Neurodynamics- and observer-based distributed robust formation control for mobile robots

2.1. Communication topology

In this study, the communication relationship of a multi-robot system is mathematically modeled using graph theory. It is defined as follows: the system contains $$ n $$ robots (nodes), and its communication network is given by an undirected graph $$ \mathcal{G}=\{\mathcal{V}, \mathcal{E} \:\mathcal{A}\} $$ with node set $$ \mathcal{V}=\{1, \:2, \:..., n\} $$ and each node represents a robot. The edge set $$ \mathcal{E} \subseteq \mathcal{V} \times \mathcal{V} $$, if the robots $$ i $$ and $$ j $$ can communicate, then there exists an edge $$ (i, j) \in \mathcal{E} $$. The adjacency matrix $$ \mathcal{A}=[a_{ij}]_{n\times n} $$ indicates access to information from $$ i $$ to $$ j $$. In addition, $$ a_{ii}=0 $$ in this study. Define the Laplacian matrix $$ L=[\mathcal{L}_{ij}]_{n\times n} $$ to describe the network connectivity, where $$ \mathcal{L}_{ii}=\begin{matrix} \sum_{j=1}^n a_{ij} \end{matrix} $$ and $$ \mathcal{L}_{ij}=-a_{ij}, \:i \ne j $$. It denotes the communication between $$ n $$ followers, and is assumed to be connected. The leader is indexed as 0 and define $$ a = {\rm{diag}}({a_{10}}, \:..., \:{a_{n0}}) \in {^{{\rm{n}} \times {\rm{n}}}} $$ to reflect the communication between the leader and the followers. For elements in $$ a $$, it is defined that $$ a_{i0}>0 $$ if the follower $$ i $$ can directly obtain the leader's states and 0 otherwise. The communication network is represented by matrix $$ H $$, which is defined as $$ H = L+a $$. We then obtain Lemma 1.

Lemma 1. Matrix $$ H $$ is positive definite if the leader is the neighbor of at least one follower and $$ \mathcal{G} $$ is connected.

2.2. Problem statement

In this study, the general structure of the mobile robot is shown in Figure 1. It is assumed that all robots share a similar structure, with the front wheel being passive and the two rear wheels being driven. There are a total of $$ n $$ robots and one (virtual) leader. The kinematics of the follower $$ i $$ are given as

Figure 1. Mobile robot communication networks.

where $$ x_i $$ and $$ y_i $$ represent the position, and its angle of direction $$ \theta_i $$ is relative to the inertial frame; $$ \upsilon_i $$ and $$ \omega_i $$ represent the linear and angular velocities of the $$ i $$th mobile robot in body fixed frame, respectively.

Then, define leader's linear and angular velocities as $$ \upsilon_0 $$ and $$ \omega_0 $$, respectively, and their time derivatives are bounded. Then, there exist positive constants $$ \lambda_\upsilon $$ and $$ \lambda_\omega $$, such that $$ \upsilon_0\le{\lambda_\upsilon} $$ and $$ \omega_0\le{\lambda_\omega} $$.

Each mobile robot follows the dynamic model formulated in the following state-space representation ^[27]

where $$ 2b_i $$ denotes the wheelbase (m) and $$ r_i $$ the wheel radius (m). The coefficient $$ c_i $$ (dimensionless) accounts for Coriolis/centripetal effects. Viscous damping is given by $$ d_{1i} $$ (N.s.m⁻¹) and $$ d_{2i} $$ (N.m.s), whereas $$ m_1 $$ and $$ m_2 $$ denote the effective mass (kg) and rotational inertia (kg.m²), respectively. The torque inputs are defined as $$ \tau_{ai}=\tau_{Li}+\tau_{Ri} $$ and $$ \tau_{bi}=\tau_{Li}-\tau_{Ri} $$, where $$ \tau_{Li} $$ and $$ \tau_{Ri} $$ are the actuation torques from the left and right wheels, respectively (all in N.m). The terms $$ \delta_{1i} $$ and $$ \delta_{2i} $$ represent disturbances with units m.s⁻² and rad.s⁻², respectively. It is assumed that $$ \lVert\delta_{1i}\rVert $$ and $$ \lVert\delta_{2i}\rVert $$ are bounded by positive constants $$ \xi_{1i} $$ and $$ \xi_{2i} $$, respectively, and that their time derivatives are integrable in $$ L_{1} $$.being the actuation torques from the left and right wheels, respectively; $$ \delta_{1i} $$ and $$ \delta_{2i} $$ are the disturbances. It is assumed that $$ \left\|\delta_{1i}\right\| $$ and $$ \left\|\delta_{2i}\right\| $$ are both bounded by positive constants $$ \xi_{1i} $$ and $$ \xi_{2i} $$, respectively, and their time derivatives are integrable in $$ L_1 $$ space.

This paper aims to present a control approach enabling multiple mobile robots to track the leader while preserving relative positions. The distance between the $$ i $$th follower and the leader is written $$ \left[ {{\triangle x_i}, \:{\triangle y_i}} \right] $$ and represents distances in the $$ x $$ and $$ y $$ directions. In addition, the orientation of each follower is the same as that of the leader.

3.1. Design of distributed estimator

First, a distributed estimation scheme is constructed using a cascaded design methodology, enabling each follower to precisely estimate the leader's position and velocity without requiring its acceleration information. To allow the estimation in a fully decentralized manner, a velocity estimator is first designed. We define the consensus error of the $$ i $$th follower as

where $$ {e_{i\upsilon }} $$ and $$ {e_{i\omega }} $$ are, respectively, consensus velocity errors for linear and angular velocities; $$ \hat \upsilon_{i0} $$ and $$ \hat \omega_{i0} $$ are respectively the estimated linear and angular velocity of the leader from $$ i $$th mobile robot; $$ i $$ and $$ j $$ are index numbers. Then, based on the defined consensus estimation error in Equation (3), the velocity estimator is designed as

where $$ {k_{a1i}} $$, $$ {k_{a2i}} $$, $$ {k_{b1i}} $$, and $$ {k_{b2i}} $$ are positive design constants; and $$ \alpha (t) = {e^{ - ct}} $$ with $$ c > 0 $$. Then, we define the positional state estimation error as

with $$ {e_{ip}} = {\left[ {{e_{ix}}, \:{e_{iy}}, \:{e_{i\theta }}} \right]^T} $$ Then, based on the estimation error defined in Equation (5), we propose the distributed estimator for each mobile robot to estimate the leader's positional state as

where $$ {{\hat P}_{i0}} $$ denotes the estimated position of the leader obtained by the $$ i $$th mobile robot, and $$ {e_{ix}} $$, $$ {e_{iy}} $$, and $$ {e_{i\theta}} $$ represent the consensus errors along the $$ x $$-axis, $$ y $$-axis, and in orientation, respectively. The dynamics of the estimation errors $$ {{\bar \upsilon}_i} $$, $$ {{\bar \omega}_i} $$, and $$ {\bar P_i} $$ are derived based on the distributed observers in Equations (4) and (6), leading to

where $$ \bar \upsilon_i = \upsilon_i-\upsilon_0 $$ and $$ \bar \omega_i = \omega_i-\omega_0 $$. Then, we further define $$ \bar x = {\left[ {{{\bar x}_1}, \:..., \:{{\bar x}_n}} \right]^T} $$, $$ \bar y = {\left[ {{{\bar y}_1}, \:..., \:{{\bar y}_n}} \right]^T} $$ and $$ \bar \theta = {\left[ {{{\bar \theta }_1}, \:..., \:{{\bar \theta }_n}} \right]^T} $$. To establish the convergence of the distributed estimator, Theorem 1 is presented to demonstrate that the estimated leader's state asymptotically approaches the actual leader's state.

Theorem 1. The estimation error dynamics given in Equations (7) and (8) are asymptotically stable if parameters in Equations (4) and (6) are properly chosen such that $$ {k_{b1i}} \ge {\sup _{t \ge 0}}\left\| {{{\dot \upsilon }_0}} \right\| $$ and $$ {k_{b2i}} \ge {\sup _{t \ge 0}}\left\| {{{\dot \omega }_0}} \right\| $$.

Since the positional state estimation process is cascaded with the velocity estimates, we first need to prove the convergence of $$ \bar{\upsilon} $$ and $$ \bar{\omega} $$. Therefore, the Lyapunov candidate function for the velocity estimation errors is proposed as

where $$ \bar \upsilon = {\left[ {{{\bar \upsilon }_1}, \:..., \:{{\bar \upsilon }_n}} \right]^T} $$ and $$ \bar \omega = {\left[ {{{\bar \omega }_1}, \:..., \:{{\bar \omega }_n}} \right]^T} $$, then, we have the time derivative of $$ V_1 $$ calculated as

where $$ {e_\upsilon } = {[{e_{1\upsilon }}, \:..., \:{e_{n\upsilon}}]^T} $$ and $$ {e_\omega } = {[{e_{1\omega }}, \:..., \:{e_{n\omega}}]^T} $$. Then, using the inequality that $$ \left| x \right| - x\tanh (\dfrac{x}{\varepsilon }) \le k\varepsilon $$ with $$ k=0.28 $$ ^[28], we can obtain the following inequalities

Combined with the results obtained in Equation (11), Equation (10) it follows from Equation (10) that the following statement holds

where $$ {\lambda _1} $$ and $$ {\lambda _2} $$ denote the minimal eigenvalues of $$ {k_{a1}} $$ and $$ {k_{a2}} $$, respectively. According to Equation (12), we derive the following inequalities as

where $$ {\lambda _3} $$ denotes the minimal eigenvalue of $$ H $$ and $$ {\lambda _4} = \min \{ {\lambda _1}, \:{\lambda _2}\} $$. Consequently, the following linear differential equation can be derived as

After that, we present Lemma 2 to assist the following analysis.

Lemma 2. Let us consider a scalar differential equation given by

where the function $$ f(t, u) $$ is continuously differentiable with respect to $$ u $$ and locally Lipschitz continuous. Suppose the solution $$ u(t) $$ exists on an interval $$ [t_0, T) $$. Furthermore, consider a continuous function $$ V_1(t) $$ satisfying the differential inequality

Then, the conclusion of this lemma holds true regarding the relation between $$ V_1(t) $$ and the solution $$ u(t) $$.

Therefore, there is $$ {V_1}(t) \le u(t) $$ for all $$ t \in [{t_0}, \:T) $$. Utilizing Lemma 2 and the expression in Equation (14), we obtain

Thus, $$ {V_1}(t) \to 0 $$ as $$ t \to \infty $$, this leads directly to the conclusion that the estimation error $$ {\bar \upsilon } $$ and $$ {\bar \omega } $$ all converge to zero as $$ t \to \infty $$. Following the established convergence of velocity estimation errors $$ {\bar \upsilon } $$ and $$ {\bar \omega } $$, we next turn to the analysis of the positional estimation error $$ {\bar P} $$. To facilitate this, define $$ {k_\theta } = \rm{diag}({k_{\theta 1}}, \:..., \:{k_{\theta n}}) $$ and propose a Lyapunov candidate function associated with the orientation error as $$ {V_2} = \dfrac{1}{2}{{\bar \theta }^T}H\bar \theta $$, considering the error dynamics specified in Equation (8), the time derivative of $$ V_2 $$ is given by

where $$ {\beta _1} \in (0, \:{\lambda _5}) $$, and $$ {\lambda _5} $$ is the minimal eigenvalue of $$ {H}{k_\theta }{H} $$. It follows from Equation (18) and $$ \left\| {\bar \omega } \right\| \to 0 $$ as $$ t \to \infty $$, we can infer that $$ \bar\theta\to0 $$. Since $$ \left\| {\bar \omega } \right\| \to 0 $$ as $$ t \to \infty $$ as well. Following the proof of $$ \bar\theta $$, we need to guarantee the co-convergence of $$ \bar x $$ and $$ \bar y $$. First, it is defined that $$ {k_x} = {\rm{diag}}({k_{x\theta }}, \:..., \:{k_{xn}}) $$, $$ {{\hat \upsilon }_0} = {[{{\hat \upsilon }_{10}}, \:..., \:{{\hat \upsilon }_{n0}}]^T} $$ and $$ \cos {{\hat \theta }_0} = {[\cos {{\hat \theta }_{10}}, \:..., \:\cos {{\hat \theta }_{n0}}]^T} $$. Then, the corresponding Lyapunov candidate function is proposed as $$ {V_3} = \dfrac{1}{2}{{\bar x}^T}H\bar x $$ and consequently, $$ \dot V_3 $$ is calculated as

where $$ {\beta _2} \in (0, \:{\lambda _6}) $$, $$ {\iota _1} = {\hat\upsilon _0}\times\cos \hat\theta {}_0 - {{\dot x}_0}{1_n} $$. Based on the results obtained in Equation (18), we infer that $$ {\iota _1} \to 0 $$ as $$ t \to \infty $$; consequently, $$ \bar x \to 0 $$ as $$ t \to \infty $$. To address the $$ y $$-axis dynamics, we construct a Lyapunov candidate function as $$ {V_4} = \dfrac{1}{2}{{\bar y}^T}H\bar y $$, and proceed to derive its time derivative as

where $$ {\beta _3} \in (0, \:{\lambda _7}) $$ and $$ {\iota _1} = {\upsilon _0}\times\sin \theta {}_0 - {{\dot x}_0}{1_n} $$; and denote $$ {k_y} = \rm{diag}({k_{y\theta }}, \:..., \:{k_{yn}}) $$ and $$ \sin {{\hat \theta }_0} = {[\sin {{\hat \theta }_{10}}, \:..., \:\sin {{\hat \theta }_{n0}}]^T} $$. Hence, one concludes that $$ \bar y \to 0 $$ as $$ t \to \infty $$. The entire analysis concludes the proof of Theorem 1.

3.2. Design of observer

In practical applications involving mobile robots, direct measurements of velocity and external disturbances are often unavailable, despite their importance for achieving desirable control performance. To address this limitation, a nonlinear extended state observer is constructed in this subsection to estimate both quantities with high accuracy. The design builds upon the system dynamics given in Equation (2). Denote $$ {X_i} = [{P_i}^T, \:{\upsilon _i}, \:{\omega _i}, \:{\delta _i}^T] $$, we can reorganize the mobile robots system as

with

To enable real-time estimation of disturbances and velocities, we construct the following nonlinear state observer as

where $$ {{\tilde Y}_i} = {{\hat Y}_i} - {Y_i} $$ and $$ {Y_i} $$ are the estimates of $$ {Y_i} $$; $$ f({{\hat X}_i}) $$ describes the idealized model without disturbance terms, $$ {K_i} \in {\mathbb{R}^{7 \times 3}} $$ denotes a positive matrix, whose form is specified as

By subtracting Equation (25) from Equation (21), the estimation error dynamics is calculated as

We next formulate a theorem to ensure the stability of the designed nonlinear observer.

Theorem 2. Based on Assumptions 2, the error dynamics defined in Equation (27) converges to zero as $$ time \to \infty $$. If $$ {{\bar A}_i} - {K_i}{C_i} $$ is made Hurwitz. Then there exists a positive symmetric matrix $$ {K_i} $$ such that

We select $$ {V_{5i}} = {\tilde X_i}^T P_i {\tilde X_i} $$ as the Lyapunov candidate function and proceed to compute its time derivative as

Thus, we can infer that as long as $$ \left\| {{{\tilde X}_i}} \right\| \le \left\| {{{\dot \delta }_i}} \right\|\left\| {{P_i}} \right\| $$, the convergence of $$ \tilde X_i $$ is ensured. Based on the definition of Assumption 2, $$ \left\| {{{\dot \delta }_i}} \right\|\left\| {{P_i}} \right\| $$ will approach $$ 0 $$. As such, $$ \tilde X_i $$ will converge to $$ 0 $$ as well.

3.3. Bioinspired kinematic controller design

A bioinspired neural-dynamics-assisted kinematic controller is proposed in this section for formation control. The tracking error of the $$ i $$th follower is given in the inertial coordinate frame as

where $$ {{\hat x}_i} $$, $$ {{\hat y}_i} $$, and $$ {{\hat \theta }_i} $$ denote the observer-based estimates of the state variables for follower $$ i $$. According to Theorem 1, the estimate of the leader's posture $$ {{\hat P}_{0i}} $$ asymptotically approaches its true value $$ {P_0} $$, and Theorem 2 guarantees that $$ {{\hat x}_i} $$, $$ {{\hat y}_i} $$, and $$ {{\hat \theta }_i} $$ converge to their respective actual states. According to the established theoretical results, the next step is to formulate a control law to compute linear and angular velocities that guarantee the preservation of the desired formation structure relative to the leader. To begin with, the tracking error of follower $$ i $$ in its local coordinate frame is obtained by transforming the global error defined in Equation (30) through an appropriate coordinate transformation, leading to

where $$ {{e_{Di}}} $$, $$ {{e_{Li}}} $$ and $$ {{e_{\theta i}}} $$ represent the tracking errors in the driving, lateral, and angular directions. Differentiating this expression yields the time evolution of the error dynamics.

where $$ {{\Lambda _{ix}}} $$, $$ {{\Lambda _{iy}}} $$ and $$ {{\Lambda _{\theta{i}}}} $$ are the dynamics introduced by distributed estimators, respectively, given as

Based on the error dynamics presented in Equation (31), the kinematic controller is subsequently formulated to regulate the formation behavior as

where $$ k_{1i} $$, $$ k_{2i} $$, and $$ k_{3i} $$ are positive design constants; $$ {V_{si}} $$ is a dynamic process, which is given as

where $$ A_i $$ and $$ B_i $$ are the positive design constants. Initially, Equation (35) was introduced to simulate the adaptive response of membrane-based systems to dynamic environmental conditions ^[29]. A key advantage of this model lies in its ability to strictly constrain the output within a predefined bound $$ B_i $$, which effectively addresses the issue of velocity saturation. Moreover, in the presence of sudden tracking errors along the driving direction, the model's inherent dynamics ensure a gradual adjustment of the velocity command. This smooth transition helps prevent abrupt changes that could otherwise result in unrealistic or infinite torque demands.

After designing the bioinspired controller, a torque controller is further developed to generate torque commands and compensate for external disturbances. Using the defined velocity error and given the dynamics of the mobile robot in Equation (2), we subsequently formulate the control law as

where $$ {{\tilde \upsilon }_i} = \upsilon _i^{cmd} - {{\hat \upsilon }_i} $$ and $$ {{\tilde \omega }_i} = \omega _i^{cmd} - {{\hat \omega }_i} $$ are the linear and angular velocity errors, respectively; $$ {k_4} $$ and $$ {k_5} $$ are positive design constants. The overall design procedure of the proposed bioinspired observer-based formation control framework is summarized in Algorithm 1, which outlines the estimation and control steps executed by each mobile robot.

3.4. Stability analysis

This section presents a comprehensive analysis of the closed-loop stability of the proposed control scheme.

Lemma 3.^[10] Define $$ V $$: $$ \mathbb{R}^+ \to \mathbb{R}^+ $$ as a continuously differentiable scalar function, and consider a uniformly continuous function $$ W $$: $$ \mathbb{R}^+ \to \mathbb{R}^+ $$. Suppose for all $$ \forall t \ge 0 $$, the following differential inequality is satisfied:

where both $$ {p_1}, {p_2}\ge0 $$ belong to integrals over $$ \left[ {0\infty } \right) $$ are finite. Then, $$ V(t) $$ remains bounded for all time, and as $$ t\to\infty $$, $$ W(t) \to 0 $$ tends to zero and $$ V(t) $$ converges to a finite nonnegative limit.

We begin by examining the convergence of the velocity error dynamics. To this end, let us define the error between the actual velocity and the commanded velocity as

where $$ \tilde\upsilon_{ai} = \upsilon_i - \hat\upsilon_i $$ and $$ \tilde\omega_{ai} = \omega_i - \hat\omega_i $$ denote the deviations between the actual and estimated translational and rotational velocities, respectively. To facilitate the convergence analysis of the proposed dynamic control strategy, we consider the following Lyapunov function candidate as $$ {V_{6i}} = \dfrac{1}{2}{{\tilde e}^2}_{\upsilon i} + \dfrac{1}{2}{{\tilde e}^2}_{\omega i} $$. By computing the time derivative of $$ {V_{6i}} $$, we obtain

where $$ {\beta _{4i}} \in (0, \:{k_{4i}}) $$ and $$ {\beta _{5i}} \in (0, \:{k_{5i}}) $$, Since both $$ {{\tilde X}_{\upsilon i}} $$ and $$ {{\tilde X}_{\omega i}} $$ have been demonstrated to converge to zero over time, the derivative of the Lyapunov function $$ \dot{V}_{6i} $$ naturally converges to zero as $$ t \rightarrow \infty $$. Accordingly, it can be inferred that the corresponding velocity error terms, $$ {{\tilde e}_{\upsilon i}} $$ and $$ {{\tilde e}_{\omega i}} $$, are also asymptotically eliminated.

Following this, we shift focus to the kinematic controller. A Lyapunov function is proposed to analyze its convergence properties, defined as

The time derivative of $$ V_{7i} $$ is obtained as

where $$ {\xi_{\upsilon i}} $$ and $$ {\xi_{\omega i}} $$ denote the deviation between the agent's velocity and the leader's reference trajectory. By applying the bioinspired control law defined in Equations (34) and (41), we have

with

Based on Equation (4) and by ensuring $$ {\upsilon}_{i0}\ge0 $$ for $$ \overline{V}\ge0 $$, its estimates $$ \tilde{\upsilon_{i0}} $$ is guaranteed to be greater or equal to zero. Then, since the parameters defined in Equation (35), it is obvious that $$ {{\rm I}_{{\rm{1i}}}} \le 0 $$, and consequently, we infer that

Then, we have $$ {{\rm I}_{{\rm{2i}}}} $$ satisfies the conditions that

Since we have proof the convergence of $$ {{\Lambda _{ix}}} $$, $$ {{\Lambda _{iy}}} $$, $$ {{\Lambda _{i\theta}}} $$, $$ {{\xi _{\upsilon i}}} $$, and $$ {{\xi _{\omega i}}} $$. Then, we infer that $$ {{V_{7i}}} $$ satisfies Lemma 4. Thus, we have $$ {x_{si}} \to 0 $$ and $$ {\sin ^2}{e_{\theta i}} \to 0 $$ as $$ t \to \infty $$. As $$ {x_{si}} \to 0 $$, we have $$ {e_{Di}} \to 0 $$ as well, then based on the dynamics defined in Equation (32). We also have $$ {e_{Li}} \to 0 $$.

Authors' contributions

Idea generation, algorithm design, and writing and editing of the original draft: Xu, Z.

Simulation and writing of the original draft: Xia, Z.

Supervision and feedback: Li, W.; Yang, S. X.

Availability of data and materials

The data supporting the conclusions of this article will be made available by the authors upon request.

AI and AI-assisted tools statement

During the preparation of this manuscript, the AI tool Chatgpt (Version 4.5, released 2025-02-27) was used solely for language editing. The tool did not influence the study design, data collection, analysis, interpretation, or the scientific content of the work.

Financial support and sponsorship

This work is supported by the Guangxi Zhuang Autonomous Region Key Research and Development Program (Project NO.AB22035023).

Conflict of interest

Yang, S. X. is the Editor-in-Chief of the journal Intelligence & Robotics. Yang, S. X. was not involved in any steps of the editorial process, notably including reviewer selection, manuscript handling, or decision-making. The other authors declare that there are no conflicts of interest.

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Copyright

© The Author(s) 2026.