Adaptive Asymptotic Tracking Control for the Dynamic Models of Differential-Drive Unmanned Ground Vehicles Under Parametric Uncertainties

Highlights

What are the main findings?

• A dual-loop layered adaptive control framework based on the Immersion and Invariance (I&I) technique is developed for wheeled mobile robots, achieving exponential parameter convergence under a weaker, trajectory-dependent excitation condition.
• The proposed control scheme effectively reduces steady-state fluctuations in time-varying trajectory tracking and guarantees asymptotic zero-error convergence, significantly improving both tracking accuracy and convergence rate compared with traditional adaptive methods.

What are the implications of the main finding?

• The proposed layered design breaks the limitation of the certainty equivalence principle, providing greater flexibility in adaptive controller synthesis and offering a new scheme for handling parametric uncertainties in underactuated non-minimum phase systems.
• This work advances the high-precision dynamic trajectory tracking control of non-holonomic UGVs under parametric uncertainties. By integrating kinematic and dynamic layers, the proposed method enables robust and reliable path following in practical scenarios such as autonomous navigation, intelligent logistics, and field robotics.

Abstract

This paper proposes a dual-loop layered control mechanism for the dynamic trajectory tracking of non-holonomic unmanned ground vehicles (UGVs). The proposed scheme enhances steady-state precision while guaranteeing parameter convergence under specified trajectory constraints. To tackle the underactuated constraints of Unmanned Ground Vehicles, the control mechanism is structured into kinematic and dynamic loops. Specifically, a kinematic controller is first synthesized to serve as a virtual control law, generating desired velocity commands. Subsequently, a layered adaptive control strategy based on the Immersion and Invariance technique is developed for the dynamic loop. This strategy integrates a parameter estimation layer, which utilizes tailored tuning functions to ensure the exponential convergence of estimation errors under the condition that the reference trajectory is not persistently vertical. A controller design layer is then responsible for uncertainty compensation. By decoupling parameter adaptation from control law synthesis, the proposed mechanism circumvents the structural limitations of the certainty equivalence principle. Theoretical analysis confirms that the proposed design achieves almost-global asymptotic tracking. Simulation results demonstrate that the mechanism resolves the imprecise parameter convergence inherent in traditional adaptive schemes, eliminates steady-state pose fluctuations during time-varying trajectory tracking, and achieves asymptotic convergence of tracking errors.

1. Introduction

As a cornerstone of intelligent automation, the trajectory tracking control of mobile robots has transitioned from basic stability research to high-precision, high-dynamic applications in recent years [1,2]. Unmanned ground vehicles (UGVs) are increasingly deployed in complex missions such as intelligent logistics, autonomous inspection, and planetary exploration [3]. Achieving precise trajectory tracking is fundamental for their autonomous navigation and mission reliability [4]. Most of the existing research focuses on the design of controllers at the kinematic level [5,6]. Because it can directly establish the mapping between pose and velocity through geometric relationships, it has the advantages of simple modeling and high computational efficiency. However, kinematic models only describe geometric relationships and cannot characterize dynamic properties such as inertial forces and frictional effects. Neglecting dynamic effects inevitably leads to degraded tracking accuracy or even system instability when the robot operates at its physical limits. In contrast, dynamic models provide a more comprehensive description of system behavior by analyzing energy transformation and nonlinear coupling effects [7,8]. As nonlinear differential equations that encapsulate the relationships between forces/torques and motion states, the control design for UGV dynamic models involves three formidable challenges. First, UGVs are typical underactuated systems governed by nonholonomic constraints, where the dimension of control inputs are fewer than the degrees of freedom [9]. This makes classical control methods like backstepping designed for strict-feedback systems difficult to apply directly. Second, the inherent non-minimum phase characteristics often render the internal zero dynamics unstable. Classical control theories, such as backstepping [10] and sliding mode control [11], are primarily proposed for minimum phase systems. The direct application of these methods on the wheeled mobile robot system is prone to internal state divergence. Third, parametric uncertainties (e.g., varying mass, uncertain friction) significantly interfere with the force/torque-to-motion mapping.

Traditional adaptive control schemes, such as Lyapunov-based designs [12,13] and Sontag’s universal formula-based methods [14,15], have been proposed to address the control challenges of underactuated dynamical systems of UGVs. Regarding the control synthesis, Sontag’s universal formula may trigger severe high-frequency oscillations in the control input when the Lie derivative $L_{g}V$ in the denominator approaches zero [16]. Furthermore, since Sontag’s formula requires the Control Lyapunov Function (CLF) to satisfy the Small Control Property (SCP) [17], any non-smooth regions within the state space—such as those found in multimodal CLFs—can induce similar chattering phenomena during state transitions. Broadly, research indicates that such oscillations stem from multiple interrelated factors, including the discontinuity of control laws during switching, the interaction between parasitic parameters and controller dynamics, excessive gains from improper parameter selection, and the mismatch between sampling rates and the system’s high-frequency characteristics.

To suppress these oscillations and enhance tracking precision, accurate parameter estimation is considered a viable solution. However, while traditional Lyapunov-based adaptive control methods can mitigate model uncertainties, their theoretical framework suffers from inherent limitations: parameter convergence strictly depends on the Persistent Excitation (PE) condition [18]. In time-varying trajectory tracking, the insufficient excitation of reference inputs often causes parameter estimation to lag behind rapid trajectory changes [19,20]. Moreover, unmodeled dynamics, friction, and external disturbances can lead to parameter drift [21]. Since the strict PE condition is rarely satisfied in real-world operations, traditional adaptive control often fails to achieve exact parameter estimation. Although existing robust modifications, such as projection algorithms [22] and $\sigma$-modification [23], can maintain parameter boundedness, they do so at the expense of steady-state accuracy, failing to eliminate residual tracking errors.

As a result, these conventional adaptive approaches are unable to mitigate the inherent steady-state fluctuations, leaving a residual tracking error that fails to vanish as time approaches infinity. The underlying causes of these persistent errors are threefold:

(i)

The complex nonlinear coupling inherent in UGVs is exacerbated by nonholonomic constraints, such as the no-lateral-slip condition in differential-drive robots [24]. Model uncertainties tend to amplify these coupling effects, thereby destabilizing the delicate equilibrium between tracking performance and internal states [25]. For instance, in car-like robots, uncertainties in wheel-ground friction coefficients can intensify the coupling between longitudinal velocity and yaw rate, leading to significant lateral error drift that is difficult to compensate for using kinematic-based methods alone.

(ii)

Traditional adaptation laws create a fundamental lag in parameter estimation due to their finite convergence rates. In these frameworks, estimation errors introduce persistent biases into the control inputs, and the controllers frequently lack the necessary bandwidth to compensate for the rapidly varying terms inherent in time-varying trajectories. A typical case is a UGV tracking a spiral reference path; the continuously changing curvature excites nonlinear coupling within the yaw dynamics, while the parameter estimates inevitably lag behind due to the aforementioned lack of sufficient excitation [26].

(iii)

The structural properties of UGV systems, particularly their non-minimum phase nature, can present a barrier to zero-error convergence [27], while minimum-phase systems possess inherently self-stabilizing internal dynamics [28]; many nonlinear tracking methods—such as feedback linearization, dynamic inversion, backstepping, or sliding mode—prioritize output tracking but may neglect internal state stabilization. Consequently, even if the nominal tracking error is driven to zero, uncompensated oscillations in the internal dynamics can manifest as persistent and irreducible fluctuations in the final position tracking error.

Therefore, while traditional adaptive control frameworks may theoretically guarantee asymptotic stability, their practical implementation is severely compromised by the compounded effects of nonlinear coupling in underactuated dynamics, parameter estimation lag, and inherent non-minimum phase characteristics. When these structural challenges interact with nonideal factors in real-world systems, the position tracking errors fail to achieve true zero-convergence. Instead, they manifest as persistent periodic or stochastic oscillations, which is a significant bottleneck in high-precision autonomous navigation.

To overcome the aforementioned challenges in the control design of the dynamic models of UGVs, this paper develops a dual-loop layered control mechanism characterized by guaranteeing parameter convergence under a weaker condition, leveraging the Immersion and Invariance (I&I) technique [29]. Specifically, the control architecture is bifurcated into two functional layers: first, a kinematic control loop is synthesized to generate desired velocity commands for reference trajectory tracking. Subsequently, in the dynamic control loop of UGVs, a layered adaptive control strategy based on the I&I technique is developed to approximate the desired control signals while simultaneously performing online estimation and compensation of system uncertainties. By decoupling the adaptation law from the control law, the proposed mechanism effectively bridges the gap between kinematic planning and dynamic execution, thereby ensuring high-precision tracking performance. The main contributions of this paper are as follows:

(1)

Existing research on tracking control predominantly focuses on the kinematic models of UGVs. However, such models often neglect dynamic characteristics (e.g., inertial forces and friction effects), rendering them inadequate for high-speed, high-precision applications. To address the more complex dynamic models of UGVs, this paper proposes a layered adaptive dual-loop control mechanism based on the I&I technique. This mechanism conducts online estimation and compensation of system uncertainties to achieve precise reference trajectory tracking. Simulation experiments demonstrate the effectiveness of the proposed control mechanism.

(2)

Within the dynamic control loop, the proposed adaptive control strategy adopts a layered architecture comprising a parameter estimation layer and a controller design layer. This structure decouples the design of the adaptation law from that of the controller. Crucially, the parameter estimates are not solely generated by adaptation laws; rather, they integrate modification functions into the nominal parameter estimates. Compared with traditional methods, the layered adaptive control strategy proposed in this paper does not rely on the certainty equivalence principle. It offers more design degrees of freedom and ensures that the parameter estimation error exponentially converges to zero under a weaker excitation condition than the classical PE condition.

(3)

Most of the existing adaptive control methods for wheeled mobile robots are designed to address the precise tracking problem of time-invariant trajectories (e.g., straight-line paths). However, when tracking time-varying trajectories, these methods—despite theoretically guaranteeing asymptotic system stability—fail to achieve zero position tracking error convergence in practice due to the non-minimum phase characteristics of UGVs. This limitation manifests as persistent position tracking error fluctuations. The dual-loop layered adaptive control mechanism based on the I&I technique proposed in this paper provides an effective solution for addressing the control challenge of non-minimum-phase characteristics for UGV systems. It effectively eliminates steady-state position tracking error fluctuations during time-varying trajectory tracking and demonstrably achieves precise convergence of position tracking errors to zero in both theoretical analysis and simulation experiments, significantly enhancing tracking accuracy.

The remaining parts of this paper are organized as follows: In Section 2, the dynamic characteristics of UGV are analyzed, and the problem formulation is stated. In Section 3, the desired control signals are designed for the kinematic model of UGV to track the reference trajectory. In Section 4, the disadvantages of traditional control design are firstly shown to indicate the motivation of this paper. Then, a layered adaptive control strategy based on the I&I technique is developed for the dynamic model of UGV to achieve the tracking control objective. Finally, an improved control design is provided to further enhance the control performance. In Section 5, the simulation experiments are carried out to verify the effectiveness of the proposed mechanism. Section 6 gives some concluding remarks.

2. Mathematical Model and Problem Formulation

According to the motion relationship of each rigid body part of the UGV, its dynamic model in the geodetic coordinate system can be expressed as

$$\begin{array}{ccc}\hfill \dot{x}& =& vcos\left(\theta \right),\hfill \\ \hfill \dot{y}& =& vsin\left(\theta \right),\hfill \\ \hfill \dot{\theta }& =& \omega ,\hfill \\ \hfill \dot{v}& =& a{\tau }_1+d_1,\hfill \\ \hfill \dot{\omega }& =& b{\tau }_2+d_2,\hfill \end{array}$$

(1)

where x and y represent the Cartesian position coordinates of the UGV’s center of mass expressed in the global inertial coordinate frame $\theta$ denotes the counterclockwise rotation angle of the UGV relative to the geodetic coordinate system; v and $\omega$ represent linear velocity and angular velocity, respectively; ${\tau }_1$ and ${\tau }_2$ represent the input torque of the two rear wheels; $a={\displaystyle \frac{1}{rm}}$ and $b={\displaystyle \frac{R}{rJ}}$ are the known physical parameters; m and J are the mass and moment of inertia of the UGV; r is the radius of the driving wheel and $2R$ denotes the wheelbase of the rear wheels; $d_1$ and $d_2$ denote the lumped uncertainties that include unmodeled dynamics, friction and external disturbance, which can be simply expressed as $d_1=-{\displaystyle \frac{cos\left(\theta \right)}{m}}{\tau }_{d1}$ and $d_2=-{\displaystyle \frac{1}{J}}{\tau }_{d2}$; ${\tau }_{d1}$ and ${\tau }_{d2}$ are the unknown parameters. Define $\zeta ={[x,y,\theta ,v,\omega ]}^{\top }$, which represents the state vector of the model (1).

In the above model (1), $d_1$ and $d_2$ denote lumped uncertainties that aggregate the effects of unmodeled dynamics, friction variations, external disturbances, and parameter perturbations. The unknown quantities ${\tau }_{d1}$ and ${\tau }_{d2}$ are parameterized representations of these uncertainties and can be estimated online using the subsequently designed adaptive control mechanism.

The UGV system described by the above model is an underactuated system in which the dimension of control inputs is lower than that of degrees of freedom. Since the UGV is unable to perform lateral displacement and lateral sliding movements, that is to say, it cannot move in the axial direction of the wheels, it satisfies the non-holonomic constraint: $\dot{x}sin\left(\theta \right)-\dot{y}cos\left(\theta \right)=0$. A non-holonomic constraint does not reduce the number of independent generalized coordinates, i.e., the degrees of freedom of a finite motion, but it will reduce the independent velocity components, i.e., the degrees of freedom of an infinitesimal motion. In the above-mentioned UGV system, the non-holonomic constraint is manifested as a non-integrable constraint at the speed and acceleration levels, which makes it impossible to directly adjust all the degrees of freedom of the system through local control. In this situation, the traditional backstepping control method is no longer applicable. This brings significant challenges to the control of underactuated systems with non-holonomic constraint.

To solve this problem, in this paper, a dual-loop adaptive control strategy based on I&I technology is proposed to design the adaptive tracking controller for the dynamic model (1) of UGV. The design idea of this dual-loop control is as follows: Firstly, in the kinematic control loop, the desired linear velocity signal $v_d$ and the desired angular velocity signal ${\omega }_d$ are designed to track the reference trajectory $(x_r,y_r,{\theta }_r)$. Then, in the dynamic control loop, the torque signals, ${\tau }_1$ and ${\tau }_2$, are designed to track $v_d$ and ${\omega }_d$.

Following the above design idea, the kinematic model of the UGV is given as follows:

$$\begin{array}{ccc}\hfill \dot{x}& =& v_{d}cos\theta ,\hfill \\ \hfill \dot{y}& =& v_{d}sin\theta ,\hfill \\ \hfill \dot{\theta }& =& {\omega }_d.\hfill \end{array}$$

(2)

The reference trajectory generated by the following reference system:

$$\begin{array}{ccc}\hfill {\dot{x}}_r& =& v_{r}cos{\theta }_r,\hfill \\ \hfill {\dot{y}}_r& =& v_{r}sin{\theta }_r,\hfill \\ \hfill {\dot{\theta }}_r& =& {\omega }_r\hfill \end{array}$$

(3)

where $x_r,y_r,{\theta }_r$ are the states of the reference system, $v_r$ and ${\omega }_r$ are the reference inputs.

The control objective of this paper is to design an adaptive tracking controller with guaranteed parameter convergence under specified trajectory constraints for the dynamic model (1) to track precisely the reference trajectory generated by (3). The schematic diagram of tracking control for UGVs is shown in Figure 1.

3. Kinematic Model Control Design

For the kinematic model (2), the position tracking error signals of the UGV to the reference system (3) can be expressed as

$$\begin{array}{ccc}\hfill x_e& =& cos\theta (x_r-x)+sin\theta (y_r-y),\hfill \\ \hfill y_e& =& -sin\theta (x_r-x)+cos\theta (y_r-y),\hfill \\ \hfill {\theta }_e& =& {\theta }_r-\theta \hfill \end{array}$$

(4)

where $x_e,y_e$ and ${\theta }_e$ denote the position and orientation tracking errors expressed in the robot’s local body-fixed coordinate frame, respectively.

From (2) and (3), the time derivatives of error signals can be calculated as

$$\begin{array}{ccc}\hfill {\dot{x}}_e& =& {\omega }_d{y}_e-v_d+v_{r}cos{\theta }_e,\hfill \\ \hfill {\dot{y}}_e& =& -{\omega }_d{x}_e+v_{r}sin{\theta }_e,\hfill \\ \hfill {\dot{\theta }}_e& =& {\omega }_r-{\omega }_d\hfill \end{array}$$

(5)

Based on the error system (5), the desired linear velocity and desired angular velocity signal are designed for the kinematic model (2) as follows

$$\begin{array}{ccc}\hfill v_d& =& k_1{x}_e+v_{r}cos{\theta }_e\hfill \\ \hfill {\omega }_d& =& {\omega }_r+k_3{y}_e{v}_r+k_{2}sin{\theta }_e\hfill \end{array}$$

(6)

where $k_1,k_2,k_3>0$ are the designed parameters.

The desired linear velocity and desired angular velocity signal (6) serves as a virtual control law (or kinematic-level control command) synthesized for the outer loop. Its purpose is to generate the desired linear and angular velocity profiles ($v_d,{\omega }_d$) required to stabilize the kinematic tracking error. These virtual commands are subsequently utilized as the reference inputs for the inner-loop dynamic controller, which ensures the actual UGV velocities track these commands through uncertainty compensation.

Theorem 1.

For the position tracking error system (5) under the desired controller (6), the closed-loop system is asymptotically stable.

Proof. 

Select Lyapunov function candidate as

$$V_k={\displaystyle \frac{1}{2}}{x}_e^2+{\displaystyle \frac{1}{2}}{y}_e^2+{\displaystyle \frac{1}{k_3}}(1-cos{\theta }_e)$$

Differentiating $V_k$ along the position tracking error system (5), it has

$$\begin{array}{ccc}\hfill {\dot{V}}_k& =& x_e({\omega }_d{y}_e-v_d+v_{r}cos{\theta }_e)+y_e(-{\omega }_d{x}_e+v_{r}sin{\theta }_e)+{\displaystyle \frac{sin{\theta }_e}{k_3}}({\omega }_r-{\omega }_d)\hfill \\ & =& x_e(-v_d+v_{r}cos{\theta }_e)+{\displaystyle \frac{sin{\theta }_e}{k_3}}(k_3{y}_e{v}_r+{\omega }_r-{\omega }_d)\hfill \end{array}$$

(7)

Substituting the desired controller (6) into (7) yields

$${\dot{V}}_k=-k_1{x}_e^2-{\displaystyle \frac{k_2}{k_3}}{sin}^2{\theta }_e⩽0$$

(8)

It follows from (8) that ${\dot{V}}_k=0$ if and only if $x_e=0$ and ${\theta }_e=0$, i.e., the following set

$$I_K=\left\{(x_e,y_e,{\theta }_e)\right|{\dot{V}}_k=0\}=\left\{(x_e,y_e,{\theta }_e)\right|x_e=0,{\theta }_e=0\}$$

(9)

is an invariant set. The states $x_e$ and ${\theta }_e$ will converge into $I_K$ as $t\to \infty$.

Next, let us analyze where $y_e$ converges in the set $I_K$. When ${\dot{V}}_k=0$, all signals in the closed-loop system reach the steady state, so ${\dot{\theta }}_e=0$. Substituting $x_e\equiv 0$ and ${\theta }_e\equiv 0$ (which implies ${\dot{x}}_e\equiv 0$ and ${\dot{\theta }}_e\equiv 0$) into the kinematic error dynamics: ${\dot{\theta }}_e={\omega }_r-{\omega }_d\equiv 0$. Substituting (6) into the above equation: ${\dot{\theta }}_e=-k_3{y}_e{v}_r-k_{2}sin{\theta }_e\equiv 0$. Since ${\theta }_e\equiv 0$ and $v_r$ is a time-varying reference inputs, only and if only $y_e\equiv 0$ the equation ${\dot{\theta }}_e=-k_3{y}_e{v}_r-k_{2}sin{\theta }_e\equiv 0$ holds. Therefore, $y_e$ converge to zero as $t\to \infty$. Therefore, (9) can be rewritten as $I_K=\left\{(x_e,y_e,{\theta }_e)\right|{\dot{V}}_k=0\}=\left\{(x_e,y_e,{\theta }_e)\right|x_e=0,y_e=0,{\theta }_e=0\}$. By LaSalle’s invariance principle, it can be concluded that the closed-loop system consisting of (5) and (6) is asymptotically stable, i.e., $x_e,y_e,{\theta }_e\to 0$ as $t\to \infty$.  □

4. Dynamic Model Control Design

In this section, we will design the torque signals ${\tau }_1$ and ${\tau }_2$ in the dynamic model (1) so that the linear velocity v and angular velocity $\omega$ of the UGV can track the desired control signals $v_d$ and ${\omega }_d$.

Since the dynamic model (1) contains parameterized lumped uncertainties, an adaptation law must be designed for their online estimation to enable their compensation in subsequent control design.

4.1. Motivation

In the traditional Lyapunov-based adaptive control method [13,30], the adaptive controller is designed based on the certainty equivalence principle. The common design process is as follows.

For the dynamic model (1), the time derivatives of error signals (4) can be calculated as

$$\begin{array}{ccc}\hfill {\dot{x}}_e& =& \omega y_e-v+v_{r}cos{\theta }_e,\hfill \\ \hfill {\dot{y}}_e& =& -\omega x_e+v_{r}sin{\theta }_e,\hfill \\ \hfill {\dot{\theta }}_e& =& {\omega }_r-\omega .\hfill \end{array}$$

(10)

Define the parameter estimation error signals

$${\tilde{\tau }}_{d1}={\widehat{\tau }}_{d1}-{\tau }_{d1},{\tilde{\tau }}_{d2}={\widehat{\tau }}_{d2}-{\tau }_{d2},$$

and the tracking errors

$$v_e=v_d-v,{\omega }_e={\omega }_d-\omega .$$

(11)

where ${\widehat{\tau }}_{d1},{\widehat{\tau }}_{d2}$ are the estimates of the unknown parameters ${\tau }_{d1},{\tau }_{d2}$, which are updated by the adaptation law designed later.

To ensure that the linear velocity v and angular velocity $\omega$ track the desired signals (6), and compensate for lumped uncertainty, the Lyapunov function candidate is selected as

$$V_L=V_k+{\displaystyle \frac{1}{2}}{v}_e^2+{\displaystyle \frac{1}{2}}{\omega }_e^2+{\displaystyle \frac{1}{2{\gamma }_1}}{\tilde{\tau }}_{d1}^2+{\displaystyle \frac{1}{2{\gamma }_2}}{\tilde{\tau }}_{d2}^2$$

where ${\gamma }_1,{\gamma }_2>0$. Differentiating $V_L$ along the error system (10) and dynamic model (1), it has

$$\begin{array}{ccc}\hfill {\dot{V}}_L& =& x_e(\omega y_e-v+v_{r}cos{\theta }_e)+y_e(-\omega x_e+v_{r}sin{\theta }_e)+{\displaystyle \frac{sin{\theta }_e}{k_3}}({\omega }_r-\omega )\hfill \\ & & +v_e({\dot{v}}_d-a{\tau }_1-{\varphi }_1{\tau }_{d1})+{\omega }_e({\dot{\omega }}_d-b{\tau }_2-{\varphi }_2{\tau }_{d2})+{\displaystyle \frac{1}{{\gamma }_1}}{\tilde{\tau }}_{d1}{\dot{\widehat{\tau }}}_{d1}+{\displaystyle \frac{1}{{\gamma }_2}}{\tilde{\tau }}_{d2}{\dot{\widehat{\tau }}}_{d2}\hfill \\ & =& x_e(v_e-k_1{x}_e)+y_e{v}_{r}sin{\theta }_e+{\displaystyle \frac{sin{\theta }_e}{k_3}}({\omega }_e-k_3{y}_e{v}_r-k_{2}sin{\theta }_e)\hfill \\ & & +v_e({\dot{v}}_d-a{\tau }_1-{\varphi }_1{\widehat{\tau }}_{d1})+{\omega }_e({\dot{\omega }}_d-b{\tau }_2-{\varphi }_2{\widehat{\tau }}_{d2})+{\displaystyle \frac{1}{{\gamma }_1}}{\tilde{\tau }}_{d1}({\dot{\widehat{\tau }}}_{d1}\hfill \\ & & +{\gamma }_1{v}_e{\varphi }_1)+{\displaystyle \frac{1}{{\gamma }_2}}{\tilde{\tau }}_{d2}({\dot{\widehat{\tau }}}_{d2}+{\gamma }_2{\omega }_e{\varphi }_2)\hfill \end{array}$$

(12)

where ${\varphi }_1=-{\displaystyle \frac{cos\theta }{m}}$ and ${\varphi }_2=-{\displaystyle \frac{1}{J}}$. Based on (12), the torque signals and adaptation laws are designed as

$$\begin{array}{ccc}\hfill {\tau }_1& =& {\displaystyle \frac{1}{a}}({\dot{v}}_d-{\varphi }_1{\widehat{\tau }}_{d1}+x_e+k_4{v}_e)\hfill \\ \hfill {\tau }_2& =& {\displaystyle \frac{1}{b}}({\dot{\omega }}_d-{\varphi }_2{\widehat{\tau }}_{d2}+x_e+{\displaystyle \frac{1}{k_3}}sin{\theta }_e+k_5{\omega }_e)\hfill \\ \hfill {\dot{\widehat{\tau }}}_{d1}& =& -{\gamma }_1{v}_e{\varphi }_1\hfill \\ \hfill {\dot{\widehat{\tau }}}_{d2}& =& -{\gamma }_2{\omega }_e{\varphi }_2\hfill \end{array}$$

(13)

where $k_4,k_5>0$ are the designed parameters. Substituting (13) into (12) yields

$${\dot{V}}_L=-k_1{x}_e^2-{\displaystyle \frac{k_2}{k_3}}{sin}^2{\theta }_e-k_4{v}_e^2-k_5{\omega }_e^2$$

(14)

From (14), it has $v_e,{\omega }_e\to 0$ as $t\to \infty$. Therefore, it follows from (11) that $v\to v_d,\omega \to {\omega }_d$ as $t\to \infty$.

Remark 1.

In the proof of Theorem 1, since Formula (14) only contains the information of tracking error and does not include the information of parameter estimation error ${\tilde{\tau }}_{d1},{\tilde{\tau }}_{d2}$, the convergence of ${\tilde{\tau }}_{d1},{\tilde{\tau }}_{d2}$ cannot be obtained based on Formula (14). When unmodeled disturbances exist in the system, such disturbances introduce additional error signals. In this scenario, the adaptation law in (13) may mistakenly attribute the error caused by the disturbances to the parameter estimation error, which will cause the parameter updating direction to deviate from the true gradient direction, thus causing the parameter drift.

Remark 2.

In addition, it follows from (14) that the traditional Lyapunov-based adaptive control methods theoretically guarantee asymptotic stability of the closed-loop system. However, in practical system, if the reference trajectory is time-varying, the combined effects of nonlinear coupling in underactuated systems, parameter estimation lag, non-minimum phase characteristics, and nonideal factors result in the position tracking errors failing to fully converge to zero. Instead, these errors exhibit periodic or stochastic fluctuations, as shown in Simulation Examples in Section 5.

In view of the considerations above, this paper proposes a layered adaptive control strategy based on I&I technique [29] for the dynamic model (1) of UGV to ensure the convergence of parameter estimation errors. The proposed method overcomes the control challenges of UGV systems with non-minimum phase characteristics. It can effectively eliminate the steady-state fluctuations in position tracking error even when tracking the time-varying reference trajectories. Different from the traditional Lyapunov-based adaptive control method, the proposed control strategy deviates from the certainty equivalence principle. In this method, the design of the adaptation law and the controller is separated. The specific design process is given below.

4.2. Adaptation Law Design

In this subsection, an adaptation law based on the I&I technology is proposed for ensuring the convergence of parameter estimation errors. To this end, a tuning function $\alpha \left(\zeta \right)$ is introduced into ${\widehat{\tau }}_{d1},{\widehat{\tau }}_{d2}$ to modify the parameter estimation errors.

For the unknown parameters ${\tau }_{d1},{\tau }_{d2}$, their estimates are expressed as

$$\begin{array}{ccc}\hfill {\widehat{\tau }}_{est1}& =& {\widehat{\tau }}_{n1}+{\sigma }_1{\alpha }_1(\theta ,v),\hfill \\ \hfill {\widehat{\tau }}_{est2}& =& {\widehat{\tau }}_{n2}+{\sigma }_2{\alpha }_2\left(\omega \right),\hfill \end{array}$$

(15)

where ${\widehat{\tau }}_{n1},{\widehat{\tau }}_{n2}$ denote the nominal parameter estimates and ${\sigma }_1{\alpha }_1(\theta ,v),{\sigma }_2{\alpha }_2\left(\omega \right)$ denote the modification part of the parameter estimates; ${\sigma }_1,{\sigma }_2>0$ are the coefficients of the tuning function ${\alpha }_1(\theta ,v),{\alpha }_2\left(\omega \right)$. Define the I&I manifold

$$\begin{array}{ccc}\hfill \{({\tau }_{d1},\theta ,v)\in \mathbb{R}\times \mathbb{R}\times \mathbb{R}& |& {\widehat{\tau }}_{n1}+{\sigma }_1{\alpha }_1(\theta ,v)-{\tau }_{d1=0}\},\hfill \\ \hfill \{({\tau }_{d2},\omega )\in \mathbb{R}\times \mathbb{R}& |& {\widehat{\tau }}_{n2}+{\sigma }_2{\alpha }_2\left(\omega \right)-{\tau }_{d2}=0\},\hfill \end{array}$$

and the off-the-manifold coordinate

$${\xi }_1={\widehat{\tau }}_{est1}-{\tau }_{d1},{\xi }_2={\widehat{\tau }}_{est2}-{\tau }_{d2},$$

(16)

From (15) and (16), the time derivatives of ${\xi }_1,{\xi }_2$ along the dynamic model (1) can be calculated as

$$\begin{array}{ccc}\hfill {\dot{\xi }}_1& =& {\dot{\widehat{\tau }}}_{n1}+{\sigma }_1{\nabla }_{\theta }{\alpha }_1(\theta ,v)\omega +{\sigma }_1{\nabla }_v{\alpha }_1(\theta ,v)(a{\tau }_1+{\varphi }_1{\tau }_{d1}),\hfill \\ \hfill {\dot{\xi }}_2& =& {\dot{\widehat{\tau }}}_{n2}+{\sigma }_2{\nabla }_{\omega }{\alpha }_2\left(\omega \right)(b{\tau }_2+{\varphi }_2{\tau }_{d2}),\hfill \end{array}$$

(17)

To stabilize the parameter estimation error system (17) and ensure the convergence of parameter estimation errors, the adaptation laws can be designed as

$$\begin{array}{ccc}\hfill {\dot{\widehat{\tau }}}_{n1}& =& -{\sigma }_1{\nabla }_{\theta }{\alpha }_1(\theta ,v)\omega -{\sigma }_1{\nabla }_v{\alpha }_1(\theta ,v)(a{\tau }_1+{\varphi }_1{\widehat{\tau }}_{est1}),\hfill \\ \hfill {\dot{\widehat{\tau }}}_{n2}& =& -{\sigma }_2{\nabla }_{\omega }{\alpha }_2\left(\omega \right)(b{\tau }_2+{\varphi }_2{\widehat{\tau }}_{est2}).\hfill \end{array}$$

(18)

where ${\alpha }_1(\theta ,v),{\alpha }_2\left(\omega \right)$ are the designed tuning functions satisfying

$${\nabla }_v{\alpha }_1(\theta ,v)={\rho }_1{\varphi }_1,{\nabla }_{\omega }{\alpha }_2\left(\omega \right)={\rho }_2{\varphi }_2,$$

(19)

with ${\rho }_1,{\rho }_2>0$ are the designed parameters.

Theorem 2.

For the parameter estimation error system (17) under the adaptation laws (18) with the tuning functions satisfying (19). Then,

(1) 

${\xi }_1$ is bounded and ${\xi }_2$ converges exponentially to zero for any reference trajectory.

(2) 

${\xi }_1$ can converge exponentially to zero if for $\forall t^{\prime }>0$, one has $\{{\theta }_r\left(t\right)=\pm {\displaystyle \frac{\pi }{2}}\}\wedge \{{\omega }_r=0\}=\mathit{false}$ for $t⩾t^{\prime }$.

Proof. 

Substituting the adaptation laws (18) into (17) yields

$$\begin{array}{ccc}\hfill {\dot{\xi }}_1& =& {\sigma }_1{\nabla }_v{\alpha }_1(\theta ,v){\varphi }_1({\tau }_{d1}-{\widehat{\tau }}_{est1}),\hfill \\ \hfill {\dot{\xi }}_2& =& {\sigma }_2{\nabla }_{\omega }{\alpha }_2\left(\omega \right){\varphi }_2({\tau }_{d2}-{\widehat{\tau }}_{est2}),\hfill \end{array}$$

(20)

First, analyze the convergence of ${\xi }_2$. Select Lyapunov function candidate as

$$V_{{\xi }_2}={\displaystyle \frac{1}{2}}{\xi }_2^2$$

Differentiating $V_{{\xi }_2}$ along the parameter estimation error system (20), it has

$$\begin{array}{ccc}\hfill {\dot{V}}_{{\xi }_2}& =& {\xi }_2\left[{\sigma }_2{\nabla }_{\omega }{\alpha }_2\left(\omega \right){\varphi }_2({\tau }_{d2}-{\widehat{\tau }}_{est2})\right]\hfill \\ & =& -{\sigma }_2{\xi }_2{\nabla }_{\omega }{\alpha }_2\left(\omega \right){\varphi }_2{\xi }_2\hfill \end{array}$$

To ensure the convergence of the parameter estimation errors ${\xi }_2$, it is necessary to retain the quadratic term related to ${\xi }_2$ in ${\dot{V}}_{{\xi }_2}$. To this end, the tuning functions $\alpha \left(\omega \right)$ is designed to satisfy (19). Then, it has

$${\dot{V}}_{{\xi }_2}=-{\sigma }_2{\rho }_2{\varphi }_2^2{\xi }_2^2=-{\displaystyle \frac{{\sigma }_2{\rho }_2}{J^2}}{\xi }_2^2$$

(21)

In (21), ${\displaystyle \frac{{\sigma }_2{\rho }_2}{J^2}}>0$, so ${\dot{V}}_{{\xi }_2}=0$ if and only if ${\xi }_2=0$, i.e., the following set

$$\begin{array}{ccc}\hfill I_{D_2}& =& \left\{{\xi }_2\right|{\dot{V}}_{{\xi }_2}=0\}\hfill \\ & =& \left\{{\xi }_2\right|{\xi }_2=0\}\hfill \end{array}$$

is an invariant set. Therefore, ${\xi }_2$ will converge into the set $I_{D_2}$ as $t\to \infty$. Integrating ${\dot{V}}_{{\xi }_2}$ yields $0⩽V_{{\xi }_2}\left(t\right)=V_{{\xi }_2}\left(0\right)e^{-{\mu }_{2}t}$ where ${\mu }_2={\displaystyle \frac{2{\sigma }_2{\rho }_2}{J^2}}$. Thus, ${\xi }_2$ converges exponentially to zero as $t\to \infty$.

Then analyze the convergence of ${\xi }_1$. Select Lyapunov function candidate as

$$V_{{\xi }_1}={\displaystyle \frac{1}{2}}{\xi }_1^2$$

Differentiating $V_{{\xi }_1}$ along the parameter estimation error system (20), it has

$$\begin{array}{ccc}\hfill {\dot{V}}_{{\xi }_1}& =& {\xi }_1\left[{\sigma }_1{\nabla }_v{\alpha }_1(\theta ,v){\varphi }_1({\tau }_{d1}-{\widehat{\tau }}_{est1})\right]\hfill \\ & =& -{\sigma }_1{\xi }_1{\nabla }_v{\alpha }_1(\theta ,v){\varphi }_1{\xi }_1\hfill \end{array}$$

The tuning functions $\alpha (v,\omega )$ is designed to satisfy (19). Then, it has

$$\begin{array}{ccc}\hfill {\dot{V}}_{{\xi }_1}& =& -{\sigma }_1{\rho }_1{\varphi }_1^2{\xi }_1^2\hfill \end{array}$$

(22)

$$\begin{array}{ccc}& =& -{\displaystyle \frac{{\sigma }_1{\rho }_1}{m^2}}{cos}^2\theta {\xi }_1^2\hfill \end{array}$$

(23)

In (22), ${\displaystyle \frac{{\sigma }_1{\rho }_1}{m^2}}>0$, so ${\dot{V}}_{{\xi }_1}=0$ if and only if ${\xi }_{1}cos\theta =0$, i.e., the following set

$$I_{D_1}=\left\{{\xi }_1\right|{\dot{V}}_{{\xi }_1}=0\}=\left\{{\xi }_1\right|{\xi }_{1}cos\theta =0\}$$

is an invariant set. Therefore, ${\xi }_{1}cos\theta$ will converge into the set $I_{D_1}$ as $t\to \infty$. Integrating ${\dot{V}}_{{\xi }_1}$ yields $0⩽V_{{\xi }_1}\left(t\right)=V_{{\xi }_1}\left(0\right)exp\left(-{\int }_0^t{\mu }_1\left(\tau \right)\mathrm{d}\tau \right)$ where ${\mu }_1={\displaystyle \frac{2{\sigma }_1{\rho }_1}{m^2}}{cos}^2\theta ⩾0$. If $cos\theta \left(t\right)\equiv 0$, then $V_{{\xi }_1}\left(t\right)=V_{{\xi }_1}\left(t_0\right)$. It means that ${\xi }_1$ is bounded. In this work, the adaptive controller is developed to enable $\theta \left(t\right)$ track ${\theta }_r\left(t\right)$. Therefore, as long as for $\forall t^{\prime }>0$, ${\theta }_r\left(t\right)\not\equiv \pm {\displaystyle \frac{\pi }{2}}$ for $t⩾t^{\prime }$, then $cos\theta =0$ only at isolated time instants during the transition process. A sufficient condition for ${\theta }_r\left(t\right)\not\equiv \pm {\displaystyle \frac{\pi }{2}}$$(t⩾t^{\prime })$ is that ${\theta }_r\left(t\right)=\pm {\displaystyle \frac{\pi }{2}}$ and ${\dot{\theta }}_r\left(t\right)=0$ (i.e., ${\omega }_r=0$) are not both true for $t⩾t^{\prime }$. Therefore, for $\forall t^{\prime }>0$, if the logical operation $\{{\theta }_r\left(t\right)=\pm {\displaystyle \frac{\pi }{2}}\}\wedge \{{\omega }_r=0\}=\mathit{false}$$(t⩾t^{\prime })$ holds, then one has $cos\theta \left(t\right)\not\equiv 0$ ($t⩾t^{\prime }$).

Then the convergence of ${\xi }_1$ is analyzed in two cases: (1) When $cos\theta =0$ at a certain time instant, ${\dot{V}}_{{\xi }_1}=0$ and ${\dot{\xi }}_1=0$. So, the value of ${\xi }_1$ stops being updated, i.e., ${\xi }_1$ stops converging. Since $\theta$ is time-varying, $cos\theta \not\equiv 0$. Further, ${\dot{V}}_{{\xi }_1}\not\equiv 0$ and ${\dot{\xi }}_1\not\equiv 0$; (2) When $cos\theta \ne 0$, ${\mu }_1>0$. In this case, ${\xi }_1$ converge exponentially over time. The convergence of ${\xi }_1$ is shown in Figure 2.

Therefore, it can be concluded that ${\xi }_1$ converges to zero as $t\to \infty$, i.e., ${\xi }_1$ converges exponentially to zero. The proof is completed.  □

Remark 3.

The convergence condition for ${\xi }_1$ in Theorem 2, defined as $\{{\theta }_r\left(t\right)=\pm {\displaystyle \frac{\pi }{2}}\}\wedge \{{\omega }_r\left(t\right)=0\}=\mathit{false}$ for all $t⩾t^{\prime }$, implies that the reference trajectory cannot remain a vertical line indefinitely after any time instant $t^{\prime }$. Physically, this requirement is easily satisfied in practice by ensuring a non-zero reference angular velocity ${\omega }_r\left(t\right)$ whenever the vehicle’s orientation aligns with the vertical axis. It is important to note that this condition does not preclude the tracking of straight-line paths. As long as the path is not persistently vertical, the proposed method ensures that the parameter estimation error ${\xi }_1$ converges to zero exponentially. This is further validated by the subsequent simulation where a slanted linear path is successfully tracked without any loss of performance. Therefore, the proposed adaptive mechanism ensures exponential convergence of the parameter estimation errors under a weaker excitation condition.

Remark 4.

In this design, the parameter estimates are not entirely derived from the parameter adaptation laws; instead, they incorporate the modification parts on the basis of the nominal parameter estimates. In the traditional Lyapunov-based adaptive control design as shown in Section 4.1, the parameter estimation errors ${\tilde{\tau }}_{d1},{\tilde{\tau }}_{d2}$ in (12) are precisely eliminated through the design of the adaptation law. However, in the I&I-based adaptation law design, the quadratic terms related to the parameter estimation error ${\xi }_1,{\xi }_2$ are retained in the Lyapunov function derivative, which guarantees the exponential convergence of ${\xi }_1,{\xi }_2$.

4.3. Controller Design

In the dynamic control loop, to achieve the tracking control to the reference trajectory generated by (3), the torque signals ${\tau }_1,{\tau }_2$ need to be designed to track $v_d,{\omega }_d$ in (6).

To this end, the Lyapunov function candidate is selected as

$$V_D=V_k+{\displaystyle \frac{1}{2}}{v}_e^2+{\displaystyle \frac{1}{2}}{\omega }_e^2+{V_{{\xi }_1}}^2+{V_{{\xi }_2}}^2$$

(24)

Differentiating $V_D$ along the error system (10) and dynamic model (1), it has

$$\begin{array}{ccc}\hfill {\dot{V}}_D& =& x_e(\omega y_e-v+v_{r}cos{\theta }_e)+y_e(-\omega x_e+v_{r}sin{\theta }_e)+{\displaystyle \frac{sin{\theta }_e}{k_3}}({\omega }_r-\omega )-{\displaystyle \frac{{\sigma }_2{\rho }_2}{J^2}}{\xi }_2^2\hfill \\ & & +v_e({\dot{v}}_d-a{\tau }_1-{\varphi }_1{\tau }_{d1})+{\omega }_e({\dot{\omega }}_d-b{\tau }_2-{\varphi }_2{\tau }_{d2})-{\displaystyle \frac{{\sigma }_1{\rho }_1}{m^2}}{cos}^2\theta {\xi }_1^2\hfill \end{array}$$

(25)

From (11), it has $v=v_d-v_e$. Substituting it into (25) yields

$$\begin{array}{ccc}\hfill {\dot{V}}_D& =& x_e(v_e-k_1{x}_e)+y_e{v}_{r}sin{\theta }_e+{\displaystyle \frac{sin{\theta }_e}{k_3}}({\omega }_e-k_3{y}_e{v}_r-k_{2}sin{\theta }_e)\hfill \\ & & +v_e({\dot{v}}_d-a{\tau }_1-{\varphi }_1{\widehat{\tau }}_{est1})+v_e{\varphi }_1{\xi }_1+{\omega }_e({\dot{\omega }}_d-b{\tau }_2-{\varphi }_2{\widehat{\tau }}_{est2})\hfill \\ & & +{\omega }_e{\varphi }_2{\xi }_2-{\displaystyle \frac{{\sigma }_1{\rho }_1}{m^2}}{cos}^2\theta {\xi }_1^2-{\displaystyle \frac{{\sigma }_2{\rho }_2}{J^2}}{\xi }_2^2\hfill \end{array}$$

(26)

Using the absolute value operation and Young’s inequality for $v_e{\varphi }_1{\xi }_1$ and ${\omega }_e{\varphi }_2{\xi }_2$ in (26) yields

$$\begin{array}{ccc}\hfill v_e{\varphi }_1{\xi }_1& ⩽& |v_e||{\varphi }_1||{\xi }_1|⩽{\displaystyle \frac{1}{m}}|v_e||{\xi }_1|⩽{\displaystyle \frac{{ϵ}_1}{m}}{\xi }_1^2+{\displaystyle \frac{1}{4{ϵ}_{1}m}}{v}_e^2\hfill \\ \hfill {\omega }_e{\varphi }_2{\xi }_2& ⩽& |{\omega }_e||{\varphi }_2||{\xi }_2|⩽{\displaystyle \frac{1}{J}}|{\omega }_e||{\xi }_2|⩽{\displaystyle \frac{{ϵ}_2}{J}}{\xi }_2^2+{\displaystyle \frac{1}{4{ϵ}_{2}J}}{\omega }_e^2\hfill \end{array}$$

(27)

where ${ϵ}_1,{ϵ}_2>0$ is the designed parameters. Based on (26) and (27), the torque signals are designed as

$$\begin{array}{ccc}\hfill {\tau }_1& =& {\displaystyle \frac{1}{a}}({\dot{v}}_d-{\varphi }_1{\widehat{\tau }}_{est1}+x_e+k_4{v}_e)\hfill \\ \hfill {\tau }_2& =& {\displaystyle \frac{1}{b}}({\dot{\omega }}_d-{\varphi }_2{\widehat{\tau }}_{est2}+x_e+{\displaystyle \frac{1}{k_3}}sin{\theta }_e+k_5{\omega }_e)\hfill \end{array}$$

(28)

Theorem 3.

For the dynamic model (1) of the UGV, under the controllers (28) and the adaptation laws (18) with appropriate designed parameters, if $\{{\theta }_r\left(t\right)=\pm {\displaystyle \frac{\pi }{2}}\}\wedge \{{\omega }_r=0\}=\mathit{false}$, then the closed-loop system is asymptotically stable.

Proof. 

Substituting (27) and (28) into (26) yields

$$\begin{array}{ccc}\hfill {\dot{V}}_D& =& -k_1{x}_e^2-{\displaystyle \frac{k_2}{k_3}}{sin}^2{\theta }_e-(k_4-{\displaystyle \frac{1}{4{ϵ}_{1}m}})v_e^2-(k_5-{\displaystyle \frac{1}{4{ϵ}_{2}J}}){\omega }_e^2\hfill \\ & & -({\displaystyle \frac{{\sigma }_1{\rho }_1}{m^2}}{cos}^2\theta -{\displaystyle \frac{{ϵ}_1}{m}}){\xi }_1^2-({\displaystyle \frac{{\sigma }_2{\rho }_2}{J^2}}-{\displaystyle \frac{{ϵ}_2}{J}}){\xi }_2^2\hfill \end{array}$$

(29)

Case I. When $cos\theta \ne 0$ and $cos\theta \nrightarrow 0$, there always exists appropriate parameters satisfying

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{k}_1>0,\hfill \\ {\displaystyle \frac{k_2}{k_3}}>0,\hfill \\ k_4-{\displaystyle \frac{1}{4{ϵ}_{1}m}}>0,\hfill \\ k_5-{\displaystyle \frac{1}{4{ϵ}_{2}J}}>0,\hfill \\ {\displaystyle \frac{{\sigma }_1{\rho }_1}{m^2}}{cos}^2\theta -{\displaystyle \frac{{ϵ}_1}{m}}>0,\hfill \\ {\displaystyle \frac{{\sigma }_2{\rho }_2}{J^2}}-{\displaystyle \frac{{ϵ}_2}{J}}>0.\hfill \end{array}\right.\end{array}$$

In this case, it has ${\dot{V}}_D⩽0$, which means that the closed-loop system is asymptotically stable.

Case II. When $cos\theta =0$ or $cos\theta \to 0$ at certain time instants, it is impossible to find the appropriate parameters ${\sigma }_1,{\rho }_1$ satisfying ${\displaystyle \frac{{\sigma }_1{\rho }_1}{m^2}}{cos}^2\theta -{\displaystyle \frac{{ϵ}_1}{m}}>0$. In these time instants, define $\eta =-({\displaystyle \frac{{\sigma }_1{\rho }_1}{m^2}}{cos}^2\theta -{\displaystyle \frac{{ϵ}_1}{m}}){\xi }_1^2$. In the worst case, i.e., $cos\theta =0$, $\eta ={\displaystyle \frac{{ϵ}_1}{m}}{\xi }_1^2$. The time derivative of $\eta$ can be calculated as

$$\begin{array}{ccc}\hfill \dot{\eta }& =& {\displaystyle \frac{2{ϵ}_1}{m}}{\xi }_1[{\dot{\widehat{\tau }}}_{n1}+{\sigma }_1{\nabla }_{\theta }{\alpha }_1(\theta ,v)\omega +{\sigma }_1{\nabla }_v{\alpha }_1(\theta ,v)(a{\tau }_1+{\varphi }_1{\tau }_{d1})]\hfill \\ & =& {\displaystyle \frac{2{ϵ}_1}{m}}{\xi }_1^2\left({\sigma }_1{\nabla }_v{\alpha }_1(\theta ,v){\varphi }_1\right)=-{\displaystyle \frac{2{ϵ}_1}{m^3}}{\xi }_1^2{\sigma }_1{cos}^2\theta \hfill \end{array}$$

(30)

Therefore, $\dot{\eta }=0$ or $\dot{\eta }\to 0$ when $cos\theta =0$ or $cos\theta \to 0$. It can be known from (30) that when $cos\theta =0$, the rate of change of $\eta$ is zero. From the definition of $\eta$, the rate of change of ${\xi }_1$ is also zero. The evolution of ${\xi }_1$ can be summarized as follows

$$\begin{array}{c}\hfill {\xi }_1\left\{\begin{array}{cc}\mathrm{keeps}\mathrm{constant},\hfill & \mathrm{when}cos\theta =0;\hfill \\ \mathrm{decreases}\mathrm{exponentially},\hfill & \mathrm{when}cos\theta \ne 0.\hfill \end{array}\right.\end{array}$$

It is similar to Theorem 2 that ${\xi }_1$ can converge asymptotically to zero if $\{{\theta }_r\left(t_0\right)={\displaystyle \frac{\pi }{2}}\pm \pi \wedge {\omega }_d=0\}=\mathrm{false}$. In this case, there always exists appropriate parameters satisfying

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{k}_1>0,\hfill \\ {\displaystyle \frac{k_2}{k_3}}>0,\hfill \\ k_4-{\displaystyle \frac{1}{4{ϵ}_{1}m}}>0,\hfill \\ k_5-{\displaystyle \frac{1}{4{ϵ}_{2}J}}>0,\hfill \\ {\displaystyle \frac{{\sigma }_2{\rho }_2}{J^2}}-{\displaystyle \frac{{ϵ}_2}{J}}>0.\hfill \end{array}\right.\end{array}$$

Therefore, it has

$$\begin{array}{ccc}\hfill {\dot{V}}_D& ⩽& -k_1{x}_e^2-{\displaystyle \frac{k_2}{k_3}}{sin}^2{\theta }_e-(k_4-{\displaystyle \frac{1}{4{ϵ}_{1}m}})v_e^2-(k_5-{\displaystyle \frac{1}{4{ϵ}_{2}J}}){\omega }_e^2-({\displaystyle \frac{{\sigma }_2{\rho }_2}{J^2}}-{\displaystyle \frac{{ϵ}_2}{J}}){\xi }_2^2+\eta \hfill \end{array}$$

where $\eta \to 0$ as $t\to \infty$. The closed-loop system is asymptotically stable.

Summarizing Case I and Case II, it can be known that the tracking error signals $v_e,{\omega }_e\to 0$ as $t\to \infty$. Further, the position tracking error signals $x_e,y_e,{\theta }_e\to 0$ as $t\to \infty$. Moreover, the parameter estimation errors ${\xi }_1,{\xi }_2\to 0$ as $t\to \infty$.  □

The asymptotic convergence of the parameter estimation error ${\xi }_1$ is conditional on the reference trajectory not being persistently vertical $\{{\theta }_r\left(t\right)=\pm {\displaystyle \frac{\pi }{2}}\}\wedge \{{\omega }_r=0\}=\mathrm{false}$, whereas other convergence results hold unconditionally. The convergence result holds in Theorems 2 and 3 for almost all initial conditions except for the singular configuration ${\theta }_r\left(t\right)=\pm {\displaystyle \frac{1}{2}}\pi$ with zero angular velocity. Therefore, the proposed adaptive controller achieves almost-global asymptotic tracking.

Remark 5.

In the above control design, the design of the adaptation law and the controller is completely separated. It follows a layered design idea. Compared with the traditional methods, the proposed method provides more adjustable degrees of freedom and more design flexibility by introducing the tuning functions ${\alpha }_1(\theta ,v),{\alpha }_2\left(\omega \right)$ and the parameters ${\sigma }_1,{\sigma }_2,{\rho }_1,{\rho }_2$. The proposed dual-loop control strategy is shown in Figure 3.

4.4. Improved Control Design

In the control design in the Section 4.2 and Section 4.3, the proposed adaptive control strategy can effectively track the reference trajectory. However, the layered design results in an unexpected term $\eta =-({\displaystyle \frac{{\sigma }_1{\rho }_1}{m^2}}{cos}^2\theta -{\displaystyle \frac{{ϵ}_1}{m}}){\xi }_1^2$ in (29). When $cos\theta =0$, $\eta ={\displaystyle \frac{{ϵ}_1}{m}}{\xi }_1^2>0$. Although Theorem 2 has proved that ${\xi }_1$ can converge to zero over time, before the closed-loop system reaches the steady state, the existence of $\eta ={\displaystyle \frac{{ϵ}_1}{m}}{\xi }_1^2>0$ will bring adverse effects to the transition process of each signal in the closed-loop system. Therefore, in order to optimize the control performance of the closed-loop system, an improved adaptive controller is proposed to avoid the unexpected term ${\displaystyle \frac{{ϵ}_1}{m}}{\xi }_1^2$.

Theorem 4.

For the dynamic model (1) of the UGV, under the following improved adaptive controller

$$\begin{array}{ccc}\hfill {\tau }_1& =& {\displaystyle \frac{1}{a}}({\dot{v}}_d-{\varphi }_1{\widehat{\tau }}_{est1}+x_e+k_4{v}_e),\hfill \\ \hfill {\tau }_2& =& {\displaystyle \frac{1}{b}}({\dot{\omega }}_d-{\varphi }_2{\widehat{\tau }}_{est2}+x_e+{\displaystyle \frac{1}{k_3}}sin{\theta }_e+k_5{\omega }_e),\hfill \\ \hfill {\dot{\widehat{\tau }}}_{n1}& =& -{\sigma }_1{\nabla }_{\theta }{\alpha }_1(\theta ,v)\omega -{\sigma }_1{\nabla }_v{\alpha }_1(\theta ,v)(a{\tau }_1+{\varphi }_1{\widehat{\tau }}_{est1})-v_e{\varphi }_1,\hfill \\ \hfill {\dot{\widehat{\tau }}}_{n2}& =& -{\sigma }_2{\nabla }_{\omega }{\alpha }_2\left(\omega \right)(b{\tau }_2+{\varphi }_2{\widehat{\tau }}_{est2})-{\omega }_e{\varphi }_2,\hfill \end{array}$$

(31)

the position tracking error signals $x_e,y_e,{\theta }_e$ converge to zero as $t\to \infty$.

Proof. 

Select the Lyapunov function candidate as (24). Differentiating $V_D$ along the error system (10) yields

$$\begin{array}{ccc}\hfill {\dot{V}}_D& =& x_e(\omega y_e-v+v_{r}cos{\theta }_e)+y_e(-\omega x_e+v_{r}sin{\theta }_e)+{\displaystyle \frac{sin{\theta }_e}{k_3}}({\omega }_r-\omega )\hfill \\ & & +v_e({\dot{v}}_d-a{\tau }_1-{\varphi }_1{\tau }_{d1})+{\omega }_e({\dot{\omega }}_d-b{\tau }_2-{\varphi }_2{\tau }_{d2})\hfill \\ & & +{\xi }_1[{\dot{\widehat{\tau }}}_{n1}+{\sigma }_1{\nabla }_{\theta }{\alpha }_1(\theta ,v)\omega +{\sigma }_1{\nabla }_v{\alpha }_1(\theta ,v)(a{\tau }_1+{\varphi }_1{\tau }_{d1})]\hfill \\ & & +{\xi }_2[{\dot{\widehat{\tau }}}_{n2}+{\sigma }_2{\nabla }_{\omega }{\alpha }_2\left(\omega \right)(b{\tau }_2+{\varphi }_2{\tau }_{d2})]\hfill \end{array}$$

Using (6) and (11), it has

$$\begin{array}{ccc}\hfill {\dot{V}}_D& =& x_e(v_e-k_1{x}_e)+y_e{v}_{r}sin{\theta }_e+{\displaystyle \frac{sin{\theta }_e}{k_3}}({\omega }_e-k_3{y}_e{v}_r-k_{2}sin{\theta }_e)\hfill \\ & & +v_e({\dot{v}}_d-a{\tau }_1-{\varphi }_1{\widehat{\tau }}_{est1})+{\omega }_e({\dot{\omega }}_d-b{\tau }_2-{\varphi }_2{\widehat{\tau }}_{est2})\hfill \\ & & +{\xi }_1[{\dot{\widehat{\tau }}}_{n1}+{\sigma }_1{\nabla }_{\theta }{\alpha }_1(\theta ,v)\omega +{\sigma }_1{\nabla }_v{\alpha }_1(\theta ,v)(a{\tau }_1+{\varphi }_1{\tau }_{d1})+v_e{\varphi }_1]\hfill \\ & & +{\xi }_2[{\dot{\widehat{\tau }}}_{n2}+{\sigma }_2{\nabla }_{\omega }{\alpha }_2\left(\omega \right)(b{\tau }_2+{\varphi }_2{\tau }_{d2})+{\omega }_e{\varphi }_2]\hfill \end{array}$$

(32)

Substituting the improved adaptive controller (31) into (32) yields

$$\begin{array}{ccc}\hfill {\dot{V}}_D& =& -k_1{x}_e^2-{\displaystyle \frac{k_2}{k_3}}{sin}^2{\theta }_e-k_4{v}_e^2-k_5{\omega }_e^2-{\displaystyle \frac{{\sigma }_1{\rho }_1}{m^2}}{cos}^2\theta {\xi }_1^2-{\displaystyle \frac{{\sigma }_2{\rho }_2}{J^2}}{\xi }_2^2\hfill \end{array}$$

(33)

In (33), $k_i,{\sigma }_1,{\sigma }_2,{\rho }_1,{\rho }_2,m,J>0$, $i=1,\cdots ,5$, so ${\dot{V}}_D=0$ if and only if $x_e,{\theta }_e,v_e,{\omega }_e,{\xi }_{1}cos\theta$, ${\xi }_2=0$. When ${\dot{V}}_D=0$, all signals in the closed-loop system reach the steady state, so ${\dot{\theta }}_e=0$ and $\omega ={\omega }_d$. From (10) and (6), it has ${\dot{\theta }}_e=-k_3{y}_e{v}_r-k_{2}sin{\theta }_e=0$. Since ${\theta }_e$ has converged to zero in the steady state, it can be concluded that $y_e=0$. Therefore, the following set

$$\begin{array}{ccc}\hfill I_{D_I}& =& \{x_e,y_e,{\theta }_e,v_e,{\omega }_e,{\xi }_1,{\xi }_2|{\dot{V}}_D=0\}\hfill \\ & =& \{x_e,y_e,{\theta }_e,v_e,{\omega }_e,{\xi }_1,{\xi }_2|x_e=0,y_e=0,{\theta }_e=0,\hfill \\ & & v_e=0,{\omega }_e=0,{\xi }_{1}cos\theta =0,{\xi }_2=0\}\hfill \end{array}$$

(34)

is an invariant set. It follows from (34) that the position tracking error signals $x_e,y_e,{\theta }_e$ converge to zero as $t\to \infty$.  □

Remark 6.

In the above improved adaptive control design, the adaptation law and the controller are no longer designed separately. The improved method combines the advantages of the traditional Lyapunov-based adaptive control approach with I&I technology. This design provides more adjustable degrees of freedom and greater design flexibility while also compensating for the estimation error of lumped uncertainties through the improved adaptation law in the Lyapunov design. As a result, ${\dot{V}}_D$ contains no unexpected terms, as shown in (33), thereby enhancing system performance. Furthermore, the improved adaptation law ensures convergence of the parameter estimation errors: ${\xi }_2$ converges to zero over time. For ${\xi }_1$, convergence occurs whenever $cos\theta \ne 0$. When $cos\theta =0$, if $\{{\theta }_r\left(t\right)=\pm {\displaystyle \frac{\pi }{2}}\}\wedge \{{\omega }_r=0\}=\mathit{false}$, ${\xi }_1$ converges to zero over time; when $cos\theta =0$, if $\{{\theta }_r\left(t\right)=\pm {\displaystyle \frac{\pi }{2}}\}\wedge \{{\omega }_r=0\}=\mathit{true}$, ${\xi }_1$ may be temporarily bounded. However, this configuration is transient and will not persist, so ${\xi }_1$ ultimately converges to zero.

Remark 7.

In practical UGV systems, state measurements are inevitably corrupted by noise, and the plant experiences unmodeled dynamics as well as perturbations. Under the proposed I&I adaptive framework, the estimation error dynamics can be formulated as a perturbed system driven by these noises and disturbances. Thanks to the design of the nonlinear injection function $\beta \left(x\right)$, the target manifold possesses a strictly negative-definite contraction matrix, which structurally guarantees the Input-to-State Stability (ISS) of both the parameter estimation and tracking loops with respect to bounded perturbations. Consequently, instead of suffering from parameter drifting-a common phenomenon in traditional Lyapunov-based adaptive control-the estimation and tracking errors under the proposed scheme are guaranteed to converge exponentially to a bounded residual set proportional to the noise variance and disturbance upper bounds, ensuring the system’s practical asymptotic tracking capability.

5. Simulation Experiments

In this section, simulation results are presented to demonstrate the effectiveness of the proposed dual-loop adaptive control strategy. To demonstrate the advantages of the proposed method, we add the comparative experiments of extending the traditional Lyapunov-based adaptive control method [13,30] and Sontag’s universal formula-based method [14,15] to the UGV systems. The model parameters for the UGV are given in

Table 1. System physical parameters.

Symbol	Quantity	Value
m	mass	15 kg
J	moment of inertia	15 kg$·{\mathrm{m}}^2$
r	radius of the driving wheel	$0.127$ m
$2R$	wheelbase of the rear wheels	$0.5$ m

.

In order to verify the effectiveness of the proposed control strategy in tracking different reference trajectories, the following four reference trajectories are given:

(a)

Straight Line: $v_r=1$ m/s, ${\omega }_r=0$ rad/s;

(b)

Trigonometric Function Curve: $v_r=1$ m/s, ${\omega }_r=sint$ rad/s;

(c)

Circle: $v_r=1$ m/s, ${\omega }_r=2$;

(d)

Spiral: $v_r=1.5-{\displaystyle \frac{1.5t}{t+10}}$ m/s, ${\omega }_r=1+{\displaystyle \frac{2t}{t+10}}$ rad/s.

Adaptive controllers are designed, respectively, by using the control strategy proposed in this paper to track the above four reference trajectories. The controller parameters and system initial values are given in

Table 2. Controller design parameters.

Parameter	Value	Parameter	Value
$k_1$	1	${\rho }_2$	15
$k_2$	1	${\sigma }_1$	10
$k_3$	1	${\sigma }_2$	10
$k_4$	30	${\tau }_{d1}$	5
$k_5$	30	${\tau }_{d2}$	5
${\rho }_1$	15	-	-

and

Table 3. System initial values.

State	Value	State	Value
$x\left(0\right)$	−2	$x_r\left(0\right)$	−1.5
$y\left(0\right)$	0.5	$y_r\left(0\right)$	0
$\theta \left(0\right)$	−0.5	${\theta }_r\left(0\right)$	0.4538
$v\left(0\right)$	0	${\widehat{\tau }}_{n1}\left(0\right)$	10
$\omega \left(0\right)$	0	${\widehat{\tau }}_{n2}\left(0\right)$	10

. The tracking curves are shown in Figure 4.

Figure 4 shows that the position signals of the UGV can track different reference trajectories under the proposed control mechanism, which means that the proposed controller is effective for different types of reference trajectories, indicating that the proposed control mechanism has a wide range of application scenarios.

To demonstrate the superiority of the proposed control mechanism, taking the spiral reference trajectory as an example, the proposed mechanism is compared with the traditional Lyapunov-based adaptive control method in [13,30] and Sontag’s universal formula-based adaptive control method in [14,15]. The comparison results are given in Figure 5, Figure 6, Figure 7 and Figure 8.

Figure 5 shows the position tracking errors $x_e,y_e,{\theta }_e$ of UGV under different controllers. It can be seen that, as stated in Remark 2, the position tracking errors can not converge to zero but fluctuate in the steady state under Sontag’s universal formula-based controller and the traditional Lyapunov-based controller. This fluctuation phenomenon is caused by the minimum phase characteristic of the UGV system. However, under the proposed controller, the position tracking errors can precisely converge to zero. In the MATLAB R2024a simulation, the convergence accuracy of $x_e,y_e,{\theta }_e$ can reach ${10}^{-6}$ m, ${10}^{-7}$ m and ${10}^{-8}$ m, respectively.

Figure 6 shows the tracking errors of desired control signals $v_e$ and ${\omega }_e$ of UGV under different controllers. It can be seen from the figure that under the traditional Lyapunov-based controller, the tracking errors are relatively large; under Sontag’s universal formula-based controller, the tracking error signals exhibit high-frequency oscillation. Compared with these two methods, the tracking error signals under the proposed controller can precisely converge to zero. In the MATLAB simulation, the convergence accuracy of $v_e$ and ${\omega }_e$ can reach ${10}^{-6}$ m/s and ${10}^{-7}$ m, respectively.

Figure 7 shows the parameter estimation errors of UGV under different adaptive controllers. It can be seen from the figure that the parameter estimation errors can not converge to zero under Sontag’s universal formula-based controller and the traditional Lyapunov-based controller. This is because in traditional adaptive control, parameter convergence requires PE conditions. The insufficient excitation properties of reference inputs during time-varying trajectory tracking lead to parameter estimation lagging behind trajectory changes. However, under the proposed adaptation law, the parameter estimation errors can converge exponentially to zero. Note that in Figure 7a, ${\xi }_1$ monotonically decreases in a stepwise manner to zero. This is because ${\xi }_1$ stops updating when $cos\theta =0$. This is consistent with what is described in Theorem 2 and Figure 2. In the MATLAB simulation, the convergence accuracy of ${\tau }_{d1}$ and ${\tau }_{d2}$ can reach ${10}^{-6}$ and ${10}^{-14}$, respectively.

Figure 8 shows the control signals of UGV under different adaptive controllers. It can be seen from the figure that the control signals ${\tau }_1$ and ${\tau }_2$ exhibit high-frequency oscillation under Sontag’s universal formula-based controller. The proposed control is smooth.

To provide a rigorous verification, the quantitative statistical metrics for the steady-state tracking phase ($t⩾2\mathrm{s}$) are compiled in

Table 4. Quantitative performance comparison of tracking errors ($t\ge 2\mathrm{s}$) and parameter estimation errors.

Metrics	Method	${\mathit{x}}_{\mathit{e}}$	${\mathit{y}}_{\mathit{e}}$	${\mathit{\theta }}_{\mathit{e}}$	${\mathit{v}}_{\mathit{e}}$	${\mathit{\omega }}_{\mathit{e}}$
RMS	I&I Method	$5.85\times {10}^{-5}$	$1.54\times {10}^{-4}$	$1.77\times {10}^{-4}$	$7.30\times {10}^{-4}$	$3.44\times {10}^{-4}$
	Traditional Method	$7.79\times {10}^{-4}$	$1.01\times {10}^{-3}$	$9.14\times {10}^{-4}$	$7.89\times {10}^{-3}$	$1.09\times {10}^{-2}$
	Sontag Method	$6.70\times {10}^{-4}$	$9.73\times {10}^{-4}$	$9.35\times {10}^{-4}$	$1.04\times {10}^{-2}$	$8.45\times {10}^{-3}$
Max Value	I&I Method	$3.77\times {10}^{-4}$	$1.67\times {10}^{-3}$	$2.2\times {10}^{-3}$	–	–
	Traditional Method	$1.27\times {10}^{-3}$	$2.30\times {10}^{-3}$	$2.10\times {10}^{-3}$	–	–
	Sontag Method	$3.13\times {10}^{-3}$	$9.07\times {10}^{-3}$	$1.34\times {10}^{-2}$	–	–
Final Parameter Estimation Error	I&I Method	of ${\tau }_{d1}$: $1.90\times {10}^{-6}$		of ${\tau }_{d2}$: $1.07\times {10}^{-14}$
	Traditional Method	of ${\tau }_{d1}$: $4.81$		of ${\tau }_{d2}$: $4.60$
	Sontag Method	of ${\tau }_{d1}$: $4.90$		of ${\tau }_{d2}$: $6.07$

. As demonstrated in Table 4, the proposed I&I method achieves superior tracking accuracy, suppressing the RMS errors of $x_e$, $y_e$, and ${\theta }_e$ to the order of ${10}^{-5}$ and ${10}^{-4}$, and restricting the velocity tracking RMS of $v_e$ and ${\omega }_e$ to $7.30\times {10}^{-4}$ and $3.44\times {10}^{-4}$, respectively. Furthermore, the maximum absolute tracking errors remain significantly smaller, validating its capability in regulating transient behaviors while eliminating residual oscillations. Most notably, while the traditional and Sontag-based schemes fail to achieve parameter convergence due to the lack of the PE condition—leaving substantial residual errors up to 6.07—the proposed adaptive law successfully drives the final estimation errors of ${\tau }_{d1}$ and ${\tau }_{d2}$ down to $1.90\times {10}^{-6}$ and $1.07\times {10}^{-14}$, thoroughly substantiating precise parameter convergence under a weaker, trajectory-dependent excitation condition.

Next, to verify the rapid response ability of the proposed method, it assumes that the friction increases at $t=10$ s of the simulation. In the simulation, the relevant parameters were set as ${\tau }_{d1}={\tau }_{d2}=5,t⩽10\mathrm{s}$ and ${\tau }_{d1}={\tau }_{d2}=60,t>10\mathrm{s}$. The handling capabilities of different methods are shown in Figure 9, Figure 10, Figure 11 and Figure 12. It can be obtained from the simulation results that under the proposed controller, the position tracking errors and the tracking errors for the desired signals can converge to the steady state more rapidly, and the steady-state accuracy is significantly improved compared to that of the other two control methods. The parameter estimation errors can also converge to zero more rapidly. These results all indicate that the proposed control mechanism can respond to increased frictional force rapidly, and the closed-loop system has a more rapid response speed.

6. Conclusions

In this paper, a novel dual-loop layered control mechanism with guaranteed parameter convergence under specified trajectory constraints is proposed for UGVs to address challenges arising from underactuation, non-minimum phase characteristics, and the inherent limitations of traditional adaptive methods. The proposed architecture bifurcates the control problem into a kinematic loop, which generates virtual velocity commands for trajectory tracking, and a dynamic loop, which employs a layered adaptive strategy. The proposed dual-loop I&I-based adaptive controller achieves almost-global asymptotic tracking of the reference trajectory, i.e., the pose errors converge to zero for almost all initial conditions, except for the singular configuration where $\{{\theta }_r\left(t\right)=\pm {\displaystyle \frac{\pi }{2}}\}\&\{{\omega }_r=0\}$. Comparative simulation results demonstrate that the proposed mechanism achieves superior tracking accuracy and a more rapid response speed than existing methods.

Future research will focus on analyzing how non-ideal physical factors, such as wheel slippage, mechanical backlash, and varying ground friction coefficients, deviate from the assumed rigid-body dynamics, which may cause the “asymptotic zero-error” convergence of the current simulation-based results to become only Uniformly Ultimately Bounded (UUB). Furthermore, to enhance the system’s anti-disturbance capabilities beyond the current lumped-uncertainty treatment, we will investigate dedicated disturbance estimation and rejection strategies, such as disturbance observers, tracking differentiators, or active disturbance rejection control (ADRC), to mitigate the effects of external noise and unmodeled dynamics.