Dynamic Cooperative Control Method for Highly Maneuverable Unmanned Vehicle Formations Based on Adaptive Multi-Mode Steering

Abstract

Traditional front-wheel-steering (FWS) unmanned vehicles frequently encounter maneuverability bottlenecks in confined spaces or during rapid formation changes due to inherent kinematic limitations. To mitigate these constraints, this study proposes an adaptive multi-mode (AMM) cooperative formation control framework tailored for four-wheel independent drive and steering (4WIDS) platforms. The methodology constructs a unified planner based on the virtual structure concept, integrated with an autonomous steering-mode selector. By synthesizing real-time mission requirements with longitudinal and lateral tracking errors, the system dynamically switches between crab steering, four-wheel counter-steering (4WCS), and conventional FWS modes to optimize spatial utilization. Validated within a seven-vehicle MATLAB/Simulink environment, simulation results demonstrate that the crab-steering mode significantly reduces relocation time for small lateral adjustments by eliminating redundant heading changes, whereas FWS and 4WCS modes are preferentially selected to ensure stability during high-speed or large-span maneuvers. These findings confirm that the proposed AMM strategy effectively reconciles the trade-off between agility and stability, providing a robust solution for complex cooperative maneuvering tasks.

1. Introduction

Unmanned ground vehicles (UGVs) are increasingly deployed across diverse sectors, ranging from logistics and environmental monitoring to emergency response and defense. While single-vehicle operations have matured, cooperative multi-vehicle platooning has emerged as a critical focal point in autonomous systems research due to its superior potential for task allocation, robustness, and energy efficiency [1,2].

Current literature on formation control typically categorizes the field into navigation, control strategies, and communication architectures [3,4]. In the domain of integrated navigation, recent work by Saikin et al. [5] and Buznikov et al. [6] focused on improving state estimation under variable environmental conditions, employing a fusion of odometry, vision, and radar to mitigate stability issues caused by path deviations. Regarding control strategies, approaches vary significantly; Jond and Plato [7] applied differential game theory to model interactions with non-cooperative vehicles, whereas Boumbarov [8] demonstrated a nonlinear adaptive controller capable of maintaining stable platooning using solely relative pose measurements, thereby removing reliance on vehicle-to-infrastructure communication. Furthermore, Gao et al. [9] successfully combined distributed graph theory with artificial potential fields to enhance obstacle avoidance in multi-lane formations.

Recent studies have also explored advanced control strategies specifically for over-actuated vehicles. For instance, Sun et al. [10] proposed a neural-network-based adaptive Model Predictive Control (MPC) framework that utilizes phase plane analysis to improve path tracking stability for 4WID vehicles under dynamic conditions. Similarly, Qiao et al. [11] developed a coordinated control strategy for 4WID-4WIS electric vehicles, optimizing trajectory tracking accuracy while simultaneously considering energy consumption economy. Additionally, Zhang et al. [12] and Zhu et al. [13] investigated stability control and versatile steering modes for distributed drive vehicles, highlighting the potential of multi-mode actuation. Parallel advancements have been made in system architecture. For example, Omefe [14] introduced a framework integrating V2X communication with AI-based decision modules to facilitate complex maneuvers like merging in mixed traffic. Li et al. [15] further elaborated on platoon control under V2X communications. Similarly, Stoll et al. [16] utilized UWB ranging coupled with vehicle dynamics modeling to achieve high-precision path replication independent of GPS. Regarding formation switching, Yu et al. [17] investigated distributed adaptive formation control with collision avoidance capabilities, and Farooq et al. [18] provided a comprehensive review on recent advancements in formation control under kinematic constraints. Moreover, Yan et al. [19] proposed a reinforcement learning-based approach for a relatively distributed formation. Additionally, the development of validation platforms, such as ConvoyNext [20] and OpenConvoy [21], alongside theoretical explorations using coalition game theory [22], has significantly standardized the benchmarking of convoy algorithms. Research on heterogeneous ground–air formations by Peng et al. [23] and wireless power transfer systems by Chang et al. [24] also contributes to the broader context of multi-agent coordination.

However, a prevalent limitation in these frameworks is their continued reliance on traditional front-wheel steering (FWS) models. The kinematic constraints inherent to FWS—specifically the minimum turning radius—severely restrict maneuverability in confined spaces or during rapid formation reconfigurations [25]. For instance, transitioning from a line to a column formation often forces FWS vehicles to execute wide turning paths. This not only increases the spatial footprint of the maneuver but also prolongs the reconfiguration time, rendering FWS platoons less effective for missions requiring high agility.

To mitigate these kinematic bottlenecks, platforms equipped with four-wheel independent drive and steering (4WIDS) offer a compelling alternative [26]. Our research group has previously engineered a 4WIDS platform capable of diverse steering modes, including front-wheel (FWS), rear-wheel (RWS), four-wheel counter-steering (4WCS), and crab steering. Related works by Hu et al. [27], Hang et al. [28], Li et al. [29], and others [30,31] have extensively studied path-tracking and stability control for such vehicles. Despite these hardware advancements, most existing approaches still optimize performance within a single mode. The coordination of such kinematic flexibility at the formation level—specifically regarding dynamic mode switching and task allocation based on maneuver difficulty—remains underexplored.

This paper presents a cooperative formation-control strategy designed explicitly to leverage multi-mode steering for UGV platoons. By integrating high-level planning with an autonomous steering-mode selection mechanism, the proposed method enables 4WIDS vehicles to execute transitions with efficiency superior to standard FWS counterparts. The primary contributions are as follows:

• A unified formation-planning framework based on the virtual-structure method is established. Anchored to the lead vehicle, this approach ensures consistent, decoupled control for all followers.
• An adaptive steering-mode selector is introduced. This mechanism autonomously determines the optimal steering mode by analyzing real-time vehicle states and the specific requirements of the assigned formation transition.
• Validation via MATLAB/Simulink simulations, which demonstrates that the proposed strategy maintains high tracking accuracy while significantly reducing both maneuver time and spatial occupation during complex tasks (e.g., column-to-line transitions) compared to conventional FWS methods [21].

Organization of the Paper: The remainder of this paper is organized as follows: Section 2 establishes the kinematic models for the 4WIDS platform. Section 3 details the proposed Adaptive Multi-Mode (AMM) cooperative control framework, including the virtual structure planner and the task-graded mode selector. Section 4 describes the simulation platform setup and scenario design, followed by a comprehensive analysis of the experimental results compared with the baseline method. Finally, Section 5 concludes the paper and discusses potential future research directions.

2. Structural and Control System Modeling for Autonomous Vehicles

2.1. Architecture of Autonomous Vehicle Platforms and Definition of Coordinate Systems

The experimental platform utilizes a four-wheel independent drive and steering (4WIDS) architecture to maximize maneuverability, as illustrated in Figure 1. The system is defined by four integrated corner modules, each housing a dedicated in-wheel motor and steering actuator. This configuration decouples the driving torque and steering angle for each wheel, facilitating not only standard Ackermann geometry but also advanced kinematic modes such as crab steering, lateral translation, and zero-radius rotation. Furthermore, the chassis is instrumented with a multi-modal sensor suite—comprising front/rear LiDARs and high-resolution cameras—to enable robust environmental perception and cooperative localization.

The vehicle’s communication architecture is founded on the Controller Area Network (CAN) standard, utilizing the CANopen application layer to streamline device interoperability. As illustrated in Figure 2, the Vehicle Control Unit (VCU) functions as the central node. To address the rigorous safety requirements of the powertrain and steering subsystems, the design incorporates a dual-redundant hybrid topology. This configuration overlays direct VCU-to-node connections (Star) with a secondary device-level ring network. Such redundancy ensures signal integrity and deterministic command transmission, effectively mitigating risks associated with link failures or high-frequency electromagnetic interference.

To precisely describe the vehicle’s motion state, this paper establishes the coordinate systems shown in Figure 3. Among these, O-XY represents the fixed global world coordinate system (World Frame). Its origin O is arbitrarily designated, with the X-axis defined as the horizontal direction and the Y-axis defined as the vertical direction, adhering to the right-hand rule. o-xy represents the vehicle coordinate system (Vehicle Frame) that moves with the vehicle. Its origin o is rigidly fixed to the vehicle’s geometric center. The x-axis aligns with the vehicle’s longitudinal symmetry axis pointing forward, while the y-axis points toward the vehicle’s left side. The planar pose of a vehicle in the world coordinate system is described by the vector P = [x, y, θ]T, where x and y are the coordinates of the origin o of the vehicle coordinate system in the world coordinate system, and θ is the vehicle’s yaw angle. This angle is defined as the angle between the x-axis of the vehicle coordinate system and the X-axis of the world coordinate system, with counterclockwise direction being positive.

2.2. Multi-Mode Steering Vehicle Kinematic Model

To enhance vehicle handling and steering efficiency, this study incorporates the Ackermann steering principle into the steering control system of the autonomous vehicle. Based on this principle, we have implemented four Ackermann steering modes as shown in Figure 4. Since the first three modes exhibit high consistency in control logic and theoretical derivation, this paper will focus on detailing the implementation methods for front-wheel Ackermann steering and four-wheel reverse Ackermann steering.

2.2.1. Front-Wheel Ackermann Steering

Front-wheel Ackermann steering is the most common type of Ackermann steering, as illustrated in Figure 5.

To achieve Ackermann steering at the front wheels, the turning centers of the inner and outer wheels must coincide. This yields the following geometric relationship:

$$cot{\beta }_1 -cot{\alpha }_1={\displaystyle \frac{K}{L}}$$

(1)

When the steering angle α₁ of the inner wheel is known, the Ackermann angle β₁ of the outer wheel can be obtained as shown in (2):

$${\beta }_1=\arctan \left({\displaystyle \frac{L\cdot \tan {\alpha }_1}{L+K\cdot \tan {\alpha }_1}}\right)$$

(2)

2.2.2. Four-Wheel Reverse Ackermann Steering

Four-wheel reverse Ackermann steering represents the most complex form of Ackermann steering, yet it enables the smallest turning radius. Under identical wheel angle conditions, its turning radius can be reduced by approximately 0.5L (where L denotes the vehicle’s wheelbase) compared to front-wheel Ackermann steering. Therefore, adopting four-wheel reverse Ackermann steering holds significant importance for enhancing vehicle maneuverability. Four-wheel reverse Ackermann steering is illustrated in Figure 6:

Based on the geometric relationships shown in Figure 6 during a vehicle’s left turn, the following equation can be derived:

$${\displaystyle \frac{L_a}{\tan {\delta }_f}}={\displaystyle \frac{L_b}{\tan {\delta }_r}}$$

(3)

$$L=L_a+L_b$$

(4)

In Equation (3), L_a denotes the vertical distance from the instantaneous pivot center during turning to the front axle, and L_b denotes the vertical distance from the instantaneous pivot center during turning to the rear axle. Solving Equations (3) and (4) simultaneously yields

$$\left\{\begin{array}{l}{L}_a={\displaystyle \frac{L\tan {\delta }_f}{\tan {\delta }_f+\tan {\delta }_r}}\\ L_b={\displaystyle \frac{L\tan {\delta }_r}{\tan {\delta }_f+\tan {\delta }_r}}\end{array}\right.$$

(5)

Based on the geometric relationships in Figure 6, the expression for the Ackermann angle of each wheel during four-wheel reverse Ackermann steering can be derived (6):

$$\left\{\begin{array}{l}{\delta }_{fl}=\mathrm{arccot}(\cot {\delta }_f-{\displaystyle \frac{K}{2L_a}})\\ {\delta }_{fr}=\mathrm{arccot}(\cot {\delta }_f+{\displaystyle \frac{K}{2L_a}})\\ {\delta }_{rl}=\mathrm{arccot}(\cot {\delta }_r-{\displaystyle \frac{K}{2L_b}})\\ {\delta }_{rr}=\mathrm{arccot}(\cot {\delta }_r+{\displaystyle \frac{K}{2L_b}})\end{array}\right.$$

(6)

Substituting Equation (5) into Equation (6) yields Equation (7):

$$\left\{\begin{array}{l}{\delta }_{fl}=\mathrm{arccot}({\displaystyle \frac{2L-K(\tan {\delta }_f+\tan {\delta }_r)}{2L\tan {\delta }_f}})\\ {\delta }_{fr}=\mathrm{arccot}({\displaystyle \frac{2L+K(\tan {\delta }_f+\tan {\delta }_r)}{2L\tan {\delta }_f}})\\ {\delta }_{rl}=\mathrm{arccot}({\displaystyle \frac{2L-K(\tan {\delta }_f+\tan {\delta }_r)}{2L\tan {\delta }_r}})\\ {\delta }_{rr}=\mathrm{arccot}({\displaystyle \frac{2L+K(\tan {\delta }_f+\tan {\delta }_r)}{2L\tan {\delta }_r}})\end{array}\right.$$

(7)

In practical applications, to facilitate the implementation of four-wheel reverse Ackermann steering, setting the basic steering angles of the front and rear wheels equal results in the Ackermann angles for each wheel in four-wheel reverse Ackermann steering as shown in Formula (8):

$$\left\{\begin{array}{l}{\delta }_{fl}={\delta }_{rl}=\arctan ({\displaystyle \frac{L\tan \delta }{-K\tan \delta +L}})\\ {\delta }_{fr}={\delta }_{rr}=\arctan ({\displaystyle \frac{L\tan \delta }{K\tan \delta +L}})\end{array}\right.$$

(8)

According to Formula (8), when a vehicle employs four-wheel reverse Ackermann steering, the Ackermann angles for each wheel can be calculated by specifying only the basic steering angle δ.

2.2.3. Crab Steering Kinematic Model

Crab steering is a unique high-mobility mode for 4WIDS vehicles, allowing the vehicle to translate along any planar direction while maintaining a constant body heading angle (θ). This mode effectively decouples the vehicle’s path from its posture, which is crucial for the “zero-radius” lateral maneuvers proposed in this study.

To achieve pure translational motion, the instantaneous velocity vectors of all four wheels must be parallel. Consequently, the steering angles of all wheels must satisfy the following coordination relationship:

$${\delta }_{fl}={\delta }_{fr}={\delta }_{rl}={\delta }_{rr}={\delta }_{crab}$$

(9)

In Equation (9), δ_crab represents the crab angle, defined as the angle between the target translation vector and the vehicle’s longitudinal axis.

In this mode, the instantaneous center of rotation (ICR) is located at infinity (R → ∞). Based on the kinematic constraints, the differential equations governing the vehicle’s motion in the global coordinate frame can be simplified as

$$\left\{\begin{array}{l}\dot{X}=v\cdot \cos (\theta +{\delta }_{crab})\hfill \\ \dot{Y}=v\cdot \sin (\theta +{\delta }_{crab})\hfill \\ \dot{\theta }=0\hfill \end{array}\right.$$

(10)

The above equations indicate that the crab steering mode eliminates the coupling of the yaw rate ($\dot{\theta }$), enabling the direct projection of longitudinal velocity v onto the lateral degree of freedom. However, in practical applications, δ_crab is subject to physical actuator limits and stability constraints, typically restricted to |δ_crab| ≤ 0.52 rad (≈30°) to prevent tire saturation and rollover risks at high speeds.

3. Adaptive Multi-Mode Steering Control

To achieve efficient and flexible unmanned vehicle formation maneuvering, this chapter details the proposed integrated control framework. Adopting a hierarchical structure, this framework decomposes complex formation tasks into three levels: formation planning at the top layer, steering mode selection at the middle layer, and trajectory tracking at the bottom layer. Through coordinated operation among these modules, the framework aims to fully leverage the vehicles’ multi-mode steering capabilities, enabling rapid, stable, and space-efficient formation transitions.

3.1. Overall Control Architecture

To implement coordinated formation control, this study establishes a hierarchical architecture (Figure 7) comprising a centralized Global Planning Layer and a distributed Tracking Loop Layer.

The Global Planning Layer resides on the lead vehicle (Vehicle 1). Its core component, the Platoon Planner, synthesizes environmental data (e.g., road width) from the Terrain Sensor with the leader’s real-time kinematic state to derive a set of desired target poses for the entire platoon. Integrated with this planner is the Adaptive Steering Mode Selector. This module evaluates specific maneuver characteristics—such as lateral displacement distance—to autonomously determine the optimal steering mode, thereby enhancing the efficiency and stability of subsequent formation transitions. The calculated target poses are subsequently broadcast to all platoon members via the communication bus.

The Tracking Loop Layer utilizes a distributed deployment strategy, where each autonomous vehicle (including the leader) operates an identical, independent tracking control loop. At this level, the Trajectory Tracker receives the specific target pose assigned by the planning layer and compares it against the actual state feedback from the Vehicle Plant. Based on the computed deviation, the controller solves for the optimal control inputs—specifically longitudinal acceleration and front-wheel steering angle—in real time to drive the actuators. This process completes the closed-loop cycle of perception, decision-making, and execution.

3.2. Formation Planner

The formation planner serves as the core of the global planning layer, primarily responsible for generating a set of desired target poses for all vehicles in the formation based on external mission commands and the real-time pose of the lead vehicle. To ensure geometric consistency of the formation, this study adopts a Virtual Structure approach, treating the entire formation as a single rigid body. The position of each vehicle in the formation (i = 1, …, N) is strictly defined by its desired relative displacement vector $d_i^{ref}={[d{x}_i,d{y}_i]}^T$ relative to the reference point of the virtual structure (in this paper, the lead vehicle).

The formation library studied in this paper contains the following four typical formations to accommodate different mission requirements:

• Linear Formation: Suitable for scenarios requiring distance maintenance and formation integrity, categorized into longitudinal and lateral types.
  Longitudinal Linear Formation:

  $$d_i^{col}={[-(i-1)\cdot L_{col},0]}^T$$

  (11)
  Horizontal Linear Formation:

  $$d_i^{row}={[0,(4-i)\cdot L_{row}]}^T$$

  (12)
  Among these, L_col and L_row represent the preset vertical and horizontal safety distances, respectively.
• V-shape Formation: Used to enhance forward visibility and reduce air resistance, with following vehicles arranged in a V-shape formation behind the lead vehicle.

  $$d_i^v=\left[\begin{array}{c}-{\displaystyle \frac{i-1}{2}}\cdot L_{vx}\\ {(-1)}^i\cdot {\displaystyle \frac{i-1}{2}}\cdot L_{vy}\end{array}\right]\hspace{1em}\mathrm{for} i>1$$

  (13)
  Among these, L_vx and L_vy control the depth and opening angle of the V-shape.
• Rectangular Formation: Suitable for patrolling or surveillance within a specific area, with vehicles arranged in multiple rows and columns to form a grid pattern.

  $$d_i^{rect}=\left[\begin{array}{c}-{\displaystyle \frac{i-1}{2}}\cdot L_{rx}\\ {\displaystyle \frac{L_{ry}}{2}}\cdot {(-1)}^i\end{array}\right]\hspace{1em}\mathrm{for} i>1$$

  (14)
  Among these, L_rx and L_ry define the row spacing and column spacing of the rectangle, respectively.
• Circular Formation: Suitable for area surveillance or specific target monitoring. To ensure the lead vehicle remains at the formation’s forefront for perception and decision-making tasks, this study defines the circular formation as follows: The lead vehicle occupies the apex of an arc, with subsequent vehicles arranged in a semicircle or arc behind it, centered on a virtual point located behind the lead vehicle. The expected relative displacement vector for the i-th follower vehicle (i > 1) is defined as:

  $$d_i^{circ}=\left[\begin{array}{c}-R\cdot (1-\cos ({\alpha }_i))\\ R\cdot \sin ({\alpha }_i)\end{array}\right]\hspace{1em}\mathrm{for} i>1$$

  (15)

  where R is the formation radius, α_i is the angle corresponding to the i-th vehicle on the arc. When vehicles are symmetrically distributed

  $${\alpha }_i=(i-4)\cdot \Delta \alpha$$

  (16)
  Among these, Δα is the preset angular spacing between vehicles.

In each control cycle, the planner first obtains the desired relative displacement vector $d_i^{ref}$ for vehicle i based on the real-time pose PL = [x_L, y_L, θ_L]T of the lead vehicle and the selected formation. Subsequently, through coordinate transformation, the absolute target position ${[x_i^{target},y_i^{target}]}^T$ of this vehicle in the global world coordinate system is calculated:

$$\left[\begin{array}{c}{x}_i^{target}\\ y_i^{target}\end{array}\right]=\left[\begin{array}{c}{x}_L\\ y_L\end{array}\right]+R({\theta }_L)\cdot d_i^{ref}$$

(17)

Among these, R(θ_L) denotes the rotation matrix of the lead vehicle:

$$R({\theta }_L)=\left[\begin{array}{cc}\cos ({\theta }_L)& -\sin ({\theta }_L)\\ \sin ({\theta }_L)& \cos ({\theta }_L)\end{array}\right]$$

(18)

To simplify control objectives, the desired heading angle ${\theta }_i^{target}$ and desired velocity $v_i^{target}$ for all vehicles are set to match the current state of the lead vehicle. Ultimately, the planner outputs the complete target pose $p_i^{target}={[x_i^{target},y_i^{target},{\theta }_i^{target}]}^T$ and target velocity $v_i^{target}$ for each vehicle, which are then distributed to the trajectory tracking controllers of each vehicle via the communication bus.

3.3. Trajectory Tracking Controller

The trajectory tracking controller serves as the core of distributed closed-loop control. Its function is to receive the desired pose $p_i^{target}$ and velocity $v_i^{target}$ from the planning layer and compute the actual physical control commands (a_i, δ_i) to minimize the error between the vehicle’s actual pose P_i and the target pose.

When a vehicle moves in a global coordinate system, its acceleration $(\ddot{x},\ddot{y})$ is simultaneously influenced by both longitudinal acceleration a and yaw angle δ, as shown in Equation (17). This coupling characteristic causes traditional PID controllers to degrade in performance or even oscillate during high-speed or high-curvature maneuvers.

$$\left[\begin{array}{c}\ddot{x}\\ \ddot{y}\end{array}\right]=\left[\begin{array}{c}\cos (\theta )\\ \sin (\theta )\end{array}\right]a+\left[\begin{array}{c}-v\sin (\theta )\\ v\cos (\theta )\end{array}\right]{\displaystyle \frac{v}{L}}\tan (\delta )$$

(19)

To address the aforementioned issues, this paper employs a nonlinear control strategy based on feedback linearization to achieve decoupling of longitudinal (X-direction) and lateral (Y-direction) control.

3.3.1. Controlled Decoupling Transformation

The core idea of this strategy is to define two virtual, decoupled control inputs (U_x, U_y), representing the vehicle’s desired acceleration in the X-direction and Y-direction within the world coordinate system, respectively.

$$\left\{\begin{array}{l}\ddot{x}=u_x\hfill \\ \ddot{y}=u_y\hfill \end{array}\right.$$

(20)

By combining the above equation with the vehicle’s second-order dynamic model, we obtain

$$\left[\begin{array}{c}{u}_x\\ u_y\end{array}\right]=\left[\begin{array}{cc}\cos (\theta )& -v\sin (\theta )\\ \sin (\theta )& v\cos (\theta )\end{array}\right]\left[\begin{array}{c}a\\ \omega \end{array}\right]$$

(21)

Among these, $\omega =\dot{\theta }={\displaystyle \frac{v}{L}}\tan (\delta )$ is the yaw rate of the vehicle.

By solving the inverse of the above simultaneous matrix equations, the decoupling control law from the desired virtual commands (U_x, U_y) to the actual physical commands (a, ω) can be obtained:

$$\left[\begin{array}{c}a\\ \omega \end{array}\right]=\left[\begin{array}{cc}\cos (\theta )& \sin (\theta )\\ -{\displaystyle \frac{\sin (\theta )}{v}}& {\displaystyle \frac{\cos (\theta )}{v}}\end{array}\right]\left[\begin{array}{c}{u}_x\\ u_y\end{array}\right]$$

(22)

From this, we can derive the formula for calculating acceleration and yaw angular velocity:

$$a=u_x\cos (\theta )+u_y\sin (\theta )$$

(23)

$$\omega ={\displaystyle \frac{1}{v}}(-u_x\sin (\theta )+u_y\cos (\theta ))$$

(24)

Finally, based on the kinematic relationship $\omega =(v/L)\tan (\delta )$, the final front wheel steering angle δ is solved for:

$$\delta =\arctan \left({\displaystyle \frac{\omega L}{v}}\right)$$

(25)

3.3.2. Decoupled Control Tracking

Through the above decoupling transformation, a complex nonlinear coupled system is converted into two independent linear systems (second-order integrators in the X and Y directions). Therefore, based on the aforementioned strategy, two completely independent PD controllers can be designed to calculate U_x and U_y, respectively.

The controller calculates the desired virtual accelerations U_x and U_y as follows:

$$u_x=K_{p,x}(x_{target}-x_{self})+K_{d,x}(v_{x,target}-v_{x,self})$$

(26)

$$u_y=K_{p,y}(y_{target}-y_{self})+K_{d,y}(v_{y,target}-v_{y,self})$$

(27)

Among these, $K_{p,x},K_{d,x},K_{p,y},K_{d,y}$ is the independently adjustable gain of the PD controller. $(x_{target},y_{target})$ and are the desired pose and velocity distributed from the planning layer.

The aforementioned decoupled control architecture significantly simplifies controller design and enables independent optimization of longitudinal and lateral tracking performance. This ensures tracking capability and stability during high-speed, high-dynamic formation transitions.

3.3.3. Stability Analysis

To verify the stability of the proposed controller, we analyze the closed-loop error dynamics. The kinematic model of the vehicle in the global frame is given by

$$\left[\begin{array}{c}\dot{X}\\ \dot{Y}\end{array}\right]=\left[\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right]v$$

(28)

Differentiating (28) with respect to time yields the acceleration relationship:

$$\left[\begin{array}{c}\ddot{X}\\ \ddot{Y}\end{array}\right]=\dot{v}\left[\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right]+v\dot{\theta }\left[\begin{array}{c}-\sin \theta \\ \cos \theta \end{array}\right]$$

(29)

Let the control inputs be the longitudinal acceleration $a=\dot{v}$ and yaw rate $\omega =\dot{\theta }$. Equation (29) can be rewritten in matrix form as

$$\left[\begin{array}{c}\ddot{X}\\ \ddot{Y}\end{array}\right]=\underset{M(\theta ,v)}{\underbrace{\left[\begin{array}{cc}\cos \theta & -v\sin \theta \\ \sin \theta & v\cos \theta \end{array}\right]}}\left[\begin{array}{c}a\\ \omega \end{array}\right]$$

(30)

The matrix M is the decoupling matrix. Assuming v ≠ 0, M is invertible. The feedback linearization control law derived in Section 3.3.1 is essentially

$$\left[\begin{array}{c}a\\ \omega \end{array}\right]=M^{-1}\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]$$

(31)

where u₁, u₂ are the virtual control inputs. Substituting (31) into (30), the nonlinearities cancel out, resulting in two decoupled linear double-integrator systems:

$$\ddot{X}=u_1,\hspace{1em}\ddot{Y}=u_2$$

(32)

We define the tracking errors as e_x = X_ref − X and e_y = Y_ref − Y. The virtual inputs u₁, u₂ are designed using PD controllers:

$$u_1={\ddot{X}}_{ref}+K_{d,x}{\dot{e}}_x+K_{p,x}{e}_x$$

(33)

$$u_2={\ddot{Y}}_{ref}+K_{d,y}{\dot{e}}_y+K_{p,y}{e}_y$$

(34)

Substituting (33) and (34) into (32), the closed-loop error dynamics become:

$${\ddot{e}}_x+K_{d,x}{\dot{e}}_x+K_{p,x}{e}_x=0$$

(35)

$${\ddot{e}}_y+K_{d,y}{\dot{e}}_y+K_{p,y}{e}_y=0$$

(36)

These are standard second-order linear homogeneous differential equations. According to the Routh-Hurwitz stability criterion, the system is asymptotically stable if and only if all coefficients of the characteristic equation s² + K_d^s + K_p = 0 are positive. Therefore, by selecting control gains K_p > 0 and K_d > 0, the roots of the characteristic equation lie in the left half of the complex plane, ensuring that $\underset{t\to \infty }{\lim }e(t)=0$. This theoretically proves the stability of the proposed AMM control strategy.

3.4. Adaptive Steering Mode Selection Strategy

To enhance the efficiency of vehicle formation changes, this paper proposes an adaptive steering mode selection strategy based on the multi-mode steering capability of unmanned vehicle platforms. This strategy aims to dynamically assign the optimal steering mode to vehicles according to instantaneous maneuver requirements, thereby significantly improving the efficiency and performance of complex formation changes without compromising system stability.

3.4.1. Intent Extraction

To enable intelligent decision-making, the mode selector must first quantify the current maneuvering requirements. This is achieved by transforming the global target pose error into the vehicle frame coordinate system:

$$\left[\begin{array}{c}{x}_{err}\\ y_{err}\end{array}\right]=R{({\theta }_i)}^T\cdot \left[\begin{array}{c}{x}_i^{target}-x_i\\ y_i^{target}-y_i\end{array}\right]$$

(37)

Among these, R(θ_i) denotes the rotation matrix of the vehicle’s current attitude. x_err and y_err represent the longitudinal and lateral translation distances required to achieve the target, respectively. Based on this error vector, the instantaneous maneuver angle α_maneuver is defined as:

$${\alpha }_{maneuver}=\mathrm{atan}2(y_{err},x_{err})$$

(38)

The instantaneous maneuver angle indicates the target point’s direction relative to the vehicle’s current heading, thereby providing a basis for mode selection.

3.4.2. Rule-Based Steering Pattern Allocation

The mode selector is a rule-based expert system that makes decisions based on the maneuver angle αmaneuver and the current vehicle speed v_i.

• Rule 1: Set to Crab Mode (CS)

When the maneuver primarily involves lateral translation—that is, when the target point is located to the side of the vehicle—the system will activate crab mode. This scenario is particularly common during formation changes from column to cross formation within a convoy. Traditional front-wheel steering requires following an S-shaped trajectory during such maneuvers, resulting in low efficiency and poor space utilization. Crab mode enables pure lateral translation without altering the vehicle’s posture, thereby achieving efficient and rapid formation transitions.

The selection criteria for this mode are as follows: When the maneuvering task primarily involves lateral movement, activating this mode causes the controller to bypass the decoupling controller and directly execute the crab-walk kinematic model. It sets the wheels to turn uniformly (${\delta }_{crab}={\alpha }_{maneuver}$) and utilizes longitudinal acceleration (a) to drive the vehicle along the maneuvering angle direction, thereby eliminating errors.

2.

Rule 2: Set to Four-Wheel Counter-Steering (4WCS)

This mode is selected when the vehicle needs to turn at an extremely tight radius or when enhanced steering response and stability are required.

When this mode is activated, the decoupled controller operates normally, and its output—the desired yaw rate ω—remains valid. However, this ω is redistributed to the underlying 4WCS kinematic model to compute front and rear wheel rotation angles with opposite signs. This enables formation maneuvers with higher yaw rate gain and stability.

3.

Rule 3: Set to front-wheel steering mode (FWS)

The FWS mode delivers optimal overall performance in terms of energy consumption, tire wear, and stability at low to medium speeds, making it suitable for most driving conditions and moderate formation adjustments.

Through the aforementioned adaptive steering mode selection strategy, the control system can dynamically reconstruct its underlying kinematic model to ensure it consistently operates within the most efficient range.

Determination of Switching Thresholds: The activation thresholds for the steering modes are determined based on the physical constraints of the 4WIDS platform and the dynamic stability conditions of the controller.

• Threshold for Crab Steering (α_th): Although the physical steering limit of the vehicle is δ_max= 60°, operating at such high angles during motion can induce excessive lateral forces and rollover risks. To ensure safety, we implemented a software saturation limit of 30° (≈0.52 rad). Consequently, the effective range for Crab Mode is defined as |ξ_maneuver| ≤ 0.5 rad (≈28°). Within this range, the vehicle performs precise linear translation. If the required angle exceeds this threshold, the system prioritizes 4WCS to utilize yaw moment for re-orientation or maintains the Crab Mode at the saturation limit, depending on the task urgency.
• Speed Threshold for Mode Switching (vth): The low-speed protection threshold is set to vth = 0.5 m/s. This value is derived from the singularity analysis of the feedback linearization control law (Equation (22)), where the term 1/v leads to numerical instability as the vehicle speed approaches zero. Below this threshold, the system locks into kinematic-based control (4WCS or FWS) to ensure smooth maneuvering during start-stop operations.

4. Simulation Results and Analysis

4.1. Simulation Platform Setup

To validate the effectiveness and advanced nature of the control framework proposed in this paper, we constructed a formation simulation platform comprising seven unmanned vehicles in the MATLAB/Simulink (R2023a) environment. The single-vehicle model is shown below.

Justification for Simulation Platform Selection: While physics engines like Gazebo or Webots offer high-fidelity environmental rendering for robotics, this study employs MATLAB/Simulink as the primary verification platform. This choice is driven by the study’s specific focus on the theoretical validation of the control logic and kinematic constraints.

• Control-Theoretic Rigor: The core contribution of this paper lies in the mathematical formulation of the AMM control law and the logical stability of the mode-switching state machine. Simulink is widely recognized as the standard environment for rigorous analysis of controller stability and dynamic response at the theoretical level.
• Kinematic Focus: Our research objective is to solve kinematic constraints and optimize formation geometry. The mathematical vehicle models built in Simulink are sufficient to capture the non-holonomic constraints and steering dynamics relevant to this scope, without the confounding factors of visual rendering or complex terrain friction found in physics-based simulators.
• Reproducibility: Using a standardized, equation-based environment facilitates the reproducibility of the control algorithms by the research community.

The entire simulation system comprises four core functional modules:

(1)

Global Reference Module: To ensure global consistency and repeatability of formation trajectories, the system introduces a Virtual Reference Generator to replace traditional physical reference vehicles as the global path planning benchmark, thereby providing a unified trajectory reference within the simulation environment.

(2)

Centralized Planning Module: This module determines the optimal formation structure for the fleet based on environmental perception data and mission requirements, while calculating the target poses for each member vehicle. Its output serves as the control reference for downstream distributed control modules.

(3)

Distributed Control Modules: Seven functionally identical vehicle control loops were constructed within the Simulink platform to simulate the independent control behavior of each vehicle in the formation. Each control loop comprises two components: a vehicle dynamics module and an adaptive trajectory tracking module. 1. The Vehicle Dynamics Module is built upon the multi-modal kinematic model proposed earlier. It dynamically responds to composite control commands (position, velocity, yaw angle, etc.) issued by the Trajectory Tracking Module and outputs the vehicle’s complete state information in real time. 2. The adaptive trajectory tracking module integrates a feedback linear decoupling controller and an adaptive steering mode selector. By comparing the target state issued by the centralized planning module with the vehicle’s own state, it achieves real-time precise control and mode switching for each vehicle.

(4)

Data Interface and Visualization Module: This module comprises the system bus and visualization subsystem. The bus system facilitates large-scale data exchange among multiple modules, ensuring standardized and scalable data transmission. The visualization subsystem collects real-time status information from each vehicle and dynamically renders two-dimensional motion animations of the fleet within the simulation environment. This provides an intuitive representation of formation behavior, aiding in result analysis.

Partial control system models are shown in Figure 8. The complete model can be found in the Supplementary Materials.

4.2. Scenario Design and Comparison Options

To validate the performance of the proposed adaptive multi-mode cooperative control strategy under complex operating conditions, particularly examining whether the controller can autonomously allocate optimal steering modes based on task difficulty (e.g., lateral/longitudinal deviation magnitude), this study designed three progressive simulation scenarios. Experiments will comprehensively evaluate the AMM strategy’s capability to overcome traditional front-wheel steering vehicle kinematic constraints, achieve efficient formation transitions, and execute flexible obstacle avoidance maneuvers through trajectory qualitative analysis and quantitative data assessment.

Scenario 1: Parallel Lane Change in Column Formation

This scenario focuses on verifying the maneuverability of the AMM framework during pure lateral translation tasks. It examines whether the system can accurately switch to crab mode under small lateral error conditions to achieve rapid lane changes without overshoot. In the simulation, a 7-vehicle formation travels uniformly along the X-direction in column formation and executes a lane change into the adjacent lane. The core objective is to verify whether vehicles can maintain body posture stability, achieving diagonal translation solely through wheel angle adjustments. This approach effectively avoids phase lag and path redundancy inherent in traditional steering methods.

Scenario 2: Multi-Formation Transition

This scenario aims to validate whether vehicles autonomously select appropriate steering modes during formation transitions based on lateral and longitudinal deviation magnitudes. The simulation configures a 7-vehicle formation traveling in a column along the X-axis, switching between transverse, longitudinal, or V-shaped formations based on planning commands. The key verification objective is to determine whether each vehicle controller exhibits differentiated decision-making: inner vehicles prioritize crab-walk mode for smooth alignment, while outer vehicles maintain front-wheel steering mode to leverage their speed advantage for rapid positioning.

Scenario 3: Mixed-Scene Testing

This scenario comprehensively evaluates the overall adaptability of the AMM framework under various typical operating conditions, including composite tasks such as lane changes with small deviations, lane changes with large deviations, and formation switching.

Comparison Group Settings:

To strictly validate the performance advantages of the proposed AMM strategy, this study explicitly establishes a “Baseline Method” for comparative analysis in the subsequent experiments. The Baseline Method adopts the traditional Front-Wheel Steering (FWS) formation control strategy, which is widely used in existing literature (e.g., [3]). In this mode, the vehicle relies solely on the front wheel steering angle δ_f to adjust the heading θ to generate lateral displacement, subject to the non-holonomic constraint. By comparing the AMM strategy with this standard Baseline, the improvements in maneuverability and trajectory efficiency can be quantitatively assessed.

4.3. Experimental Results and Analysis

To validate the effectiveness of the AMM cooperative control strategy, this paper constructed a formation system comprising seven 4WIDS unmanned vehicles on the MATLAB/Simulink simulation platform. The simulation step size was set to 0.01 s, with a vehicle wheelbase L = 2.0 m. Detailed results analysis is presented below for the three typical scenarios designed in the previous section.

4.3.1. Analysis of the Experiment on Coordinated Parallel Lane Changes in Convoys

This experiment is designed to evaluate whether the AMM cooperative control strategy can autonomously select the appropriate steering mode under varying magnitudes of lateral deviation. Specifically, it examines whether the controller can correctly engage the crab-steering mode to rapidly converge to the target position when the lateral deviation is small, and whether it can switch to the front-wheel steering mode to accomplish larger lateral lane changes while minimizing longitudinal speed loss.

• Trajectory Analysis and Baseline Comparison: As illustrated in Figure 9, the trajectory of the proposed AMM strategy (utilizing Crab Mode) exhibits a distinct “trapezoidal” profile. Upon receiving the lane-change command (lateral offset = 5 m), each vehicle locks its body orientation (θ ≈ 0, as shown in Figure 10) and generates lateral velocity directly through coordinated wheel-angle adjustments. In contrast, the Baseline FWS method (shown as the blue dashed line in Figure 9) exhibits a typical “S-shaped” lag. Constrained by non-holonomic kinematics, FWS vehicles must first accumulate yaw angle to generate a lateral velocity component, introducing significant phase lag and path redundancy. The simulation results visually confirm that the AMM strategy achieves an almost instantaneous response to abrupt lateral reference changes with zero overshoot, whereas the baseline method requires a much longer adjustment period.
• Quantitative Performance Assessment: To rigorously quantify the advantages of the proposed method,

  Table 1. Lane change maneuverability comparison.

  Evaluation Indicators	CS	FWS	Enhancement Effect
  Actuation Delay	0.1	0.4	75%
  time-consuming (s)	1.01	7.68	86.8%
  Position deviation (m)	0.003	0.005	40%

  Note: Time taken refers to the duration from receiving the lane change command to achieving a deviation of less than 2% from the target position.

  summarizes the performance metrics of both modes under identical lateral offset requirements. The results demonstrate significant improvements:
  • Actuation Delay: The AMM strategy reduces the actuation delay by 75% (from 0.4 s to 0.1 s) compared to the baseline.
  • Settling Time: The total settling time is shortened by 86.8% (from 7.68 s to 1.01 s), indicating a drastic increase in maneuvering efficiency in confined spaces.
  • Steady-State Error: The position deviation is reduced by 40% (from 0.005 m to 0.003 m).
• Longitudinal Dynamics and Mode Selection Logic: Furthermore, analysis of the X-axis trajectories reveals the rationale behind the mode-switching strategy (as shown in Figure 11). The formation adopting the Crab Mode experiences a noticeable reduction in longitudinal velocity (v_x = v × cos α) during the maneuver. In contrast, the formation using FWS maintains its longitudinal speed better at the onset. This physical trade-off validates the proposed adaptive logic: utilizing Crab Mode for small offsets to prioritize path efficiency and precision, while switching to Front-Wheel Steering for larger displacements to maintain longitudinal momentum and overall operational efficiency.

4.3.2. Analysis of Multi-Formation Transition and Adaptive Hierarchical Control

This experiment serves as a critical component in validating the core contribution of this paper: the Task-Graded Adaptive Decision mechanism. The simulation replicates a complex reconfiguration process where the formation transitions from a column to a line, and subsequently to a V-shape.

• Mechanism Analysis of Differentiated Decision-Making: The trajectory results (Figure 12 and Figure 13) explicitly demonstrate how the proposed controller assigns differentiated strategies based on the “Task Difficulty” (defined by lateral deviation magnitude):
  • Inner Vehicles (e.g., C2, C3): The controller identifies these vehicles as performing “fine-tuning tasks” (Lateral deviation < 6 m). Consequently, it automatically activates Crab Mode (Mode 1). As seen in the trajectory, C2 and C3 execute precise linear oblique translations. This strategy utilizes the holonomic property of crab steering to take the shortest path, avoiding the redundant heading adjustments required by traditional Ackerman steering.
  • Outer Vehicles (e.g., C6, C7): Due to the substantial lateral displacement (>10 m) and the need for significant longitudinal acceleration to catch up with the formation, the controller classifies this as a “high-dynamic maneuver.” The system forcibly locks these vehicles into Front-Wheel Steering (FWS) Mode. This decision leverages the high-speed stability of FWS, preventing the potential body oscillations and tire saturation that can occur when attempting large-angle crab maneuvers at high speeds.
• Stability and Anti-Chattering Analysis: A critical challenge in multi-mode control is the potential for “chattering” (rapid mode flickering) at the decision boundaries. As observed in the smooth mode transitions in Figure 13, the proposed Hysteresis Logic and Task Latch Mechanism effectively eliminate this issue.
  • During the transition phase, even if the instantaneous error of the outer vehicles (C6, C7) fluctuates near the threshold, the hysteresis logic ensures the mode remains locked in FWS until the maneuver is substantially complete.
  • The trajectory forms a smooth, continuous S-curve without the step-like discontinuities often seen in discrete switching logic. This confirms that the proposed framework not only optimizes spatial efficiency but also guarantees the dynamic stability of the closed-loop system during complex topology changes.

5. Conclusions and Future Work

5.1. Conclusions

To address the limitations of traditional front-wheel steering (FWS) unmanned vehicle formations in high-mobility missions, this paper proposes an adaptive multi-mode (AMM) steering coordination control strategy. This strategy effectively enhances the coordination efficiency of multi-vehicle systems by integrating virtual structuring methods, task-level decision-making, and anti-shudder logic. The main innovations of this work are reflected in the following three aspects:

(1)

Overcoming kinematic constraints to enhance maneuver efficiency: The developed AMM framework enables seamless transitions between front-wheel steering, four-wheel counter-steering (4WCS), and crab-walk modes. Simulation results demonstrate that during pure lateral translation tasks, the crab-walk mode effectively avoids path redundancy caused by heading angle adjustments. Compared to traditional front-wheel steering systems, response time is reduced by approximately 75%, enabling smooth maneuvers without overshoot.

(2)

Establishing a hierarchical decision-making mechanism for intelligent coordination: Addressing differentiated requirements during formation reconstruction, a task-hierarchy logic based on lateral displacement is proposed. The system automatically classifies vehicles into “Fine Adjustment Groups” (using crab mode) and “Large-Range Maneuver Groups” (maintaining FWS mode). Through V-shaped and lateral deployment experiments, this mechanism ensures precise alignment of the inner vehicles while fully leveraging the speed advantage of the outer vehicles, significantly enhancing the overall formation reconfiguration efficiency.

(3)

Overcoming Stability Challenges During Mode Transition: To address control jitter and overshoot issues prone to occur during discrete mode switching, we introduced hysteresis comparison logic, anti-jitter timers, and cascaded velocity planning methods. Experimental results demonstrate that all vehicles exhibit smooth trajectories under complex conditions, with decisive and stable mode transitions. This method completely eliminates the step-like oscillations commonly found in traditional control systems, fully validating the robust performance of the proposed control system.

5.2. Limitations and Future Work

Although simulation results validate the effectiveness of the AMM strategy, this study has several limitations that require further refinement in subsequent work:

(1)

The vehicle dynamics and nonlinear disturbance factors have not been fully considered. Current research primarily relies on kinematic models and has not yet addressed dynamic characteristics such as tire lateral deflection and ground adhesion coefficients. Future work will incorporate high-fidelity dynamic models and design corresponding robust controllers to compensate for these nonlinear disturbances.

(2)

The stability issue under constrained communication conditions remains unresolved. Existing control frameworks assume ideal communication, neglecting the actual delays and packet loss that exist. Future work will focus on developing formation control methods for constrained communication environments and designing observer-based state compensation algorithms to enhance system robustness.

(3)

The dynamic obstacle avoidance capability remains to be validated, and real-vehicle testing is currently lacking. The current avoidance scenarios are relatively simple. Future work will integrate model predictive control methods to investigate real-time obstacle avoidance in complex dynamic environments. This algorithm will be deployed on the 4WIDS unmanned vehicle test platform, with its engineering feasibility validated through real-vehicle experiments.