###### Abstract  
In this paper, we study the finite-time formation control problem of uncertain nonholonomic mobile robots following a parameterized path. A path-guided formation control scheme based on an extended state observer is proposed. To compensate for unmeasured velocities and disturbances simultaneously, a third-order fast finite-time extended state observer (FFTESO) is proposed. Then, a distributed formation control solution is developed to achieve the kinematic and dynamic system coordination control of the nonholonomic mobile robots. For the kinematic system control, a path-parameter-updating law is developed for a virtual leader, and the desired linear velocity and heading angle are developed for the mobile robots. For the kinetic control, an anti-disturbance control protocol is designed based on the estimated signals. The salient features of the proposed algorithms lie in that the estimations of disturbances and unmeasured velocities are achieved against the system's nonholonomic constraints, and the path following the control and cooperative control is synthesized together for path-guided formation, which reduces the complexity of the controller design. Finally, simulation studies are conducted to demonstrate the effectiveness of the proposed algorithm. onholonomic mobile robot; finite-time convergence; formation control; path-guided; extended state observer +
Footnote †: journal: Applied  
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where \(\|Y_{e}\|=\max\limits_{1\leq i\leq N}\{s_{i}\|\delta_{ie}\|+s_{i}\|\xi_{ie}\|+\sum \limits_{j=1}^{N}a_{ij}\|\xi_{je}\|\},K_{1}^{*}=\min\limits_{i=1,\dots,N}\{k_{ i}^{p},\frac{1}{\rho_{2}}\},\beta_{1}^{*}=\min\limits_{i=1,\dots,N}\{\beta_{i}^{p}, \beta^{*}\}\), and \(\kappa_{1}=k_{1}^{*}(2N+1)^{-\beta_{1}^{*}}\). The errors \(\|\xi_{ie}\|\), and \(\|\xi_{je}\|\) are bounded since the estimation errors are finite-time uniformly ultimately bounded. We assume that \(\|\delta_{ie}\|\) is bounded, which will be proved in the next theorem. Then, there exists a positive constant \(Y^{*}\) such that \(\|Y_{e}\|\leq Y^{*}\). One has  
\[\dot{V}_{3}\leq-(\kappa_{1}-\frac{\|Y^{*}\|}{\|E_{1}\|^{\beta_{1}^{*}}})\|E_{1 }\|^{\beta_{1}^{*}+1}\] (41)  
According to Lemma 1, \(\|E_{1}\|\leq\frac{\|Y^{*}\|}{\kappa_{1}}\) can be reached in a finite time. The convergence time is  
\[T_{4}\leq\frac{V_{4}^{\frac{1-\beta_{1}^{*}}{2}}}{\kappa_{1}(1-\beta_{1}^{*})}.\] (42)  
\(\square\)  
**Theorem 4**. : _Consider a network of nonholonomic mobile robots with dynamics (1) and (2) with the dynamic control laws (30), (33), and (35). If Assumptions 1-3 are satisfied, subsystem (37) is finite-time stable._  
Proof. : Choose the Lyapunov function candidate as follows:  
\[V_{4}=\frac{1}{2}\underset{i=1}{\overset{N}{\sum}}[\left(e_{i}^{v}\right)^{2 }+\left(e_{i}^{\theta}\right)^{2}+\left(e_{i}^{\omega}\right)^{2}]\] (43)  
Taking the time derivative of \(V_{4}\) yields  
\[ \begin{array}{ll}\dot{V}_{4}&=\underset{i=1}{\overset{N}{\sum}}[e_{i}^{v}(- k_{i}^{v}\text{sig}^{\theta_{i}^{v}}(e_{i}^{v}))+e_{i}^{\theta}[-k_{i}^{\theta} \text{sig}^{\theta_{i}^{\theta}}(e_{i}^{\theta})+e_{i}^{\omega}]+e_{i}^{ \omega}[-k_{i}^{\omega}\text{sig}^{\theta_{i}^{\omega}}(e_{i}^{\omega})-e_{i} ^{\theta}]]\\ &=\underset{i=1}{\overset{N}{\sum}}[\left.-k_{i}^{v}\right|e_{i}^{v}\right|^{ \beta_{1}^{*}+1}-k_{i}^{\theta}|e_{i}^{\theta}|^{\beta_{1}^{*}+1}-k_{i}^{ \omega}|e_{i}^{\omega}|^{\beta_{i}^{*}+1}_{i}]\end{array} \] (44)  
Denote \(E_{i2}=[e_{i}^{v},e_{i}^{\theta},e_{i}^{\omega}]^{T}\). According to inequality \(\underset{i=1}{\overset{N}{\sum}}(z_{i})^{p}\geq N^{1-p}(\underset{i=1}{ \overset{N}{\sum}}z_{i})^{p}\) with the positive real numbers \(z_{1},\dots,z_{N}>0\) and \(p>1\), we have  
\[ \begin{array}{ll}\dot{V}_{4}&\leq-k_{2}^{*}(3N)^{-\beta_{2}^{*}}[\underset{ i=1}{\overset{N}{\sum}}(|e_{i}^{v}|+|e_{i}^{\theta}|+|e_{i}^{\omega}|)]^{1+ \beta_{2}^{*}}\\ &\leq-\kappa_{2}\underset{i=1}{\overset{N}{\sum}}\|E_{i2}\|^{\beta_{2}^{*}+ 1}_{i}\end{array}