative tracking ren-dezvous of nonholonomic mobile robots (NMRs) subject to uncertain and unmodelled dynamics. A hierarchical coop-erative control framework is proposed, which consists of a novel distributed estimator along with local neuro-adaptive tracking controllers. Rigorous stability analysis as well as simulation experiments illustrate the proposed method.
Index Terms—Nonholonomic mobile robot, cooperative rendezvous, neuro-adaptive control, distributed estimator.
I. INTRODUCTION
Cooperative control of nonholonomic mobile robots
(NMRs) is a challenging research topic with applications in tracking, rendezvous and formation of autonomous vehicles [1]. When designing controllers to achieve such tasks, the NMR dynamics are usually assumed to be known [2]–[4]. However, more challenges arise if the NMR dynamics are uncertain. Uncertainties in NMRs arise from unmodelled dy-namics, friction, resistance, velocity-controlled motor dynam-ics, and so on. For a single uncertain mobile robot, various adaptive and neural-adaptive designs have been proposed [5]– [7]. However, teams of uncertain NMRs have seldom been considered: research has mostly been focused on first-order [8], [9], second-order [10], or higher-order uncertain integra-tors [11]–[13], which do not exhibit all the challenges of nonholonomic dynamics.
This work was supported by the Fundamental Research Funds for the Central Universities and Postgraduate Research & Practice Innovation Program of Jiangsu Province (NO. KYCX17 0040), the State Key Lab-oratory of Intelligent Control and Decision of Complex Systems (Beijing Institute of Technology, China), the National Natural Science Foundation of China (Grant No. 61703099, 61673107), the China Postdoctoral Sci-ence Foundation (Grant No. 2017M621589), the National Ten Thousand Talent Program for Young Top-notch Talents (Grant No. W2070082), the Cheung Kong Scholars Programme of China for Young Scholars (Grant No. Q2016109), the Jiangsu Provincial Key Laboratory of Networked Collective Intelligence (Grant No. BM2017002), the Fundamental Re-search Funds for the Central Universities (Grant No. 4007019109), and the special guiding funds for double first-class (Grant No. 4007019201). (Corresponding author: S. Baldi)
P. Lu and W. Yu are with School of Mathematics, Southeast Uni-versity, Nanjing 210096, P.R. China (e-mail: lupeifen2014@163.com; wwyu@seu.edu.cn).
S. Baldi is with School of Mathematics, Southeast University, 211189 Nanjing, China, and guest with Delft Center for Systems and Control, TU Delft, 2628CD Delft, The Netherlands (e-mail: s.baldi@tudelft.nl).
G. Chen is with the Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, P.R. China (e-mail: eegchen@cityu.edu.hk).
The main contribution of this paper is to address and solve the cooperative rendezvous problem for uncertain NMRs. A hierarchical cooperative control framework is proposed, con-sisting of a distributed estimator to asymptotically reconstruct the leader NMR state information, and local neuro-adaptive tracking controllers to approximate the uncertain nonlinear-ities. Although the estimator/observer idea was adopted in [14] (fractional-order integrators), [12], [15], [16] (first-order integrators), [10], [11], [17] (second-order integrators), [18]– [21] (high-order linear dynamics), the novelty of the proposed estimator is to deal with the nonholonomic dynamics. Due to the nonholonomic dynamics, the adaptive controllers, error systems and Lyapunov functions are more challenging to deal with than those in [10]–[12], [14]–[19].
Notations: R and R+denote the sets of real and positive real
numbers, respectively. For a column vector x ∈ Rn, denote
||x||1 and ||x|| the 1- and 2-norm, respectively. For a matrix
M , define ||M ||F =
p
tr{MTM }. Denote ¯σ(M ) and σ(M )
the maximum and minimum singular value. 1nand 0n are the
n × 1 all-one vector and the n × 1 all-zero vector, respectively. max f (·) represents the maximum of the function f (·).
Consider a directed graph G = (V(G), ε(G)) with node set V = {1, 2, · · · , N } and edge set ε ⊂ V(G) × V(G).
The adjacency matrix A = (aij)N ×N of G is defined as:
aij 6= 0 if (j, i) ∈ ε(G) and 0 otherwise; aii = 0 for
each i ∈ V(G). The Laplacian matrix L = (lij)N ×N is
defined as: lij = −aij, i 6= j, and lii = P
j∈Ni
aij for
i = 1, 2, · · · , N . The set of neighbors for node i is represented
by Ni= {j ∈ V|(j, i) ∈ ε(G)}. In a leader-follower graph,
the leader is denoted as node 0 and the followers as nodes {1, · · · , N }. Including the leader as node 0 results in the
augmented graph ¯G [22]. The leader adjacency matrix is a
diagonal matrix B = diag(b1, · · · , bN), where bi > 0 if the
follower agent i has access to the leader’s information (state)
and bi= 0 otherwise. Denote H = L + B.
II. PROBLEMSTATEMENT
Consider a group of N (≥ 2) NMRs described by ˙ xi(t) =(vi(t)+fi(xi(t), yi(t), θi(t))+ξ1i(t)) cos θi(t), ˙ yi(t) =(vi(t)+fi(xi(t), yi(t), θi(t))+ξ1i(t)) sin θi(t), ˙ θi(t) =ωi(t)+gi(xi(t), yi(t), θi(t))+ξ2i(t), (1) where i = 1, 2, · · · , N , pi(t) = (xi(t), yi(t))T ∈ R2 is the
position of the ith NMR in the global coordinate, θi(t) is its
orientation, the linear and angular velocity, vi(t) ∈ R and
ωi(t) ∈ R are control inputs to be designed; the unknown
© 2020 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
and unmodeled nonlinear functions fi(xi(t), yi(t), θi(t)) and
gi(xi(t), yi(t), θi(t)) will be denoted as fi(t) and gi(t) for
compactness; ξ1i(t) ∈ R and ξ2i(t) ∈ R are external bounded
disturbance signals, with known upper bounds kξ1(t)k ≤ ¯ξ1
and kξ2(t)k ≤ ¯ξ2, ∀t, with ξ1(t) = (ξ11(t), · · · , ξ1N(t)),
ξ2(t) = (ξ21(t), · · · , ξ2N(t)).
Using neuro-adaptive ideas [8], [10], [11], consider the approximations of the nonlinearities in (1) on a compact set
Ωi∈ R3, i.e. fi(Λi(t)) = ΓT1iϕ1i(Λi(t)) + ε1i(t), gi(Λi(t)) = ΓT2iϕ2i(Λi(t)) + ε2i(t), where Λi(t) = (xi(t), yi(t), θi(t))T, ϕ1i(Λi(t)) ∈ R v1i and ϕ2i(Λi(t)) ∈ R
v2i are basis sets for node i, Γ
1i ∈
Rv1i and Γ2i ∈ R
v2i are unknown coefficients, and ε
1i,
ε2i are bounded approximation errors. For compactness,
let us define the global network nonlinearities: f (Λ(t)) =
ΓT 1ϕ1(Λ(t)) + ε1(t) and g(Λ(t)) = ΓT2ϕ2(Λ(t)) + ε2(t), where Λ(t) = (ΛT 1(t), · · · , ΛTN(t))T, ϕ1= (ϕT11, · · · , ϕ T 1N) T, ϕ2 = (ϕT21, · · · , ϕ T 2N) T, ΓT 1 = diag ΓT11, · · · , Γ T 1N, Γ T 2 = diag ΓT 21, · · · , Γ T 2N, ε1(t) = (ε11(t), · · · , ε1N(t)) T and ε2(t) = (ε21(t), · · · , ε2N(t))
T. Moreover, define the unknown
upper bounds kΓ1kF ≤ Γ1m, kΓ2kF ≤ Γ2m, kϕ1(Λ(t))k ≤
φ1m and kϕ2(Λ(t))k ≤ φ2m.
The NMRs in (1) are required to track a leader NMR ˙ x0(t) = v0(t) cos θ0(t), (2a) ˙ y0(t) = v0(t) sin θ0(t), (2b) ˙ θ0(t) = ω0(t), (2c)
where p0(t) = (x0(t), y0(t))T ∈ R2is the reference trajectory,
θ0(t) is the reference orientation. The reference linear and
angular velocity are v0(t), and ω0(t), satisfying |v0(t)| ≤ ¯v0,
|ω0(t)| ≤ ¯ω0, | ˙v0(t)| < ρ1 and | ˙ω0(t)| < ρ2, for known
bounds ¯v0, ¯ω0, ρ1 and ρ2.
Assumption 2.1 The augmented graph ¯G describing the
interconnections of (1) and (2) is strongly connected, implying that it contains a directed spanning tree with the root node being the leader NMR.
Remark 2.1 It was proven that Assumption 2.1 guarantees that H = (L + B) is nonsingular [23]. A detailed explanation for the non-singularity of matrix H is in [22].
Before moving on, the following lemma is needed. Lemma 2.1 [24]: Suppose that h(x(t), t) is essentially bounded and that the origin is a Filippov solution (denoted
with 0n∈ F [h(0n, t)]) of the differential system
˙
x = h(x(t), t). (3)
Let V (·) : Rn → R satisfy V (0
n) = 0 and 0 < V1(kx(t)k) ≤
V (x(t)) ≤ V2(kx(t)k) for x(t) 6= 0n, where V1(·) and
V2(·) are K-class functions. System (3) is uniformly
asymp-totically stable if there exists a K-class function k(·) such
that maxV (x(t)) ≤ −k(x(t)) < 0 for all x(t) 6= 0˙˜ n, where
˙˜
V (x(t)) is the set-valued Lie derivative along the trajectories of (3).
III. THE PROPOSEDDISTRIBUTEDESTIMATOR
Since some of the follower NMRs cannot directly obtain the leader information necessary to solve the tracking rendezvous, we propose the following novel distributed estimator to
esti-mate the position (x0(t), y0(t))T and the orientation θ0(t) of
the leader NMR by the ith follower, i = 1, ..., N . ˙ˆxi(t) =ˆvi(t) cos ˆθi(t)+bi(x0(t)− ˆxi(t))+ X j∈Ni (ˆxj(t)− ˆxi(t)), ˙ˆyi(t) =ˆvi(t) sin ˆθi(t)+bi(y0(t)− ˆyi(t))+ X j∈Ni (ˆyj(t)− ˆyi(t)), ˙ˆ θi(t) =ˆωi(t) + bi θ0(t) − ˆθi(t) + X j∈Ni ˆθj(t) − ˆθi(t) , ˙ˆvi(t) =α1 bi(v0(t) − ˆvi(t)) + X j∈Ni (ˆvj(t) − ˆvi(t)) + β1sgn bi(v0(t)− ˆvi(t))+ X j∈Ni (ˆvj(t)− ˆvi(t)) , ˙ˆ ωi(t) =α2 bi(ω0(t) − ˆωi(t)) + X j∈Ni (ˆωj(t) − ˆωi(t)) + β2sgn bi(ω0(t)− ˆωi(t))+ X j∈Ni (ˆωj(t)− ˆωi(t)) . (4) Lemma 2.2 Consider the NMRs given by (1) and (2). Under
Assumption 2.1, if the gains β1 and β2 are chosen such that
β1> ρ1, β2> ρ2, then, the distributed estimators (4) converge
asymptotically to the state of the leader NMR.
Proof. Define ˜x(t) := ˆx(t) − x0(t)1N, ˜y(t) := ˆy(t) −
y0(t)1N, ˜θ(t) := ˆθ(t) − θ0(t)1N, ˜v(t) := ˆv(t) − v0(t)1N and ˜ ω(t) := ˆω(t) − ω0(t)1N, with ˆx(t) = (ˆx1(t), · · · , ˆxN(t))T, ˆ y(t) = (ˆy1(t), · · · , ˆyN(t))T, ˆθ(t) = (ˆθ1(t), · · · , ˆθN(t))T, ˆ v(t) = (ˆv1(t), · · · , ˆvN(t))T and ˆω(t) = (ˆω1(t), · · · , ˆωN(t))T. Let ξ(t) := H ˜v(t) and Ξ(t) := H ˜ω(t).
Consider the Lyapunov function
V1(Ξ(t)) = ΞT(t)KΞ(t), (5)
where K = diag (ki) ≡ diag (1/qi), with q =
(q1, · · · , qN)T = (L + B)−11N. One can thus get the
set-valued derivative of V1(Ξ(t)) along ˙Ξ(t) as
˙ V1(Ξ(t)) =−α2ΞT(t)QΞ(t)−β2Fsgn(Ξ(t))THTKΞ(t) −β2FΞT(t)KHsgn(Ξ(t))−˙ω0(t)1TNH TKΞ(t) − ˙ω0(t)ΞT(t)KH1N. (6)
Meanwhile, one has
− β2Fsgn(Ξ(t))THTKΞ(t) = − β2Fsgn(Ξ(t))T(L + diag(b1, · · · , bN))TKΞ(t) =β2 X i=1 ki X j=1,i6=j lijF [kΞi(t)k1− sgn(Ξj(t))Ξi(t)] − β2 X i=1 bikikΞi(t)k1. (7)
According to the property of 1-norm, it is trival to show that
−β2Fsgn(Ξ(t))THTKΞ(t) ≤ −β2P
i=1
bikikΞi(t)k1. (8)
In addition, because | ˙ω0(t)| ≤ ρ2, one has
− ˙ω0(t)1TNH TKΞ(t) = − ˙ω0(t)1TN(KL + diag(b1k1, · · · , bNkN)) T Ξ(t) ≤| ˙ω0(t)| X i=1 bikikΞi(t)k1≤ ρ2 X i=1 bikikΞi(t)k1. (9)
it follows that max ˙V1(Ξ(t)) ≤ −α2ΞT(t)QΞ(t) ≤ 0, where
Q = K(L + B) + (L + B)TK. This implies that Ξ(t), and
thus ˜ω(t) converge to 0N asymptotically. Similarly, under the
assumption of | ˙v0(t)| < ρ1, consider the Lyapunov function
V2(ξ(t)) = ξT(t)Kξ(t), (10)
one can verify that the derivative of V2 along ˙ξ(t) satisfies
max ˙V2(ξ(t)) ≤ −α1ξT(t)Qξ(t) ≤ 0. So, ˜v(t) converges to 0
asymptotically as well.
The time derivative of ˜θ(t) along with (2c) and the
estima-tion dynamics is given by ˙˜
θ(t) = −H ˜θ(t) + ˆω(t) − ω0(t)1N = −H ˜θ(t) + ˜ω(t).
It is easy to get the solution θ(t)˜ = e−Htθ(0) +˜
Rt
0e
−H(t−τ )ω(τ )dτ. Since ˜˜ ω(t) converges to 0
N
asymptot-ically and H is a Hurwitz matrix, one has lim t→∞ ˜ θ(t) = lim t→∞e −Htθ(0) + lim˜ t→∞ Rt 0e −H(t−τ )ω(τ )dτ = 0˜ N.
In addition, one obtains ˙˜ x(t) =diagcos ˆθi(t) (ˆv(t) − v0(t)1N) + v0(t)diag cos ˆθi(t) − cos θ0(t) 1N− H ˜x(t).
Similarly, let $(t) := diagcos ˆθi(t)
(ˆv(t) − v0(t)1N) + v0(t)diag cos ˆθi(t) − cos θ0(t) 1N. Then, ˜ x(t) = e−Htx(0) +˜ Z t 0 e−H(t−τ )$(τ )dτ.
Since ˜θ(t) and ˜v(t) converge to 0N asymptotically, one gets
lim t→∞$(t) = limt→∞diag cos ˆθi(t) (ˆv(t) − v0(t)1N) + lim t→∞v0(t)diag cos ˆθi(t)−cos θ0(t) 1N= 0N.
Therefore, it can be verified that lim
t→∞(ˆx(t) − x0(t)1N) = 0N.
In a similar way, one can verify lim
t→∞(ˆy(t) − y0(t)1N) = 0N.
IV. DISTRIBUTEDPROTOCOLDESIGN
A distributed neuro-adaptive controller is adopted to deal with the uncertainties in (1).
(
v(t) = − ˆf (t) + c1xe(t) + diag (cos θei(t)) ˆv(t), (11a)
ω(t) = ˆω(t)− ˆg(t)+c2sin θe(t)+diag (yei(t)) ˆv(t), (11b)
where c1, c2 > 0, v(t) = (v1(t), · · · , vN(t))T,
ω(t) = (ω1(t), · · · , ωN(t))T, xe(t) = (xe1(t), · · · , xeN(t))
T,
sin θe(t) = (sin θe1(t), · · · , sin θeN(t))
T, f (t)ˆ = ˆf1(Λ1(t)), · · · , ˆfN(ΛN(t))T and ˆg(t) = (ˆg1(Λ1(t)), · · · , ˆgN(ΛN(t))) T with xei(t) yei(t) θei(t) = cos θi(t) sin θi(t) 0 − sin θi(t) cos θi(t) 0 0 0 1 ˆ xi(t)−xi(t) ˆ yi(t)−yi(t) ˆ θi(t)−θi(t) . (12)
and the neuro-adaptive approximators fˆi(Λi(t)) =
ˆ ΓT 1i(t)ϕ1i(Λi(t)), ˆgi(Λi(t)) = ˆΓ T 2i(t)ϕ2i(Λi(t)), where ˆ Γ1i(t) ∈ R v1i and ˆΓ 2i(t) ∈ R v2i are designed as ˙ˆΓ1i(t) = −F1iϕ1i(Λi(t))xei(t) − κ1F1iΓˆ1i(t), ˙ˆΓ2i(t) = −F2iϕ2i(Λi(t)) sin θei(t) − κ2F2iΓˆ2i(t), (13) where F1i = ρ1iIv1i, F2i = ρ2iIv2i and ρ1i > 0, ρ2i > 0,
κ1> 0 and κ2> 0 are scalar tuning gains.
For the global network, the estimators can be written
as ˆf (Λ(t)) = ˆΓT 1(t)ϕ1(Λ(t)), ˆg(Λ(t)) = ˆΓT2(t)ϕ2(Λ(t)), where ˆΓT 1(t) = diag ˆΓT11(t), · · · , ˆΓ T 1N(t) and ˆΓT 2(t) = diag ˆΓT 21(t), · · · , ˆΓ T 2N(t) .
Remark 4.1 The sign function in (4) allows to reconstruct the information of the leader asymptotically (sliding mode-observer). If the sign function is replaced with a sigmoid or a saturation function, chattering will be avoided at the price of losing asymptotic reconstruction of the leader’s state.
V. MAINRESULTS
Theorem 3.1 For a sufficiently large number of neurons ¯v1i
and ¯v2i, i = 1, · · · , N, the distributed control protocol (11)
with adaptive law (13) guarantees the overall cooperative error
vectors xe(t), ye(t), sin θe(t) and the parametric estimation
errors ˜Γ(t) to be uniformly ultimately bounded.
Proof. Based on (12), one has ˙ xei(t) =(ωi(t) + gi(t) + ξ2i(t))yei(t) +vˆi(t) cos ˆθi(t)−(vi(t)+fi(t)+ξ1i(t)) cos θi(t) +bi(x0(t)− ˆxi(t))+ X j∈Ni (ˆxj(t)− ˆxi(t)) cos θi(t) +vˆi(t) sin ˆθi(t)−(vi(t)+fi(t)+ξ1i(t)) sin θi(t) +bi(y0(t)− ˆyi(t))+ X j∈Ni (ˆyj(t)− ˆyi(t)) sin θi(t), ˙ yei(t) = − (ωi(t) + gi(t) + ξ2i(t))xei(t) −vˆi(t) cos ˆθi(t)−(vi(t)+fi(t)+ξ1i(t)) cos θi(t) +bi(x0(t)− ˆxi(t))+ X j∈Ni (ˆxj(t)− ˆxi(t)) sin θi(t) +vˆi(t) sin ˆθi(t)−(vi(t) + fi(t)+ξ1i(t)) sin θi(t) +bi(y0(t)− ˆyi(t))+ X j∈Ni (ˆyj(t)− ˆyi(t)) cos θi(t), ˙ θei(t) =ˆωi(t) + bi θ0(t) − ˆθi(t) − ωi(t) − gi(t) − ξ2i(t) + X j∈Ni ˆθj(t) − ˆθi(t) . (14) Combining (1) and (4), the system (14) can be written as
˙ xe(t) =diag (ωi(t) + gi(t) + ξ2i(t)) ye(t) − (v(t) + f (t) + ξ1(t)) + diag (cos θei(t)) ˆv(t)
−diag (cos θi(t)) H ˜x(t)−diag (sin θi(t)) H ˜y(t),
˙
ye(t) = − diag (ωi(t) + gi(t) + ξ2i(t)) xe(t)
+diag (sin θei(t)) ˆv(t)+diag (sin θi(t)) H ˜x(t)
− diag (cos θi(t)) H ˜y(t),
˙
θe(t) = − H ˜θ(t) + ˆω(t) − ω(t) − g(t) − ξ2(t).
(15) Now, the function estimation error is
( ˆf (Λ(t)) − f (Λ(t)) = ˜ΓT
1(t)ϕ1(Λ(t)) − ε1,
ˆ
g(Λ(t)) − g(Λ(t)) = ˜ΓT2(t)ϕ2(Λ(t)) − ε2,
where ˜ΓT
1(t) = ˆΓT1(t) − ΓT1 and ˜ΓT2(t) = ˆΓT2(t) − ΓT2 are the
parameter estimation errors. Consider the Lyapunov function
V3(t) = 1 2x T e(t)xe(t) + 1 2y T e(t)ye(t) + 1TN(1N − cos θe(t)) +1 2tr n ˜ΓT 1(t)F1−1Γ˜1(t) o +1 2tr n ˜ΓT 2(t)F2−1Γ˜2(t) o +1 2x˜ T(t)˜x(t)+1 2y˜ T(t)˜y(t)+1 2 ˜ θT(t)˜θ(t)+ξT(t)P ξ(t) + ΞT(t)P Ξ(t), (17)
where F−1 is the inverse of F = diag(F1, · · · , FN).
Using (13) and (16), one gets
xTe(t) ˆf (t) − f (t) − ξ(t)+ trn ˜ΓT1(t)F1−1˙˜Γ1(t) o ≤ − κ1 ˜ Γ1(t) 2 F − κ1Γ1m ˜ Γ1(t) F+ ε1m+ ¯ξ1 kxe(t)k and (sin θe(t))T(ˆg(t) − g(t) − ξ2(t)) + trn ˜ΓT2(t)F −1 2 ˙˜Γ2(t) o ≤ −κ2 ˜ Γ2(t) 2 F−κ2 Γ2m ˜ Γ2(t) F+ ε2m+ ¯ξ2 k sin θe(t)k.
By Lemma 2.2, it follows that maxV˙˜1(Ξ(t)) ≤
−α2ΞT(t)QΞ(t), maxV˙˜2(ξ(t)) ≤ −α1ξT(t)Qξ(t),
k˜ω(t)k ≤ 1
σ(H)kΞ(t)k and k˜v(t)k ≤
1
σ(H)kξ(t)k. By
calculating the set-valued Lie derivative of V3(t), one has
˙
V3(t) ≤
− c1kxe(t)k2+ ¯σ(H)kxe(t)kk˜x(t)k + ¯σ(H)kxe(t)kk˜y(t)k
+ ¯σ(H)kye(t)kk˜x(t)k + ¯σ(H)kye(t)kk˜y(t)k
+ ¯σ(H)k sin θe(t)k ˜ θ(t) − c2k sin θe(t)k 2− κ 1 ˜ Γ1(t) 2 F − κ1Γ1m ˜ Γ1(t) F+ ε1m+ ¯ξ1 kxe(t)k − κ2 ˜ Γ2(t) 2 F −κ2Γ2m ˜ Γ2(t) F+ ε2m+ ¯ξ2ksin θe(t)k+ 1 σ(H)k˜x(t)kkξ(t)k + k˜x(t)k2¯v0 √ N − σ(H)k˜x(t)k2+ 1 σ(H)k˜y(t)kkξ(t)k + k˜y(t)k2¯v0 √ N − σ(H)k˜y(t)k2− σ(H) ˜ θ(t) 2 + 1 σ(H) ˜ θ(t) kΞ(t)k − α2σ(Q)kΞ(t)k 2− α 1σ(Q)kξ(t)k2. (18) Let z(t) =kxe(t)k, ksin θe(t)k, ˜ Γ1(t) F, ˜ Γ2(t) F, k˜x(t)k, k˜y(t)k, ˜ θ(t) , kΞ(t)k, kξ(t)k T
. Then, (18) can be written as ˙ V3(t) ≤−zT(t)Rz(t)+rTz(t)+ ¯σ(H)(k˜x(t)k+k˜y(t)k) p 2V3(t), (19) where R = c1 0 0 0 c c 0 0 0 0 c2 0 0 0 0 c 0 0 0 0 κ1 0 0 0 0 0 0 0 0 0 κ2 0 0 0 0 0 ∗ 0 0 0 d 0 0 0 − 1 2σ(H) ∗ 0 0 0 0 d 0 0 − 1 2σ(H) 0 ∗ 0 0 0 0 d − 1 2σ(H) 0 0 0 0 0 0 0 ∗ α2σ(Q) 0 0 0 0 0 ∗ ∗ 0 0 α1σ(Q) , c = −12σ(H),¯ d = σ(H) and r = ε1m + ξ¯1, ε2m + ¯ ξ2, −κ1Γ1m, −κ2Γ2m, 2¯v0 √ N , 2¯v0 √ N , 0, 0, 0.
Define Vz(z) = zTRz + rTz, which is positive definite if
R is positive definite, and kzk > σ(R)krk , i.e. far enough from
the origin. According to the Sylvester criterion, R is positive definite when c1> 0, c1c2> 0, c1c2κ1> 0, c1c2κ1κ2> 0, c1c2κ1κ2d > 0, c1c2κ1κ2dg > 0, c1c2κ1κ2dgh > 0, c1c2κ1κ2dghα2σ(Q) > 0, c1c2κ1κ2dghα2σ(Q)α1σ(Q) > 0. By Lemma 2.1,R0tσ(H)k˜¯ x(t)kds andRt 0¯σ(H)k˜y(t)kds are
all bounded. Hence, ¯σ(H)k˜x(t)k ∈ L1 and ¯σ(H)k˜y(t)k ∈ L1
are nonnegative. Combining kye(t)k ≤p2V3(t), one obtains
¯
σ(H)kye(t)kk˜x(t)k + ¯σ(H)kye(t)kk˜y(t)k ≤ ¯σ(H)(k˜x(t)k +
k˜y(t)k)p2V3(t). By using (19), one has ˙V3(t) ≤ p(t)pV3(t),
where p(t) := √2¯σ(H)(k˜x(t)k + k˜y(t)k) and p(t) ∈ L1,
implying d √ V3(t) dt ≤ p(t) 2 , or equivalently pV3(t) ≤ pV3(0) + Rt 0 p(s) 2 ds. Since p(t) ∈ L1, V3(t) is bounded,
implying the existence of a positive constant δ such that,
for each l > 0, pV3(t) ≤ δ, ∀pV3(0) ≤ l. Then, from
(19), for pV3(0) ≤ l, one has ˙V3(t) ≤ −Vz(z(t)) +
p(t)δ, which implies the following nonincreasing relation d dt V3(t) − δR t 0p(s)ds ≤ 0.
Since V3(t) is bounded from below by zero, V3(t) converges
to some nonnegative constants.
According to (17), kxe(t)k,kye(t)k,k˜x(t)k,k˜y(t)k and
˜ θ(t)
are all bounded and their same upper bound are
√ 2δ. Besides, ˜ Γ1(t) F and ˜ Γ2(t)
F have the same upper bound
δp2Π1max. Also, kξ(t)k and kΞ(t)k can obtain their same
upper bound δqσ(P )1 .
VI. NUMERICAL SIMULATION
The communication network of 4 NMRs is depicted in Fig. 1. The linear velocity and the angular velocity of the
leader NMR are set as v0 = 100|ω0| and ω0 = −1+(cos t)sin t 2,
respectively. The initial states of the leader NMR are
(x0, y0, θ0)T = (10, 30, 900/π)T, and the initial states of
the follower NMRs are (x1, y1, θ1, x2, y2, θ2, x3, y3, θ3)T =
(15, −10, −360/π, 10, −20, 1440/π, 4, −6, −900/π)T. The
parameters are chosen to be α1 = 100, β1 = 0.5, α2 = 30,
β2= 1 and b1= 35.
Let Λi = (xi, yi, θi)T, fi(Λi) = 1−exp
−Λi
1+exp−Λi and ξi be
uniformly distributed noise. The local NMR NN tuning laws
are given by (13), with ϕi(Λi) = exp
−kΛ i−πik2 η2 i , where πi = (πi1, πi2, · · · , πiq) T and η
i are the centers of the
receptive fields and the widths of the Gaussian function,
re-spectively. Set c1= 40, c2= 40, κ = 0.01, F1= 1500, F2=
1500, F3 = 1500, π1 = (30, 30, 30)T, π2 = (25, 25, 25)T,
π3 = (20, 20, 20)T, η1 = 8, η2 = 8 and η3 = 8. The
position and orientation observer tracking errors between the observers and the leader NMR are in Figs. 2-4 showing asymptotic reconstruction of the leader’s state. Fig. 5 shows that cooperative tracking rendezvous is achieved successfully,
Fig.1: Communication topology with three follower NMRs and one leader NMR.
0 2 4 6 8 10 12 time 0 100 200 300 400 500 600 700 x-coordinate 0 0.5 1 1.5 2 2.5 time 0 50 100 150
Fig.2: Evolution of the leader NMR and the three observers in the x coordinates.
0 2 4 6 8 10 12 time -40 -20 0 20 40 60 80 100 120 140 y-coordinate
Fig.3: Evolution of the leader NMR and the three observers in the y coordinates.
0 2 4 6 8 10 12 time -15 -10 -5 0 5 10 15 20 25 orientation
Fig.4: Evolution of the leader NMR and the three observers about the angle.
0 2 4 6 8 10 12 time -10 0 10 0 2 4 6 8 10 12 time -20 -10 0 0 2 4 6 8 10 12 time -0.5 0 0.5 1
Fig.5: Evolution of the tracking errors.
-100 0 100 200 300 400 500 x-coordinate -40 -20 0 20 40 60 80 100 y-coordinate Leader Follower 1 Follower 2 Follower 3
Fig.6: 2D trajectories of the leader robot and the three follower robots.
i.e. all followers meet and follow the leader. This is further shown in the x − y plane of Fig. 6.
VII. CONCLUSIONS
The problem of tracking rendezvous for multiple NMRs with unmodelled uncertain nonlinearities and disturbances has been addressed and solved in this work. The proposed protocol relies on distributed estimation of the leader’s state and neuro-adaptive local controllers. Future research could include the study of switching topologies and second-order nonholonomic dynamics.
REFERENCES
[1] Z. P. Jiang, “Controlling underactuated mechanical systems: A review and open problems,” Advances in the Theory of Control, Signals and Systems With Physical Modeling, vol. 407, no. 77-88, 2015.
[2] Y. Wu, B. Wang, and G. Zong, “Finite-time tracking controller design for nonholonomic systems with extended chained form,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 52, no. 11, pp. 798–802, 2005.
[3] X. Peng, K. Guo, and Z. Geng, “Moving target circular formation control of multiple non-holonomic vehicles without global position measurements,” IEEE Transactions on Circuits and Systems II: Express Briefs, 2019.
[5] M.-S. Kim, J.-H. Shin, S.-G. Hong, and J.-J. Lee, “Designing a robust adaptive dynamic controller for nonholonomic mobile robots under modeling uncertainty and disturbances,” Mechatronics, vol. 13, no. 5, pp. 507–519, 2003.
[4] P. Rao and X. Li, “Cooperative formation of self-propelled vehicles with directed communications,” IEEE Transactions on Circuits and Systems II: Express Briefs, 2019.
[6] B. S. Park, S. J. Yoo, J. B. Park, and Y. H. Choi, “Adaptive neural sliding mode control of nonholonomic wheeled mobile robots with model uncertainty,” IEEE Transactions on Control Systems Technology, vol. 17, no. 1, pp. 207–214, 2009.
[7] J. Huang, C. Wen, W. Wang, and Z.-P. Jiang, “Adaptive stabilization and tracking control of a nonholonomic mobile robot with input saturation and disturbance,” Systems & Control Letters, vol. 62, no. 3, pp. 234–241, 2013.
[8] A. Das and F. L. Lewis, “Distributed adaptive control for synchronization of unknown nonlinear networked systems,” Automatica, vol. 46, no. 12, pp. 2014–2021, 2010.
[9] X. Liu, J. Lam, W. Yu, and G. Chen, “Finite-time consensus of multiagent systems with a switching protocol,” IEEE Transactions on Neural Networks and Learning Systems, vol. 27, no. 4, pp. 853–862, 2015.
[10] A. Das and F. L. Lewis, “Cooperative adaptive control for synchroniza-tion of second-order systems with unknown nonlinearities,” Internasynchroniza-tional Journal of Robust and Nonlinear Control, vol. 21, no. 13, pp. 1509– 1524, 2011.
[11] S. El-Ferik, A. Qureshi, and F. L. Lewis, “Neuro-adaptive cooperative tracking control of unknown higher-order affine nonlinear systems,” Automatica, vol. 50, no. 3, pp. 798–808, 2014.
[12] G. Wen, Y. Zhao, Z. Duan, W. Yu, and G. Chen, “Containment of higher-order multi-leader multi-agent systems: A dynamic output approach,” IEEE Transactions on Automatic Control, vol. 61, no. 4, pp. 1135–1140, 2015.
[13] Y. A. Harfouch, S. Yuan, and S. Baldi, “An adaptive switched control approach to heterogeneous platooning with intervehicle communication losses,” IEEE Transactions on Control of Network Systems, vol. 5, no. 3, pp. 1434–1444, 2017.
[14] H. Pan, X. Yu, and L. Guo, “Admissible leader-following consensus of fractional-order singular multi-agent system via observer-based pro-tocol,” IEEE Transactions on Circuits and Systems II: Express Briefs, 2018.
[15] G. Wen, W. Yu, Y. Xia, X. Yu, and J. Hu, “Distributed tracking of nonlinear multiagent systems under directed switching topology: An observer-based protocol,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 47, no. 5, pp. 869–881, 2016.
[16] H. Liang, Y. Zhou, H. Ma, and Q. Zhou, “Adaptive distributed observer approach for cooperative containment control of nonidentical networks,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 49, no. 2, pp. 299–307, 2018.
[17] H. Zhang and F. L. Lewis, “Adaptive cooperative tracking control of higher-order nonlinear systems with unknown dynamics,” Automatica, vol. 48, no. 7, pp. 1432–1439, 2012.
[18] I. Zaidi, M. Chaabane, F. Tadeo, and A. Benzaouia, “Static state-feedback controller and observer design for interval positive systems with time delay,” IEEE Transactions on circuits and systems II: Express Briefs, vol. 62, no. 5, pp. 506–510, 2014.
[19] L. Zhao, H. Ma, L. Xu, and X. Wang, “Observer-based adaptive sampled-data event-triggered distributed control for multi-agent sys-tems,” IEEE Transactions on Circuits and Systems II: Express Briefs, 2019.
[20] R. Sakthivel, A. Parivallal, B. Kaviarasan, H. Lee, and Y. Lim, “Finite-time consensus of markov jumping multi-agent systems with “Finite- time-varying actuator faults and input saturation,” ISA transactions, vol. 83, pp. 89–99, 2018.
[21] R. Sakthivel, R. Sakthivel, B. Kaviarasan, H. Lee, and Y. Lim, “Finite-time leaderless consensus of uncertain multi-agent systems against “Finite- time-varying actuator faults,” Neurocomputing, vol. 325, pp. 159–171, 2019. [22] H. Zhang, Z. Li, Z. Qu, and F. L. Lewis, “On constructing lyapunov functions for multi-agent systems,” Automatica, vol. 58, pp. 39–42, 2015. [23] Z. Qu, Cooperative control of dynamical systems: applications to
autonomous vehicles. Springer Science & Business Media, 2009. [24] D. Shevitz and B. Paden, “Lyapunov stability theory of nonsmooth
systems,” IEEE Transactions on Automatic Control, vol. 39, no. 9, pp. 1910–1914, 1994.
View publication stats View publication stats
