Dynamic Threshold Global Performance-Guaranteed Formation Control for Wheeled Mobile Robots with Smooth Extended State Observer

Authors: minjing  wang
yibo  zhang
Venue: IEEE_org_ICIST_2024_Conference
Published: 2024-08-01 09:45:26

Abstract: In this paper, a dynamic threshold global performance-guaranteed formation control method is proposed for wheeled mobile robots (WMRs). Unlike existing prescribed performance formation control methods that are constrained by initial values, we design a dynamic threshold global performance-guaranteed (DTGPG) function to address the initial value constraints while being able to secondary adjust the steady state performance boundaries. Moreover, we design a smooth extended state observer (SESO) based on a sigmoid-like function to mitigate the chattering problem of the existing event-triggered ESO. Then a DTGPG-based guidance law and a SESO-based control law are designed to implement the formation control. The proof shows that the total closed-loop system is input-to-state stable (ISS). Through simulation, the benefits and validity of the proposed control methodology are confirmed.

Main paper: 
 i. introduction: 
 Multi-wheeled mobile robots (WMRs) formation control with extremely high demands on transient and steady state performance. In the transient phase, small overshoots and fast convergence can avoid collisions between WMRs. In the steady state phase, high accuracy tracking performance can significantly improve the overall coordination and task execution efficiency. Therefore, it is crucial to prescribe the performance of the multi-WMRs system. In [1] , a collision avoidance prescribed performance control (PPC) method is proposed for WMR formations, which guarantees the performance of the multi-WMR system by adding communication limits and collision limits to the prescribed performance This work is partly distributed under the "South China Sea Rising Star" Education Platform Foundation of Hainan Province (JYNHXX2023-17G), the Natural Science Foundation of Hainan Province (624MS036). (Corresponding author: Di Wu) function. In [2] , a fixed-time performance-guaranteed formation control problem for multi-WMRs is investigated, which achieves fixed-time convergence by introducing a segmented time-varying function into the performance function. In [3] , a field-of-view constrained performance-guaranteed formation control method is proposed for multi-WMRs, which designs a guaranteed performance function that considers leader and follower distance maintenance to avoid collisions. Although the above work [1] - [3] can effectively improve the performance of multi-WMRs, there are still two points that need to be improved: 1. They are all subject to initial conditions, which will increase the human intervention in practical applications, i.e., calculating the starting position of the WMRs in advance. 2. The standard PPC cannot perform a secondary adjustment of the performance boundaries after reaching the steady state.On the other hand, when performing tasks in complex environments, frozen and uneven road surfaces are usually encountered. These disturbances may affect the stability of WMR formations. Therefore, how to quickly and accurately estimate the external disturbances is also crucial. In [4] , a nonlinear extended state observer (ESO) is proposed to estimate the external disturbance, which recovers the velocity and estimates the external disturbance through position and heading errors. Then to improve the estimation rate, a finite time ESO is designed. In [5] , an event-triggered ESO is designed to adjust the allocation of resources. Note that event-triggered ESO [5] can save resources when estimating disturbances, but will inevitably have chattering problems.Inspired by the aforementioned observations, we propose a dynamic threshold global performance-guaranteed (DTGPG) formation control method for WMRs with a smooth extended state observer (SESO). The key contributions of this work are: Unlike the standard PPC methods described in [6] and the TPP methods in [7] - [9] , this paper proposes DTGPG capable of solving the initial value constraints problem and secondary adjustment of the steady state performance bounds. In contrast to event-triggered ESO [5] , we design the SESO to mitigate chattering by introducing a sigmoid-like function to smooth the estimation error. The total closed-loop system is proved to be input-to-state stable (ISS). Some of the symbols in this paper are defined in Table I .To describe the communication among the virtual leader and WMRs, a directed graph is described as G = {V, M}. V = {n 1 , . . . , n M } and M = {(n i , n j ) ∈ V × V} represent a vertex set and an edge set, respectively. An adjacency matrix associated with G is defined as A = [a ij ] ∈ R M ×M . Correspondingly, a degree matrix connected with G is characterized asD = diag{d i } ∈ R M ×M with d i = M j=1 a ij .Additionally, a Laplacian matrix associated with G is defined as L = D-A. Note that here i = 1, ..., M, j = 1, ..., M .Suppose that there exist N followers, labeled as agents n 1 to n N , and M -N leaders, labeled as agents n N +1 to n M , under a communication topology graph. A group of followers consisting of N wheeled mobile robots is modelled as follows⎧ ⎪ ⎨ ⎪ ⎩ ηi = R i ν i νi = r i J + i M -1 i τ i + r i J + i M -1 i T i -D iθ r 2 i J + i M -1 i J i R -1 i ηi -J + i M -1 i F i r 2 i (1)where i=1, ..., N . η i = [x i , y i , ψ i ] T ∈ R 3 denotes the position and yaw angle.ν i = [u i , v i , w i ] T ∈ R 3 denotes the velocity vector. τ i = [τ i1 , τ i2 , τ i3 , τ i4 ] T ∈ R 4 denotes the control input. T i = [T i1 , T i2 , T i3 , T i4 ] T ∈ R 4 denotes the external disturbance.The kinetic parameters and matrices of this WMR can be found in [10] .J i ∈ R 4×3 and J + i ∈ R 3×4 satisfy the relationship J + i J i = I 3 . Assumption 1:The graph G contains a spanning tree with the virtual leader as the root node.We define the distributed error as followsEQUATIONwhere η jr = [η jx , η jy , η jψ ] T ∈ R 3 represents the trajectory of the virtual leader. The coefficient a ij is defined in [11] . To ensure that the developed control is free from the influence of initial conditions and can dynamically adjust prescribed thresholds, the error is constrained within the following prescribed regionsEQUATIONwhere I ik (W ik ) is a dynamic threshold global performanceguaranteed (DTGPG) function similar to the [12] , and is defined as followsEQUATION)with W ik = 1/P ik . P ik is a dynamic threshold finite-time prescribed function similar to the [13]EQUATIONwhere l ik and ω ik are positive constants. Θ ik,∞ = lim t→∞ Θ ik (t) is the steady-state value. ik > 0 represents the convergence rate. T ik,a is the settling time to reach steady state. c ik = T ik,b -T ik,a is the duration of the dynamic adjustment.Then, we employ the following barrier function to implement the error constraint in (3)Z ik = J ik 1 -J 2 ik (6)whereJ ik = P ik H ik with H ik = E ik / E 2 ik + l ik .The properties of the barrier function are described in [12] .To facilitate the subsequent strategy design, defineΛ i = r i J + i M -1 i T i -D iθ r 2 i J + i M -1 i J i R -1 i ηi -J + i M -1 i F i r 2i to denote internal uncertainty and external disturbances suffered by the ith WMR. (1) can be reformulated asEQUATION)Assumption 2: For the multi-WMR system, the unknown total disturbance Λ i is smooth and continuous.Then, we regard the total disturbances Λ i as an extended state, and to avoid unnecessary waste of resources when approximating the disturbances, an ESO based on event-triggered mechanism is designed as [5] ⎧ ⎪ ⎨ ⎪ ⎩EQUATION)where ε i1 and ε i2 ∈R 3×3 denote positive diagonal matrices.The variables νi =[û i , vi , ŵi ] T ∈R 3 and Λi =[ Λiu , Λiv , Λiw ] T ∈ R 3 denote the estimates of ν i and Λ i , respectively. ν i ∈ R 3 represents the aperiodic sampling of ν i . The event-triggered mechanism is defined asν i (t)=ν i (t νi ),∀t∈[t νi , t νi +1 ), νis (t)=ν i (t)-ν i (t) t νi +1 = inf{t ∈ R | νis (t) ≥ X i } (9)where X i ∈ R + denotes the event triggering threshold, and νis (t) denotes the aperiodic sampling error. When νis (t) ≥ X i , update ν i (t); otherwise, maintain the last updated value.Remark 1: In addition to using ESO to estimate the external disturbances, the neural network [14] and the neural predictor [15] also achieve the same objective.Existing ESO based on event-triggered mechanism [5] suffers from unavoidable chattering when approximating the disturbances. To solve the chattering problem, we design the SESO as follows νi = -ε i1EQUATIONwhere B(ν s i ) = col(B(ν s iΞ )), Ξ = u, v, w ∈ R 3 is the sigmoid-like function vector, defined as followsEQUATIONNext, to facilitate the stability analysis of the SESO, define a positive vectorEQUATIONThe ( 10) can be rewritten asEQUATIONwhere νi = νiEQUATIONwhereA i1 = -ε i1 I 3 I 3 -ε i2 V i O 3 B i1 = ε i1 I 3 ε i2 V i C i1 = O 3 I 3 .Note that the matrix A i1 is a Hurwitz matrix. There exists a positive-definite matrix P i1 satisfying the following inequalityEQUATION)Lemma 1: The system ( 14) is ISS.Proof: Consider a Lyapunov function candidate as followsEQUATION)The time derivative V 1 based on ( 14) and ( 15) satisfiesEQUATIONwherej 1 =min i=1,...,N (j i1 ),N 1 =[N T 11 , ..., N T N 1 ] T ∈R 6N , νs = [ν T 1s , ..., νT Ns ] T ∈ R 3N , Λ = [ ΛT 1 , ..., ΛT N ] T ∈ R 3N , P 1 = diag{P 11 , ..., P N 1 } ∈ R 6N ×6N , B 1 =diag{B 11 , ..., B N 1 } ∈ R 6N ×3N , and C 1 = diag{C 11 , ..., C N 1 } ∈ R 6N ×3N . Since N 1 ≥ 2( P 1 B 1 νs + P 1 C 1 Λ )/j 1 σ 1 , one has V1 ≤ -j 1 (1-σ 1 ) N 1 2 /2, where 0 < σ 1 < 1. It follows that the subsystem (14) is ISS. There exists a KL function Y 1 (•) and K ∞ function C νs (•) and C Λ(•) satisfying N 1 (t) ≤ Y 1 ( N 1 (0) , t) + C νs ( νs ) + C Λ( Λ ), where C νs (s) = ((2s P 1 B 1 λ max (P 1 ))/(j 1 σ 1 λ min (P 1 ))) and C Λ(s) = ((2s P 1 C 1 λ max (P 1 ))/(j 1 σ 1 λ min (P 1 ))).In this section, we design the DTGPG-based guidance law and the SESO-based control law. First, we design the guidance law. The time derivative of ( 6) is represented byEQUATION)whereμ ik = (1 + J 2 ik )/(1 -J 2 ik ) 2 and ρ ik = l ik /( E 2 ik + l ik (E 2 ik + l ik )).Next, to simplify the design of the controller, we rewrite (18) in a vector formEQUATION)whereZ i = [Z ix , Z iy , Z iψ ] T ∈ R 3 , E i = [E ix , E iy , E iψ ] T ∈ R 3 , μ i1 = diag{μ ix P ix ρ ix , μ iy P iy ρ iy , μ iψ P iψ ρ iψ } ∈ R 3×3 , and μ i2 = diag{μ ix Ṗix H ix , μ iy Ṗiy H iy , μ iψ Ṗiψ H iψ } ∈ R 3×3 .Take the time derivative of (2) based on (1) satisfiesEQUATIONwhereι i = N j=1 a ij + M j=N +1 a ij . Substituting (20) into (19) results in Żi = μ i1 (ι i R i ν i - N j=1 a ij R j ν j - M j=N +1 a ij ηjr ) + μ i2 . (21)From ( 21), the DTGPG-based guidance law is chosen asEQUATION)We substitute ( 22) into (21), and it follows thatŻi = μ i1 N j=1 a ij R j νj -κ i1 Z i (23)with κ i1 ∈ R 3×3 being a positive diagonal matrix. Differing from the first-order low-pass filtering method in the traditional DSC, a second-order linear tracking differentiator (LTD) with respect to α i is introducedEQUATIONwhere α * if ∈ R 3 is the filtered value of αi , and γ i ∈ R + . Second, we design the control law. Defining a velocity errorEQUATION)Then, we designed the SESO-based control law to stabilize (25)EQUATION)with κ i2 ∈ R 3×3 being a positive diagonal matrix. The dynamics of Z ie is further obtained by substituting (26) into (25)EQUATION)where 23) and ( 27), we can obtain the following subsystemsα * i = α * if -αi . From (EQUATION)Lemma 2: The system (28) is ISS.Proof: Consider a Lyapunov function candidate asEQUATIONwhere Theorem 1: For multi-WMRs (1) subject to initial conditions, the closed-loop system is ISS consisting of SESO (10) , the DTGPG-based guidance law (22), and the SESO-based control law (26). Moreover, Zeno behavior can be avoided.EQUATIONThe ISS properties of subsystems ( 14) and (28) are proven through Lemma 1 and Lemma 2, respectively. The state of the subsystem ( 14), ν, and Λ are inputs of the subsystem (28). Under Assumptions 1-2, according to the cascade stability theorem, the closed-loop system is ISS. It yields that the ultimate boundedness of N 2 (t) as t → ∞EQUATIONwith H * = (4(n 3 n * + 1) λ max (P 1 ))/(nj 1 σ 1 λ min (P 1 )).The detailed proof process of the Zeno behavior can be referred to [5] . The proof of Theorem 1 is complete.From Fig. 1 , it can be seen that we consider a communication topology consisting of three followers n 1 , n 2 , and n 3 , as well as two virtual leaders n 4 and n 5 to verify the effectiveness of the proposed controller. The physical parameters of the WMR can refer to [10] . This external disturbance is similar to [16] . The initial values of three followers are chosen asη 1 (0) = [0, 0, 3π/2] T , η 2 (0) = [2, -10, π/2] T , η 3 (0) = [2, -17, 4π/3] T .The trajectories of the two virtual leaders are chosen as η 4r = [-5 sin(0.2t), -5 cos(0.2t), atan2( η4y , η4x )] T η 5r = [-15 sin(0.2t), -15 cos(0.2t), atan2( η5y , η5x )] T .The main design parameters are set as κ 11 =diag{12, 7, 10}, κ 21 =diag{7, 7, 10},κ 31 =diag{12, 9, 10}, κ i2 =diag{20, 20, 20}, ε i1 =diag{2, 2, 2}, ε i2 =diag{40, 40, 40}, T 1x,a = T 1ψ,a = T 2x,a = T 2ψ,a = T 3x,a = T 3ψ,a = 0.5, T 1y,a = T 2y,a = 4 . Fig. 1 demonstrates these three vehicles forming a circular formation guided by two virtual leaders. Fig. 2 shows that the tracking profile is not constrained by the initial value and is able to dynamically adjust the performance boundaries using the proposed DTGPG control scheme. Fig. 3 shows that SESO is not only able to estimate internal uncertainties and external disturbances but also to reduce chattering. Fig. 4 shows the number of triggering events. ν 1 , ν 2 , and ν 3 are triggered 179, 213, and 211 times respectively. Compared to time triggering 2800 times, it effectively saves resources.In this paper, the dynamic threshold global prescribed performance formation control problem was investigated for WMRs in the presence of unknown total disturbances. A dynamic threshold global performance-guaranteed formation control method based on SESO was proposed, which had three advantages: 1) it could adjust the steady-state performance boundary twice, 2) it resolved the initial value constraints present in standard PPC, and 3) it mitigated the chattering problem in event-triggered ESO. This cascade system consisting of the SESO, the DTGPG-based guidance law, and the SESO-based control law was proved to be ISS. The main results were demonstrated by the simulation examples. 
 
 Appendix:  



Reviews: review: title: title: This paper can be accepted.
main_review: main_review: 1. Format and alignment of author information, omit 1st and 2th.
2. The fonts of legends and labels are small. It is recommended to increase the font size and image resolution.
3. When displaying multiple data sets in the same chart, use more distinct line styles and colors.
4. The experimental settings can be described in more detail, including the settings of the simulation environment, the simulation of random interference, and the selection criteria of control parameters, etc.
5. The article mainly compares with traditional methods. Comparative experiments can be added to compare with existing technology, the latest, different backgrounds or other improvement methods to highlight the innovation and adaptability of the proposed method.
6. Add explanations of some key steps in practical applications. For example, when proving input-to-state stable (ISS), you can briefly explain the significance and impact of ISS in practical applications; in the section discussing the number of trigger events, you can further explain The significance of these triggering events in practical applications (including the relationship between resource consumption and real-time performance), etc.
7. In the final summary part, the practical application value of these innovations can be further emphasized, such as the application prospects in actual robot control, etc.
title: title: Accept after modification
main_review: main_review: This paper primarily proposes a dynamic threshold global performance-guaranteed formation control method for wheeled mobile robots. Overall, the manuscript provides a thorough exposition. However, a few issues warrant attention:

1.The manuscript employs two different notations for equations: "( )" and "equations ( )." It is recommended to review the entire document and standardize the notation to a single, consistent format.
2.In Equation 1, the variables R, M, F, r, and D lack corresponding nomenclature. It is suggested that the authors provide their definitions and explain their practical significance.
3.The text currently uses two variations of the term for the event triggering mechanism—"event triggering mechanism" and "event-triggered mechanism." It would be beneficial to standardize this terminology throughout the manuscript.
4.The vertical axis labels in Figures 2 and 3 should include more information than just units; please ensure that they are fully descriptive.
5.In Figure 4, the horizontal axis represents time, while the vertical axis denotes the duration of each event triggering. However, the figure is titled ‘The number of triggering events.’ This title may not accurately reflect the content of the graph. It is recommended to revise the figure title to better represent the data presented.
title: title: In this paper, the dynamic threshold global prescribed performance formation control problem was investigated for WMRs in the presence of unknown total disturbances. A dynamic threshold global performance-guaranteed formation control method based on SESO was proposed, which had three advantages: 1) it could adjust the steady-state performance boundary twice, 2) it resolved the initial value constraints present in standard PPC, and 3) it mitigated the chattering problem in event-triggered ESO.
main_review: main_review: I think the conclusion section is too brief, and I believe the author has a clear understanding of the limitations and future developments of the methods proposed in the article.