ESNO-based predefined-time trajectory tracking control for mobile manipulator with full-state constraints

Abstract

In this study, the problem of adaptive predefined-time trajectory tracking for mobile manipulator (MMR) subject to external disturbances and full-state constraints is addressed. Firstly, a novel ESN observer (ESNO) with guaranteed predefined-time convergence is proposed, incorporating an adaptive error compensation mechanism to enhance disturbance estimation accuracy. Secondly, an interval-based asymmetric prescribed performance control strategy with a tunable predefined-time prescribed performance function is developed, and its simplified structure reduces computational burden. Thirdly, a predefined-time adaptive filter is designed to avoid the “complexity explosion” problem. Finally, Lyapunov stability analysis verifies predefined-time convergence of all errors, while numerical simulations and physical MMR platform experiments verify the feasibility and superiority of the algorithm.

Introduction

In recent years, mobile manipulator (MMR), as a kind of nonlinear system with highly coupled dynamics and non-complete constraints, has been widely applied in the fields of industry, agriculture, and healthcare.^1–3 Unlike fixed-base robotic arms, MMR consist of a mobile base and a multi-jointed arm, providing greater mobility and a larger workspace area, which allows the mobile arm to be used in a wider range of tasks. ² However, high-precision interactive tasks coupled with modeling uncertainties and external disturbances impose considerable challenges to the robust performance of MMR. ⁴ To address the trajectory tracking problem of mobile robotic arms, some classical nonlinear control methods have been applied so far, such as backstepping, neural network adaptive control, ⁵ computed torque control, ⁶ sliding mode control,^4,7,8 and model predictive control. ⁹ With the increasing complexity of tasks in precision manufacturing and service robotics, higher demands are placed on both transient performance and operational robustness of mobile manipulator arms. While existing control methods can guarantee steady-state tracking accuracy, key challenges persist in simultaneously achieving overshoot suppression and disturbances rejection during dynamic transitions under coupled uncertainties.

Commonly, tracking control strategies for mobile robotic arms are classified into integral control and modular control. ¹⁰ In the literature, the MMR non-fully mobile platform and the fully mobile robotic arm are viewed as separate entities, and separate controllers are designed for each subsystem. ¹¹ This method of treating the system coupling between the MMR mobile platform and the robotic arm as a perturbation provides an effective solution to the overall control problem of MMRs. By integrating observer and adaptive techniques, a finite time convergence controller is designed. ¹² Nevertheless, observation error and adaptive error cannot be avoided simultaneously. Subsequently, the integration of adaptive control with backstepping techniques was explored, addressing the MMR tracking problem from different perspectives.^13,14 To further improve the tracking performance, some scholars have also considered compensating for the effect of estimation error on the system by adding a compensator. ¹⁵ Note that all the above methods can only realize the finite time or asymptotic stabilization of the system. Crucially, for both finite-time and asymptotic convergence, a large initial error inevitably leads to extended convergence times and can even induce system singularity.^8,11,16 Based on this, fixed-time convergence methods, which require less initial state of the system and improve the convergence speed and accuracy of the system, have been widely employed.^17,18 Unfortunately, the convergence time is heavily dependent on the controller parameters and is prone to potential singularity problems. The above problem is avoided by utilizing inequality deflation technique and power finger transformation technique. ¹⁹ Meanwhile, utilizing the inequality scaling technique also brings about a more conservative system convergence time. After that, the study of predefined time convergence well compensates the shortcomings of fixed time convergence theories.^20,21 It has rarely been considered for the control of MMR.

Note that pursuing a faster convergence rate inherently amplifies overshoot in the controller’s torque output, thereby resulting in degraded tracking accuracy due to error overshoot propagation through the closed-loop system. ²² Fortunately, scholars’ extensive research on output constraints has enabled the overshooting of the tracking error to be kept within the prescribed interval in time when the control inputs become too large. Currently, conventional barrier Lyapunov function (BLF)-based methods, prescribed performance control (PPC)^23–25 and model predictive control (MPC) ²⁶ are the main strategies. A trajectory tracking control strategy for mobile manipulators based on the adaptive PD method is proposed by using prescribed performance function (PPF), and decent control performance is obtained. ²⁷ A controller for MMR by incorporating a time-varying prescribed performance function within a predetermined-time stabilization strategy, thereby enhancing both steady-state and transient performance. ²⁸ However, only constraining the position tracking error was considered while neglecting coupled velocity-state limitations.

As a recurrent neural network (NN), the echo state network (ESN) has been widely used in the control of nonlinear systems due to its excellent approximation performance for nonlinear functions.^29,30 Although ESN exhibits potential in dynamic coupling approximation and disturbance observers excel in uncertainty estimation, their synergistic integration in MMR control remains underexplored, especially for scenarios demanding simultaneous multi-state constraints and non-differentiable disturbance rejection. This motivates a novel hybrid observer in this paper for realizing high-precision control of MMR.

Inspired by previous studies, this paper systematically addresses the trajectory tracking control problem of the MMR system subject to external disturbances, all-state performance constraints, and operational scenarios. The primary contributions of this work are summarized as follows:

• (1) A novel echo state network observer (ESNO) with guaranteed predefined-time convergence is proposed to approximate the upper bound of the external disturbances and modeling uncertainties. Compared with Xing et al. ²⁶ and Wang and Su, ³¹ the ESNO has a faster convergence speed.

• (2) Unlike prior studies focusing solely on position constraints,^27,28 an interval-based asymmetric PPC strategy with adjustable predefined-time PPF (PTPPF) is developed, enabling simultaneous enforcement of both position and velocity tracking error bounds. This enhances tracking accuracy and is particularly beneficial for MMR applications.

• (3) A practical predefined-time stability theorem is established under the proposed control framework, ensuring that both system states and tracking errors converge to a predefined compact set within a predefined time. This approach eliminates initialization sensitivity and avoids potential singularity issues found in fixed-time and predefined-time controllers. ³²

• The effectiveness of the proposed framework is validated through detailed simulations and physical experiments on an MMR prototype, demonstrating its robustness in real-world operational scenarios.

The rest of this paper is designed as follows. The model description and some premises of MMR are given in Section II. In Section III, predefined time converged ESNO and predefined time trajectory tracking controller with full state constraints are given. Numerical simulations and physical experiments are given in Section IV. Finally, Section V gives the conclusion of this work.

• Notation. $\mathbb{R}$ is the set of real numbers, ||·|| denotes the Euclidean norm, and |·| represents the absolute value, the exponential function $\exp (x)=e^x$ , where $e$ is the natural base. $I_n$ is an $n\times n$ identity matrix, ${\mathrm{diag}}_{J=1}^m[y_J]$ denotes a diagonal matrix with elements $y_1,\dots ,y_j,\dots ,y_N$ , and $\mathrm{dia}{g}_m[b]\in {\mathbb{R}}^{m\times m}$ is all diagonal elements as $b$ .

Problem formulation and preliminaries

A two-link mobile manipulator is shown in Figure 1. ³³ The two rear wheels of the mobile platform are driven by independent motors, and a two-link robot arm is installed at the center of mass $O$ . Since the mobile platform is driven by two motors, which results the MMR is subject to a nonholonomic constraint between the two driving wheels and the ground. The MMR dynamic model can be expressed by the Euler-Lagrangian method

$$M\stackrel{··}{q}=-C\stackrel{·}{q}-G+{\tau }_u+A^T{u}_f+{\tau }_d,$$

(1)

where $M\in {\mathbb{R}}^{5\times 5}$ is the nominal inertia matrix; $C\in {\mathbb{R}}^{5\times 5}$ is the Coriolis and centripetal forces; $q=[o_x,o_y,{\phi }_o,{\theta }_1,{\theta }_2{]}^T$ is the generalized coordinates, where ${\theta }_1$ , ${\theta }_2$ , represent the angles that govern the motor rotation of robotic arm links 1 and 2, respectively, $(o_x,o_y)$ is the center of mass coordinates, ${\phi }_o$ denotes platform orientation angle. $A=[-\sin {\phi }_o,\cos {\phi }_o,-d,0,0{]}^T$ represents the matrix associated with the constraints, $G$ is the gravitational vector. $u_f$ denotes the Lagrange multiplier, ${\tau }_u=B_o{u}_{\tau }$ is the input vector, which consists of the wheel torque of the mobile base and the joint torque of the robotic arm. $B_o\in {\mathbb{R}}^{4\times 4}$ denotes input transformation matrix, and ${\tau }_d$ is the external disturbances. By taking $O$ as the reference point, the following equation is obtained

$${\stackrel{·}{o}}_x\cos ({\phi }_o)={\stackrel{·}{o}}_y\sin ({\phi }_o)+{\stackrel{·}{\phi }}_o,$$

(2)

and (2) can be rewritten in the following vector form:

$$A\stackrel{·}{q}=0,$$

(3)

where $d$ is the wheelbase offset from $O$ . Defining independent velocity coordinates $\Theta =[{\theta }_{\mathrm{ol}},{\theta }_{\mathrm{or}},{\theta }_1,{\theta }_2{]}^T$ , where ${\theta }_{\mathrm{ol}},{\theta }_{\mathrm{or}}$ represent the platform linear and angular velocity, respectively, it follows that

$$\stackrel{·}{q}=S\stackrel{·}{\Theta },$$

(4)

where the matrix $S\in {\mathbb{R}}^{5\times 4}$ is set of basis in null space of vector $A$ , leading to the relation $S^T{A}^T=0$ . From (4), $\stackrel{·}{\Theta }$ can be obtained

$$\stackrel{·}{\Theta }=\overline{S}\stackrel{·}{q},$$

where $\overline{S}=(S^{T}S{)}^{-1}{S}^T\in {\mathbb{R}}^{4\times 5}$ is fundamentally employed to address the dimensionality reduction imposed by the nonholonomic constraint (4), which reduces the system’s independent degrees of freedom from 5 to 4. By providing the minimum-norm solution that maps the constrained velocity $\stackrel{·}{q}$ to the independent velocity space $\stackrel{·}{\Theta }$ , it exactly satisfies (4) while preserving kinematic consistency through the inverse relationship $\overline{S}S=I_4$ -a critical property for non-square $S$ that prevents artificial singularities inherent in naive matrix inversion approaches.

Figure 1.

Two-link MMRs in the coordinate system.

Then the MMR dynamic model (4) and (1) can be rewritten as

$$\stackrel{··}{\Theta }={\tilde{M}}^{-1}(-\tilde{C}\stackrel{·}{\Theta }-\tilde{G}+{\tau }_u+{\tau }_d),$$

(5)

in which

$$\begin{array}{l}M=S^{\text{T}}MS,\\ \tilde{C}n=S^{\text{T}}M\stackrel{.}{\text{S}}+S^{\text{T}}CS,\\ {\tau }_{u}n=S^{\text{T}}{B}_o{u}_{\tau },\\ \tilde{G}n\triangleq S^{T}G.\end{array}$$

In the generalized coordinate system, the chassis position vector of the MMR and the trajectory of the end manipulator arm $X_1=[o_x,o_y,o_{\theta x},o_{\theta y}{]}^T$ have the following relationship with the rotation angle of the motors $\Theta$ :

$${\stackrel{·}{X}}_1=J(q)\stackrel{·}{\Theta },$$

where $(o_{\theta x},o_{\theta y})$ is the end manipulator arm 2D position coordinates, $J(q)$ is the Jacobi transformation matrix. To enable the MMR joint motor position to track the desired trajectory ${\Theta }_d$ , the following assumptions and lemmas are introduced to establish a rigorous theoretical foundation for the control design.

• Lemma 1. ³⁴ There are constants $0<{\epsilon }_1<1<{\epsilon }_2<+\infty$ , for any positive integer $\mathfrak{M}$ , variables $T_i\in \mathbb{R}(i=1,2,\cdots ,\mathfrak{M})$ , the following inequalities hold

$${\left(\sum _{i=1}^{\mathfrak{M}}\left|T_i\right|\right)}^{{\epsilon }_1}\le \sum _{i=1}^M{\left|T_i\right|}^{{\epsilon }_1};{\mathfrak{M}}^{1-{\epsilon }_2}{\left(\sum _{i=1}^{\mathfrak{M}}\left|T_i\right|\right)}^{{\epsilon }_2}\le \sum _{i=1}^M{\left|T_i\right|}^{{\epsilon }_2}.$$

• Lemma 2. ³² For the MMR system (5), if there exists a continuous radially unbounded function $V_o(\wp (t))$ and constants $n=(2-\varsigma )/2$ , $m=(2+\varsigma )/2$ and $\Re >0$ , satisfying

$$V_o(\wp (t))+\frac{\pi }{\varsigma T_v}(V_o^n(\wp (t))+V_o^m(\wp (t)))\le \Re ,t\in (t_0,\infty ),$$

where $t_0$ is the initial time, $\varsigma \in (0,2)$ is convergence exponent, then the MMR system is said to be predefined-time stable (PTS) with the setting time bounded by $T_v$ .

• Lemma 3. ³⁵ For any constants ${\epsilon }_{\mathrm{\Delta }}\in \mathbb{R}$ , $\varpi >0$ and $g\le \varpi$ are satisfied, the following holds:

$$|{\epsilon }_{\mathrm{\Delta }}|-\varpi <\frac{{\epsilon }_{\mathrm{\Delta }}^2}{\sqrt{{\epsilon }_{\mathrm{\Delta }}^2+{\varpi }^2}}<|{\epsilon }_{\mathrm{\Delta }}|.$$

• Lemma 4. ³⁶ For all real variable $\delta$ and constants $\hslash >0$ and ${\kappa }_{\hslash }=0.2785$ , the relation

$$\left|\delta \right|-\delta \tanh (\frac{\delta }{\hslash })\le \hslash {\kappa }_{\hslash },$$

is satisfied.

• Lemma 5. ³⁷ Let $y>0$ , $h\le y$ and $k>0$ . then $y^k(h-y)\le \frac{1}{1+k}(h^{1+k}-y^{1+k})$ holds. Furthermore, for any $y>0$ , $h\le y$ , $k>1$ , the inequality ${(h-y)}^k\le y^k-h^k$ holds.

• Assumption 1. The external disturbance ${\tau }_d$ is bounded and has an upper bound ${\tau }_{\mathrm{dm}}$ , satisfying ${\tau }_{\mathrm{dm}}=max\{|{\tau }_d|\}$ .

• Assumption 2. The reference trajectory $X_d=J(·){\stackrel{·}{\Theta }}_d$ is bounded and continuous, that is, ${\stackrel{·}{X}}_d$ and ${\stackrel{··}{X}}_d$ exist and are bounded.

The position tracking error $e_1$ and velocity tracking error $z_2$ are defined as

$$e_1=X_1-X_d;z_2=\stackrel{·}{\Theta }-z_v.$$

(6)

where $z_v$ is the output of the virtual control law ${\alpha }_{\Theta ,i}$ after filtering.

To make the tracking performance have a specified transient performance, the performance constraint on the error $e_1=[e_{1,1},e_{1,2},e_{1,3},e_{1,4}{]}^T$ and $z_2=[z_{2,1},z_{2,2},z_{2,3},z_{2,4}{]}^T$ are considered in this work. The error $e_{1,i}$ satisfies the following constraint

$${\rho }_{i,2}(t)\le e_{1,i}\le {\rho }_{i,1}(t),$$

(7)

where ${\rho }_{i,2}(t)$ and ${\rho }_{i,1}(t)$ are PPFs. In this work, a novel PTPPF is introduced which enables fast convergence ³⁸ :

$$\begin{array}{c}{\rho }_{i,j}(t)=\{\begin{array}{cc}({{\rho }_{0,}}_{i,j}-{\rho }_{\infty ,i,j})\exp ({\chi }_{i,j}(t))+{\rho }_{\infty ,i,j},\hfill & t<T_{i,j}\hfill \\ \multicolumn{1}{c}{{\rho }_{\infty ,i,j},}& t\ge T_{i,j}\hfill \end{array},\hfill \\ {\chi }_{i,j}(t)=\frac{-t((1+{\sigma }_{i,j})\sin (\pi t/(2T_{i,j}))+(1-{\sigma }_{i,j}))}{(1+{\sigma }_{i,j})\sin (\pi t/(2T_{i,j}))+{\sigma }_{i,j}-1},\hfill \end{array}$$

(8)

in which $T_{i,j}$ is an artificially set convergence time, ${\sigma }_{i,j}\in (0,(4-\pi )/\pi )$ can change the rate of convergence of the PPF. ${{\rho }_{0,}}_{i,j}$ and ${\rho }_{\infty ,i,j}$ are the convergence of the steady state and initial values boundaries of the performance function, respectively. ${\rho }_{i,1}(t)$ and ${\rho }_{i,2}(t)$ denote the upper and lower bounds.

• Remark 1. In contrast to the traditional exponential function (TEF) $({{\rho }_{0,}}_{i,j}-{\rho }_{\infty ,i,j})\exp (t)+{\rho }_{\infty ,i,j}$ , the adopted PTPPF provides tunable convergence characteristics through adjustment parameter ${\sigma }_{i,j}$ . As visualized in Figure 2, the TEF exhibits fixed convergence behavior regardless of its initial error. While this property prevents false triggering in high-precision systems, it lacks flexibility for scenarios requiring dynamic convergence control.

Figure 2.

Performance comparison between PTPPF and TEF.

To transform the system with constraint condition into an equivalent unconstrained system, a transformation function is designed as

$$z_{1,i}=\ln ((e_{z,i}+{\eta }_{2,i})/({\eta }_{1,i}-e_{z,i})),i=1,2,3,4,$$

(9)

in which ${\eta }_{1,i}=2{\ell }_i{\rho }_{i,2}/({\rho }_{i,2}{+}_i{\rho }_{i,1}),{\eta }_{i,1}=2{\ell }_i{\rho }_{i,1}/({\rho }_{i,2}{+}_i{\rho }_{i,1})$ and $e_{z,i}$ is the converted error, which is expressed as

$$e_{z,i}=2{\ell }_i{e}_{1,i}/({\rho }_{i,2}+{\rho }_{i,1}),$$

(10)

where ${\ell }_i$ is a constant factor taking values in the range of (0,1), which allows the constraint bounds of $e_{z,i}$ to be adjusted. Differentiating $e_{z,i}$ , one obtains

$${\stackrel{·}{e}}_{z,i}={ƛ}_{1,i}{\stackrel{·}{e}}_{1,i}+{ƛ}_{2,i},$$

(11)

with

$$\begin{array}{c}{ƛ}_{1,i}=2b_i/({\rho }_{i,2}{+}_i{\rho }_{i,1})>0,\hfill \\ {ƛ}_{2,i}=2b_i{e}_{1,i}({\stackrel{·}{\rho }}_{i,2}+{\stackrel{·}{\rho }}_{i,1})/({\rho }_{i,2}+{\rho }_{i,1}{)}^2.\hfill \end{array}$$

(12)

Unlike the traditional PPC control proposed by Dou and Wen, ³⁹ the interval-based error transformation (10) enables constraints to be imposed on both the tracking error $e_{1,i}$ and the transformation-induced error $e_{z,i}$ . In light of (7), the constraint range of $e_{z,i}$ is $e_{z,i}\in (-2b_i{\rho }_{i,2}/({\rho }_{i,2}+{\rho }_{i,1}),2b_i{\rho }_{i,1}/({\rho }_{i,2}+{\rho }_{i,1}))$ .

The differential of $e_{z,i}$ can be get

$${\stackrel{·}{z}}_{1,i}={\lambda }_{1,i}{\stackrel{·}{e}}_{1,i}+{\lambda }_{2,i},$$

where

$$\begin{array}{c}{\lambda }_{1,i}={ƛ}_{1,i}(\frac{1}{e_{z,i}+{\eta }_{i,2}}+\frac{1}{{\eta }_{i,1}-e_{z,i}}),\hfill \\ {\lambda }_{2,i}={ƛ}_{2,i}\left(\frac{1}{e_{z,i}+{\eta }_{i,2}}\right)+\left(\frac{1}{{\eta }_{i,1}-e_{z,i}}\right)\hfill \\ \multicolumn{1}{c}{+\frac{{\stackrel{·}{\rho }}_{i,2}}{e_{z,i}+{\eta }_{i,2}}-\frac{{\stackrel{·}{\rho }}_{i,1}}{{\eta }_{i,1}-e_{z,i}}.}\end{array}$$

Controller design

Design adaptive predefined-time ESNO

According to the ability of ESN to approximate the unknown function, the ESNO is developed in this work to approximate the disturbance. Defining the observation disturbance as

$${\tau }_{d,i}(W_i^{*})=(W_i^{*}{)}^T{\Phi }_i+{\nabla }_i\le {\overline{W}}_i^T{\Phi }_i+{\overline{\nabla }}_i,$$

(13)

where ${\overline{W}}_i=\Vert W_i^{*}\Vert$ and ${\overline{\nabla }}_i=\Vert {\nabla }_{i,j}\Vert$ are scalar functions and denote as the upper bounds on the ESNO output weights and approximation errors in the $i_{\mathrm{th}}$ direction, respectively.

The approximation error of ESNO is defined as ${\tilde{W}}_i=W_i^{*}-{\hat{W}}_i$ , ${\tilde{\nabla }}_i={\overline{\nabla }}_i-{\hat{\nabla }}_i$ , and ${\hat{\tau }}_{d,i}({\hat{W}}_i)$ is estimated by ${\hat{\tau }}_{d,i}({\hat{W}}_i)={\hat{W}}_i^T{\Phi }_i$ .

The observer is designed as:

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{O}=M^{-1}(-\tilde{C}\stackrel{·}{\Theta }+{\tau }_u)+k_o(\Theta -O{)}^2}\\ \multicolumn{1}{c}{+\frac{\pi }{r{T}_v}[\frac{1}{2^{1-(r/2)}}(\Theta -O{)}^{1-r}+\frac{1}{2^{1+(r/2)}}(\Theta -O{)}^{1+r}]}\\ \multicolumn{1}{c}{+(\Theta -O)\hat{\nabla }/\sqrt{{(\Theta -O)}^2+k_{\mathrm{hk}}^2}+{\hat{\tau }}_d(\hat{W}).}\end{array}$$

(14)

The disturbance observation error along the $i_{\mathrm{th}}$ direction is defined as $e_{o,i}={\Theta }_i-O_i$ , which is incorporated into the controller’s adaptive law to estimate disturbance bounds, enabling the predefined-time convergence of tracking errors.

Consider the derivative of $e_{o,i}$ as follows:

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{e}}_{o,i}=(W_i^{*}{)}^T{\Phi }_i+{\nabla }_i-{\hat{W}}_i^T{\Phi }_i-k_{o,i}{e}_{o,i}^2}\\ \multicolumn{1}{c}{-\frac{\pi }{r{T}_v}(\frac{1}{2^{1-(r/2)}}{e}_{o,i}^{1-r}+\frac{1}{2^{1+(r/2)}}{e}_{o,i}^{1+r}).}\end{array}$$

• Theorem 1: Consider the MMR system (1)–(5), under the assumptions 1-2. Select the candidate Lyapunov function $V_1=\frac{1}{2}\sum _{i=1}^4(e_{o,i}^2+{\tilde{W}}_i^T{\tilde{W}}_i/{\mu }_{1,i}+{\tilde{\nabla }}_i^2/{\mu }_{2,i}+k_{\mathrm{hk},i}^2/{\mu }_{3,i})$ , where ${\mu }_{1,i}$ , ${\mu }_{2,i}$ , ${\mu }_{3,i}>0$ are adaptation gain. If the ESNO weight adaptive laws are employed as (15), then the disturbance ${\tau }_d$ is approximated by the observer (13), and the error signals ${\tilde{W}}_i$ , ${\tilde{\nabla }}_i$ and adaptive parameter $k_{\mathrm{hk},i}$ can be converged within predefined time $T_v$ .

$$\begin{array}{l}{\stackrel{·}{\widehat{W}}}_i=u_{1,i}\left({\Phi }_i\left|e_{o,i}\right|-\frac{\pi }{r{T}_v}(\frac{2-r}{2^{1-\left(r/2\right)}}{\widehat{W}}_i^{1-r}+\frac{2+r}{2^{1+\left(r/2\right)}}{\widehat{W}}_i^{1+r}\right)),\\ {\stackrel{·}{\widehat{\nabla }}}_i=u_{2,i}\left[\frac{e_{o,i}^2}{\sqrt{e_{o,i}^2+k_{hk,i}^2}}-\frac{\pi }{r{T}_v}\left(\frac{2-r}{2^{1-(r/2)}}{\widehat{\nabla }}_i^{1-r}+\frac{2+r}{2^{1+(r/2)}}{\widehat{\nabla }}_i^{r+1}\right)\right],\\ {\stackrel{·}{k}}_{hk,i}=u_{3,i}\left[{\kappa }_{h1,i}\left|e_{o,i}\right|-{\kappa }_{h2,i}{k}_{hk,i}-\frac{\pi }{r{T}_v}\left(\frac{1}{2^{1+\left(r/2\right)}}{k}_{hk,i}^{r+1}+\frac{1}{2^{1-\left(r/2\right)}}{k}_{hk,i}^{1-r}\right)\right].\end{array}$$

(15)

• Proof: To enhance the readability of the main text, the detailed proof of this process is provided in Appendix A.

Design adaptive predefined time controller

• Theorem 2: Under the assumptions 1-2 are valid. Consider the MMR system (1)–(5) and the controller (16) and virtual control law (17), it is demonstrated that the leader can achieve the specified trajectory tracking control performance within predefined time $T_v$ .

$$\begin{array}{l}{\tau }_u=M({\psi }_1^{-1}({\stackrel{·}{z}}_v-k_2{z}_4-J(q){\lambda }_1(z_1+e_v)-{\widehat{\tau }}_d-e_m-\frac{\pi }{r{T}_v}(\frac{1}{2^{1-(r/2)}}{z}_4^{1-r}+\frac{1}{2^{1+(r/2)}}{z}_4^{1+r}))-{\psi }_2))\\ +\tilde{C}\stackrel{·}{\Theta }+\tilde{G},\end{array}$$

(16)

$${\alpha }_{\Theta ,i}=\left({\lambda }_{1,i}^{-1}-k_{\alpha }{z}_{1,i}-\frac{\pi }{r{T}_v}\left(\frac{1}{2^{1-\left(r/2\right)}}{z}_1^{1-r}+\frac{1}{2^{1+\left(r/2\right)}}{z}_1^{1+r}\right)-{\lambda }_{2,i}\right)+{\stackrel{·}{X}}_{d,i},$$

(17)

where $e_m=[e_{m,1},e_{m,2},e_{m,3},e_{m,4}{]}^T$ , ${e_{m,}}_i=z_{4,i}{\hat{\nabla }}_i/\sqrt{e_{o,i}^2+k_{\mathrm{hk},i}^2}$ . The control algorithm logic diagram is shown in Figure 3.

Figure 3.

Predefined control algorithm logic diagram.

The following error auxiliary variable is constructed

$$e_{v,i}=z_{v,i}-{\alpha }_{\theta ,i}.$$

where $e_{v,i}$ denoted the filter error and $z_{v,i}$ is the filtered output variable. To avoid the problem of complex explosions, a second-order filter that can converge in a predefined time is designed:

$$\begin{array}{c}{\stackrel{·}{z}}_{v,i}=-\frac{\pi }{r{T}_v}(e_{v,i}^{1+r}+e_{v,i}^{1-r})-k_{\mathrm{av},i}{e}_{v,i}-{\hat{z}}_{u,i}\tanh \left(\frac{e_{v,i}}{{\kappa }_{h,i}}\right),\hfill \\ {\stackrel{·}{\hat{z}}}_{u,i}=e_{v,i}\tanh (e_{v,i}/{\kappa }_{h,i})-k_{z,i}{\hat{z}}_{u,i}-\frac{\pi }{r{T}_v}(\frac{2-r}{2^{1-(r/2)}}{\hat{z}}_{u,i}^{1-r}\hfill \\ +\frac{2+r}{2^{1+(r/2)}}{\hat{z}}_{u,i}^{1+r}).\hfill \end{array}$$

(18)

in which $z_{v,i}\ge \left|{\stackrel{·}{\alpha }}_{\theta ,i}\right|$ .

A detailed proof of the controller is given below in three steps.

Firstly, another Lyapunov function $V_D=\sum _{i=1}^4\frac{1}{2}(e_{o,i}^2+z_{1,i}^2)$ is developed and its the first-order time derivative with respect to $V_D$ , one gets

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_D\le \sum _{i=1}^4{e}_{o,i}[{\tilde{W}}_i^T{\Phi }_i+{\overline{\nabla }}_i-k_{o,i}{e}_{o,i}^2-\frac{\pi }{r{T}_v}(\frac{1}{2^{1-(r/2)}}{e}_{o,i}^{1-r}}\end{array}+\frac{1}{2^{1+(r/2)}}{e}_{o,i}^{1+r})-\frac{e_{o,i}{\hat{\nabla }}_i}{\sqrt{e_{o,i}^2+k_{\mathrm{hk},i}^2}}]+z_1^T[{\lambda }_1(J(q)(z_2+e_v+{\alpha }_{\theta }-{\stackrel{·}{X}}_d)+{\lambda }_2].$$

(19)

Invoking (17) into (19), one yields

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_D\le e_{o,i}[{\tilde{W}}_i^T{\Phi }_i+{\overline{\nabla }}_i-k_{o,i}{e}_{o,i}}\end{array}-\frac{\pi }{r{T}_v}(\frac{1}{2^{1-(r/2)}}{e}_{o,i}^{1-r}+\frac{1}{2^{1+(r/2)}}{e}_{o,i}^{1+r})-\frac{e_{o,i}{\hat{\nabla }}_i}{\sqrt{e_{o,i}^2+k_{\mathrm{hk},i}^2}}]k_{\alpha }{z}_1^T{z}_1-\frac{\pi }{r{T}_v}{z}_1^T(\frac{1}{2^{1-(r/2)}}{z}_1^{1-r}+\frac{1}{2^{1+(r/2)}}{z}_1^{1+r})+z_1^T{\lambda }_{1}J(q)(z_2+e_v).$$

(20)

Secondly, choose another Lyapunov function $V_2=V_D+\frac{1}{2}{e}_v^T{e}_v+\frac{1}{2}{\tilde{z}}_u^T{\tilde{z}}_u$ , and the time derivative of $V_2$ , one gets

$$\begin{array}{l}{\stackrel{·}{V}}_2\le \sum _{i=1}^4[-k_{o,i}{e}_{o,i}^2+e_{o.i}{\tilde{W}}_i^T{\Phi }_i+e_{o.i}{\overline{\nabla }}_i-\frac{\pi }{r{T}_v}\left(\frac{1}{2^{1-\left(r/2\right)}}{e}_{o,i}^{2-r}+\frac{1}{2^{1+\left(r/2\right)}}{e}_{o,i}^{2+r}\right)-e_{o,i}^2{\widehat{\nabla }}_i/\sqrt{e_{o,i}^2+k_{hk,i}^2}-\frac{\pi }{r{T}_v}\left(\frac{1}{2^{1-\left(r/2\right)}}{z}_{1,i}^{2-r}+\frac{1}{2^{1+\left(r/2\right)}}{z}_{1,i}^{2+r}\right)]\\ +z_1^T{\lambda }_{1}J(q)(z_2+e_v)+z_1^T{\lambda }_{1}J(q)(z_2+e_v)+e_v^T-\left(\frac{\pi }{r{T}_v}\left(e_v^{1+r}+e_v^{1-r}\right)-k_{av}{e}_v-{\widehat{z}}_{u}tanh\left(\frac{e_v}{{\kappa }_h}\right)-{\stackrel{·}{\alpha }}_{\theta }\right)\\ -{\tilde{z}}_u^T\left(e_{v}tanh\left(\frac{e_v}{{\kappa }_h}\right)-k_z{\widehat{z}}_u-\frac{\pi }{r{T}_v}\left(\frac{2-r}{2^{1-(r/2)}}{\widehat{z}}_u^{1-r}+\frac{2+r}{2^{1+(r/2)}}{\widehat{z}}_u^{1+r}\right)\right).\end{array}$$

According to the Lemma 5, the following inequalities hold

$$\begin{array}{c}(2z_{u,i}^{2+r}-{\tilde{z}}_{u,i}^{2+r})/(2+r)\ge {\tilde{z}}_{u,i}{\hat{z}}_{u,i}^{1+r},\hfill \\ (2z_{u,i}^{2-r}-{\tilde{z}}_{u,i}^{2-r})/(2-r)\ge {\tilde{z}}_{u,i}{\hat{z}}_{u,i}^{1-r}.\hfill \end{array}$$

(21)

Based on Lemma 4 and $z_{u,i}\ge \left|{\stackrel{·}{\alpha }}_{\theta ,i}\right|$ , equation (21) is rewritten as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2\le \sum _{i=1}^4[e_{o.i}{\tilde{W}}_i^T{\Phi }_i+\left|e_{o.i}\right|{\overline{\nabla }}_i-k_{o,i}{e}_{o,i}^2}\\ \multicolumn{1}{c}{-\frac{\pi }{r{T}_v}(\frac{1}{2^{1-(r/2)}}{e}_{o,i}^{2-r}+\frac{1}{2^{1+(r/2)}}{e}_{o,i}^{2+r})}\\ -e_{o,i}^2{\hat{\nabla }}_i/\sqrt{e_{o,i}^2+k_{\mathrm{hk},i}^2}]-\frac{\pi }{r{T}_v}{z}_1^T(\frac{1}{2^{1-(r/2)}}{z}_1^{1-r}+\frac{1}{2^{1+(r/2)}}{z}_1^{1+r})\\ -\frac{\pi }{r{T}_v}{e}_v^T(e_v^{1+r}+e_v^{1-r})-k_{\mathrm{av}}{e}_v^T{e}_v-e_v^T({\stackrel{·}{\alpha }}_{\theta }+z_u^T\tanh (e_v/{\kappa }_h))\\ -\frac{\pi }{r{T}_v}(\frac{2-r}{2^{1-(r/2)}}{\hat{z}}_u^{2-r}+\frac{2+r}{2^{1+(r/2)}}{\hat{z}}_u^{2+r})+\frac{\pi }{r{T}_v}(\frac{z_u^{2-r}}{2^{-(r/2)}}+\frac{z_u^{2+r}}{2^{(r/2)}})+z_1^{T}J(q){\lambda }_1(z_1+e_v).\end{array}$$

(22)

Then, (22) can be rewritten as:

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2\le \sum _{i=1}^4[e_{o.i}{\tilde{W}}_i^T{\Phi }_i-e_{o,i}^2{\hat{\nabla }}_i/\sqrt{e_{o,i}^2+k_{\mathrm{hk},i}^2}]}\\ \multicolumn{1}{c}{-\frac{\pi }{r{T}_v}{e}_o^T(\frac{1}{2^{1-(r/2)}}{e}_o^{1-n}+\frac{1}{2^{1+(r/2)}}{e}_o^{1+n})}\\ \multicolumn{1}{c}{-\frac{\pi }{r{T}_v}{e}_v^T(e_v^{1+r}+e_v^{1-r})-\frac{\pi }{r{T}_v}{z}_1^T(\frac{1}{2^{1-(r/2)}}{z}_1^{1-r}}\\ \multicolumn{1}{c}{+\frac{1}{2^{1+(r/2)}}{z}_1^{1+r})-\frac{\pi }{r{T}_v}{\hat{z}}_u^T(\frac{2-r}{2^{1-(r/2)}}{\hat{z}}_u^{1-r}}\\ \multicolumn{1}{c}{+\frac{2+r}{2^{1+(r/2)}}{\hat{z}}_u^{1+r})+z_1^{T}J(q){\lambda }_1(z_1+e_v)+R,}\end{array}$$

(23)

in which

$$R=z_u\hslash {\kappa }_{\hslash }+\frac{\pi }{r{T}_v}(\frac{z_u^{2-r}}{2^{-(r/2)}}+\frac{z_u^{2+r}}{2^{(r/2)}}).$$

In additions, to enhance the robustness of the formation system, the velocity error $z_{2,i}$ is constrained for transient performance.

$$z_{3,i}=2b_i{z}_{2,i}/({\overline{p}}_i(t)+{\underset{¯}{p}}_i(t)).$$

(24)

Similarly to (10), where ${\overline{p}}_i(t)$ and ${\underset{¯}{p}}_i(t)$ are the PPF for the upper and lower constraint bounds, denoting the ${\overline{p}}_{\infty ,i}(t)$ and ${\underset{¯}{p}}_{\infty ,i}(t)$ are the convergence value of the PPF ${\overline{p}}_i(t)$ and ${\underset{¯}{p}}_i(t)$ , respectively. The $z_{3,i}$ is the transformed variable of velocity error $z_{2,i}$ with the following range of values $z_{3,i}\in (-U_i,D_i)$ , where $U_i=2b_i{\underset{¯}{p}}_i(t)/({\overline{p}}_i(t)+{\underset{¯}{p}}_i(t)),D_i=2b_i{\overline{p}}_i(t)/({\overline{p}}_i(t)+{\underset{¯}{p}}_i(t))$ .

Differentiating $z_{3,i}$ with respect to time, one obtains

$${\stackrel{·}{z}}_{3,i}={\overline{\lambda }}_{1,i}{\stackrel{·}{z}}_{2,i}+{\overline{\lambda }}_{2,i},$$

(25)

in which

$$\begin{array}{c}{\overline{\lambda }}_{1,i}={ƛ}_{1,i}>0,\hfill \\ {\overline{\lambda }}_{2,i}=2b_i{z}_{2,i}({\stackrel{·}{\overline{p}}}_i+{\stackrel{·}{\underset{¯}{p}}}_i)/({\overline{p}}_i+{\underset{¯}{p}}_i{)}^2.\hfill \end{array}$$

According to (9), another transformation function designed for $z_{3,i}$ is

$$z_{4,i}=\ln ((z_{3,i}+{\underset{¯}{\eta }}_i)/({\overline{\eta }}_i-z_{3,i})),$$

(26)

and differentiate it with respect to time, it has ${\stackrel{·}{z}}_{4,i}={\psi }_{1,i}{\stackrel{·}{z}}_{2,i}+{\psi }_{2,i}$ with

$$\begin{array}{c}{\psi }_{1,i}={\overline{\lambda }}_{1,i}(\frac{1}{z_{3,i}+{\underset{¯}{\eta }}_i}+\frac{1}{{\overline{\eta }}_i-z_{3,i}}),\hfill \\ {\psi }_{2,i}={\overline{\lambda }}_{2,i}(\frac{1}{z_{3,i}+{\underset{¯}{\eta }}_i}+\frac{1}{{\overline{\eta }}_i-z_{3,i}})\hfill \\ \multicolumn{1}{c}{+\frac{{\stackrel{·}{\rho }}_{i,2}}{z_{3,i}+{\underset{¯}{\eta }}_i}-\frac{{\stackrel{·}{\rho }}_{i,1}}{{\overline{\eta }}_i-z_{3,i}}.}\end{array}$$

Thirdly, construct the positive definite Lyapunov function

$$V_3=V_2+\frac{1}{2}{z}_{4,i}^2.$$

(27)

By taking the first-order derivative of $V_3$ and substituting the control law (16), ones yields

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_3\le \sum _{i=1}^4[-k_{o,i}{e}_{o,i}^2+e_{o,i}{\tilde{W}}_i^T{\Phi }_i+\left|e_{o.i}\right|{\overline{\nabla }}_i+\left|z_{4,i}\right|{\overline{\nabla }}_i}\\ \multicolumn{1}{c}{-\frac{e_{o,i}^2{\hat{\nabla }}_i}{\sqrt{e_{o,i}^2+k_{\mathrm{hk},i}^2}}-z_{4,i}{\tilde{W}}_i^T{\Phi }_i-\frac{z_{4,i}^2{\hat{\nabla }}_i}{\sqrt{z_{4,i}^2+k_{\mathrm{hk},i}^2}}]}\\ \multicolumn{1}{c}{-\frac{\pi }{r{T}_v}{e}_o^T(\frac{1}{2^{1-(r/2)}}{e}_o^{1-r}+\frac{1}{2^{1+(r/2)}}{e}_o^{1+r})-\frac{\pi }{r{T}_v}{z}_1^T(\frac{1}{2^{1-(r/2)}}{z}_1^{1-r}+\frac{1}{2^{1+(r/2)}}{z}_1^{1+r})}\\ \multicolumn{1}{c}{-\frac{\pi }{r{T}_v}{e}_v^T(e_v^{1+r}+e_v^{1-r})-\frac{\pi }{r{T}_v}{\tilde{z}}_u^T(\frac{2-r}{2^{1-(r/2)}}{\tilde{z}}_u^{1-r}+\frac{2+r}{2^{1+(r/2)}}{\hat{z}}_u^{1+r})}\\ \multicolumn{1}{c}{-\frac{\pi }{r{T}_v}{z}_4^T(\frac{1}{2^{1-(r/2)}}{z}_4^{1-r}+\frac{1}{2^{1+(r/2)}}{z}_4^{1+r})+R.}\end{array}$$

(28)

In light of Lemma 3, (28) can be rewritten as

$$\begin{array}{c}{\stackrel{·}{V}}_3\le \sum _{i=1}^4[-k_{o,i}{e}_{o,i}^2+e_{o,i}{\tilde{W}}_i^T{\Phi }_i+\frac{e_{o,i}^2{\tilde{\nabla }}_i}{\sqrt{e_{o,i}^2+k_{\mathrm{hk},i}^2}}\hfill \\ +\frac{z_{4,i}^2{\tilde{\nabla }}_i}{\sqrt{z_{4,i}^2+k_{\mathrm{hk},i}^2}}+\frac{{\overline{\nabla }}_i^2}{h_{m,i}}+h_{m,i}{k}_{\mathrm{hk},i}^2-z_{4,i}{\tilde{W}}_i^T{\Phi }_i]\hfill \\ -\frac{\pi }{r{T}_v}{e}_o^T(\frac{1}{2^{1-(r/2)}}{e}_o^{1-r}+\frac{1}{2^{1+(r/2)}}{e}_o^{1+r})\hfill \\ -\frac{\pi }{r{T}_v}{z}_1^T(\frac{1}{2^{1-(r/2)}}{z}_1^{2-r}+\frac{1}{2^{1+(r/2)}}{z}_1^{2+r})\hfill \\ -\frac{\pi }{r{T}_v}{e}_v^T\left(\begin{array}{c}{e}_v^{1+r}\hfill \\ \multicolumn{1}{c}{+e_v^{1-r}}\end{array}\right)-\frac{\pi }{r{T}_v}{\tilde{z}}_u^T(\frac{2-r}{2^{1-(r/2)}}{\tilde{z}}_u^{1-r}+\frac{2+r}{2^{1+(r/2)}}{\hat{z}}_u^{1+r}),\hfill \\ -\frac{\pi }{r{T}_v}{z}_4^T(\frac{1}{2^{1-(r/2)}}{z}_4^{1-r}+\frac{1}{2^{1+(r/2)}}{z}_4^{1+r})+R,\hfill \end{array}$$

According to Theorem 1, the following Lyapunov function $V_E$ is constructed

$$V_E=V_3+\frac{1}{2}\sum _{i=1}^4(\frac{1}{u_{1,i}}{\tilde{W}}_i^T{\tilde{W}}_i+\frac{{\tilde{\nabla }}_i^2}{u_{2,i}}+\frac{k_{\mathrm{hk},i}^2}{u_{3,i}}),$$

and the time derivative of $V_E$ is

$${\stackrel{·}{V}}_E={\stackrel{·}{V}}_3+\sum _{i=1}^4(\frac{1}{u_{1,i}}{\tilde{W}}_i^T{\stackrel{·}{\hat{W}}}_i-\frac{1}{u_{2,i}}{\tilde{\nabla }}_i^T{\stackrel{·}{\hat{\nabla }}}_i+\frac{1}{u_{3,i}}{k}_{\mathrm{hk},i}^T{\stackrel{·}{k}}_{\mathrm{hk},i}).$$

(29)

Based on the adaptive law (15), the novel adaptive law of the ESNO is designed as

$$\begin{array}{l}{\stackrel{·}{\widehat{W}}}_i=u_{1,i}({\Phi }_i|e_{o,i}|+{\Phi }_i|z_{4,i}|-\frac{\pi }{r{T}_v}(\frac{2-r}{2^{1-(r/2)}}{\widehat{W}}_i^{1-r}+\frac{2+r}{2^{1+(r/2)}}{\widehat{W}}_i^{1+r})),\\ {\stackrel{·}{\widehat{\nabla }}}_i=u_{2,i}[\frac{e_{o,i}^2{\widehat{\nabla }}_i}{\sqrt{e_{o,i}^2+k_{hk,i}^2}}+\frac{z_{4,i}^2{\widehat{\nabla }}_i}{\sqrt{z_{4,i}^2+k_{hk,i}^2}}-\frac{\pi }{r{T}_v}(\frac{2-r}{2^{1-(r/2)}}{\widehat{\nabla }}_i^{1-r}+\frac{2+r}{2^{1+(r/2)}}{\widehat{\nabla }}_i^{r+1})],\\ {\stackrel{·}{k}}_{hk,i}=u_{3,i}[{\kappa }_{h1,i}|e_{o,i}|+{\kappa }_{v1,i}|z_{4,i}|-{\kappa }_{h2,i}{k}_{hk,i}-\frac{\pi }{r{T}_v}(\frac{1}{2^{1+(r/2)}}{k}_{hk,i}^{r+1}+\frac{1}{2^{1-(r/2)}}{k}_{hk,i}^{1-r})].\end{array}$$

(30)

Substituting (30) into (29) yields the following simplified expression:

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_E\le \sum _{i=1}^4-k_{o,i}{e}_{o,i}^2-{\kappa }_{h2,i}{k}_{\mathrm{hk},i}^2+[-\frac{\pi }{r{T}_v}(\frac{1}{2^{1-n/2}}{e}_{o,i}^{2-r}}\end{array}+\frac{1}{2^{1+n/2}}{e}_{o,i}^{2+r})+\frac{{\overline{\nabla }}_i^2}{h_{m,i}}-\frac{e_{o,i}{\hat{\nabla }}_i}{\sqrt{e_{o,i}^2+k_{\mathrm{hk},i}^2}}]$$

(31)

$$\begin{array}{l}-\frac{\pi }{r{T}_v}(\frac{1}{2^{1-r/2}}{z}_{1,i}^{2-r}+\frac{1}{2^{1+r/2}}{z}_{1,i}^{2+r})-\frac{\pi }{r{T}_v}(e_{v,i}^{2+r}+e_{v,i}^{2-r})-\frac{\pi }{r{T}_v}(\frac{2-r}{2^{1-r/2}}{\tilde{z}}_{u,i}^{2-r}+\frac{2+r}{2^{1+r/2}}{\widehat{z}}_{u,i}^{2+r})\\ -\frac{\pi }{r{T}_v}(\frac{1}{2^{1-r/2}}{z}_{4,i}^{2-r}+\frac{1}{2^{1+r/2}}{z}_{4,i}^{2+r})]-\sum _{i=1}^4{\tilde{W}}_i^T(-\frac{\pi }{r{T}_v}(\frac{2-r}{2^{1-r/2}}{\widehat{W}}_i^{1-r}+\frac{2+r}{2^{1+r/2}}{\widehat{W}}_i^{1+r}))\\ [\frac{\pi }{r{T}_v}(\frac{2-r}{2^{1-r/2}}{\widehat{\nabla }}_i^{2-r}+\frac{2+r}{2^{1+r/2}}{\widehat{\nabla }}_i^{r+2})]-\sum _{i=1}^4{k}_{h,i}{\kappa }_{h1,i}\left|e_{o,i}\right|+{\kappa }_{v1,i}\left|z_{4,i}\right|-\frac{\pi }{r{T}_v}(\frac{1}{2^{1+r/2}}{k}_{h,i}^{r+1}+\frac{1}{2^{1-r/2}}{k}_{h,i}^{1-r})]+R.\end{array}$$

In light of (21) and (34), (31) is rewritten as

$$\begin{array}{c}{\stackrel{·}{V}}_E\le \sum _{i=1}^4[-(k_{o,i}-\frac{h_{2,i}{\kappa }_{h1,i}+h_{3,i}{\kappa }_{v1,i}}{2})e_{o,i}^2\hfill \\ \multicolumn{1}{c}{-({\kappa }_{h2,i}-h_{m,i})k_{\mathrm{hk},i}^2-\frac{\pi }{r{T}_v}(\frac{1}{2^{1-(r/2)}}{e}_{o,i}^{2-r}+\frac{1}{2^{1+(r/2)}}{e}_{o,i}^{2+r})}\\ \multicolumn{1}{c}{-\frac{\pi }{r{T}_v}(\frac{1}{2^{1-(r/2)}}{z}_{1,i}^{1-r}+\frac{1}{2^{1+(r/2)}}{z}_{1,i}^{1+r})-\frac{\pi }{r{T}_v}(e_{v,i}^{2+r}+e_{v,i}^{2-r})}\\ -\frac{\pi }{r{T}_v}(\frac{2-r}{2^{1-(r/2)}}{\tilde{z}}_{u,i}^{2-r}+\frac{2+r}{2^{1+(r/2)}}{\hat{z}}_{u,i}^{2+r})\hfill \\ \multicolumn{1}{c}{-\frac{\pi }{r{T}_v}{\hat{W}}_i^T(\frac{2-r}{2^{1-(r/2)}}{\hat{W}}_i^{1-r}+\frac{2+r}{2^{1+(r/2)}}{\hat{W}}_i^{1+r})}\\ -\frac{\pi }{r{T}_v}(\frac{1}{2^{1-(r/2)}}{z}_{4,i}^{2-r}+\frac{1}{2^{1+(r/2)}}{z}_{4,i}^{2+r})\hfill \\ \multicolumn{1}{c}{-\frac{\pi }{r{T}_v}(\frac{2-r}{2^{1-(r/2)}}{\hat{\nabla }}_i^{1-r}+\frac{2+r}{2^{1+(r/2)}}{\hat{\nabla }}_i^{r+1})}\\ -\frac{\pi }{r{T}_v}(\frac{1}{2^{1+(r/2)}}{k}_{\mathrm{hk},i}^{r+2}+\frac{1}{2^{1-(r/2)}}{k}_{\mathrm{hk},i}^{2-r})]+R_1,\hfill \end{array}$$

(32)

where $2k_{o,i}>h_{2,i}{\kappa }_{h1,i}+h_{3,i}{\kappa }_{v1,i}$ , ${\kappa }_{h2,i}>h_{m,i}$ and $R_1=R+R_{\mathrm{\Delta }2}+\sum _{i=1}^4\frac{{\overline{\nabla }}_i^2}{h_{m,i}}$ . According to Lemma 1 and Lemma 2, (32) can be rewritten as ${\stackrel{·}{V}}_E\le -\frac{\pi }{r{T}_v}(V_E^{1-r/2}+V_E^{1+r/2})+R_1$ . If the Assumptions 1–3 are satisfied, the designed controller satisfies $\underset{t\to +\infty }{\lim }{\stackrel{·}{V}}_E=R_1$ , which implies that the errors $z_{4,i}$ , $e_{v,i}$ , ${\tilde{z}}_u$ , $z_{1,i}$ , $z_{2,i}$ and $e_{o,i}$ associated with the controller design can converge to a small neighborhood of $R_1$ within a predetermined time $T_v$ . Hence, the system is predefine-time stable.▪

• Remark 2. Although the predefined-time convergence is theoretically guaranteed by satisfying inequality constraints (36) and (32), practical performance optimization requires empirical parameter tuning. Through iterative experiments, increasing parameters $k_2$ , $k_{\alpha }$ , $k_{\mathrm{av},i}$ , $k_{z,i}$ , $r$ , $u_{2,i}$ and $u_{3,i}$ generally enhances transient performance and convergence speed. For scenarios demanding higher trajectory tracking accuracy under stricter performance constraints or larger disturbances, jointly increasing ${\rho }_{\infty ,i,j}$ , $T_v$ , $k_{o,i}$ while decreasing $u_{1,i}$ , $r$ is recommended. Crucially, parameters must be co-tuned rather than adjusted individually to meet multi-objective requirements.

Numerical simulations and physical experiments

In this section, to verify the robustness and transient performance of the proposed controller, simulation and physical experiments were conducted on the above model, which are named as Case Study 1 and Case Study 2, respectively. The controller and observer parameters are selected based on the stability conditions derived in Theorem 1 and Theorem 2 and the desired performance specifications.

Case Study 1: The end-effector of the mobile robotic arm and the ideal trajectory of the chassis are set as $X_d=[0.2t,0.2t,0.2t,0.2t+0.1\sin (0.5t)+0.15{]}^T$ . $X_1(0)=[\frac{\pi }{2},\frac{\pi }{6},0,0{]}^T$ is randomly chosen as the initial positions of the MMR, which for brevity is defined as $\mathrm{actual}$ . The external disturbance is selected as ${\tau }_d=[0.5,0.5,0.5,0.5{]}^T$ and the model parameters associated with system (1) are adopted from Wu et al. ³³ The controller and ESNO parameters are designed to constrain the tracking errors within the specified asymmetric bounds and their velocity counterparts, which are summarized in Table 1, while the PTPPF parameters are listed in Table 2.

Table 1.

Controller and observer parameters (param.).

Param.	Value	Param.	Value	Param.	Value	Param.	Value
$k_a$	$\mathrm{diag}\{3.9,3.7,3.5,3.5\}$	$k_{\mathrm{av},2}$	$0.025$	${\kappa }_{h,i}$	$1.4$	$u_{3,1}$	$1.31$
$k_z$	$\mathrm{dia}{g}_4[4.5]$	$k_{\mathrm{av},4}$	$0.03$	$u_{1,i},u_{2,3},u_{2,4}$	$0.01$	$u_{3,2}$	$2.31$
$k_{\mathrm{av},1},k_{\mathrm{av},3}$	$0.02$	$k_{z,i}$	$1.5$	$u_{2,1},u_{2,2}$	$0.1$	$u_{3,3},u_{3,4}$	$1.01$
$c_i$	$[-0.1,0.1]$	$b_i$	$[0,1]$	$k_{h2,i}$	$1.96$	$k_{o,i}$	$0.4$

Table 2.

PTPPF parameters.

Param.	Value	Param.	Value	Param.	Value	Param.	Value
${\rho }_{i,1},{\rho }_{2,1}$	$3.5$	${\rho }_{\infty ,1,1}$	$0.28$	${\rho }_{\infty ,4,1},{\rho }_{\infty ,4,2}$	$0.25$	${\overline{p}}_{\infty ,3},{\overline{p}}_{\infty ,4}$	$1.05$
${\rho }_{1,2},{\rho }_{4,2}$	$3.6$	${\rho }_{\infty ,1,2}$	$0.27$	$b_1,b_2,b_4$	$0.35$	${\underset{¯}{p}}_{\infty ,1}$	$1.47$
${\rho }_{2,2}$	$0.02$	${\rho }_{\infty ,2,i},{\rho }_{\infty ,3,1}$	$0.25$	$b_3$	$0.3$	${\underset{¯}{p}}_{\infty ,2}$	$1.57$
${\rho }_{4,2}$	$3.6$	${\rho }_{\infty ,3,2}$	$0.23$	${\overline{p}}_{\infty ,1},{\overline{p}}_{\infty ,2}$	$1.08$	${\underset{¯}{p}}_{\infty ,3},{\underset{¯}{p}}_{\infty ,4}$	$1.53$
${\overline{p}}_1,{\overline{p}}_2,{\overline{p}}_3$	$3.5$	${\overline{p}}_4(t)$	$3.53$	${\underset{¯}{p}}_1,{\underset{¯}{p}}_4$	$3.6$	${\underset{¯}{p}}_2$	$3.3$
${\underset{¯}{p}}_3$	$3.4$	${\sigma }_{i,j}$	$0.25$	${\ell }_i$	1	-	-

The external disturbance is designed as ${\tau }_d=[0.88,0.3,0.2,0.2{]}^T$ , the predefined convergence times are set uniformly as $T_v=T_{i,j}=3s$ , and exponent parameters are chosen as $r=0.6$ . For comparative analysis, two additional initial positions are tested: $\mathrm{actual}2:[\frac{0.5\pi }{2},\frac{0.5\pi }{6},0,0{]}^T$ and $\mathrm{actual}3:[\frac{\pi }{2},\frac{\pi }{6},0.1,0.08{]}^T$ . Regarding the designed control algorithm, all the results based on numerical simulation are exhibited in Figures 4 to 13.

Figure 4.

Trajectory tracking of mobile platform and end-effector.

Figure 5.

Trajectory comparison under different initial conditions.

Figure 6.

Mobile platform and End effector position tracking errors.

Figure 7.

Transformed mobile platform and end-effector position error.

Figure 8.

Mobile platform and end-effector velocity error.

Figure 9.

Transformed mobile platform and end-effector velocity error.

Figure 10.

Estimates of the ESNO weight $||\hat{W}||$ for each compontent.

Figure 11.

Output of the compensation system and the value of estimation error $e_o$ .

Figure 12.

The adaptive parameters ${\hat{\tau }}_d$ , ${\hat{\nabla }}_i$ and $k_{h,i}$ for ESNO.

Figure 13.

The controller output torque.

From Figure 4, it is easy to see that the end of the MMR and also the chassis are able to track the desired trajectory. A comparison by selecting any initial position in Figure 5 shows that the MMR can track the desired trajectory quickly for different initial positions. This is consistent with the advantage of predefined time stabilization independent of initial position. As shown in Figures 6 and 7, the position tracking error is constrained to the specified range and converges to within ±0.01 m in approximately $2.5s$ , the angular tracking error converges to within ±0.5° in the same period. This demonstrates compliance with the prescribed performance bounds. Meanwhile, the transformed error $e_z$ rapidly decreases and stabilizes within the predefined envelope by $t=3s$ . As can be seen from Figures 7 and 8, the velocity errors $z_2$ are larger in magnitude compared to the position errors, both the velocity errors and the transformed errors $z_3$ are still constrained within the specified range. This confirms that the proposed method effectively bounds both tracking and velocity errors, consistent with its theoretical advantages. Furthermore, the trajectory tracking errors and velocity errors converge rapidly, with all errors achieving convergence within a predefined time. As observed in Figures 9 and 10, the estimates of the ESNO weights and the output of the compensation system converge quickly. Figure 10 presents that the adaptive parameters ${\hat{\nabla }}_i$ and $k_{h,i}$ for ESNO can converge with a high rate, exemplifying the superiority of the compensation system. In Figure 11, it can be seen that the output of the ESN also converges very quickly, except for a slight overshoot in the initial stage. The control torque output is shown in Figure 12. All the simulation results demonstrate the strong transient performance and robustness of the designed controller.

Case Study 2: The actual MMR and its control hardware are presented in Figure 14. The initial positions of the mobile platform and the end-effect were designed as $[0,0,0.05{]}^T$ and $[0.15,0.1,0.15{]}^T$ , while their desired trajectories were set to $X_d=[0.16t,0,0.16t+0.05\sin (2t),0.05\sin (2t{)]}^T$ . To demonstrate the superiority of the proposed method, comparative experiments were conducted under identical conditions against conventional PD and PID control algorithms. All experimental results are illustrated in Figures 14 to 17.

Figure 14.

The MMR and the control hardware.

Figure 15.

The real-world testing scenario.

Figure 16.

3D trajectory of the mobile platform and end-effector.

Figure 17.

2D trajectory comparison for the MMR.

Figure 15 shows the real-world testing scenario. Figure 16 displays 3D operational trajectory diagrams of the mobile platform and end-effector derived from experimental data. It can be observed that the proposed method enables both the mobile platform and robotic arm to closely track their respective desired trajectories during actual operation. As show in Table 3, compared to conventional PID control, the proposed controller demonstrates significantly enhanced robustness, achieving a more than 75% reduction in position error and 60% lower velocity error. Figure 17 provides a 2D trajectory comparison for the MMR. Despite limitations in hardware resources leading to suboptimal tracking performance, the experimental results partially validate the advantages of the proposed algorithm.

Table 3.

Tracking accuracy comparison.

Controller	Average position error	Average velocity error
Proposed	0.0183 m	0.0527 m/s
PID	0.0752 m	0.1605 m/s
PD	0.0921 m	0.160 m/s

Conclusions

For the trajectory tracking control a predefined time-stabilized control strategy with full state constraints is proposed based on the ESNO for the MMR under external disturbances. Firstly, an ESN-based adaptive predefined time-convergent disturbance observer is proposed to approximate the external disturbance. Secondly, a controller with tracking position and velocity constraints is constructed based on an interval-based concept of the PPC. The introduction of a PTPPF and the design of the PPC with a simple form enable the tracking error and the transformation error of the PPC to be taken into account and converge within a predefined time. Thirdly, a predefined time convergence second order filter is proposed to solve the complexity explosion problem. Finally, through numerical simulations and comparative physical experiments, the proposed controller has demonstrated that the control architecture exhibits improvements in both dynamic response speed and steady-state tracking accuracy. However, three key limitations warrant attention: (1) The computational burden exhibits cubic scaling with degrees of freedom (DOF), limiting real-time implementation beyond 6-DOF systems. (2) Coordination of MMR introduces communication delays not currently addressed in our stability framework. (3) barrier function saturation may occur under extreme impact disturbances exceeding 15 Nm. To overcome these challenges, future research will develop lightweight ESN architectures via neuromorphic computing, design delay-robust consensus protocols for distributed multi-agent systems, and integrate impedance-controlled barrier functions for impact-resilient performance. These directions will extend the method’s applicability to complex industrial scenarios while preserving its predefined-time convergence guarantees.