Intelligent Leader–Follower Formation Control Using an Interval Type-2 Fuzzy Logic Controller and Adaptive Radial Basis Function Neural Networks

Abstract

This work proposes a leader–follower formation framework comprising three differential-drive mobile robots that are equipped with geometry-aware kinematics coupled with adaptive dynamics. The leader robot is directed by a nonlinear body frame tracking controller that is enhanced with an online radial basis function neural network. This network compensates for residual error in velocity-domain effects from the current position error, which enhances feedforward compensation without affecting stability. The dynamics of each robot are driven by an interval type-2 adaptive fuzzy-PID controller that adapts membership-function centers and footprint of uncertainty bounds in real time to deal with model mismatch and disturbances. As a result, the controller yields low chatter and ensures smooth control actions. The followers preserve formation using a geometric partial feedback linearization (PFL) design that integrates effectively with the adaptive layers. Controller gains for the leader and PFL formation kinematic controllers are tuned using the Secretary Bird Optimization Algorithm. A Lyapunov analysis for all the proposed controllers guarantees convergence and boundedness under nonholonomic constraints. Overall, the framework exhibits reliable path tracking and formation preservation, enhanced design for disturbance rejection, and smooth-behaved control effort through diverse operating conditions, such as robot dynamic model parameter uncertainties and control signals disturbances. Results show that all controllers ensured system stability and achieved low root mean square error values. Compared with existing works, the path accuracy for the leader was enhanced by 57%, while followers 1 and 2 achieved accuracy gains of 42.25% and 37.33%, respectively.

Keywords

formation control leader–follower formation online RBF adaptive Type-2 FLC Lyapunov stability

1. Introduction

Leader–follower formation control is widely used in multirobot navigation for tasks such as oil exploration and search-and-rescue (Shukla & Karki, 2016). It reduces the time, energy, and computational effort required compared with controlling each robot individually. The five main paradigms are described as follows: leader–follower, in which leaders define the path and followers keep set positions but risk failure if the leader is lost (Hirata-Acosta et al., 2021); leader-obstacle, which regards obstacles as temporary leaders to guide safe detours (Parvareh et al., 2023); virtual structure, which maintains precise geometric shapes through global coordination (Wang et al., 2024); behavior-based, which is inspired by swarms using simple local rules for flexible but less precise movement (Aldana-López et al., 2022); and consensus-based, which employs multiple leaders as shared references for improved accuracy and robustness (Xu et al., 2024).

Formation approaches manage multirobot tracking by designating one or more leaders, with followers maintaining specified distances and orientations. Since 2020, studies have explored robust, adaptive, observer-based, sliding-mode, neural-dynamics, fuzzy/adaptive, and deep-learning approaches. In observer-based control, Hassan and Hammuda (2020) proposed a least-squares smoothed regularized observer for real-time follower state prediction without predefined paths, which enables adaptive path changes (Hassan & Hammuda, 2020). Proportional–integral observers with active disturbance rejection control have also been applied to omnidirectional robots, which achieve robust distance-bearing regulation under disturbances (Nasser et al., 2021; Ramírez-Neria et al., 2023). A robust adaptive controller developed in 2023 enabled virtual leader switching for obstacle avoidance while estimating uncertainty bounds using model-free adaptation (Parvareh et al., 2023).

The hybrid sliding-mode and neural-dynamics designs by Wang et al. (2024) include combined sliding-mode control with attractive–repulsive forces for collision avoidance and dynamic gain scheduling to maintain formation (Wang et al., 2024). Xu et al. (2024) proposed a bio-inspired neural-dynamics strategy with backstepping–sliding-mode integration and an adaptive sliding filter, ensuring distributed normalized inputs, chattering suppression, and noise-robust state estimation (Xu et al., 2024). Other notable methods include bearing-based control, such as that of Trinh and Ahn (2022), who achieved finite-time convergence for acyclic formations via state-dependent gains (Trinh & Ahn, 2022), and fuzzy-adaptive control, as demonstrated by Nourizadeh et al. (2022), who developed a type-1 fuzzy-adaptive backstepping controller to manage large initial spikes and ensure global stability through Lyapunov analysis (Al Mhdawi et al., 2023; Nourizadeh et al., 2022). Lastly, Zhang et al. (2020) combined a disturbance observer with super-twisting second-order sliding-mode control to reject disturbances, eliminate chattering, and enhance leader–follower guidance under uncertainties (Zhang et al., 2020).

Notable controllers in this field include that of Ali et al. (2021), who developed an adaptive fuzzy-PID hybrid scheme for leader–follower formations under communication delays. A classical PID core was augmented with a fuzzy layer for real-time gain tuning, using trajectory error feedback to improve responsiveness and disturbance rejection while maintaining tight position–velocity coordination despite delays (Ali et al., 2021). Bui and Phung (2024) proposed a radial basis function neural network (RBFNN)-enhanced sliding-mode backstepping controller for UAV formations driven by a virtual leader. The RBFNN estimates external disturbances and unmodeled dynamics, while the nominal backstepping–sliding structure provides robust tracking with PID-like adaptive gain tuning (Bui & Phung, 2024).

Aryankia and Selmic (2021) applied neuro-adaptive backstepping with RBFNN for nonlinear multiagent swarms under delays; this method uses RBF approximation to handle unknown nonlinearities and disturbances with online weight adaptation (Aryankia & Selmic, 2021). Goto et al. (2024) employed an adaptive immune–fuzzy quasi-sliding mode controller that combines a PD core with a fuzzy–immune modulation layer to adaptively adjust control effort and suppress chattering (Goto et al., 2024). Recent studies have further explored advanced adaptive and learning-based schemes. A neural adaptive prescribed-performance controller presented in (Xie et al., 2024) integrates graph-based guidance via RBFNN approximation and backstepping, improving tracking accuracy but exhibiting structural complexity and some chattering. An adaptive distributed controller based on a recurrent soft actor–critic algorithm was proposed in (Li et al., 2024), which enables formation coordination with only position of fast adaptation in simulation. However, it lacks real-world validation and faces challenges related to data interpretability.

Despite notable progress in leader–follower formation control, ensuring accurate trajectory tracking and stable formation maintenance under nonlinear dynamics, modeling uncertainties, and unpredictable external disturbances remains a major challenge. Existing techniques often confine adaptation to the kinematic level while neglecting the intricate coupling with system dynamics. Meanwhile, dynamic-level controllers are frequently implemented without adequate mechanisms to mitigate residual tracking errors or counteract time-varying disturbances. Moreover, controller gains are commonly determined through heuristic tuning, which can result in performance degradation when operating conditions deviate from nominal scenarios. These limitations highlight the need for a unified, adaptive, and disturbance-resilient control framework that combines kinematic and dynamic considerations, ensuring robustness and high-precision formation performance across diverse and uncertain environments.

The proposed intelligence framework integrates three layers. First, an online RBFNN augments the leader's kinematic law by compensating residual errors in the velocity control signal. Second, an interval type-2 adaptive fuzzy-PID (IT2AFPID) dynamic controller adapts its membership-function centers and footprint of uncertainty (FoU) in real time. Third, gains in the nominal leader controller and the followers’ partial feedback linearization (PFL) law are refined via the Secretary Bird Optimization Algorithm (SBOA) using a root mean square error (RMSE)-based objective. Overall, these elements yield a cohesive, learning-enabled control architecture. The contributions are in three parts:

Leader kinematic tracking controller - A nonlinear tracking law is augmented with an online RBFNN to compensate for residual errors in leader controller's velocities. The RBFNN modifies the nominal velocity commands in real time using online stochastic gradient descent learning laws, ensuring stability while eradicating the need for integral action. SBOA-based optimized gains are applied to the nominal nonlinear tracking law. A filtered orientation error formulation is used to achieve fast convergence without overshoot.

Dynamic-level controller for leader and followers - A feedback linearization scheme is integrated with an IT2AFPID controller in the feedback loop as an inner velocity loop. The IT2AFPID adaptively tunes the centers and spreads of Gaussian membership functions (MFs, the adaptive FoU) to maintain robustness under varying dynamics, while nonlinear gain computation allows precise tracking despite time-varying uncertainties.

Follower formation controller - A partial geometric nonlinear formation law with PFL, whose gains are optimized using the SBOA. The optimization minimizes formation errors, ensuring smooth orientation and distance regulation between robots.

The paper is organized as follows. Section 2 presents the mathematical kinematic and dynamic models. Section 3 illustrates the design of the proposed controllers. Section 4 introduces the Lyapunov stability analysis for the proposed controller by lemmas and their proofs. Section 5 presents the SBOA. Subsequently, Section 6 presents an analysis of the simulation results of the proposed controllers and compares them with other formation algorithms for leader–follower tracking. Lastly, Section 7 provides conclusions and opportunities for future research.

2. Methodology

This section presents the mobile robot kinematic and dynamic models.

2.1 Differential-Drive Mobile Robot Kinematics

The schematic of the nonholonomic differential-drive mobile robot is displayed in Figure 1, where the two axle wheels control rotational motion and the passive wheel at the head provides directional motion. Here, $P_{i}$ denotes the axle center point, c represents the robot's mass center of gravity, $φ_{l}$ and $φ_{r}$ represent the two angular velocities on both sides, the length a signifies the distance between points $P_{i}$ and c, $2 b$ is the width of the robot body, r denotes the wheel radius, $θ$ refers to the heading angle with the x-axis, and $P_{i} = (x y θ)^{T}$ represents the robot position in the $(x, y)$ axis. The mobile robot has non-sliding rolling constraints for the wheels on the motion ground, which establish the relationship between the robot's coordinates and motion velocities (Wang et al., 2023).

Equation (1) presents the matrix form of the system constraint (Wang et al., 2023):

\begin{aligned} A^{T} (q) \dot{q} = 0 \end{aligned}

(1)

where

A (q)

denotes the constraint matrix of n constraints and m coordinates (Wang et al., 2023):

\begin{aligned} A^{T} (q) = (sin θ - cos θ) \end{aligned}

(2)

Let $S (q)$ be the full-rank matrix formed by a set of smooth and linearly independent vectors spanning the null space of the constraint matrix, as follows (Kadhim & Hassan, 2020):

\begin{aligned} S^{T} (q) A^{T} (q) = 0 \end{aligned}

(3)

Then, $S (q)$ for a differential-drive mobile robot is:

\begin{aligned} S (q) = [\begin{array}{cc} \cos θ & 0 \\ \sin θ & 0 \\ 0 & 1 \end{array}] \end{aligned}

(4)

Figure 1.

Differential drive mobile robot platform.

The kinematic model is represented as (Wang et al., 2023):

\begin{aligned} \begin{array}{cc} \dot{q} = S (q) v \end{array} \end{aligned}

(5)

where

v = (u, w)

is the kinematic input velocity vector derived from the robot dynamics model.

2.2 Differential-Drive Mobile Robot Dynamics

The Lagrange method used to format the dynamical equation is as follows (Fierro & Lewis, 1998):

\begin{aligned} M (q) \ddot{q} + C (q, \dot{q}) + τ_{d} = B (q) τ + A^{T} (q) λ \end{aligned}

(6)

where

M (q) \in R^{n * n}

as the positive definite robot inertia matrix,

C (q, \dot{q}) \in R^{n * n}

as the centripetal and Coriolis matrix,

τ_{d} \in R^{n * 1}

as the bounded unknown disturbances.

B (q) R^{n * 2}

as the transformation matrix in the inputs.

τ \in R^{n * 1}

as the controlled torque input vector, and

λ

This is the Lagrange multiplier. The matrices of dynamics are (Fierro & Lewis, 1998):

\begin{aligned} M (q) & = [\begin{array}{ccc} m & 0 & - m a \sin θ \\ 0 & m & m a \cos θ \\ - m a \sin θ & m a \cos θ & I \end{array}] \end{aligned}

(7)

\begin{aligned} C (q, \dot{q}) & = [\begin{array}{l} - m a {\dot{θ}}^{2} \cos θ \\ - m a {\dot{θ}}^{2} \sin θ \\ 0 \end{array}] \end{aligned}

(8)

\begin{aligned} B (q) & = \frac{1}{r} [\begin{array}{cc} \cos θ & \cos θ \\ \sin θ & \sin θ \\ b & - b \end{array}] \end{aligned}

(9)

where I represent the moment of inertia for the robot mass. To remove the Lagrange multiplier, Equation (6) is multiplied by

S^{T} (q)

This gets (Hasana & Alwan, 2021):

\begin{aligned} M_{1} (q) \dot{v} + M_{2} (q) v + \bar{C} (q, \dot{q}) + {\bar{τ}}_{d} = \bar{B} (q) τ \end{aligned}

(10)

and

\begin{aligned} \begin{aligned} M_{1} (q) & = S^{T} (q) M (q) S (q), \\ M_{2} (q) & = S^{T} (q) M (q) \dot{S} (q), \\ \bar{C} (q, \dot{q}) = S^{T} (q) C (q, \dot{q}), {\bar{τ}}_{d} & = S^{T} (q) τ_{d} and \bar{B} (q) = S^{T} (q) B (q) S (q) \end{aligned} \end{aligned}

(11)

By multiplying Equation (11) by $\bar{B} (q)^{- 1}$ :

\begin{aligned} τ = \hat{M_{1}} (q) \dot{v} + \hat{M_{2}} (q) v + \hat{C} (q, \dot{q}) + \hat{τ_{d}} \end{aligned}

(12)

\begin{aligned} \begin{aligned} \hat{M_{1}} & = \bar{B} (q)^{- 1} M_{1} (q), \hat{M_{2}} (q) = \bar{B} (q)^{- 1} M_{2} (q) \\ \hat{C} (q, \dot{q}) & = \bar{B} (q)^{- 1} \bar{C} (q, \dot{q}) and \hat{τ_{d}} = \bar{B} (q)^{- 1} {\bar{τ}}_{d} \end{aligned} \end{aligned}

(13)

3. Control Framework

This section presents the leader's kinematic controller, the IT2AFPID dynamic controller for each robot, and the followers’ PFL-based formation law.

3.1 Leader Kinematic Controller (Path Tracking)

The tracking of the specific path problem involves guiding the robot to follow a specific path in Cartesian space under defined circumstances. The objective is to design a control signal that guides the tracking error toward zero. The goal is to follow a reference trajectory defined by a point $q_{r} = (x_{r} y_{r} θ_{r})^{T}$ with corresponding time derivatives $({\dot{x}}_{r} {\dot{y}}_{r} {\dot{θ}}_{r})^{T}$ , where the trajectory is generated as (Wang et al., 2023):

\begin{aligned} v_{r} & = \pm \sqrt{{\dot{x}}_{r}^{2} + {\dot{y}}_{r}^{2}} \end{aligned}

(14)

\begin{aligned} θ_{r} & = a t a n 2 ({\dot{y}}_{r}, {\dot{x}}_{r}); w_{r} = {\dot{θ}}_{r} = \frac{{\ddot{y}}_{r} {\dot{x}}_{r} - {\ddot{x}}_{r} {\dot{y}}_{r}}{v_{r}^{2}} \end{aligned}

(15)

where

v_{r} and w_{r}

are the linear and angular velocities of the reference path, respectively, and

θ_{r}

is the reference heading angle. The feedback control objective is to design a control vector

u = u (q, \dot{q}, q_{r}, v)

such that the error vector between the leader and the reference path,

e = q - q_{r},

is stabilized around zero. The proposed controller operates at the kinematic-level nonlinear tracking controller-based error dynamics and incorporates an RBFNN to adaptively compensate for model uncertainties (Waheed et al., 2020). Let the actual state be:

\begin{aligned} q = [\begin{matrix} x \\ y \\ θ \end{matrix}], q_{r} = [\begin{matrix} x_{r} \\ y_{r} \\ θ_{r} \end{matrix}] \end{aligned}

(16)

The robot frame tracking error after multiplication by the transformation matrix T is:

\begin{aligned} [\begin{matrix} e_{x} \\ e_{y} \\ e_{θ} \end{matrix}] = [\begin{array}{ccc} cos θ_{r} & sin θ_{r} & 0 \\ - sin θ_{r} & cos θ_{r} & 0 \\ 0 & 0 & 1 \end{array}] [\begin{matrix} x - x_{r} \\ y - y_{r} \\ θ - θ_{r} \end{matrix}] \end{aligned}

(17)

It is simplified to:

\begin{aligned} \begin{aligned} e_{x} & = \cos (θ) (x - x_{r}) + \sin (θ) (y - y_{r}) \\ e_{y} & = - \sin (θ) (x - x_{r}) + \cos (θ) (y - y_{r}) \\ e_{θ} & = a tan 2 (sin (θ - θ r), cos (θ - θ r)) \end{aligned} \end{aligned}

(18)

where the mod function takes the raw angle difference and maps it into the interval

[- π, π]

. Then, the nominal part of the proposed nonlinear tracking controller is (Kanayama et al., 1990)

\begin{aligned} \begin{aligned} v_{n o m} & = v_{r} \cos (e_{θ}) - k_{1} e_{x} \\ w_{n o m} & = w_{r} - k_{2} e_{y} - k_{3} \sin (e_{θ}) \end{aligned} \end{aligned}

(19)

where

k_{1}, k_{2}, and k_{3}

are the positive design gains.

v_{n o m}

and

w_{n o m}

are the nominal part velocities control signals. To learn and compensate for error tracking residual (due to modeling inaccuracies or disturbances) in the nominal kinematic controller, an online adaptive correction term is added to the control signals

v_{n o m}

and

w_{n o m}

. Thus, the RBF network estimates a correction term for the residual error based on the current error vector

e = (e_{x}, e_{y}, e_{θ})

. To refine the final control inputs, the RBF nodes are defined as:

\begin{aligned} ϕ_{i} (e) = \exp (- \frac{‖ e - c_{i} ‖^{2}}{2 σ^{2}}), i = 1, \dots, N \end{aligned}

(20)

where N is the number of nodes, and the outputs of the networks are:

\begin{aligned} {\hat{f}}_{v} = W_{v}^{⊤} Φ (e); {\hat{f}}_{w} = W_{w}^{⊤} Φ (e) \end{aligned}

(21)

where

{\hat{f}}_{v}

and

{\hat{f}}_{w}

represent the estimated residuals in each velocity. The total control input will then be:

\begin{aligned} v_{c} = v_{n o m} + {\hat{f}}_{v}; w_{c} = w_{n o m} + {\hat{f}}_{w} \end{aligned}

(22)

where

Φ \in R^{N}

is the vector of basis function activations, and

W_{v}, W_{w} \in R^{N}

are weight vectors for v and

ω

. To enhance the compensation robustness, the online adaptation for RBF weights is designed as:

\begin{aligned} \begin{aligned} {\dot{W}}_{v} & = η Φ (e) e_{v} \\ {\dot{W}}_{w} & = η Φ (e) e_{w} \end{aligned} \end{aligned}

(23)

where

e_{v} = - k_{1} e_{x}

and

e_{w} = - k_{2} e_{y} - k_{3} \sin (e_{θ})

. The weights are updated using gradient descent based on nominal tracking errors:

\begin{aligned} \begin{aligned} W_{v} & \leftarrow W_{v} + η \cdot Φ \cdot e_{v} \cdot Δ t \\ W_{w} & \leftarrow W_{w} + η \cdot Φ \cdot e_{w} \cdot Δ t \end{aligned} \end{aligned}

(24)

where

η

is the learning rate and

Δ t

is the control time step. The errors

e_{v}

and

e_{w}

work as teaching signals. When

e_{x} > 0

, then

e_{v} = - k_{1} e_{x} < 0

, which motivates

(W_{v}^{⊤} Φ (e))

in a way that reduces reliance on large

- k_{1} e_{x}

in feedback. However,

Φ

localizes the update such that only the basis functions near the current

error

adapt significantly. In real time, the network learns a map

e \mapsto ({\hat{f}}_{v}, {\hat{f}}_{w})

, who compensates for consistent, repeatable error residuals treated as unmodeled terms. This strategy allows the feedback control terms to handle non-repeatable disturbances. In general, the RBFNN augments the nominal controller with a learned, state-dependent bias to improve the tracking performance whenever the nominal model underperforms (e.g., residual velocity errors). It operates online to reduce residual tracking effort left to the proportional terms by adding an adaptive feedforward/compensation term. This structure provides the stability of the nominal controller (outer loop) while allowing the RBF network to compensate for predicted residuals. As a result, steady-state and control job errors are reduced, and the transient tracking performance is enhanced. Figure 2 shows the integration of RBF compensation into the kinematic controller. Table 1 illustrates all the RBF parameters.

Figure 2.

RBFNN proposed to compensate for the global leader kinematic control signal.

Table 1.

Parameters of RBF and the Kinematic Controller.

Parameter	Symbol	Description
RBF center	$c_{i}$	Center of the $i$ -th RBF unit
RBF width	$σ$	Spread of the Gaussian basis functions
Number of RBF nodes	$N$	Number of basic functions
Activation function	$ϕ_{i} (e)$	Output of the $i$ -th Gaussian
Activation vector	$Φ$	Vector of all $ϕ_{i} (e)$
RBF output weight	$W_{v}, W_{w}$	Output weights for $v, ω$
Learning rate	$η$	Adaptation gains for online learning
Time step	$Δ t$	Time difference between updates

3.2 Dynamic Controller Design

The dynamic controller design is based on feedback linearization of the robot dynamics, which is equipped with an IT2AFPID controller (Karam et al., 2019) and (Hassan et al., 2020). This integration aims to design a robust controller. The rules of the IT2AFPID controller are suited for nonlinear systems, where it has an FoU in its MF (Ghintab & Hassan, 2025) and (Boulkroune et al., 2025). This feature enables the controller to effectively handle uncertain systems, time-varying systems, and missing dynamics (Sabeeh et al., 2022) and (Rigatos et al., 2020).

The dynamic controller's objective is to track the desired velocity, $v_{c} = [v_{c}, ω_{c}]^{⊤},$ which is generated from the leader kinematic controller. The velocity-tracking error is defined as:

\begin{aligned} e_{v} = v_{c} - v \end{aligned}

(25)

Then, the control rule is

\begin{aligned} τ = \bar{B} {(q)}^{- 1} (M_{2} (q) v + M_{1} (q) {\dot{v}}_{c} + \bar{C} (q, \dot{q}) + {\bar{τ}}_{d} + M_{1} (q) u_{f}) \end{aligned}

(26)

where

u_{f} = [u_{1}; u_{2}]

is the output control variable of the IT2AFPID controller. The integration of a fuzzy system with a dynamic model, as presented in (Abdullsahib & Alwan, 2025), results in the following controller:

\begin{aligned} u_{f} = f_{IT 2} (e_{v}, {\dot{e}}_{v}, \int_{0}^{t} e_{v} d t) \end{aligned}

(27)

This approach ensures adaptive compensation (Merazka et al., 2017) and bounded behavior near the origin. Based on the inverse of the system dynamics, the actual velocity is given by:

\begin{aligned} \dot{v} & = M_{1}^{- 1} (q) (\bar{B} (q) τ - M_{2} (q) v - \bar{C} (q, \dot{q}) - {\bar{τ}}_{d}) = {\dot{v}}_{c} + u_{f} \end{aligned}

(28)

\begin{aligned} \Rightarrow {\dot{e}}_{v} = - u_{f} \end{aligned}

(29)

The proposed IT2AFPID adapts the Gaussian MFs and integral-enhanced output gain tuning (Challoob et al., 2024), providing improved convergence and robustness in error tracking. The following novel elements distinguish our IT2AFPID controller:

Adaptive interval type-2 fuzzy inference, where each MF learns independent upper sigma $σ_{u}$ and lower sigma $σ_{l}$ MF values.

Aggressive gain modulation via adaptive defuzzification gain ( $k_{f}$ ), scaled with uncertainty.

Integral action with a dead-zone reset to suppress chattering and mitigate windup.

Real-time adaptation of MF centers based on online error signals.

The controller receives an input vector of dimension 18, representing 6 channels of error signals for each robot, as shown in Equation (30). This vector contains the proportional error $e_{i}$ , the derivative of error ${\dot{e}}_{i}$ , and the integral of error ( $\int_{0}^{t} e_{i}$ ), where t is the simulation time:

\begin{aligned} input\; = {[e_{1}, {\dot{e}}_{1}, \int_{0}^{t} e_{1}, \dots, e_{6}, {\dot{e}}_{6}, \int_{0}^{t} e_{6}]}^{T} \in R^{18} \end{aligned}

(30)

where

e_{v} = (e_{v}, e_{w})

represents the leader error and is formatted as (

e_{1}, e_{2}

). When entered into the fuzzy system, along with the indexed errors from the two other follower robots, each channel i ∈ {1, …,

n_{Vars}

} is processed independently to produce an output control signal

u_{f} (u_{i})

, as described in the algorithm. The input vector is defined as

z_{i} = {[e_{i}, {\dot{e}}_{i}, \int_{0}^{t} e_{i}]}^{T}

. Each channel uses

n_{MF\;}

Gaussian MFs with centers

c_{i, j}

and interval type-2 spreads

σ_{u, i, j}, σ_{l, i, j}

for

j

= 1, …,

n_{MF}

. A smoothed activity/uncertainty scalar

{\tilde{u}}_{i}

modulates the FoU. A comprehensive explanation of the proposed adaptive IT2AFPID controller structure is provided in Table 2.

Table 2.

Pseudo IT2AFPID Controller Design Algorithm.

where $k_{u}$ and $k_{l}$ represent the upper/lower gains for the upper/lower MF of FoU. $μ_{u} a n d μ_{l}$ are the upper/lower MF activations, and $g_{m}$ denotes the bounds of activated gain. Steps 1–5 apply clamping to avoid large error spikes from destabilizing adaptation, and integral action is disabled near the target to prevent windup and overshoot. Step 6 estimates system uncertainty by assessing error, error rate, and integral error, normalizing the result to a stable range. Steps 7–8 smooth this uncertainty estimate over time by utilizing a smoothing parameter $α_{σ}$ ∈ [0.01,0.1] to avoid abrupt changes in MFs. In Steps 9–12, MF centers shift adaptively with error direction to track dynamic drifts, reducing mismatch between static MFs and varying operating conditions. Steps 13–21 adjust MF shapes dynamically, widening intervals when uncertainty grows and narrowing them as stability returns, forming the core of the adaptive type-2 fuzzy logic mechanism. Steps 22–24 calculate the upper and lower membership values to form a fuzzy uncertainty envelope based on the current input. Steps 25–28 compute activation strengths by selecting the dominant match from error, error derivative, or integral, ensuring robust and rapid inference. Steps 29–30 map these strengths into crisp control gains via weighted averaging, with gains $y_{j}$ ∈ [0.9,3], and $k_{f}$ as the average gain. Steps 31–32 use the adaptive gain $k_{f}$ to modulate PID terms, boosting the integral term nonlinearly by $1 + | e |$ to enable rapid and smooth correction during large deviations, and Step 33 updates the control input for the next cycle.

3.3 Formation Controller (Kinematic Level for Followers)

The block diagram shown in Figure 3 illustrates the proposed leader–follower formation strategies with controllers for one leader and two followers (Do et al., 2022). Each follower maintains a specified distance and orientation from the leader robot, as shown in Figure 4. The measurement of position for each follower with respect to the leader, along with the horizontal angle of the midline connection, is necessary for optimal coordination. In the mathematical formulation, the robot's center of gravity serves as the reference point for control. Based on Figure 4, the i^th follower with position $P_{i} = (x_{i} y_{i} θ_{i})^{T}$ is designed to track the leader position $P = (x y θ)^{T}$ . The desired formation for the followers is defined as $P_{i}^{d} = (x_{i}^{d} y_{i}^{d} θ_{i}^{d})^{T}$ , which represents the formation controller objective to be satisfied by the followers (Riahifard et al., 2019). From Figure 4, the generalized coordinate for the followers is:

\begin{aligned} P_{i} = [\begin{matrix} x_{i} \\ y_{i} \\ θ_{i} \end{matrix}] = [\begin{matrix} x + λ_{i} \cos (ψ_{i} + θ) + acos θ \\ y + λ_{i} \sin (ψ_{i} + θ) + asin θ \\ θ_{i} \end{matrix}] \end{aligned}

(31)

and the desired formation is:

\begin{aligned} P_{i}^{d} = [\begin{matrix} x_{i}^{d} \\ y_{i}^{d} \\ θ_{i}^{d} \end{matrix}] = [\begin{matrix} x + λ_{i}^{d} \cos (ψ_{i}^{d} + θ) + acos θ \\ y + λ_{i}^{d} \sin (ψ_{i}^{d} + θ) + asin θ \\ θ \end{matrix}] \end{aligned}

(32)

where

ψ_{i}^{d}

is the desired heading angle, and

λ_{i}^{d}

represents the desired distance. The desired distance between the leader and followers is (Poonawala et al., 2013):

\begin{aligned} λ_{i} & = \sqrt{λ_{i x}^{2} + λ_{i y}^{2}} \end{aligned}

(33)

where

λ_{i x}

and

λ_{i y}

represent the distance projections along the axes, with the variables defined in Figure 3:

\begin{aligned} λ_{i y} & = y - a sin (θ_{i}) - y_{i} = - λ_{i} \sin (ψ_{i} + θ) \end{aligned}

(34)

\begin{aligned} λ_{i x} & = x - a \cos (θ_{i}) - x_{i} = - λ_{i} \cos (ψ_{i} + θ) \end{aligned}

(35)

Figure 3.

Leader-follower proposed formation control of i^th- follower block diagram.

Figure 4.

Leader-follower mobile robot formation.

The separation angle $ψ_{i}$ is rewritten as:

\begin{aligned} ψ_{i} = a t a n 2 {λ_{i y}, λ_{i x}} - θ + π \end{aligned}

(36)

The system formation dynamics can be calculated and simplified with derivatives of $λ_{i}$ and $ψ_{i}$ as (Tutuko et al., 2018)

\begin{aligned} \frac{d ψ_{i}}{d t} = \frac{1}{λ_{i}} {v \sin ψ_{i} - v_{i} \sin ξ_{i} + a ω_{i} \cos ξ_{i}} - ω \end{aligned}

(37)

where

v and ω

are the leader velocities, and

v_{i}

and

ω_{i}

are the follower

i

velocities.

ξ_{i} = ψ_{i} + θ - θ_{i}

. Then, the error vector of the formation control is defined as:

\begin{aligned} \begin{array}{cc} E_{i} = T (θ_{i}) (P_{i}^{d} - P) \end{array} \end{aligned}

(38)

Using Equation (33), the formation control error is rewritten as:

\begin{aligned} E_{i} = [\begin{matrix} E_{i x} \\ E_{i y} \\ E_{i θ} \end{matrix}] = [\begin{matrix} λ_{i}^{d} \cos (ψ_{i}^{d} + θ - θ_{i}) - λ_{i}^{d} \cos ξ_{i} \\ λ_{i}^{d} \sin (ψ_{i}^{d} + θ - θ_{i}) - λ_{i}^{d} \sin ξ_{i} \\ θ_{i}^{d} - θ_{i} \end{matrix}] \end{aligned}

(39)

Then, the obtained error dynamics is expressed as:

\begin{aligned} \frac{d E_{i}}{d t} = \dot{T} (θ_{i}) (P_{i}^{d} - P) + T (θ_{i}) ({\dot{P}}_{i}^{d} - \dot{P}) \end{aligned}

(40)

The simplified form is

\begin{aligned} \frac{d E_{i}}{d t} = [\begin{array}{cc} \cos (θ - θ_{i}) & - λ_{i}^{d} \sin (ψ_{i}^{d} + θ - θ_{i}) \\ \sin (θ - θ_{i}) & λ_{i}^{d} \cos (ψ_{i}^{d} + θ - θ_{i}) \\ 0 & 0 \end{array}] v + [\begin{array}{cc} - 1 & E_{i y} \\ 0 & - (a + E_{i x}) \\ 0 & - 1 \end{array}] v_{i} + [\begin{matrix} 0 \\ 0 \\ ω_{i}^{d} \end{matrix}] \end{aligned}

(41)

The formation controller inputs at the kinematic level $v_{c i} = [v_{c i}; ω_{c i}]$ are designed to drive the error to zero asymptotically through dynamic convergence. Moreover, the feedback linearization technique is applied to eliminate nonlinearities. Therefore, the control equations are given by:

\begin{aligned} v_{c i} = (\begin{array}{cc} \cos (θ - θ_{i}) & - λ_{i}^{d} \sin (ψ_{i}^{d} + θ - θ_{i}) \\ \frac{1}{a} \sin (θ - θ_{i}) & \frac{λ_{i}^{d}}{a} \cos (ψ_{i}^{d} + θ - θ_{i}) \end{array}) v + (\begin{array}{ccc} ρ_{1} & 0 & 0 \\ 0 & \frac{ρ_{2}}{a} & \frac{ρ_{3}}{a} \end{array}) E_{i} \end{aligned}

(42)

where

ρ_{i} (i : 1 \to 3)

represents the positive controller gains. Then, the error dynamics in terms of the control inputs are

\begin{aligned} \frac{d E_{i}}{d t} = [\begin{array}{ccc} - ρ_{1} & ω_{i} & 0 \\ - ω_{i} & - ρ_{2} & - ρ_{3} \\ 0 & \frac{- ρ_{2}}{a} & \frac{- ρ_{3}}{a} \end{array}] E_{i} + [\begin{array}{cc} 0 & 0 \\ 0 & 0 \\ \frac{- 1}{a} \sin (θ - θ_{i}) & \frac{- λ_{i}^{d}}{a} \cos (ψ_{i}^{d} + θ - θ_{i}) \end{array}] v + [\begin{array}{cc} 0 \\ 0 \\ ω_{i}^{d} \end{array}] \end{aligned}

(43)

Owing to nonholonomic constraints, the angle of each follower cannot remain identical during the robot maneuvers. Thus, the desired orientation cannot be assigned as $θ_{i}^{d} = θ$ . Therefore, the reference orientation must be derived and chosen as:

\begin{aligned} ω_{i}^{d} = \frac{(\begin{array}{cc} v \sin (θ - θ_{i}) + λ_{i}^{d} ω \cos (ψ_{i}^{d} + θ - θ_{i}) + 2 ρ_{2} E_{i y} \end{array})}{a} \end{aligned}

(44)

Applying it to the dynamics yields:

\begin{aligned} \frac{d E_{i}}{d t} = (\begin{array}{ccc} - ρ_{1} & ω_{i} & 0 \\ - ω_{i} & - ρ_{2} & - ρ_{3} \\ 0 & \frac{- ρ_{2}}{a} & \frac{- ρ_{3}}{a} \end{array}) E_{i} \end{aligned}

(45)

4. Lyapunov Stability Proof for the Proposed Controllers

Lyapunov stability analysis will be presented for the kinematic controller of the leader, the dynamic controllers of all robots, and the formation-level kinematic controller. The analysis will be structured in lemmas and their proofs.

4.1 Stability Analysis of the Leader Kinematic Controller

To ensure the robustness of the proposed leader kinematic controller, it should be subjected to a candidate Lyapunov function. It should provide asymptotic stability with this controller. The proposed Lyapunov function is:

\begin{aligned} V_{1} = \frac{1}{2} e^{⊤} e + \frac{1}{2 η} (∥ {\tilde{W}}_{v} ∥^{2} + ∥ {\tilde{W}}_{w} ∥^{2}) \end{aligned}

(46)

Let $e := {[\begin{array}{ccc} e_{x} & e_{y} & e_{θ} \end{array}]}^{⊤}$ be the tracking error for the body frame. $W_{v}, W_{w} \in R^{N}$ be the weight vectors for the linear and angular velocities, with constants $W_{v}^{⋆}, W_{w}^{⋆}$ , the formulated weight errors ${\tilde{W}}_{v} := W_{v} - W_{v}^{⋆}$ , ${\overset{\leftarrow}{W}}_{w} := W_{w} - W_{w}^{⋆}$ , $‖ \cdot ‖$ is the Euclidean second norm and $η > 0$ is selected as the design constant. The following lemma will validate the proposed Lyapunov function.

Lemma 1:

Under the proposed control law, the tracking errors $e_{x}, e_{y}$ , and $e_{θ}$ are uniformly ultimately bounded (UUB), and the closed-loop system is stable in the sense of Lyapunov.

Proof:

Taking the derivative of $V_{1}$ and substituting the controller and update laws yields:

\begin{aligned} {\dot{V}}_{1} = e^{⊤} \dot{e} + \frac{1}{η} {\tilde{W}}_{v}^{⊤} {\dot{W}}_{v} + \frac{1}{η} {\tilde{W}}_{w}^{⊤} {\dot{W}}_{w} \end{aligned}

(47)

The adaptation law cancels the approximation residuals under ideal online learning; the derivative is negative semi-definite:

\begin{aligned} {\dot{V}}_{1} \leq - k_{1} e_{x}^{2} - k_{2} e_{y}^{2} - k_{3} \sin^{2} (e_{θ}) \end{aligned}

(48)

Thus, $e_{x}, e_{y}, e_{θ} \to 0$ as $t \to \infty$ are within a bounded region.

4.2 Dynamic Control Stability Analysis

The following lemma summarizes the stability condition and its proof.

Lemma 2:

Global asymptotic stability by an IT2AFPID controller-based velocity tracking controller can be guaranteed. Based on the dynamics of Equation (28), $M_{1} \in R^{2 \times 2}$ is the positive-definite inertia matrix, $v \in R^{2}$ is the actual velocity, and $τ \in R^{2}$ is the control torque presented in Equation (26), with $u_{f} \in R^{2}$ Represent the output of the proposed controller. Then, if the fuzzy control law satisfies:

\begin{aligned} ‖ u_{f} ‖ \geq \frac{μ_{1}}{2 λ_{min} (M_{1})} ‖ e_{v} ‖ \end{aligned}

(49)

and

‖ {\dot{M}}_{1} ‖ \leq μ_{1}

for a known scalar value

μ_{1} > 0

λ_{min} as minimium egain value of M_{1},

The closed-loop system is globally asymptotically stable. That is,

e_{v} \to 0

t \to \infty

Proof:

Based velocity error of Equation (25), and substituting Equation (26) into Equation (28) results in:

\begin{aligned} \dot{v} = M_{1}^{- 1} (q) (\begin{matrix} \bar{B} (q) \cdot \bar{B} {(q)}^{- 1} (M_{2} (q) v + \bar{C} (q, \dot{q}) + {\bar{τ}}_{d} + M_{1} (q) u_{f}) \\ - M_{2} (q) v - \bar{C} (q, \dot{q}) - {\bar{τ}}_{d} \end{matrix}) \end{aligned}

(50)

Then

\begin{aligned} \dot{v} \approx M_{1}^{- 1} (q) (M_{2} (q) v + \bar{C} (q, \dot{q}) + {\bar{τ}}_{d} + M_{1} u_{f} - M_{2} (q) v - \bar{C} (q, \dot{q}) - {\bar{τ}}_{d}) = u_{f} \end{aligned}

(51)

Therefore, the derivation of velocity error dynamics:

\begin{aligned} {\dot{e}}_{v} = \dot{v} - {\dot{v}}_{c} = u_{f} - {\dot{v}}_{c} \end{aligned}

(52)

However, for stability analysis and the assumption of the desired velocity $v_{c}$ It is slowly varying; the following approximation is assumed:

\begin{aligned} {\dot{e}}_{v} \approx u_{f} \end{aligned}

(53)

Now, the candidate Lyapunov function is:

\begin{aligned} V_{2} = \frac{1}{2} e_{v}^{⊤} M_{1} e_{v} \end{aligned}

(54)

Since, $M_{1}$ and $V_{2}$ are positive definite and radially unbounded, then the computed time derivative is:

\begin{aligned} {\dot{V}}_{2} = \frac{d}{d t} (\frac{1}{2} e_{v}^{⊤} M_{1} e_{v}) = e_{v}^{⊤} M_{1} {\dot{e}}_{v} + \frac{1}{2} e_{v}^{⊤} {\dot{M}}_{1} e_{v} \end{aligned}

(55)

Substitute ${\dot{e}}_{v} = - u_{f}$ , then:

\begin{aligned} {\dot{V}}_{2} = - e_{v}^{⊤} M_{1} u_{f} + \frac{1}{2} e_{v}^{⊤} {\dot{M}}_{1} e_{v} . \end{aligned}

(56)

By using norm inequalities, the result is:

\begin{aligned} {\dot{V}}_{2} \leq - λ_{min} (M_{1}) ‖ e_{v} ‖ \cdot ‖ u_{f} ‖ + \frac{1}{2} ‖ {\dot{M}}_{1} ‖ \cdot {‖ e_{v} ‖}^{2} \end{aligned}

(57)

Under the assumption of $‖ {\dot{M}}_{1} ‖ \leq μ_{1}$ , then:

\begin{aligned} {\dot{V}}_{2} \leq - λ_{min} (M_{1}) ‖ e_{v} ‖ \cdot ‖ u_{f} ‖ + \frac{μ_{1}}{2} {‖ e_{v} ‖}^{2} \end{aligned}

(58)

Then, for ensuring ${\dot{V}}_{2} \leq 0$ , impose:

\begin{aligned} ‖ u_{f} ‖ \geq \frac{μ_{1}}{2 λ_{min} (M_{1})} ‖ e_{v} ‖ \end{aligned}

(59)

led to:

\begin{aligned} {\dot{V}}_{2} \leq - (λ_{min} (M_{1}) ‖ e_{v} ‖ \cdot ‖ u_{f} ‖ - \frac{μ_{1}}{2} {‖ e_{v} ‖}^{2}) \leq 0 \end{aligned}

(60)

Then, ${\dot{V}}_{2} = 0$ only when $e_{v} = 0$ , based on LaSalle's Invariance Principle, ensures all trajectories converge to the origin as $lim_{t \to \infty} e_{v} (t) = 0$ . Therefore, the velocity tracking error will converge asymptotically and the origin. $e_{v} = 0$ It is proven globally asymptotically stable.

4.3 Formation Controller Stability Analysis

Lemma 3 provides asymptotic stability of the formation tracking error system.

Lemma 3:

Consider the error dynamics of Equation (45) with $ρ_{1}, ρ_{2}, ρ_{3} > 0$ Being a positive controller gain. Define the formation error vector as $E_{i} = [E_{i x}, E_{i y}, E_{i θ}]^{⊤}$ . Then, under the control law Equation (42), the origin $E_{i} = 0$ is globally asymptotically stable.

Proof:

The proposed positive definite Lyapunov function is:

\begin{aligned} V_{3} = \frac{1}{2} E_{i}^{⊤} P E_{i} \end{aligned}

(61)

where

P \in R^{3 \times 3}

It is a symmetric positive definite matrix. For simplicity, take

P = I_{3 \times 3}

, so:

\begin{aligned} V_{3} = \frac{1}{2} E_{i x}^{2} + \frac{1}{2} E_{i y}^{2} + \frac{1}{2} E_{i θ}^{2} \end{aligned}

(62)

Taking the time derivative as:

\begin{aligned} {\dot{V}}_{3} = E_{i}^{⊤} {\dot{E}}_{i} \end{aligned}

(63)

And substituting Equation (44), then:

\begin{aligned} {\dot{V}}_{3} & = [\begin{array}{ccc} E_{i x} & E_{i y} & E_{i θ} \end{array}] [\begin{array}{ccc} - ρ_{1} & ω_{i} & 0 \\ - ω_{i} & - \frac{ρ_{2}}{a} & - \frac{ρ_{3}}{a} \\ 0 & \frac{ρ_{2}}{a} & \frac{ρ_{3}}{a} \end{array}] [\begin{array}{cc} E_{i x} \\ E_{i y} \\ E_{i θ} \end{array}] \\ = - ρ_{1} E_{i x}^{2} - \frac{ρ_{2}}{a} E_{i y}^{2} + \frac{ρ_{3}}{a} E_{i y} E_{i θ} - \frac{ρ_{3}}{a} E_{i y} E_{i θ} \end{aligned}

(64)

\begin{aligned} = - ρ_{1} E_{i x}^{2} - \frac{ρ_{2}}{a} E_{i y}^{2} \end{aligned}

(65)

This derivative is negative semi-definite since $ρ_{1} > 0$ and $ρ_{2} > 0$ , and ${\dot{V}}_{3} \leq 0$ . This implies that $V_{3} (E_{i} (t))$ is non-increasing over time, i.e., $E_{i} (t)$ Remains bounded. To conclude asymptotic stability, LaSalle's Invariance Principle is invoked. The set when ${\dot{V}}_{3} = 0$ corresponds to:

\begin{aligned} E_{i x} = 0, E_{i y} = 0 \end{aligned}

(66)

Substituting equation (65) into the error dynamics, then:

\begin{aligned} {\dot{E}}_{i θ} = \frac{ρ_{2}}{a} E_{i y} + \frac{ρ_{3}}{a} E_{i θ} = \frac{ρ_{3}}{a} E_{i θ} \end{aligned}

(67)

Since $E_{i y} = 0$ , then ${\dot{E}}_{i θ} = \frac{ρ_{3}}{a} E_{i θ}$ . For $\dot{V} = 0$ , and ${\dot{E}}_{i θ} = 0$ , implies $E_{i θ} = 0$ . Therefore, the largest invariant set where ${\dot{V}}_{3} = 0$ is:

\begin{aligned} {E_{i} \in R^{3} : E_{i x} = 0, E_{i y} = 0, E_{i θ} = 0} \end{aligned}

(68)

By LaSalle's Invariance Principle, the origin $E_{i} = 0$ is globally asymptotically stable.

5. Secretary Bird Optimization Algorithm (SBOA)

The SBOA is a nature-inspired schema that involves the hunting mechanism of secretary birds, designed by Fu et al. (2024). SBOA iteratively improves optimal solutions by alternating between exploration (jumping attacks) and exploitation (walking and stalking). The following are the details of the algorithm's working. (Fu et al., 2024)

1. Initialization: Initialize population N For candidate solutions (agents), each population has its position vector in the D -dimensional search space:

\begin{aligned} X_{i} = [x_{i 1}, x_{i 2}, \dots, x_{i D}], i = 1, \dots, N . \end{aligned}

(69)

where N is the Population size, bounded initialization for each design variable (gain)

x_{i j}

Is randomly as follows:

\begin{aligned} x_{i j} = U (x_{j}^{min}, x_{j}^{max}), j = 1, \dots, D . \end{aligned}

(70)

where D is the dimension of the design variables.

2. Evaluate Fitness: Calculate the designed objective function $f (X_{i})$ for each candidate i. Which represents the current best solution $X_{best}$ :

\begin{aligned} X_{best} = \arg min_{X_{i}} f (X_{i}) \end{aligned}

(71)

3. Iterative Position Update: In each iteration ( $t = 1, 2, \dots, T$ ) and for each agent i, update positions:

Exploration scenario (Jumping Attack): This mechanism is used to explore the search space with large jumps to avoid falls in local minima:

\begin{aligned} X_{i} (t + 1) = X_{i} (t) + r_{1} \cdot (X_{best} (t) - X_{i} (t)) + r_{2} \cdot S \end{aligned}

(72)

where

r_{1}, r_{2} = U (0, 1)

Represent uniformly random variables. S is a random vector; its objective is to provide a sudden jump, often drawn based on a uniform or Gaussian distribution, to reach diversity. Exploitation scenario (Walking and Stalking): This mechanism updates the solution by fine movements toward the best-obtained position:

\begin{aligned} X_{i} (t + 1) = X_{i} (t) + a (t) \cdot (X_{best} (t) - X_{i} (t)) \end{aligned}

(73)

and

a (t)

Is computed as follows:

\begin{aligned} a (t) = 2 (1 - \frac{t}{T_{max}}) . \end{aligned}

(74)

It is used to control the step size, minimizing linearly from 2 to 0 for balancing in the exploration and exploitation, and $T_{max}$ is the maximum iteration number.

4. Boundary Control: The next step after position is to update components. This guarantees that each component conforms search space limits:

\begin{aligned} x_{i j} (t + 1) = {\begin{array}{cc} x_{j}^{min} & if x_{i j} (t + 1) < x_{j}^{min} \\ x_{j}^{max} & if x_{i j} (t + 1) > x_{j}^{max} \\ x_{i j} (t + 1) & otherwise \end{array} \end{aligned}

(75)

5. Fitness Re-evaluation: Evaluate $f (X_{i} (t + 1))$ . To check if any agent reaches the best solution, update:

\begin{aligned} X_{b e s t} (t + 1) = \arg min_{X_{i}} f (X_{i} (t + 1)) \end{aligned}

(76)

6. Termination: Repeat steps 3 to 5 until the maximum iteration $T$ The reached or convergence limit is met.

The SBOA, which outperforms in the minimization objective function over 30 optimization algorithms using various penalty functions, is used to tune the leader's kinematic controller gains based on the designed objective function $J_{1} (k_{1}, k_{2}, k_{3})$ . This function represents the sum of the root mean square (RMS) kinematic tracking errors and the RMS dynamic velocity-tracking errors of the leader robot (Kamil et al., 2020).

Let $e_{x} (t), e_{y} (t)$ , and $e_{θ} (t)$ denote the leader's kinematic tracking errors. Equation (25) defines the linear and angular velocity-tracking errors, where $v_{c} (t)$ and $ω_{c} (t)$ are the commanded velocities, and $v (t)$ and $ω (t)$ are the actual dynamic outputs. The RMSEs are defined as (Karam et al., 2018):

\begin{aligned} RM S_{x, y, z} = \sqrt{\frac{1}{T} \int_{0}^{T} e_{x y z}^{2} (t) d t}; RM S_{v, w} = \sqrt{\frac{1}{T} \int_{0}^{T} e_{v w}^{2} (t) d t} \end{aligned}

(77)

$T$ denotes the maximum iteration number. Thus, the objective function is given by:

\begin{aligned} J_{1} (k_{1}, k_{2}, k_{3}) = RM S_{x, y, z} + RM S_{x, y, z} \end{aligned}

(78)

According to the kinematic-level formation controller, the objective function is based on the i^th follower by finding the optimal gains. $J_{2} (ρ_{1}, ρ_{2}, ρ_{3})$ defined in Equation (45), minimizes the kinematic-level error in Equation (39) and the dynamic-level error in Equation (25).

\begin{aligned} RM S_{i, x, y, z} = \sqrt{\frac{1}{T} \int_{0}^{T} E_{i, x y z}^{2} (t) d t}, RM S_{i, v, w} = \sqrt{\frac{1}{T} \int_{0}^{T} e_{i, v w}^{2} (t) d t} \end{aligned}

(79)

6. Simulation Results and Analysis

Two trajectory scenarios—star-shaped path and sinusoidal path—are used to evaluate the controllers under sharp-turn conditions that test controller performance. The controllers must optimally adjust each robot's linear and angular velocities to minimize path tracking errors, dynamic deviations, and formation discrepancies. The star-shaped path is mathematically defined as follows:

\begin{aligned} x_{r} (t) = F . (Z + \cos (36 t / Y)) . \cos (6 t / Y), y_{r} (t) = F . (Z + \cos (36 t / Y)) . s i n 6 t / Y \end{aligned}

(80)

where F is the radius of the trajectory, Y is the time scaling parameter, and Z is the trajectory-scaling factor. For the sinusoidal path,

\begin{aligned} x_{r} (t) = t, y_{r} (t) = n \times \sin (t / f) \end{aligned}

(81)

where n is the amplitude and f represents the frequency-scaling factor. The differential-drive mobile robot discussed in Section 3.2 uses the following dynamic parameters:

M = 0.9 k g, I = 0.0034 k g . m^{2}, r = 0.027 m, 2 b = 0.1191,

and

a = 0.028

(Wang et al., 2023). The optimal controller gains are obtained using SBOA based on the minimized error for each controller. For the kinematic-level leader, the adaptive RBFNN is configured with 9 RBF nodes, Gaussian RBFs with a width of 0.3, and a learning/adaptation rate of 0.001. For the dynamic-level IT2AFPID controller, Table 3 presents the used and initial parameters.

Table 3.

Parameters of IT2AFPID Controller.

Parameter	Description	Value
$n_{Vars}$	Number of control channels	6
$n_{MF}$	Number of membership functions per variable	9
$σ_{0}$	Initial standard deviation matrix	$1.5$
$σ_{u}$	Upper bound standard deviation matrix	$1.5 \times I_{6 \times 9}$
$σ_{l}$	Lower bound standard deviation matrix	$1.2 \times I_{6 \times 9}$
$σ$	Lower bound for the updated sigma	0.05
C	Centers of Gaussian MFs (shared across all vars)	Linspace $(- 5, 5, 9)$
$k_{upper}$	Upper bound gain for adaptive $σ_{u}$	0.4
$k_{lower}$	Lower bound gain for adaptive $σ_{l}$	0.5
$α_{c}$	Learning rate for center adaptation	0.01
$α_{σ}$	Learning rate for sigma adaptation	0.01
$O$	Rule output range vector	Linspace (0.9, 3, 9)

The parameters of SBOA are as follows: population size (30–50), number of iterations (10–50), flight scaling factor (0.1–0.5), and Levy exponent of 1.5. Based on these settings, the optimal control gains based on SBOA for star and sinusoidal trajectories are: ( $k_{1}, k_{2}, k_{3}, ρ_{1}, ρ_{2}, ρ_{3}$ )=(16.3222, 7.3958, 12.1080, 2.6493, 2.1684, 11.0255). The behavior of the objective function for SBOA is depicted in Figure 5.

Figure 5.

Objective functions minimization by SOBA.

The desired formation is defined by a bearing angle $ψ_{i}^{d} = \pm 90^{\circ}$ and a separation distance $λ_{i}^{d} = 5 cm$ , with initial conditions as: [x, y, $θ$ ] = [8; −4; $90 \circ$ ], [6; −3; $60 \circ$ ] and [5; −3; $90 \circ$ ] for the leader and followers. The star-path tracking results, errors along each axis, control torques, and corresponding velocities are shown in Figures 6–9, respectively, while Figure 10 illustrates the convergence of the velocity-tracking error.

Figure 6.

Star path formation control.

Figure 7.

Error in axes for each robot-based formation.

Figure 8.

Controlling torque signals for each robot-based formation.

Figure 9.

Velocities for each robot-based formation.

Figure 10.

Velocities error $v_{c} - v$ for each robot-based formation.

The tracking results for the second sinusoidal path, errors along each axis, control torques, and corresponding velocities are shown in Figures 11–14, respectively, while Figure 15 illustrates the convergence of $v_{c} - v$ error to zero.

Figure 11.

Sinusoidal path formation control.

Figure 12.

Error in axes for each robot- sinusoidal path.

Figure 13.

Controlling torque signals for each robot- sinusoidal path.

Figure 14.

Velocities for each robot- sinusoidal path.

Figure 15.

Velocities error $v_{c} - v$ for each robot- sinusoidal path.

The abovementioned figures demonstrate that the tracking error for the leader relative to the reference path, as well as the formation-tracking error for the followers relative to the leader, converges to zero. Table 4 presents the RMS achieved by the proposed controllers compared to previous works using the same dynamic model for formation control-based leader–follower robots.

Table 4.

Leader-Follower Formation Results.

	Leader tracking RMS error			Follower 1/ Follower 2 formation tracking RMS error
References	$e_{x}$ (m)	$e_{y}$ (m)	$e_{θ}$ (red)	$e_{x}$ (m)	$e_{y}$ (m)	$e_{θ}$ (red)
Ref. (Wang et al., 2023)	15×10⁻⁵	5×10⁻⁵	4×10⁻⁵	0.0031	0.0119	0.0122
Ref. (Wang et al., 2023)	15×10⁻⁵	5×10⁻⁵	4×10⁻⁵	−	−	−
Ref. (Velasco-Villa et al., 2024)	−	−	−	0.015	0.04	−
Proposed controllers -first trajectory	3.4×10⁻⁵	1.8×10⁻⁵	2.8×10⁻⁵	0.0022	0.0096	0.0019
Proposed controllers -first trajectory	3.4×10⁻⁵	1.8×10⁻⁵	2.8×10⁻⁵	0.0003	0.0102	0.0111
Proposed controllers -second trajectory	2×10⁻⁵	7×10⁻⁵	1.2×10⁻⁵	8×10⁻⁵	0.0016	0.0003
Proposed controllers -second trajectory	2×10⁻⁵	7×10⁻⁵	1.2×10⁻⁵	2×10⁻⁵	0.0035	0.0047

Figure 16.

First path tracking under robot mass uncertainty of 65%.

Figure 17.

Axes error under robot mass uncertainty.

Figure 18.

First path tracking under robot torque disturbance by 42.222%.

Figure 19.

Torque control signal response under disturbance by 36.222%.

Figure 20.

Tracking under both uncertainty and disturbance.

Figure 21.

Axes error under both uncertainty and disturbance.

The work cited in reference 19 adopts feedback linearization controllers, which results in a linearized model and light linear control. A second study, which also used the same robot parameters, controlled the formation using real-time distance measurements. By contrast, our framework employs a geometric nonlinear controller for the followers and enhances the leader's kinematic controller with an online RBFNN for residual compensation of velocity errors. At the dynamic level, each robot utilizes an IT2AFPID in feedback to address model nonlinearities, parametric uncertainty, and disturbances in torque control signals. Using identical robot parameters, initial conditions, and desired formation, our intelligent, adaptive design achieves superior tracking performance and smoother control effort, as summarized in Table 4.

The proposed control framework exhibits significant improvements in path tracking accuracy, with leader position errors reduced by 77.33%, 64%, and 30% along the x, y, and θ axes, respectively. Followers also show notable enhancements, with follower 1 achieving up to 84.42% improvement in orientation tracking and follower 2 showing a 90.03% reduction in x-axis error. Compared with prior approaches, these results highlight smoother torque inputs and well-profiled velocity generation without overshoot or oscillations, confirming the effectiveness of the IT2AFPID controller in minimizing dynamic velocity errors.

Robustness tests involving mass uncertainties and control signal disturbances applied during 20 s to 30 s of a 65 s trial were conducted to further validate the resilience of the controller. Specifically, a 65% mass uncertainty coupled with a 42.67% torque disturbance and combined disturbances (55% mass uncertainty, 36.22% torque disturbance) were introduced. Despite these perturbations, the tracking performance and axis errors remained within acceptable bounds, as illustrated in Figures 16 and 17. These findings confirm the controller's ability to sustain formation accuracy under high uncertainties and abundant external perturbations.

Figure 18 depicts the tracking performance under torque signal disturbance, while Figure 19 displays the control torque signal response.

The case involving both effects simultaneously is displayed in Figure 20, while the axis-wise errors is are depicted in Figure 21.

Table 5 summarizes the RMSEs along the axes for the leader and followers based on the aforementioned three cases.

Table 5.

Leader-Follower Formation Results Under Uncertainty and Disturbances.

	Leader tracking RMS error			Follower 1/ Follower 2 formation tracking RMS error
Cases	$e_{x}$ (m)	$e_{y}$ (m)	$e_{θ}$ (red)	$e_{x}$ (m)	$e_{y}$ (m)	$e_{θ}$ (red)
Uncertainty	1.8×10⁻⁵	5.2×10⁻⁵	4.4×10⁻⁵	0.0022	0.0096	0.0019
Uncertainty	1.8×10⁻⁵	5.2×10⁻⁵	4.4×10⁻⁵	0.0003	0.0102	0.0111
Disturbances	1.4×10⁻⁵	2.3×10⁻⁵	5.1×10⁻⁵	0.0022	0.0096	0.0019
Disturbances	1.4×10⁻⁵	2.3×10⁻⁵	5.1×10⁻⁵	0.0003	0.0102	0.0111
Both	2×10⁻⁵	5.3×10⁻⁵	4.3×10⁻⁵	0.0022	0.0096	0.0019
Both	2×10⁻⁵	5.3×10⁻⁵	4.3×10⁻⁵	0.0003	0.0102	0.0111

Under the uncertainty and disturbance cases, the leader's RMS tracking error remained essentially zero, while the axis-wise RMSEs of the followers in formation tracking show only minor variation. When compared with the references listed in Table 4, the proposed controllers exhibited strong robustness. Specifically, the RMSEs remained nearly invariant despite the introduction of a 55% mass uncertainty and a 36.22% torque disturbance. This resilience is attributed to the leader's robust tracking law, which incorporates online velocity-error compensation by RBFNN, and the real-time nonlinear IT2AFPID dynamic controller, which adapts effectively to model uncertainty and external disturbances.

7. Conclusion

This study developed a comprehensive leader–follower formation control framework featuring a kinematic-level RBFNN-enhanced tracking controller for the leader and a novel IT2AFPID dynamic controller for all robots. The IT2AFPID leverages adaptive Gaussian MFs and refined uncertainty handling, operating in parallel across the three robots through a vectorized structure. Formation control for the followers utilizes PFL, with gains tuned via the SBOA. When tested on star-shaped and sinusoidal trajectories involving sharp maneuvers, the controllers achieved substantial improvements over prior works. Specifically, the leader's RMSE decreased by 77.33%, 64%, and 30% along the x, y, and θ axes, respectively. Moreover, follower 1 exhibited enhancements of 29.03%, 19.32%, and 84.42%, while follower 2 showed enhancements of 90.03%, 14.28%, and 9.01% along the same axes. Collectively, these improvements translate to an average tracking-accuracy gain of over 57% for the leader robot, and formation-tracking enhancements of 44.25% and 37.77% for followers 1 and 2, respectively. The framework also demonstrated robustness under significant uncertainties, tolerating up to 65% mass variation and 42.67% torque disturbance individually and maintaining formation performance under combined uncertainties of 55% mass and 36.22% torque disturbance.

Footnotes

ORCID iDs

Zeyad A Karam

Mohammed Y Hassan

Amjad J Humaidi

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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