Formation Control of Wheeled Mobile Robots with Fault-Tolerance Capabilities

Abstract

This research investigates the impact of actuator faults on the formation control of multiple-wheeled mobile robots—a critical aspect in coordinating multi-robot systems for applications such as surveillance, exploration, and transportation. When a fault occurs in any of the robots, it can disrupt the formation and adversely affect the system’s performance, thereby compromising system efficiency and reliability. While numerous studies have focused on fault-tolerant control strategies to maintain formation integrity, there is a notable gap in the literature regarding the relationship between controller gains and settling time under varying degrees of actuator loss. In this paper, we develop a kinematic model of wheeled mobile robots and implement a leader–follower-based formation control strategy. Actuator faults are systematically introduced with varying levels of effectiveness (e.g., 80%, 60%, and 40% of full capacity) to observe their effects on formation maintenance. We generate data correlating controller gains with settling time under different actuator loss conditions and fit a polynomial curve to derive an equation describing this relationship. Comprehensive MATLAB simulations are conducted to evaluate the proposed methodology. The results demonstrate the influence of actuator faults on the formation control system and provide valuable insights into optimizing controller gains for improved fault tolerance. These findings contribute to the development of more robust multi-robot systems capable of maintaining formation and performance despite the presence of actuator failures.

1. Introduction

Formation control of multi-robot systems has emerged as a vital area of research, driven by the rapid proliferation of robotics in surveillance, exploration, transportation, and various industrial applications. Coordinating multi-wheeled mobile robots (WMRs) to achieve a desired formation pattern not only improves task efficiency but also enhances robustness and adaptability in complex and dynamic environments [1,2,3,4]. By maintaining a formation, a group of mobile robots can execute distributed tasks more effectively, minimize detection risk in surveillance missions, and provide extensive coverage in large-scale exploration. Formation control strategies commonly rely on decentralized or leader–follower approaches, where a designated leader sets the trajectory while the followers maintain specific relative positions [5,6,7].

Despite considerable progress in formation control techniques, one critical challenge remains, that is, the system’s ability to maintain formation in the presence of faults. Actuator faults—arising from component degradation, sensor inaccuracies, mechanical wear, or unexpected collisions—can significantly compromise a robot’s mobility, speed, or steering capabilities [8,9,10,11]. When manifested in one or more robots within a formation, such faults often lead to deviations from the desired configuration and ultimately impact settling time, position accuracy, and overall formation integrity. Prolonged recovery periods can be particularly detrimental in time-sensitive applications, such as search-and-rescue operations, where delayed responses may compromise mission success [12,13].

The field of fault-tolerant control (FTC) has addressed many of these issues, focusing on robust and adaptive controllers designed to accommodate faults while maintaining system stability and performance [14,15,16,17]. Numerous strategies have been proposed, including control reconfiguration, gain-scheduling, and adaptive estimation of faulty actuators to ensure that robots remain operational despite degraded capabilities [18,19,20,21]. Research efforts have explored both model-based and model-free approaches, leveraging techniques from control theory, artificial intelligence, and hybrid systems analysis [22,23,24].

However, despite these advances, a notable gap persists in the current literature—the quantification and explicit characterization of the relationship between controller gains and settling time under varying degrees of actuator loss. While it is generally understood that adjusting controller parameters can mitigate the effects of faults, the exact nature of this relationship and its implications for optimizing control strategies remain insufficiently explored. Understanding how the controller gains influence the transient response of the system—specifically the time required for the formation to re-establish itself after a fault—could enable more informed design choices and improve the overall resilience of multi-robot systems.

Existing studies often focus on high-level strategies and qualitative metrics, neglecting the finer quantitative details necessary for the systematic tuning of controller parameters. For instance, while extensive research has addressed formation control under nominal conditions [25,26], and leader–follower schemes have been studied rigorously [2,27,28], less attention has been paid to how these schemes respond when a robot’s actuator efficiency is reduced to fractions of its nominal capacity (e.g., 80%, 60%, or 40%). This gap in the literature hinders the development of analytical tools and practical guidelines for optimizing controller gains to minimize performance degradation.

This paper aims to fill this gap by presenting a comprehensive analysis of how actuator faults affect formation maintenance and settling time in the leader–follower formation control of WMRs. We build on foundational kinematic models of WMRs [29,30,31] and integrate actuator fault scenarios directly into the formation control framework. We generate empirical data that correlate controller gains with the resulting settling times by simulating varying degrees of actuator effectiveness. A polynomial curve fitting approach is then used to derive a functional equation describing this relationship.

The findings are expected to inform the design of more robust, fault-tolerant formation controllers and facilitate the development of guidelines for selecting controller parameters that minimize the performance degradation associated with actuator faults. This work contributes to the broader pursuit of autonomous, resilient multi-robot systems capable of operating in uncertain, real-world conditions. Through extensive MATLAB simulations and analysis, we provide valuable insights that can guide future research and development efforts in achieving more robust formation control strategies.

2. Literature Review and Problem Formulation

Formation control and fault-tolerant strategies in multi-robot systems have been extensively studied, forming a robust body of knowledge. Researchers have addressed various aspects, ranging from basic formation maintenance to sophisticated distributed algorithms that handle uncertainties, disturbances, and faults. This section reviews relevant literature on (1) formation control strategies for WMRs, (2) fault detection and isolation (FDI) methods, FTC approaches, and (3) the interplay between controller parameter selection and performance under fault conditions. Furthermore, the research gap has been highlighted at the end of the section, and the consequent formulation of the research problem has been established.

2.1. Formation Control Strategies for Wheeled Mobile Robots

Multi-robot formation control has garnered significant attention due to extensive applications in cooperative tasks such as surveillance, mapping, and transportation [1,2,3,32]. Classic approaches rely on leader–follower schemes, virtual structures, and behavior-based methods. In leader–follower paradigms, one or more robots (leaders) define the trajectory, and followers maintain predefined relative positions [5,6,7,33]. Early works focused on kinematic models of WMRs and control laws for trajectory tracking [29], while more recent studies addressed time-varying communication topologies and nonlinearities in nonholonomic WMRs [27,34].

Olfati-Saber and Murray [4] introduced consensus-based approaches that influenced subsequent studies integrating graph-theoretic methods [26,28]. Cao, Ren, and Egerstedt explored distributed containment control and cooperative tracking for formation maintenance [2,35]. Lyapunov-based methods [13], decentralized algorithms [36], and predictive control techniques [37] have been proposed to improve convergence rates and robustness. In the context of multi-robot systems, approaches such as those by Pham et al. [38] emphasize passive adaptive FTC that does not require prior knowledge of actuator faults or dead zones. Similarly, Phuong et al. [39] employed Lyapunov stability theory to develop controllers that ensure stability in the presence of actuator faults.

Experimental studies have validated these strategies through real-time implementations on small-wheeled robots in uncertain environments [40,41]. However, these works do not address actuator faults, which are critical considerations in real-world applications. Our research builds upon these by integrating actuator fault scenarios into the leader–follower framework, assessing their impact on formation maintenance.

2.2. Fault Detection, Isolation, and Fault-Tolerant Control

As robotic teams operate in complex environments, reliability and fault resilience have become critical. Actuator faults may arise from degradation, drift, or environmental interactions [8,9,10]. FDI methods use analytical redundancy, parity equations, or estimators (e.g., Kalman filters, unknown input observers) to detect faults early [18,42].

FTC strategies are either passive or active. Passive FTC designs robust controllers for a known class of faults, while active FTC reconfigures control laws after fault detection [14,16]. Gain-scheduling techniques [15,17,19] adjust controller parameters dynamically. Adaptive and sliding-mode controls have also been developed for nonlinear WMR models to ensure stability under faults [20,43]. Recent advancements in fault-tolerant control (FTC) have enhanced robotic system reliability. Ref. [44] used the extended Kalman filter (EKF) and graph theory to detect and isolate faults in a master-slave network of wheeled mobile robots, ensuring functionality by isolating faulty units. Ngo et al. [45] addressed fault-tolerant control allocation for cooperative transportation with multiple omnidirectional robots, integrating fault detection, isolation, and reconfiguration (FDIR) with a distributed controller to redistribute efforts among healthy actuators. Together, these studies improve fault detection, isolation, and control reconfiguration, bolstering robotic reliability across diverse setups.

2.3. Controller Parameter Selection and Performance Under Fault Conditions

While formation control and FTC have advanced, few works examine the direct relationship between controller gains and system performance under partial actuator faults. Typically, gains are chosen for stability and nominal robustness [34,35,36], with little focus on fault severity and its influence on transient responses. The work by Ying and Zhu (2023) on integrating model predictive control (MPC) with an intermediate estimator for fault-tolerant tracking control provides valuable insights into how controller gains can be adjusted to compensate for actuator faults [46]. Their approach highlights the importance of estimation techniques for dealing with unknown faults and dynamically adapting controller parameters to maintain system performance.

Some optimization-based approaches use genetic algorithms or heuristic methods to tune gains [47,48], but these seldom consider varying actuator efficiencies. Data-driven models and polynomial fitting techniques to relate gains and performance are underexplored in fault scenarios, leaving a gap in establishing explicit quantitative relationships.

The existing literature offers a robust foundation in formation control, FTC methods, and distributed strategies for WMRs. However, it falls short in providing a clear, quantitative framework that connects controller gains, actuator fault severity, and transient system performance, such as settling time. This gap in the literature is addressed in the present study, which systematically examines the impact of controller gains on the transient behavior of formation control systems under different fault conditions. By doing so, this research aims to inform and guide more effective design decisions for robust multi-robot formations, ultimately improving their performance in real-world scenarios.

2.4. Research Gap and Original Contributions

While the literature provides robust strategies for formation control [2,4] and fault-tolerant control (FTC) [14,20,22], it lacks a quantitative analysis of how actuator faults affect transient performance, such as settling time, and how controller gains can be tuned to mitigate these effects. Our study addresses this gap by systematically analyzing the impact of actuator faults (e.g., 80%-40% efficiency) on formation maintenance and deriving a polynomial relationship between controller gains and settling time, as detailed in Section 11 and Section 13.

The relationship between controller gains and the system performance under varying levels of actuator effectiveness remains underexplored in the context of formation control for WMRs. This study is motivated by the need to address key questions related to system performance and stability when actuator faults occur, such as the following:

• To what extent can compensation for or mitigation of the loss in actuator effectiveness be achieved by enhancing controller gains?
• What is the relationship between performance degradation and the degree of actuator effectiveness loss in fault-prone environments?
• How do actuator faults impact the overall stability and responsiveness of the system?

Understanding these dynamics is crucial for improving the reliability and fault tolerance of multi-robot formations operating in uncertain environments, thereby enabling more robust and adaptable systems. This research addresses key gaps by systematically analyzing actuator faults and their effects on formation control. The main contributions are as follows:

• Fault-tolerant formation control design: A robust control strategy that maintains formation stability under varying actuator fault conditions.
• Actuator fault analysis: A systematic study on the effects of partial actuator failures (80–40% efficiency) on multi-robot formations.
• Gain–settling time relationship: A quantitative model linking controller gains with settling time, providing guidelines for optimal tuning.
• Leader–follower control with actuator faults: Extension of traditional formation control by integrating actuator loss into the system model.
• Residual-based fault detection: A real-time fault detection mechanism using residual computation for early identification and mitigation.
• Comprehensive MATLAB simulations: Extensive testing of different fault levels to validate performance, stability, and recovery behavior.

These contributions enhance the fault resilience of formation control strategies and provide practical insights for real-world multi-robot applications.

3. Kinematic Model of the Multi-Robot System

This section provides a detailed description of the mathematical models involved in our study. The proposed system consists of $N+1$ differential-drive (unicycle-type) wheeled mobile robots (WMRs). One of these robots is designated as the leader (indexed by $i=1$), while the remaining N robots (indexed by $i=2,3,\dots ,N+1$) act as followers. Despite each robot being controlled independently (with its own velocity inputs), their motions are coordinated to maintain a prescribed formation.

3.1. Individual Robot Kinematics

Each robot i has a local body-fixed frame attached at its center (or another convenient reference point). Let us have the following:

$${\mathbf{q}}_i\left(t\right)=\left[\begin{array}{c}{x}_i\left(t\right)\\ y_i\left(t\right)\\ {\theta }_i\left(t\right)\end{array}\right]$$

Denote the configuration of robot i, where $x_i\left(t\right)$ and $y_i\left(t\right)$ are the coordinates of the robot i in the global (inertial) $XY$ frame, ${\theta }_i\left(t\right)$ is the robot’s orientation (heading angle) measured with respect to the global X-axis. For a differential-drive WMR (also known as a unicycle model), the linear velocity $v_i\left(t\right)$ acts along the robot’s forward direction, while the angular velocity ${\omega }_i\left(t\right)$ governs rotation about its vertical axis. Under the assumption of ideal rolling without slipping, the single-robot kinematics are as follows:

$$\begin{array}{cc}\hfill {\dot{x}}_i\left(t\right)& =v_i\left(t\right)cos\left({\theta }_i\left(t\right)\right),\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill {\dot{y}}_i\left(t\right)& =v_i\left(t\right)sin\left({\theta }_i\left(t\right)\right),\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\dot{\theta }}_i\left(t\right)& ={\omega }_i\left(t\right).\hfill \end{array}$$

(3)

Here, ${\dot{x}}_i$ and ${\dot{y}}_i$ represent the translational velocities of robot i in the global X and Y directions, respectively, and ${\dot{\theta }}_i$ is the time derivative of its orientation.

3.1.1. Non-Holonomic Constraint

Due to the wheel arrangement and the no-slip assumption, each robot is subject to a non-holonomic constraint:

$$-sin\left({\theta }_i\left(t\right)\right){\dot{x}}_i\left(t\right)+cos\left({\theta }_i\left(t\right)\right){\dot{y}}_i\left(t\right)=0,$$

Which implies that the robot cannot move laterally (sideways). Motion is, therefore, constrained to the direction of the robot’s heading ${\theta }_i$.

3.1.2. Wheel Velocities and Robot Velocities (Optional Detail)

If one desires to model the left and right wheel speeds explicitly, denote them by $v_{L_i}$ and $v_{R_i}$. With wheel radius r and wheelbase L, the relationship between $\left(v_i,{\omega }_i\right)$ and $\left(v_{L_i},v_{R_i}\right)$ is as follows:

$$v_i={\displaystyle \frac{r}{2}}\left(v_{R_i}+v_{L_i}\right),{\omega }_i={\displaystyle \frac{r}{L}}\left(v_{R_i}-v_{L_i}\right).$$

These equations allow direct control of the robot’s linear and angular motions via the two independent wheel speeds.

3.2. Basic Control Law for the i-th Robot

Let each robot i track a desired reference trajectory ${\mathbf{q}}_{d_i}\left(t\right)$. Define the tracking error as follows:

$${\mathbf{e}}_i\left(t\right)={\mathbf{q}}_{d_i}\left(t\right)-{\mathbf{q}}_i\left(t\right),$$

where ${\mathbf{q}}_i\left(t\right)$ is the actual configuration of the i-th robot. A common approach to ensure ${\mathbf{e}}_i\left(t\right)$ converges to zero is to impose exponential decay.

$${\dot{\mathbf{e}}}_i\left(t\right)=-\mathsf{\Lambda }{\mathbf{e}}_i\left(t\right),$$

where $\mathsf{\Lambda }$ is a positive diagonal matrix, e.g., $\mathsf{\Lambda }=\mathrm{diag}({\lambda }_1,{\lambda }_2,{\lambda }_3)$.

Given the single-robot kinematics ${\dot{\mathbf{q}}}_i=\mathbf{J}\left({\theta }_i\right){\mathit{\zeta }}_i$, where we have the following:

$${\mathit{\zeta }}_i=\left[\begin{array}{c}{v}_i\\ {\omega }_i\end{array}\right],\mathbf{J}\left({\theta }_i\right)=\left[\begin{array}{cc}cos\left({\theta }_i\right)& 0\\ sin\left({\theta }_i\right)& 0\\ 0& 1\end{array}\right],$$

We can impose the desired dynamics by setting the following:

$$\mathbf{J}\left({\theta }_i\right){\mathit{\zeta }}_i={\dot{\mathbf{q}}}_{d_i}\left(t\right)+\mathsf{\Lambda }\left({\mathbf{q}}_{d_i}\left(t\right)-{\mathbf{q}}_i\left(t\right)\right).$$

Solving for ${\mathit{\zeta }}_i$ gives the computed velocity (or inverse differential kinematics) control law for the i-th robot:

$${\mathit{\zeta }}_i={\mathbf{J}}^{-1}\left({\theta }_i\right)\left[{\dot{\mathbf{q}}}_{d_i}\left(t\right)+\mathsf{\Lambda }\left({\mathbf{q}}_{d_i}\left(t\right)-{\mathbf{q}}_i\left(t\right)\right)\right].$$

(4)

Under appropriate gain selection in $\mathsf{\Lambda }$, this ensures that ${\mathbf{e}}_i\left(t\right)$ converges exponentially to zero. In the multi-robot case, each follower robot applies a similar local feedback strategy, with ${\mathbf{q}}_{d_i}$ defined to maintain formation requirements relative to the leader or neighboring robots.

Each ${\mathit{\zeta }}_i$ influences the position and orientation of robot i according to (1)–(3), but the overall formation control will typically require coupling or coordination among all robot velocities.

3.3. Global System Configuration

The collective state of the multi-robot system at time t can be written by stacking the individual robot states:

$$\mathbf{Q}\left(t\right)=\left[\begin{array}{c}{\mathbf{q}}_1\left(t\right)\\ {\mathbf{q}}_2\left(t\right)\\ ⋮\\ {\mathbf{q}}_{N+1}\left(t\right)\end{array}\right],\mathrm{where}{\mathbf{q}}_i\left(t\right)=\left[\begin{array}{c}{x}_i\left(t\right)\\ y_i\left(t\right)\\ {\theta }_i\left(t\right)\end{array}\right].$$

Similarly, the complete set of control inputs is gathered as follows:

$$\mathit{\zeta }\left(t\right)=\left[\begin{array}{c}{\mathit{\zeta }}_1\left(t\right)\\ {\mathit{\zeta }}_2\left(t\right)\\ ⋮\\ {\mathit{\zeta }}_{N+1}\left(t\right)\end{array}\right],{\mathit{\zeta }}_i\left(t\right)=\left[\begin{array}{c}{v}_i\left(t\right)\\ {\omega }_i\left(t\right)\end{array}\right].$$

Even though each robot’s motion is governed by ${\mathit{\zeta }}_i$ independently, maintaining a desired formation requires that these control inputs be carefully orchestrated. In subsequent sections, we describe how the leader and follower robots coordinate their velocities so that the team achieves and preserves the specified formation shape while navigating in the environment.

4. Loss of Effectiveness in Actuators

In this work, we investigate the loss of effectiveness in actuators, which is one of the most common types of actuator faults. This fault refers to a situation where an actuator does not perform at its full capacity but still provides some level of control. The extent of the loss is quantified by a factor ${\gamma }_k^{ji}$ as shown in

Table 1. Effect of fault factor ${\gamma }_k^{ji}$ on the actuator functionality.

Fault Factor	Condition	Effect on Control Input
${\gamma }_k^{ji}=0$	Actuator functioning normally	Control input remains unaffected
$0<{\gamma }_k^{ji}<1$	Partial loss of effectiveness	Reduces control ability but not completely disabled
${\gamma }_k^{ji}=1$	Actuator completely failed	Control input is fully disabled

${\gamma }_k^{ji}$ represents the fault factor affecting actuator functionality.

, which represents the reduction in effectiveness of the i-th actuator of the j-th follower robot at time step k. The control input vector for the i-th actuator of the j-th robot, considering this loss of effectiveness, can be expressed as follows:

$$v_F^{ji}=v_i^{ji}(1-{\gamma }_k^{ji}),0\le {\gamma }_k^{ji}\le 1$$

Here, $v_F^{ji}$ is the control input after the fault occurs; $u_i^{ji}$ is the normal control input when there is no fault; ${\gamma }_k^{ji}$ is the fault factor that indicates the loss of effectiveness.

In cases where there is only a partial loss of effectiveness ($0<{\gamma }_k^{ji}<1$), the post-fault model of the system can be described by modifying the control inputs accordingly to account for the fault. This approach allows the system to continue operating with reduced control capabilities while compensating for the actuator fault. The model and its behavior in these conditions are discussed in detail in works such as [49,50].

5. Leader–Follower Formation Control Under Loss of Actuation Fault

Formation control involves designing control laws that enable a group of robots to move cohesively while maintaining a specified geometric arrangement. The leader–follower approach designates one robot as the leader, which dictates the movement of the formation, while the followers adjust their positions relative to the leader or other followers.

The hierarchical leader–follower structure was adopted for this research due to its significant advantages in addressing actuator faults within multi-agent formation control systems, outperforming alternative architectures such as fully decentralized systems [51], star topologies [52], or ring/mesh configurations [53]. Specifically, in managing actuator faults—modeled here as a loss of effectiveness (e.g., 80%, 60%, 40% capacity)—the hierarchical structure excels in fault isolation and containment. Unlike fully decentralized systems, where fault detection depends on global consensus and can be slow or resource-intensive, the tiered hierarchy facilitates rapid, localized fault identification through explicit error terms at each level (e.g., $e_{F1}$, $e_{F3}$), as implemented in our residual-based approach. This capability directly supports our objective of minimizing performance degradation under varying fault severities. In contrast, a star topology risks complete destabilization if the single leader fails, whereas the hierarchical structure distributes dependencies across multiple tiers, enhancing resilience. Additionally, while ring or mesh topologies offer robust communication, they lack the directed control flow required for efficient fault isolation and compensation, making them less compatible with the leader–follower framework central to our study. Therefore, the hierarchical structure provides an optimal balance of scalability, fault tolerance, and control efficiency, making it the ideal choice for developing robust, fault-tolerant multi-robot systems capable of maintaining formation integrity despite actuator failures.

Figure 1 illustrates the hierarchical leader–follower formation control structure adopted in this study, comprising one leader robot (L) and four follower robots ($F_1$, $F_2$, $F_3$, $F_4$), arranged in a two-tier configuration. The leader, situated at the top of the hierarchy, is tasked with tracking a predefined desired state, $q_{dL}={[x_{d,L},y_{d,L},{\theta }_{d,L}]}^T$, representing its target position and orientation in the global frame. The leader’s state evolves according to $e_L=q_{d,L}-q_L$, where $e_L$ denotes the tracking error as defined in Equation (5). The first-tier followers, $F_1$ and $F_2$, are directly coupled to the leader via directed communication links, denoted as $d_{L1}$ and $d_{L2}$, which represent desired offsets or control dependencies (e.g., ${\Delta }_2$ and ${\Delta }_3$ in Equation (8). Their states are expressed as $e_{F1}=q_L+{\Delta }_2-q_{F1}$ and $e_{F2}=q_L+{\Delta }_3-q_{F2}$, where $e_{F1}$ and $e_{F2}$ are local error terms. The second-tier followers, $F_3$ and $F_4$, depend on the first-tier followers through links $d_{F13}$ and $d_{F24}$, with states defined as $e_{F3}=q_{F1}+{\Delta }_4-q_{F3}$ and $e_{F4}=q_{F2}+{\Delta }_5-q_{F4}$, respectively. Visually, the leader is depicted in peach, first-tier followers in pink, and second-tier followers in green, emphasizing the tiered organization and control flow. This structure facilitates a directed propagation of control signals from the leader through the tiers, enabling scalable coordination and localized error management. In the context of actuator faults, such as the loss of effectiveness modeled in Section 4, this hierarchy enhances fault tolerance by isolating fault impacts to specific branches, thereby supporting the residual-based fault detection and compensation strategies detailed in Section 8 and Section 9.

5.1. Control Objectives

• Leader: Navigate from an initial pose $q_{\mathrm{L}}\left(0\right)$ to a desired final pose $q_{d,\mathrm{L}}$.
• Followers: Maintain predefined relative positions and orientations with respect to their designated reference robots (either the leader or another follower) to sustain the desired geometric formation while following the leader’s trajectory.

5.2. Leader Control Law

The leader’s control strategy ensures that it moves toward its desired pose, acting as the reference for the followers.

5.2.1. Error Definition

The positional and orientational discrepancies between the leader’s current pose and its desired pose are defined as follows:

$$e_{\mathrm{L}}=q_{d,\mathrm{L}}-q_{\mathrm{L}}$$

(5)

where:

$$q_{\mathrm{L}}=\left[\begin{array}{c}{x}_{\mathrm{L}}\\ y_{\mathrm{L}}\\ {\theta }_{\mathrm{L}}\end{array}\right],q_{d,\mathrm{L}}=\left[\begin{array}{c}{x}_{d,\mathrm{L}}\\ y_{d,\mathrm{L}}\\ {\theta }_{d,\mathrm{L}}\end{array}\right]$$

5.2.2. Control Input Calculation

To drive the leader toward its desired pose, we employ a feedback control law based on the inverse Jacobian:

$$\begin{array}{c}\hfill {\mathit{\zeta }}_{\mathrm{L}}={\mathbf{J}}_{\mathrm{L}}^{-1}{\mathbf{K}}_{\mathrm{L}}{e}_{\mathrm{L}}\end{array}$$

(6)

where:

$${\mathbf{J}}_{\mathrm{L}}=\left[\begin{array}{ccc}cos\left({\theta }_{\mathrm{L}}\right)& -sin\left({\theta }_{\mathrm{L}}\right)& 0\\ sin\left({\theta }_{\mathrm{L}}\right)& cos\left({\theta }_{\mathrm{L}}\right)& 0\\ 0& 0& 1\end{array}\right]$$

is the Jacobian matrix of the leader, and ${\mathbf{K}}_{\mathrm{L}}=diag(k_{l1},k_{l2},k_{l3})$ is the gain matrix, with positive entries $k_{l1},k_{l2},k_{l3}$.

5.2.3. Actuation Loss Modeling

To simulate real-world scenarios where actuation capabilities may degrade over time, we introduce an actuation loss factor $k_{\mathrm{l}\_\mathrm{loss}}$:

$$v_{\mathrm{L}}=\left\{\begin{array}{cc}{\zeta }_{\mathrm{L},1},\hfill & t<t_{\mathrm{loss}}\hfill \\ k_{\mathrm{l}\_\mathrm{loss}}{\zeta }_{\mathrm{L},1},\hfill & t\ge t_{\mathrm{loss}}\hfill \end{array}\right.$$

(7)

The angular velocity remains unaffected:

$${\omega }_{\mathrm{L}}={\zeta }_{\mathrm{L},3}$$

5.3. Follower Control Law

Each follower aims to maintain a specific offset relative to its reference robot to preserve the desired formation. The control law for the followers ensures that positional and orientational errors are minimized.

5.3.1. Desired Pose of Followers

For follower i ($i=2,3,\dots ,N+1$), the desired pose is defined relative to its reference robot (which could be the leader or another follower):

$$q_{d,i}=q_{\mathrm{ref},i}+{\Delta }_i$$

(8)

where:

• $q_{\mathrm{ref},i}$ is the pose of the reference robot for follower i.
• ${\Delta }_i=\left[\begin{array}{c}\Delta x_i\\ \Delta y_i\\ \Delta {\theta }_i\end{array}\right]$ is the desired offset, designed to achieve the desired geometric formation (e.g., V-shape).

The offsets for the formation can be defined based on the desired geometric arrangement. For a V-shaped formation, the offsets might be as follows:

$$\begin{array}{cc}\hfill {\Delta }_i& =\left[\begin{array}{c}{d}_{x,i}\\ -d_{y,i}\\ -{\varphi }_i\end{array}\right]\mathrm{for}i=2,3,\dots ,N+1\hfill \end{array}$$

where $d_{x,i}$ and $d_{y,i}$ are the horizontal and vertical offsets, and ${\varphi }_i$ is the formation angle for follower i.

5.3.2. Error Calculation

The error for follower i is the difference between its desired pose and current pose:

$$e_i=q_{d,i}-q_i=\left[\begin{array}{c}{e}_{i,x}\\ e_{i,y}\\ e_{i,\theta }\end{array}\right]=\left[\begin{array}{c}{x}_{d,i}-x_i\\ y_{d,i}-y_i\\ {\theta }_{d,i}-{\theta }_i\end{array}\right]$$

(9)

5.3.3. Orientation Error Adjustment

To ensure smooth convergence, the orientation error ${\psi }_i$ is adjusted based on the magnitude of the positional error:

$${\psi }_i=\left\{\begin{array}{cc}atan2(e_{i,y},e_{i,x})-{\theta }_i,\hfill & \mathrm{if}\parallel e_{i,1:2}\parallel \ge ϵ\hfill \\ -{\varphi }_i-{\theta }_i,\hfill & \mathrm{if}\parallel e_{i,1:2}\parallel <ϵ\hfill \end{array}\right.$$

(10)

where $ϵ$ is a small threshold value to determine when the follower is sufficiently close to its desired position.

5.3.4. Control Input Calculation

The control inputs for follower i are computed using a feedback control law:

$${\mathit{\zeta }}_i={\mathbf{J}}_i^{-1}{\mathbf{K}}_i\left[\begin{array}{c}{e}_{i,x}\\ e_{i,y}\\ {\psi }_i\end{array}\right]$$

(11)

where:

$${\mathbf{J}}_i=\left[\begin{array}{ccc}cos\left({\theta }_i\right)& -sin\left({\theta }_i\right)& 0\\ sin\left({\theta }_i\right)& cos\left({\theta }_i\right)& 0\\ 0& 0& 1\end{array}\right]$$

is the Jacobian matrix for follower i, and ${\mathbf{K}}_i=diag(k_{i1},k_{i2},k_{i3})$ is the gain matrix with positive entries $k_{i1},k_{i2},k_{i3}$.

5.3.5. Actuation Loss Modeling

Similar to the leader, each follower i experiences an actuation loss factor $k_{\mathrm{f}\_\mathrm{loss},i}$ after a certain time $t_{\mathrm{loss}}$:

$$v_i=\left\{\begin{array}{cc}{\zeta }_{i,1},\hfill & t<t_{\mathrm{loss}}\hfill \\ k_{\mathrm{f}\_\mathrm{loss},i}{\zeta }_{i,1},\hfill & t\ge t_{\mathrm{loss}}\hfill \end{array}\right.$$

(12)

The angular velocity remains unaffected:

$${\omega }_i={\zeta }_{i,3}$$

5.4. Gain Scheduling and Actuation Loss Parameters

5.4.1. Gain Scheduling

To enhance the adaptability of the control system, gains are scheduled based on a gain vector G:

$${\mathbf{K}}_{\mathrm{robot}}=G\left(n\right)·diag(k_1,k_2,k_3)$$

where n corresponds to a specific index in G, and ${\mathbf{K}}_{\mathrm{robot}}$ can be either ${\mathbf{K}}_{\mathrm{L}}$ or ${\mathbf{K}}_i$ for the followers.

5.4.2. Actuation Loss Factors

Actuation loss factors are defined for both the leader and each follower:

$$k_{\mathrm{l}\_\mathrm{loss}}=L\left(1\right),k_{\mathrm{f}\_\mathrm{loss},i}=L(i+1)\mathrm{for}i=1,2,\dots ,N$$

where L is a vector containing the loss factors for the leader and each of the followers.

6. Control Law Derivation

To derive the control laws for both the leader and followers, we start from the kinematic equations and incorporate feedback terms to ensure convergence to the desired states.

6.1. Leader Control Law Derivation

Starting from the leader’s kinematic equation:

$${\dot{q}}_{\mathrm{L}}={\mathbf{J}}_{\mathrm{L}}{\mathit{\zeta }}_{\mathrm{L}}$$

(13)

To achieve asymptotic stability, we define the error dynamics such that the error $e_{\mathrm{L}}$ converges to zero exponentially. Specifically, we desire the following:

$$e_{\mathrm{L}}\left(t\right)=e^{-\mathsf{\Lambda }t}{e}_{\mathrm{L}}\left(0\right)$$

(14)

where $\mathsf{\Lambda }=diag({\lambda }_1,{\lambda }_2,{\lambda }_3)$ is a positive definite diagonal matrix.

Differentiating the error:

$${\dot{e}}_{\mathrm{L}}\left(t\right)=-\mathsf{\Lambda }{e}_{\mathrm{L}}\left(t\right)$$

(15)

Substituting the kinematic equation:

$${\dot{e}}_{\mathrm{L}}\left(t\right)={\dot{q}}_{d,\mathrm{L}}\left(t\right)-{\mathbf{J}}_{\mathrm{L}}{\mathit{\zeta }}_{\mathrm{L}}=-\mathsf{\Lambda }{e}_{\mathrm{L}}\left(t\right)$$

(16)

Assuming that ${\dot{q}}_{d,\mathrm{L}}\left(t\right)=0$ for set-point control, the control input is designed to satisfy:

$${\mathbf{J}}_{\mathrm{L}}{\mathit{\zeta }}_{\mathrm{L}}=\mathsf{\Lambda }{e}_{\mathrm{L}}\left(t\right)$$

(17)

Solving for ${\mathit{\zeta }}_{\mathrm{L}}$:

$${\mathit{\zeta }}_{\mathrm{L}}={\mathbf{J}}_{\mathrm{L}}^{-1}\mathsf{\Lambda }{e}_{\mathrm{L}}\left(t\right)$$

(18)

This feedback control law ensures that the leader’s pose converges to the desired pose exponentially.

6.2. Follower Control Law Derivation

For each follower i ($i=2,3,\dots ,N+1$), the control law is derived similarly to the leader’s control law, ensuring that the positional and orientational errors relative to the desired formation are minimized.

Starting from the follower’s kinematic equation:

$${\dot{q}}_i={\mathbf{J}}_i{\mathit{\zeta }}_i$$

(19)

Defining the error dynamics for follower i:

$$e_i\left(t\right)=q_{d,i}\left(t\right)-q_i\left(t\right)$$

(20)

Desired exponential decay of the error:

$$e_i\left(t\right)=e^{-{\mathsf{\Lambda }}_{i}t}{e}_i\left(0\right)$$

(21)

Differentiating the error:

$${\dot{e}}_i\left(t\right)=-{\mathsf{\Lambda }}_i{e}_i\left(t\right)$$

(22)

Substituting the follower’s kinematic equation:

$${\dot{e}}_i\left(t\right)={\dot{q}}_{d,i}\left(t\right)-{\mathbf{J}}_i{\mathit{\zeta }}_i=-{\mathsf{\Lambda }}_i{e}_i\left(t\right)$$

(23)

Assuming ${\dot{q}}_{d,i}\left(t\right)=0$ for set-point control, the control input is designed to satisfy the following:

$${\mathbf{J}}_i{\mathit{\zeta }}_i={\mathsf{\Lambda }}_i{e}_i\left(t\right)$$

(24)

Solving for ${\mathit{\zeta }}_i$:

$${\mathit{\zeta }}_i={\mathbf{J}}_i^{-1}{\mathsf{\Lambda }}_i{e}_i\left(t\right)$$

(25)

This control law ensures that each follower converges to its desired pose relative to its reference robot, maintaining the formation.

7. Stability Analysis

Ensuring the stability of the multi-robot formation control system is paramount. We perform a Lyapunov stability analysis for both the leader and the followers to confirm that the designed control laws lead to the asymptotic convergence of the errors.

7.1. Lyapunov Function for the Leader

Consider the Lyapunov function candidate for the leader:

$$V_{\mathrm{L}}={\displaystyle \frac{1}{2}}{e}_{\mathrm{L}}^{\top }{e}_{\mathrm{L}}$$

(26)

Taking the time derivative along the trajectories of the system:

$${\dot{V}}_{\mathrm{L}}=e_{\mathrm{L}}^{\top }{\dot{e}}_{\mathrm{L}}$$

(27)

$$=e_{\mathrm{L}}^{\top }(-\mathsf{\Lambda }{e}_{\mathrm{L}})$$

(28)

$$=-e_{\mathrm{L}}^{\top }\mathsf{\Lambda }{e}_{\mathrm{L}}\le 0$$

(29)

Since $\mathsf{\Lambda }$ is positive definite, ${\dot{V}}_{\mathrm{L}}\le 0$, and ${\dot{V}}_{\mathrm{L}}=0$ only when $e_{\mathrm{L}}=\mathbf{0}$. By LaSalle’s invariance principle, the system is asymptotically stable, ensuring that the leader’s pose converges to the desired pose.

7.2. Lyapunov Function for Each Follower

Similarly, for each follower i ($i=2,3,\dots ,N+1$), we define the Lyapunov function:

$$V_i={\displaystyle \frac{1}{2}}{e}_i^{\top }{e}_i$$

(30)

Taking the time derivative, we have the following:

$${\dot{V}}_i=e_i^{\top }{\dot{e}}_i$$

(31)

$$=e_i^{\top }(-{\mathsf{\Lambda }}_i{e}_i)$$

(32)

$$=-e_i^{\top }{\mathsf{\Lambda }}_i{e}_i\le 0$$

(33)

Given that ${\mathsf{\Lambda }}_i$ is positive definite, ${\dot{V}}_i\le 0$, and ${\dot{V}}_i=0$ only when $e_i=\mathbf{0}$. Applying LaSalle’s invariance principle, each follower’s error $e_i$ asymptotically converges to zero, ensuring that the followers maintain their desired formation relative to their reference robots.

7.3. Overall System Stability

Considering all robots collectively, the overall Lyapunov function for the multi-robot system is the sum of the individual Lyapunov functions:

$$V_{\mathrm{total}}=V_{\mathrm{L}}+\sum _{i=2}^{N+1}{V}_i$$

(34)

The time derivative of the total Lyapunov function is as follows:

$$\begin{array}{cc}{\dot{V}}_{\mathrm{total}}\hfill & ={\dot{V}}_{\mathrm{L}}+{\sum }_{i=2}^{N+1}{\dot{V}}_i\hfill \\ \hfill & =-e_{\mathrm{L}}^{\top }\mathsf{\Lambda }{e}_{\mathrm{L}}-{\sum }_{i=2}^{N+1}{e}_i^{\top }{\mathsf{\Lambda }}_i{e}_i\hfill \\ \hfill & \le 0\hfill \end{array}$$

(35)

Since each term is non-positive and only zero when all errors are zero, the multi-robot formation control system is asymptotically stable. This ensures that both the leader and all followers converge to their desired poses, maintaining the specified formation.

8. Fault Detection via Residual Generation

As described in Section 3 (see Equations (1)–(6)), the kinematic model of the mobile robot is given by the following:

$$q(t+dt)=q\left(t\right)+J\left(\theta \left(t\right)\right)\left[\begin{array}{c}v\left(t\right)\\ r\left(t\right)\end{array}\right]dt,$$

(36)

with

$$q\left(t\right)=\left[\begin{array}{c}x\left(t\right)\\ y\left(t\right)\\ \theta \left(t\right)\end{array}\right],J\left(\theta \left(t\right)\right)=\left[\begin{array}{cc}cos\theta \left(t\right)& 0\\ sin\theta \left(t\right)& 0\\ 0& 1\end{array}\right],$$

(37)

and where $v\left(t\right)$ and $r\left(t\right)$ denote the translational and rotational control inputs, respectively. In the nominal (fault-free) case, the controller computes a nominal control input $v_{\mathrm{nom}}\left(t\right)$ (with the corresponding rotational command $r\left(t\right)$), and the predicted state is given by the following:

$$\widehat{q}(t+dt)=q\left(t\right)+J\left(\theta \left(t\right)\right)\left[\begin{array}{c}{v}_{\mathrm{nom}}\left(t\right)\\ r\left(t\right)\end{array}\right]dt.$$

(38)

This is the expected state if everything works perfectly.

8.1. Fault Injection and Residual Computation

When a fault (e.g., an actuator loss) occurs, it is modeled by a multiplicative fault factor $\gamma$ from Section 4. Thus, the actual translational control input is as follows:

$$v_{\mathrm{act}}\left(t\right)=(1-\gamma )v_{\mathrm{nom}}\left(t\right)$$

(39)

The rotational input $r\left(t\right)$ remains unaffected by this fault. Thus, the actual state evolution under fault is as follows:

$$q(t+dt)=q\left(t\right)+J\left(\theta \left(t\right)\right)\left[\begin{array}{c}{v}_{\mathrm{act}}\left(t\right)\\ r\left(t\right)\end{array}\right]dt$$

(40)

Substituting $v_{\mathrm{act}}\left(t\right)$:

$$q(t+dt)=q\left(t\right)+J\left(\theta \left(t\right)\right)\left[\begin{array}{c}(1-\gamma )v_{\mathrm{nom}}\left(t\right)\\ r\left(t\right)\end{array}\right]dt$$

(41)

This shows how the fault reduces the robot’s translational movement, causing the actual state to deviate from the predicted state. Similarly, the residual is the key to detecting this deviation. It is defined as the difference between the actual state and the nominal prediction:

$$r_{\mathrm{res}}(t+dt)=q(t+dt)-\widehat{q}(t+dt)$$

(42)

Substitute the expressions for $q(t+dt)$ and $\widehat{q}(t+dt)$:

$$\begin{array}{c}{r}_{\mathrm{res}}(t+dt)=\left[q\left(t\right)+J\left(\theta \left(t\right)\right)\left[\begin{array}{c}{v}_{\mathrm{act}}\left(t\right)\\ r\left(t\right)\end{array}\right]dt\right]-\left[q\left(t\right)+J\left(\theta \left(t\right)\right)\left[\begin{array}{c}{v}_{\mathrm{nom}}\left(t\right)\\ r\left(t\right)\end{array}\right]dt\right]\\ r_{\mathrm{res}}(t+dt)=J\left(\theta \left(t\right)\right)\left(\left[\begin{array}{c}{v}_{\mathrm{act}}\left(t\right)\\ r\left(t\right)\end{array}\right]-\left[\begin{array}{c}{v}_{\mathrm{nom}}\left(t\right)\\ r\left(t\right)\end{array}\right]\right)dt\\ =J\left(\theta \left(t\right)\right)\left[\begin{array}{c}{v}_{\mathrm{act}}\left(t\right)-v_{\mathrm{nom}}\left(t\right)\\ r\left(t\right)-r\left(t\right)\end{array}\right]dt=J\left(\theta \left(t\right)\right)\left[\begin{array}{c}{v}_{\mathrm{act}}\left(t\right)-v_{\mathrm{nom}}\left(t\right)\\ 0\end{array}\right]dt\end{array}$$

(43)

Now, substitute $v_{\mathrm{act}}\left(t\right)=(1-\gamma )v_{\mathrm{nom}}\left(t\right)$:

$$v_{\mathrm{act}}\left(t\right)-v_{\mathrm{nom}}\left(t\right)=(1-\gamma )v_{\mathrm{nom}}\left(t\right)-v_{\mathrm{nom}}\left(t\right)=-\gamma v_{\mathrm{nom}}\left(t\right)$$

So:

$$r_{\mathrm{res}}(t+dt)=J\left(\theta \left(t\right)\right)\left[\begin{array}{c}-\gamma v_{\mathrm{nom}}\left(t\right)\\ 0\end{array}\right]dt$$

(44)

Multiply the matrix:

$$r_{\mathrm{res}}(t+dt)=\left[\begin{array}{cc}cos\theta \left(t\right)& 0\\ sin\theta \left(t\right)& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}-\gamma v_{\mathrm{nom}}\left(t\right)\\ 0\end{array}\right]dt=\left[\begin{array}{c}-\gamma v_{\mathrm{nom}}\left(t\right)cos\theta \left(t\right)\\ -\gamma v_{\mathrm{nom}}\left(t\right)sin\theta \left(t\right)\\ 0\end{array}\right]dt$$

(45)

In Equation (45) x and y components reflect the positional error caused by the fault, scaled according to the robot’s orientation through ($cos\theta \left(t\right)$ and $sin\theta \left(t\right)$). The component $\theta$ remains zero because the fault influences only translational velocity and does not affect rotation.

8.2. Fault Detection Criterion and Fault Factor Estimation

To detect a fault, compute the norm of the residual (typically the Euclidean norm) and compare it to a threshold:

$$\begin{array}{c}\parallel r_{\mathrm{res}}(t+dt)\parallel =\sqrt{{(-\gamma v_{\mathrm{nom}}\left(t\right)cos\theta \left(t\right)dt)}^2+{(-\gamma v_{\mathrm{nom}}\left(t\right)sin\theta \left(t\right)dt)}^2+0^2}\\ =\sqrt{{\gamma }^2{v}_{\mathrm{nom}}^2\left(t\right)d{t}^2({cos}^2\theta \left(t\right)+{sin}^2\theta \left(t\right))}=\sqrt{{\gamma }^2{v}_{\mathrm{nom}}^2\left(t\right)d{t}^2}=\left|\gamma \right||v_{\mathrm{nom}}\left(t\right)|dt\end{array}$$

(46)

A fault is declared if the norm of the residual exceeds a predefined threshold, i.e.,

$$\parallel r_{\mathrm{res}}(t+dt)\parallel >\mathrm{Threshold}$$

(47)

Furthermore, an estimate of the fault factor is obtained by

$${\gamma }_{\mathrm{est}}\left(t\right)=1-\left({\displaystyle \frac{v_{\mathrm{act}}\left(t\right)}{v_{\mathrm{nom}}\left(t\right)}}\right)$$

Since $v_{\mathrm{act}}\left(t\right)=(1-\gamma )v_{\mathrm{nom}}\left(t\right)$:

$${\gamma }_{\mathrm{est}}\left(t\right)=1-\left({\displaystyle \frac{(1-\gamma )v_{\mathrm{nom}}\left(t\right)}{v_{\mathrm{nom}}\left(t\right)}}\right)=1-(1-\gamma )=\gamma$$

So, ${\gamma }_{\mathrm{est}}\left(t\right)\approx \gamma$. However, $v_{\mathrm{act}}\left(t\right)$ isn’t directly measurable. Instead, use the residual:

$$r_{\mathrm{res}}(t+dt)=\left[\begin{array}{c}-\gamma v_{\mathrm{nom}}\left(t\right)cos\theta \left(t\right)dt\\ -\gamma v_{\mathrm{nom}}\left(t\right)sin\theta \left(t\right)dt\\ 0\end{array}\right]$$

Project the residual onto the direction of motion ($\left[\begin{array}{c}cos\theta \left(t\right)\\ sin\theta \left(t\right)\\ 0\end{array}\right]$):

$$r_{\mathrm{res}}·\left[\begin{array}{c}cos\theta \left(t\right)\\ sin\theta \left(t\right)\\ 0\end{array}\right]=(-\gamma v_{\mathrm{nom}}\left(t\right)cos\theta \left(t\right)dt)cos\theta \left(t\right)+(-\gamma v_{\mathrm{nom}}\left(t\right)sin\theta \left(t\right)dt)sin\theta \left(t\right)$$

$$=-\gamma v_{\mathrm{nom}}\left(t\right)dt({cos}^2\theta \left(t\right)+{sin}^2\theta \left(t\right))=-\gamma v_{\mathrm{nom}}\left(t\right)dt$$

Solve for $\gamma$:

$${\gamma }_{\mathrm{est}}=-{\displaystyle \frac{r_{\mathrm{res},x}cos\theta \left(t\right)+r_{\mathrm{res},y}sin\theta \left(t\right)}{v_{\mathrm{nom}}\left(t\right)dt}}$$

(48)

• Nominal Case: $\gamma =0$, so $r_{\mathrm{res}}=0$, and ${\gamma }_{\mathrm{est}}\approx 0$.
• Fault Case: $\gamma >0$, so ${\gamma }_{\mathrm{est}}>0$.

Here, $r_{\mathrm{res},x}(t+dt)$ and $r_{\mathrm{res},y}(t+dt)$ represent the residual components measured at time $t+dt$, while $cos\theta \left(t\right)$ and $sin\theta \left(t\right)$ define the robot’s direction at time t. Additionally, $v_{\mathrm{nom}}\left(t\right)$ denotes the nominal velocity input, and $dt$ is the time step over which the residual is computed. The negative sign explicitly accounts for the residual being opposite to the intended motion due to the fault. This formulation enables direct computation of ${\gamma }_{\mathrm{est}}$ from the residual, clearly illustrating how the fault factor relates to the observed deviation in the robot’s state. Given measurements of $r_{\mathrm{res}}$, $v_{\mathrm{nom}}$, and $\theta$, this expression can be used to analyze or plot ${\gamma }_{\mathrm{est}}$ over time.

Figure 2 and Figure 3 validate the fault detection and estimation mechanism employed in fault-tolerant formation control. Figure 2 presents the estimated fault factors (${\gamma }_{\mathrm{est}}$) for the leader and four followers over a 10-second interval. The leader experiences a significant fault factor increase to 0.5 at $t=1s$, while the followers encounter less severe faults at subsequent times: Follower 1 reaches 0.1 at $t=2s$, Follower 2 reaches 0.2 at $t=3s$, Follower 3 reaches 0.1 at $t=4s$, and Follower 4 reaches 0.5 at $t=5s$. Complementarily, Figure 3 illustrates the normalized residual norms ($\parallel r_{\mathrm{res}}\parallel$), highlighting clear threshold violations (threshold = 0.01) at each fault onset. These residual spikes precisely correspond to the fault occurrences, confirming the reliability of the detection strategy and its capability to quantify fault severity for adaptive formation control.

8.3. Determining Fault Levels and Model Selection

In fault-tolerant formation control for mobile robots, maintaining a desired formation despite actuator faults—like a reduction in translational velocity—is critical. This process hinges on detecting and adapting to faults using two key steps: (1) determining the fault level from the residual, and (2) selecting an appropriate model to adjust the control strategy based on that fault level.

8.3.1. Selecting a Model for the Fault Level

Once we know the fault level $\gamma$, we need a model that describes how the robot behaves under that specific fault condition. In this context, we’ve developed polynomial curve models for specific fault levels—like 20%, 40%, 60%, as shown in

Table 2. Summary of curve-fitting results for different fault levels.

Fault Level	Equation	Coefficients (a, b, c)	Fit Metrics
20%	$y=a{e}^{-bx}+c$	$a=16.0635,$ $b=1.1144,$ $c=1.5081$	$SSE=2.3525$ $RMSE=0.3720$
40%	$y=a{e}^{-bx}+c$	$a=21.2821,$ $b=1.1316,$ $c=1.6965$	$SSE=4.0741$ $RMSE=0.4895$
60%	$y=a{e}^{-bx}+c$	$a=31.7079,$ $b=1.1496,$ $c=2.0391$	$SSE=8.3241$ $RMSE=0.6992$
80%	$y=a{e}^{-bx}+c$	$a=31.9441,$ $b=0.6521,$ $c=2.1744$	$SSE=5.1503$ $RMSE=0.5504$

—that relate control parameters (gain) to performance metrics (settling time). If $\gamma$ matches a predefined fault level (e.g., $\gamma =0.2$ matches the 20% fault model), we use that model directly. The selected model then guides adjustments to the control strategy—such as increasing the control gain—to compensate for the fault and keep the formation intact. This adaptability ensures the system performs as close as possible to its fault-free behavior.

8.3.2. Residual Values Under Different Faults

The residual magnitude $\parallel r_{\mathrm{res},xy}\parallel$ from 46 scales linearly with the fault level $\gamma$, as shown by the following:

$$\parallel r_{\mathrm{res},xy}\parallel =\gamma v_{\mathrm{nom}}\left(t\right)dt$$

The correspondence levels between fault level $\gamma$ and residual magnitude are shown in

Table 3. Residual magnitudes for different fault levels.

Fault Level ($\mathit{\gamma }$)	Residual Magnitude ($\parallel {\mathit{r}}_{\mathbf{res},\mathit{xy}}\parallel$)
0% (no fault)	0 m
10% (0.1)	0.1 m
20% (0.2)	0.2 m
30% (0.3)	0.3 m
40% (0.4)	0.4 m

.

This table shows that the residual grows with fault severity, providing a clear signal for detection and estimation. For instance, a 30% fault yields a residual of 0.3 m, three times larger than a 10% fault’s 0.1 m.

In fault-tolerant formation control, the residual acts as a fault detector and quantifier. By comparing the robot’s actual and predicted states, we compute $\gamma$ to determine the fault’s severity (e.g., a 15% velocity loss). Then, we select or interpolate a model tailored to that fault level, adjusting the control approach to mitigate the fault’s impact. The residual’s sensitivity to fault size—demonstrated in

Table 4. Curve-fitting coefficients for various fault levels.

Fault Level ($\mathit{\gamma }$)	a	b	c
20%	16.0635	1.1144	1.5081
40%	21.2821	1.1316	1.6965
60%	31.7079	1.1496	2.0391
80%	31.9441	0.6521	2.1744

—ensures faults are detectable and measurable. This approach ensures robots maintain their formation despite faults, blending detection, estimation, and adaptation into a robust control framework.

9. Methodology for Fault Compensation

9.1. Quantifying Fault Severity from the Residual

The first step in compensating for the robot’s loss of effectiveness involves quantifying fault severity based on a measured residual, $r_{\mathrm{res}}$. A larger residual value signifies a more pronounced deviation between the robot’s actual and nominal behaviors, thereby indicating higher fault severity. Following established approaches in fault-tolerant control, the fault severity parameter $\gamma$ can be estimated by normalizing the residual with respect to a nominal value:

$${\gamma }_{\mathrm{est}}={\displaystyle \frac{\parallel r_{\mathrm{res}}\parallel }{v_{\mathrm{nom}}dt}},$$

(49)

where $\parallel r_{\mathrm{res}}\parallel$ denotes the norm of the residual, $v_{\mathrm{nom}}$ is a characteristic nominal value (e.g., nominal velocity), and $dt$ is a time-scaling factor. The resulting ${\gamma }_{\mathrm{est}}$ typically lies in the interval $[0,1]$, encompassing fault levels of increasing severity (e.g., 20%, 40%, 60%, and 80%).

9.2. Relating Gain to Settling Time

To account for faults on the robot’s settling time, an exponential decay model of the form

$$y=a{e}^{-bx}+c$$

(50)

is used, where y is the settling time, x is the control gain, and a, b, and c are coefficients empirically determined for each fault level. Table 4 summarizes these coefficients.

These results indicate that the minimum achievable settling time under a given fault level is bounded below by c, with higher fault severity leading to increased settling times.

9.3. Compensation Strategy

The term “loss of effectiveness” refers to the reduction in control performance due to the fault. The objective of fault-tolerant control designs is often to restore settling time to nominal (no-fault) conditions. Let $y_{\mathrm{target}}$ represent this desired settling time. The compensation strategy involves adjusting the control gain x such that the actual settling time y under fault approaches $y_{\mathrm{target}}$.

9.4. Gain Selection Procedure

• Estimate fault severity: Measure residual $\parallel r_{\mathrm{res}}\parallel$ and compute ${\gamma }_{\mathrm{est}}$.
• Obtain or interpolate model coefficients: If ${\gamma }_{\mathrm{est}}$ matches a discrete fault level, use corresponding coefficients (Table 4). For intermediate values, interpolate coefficients linearly.
• Set a target settling time: Select $y_{\mathrm{target}}$ based on the nominal or required performance.
• Solve for the gain: Insert $y_{\mathrm{target}}$ into the exponential model:

  $$y=a{e}^{-bx}+c⟹x=-{\displaystyle \frac{1}{b}}ln\left({\displaystyle \frac{y_{\mathrm{target}}-c}{a}}\right).$$

  (51)
  Verify that $y_{\mathrm{target}}\ge c$; otherwise, no finite x will achieve the desired settling time.

By mapping residuals to a fault severity parameter and interpolating coefficients for the exponential settling-time model, one systematically selects a control gain to compensate for fault-induced performance degradation. This framework maintains robot performance close to nominal conditions, effectively handling varying fault intensities.

10. Simulation and Implementation

To validate the theoretical control laws and stability analysis, MATLAB simulations were implemented. The simulation environment models the kinematics of the leader and followers, applying the derived control laws, and observing the convergence behavior of the formation.

10.1. Simulation Setup

The simulation involves one leader and N followers, each initialized from random poses to emulate diverse starting conditions. The leader is assigned a predefined final pose, serving as the primary reference for the formation. The followers are designed to maintain specific offsets relative to their designated reference robots, thereby achieving the desired geometric formation, such as a V-shape. Control gains $k_{l1},k_{l2},k_{l3}$ for the leader and $k_{i1},k_{i2},k_{i3}$ for each follower are carefully selected to ensure rapid convergence to the desired states without overshooting. To simulate real-world scenarios where the actuation capabilities can degrade over time, a residual effectiveness $k_{\mathrm{loss},i}$ is introduced. At a specified time $t=t_{\mathrm{loss}}$, both the leader and the followers experience a reduction in their linear velocities in accordance with their respective loss factors, while their angular velocities remain unaffected.

10.2. Simulation Results and Performance Evaluation

The MATLAB simulations demonstrate the effectiveness of the proposed control strategies in maintaining the desired formation. Specifically, the leader successfully navigates to its predefined pose following the control law. Concurrently, the followers adjust their positions and orientations to uphold the desired geometric formation relative to the leader and among themselves. Remarkably, even after the introduction of actuation losses at $t=t_{\mathrm{loss}}$, the formation remains stable, albeit with reduced linear velocities, highlighting the robustness of the control framework.

The simulation results also corroborate the Lyapunov-based stability analysis, as evidenced by the exponential decay of the positional and orientational errors over time. The performance evaluation encompasses several key metrics:

• Tracking error: Measures the average distance between the followers and their desired positions, indicating the precision of formation maintenance.
• Stability: Assesses the time required for the followers to achieve and maintain the desired formation after initialization or disturbances.
• Robustness: Evaluates the formation’s ability to preserve its shape under various disturbances or alterations in the leader’s trajectory.

Control strategies effectively maintain the desired formation with minimal tracking error and exhibit strong stability characteristics. The system demonstrates resilience against actuation losses, ensuring sustained formation integrity even under degraded actuation conditions.

11. Formation Control Fault Analysis

Actuation Loss Simulation and Settling Time Analysis

This simulation assesses the impact of varying actuation loss on the settling time within a leader–follower system. Actuation loss factors are introduced to emulate reductions in control capabilities for both the leader and the followers. During the simulation, the settling time for each agent is defined as the moment when the error diminishes below 2% of its initial value and remains stable thereafter. This methodology facilitates a comprehensive analysis of how actuation loss influences the system’s stabilization capabilities.

The recorded data indicate that increasing actuation loss (i.e., decreasing control capability) results in longer settling times for all agents, with followers exhibiting greater sensitivity compared to the leader. At higher levels of actuation loss, some followers fail to stabilize, leading to undefined settling times. Additionally, control gains significantly affect system performance; higher gains can reduce settling times but may also induce overshoot or instability if not properly tuned.

Figure 4 illustrates the relationship between actuation loss and settling time for both the leader and the followers. The leader, depicted by the red solid line, consistently demonstrates shorter settling times compared to the followers, even as actuation loss increases. In contrast, the followers, represented by distinct markers and line styles, show varying degrees of performance degradation. For example, Follower 4 stabilizes only up to an actuation loss factor of 0.60, beyond which its settling time becomes undefined, as evidenced by missing data points.

Table 5. Settling times for leaders and followers at different actuation loss factors.

Actuation Loss Factor	Leader (s)	Follower 1 (s)	Follower 2 (s)	Follower 3 (s)	Follower 4 (s)
0	2.00	3.40	3.60	4.50	5.50
0.20	2.30	4.10	4.30	5.30	6.80
0.40	2.70	5.20	5.50	6.90	NaN
0.60	3.70	7.40	7.80	9.90	NaN
0.80	6.50	NaN	NaN	NaN	NaN

NaN indicates that the settling time is not applicable, and 0 means fully actuated.

provides a detailed numerical analysis, presenting settling times for actuation loss factors ranging from 0 (no loss) to 0.80 (significant loss). At an actuation loss factor of 0, all agents stabilize within 6 s, with the leader achieving the fastest settling time of 2.00 s. However, as the actuation loss factor decreases to 0.80, the leader’s settling time increases to 6.50 s, while the followers fail to stabilize, indicated by undefined settling times (‘NaN’).

This analysis underscores the critical influence of actuation loss on system stability and emphasizes the importance of optimizing control parameters to maintain robust performance in leader–follower systems. The findings highlight the necessity for adaptive control strategies that can mitigate the adverse effects of actuation losses, thereby ensuring sustained system reliability and effectiveness.

12. Formation Control Results Under Fully and Partially Actuated

This section presents the simulation results for the multi-robot formation when all robots are fully actuated (100% actuator capacity) and partially actuated. Figure 5 shows the trajectories, orientations, distances, and error plots of the formation of the mobile robot when fully actuated. Similarly, Figure 6 shows the response of the formation when a fault occurs or when it is partially actuated (50% actuated). Notice how the formation converges smoothly and maintains the desired offsets.

Discussion:

• Figure 5a shows each robot’s path, highlighting that the formation converges to a stable configuration.
• Figure 5b indicates that orientation $\varphi$ for all robots synchronizes around 0–1 rad at steady state.
• Figure 5c,d confirm that each follower tracks the leader’s motion with minimal offset after the initial transient.
• Figure 5e,f quantify inter-robot distances, settling near 2–$3\mathrm{m}$ after an overshoot around $t=2\mathrm{s}$.
• Figure 5g,h display error convergence. Both position and orientation errors decay to near zero by $t=5\mathrm{s}$, reflecting successful formation control under full actuator capacity.

With this 100% actuation baseline established, the next sections analyze the system performance under reduced actuator capacity (80%, 60%, 40%, and 20%). Each scenario follows the same format of plots, allowing direct comparison of settling time, error magnitude, and steady-state offsets.

Discussion:

• Figure 6a shows each robot’s path, highlighting that the formation converges to a stable configuration.
• Figure 6b indicates that orientation $\varphi$ for all robots synchronizes around 0–1 rad at steady state.
• Figure 6c,d confirm that each follower tracks the leader’s motion with minimal offset after the initial transient.
• Figure 6e,f quantify inter-robot distances, settling near 2–$3\mathrm{m}$ after an overshoot around $t=2\mathrm{s}$.
• Figure 6g,h display error convergence. Both position and orientation errors decay to near zero by $t=5\mathrm{s}$, reflecting successful formation control under 50% actuation capacity.

Comparative Analysis of 100% vs. 50% Actuation

The comparative analysis between the 100% actuation and 50% actuation scenarios provides insight into the system’s robustness and performance under varying actuation capacities.

• Trajectories and Orientation: Both actuation levels demonstrate the ability to achieve stable formations, as evidenced by the convergence of trajectories in Figure 5 (100%) and Figure 6 (50%). The fully actuated system exhibits faster convergence with smoother trajectories, while the partially actuated system shows slightly delayed stabilization and minor deviations during the transient phase.
• Position Tracking: The tracking performance, illustrated in Figure 5c,d (100%) and Figure 6c,d (50%), indicates that both systems effectively follow the leader’s motion. However, the 50% actuation system may experience increased tracking errors or longer stabilization times due to reduced control authority.
• Inter-Robot Distances: Inter-robot distances remain within the desired range of 2–$3\mathrm{m}$ in both scenarios, as shown in Figure 5 (100%) and Figure 6 (50%). The partially actuated system demonstrates a slightly higher overshoot and longer settling time, reflecting the impact of limited actuation on distance regulation.
• Error Convergence: Error convergence to near-zero values is achieved in both actuation levels, as depicted in Figure 5 and Figure 6. The fully actuated system converges more rapidly, whereas the partially actuated system, while still effective, requires a marginally extended period to reach comparable error levels.
• Implications for Formation Control: The results indicate that while full actuation offers superior performance in terms of faster stabilization and minimal transient deviations, partial actuation still maintains effective formation control, albeit with slightly reduced performance metrics. This suggests that the system is robust to variations in actuation capacity, making it adaptable to scenarios where full actuation may not be feasible.

13. Effect of Fault on Settling Time, Exponential Curve Fitting, and System Efficiency

In order to analyze how varying fault severities influence the settling time of a multi-wheeled mobile robot formation, four distinct fault levels of 20%, 40%, 60%, and 80% were systematically introduced. After injecting each fault level, the control gain k was increased stepwise from low to high, and the resulting settling times were recorded.

• Exponential Curve Fitting of Settling Time

Figure 7, Figure 8, Figure 9 and Figure 10 present the measured settling times for each fault scenario, plotted as a function of the control gain k. The comparison of all fault severity levels is shown in Figure 11. Overlaid on each dataset are best-fit exponential curves of the form

$$y=a{e}^{-bx}+c,$$

where y is the measured settling time, x is the control gain, and a, b, and c are parameters identified via least-squares fitting.

• Coefficient a (initial offset): Represents how large the settling time is when x (i.e., the gain) is relatively small.
• Coefficient b (decay rate): Governs how rapidly the settling time decreases as x grows. A larger b generally indicates a faster drop in settling time.
• Constant c (residual value): Sets a lower bound on the settling time; even at large gains, y cannot fall below c.

As summarized in Table 2, higher fault levels tend to exhibit larger a (indicating greater initial settling times) and higher c (indicating a larger residual settling time). Nonetheless, in every scenario, increasing the gain k still substantially reduces the settling time, confirming that the proposed control approach can compensate effectively for actuator faults.

13.1. System Efficiency and Performance

In addition to capturing the reduction in settling time, it is useful to define a metric that reflects the “efficiency” of using higher gains. One such measure is as follows:

$$E\left(x\right)={\displaystyle \frac{\mathrm{Gain}}{\mathrm{Settling}\mathrm{Time}}}={\displaystyle \frac{x}{y\left(x\right)}}.$$

Under the exponential model $y\left(x\right)=a{e}^{-bx}+c$, this becomes

$$E\left(x\right)={\displaystyle \frac{x}{a{e}^{-bx}+c}}.$$

A larger $E\left(x\right)$ suggests that increasing the control gain yields greater (and faster) improvement in settling time, indicating a more “efficient” use of control effort. Conversely, a small $E\left(x\right)$ implies that the gain is insufficient or the fault level is so severe that the payoff in settling-time reduction is limited.

13.2. Identifying a “Plateau” Region and Efficiency Trade-Off

Although mathematically $y\left(x\right)$ keeps decreasing as x increases, in practice, there is often a point after which further increases in x yield only marginal improvements in settling time. Two common methods can help locate this “plateau” or “diminishing returns” region.

Numerical threshold on $y\left(x\right)-c$: Let $\epsilon$ be a small tolerance, and $x^{★}$ such that we have the following:

$$a{e}^{-b{x}^{★}}<\epsilon .$$

Solving yields

$$x^{★}>{\displaystyle \frac{1}{b}}ln\left({\displaystyle \frac{a}{\epsilon }}\right).$$

Beyond $x^{★}$, the settling time $y\left(x\right)$ is already within $\epsilon$ of its lower asymptote c.

Small Derivative Threshold: Alternatively, consider the slope:

$${\displaystyle \frac{dy}{dx}}=-ab{e}^{-bx}.$$

Choose a small $\delta$ and find $x^{★}$ where

$$ab{e}^{-b{x}^{★}}<\delta ,$$

implying

$$x^{★}>{\displaystyle \frac{1}{b}}ln\left({\displaystyle \frac{ab}{\delta }}\right).$$

For $x>x^{★}$, the curve is “flat” enough that further gains yield negligible reductions in y.

Efficiency at the Plateau Point: Once $x^{★}$ is identified (via either approach), you can compute the following:

$$E\left(x^{★}\right)={\displaystyle \frac{x^{★}}{a{e}^{-b{x}^{★}}+c}}.$$

This value of $E\left(x^{★}\right)$ demonstrates how efficiently the control effort is applied at the onset of diminishing returns. To illustrate a good versus bad point:

• Moderate gain, near the plateau: Shows most of the benefit in settling-time reduction without requiring excessive control effort.
• High gain, far beyond the plateau: May further reduce y by only a fraction of a second, yet requires a disproportionately larger gain (and possibly higher actuator stress or noise).

13.3. Discussion and Practical Implications

Analyzing these results reveals that more severe faults (60% and 80%) begin with higher initial settling times but still respond favorably as the control gain is increased. The coefficient a tends to grow with fault severity, reflecting a larger initial “offset” in settling time. Likewise, the constant term c also rises under higher fault conditions, indicating a larger residual settling time even at elevated gains. Despite these increases in a and c, the exponential curves confirm that the proposed control strategy significantly mitigates the impact of faults, reducing the settling time to a manageable range.

Moreover, as the control gain x becomes sufficiently large, the settling time $y\left(x\right)$ approaches the residual value c, beyond which further increases in x yield only marginal improvements. This highlights the importance of carefully tuning the gain—while higher gains can indeed counteract fault-induced performance degradation, excessively large gains may lead to diminishing returns in settling-time reduction and could exacerbate noise, actuator wear, or stability concerns.

A synthesis of key insights is outlined below:

• Fault tolerance at higher gains: Even severe faults (60% and 80%) can be effectively counteracted by sufficiently high controller gains, resulting in settling times only moderately above those observed in lower-fault scenarios.
• Parameter trends: Larger faults elevate both a and c, implying higher initial and final settling times. Nonetheless, the reduction in y remains significant once the gain surpasses a moderate threshold.
• Efficiency metric: Defining an efficiency measure, such as $E\left(x\right)=x/y\left(x\right)$, helps quantify the trade-off between higher gain and shorter settling time. This metric illustrates where gains are most productively employed and where they become less beneficial.
• Plateau or threshold gain: Establishing a plateau point $x^{★}$ (e.g., via a small threshold on $y\left(x\right)-c$ or on the derivative $\frac{dy}{dx}$) clarifies when additional gain produces minimal improvements in settling time. Beyond $x^{★}$, the marginal benefit of further increasing x diminishes rapidly.

By identifying a gain value near or slightly exceeding $x^{★}$ and comparing it against a substantially larger gain that provides only negligible further benefits, it becomes possible to discern an optimal compromise between performance, actuator demand, and fault tolerance. These results underscore the importance of employing adaptive or fault-tolerant control strategies in multi-robot systems, where partial actuator or sensor failures can be anticipated. When combined with robust control methodologies, a suitably chosen gain ensures that stable formation control can be maintained across a broad spectrum of fault conditions.

14. Conclusions

In this paper, we presented a comprehensive investigation into the formation control of multi-wheeled mobile robots under varying levels of actuator faults. By integrating a leader–follower control strategy with kinematic modeling, Lyapunov-based stability analysis, and systematic fault modeling, we demonstrated the effectiveness of our approach in maintaining formation integrity despite reduced actuator capabilities. The simulation results highlighted how increasing control gains can offset the degradation caused by partial actuator failures, thereby improving settling times and preserving the desired geometric configuration of the robot group. We further employed exponential (or polynomial) curve fitting to quantify the relationship between control gains and settling times under different fault severities, providing actionable insights into optimal gain selection and revealing the region of diminishing returns, where further gain increases offer minimal performance benefits.

These findings underscore the robustness and adaptability of the proposed control framework. Even when faults significantly limit actuator performance, well-tuned gains ensure stable and efficient formation convergence. Nonetheless, extremely severe fault levels pose challenges to complete stabilization, emphasizing the necessity of adaptive or reconfigurable control architectures.

For future work, several directions present themselves. First, real-world experiments in unstructured, dynamic environments would validate the proposed controllers’ performance outside of the simulation. Second, incorporating actuator dynamics and sensor noise into the modeling process can capture additional real-world complexities. Third, adaptive control methods, such as gain scheduling or online parameter tuning, could further enhance resiliency to abrupt or evolving fault conditions. Finally, extending this framework to larger teams of heterogeneous robots and integrating advanced cooperative decision-making strategies would open pathways to more robust, large-scale multi-robot deployments in diverse applications.