Open AccessArticle

Backstepping-Based Nonsingular Terminal Sliding Mode Control for Finite-Time Trajectory Tracking of a Skid-Steer Mobile Robot

Mulugeta Debebe Teji

^1,2,†

Ting Zou

^2,*,†

and

Dinku Seyoum Zeleke

Department of Electrical and Computer Engineering, Addis Ababa Science and Technology University, Akaki Kality P.O. Box 16417, Ethiopia

Department of Mechanical and Mechatronics Engineering, Memorial University of Newfoundland, St. John’s, NL A1B 3X5, Canada

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Robotics 2024, 13(12), 180; https://doi.org/10.3390/robotics13120180

Submission received: 23 October 2024 / Revised: 6 December 2024 / Accepted: 13 December 2024 / Published: 16 December 2024

(This article belongs to the Special Issue Navigation Systems of Autonomous Underwater and Surface Vehicles)

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Abstract

Skid-steer mobile robots (SSMRs) are ubiquitous in indoor and outdoor applications. Their accurate trajectory tracking control is quite challenging due to the uncertainties arising from the complex behavior of frictional force, external disturbances, and fluctuations in the instantaneous center of rotation (ICR) during turning maneuvers. These uncertainties directly disturb velocities, hindering the robot from tracking the velocity command. This paper proposes a nonsingular terminal sliding mode control (NTSMC) based on backstepping for a four-wheel SSMR to cope with the aforementioned challenges. The strategy seeks to mitigate the impacts of external disturbances and model uncertainties by developing an adaptive law to estimate the integrated lumped outcome. The finite time stability of the closed-loop system is proven using Lyapunov’s theory. The designed NTSMC input is continuous and avoids noticeable chattering. It was noted in the simulation analysis that the proposed control strategy is strongly robust against disturbance and modeling uncertainties, demonstrating effective trajectory tracking performance in the presence of disturbance and modeling uncertainties.

Keywords:

skid-steer mobile robot; instantaneous center of rotation; uncertainty; nonsingular terminal sliding mode control; finite-time stability

1. Introduction

With the rapid advancement of technology, the potential application of autonomous mobile robots is enormous, revolutionizing society’s quality of life by improving human well-being and assisting human activities in general. Mobile robot applications include surveillance, planetary exploration, emergency rescue operations, mining, entertainment, precision agriculture, and transportation. The essential feature of a mobile robot is its capacity to navigate and maneuver efficiently within its surroundings, which primarily depends on its steering systems [1]. Skid-steer mobile robots (SSMRs) are widely used ground mobile robots owing to their simple mechanical structure, remarkable flexibility, and ability to produce enough traction forces. The concept of planar motion of a rigid body states that the pure rotation of a rigid body occurs around the instantaneous center of rotation (ICR).The application of this principle can also be extended to skid-steer vehicles, where the turning maneuvers are solely dependent on the slipping and lateral skidding of each wheel on the ground. Although these actions are not energy-efficient, they are necessary and crucial for the robot to alter its travel direction. An SSMR is steered by initiating lateral skidding, which results from unequal sideway forces generated on each side of the robot’s wheel. Namely, the robot laterally skids due to the moment generated by unequal forces exceeding the frictional moment [2].

Capturing and incorporating the friction force between the ground and the wheel into the dynamics of an SSMR is challenging, making the accurate motion modeling of SSMRs difficult due to the diverse nonlinear nature of the coefficients of friction. In recent years, model-free control algorithms have been proposed to improve the performance of SSMRs. For instance, a learning-based nonlinear model-predictive control was used by formulating disturbances as a Gaussian process in [3], a deep reinforcement learning algorithm was proposed for path tracking in [4], and an enhanced self-organizing incremental neural network was reported in [5]. Model-free approaches do not require a prior understanding of the robot’s dynamic or kinematic models. However, they require large amounts of data and computational complexity for real-time applications. In a more recent study, in contrast to model-free control, Zhu et al. [6] conducted a comparative study of six different control frameworks based on a kinematic model to counteract disturbances associated with unmodeled dynamics of SSMRs in outdoor applications. The control architectures including proportional-derivative, sliding mode (SMC), Lyapunov function, nonlinear model-predictive (NMPC), tube-based NMPC, and model-predictive sliding mode control have been studied. It has been shown that the tube-based NMPC exhibits a superior disturbance rejection capability compared to other control architectures. So far, several dynamics-based nonlinear controls have been proposed for SSMRs, such as adaptive control [7], dynamic control law [8], etc. A robust tube-based NMPC was presented in [9] to regulate the nonlinear tire–ground dynamics of SSMRs over deformable terrain. Prado et al. [10] proposed a tube-based NMPC to mitigate the impact of terrain disturbances originating from traction losses of SSMRs. A combined MPC with a fuzzy system was used in [11] to adapt the slope angle variation of SSMRs during sloppy maneuvering. However, despite its distinctive advantage of handling system constraints, an NMPC leads to computational complexity in executing real-time optimization problems. An adaptive fault-tolerant control utilizing the fuzzy sliding mode control technique was introduced in [12] for robotic manipulator systems subject to composite actuator failures, unknown disturbances, and joint friction.

Another critical challenge SSMRs face is the change in the location of

ICR

during turning maneuvers due to lateral skidding. When an SSMR traverses a curved path, the location of the

ICR

experiences a backward shift by a distance of

X_{ICR}

from point P due to lateral skidding, as shown in Figure 1. The distance (

X_{ICR}

) varies along the

X_{p}

direction depending on the amount of forces exerted by the robot’s wheel in the lateral direction, which results in lateral skidding. The nonlinear dynamic wheel–terrain interactions, instantaneous lateral velocity, and angular velocity during the curved path motion determine to what extent

ICR

shifts along the

X_{p}

direction. The worst-case scenario occurs if

X_{ICR}

extends beyond the robot’s wheelbase, which can cause lateral skidding and the loss of motion stability. Incorporating such behavior into SSMR modeling is significantly challenging. Thus, most of the work carried out on SSMRs ignores the effect of lateral skidding during turning maneuvers to simplify the modeling and control complexity. So far, however, very few studies have examined the role of lateral skidding, commonly keeping

X_{ICR}

at a constant value during turning maneuvers [8,13]. The varying nature of

X_{ICR}

gives rise to alternations in model parameters, resulting in the need for a robust control method dealing with modeling uncertainties.

SMC is a well-known robust control strategy that guarantees performance against uncertainties and disturbances. The design of an SMC contains two sequential steps that need to be undertaken. First, a sliding manifold is formulated based on the desired error dynamics. Subsequently, the control input is designed to ensure that the system trajectories remain on the sliding manifold, regardless of model uncertainties, nonlinearities, and external disturbances [14,15,16]. An important feature of a linear sliding manifold is its simplicity, which guarantees asymptotic stability despite the slower convergence time of the states closer to equilibrium. That means the system states cannot reach the equilibrium in a finite time [16,17]. A terminal sliding mode control (TSMC) based on the nonlinear switching manifold has emerged to deal with this problem [17,18,19]. TSMC offers the benefit of a fast finite-time convergence of the states to equilibrium. As indicated by Wu et al. [19], however, uncertainties can potentially impact the robustness of TSMC. Singularities may occur when the sliding manifold’s time derivative occurs, leading to infinite control input. In addition, the presence of uncertainties and disturbances may cause the system states to deviate from the sliding surface. This unwanted behavior is referred to as the chattering phenomenon, which causes the controlled system to exhibit high-frequency oscillations.

To tackle the singularity problem of TSMC, a nonsingular terminal sliding mode control (NTSMC) has been introduced in recent years. An adaptive, fast nonsingular terminal sliding mode control (AFNTSM) was proposed by Sun et al. [20] to ensure the precise and steady steering performance of a vehicle steer-by-wire system. The effectiveness of NTSMC was verified via an experimental test, which showed a fast finite convergence and robust performance against different road conditions. In another study, an AFNTSM was developed to realize attitude regulation of a quadrotor, and its effectiveness was validated through experimental testing [21]. The proposed AFNTSM mitigates the chattering phenomenon and can guarantee finite-time convergence. A novel fractional-order nonsingular terminal sliding mode control approach was introduced in [22] for tether satellite deployment, effectively addressing the singularity issue associated with the terminal sliding mode and achieving finite-time convergence. Furthermore, a fractional-order disturbance observer was proposed to cope with time-varying disturbances. In [23], a hybrid control method combining passivity-based control and nonsingular terminal sliding mode control (PBC-NTSMC) was proposed to suppress the resonance of the LC filter and enhance the system’s robustness to parameter and load variations in the current source converter system. This approach improves dynamic performance by minimizing chattering while maintaining passivity. Liang et al. [24] proposed a second-order fast NTSMC combined with an ANN for rehabilitation robots to achieve a rapid convergence rate and robustness against uncertain parameters and disturbances estimated by the ANN. A cascaded NTSM-PID controller scheme was presented in [25] to achieve robustness against disturbance and improve the tracking accuracy of a four-wheel independently driven skid-steer robotic vehicle. In [18,26,27,28], NTSMC was designed for trajectory tracking control of robotic manipulators to obtain finite convergence of states to equilibrium and robust performance against uncertainties and external disturbances. In view of all that has been mentioned so far, some of the challenges associated with SSMR motion modeling and its control arise from the nonlinear behavior of coefficients of friction and change in the location of the ICR, which poses modeling uncertainties. Therefore, well-calibrated, robust control laws are needed to address these challenges so that SSMRs can carry out their assigned tasks appropriately.

The main contributions of this paper are threefold: (a) This study formulates the uncertainties originating from friction forces due to wheel–ground interaction, changes in

X_{ICR}

location, and external disturbances combined as lumped parametric uncertainties. Subsequently, an adaptive law is designed that tracks the occurrence uncertainties and estimates the lumped parametric uncertainties. (b) A continuous and chattering-free robust nonsingular terminal sliding mode control (NTSMC) based on a backstepping method is developed. (c) In addition, the finite-time stability of the closed-loop system is proven using Lyapunov’s theory.

The remainder of this paper is outlined as follows. Section 2 and Section 3 cover the kinematic and dynamic models of the SSMR, respectively. Section 4 proposes the formulation of robust control and the synthesis of finite-time stability. Section 5 presents some numerical simulation results. Finally, Section 6 presents some concluding remarks.

2. Kinematic Model

Typically, an SSMR is equipped with wheels that cannot be steered, and it achieves steering by moving one side of the wheels at a faster speed than the other side [11,29]. As shown in Figure 1, two coordinate frames are defined to model the maneuverability of the SSMR: a fixed reference coordinate frame (

X_{f} O Y_{f}

) and a moving local coordinate frame attached at the robot’s center of mass (COM) (

X_{p} P Y_{p}

). The robot’s heading angle and angular velocity are represented by

θ

and

ω

, respectively. The actual robot pose (

q = {[X Y θ]}^{T}

) is used to define the robot’s configuration, while the reference (target) pose is represented by

q_{ref}

, which defines

q_{ref} = {[X_{r e f} Y_{r e f} r e f]}^{T}

. In a skid-steered wheeled mobile robot with an instantaneous center of rotation (

ICR

), the longitudinal and the lateral velocity components at the COM are

υ

and

υ_{y}

, respectively. Likewise, the velocity components (

υ_{i x}

and

υ_{i y}

) describe the longitudinal and lateral velocity of the

i th

wheel for

i = 1, 2, 3, 4

υ_{i}

is the total velocity vector of the

i th

wheel. The kinematic parameter (

2 c

) represents the width of the SSMR, while the distances from the COM to the rear wheels and from the COM to the front wheel are denoted by a and b, respectively. If we define

z

as a displacement vector, then

\dot{z} = {[υ ω]}^{T}

. Further details of SSMR modeling are found in [8,13]. The SSMR’s wheel velocities can be determined based on the robot’s velocity at the COM and its kinematic parameters through the equations provided below.

\begin{matrix} υ_{1 x} & = υ_{2 x} = υ - c ω \\ υ_{3 x} & = υ_{4 x} = υ + c ω \\ υ_{2 y} & = υ_{3 y} = (- X_{I C R} + b) ω \\ υ_{1 y} & = υ_{4 y} = (- X_{I C R} - a) ω, \end{matrix}

(1)

As mentioned before, SSMR motion is characterized by a pure rotation about

ICR

at each instant, involving a backward shift of a distance (

X_{ICR}

) from point P. This distance is calculated as

X_{ICR} = \frac{- υ_{y}}{\dot{θ}}

, which leads to the following velocity constraint introduced by Caracciolo et al. [13]:

υ_{y} + X_{ICR} \dot{θ} = 0,

(2)

where

υ_{y}

is the lateral velocity and

0 \leq X_{ICR} \leq a

. This implies that the distance (

X_{ICR}

) must not exceed a length of a, as shown in Figure 1, in order to prevent excessive skidding of the SSMR, which could lead to a total loss of control.

It is essential to ensure the boundedness of

X_{ICR}

to maintain lateral stability and reduce the robot’s side-slip angle during sharp turns [30]. The non-holonomic constraint can be rewritten in the Pfaffian form as follows:

[- sin θ cos θ X_{ICR}] {[\dot{X} \dot{Y} \dot{θ}]}^{T} = A (q) \dot{q} = 0 .

(3)

If we let

S (q) \in R^{3 \times 2}

be a full-rank matrix with columns orthogonal to

A (q)

, then we can write the following:

S^{T} (q) A^{T} (q) = 0,

(4)

where

S (q)

is defined in the following equation.

S (q) = [\begin{matrix} cos θ & X_{ICR} sin θ \\ sin θ & - X_{ICR} cos θ \\ 0 & 1 \end{matrix}] .

(5)

Now, the generalized velocity vector can be written as follows:

\dot{q} = S (q) \dot{z} .

(6)

It is noted from Equation (6) that the generalized velocity vector is dependent on

X_{ICR}

. Differentiating both sides of Equation (6) leads to

\ddot{q} = S (q) \ddot{z} + \dot{S} (q) \dot{z} .

(7)

3. Dynamic Model

The dynamic model provides a better understanding of the uncertainties and external disturbances affecting the system’s dynamic responses. Lateral skidding is an inevitable phenomenon, and determining the exact location of

X_{ICR}

is difficult due to the nonlinear, dynamic nature of the wheel–ground interaction during turning maneuvers. The unknown lateral skidding and ground interaction forces significantly affect the dynamic model of the SSMR. The dynamic model of the SSMR is formulated as follows [8]:

M (q) \ddot{q} + R (\dot{q}) + A^{T} (q) λ + τ_{d} = B (q) τ .

(8)

where

M (q) \in R^{3 \times 3}

is a symmetric positive definite inertia matrix,

R (\dot{q}) \in R^{3 \times 1}

is the generalized resistive force,

B (q) \in R^{3 \times 2}

is the input transformation matrix,

τ_{d} \in R^{3 \times 1}

represents torque disturbances, and

τ \in R^{2 \times 1}

is the input torque produced by the wheels. The terms

M (q)

R (\dot{q})

τ_{d}

B (q)

, and

τ

defined as follows:

\begin{matrix} M (q) = [\begin{matrix} m & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & I \end{matrix}], R (\dot{q}) = {[\begin{matrix} F_{r x} (\dot{q}) & F_{r y} (\dot{q}) & M_{r} (\dot{q}) \end{matrix}]}^{T} \\ τ_{d} = {[\begin{matrix} τ_{d 1} & τ_{d 2} & τ_{d 3} \end{matrix}]}^{T}, B (q) = \frac{1}{r} [\begin{matrix} cos θ & cos θ \\ sin θ & sin θ \\ - c & c \end{matrix}], τ = [\begin{matrix} τ_{L} \\ τ_{R} \end{matrix}] = [\begin{matrix} τ_{1} + τ_{2} \\ τ_{3} + τ_{4} \end{matrix}] \end{matrix}

where

τ_{L}

and

τ_{R}

are the input torque produced by the left and right wheels, respectively.

τ_{i}

is the input torque produced by the

i th

wheel for

i = 1, 2, 3, 4

A (q)

is defined in Equation (3) and given by

A (q) = [- sin θ cos θ X_{ICR}]

λ

represents the Lagrange multipliers or constraint forces.

For the sake of control, it is more appropriate to rewrite Equation (8) in relation to

\dot{z}

. Substituting Equation (7) into Equation (8) and multiplying both sides by

S^{T}

yields the following:

\tilde{M} \ddot{z} + \tilde{C} \dot{z} + \tilde{R} + {\tilde{τ}}_{d} = \tilde{B} τ .

(9)

The terms

\tilde{M}

\tilde{C}

\tilde{R}

{\tilde{τ}}_{d}

, and

\tilde{B}

are defined as follows:

\begin{matrix} \tilde{M} = S^{T} MS = [\begin{matrix} m & 0 \\ 0 & m X_{ICR}^{2} + I \end{matrix}], \\ \tilde{C} = S^{T} M \dot{S} = m X_{ICR} [\begin{matrix} 0 & \dot{θ} \\ - \dot{θ} & {\dot{X}}_{ICR} \end{matrix}], \\ \tilde{R} = S^{T} R = [\begin{matrix} F_{r x} (\dot{q}) \\ X_{ICR} F_{r y} (\dot{q}) + M_{r} (\dot{q}) \end{matrix}], \\ {\tilde{τ}}_{d} = S^{T} τ_{d} \\ \tilde{B} = S^{T} B = 1 / r [\begin{matrix} 1 & 1 \\ - c & c \end{matrix}], \end{matrix}

where m, I, and r denote the robot mass, moment of inertia, and wheel radius, respectively. It is important to note that

\tilde{M}

\tilde{C}

, and

\tilde{R}

, as defined earlier, are entirely dependent on the location of

X_{ICR}

. Any change in the location of

X_{ICR}

results in model parameter variations, thereby introducing model uncertainties to the dynamic system and posing a significant challenge in robot control.

F_{r x} (\dot{q})

and

F_{r y} (\dot{q})

represent the resistive forces in the inertial frame along the longitudinal and lateral directions, respectively, while

M_{r} (\dot{q})

denotes the resistant moment around the center of mass. These forces and moments can be computed as follows [8]:

\begin{matrix} F_{r x} (\dot{q}) = & \frac{m g μ_{x} cos θ [s g n (v_{2 x}) + s g n (v_{4 x})]}{2} - \frac{m g μ_{y} sin θ [a s g n (v_{3 y}) + b s g n (v_{4 y})]}{a + b} . \end{matrix}

(10)

\begin{matrix} F_{r y} (\dot{q}) = & \frac{m g μ_{x} sin θ [s g n (v_{2 x}) + s g n (v_{4 x})]}{2} + \frac{m g μ_{y} cos θ [a s g n (v_{3 y}) + b s g n (v_{4 y})]}{a + b} . \end{matrix}

(11)

\begin{matrix} M_{r} (\dot{q}) = & \frac{m g [2 a b μ_{y} s g n (v_{3 y}) - 2 a b μ_{y} s g n (v_{4 y})]}{2 (a + b)} - \frac{m g [a c μ_{x} s g n (v_{2 x}) + a c μ_{x} s g n (v_{4 x})]}{2 (a + b)} \\ - \frac{m g [b c μ_{x} s g n (v_{2 x}) + b c μ_{x} s g n (v_{4 x})]}{2 (a + b)}, \end{matrix}

(12)

where g,

μ_{x}

, and

μ_{y}

represent the gravity of the acceleration, longitudinal, and lateral coefficients of friction forces, respectively, and

s g n (\cdot)

denotes the signum function. Furthermore, it is essential to emphasize that the robot’s dynamic model of Equation (9) is also affected by model uncertainty due to the varying coefficients of friction forces, as denoted in Equations (10)–(12). These friction force coefficients depend primarily upon the dynamic nature of the surface of the wheel–ground interaction contact point.

4. Control Problem Formulation

For the trajectory tracking problem, the desired reference state vector is generated as follows.

{\dot{q}}_{ref} = [\begin{matrix} {\dot{X}}_{r e f} \\ {\dot{Y}}_{r e f} \\ {\dot{θ}}_{r e f} \end{matrix}] = [\begin{matrix} cos θ & 0 \\ sin θ & 0 \\ 0 & 1 \end{matrix}] [\begin{matrix} υ_{r e f} \\ ω_{r e f} \end{matrix}] .

(13)

where

υ_{r e f}

and

ω_{r e f}

are the reference velocity and angular velocity, respectively.

To ensure that the robot poses to track the intended reference trajectory, the trajectory tracking error can be described by the following equation:

e = T (θ) (q_{ref} - q),

(14)

where

e = {[e_{x} e_{y} e_{θ}]}^{T}

and

T (θ)

is the transformation matrix that transforms the local frame into a fixed reference coordinate frame.

T (θ) = [\begin{matrix} cos θ & sin θ & 0 \\ - sin θ & cos θ & 0 \\ 0 & 0 & 1 \end{matrix}] .

(15)

The corresponding tracking error derivative (

\dot{e} = {[{\dot{e}}_{x} {\dot{e}}_{y} {\dot{e}}_{θ}]}^{T}

) is, thus, readily obtained as follows:

[\begin{matrix} {\dot{e}}_{x} \\ {\dot{e}}_{y} \\ {\dot{e}}_{θ} \end{matrix}] = [\begin{matrix} υ_{r e f} cos e_{θ} \\ υ_{r e f} sin e_{θ} \\ ω_{r e f} \end{matrix}] + [\begin{matrix} - 1 & e_{y} \\ 0 & - e_{x} \\ 0 & - 1 \end{matrix}] [\begin{matrix} υ \\ ω \end{matrix}] .

(16)

The popularity of the backstepping control approach has grown among researchers in recent decades due to its application to nonlinear systems. This approach systematically defines the Lyapunov function for stability analysis and the design of a feedback control law [31,32]. Moreover, it has the potential to be integrated with other control techniques, including neural networks [33], adaptive control [34], reinforcement learning [35], sliding mode control [36], and fuzzy control [37], to improve the system’s overall performance. By virtue of its distinctive advantages, a backstepping kinematic controller is proposed in this study to eliminate the pose tracking error(

e

). Subsequently, the control input,

{\dot{z}}_{c} = {[υ_{c} ω_{c}]}^{T}

, can also be used as a command velocity for NTSMC. As proposed by Kanayama et al. [38],

{\dot{z}}_{c}

can be given by

\begin{matrix} [\begin{matrix} υ_{c} \\ ω_{c} \end{matrix}] = [\begin{matrix} k_{x} e_{x} + υ_{r e f} cos e_{θ} \\ ω_{r e f} + k_{y} υ_{r e f} e_{y} + k_{θ} υ_{r e f} sin e_{θ} \end{matrix}], \end{matrix}

(17)

where

k_{x}

k_{y}

, and

k_{θ}

are positive constant numbers.

The time derivative of

{\dot{z}}_{c}

can be determined as follows [39]:

\begin{matrix} [\begin{matrix} {\dot{υ}}_{c} \\ {\dot{ω}}_{c} \end{matrix}] = [\begin{matrix} k_{x} {\dot{e}}_{x} + {\dot{υ}}_{r e f} cos e_{θ} - υ_{r e f} {\dot{e}}_{θ} sin e_{θ} \\ {\dot{ω}}_{r e f} + k_{y} {\dot{υ}}_{r e f} e_{y} + k_{y} υ_{r e f} {\dot{e}}_{y} + k_{θ} {\dot{υ}}_{r e f} sin e_{θ} + k_{θ} υ_{r e f} {\dot{e}}_{θ} cos e_{θ} \end{matrix}] . \end{matrix}

(18)

Assuming that

υ_{r e f}

and

ω_{r e f}

are constant, Equation (18) becomes

\begin{matrix} [\begin{matrix} {\dot{υ}}_{c} \\ {\dot{ω}}_{c} \end{matrix}] = [\begin{matrix} k_{x} {\dot{e}}_{x} - υ_{r e f} {\dot{e}}_{θ} sin e_{θ} \\ k_{y} υ_{r e f} {\dot{e}}_{y} + k_{θ} υ_{r e f} {\dot{e}}_{θ} cos e_{θ} \end{matrix}] . \end{matrix}

(19)

Definition 1.

If we define the signal as

e_{z} = z_{c} - z

, the velocity tracking error (

{\dot{e}}_{z}

) can be defined as the difference between the command velocity control inputs (

{\dot{z}}_{c}

) and the true velocity (

\dot{z}

{\dot{e}}_{z} = {[{\dot{e}}_{z 1} {\dot{e}}_{z 2}]}^{T} = {\dot{z}}_{c} - \dot{z} .

(20)

It is well known that the coefficient of friction is highly nonlinear, complex, and difficult to model. Furthermore, accurately determining

X_{ICR}

while executing a turning maneuver poses a considerable challenge. The dynamic equation of Equation (9) can be rewritten as

({\tilde{M}}_{0} + Δ \tilde{M}) \ddot{z} + ({\tilde{C}}_{0} + Δ \tilde{C}) \dot{z} + ({\tilde{R}}_{0} + Δ \tilde{R}) + {\tilde{τ}}_{d} = \tilde{B} τ,

(21)

where

{\tilde{M}}_{0}

{\tilde{C}}_{0}

, and

{\tilde{R}}_{0}

denote the nominal model parameters and

Δ \tilde{M}

Δ \tilde{C}

, and

Δ \tilde{R}

are the corresponding uncertainties. Now, we introduce a lumped uncertainty, which denotes a collective source of uncertainties arising from model parameters and external disturbances combined with a single lumped perturbation of a specific structure. If we define the lumped parametric uncertainties as

ρ = Δ \tilde{M} \ddot{z} + Δ \tilde{C} \dot{z} + Δ \tilde{R} + {\tilde{τ}}_{d}

, then Equation (21) can be written as follows:

{\tilde{M}}_{0} \ddot{z} + {\tilde{C}}_{0} \dot{z} + {\tilde{R}}_{0} = \tilde{B} τ - ρ,

(22)

The parametric uncertainties are upper-bounded by positive vectors (

b_{0}

and

b_{1}

) but are assumed to be unknown.

ρ \leq b_{0} + {\dot{z}}_{ρ} b_{1},

(23)

where

{\dot{z}}_{ρ} = d i a g (|υ|, |ω|)

b_{0} = {[b_{01} b_{02}]}^{T}

, and

b_{1} = {[b_{11} b_{12}]}^{T}

Taking the time derivative of Equation (20) and using Equation (22) yields the following error dynamics:

{\ddot{e}}_{z} = {\ddot{z}}_{c} - {\tilde{M}}_{0}^{- 1} (\tilde{B} τ - ρ - {\tilde{C}}_{0} \dot{z} - {\tilde{R}}_{0}) .

(24)

As discussed above, a nonsingular terminal sliding manifold is selected to avoid the singularity problem [17] as follows:

S (t) = β e_{z} + {|{\dot{e}}_{z}|}^{γ} s g n ({\dot{e}}_{z}),

(25)

where

S (t) = {[S_{1} (t) S_{2} (t)]}^{T}

β = d i a g (β_{1}, β_{2})

, and

{|{\dot{e}}_{z}|}^{γ} s g n ({\dot{e}}_{z}) = {[{|{\dot{e}}_{z 1}|}^{γ_{1}} s g n ({\dot{e}}_{z 1}) {|{\dot{e}}_{z 2}|}^{γ_{2}} s g n ({\dot{e}}_{z 2})]}^{T}

. In general,

β

and

γ

are used to adjust the dynamic convergence behaviors of the sliding surface.

β_{i}

is chosen as

β_{i} > 0

for

i = 1, 2

, and

γ_{i}

is chosen to satisfy the condition of

1 < γ_{i} < 2

for

i = 1, 2

to mitigate the singularity problem associated with the derivative of

S

along the system dynamics.

To ensure the sliding motion occurs, the derivative of sliding surface

\dot{S} (t) = 0

should be satisfied.

\dot{S} = β {\dot{e}}_{z} + γ {|{\dot{e}}_{z}|}^{γ - 1} {\ddot{e}}_{z},

(26)

where

γ {|{\dot{e}}_{z}|}^{γ - 1} = d i a g (γ_{1} {|{\dot{e}}_{z}|}^{γ_{1} - 1}, γ_{2} {|{\dot{e}}_{z}|}^{γ_{2} - 1})

For the tracking problem, substituting Equation (24) into Equation (26) leads to

\dot{S} = β {\dot{e}}_{z} + γ {|{\dot{e}}_{z}|}^{γ - 1} [{\ddot{z}}_{c} - {\tilde{M}}_{0}^{- 1} (\tilde{B} τ - ρ - {\tilde{C}}_{0} \dot{z} - {\tilde{R}}_{0})] .

(27)

The design of feedback control is vital to stabilize

S = 0

to ensure the attractive property of the manifold. According to the sliding mode control design, the control input can be defined as

τ = τ_{e q} + τ_{r},

(28)

where

τ_{e q}

is the equivalent control and

τ_{r}

is reaching control to ensure the reachability of the sliding manifold. The system’s state trajectory is pulled towards the sliding manifold at

S = 0

. Hence, the equivalent control aims to maintain the system’s state on a sliding surface. As reported in [40],

τ_{e q}

can be obtained by setting

\dot{S} = 0

and

ρ = 0

in Equation (27).

\begin{matrix} τ_{e q} = & {({\tilde{M}}_{0}^{- 1} \tilde{B})}^{- 1} [β γ^{- 1} {|{\dot{e}}_{z}|}^{2 - γ} s g n ({\dot{e}}_{z}) + {\ddot{z}}_{c} + {\tilde{M}}_{0}^{- 1} {\tilde{C}}_{0} \dot{z} + {\tilde{M}}_{0}^{- 1} {\tilde{R}}_{0}] . \end{matrix}

(29)

Referring to Equation (23), let

{\hat{b}}_{i}

be the estimation of the upper bound and

b_{i}

be the true upper bound, assuming that

{\hat{b}}_{i} \leq b_{i}

for

i = 0, 1

. The reaching control,

τ_{r}

, can be designed as [40]

\begin{matrix} τ_{r} = & {({\tilde{M}}_{0}^{- 1} \tilde{B})}^{- 1} [k_{1} S + k_{2} {|S|}^{σ} s g n (S) + ({\hat{b}}_{0} + {\dot{z}}_{ρ} {\hat{b}}_{1})], \end{matrix}

(30)

where

k_{1} = d i a g (k_{11}, k_{12})

k_{2} = d i a g (k_{21}, k_{22})

{|S|}^{σ} s g n (S) = {[{|S_{1}|}^{σ_{1}} s g n (S_{1}) {|S_{2}|}^{σ_{2}} s g n (S_{2})]}^{T}

k_{1 i}

k_{2 i} > 0

, and

0 < σ_{i} < 1

for

i = 0, 1

. The parameter

σ

plays a crucial role in balancing the chattering of control signals with the system’s robustness. Therefore, the total NTSMC law for the system is expressed as follows:

\begin{matrix} τ = & {({\tilde{M}}_{0}^{- 1} \tilde{B})}^{- 1} [β γ^{- 1} {|{\dot{e}}_{z}|}^{2 - γ} s g n ({\dot{e}}_{z}) + {\ddot{z}}_{c} + {\tilde{M}}_{0}^{- 1} {\tilde{C}}_{0} \dot{z} + {\tilde{M}}_{0}^{- 1} {\tilde{R}}_{0} + k_{1} S + \\ k_{2} {|S|}^{σ} s g n (S) + ({\hat{b}}_{0} + {\dot{z}}_{ρ} {\hat{b}}_{1})] . \end{matrix}

(31)

\dot{S}

in Equation (27) is simplified further by substituting Equation (31) into Equation (27).

\dot{S} = γ {|{\dot{e}}_{z}|}^{γ - 1} [- k_{1} S - k_{2} {|S|}^{σ} s g n (S) + ({\bar{b}}_{0} + {\dot{z}}_{ρ} {\bar{b}}_{1})],

(32)

where

{\bar{b}}_{0}

and

{\bar{b}}_{1}

are defined as

{\bar{b}}_{0} = {\hat{b}}_{0} - b_{0}

and

{\bar{b}}_{1} = {\hat{b}}_{1} - b_{1}

, respectively.

ψ = {[S {\bar{b}}_{0} {\bar{b}}_{1}]}^{T}

is defined as the state vector, then the Lyapunov function (

V (ψ)

) is expressed as follows:

V (ψ) = \frac{1}{2} S^{T} S + \frac{1}{2} α_{0} {\bar{b}}_{0}^{T} {\bar{b}}_{0} + \frac{1}{2} α_{1} {\bar{b}}_{1}^{T} {\bar{b}}_{1},

(33)

where

α_{0}

and

α_{1}

are positive constants. The time derivative of

V (ψ)

yields

\dot{V} = S^{T} \dot{S} + α_{0} {\dot{\hat{b}}}_{0}^{T} {\bar{b}}_{0} + α_{1} {\dot{\hat{b}}}_{1}^{T} {\bar{b}}_{1} .

(34)

Substituting Equation (32) into Equation (34) yields

\begin{matrix} \dot{V} = & S^{T} (γ {|{\dot{e}}_{z}|}^{γ - 1} [- k_{1} S - k_{2} {|S|}^{σ} s g n (S) + ({\bar{b}}_{0} + {\dot{z}}_{ρ} {\bar{b}}_{1})]) + α_{0} {\dot{\hat{b}}}_{0}^{T} {\bar{b}}_{0} + α_{1} {\dot{\hat{b}}}_{1}^{T} {\bar{b}}_{1} . \end{matrix}

(35)

Rearranging Equation (35) yields

\begin{matrix} \dot{V} = & - S^{T} γ {|{\dot{e}}_{z}|}^{γ - 1} k_{1} S - S^{T} γ {|{\dot{e}}_{z}|}^{γ - 1} k_{2} {|S|}^{σ} s g n (S) + S^{T} γ {|{\dot{e}}_{z}|}^{γ - 1} {\bar{b}}_{0} + \\ S^{T} γ {|{\dot{e}}_{z}|}^{γ - 1} {\dot{z}}_{ρ} {\bar{b}}_{1} + α_{0} {\dot{\hat{b}}}_{0}^{T} {\bar{b}}_{0} + α_{1} {\dot{\hat{b}}}_{1}^{T} {\bar{b}}_{1} . \end{matrix}

(36)

The following inequality holds, since

S^{T} γ {|{\dot{e}}_{z}|}^{γ - 1} k_{2} {|S|}^{σ} s g n (S) \geq 0

. In addition, the assumption of

{\hat{b}}_{i} \leq b_{i}

leads to

{\bar{b}}_{i} \leq 0

\begin{matrix} \dot{V} \leq & - S^{T} γ {|{\dot{e}}_{z}|}^{γ - 1} k_{1} S + S^{T} γ {|{\dot{e}}_{z}|}^{γ - 1} {\bar{b}}_{0} + S^{T} γ {|{\dot{e}}_{z}|}^{γ - 1} {\dot{z}}_{ρ} {\bar{b}}_{1} + α_{0} {\dot{\hat{b}}}_{0}^{T} {\bar{b}}_{0} + α_{1} {\dot{\hat{b}}}_{1}^{T} {\bar{b}}_{1} \\ = - S^{T} γ {|{\dot{e}}_{z}|}^{γ - 1} k_{1} S - S^{T} γ {|{\dot{e}}_{z}|}^{γ - 1} |{\bar{b}}_{0}| - S^{T} γ {|{\dot{e}}_{z}|}^{γ - 1} {\dot{z}}_{ρ} |{\bar{b}}_{1}| - 2 α_{0} {\dot{\hat{b}}}_{0}^{T} |{\bar{b}}_{0}| + \\ α_{0} {\dot{\hat{b}}}_{0}^{T} |{\bar{b}}_{0}| - 2 α_{1} {\dot{\hat{b}}}_{1}^{T} |{\bar{b}}_{1}| + α_{1} {\dot{\hat{b}}}_{1}^{T} |{\bar{b}}_{1}|, \end{matrix}

(37)

where

|{\bar{b}}_{0}| = {[|{\bar{b}}_{01}| |{\bar{b}}_{02}|]}^{T}

and

|{\bar{b}}_{1}| = {[|{\bar{b}}_{11}| |{\bar{b}}_{12}|]}^{T}

. We know that

{\hat{b}}_{i}

is assumed to be positive. In order to develop an adaptive law to estimate the lumped parametric uncertainties, Equation (37) can be written as

\begin{matrix} \dot{V} \leq & - S^{T} γ {|{\dot{e}}_{z}|}^{γ - 1} k_{1} S + (α_{0} {\dot{\hat{b}}}_{0}^{T} - {|S|}^{T} γ {|{\dot{e}}_{z}|}^{γ - 1}) |{\bar{b}}_{0}| \\ + (α_{1} {\dot{\hat{b}}}_{1}^{T} - {|S|}^{T} γ {|{\dot{e}}_{z}|}^{γ - 1} {\dot{z}}_{ρ}) |{\bar{b}}_{1}| - 2 α_{0} {\dot{\hat{b}}}_{0}^{T} |{\bar{b}}_{0}| - 2 α_{1} {\dot{\hat{b}}}_{1}^{T} |{\bar{b}}_{1}|, \end{matrix}

(38)

where

{|S|}^{T} = [|S_{1}| |S_{2}|]

. Therefore, the estimated adaptive law of the upper bound (

{\dot{\hat{b}}}_{0}

) and

{\dot{\hat{b}}}_{1}

can be explicitly defined as

{\dot{\hat{b}}}_{0} = {[{\dot{\hat{b}}}_{01} {\dot{\hat{b}}}_{02}]}^{T}

and

{\dot{\hat{b}}}_{1} = {[{\dot{\hat{b}}}_{11} {\dot{\hat{b}}}_{12}]}^{T}

, respectively. Note:

γ_{01} = γ_{11} = γ_{1}

and

γ_{02} = γ_{12} = γ_{2}

. Similarly,

α_{01} = α_{02} = α_{0}

and

α_{11} = α_{12} = α_{1}

. From Equation (38), it is evident that the absolute values of adaptive estimation errors always satisfy the following inequalities:

|{\bar{b}}_{0}| \geq 0

and

|{\bar{b}}_{1}| \geq 0

. Therefore, it is necessary to ensure the condition (

\dot{V} \leq 0

) by setting the second and third terms inside the brackets to zero to achieve a stable estimation update rule. Thus, the estimation update rule becomes

{\dot{\hat{b}}}_{01} = \frac{γ_{01}}{α_{01}} |s_{1}| {|υ_{c} - υ|}^{γ_{01} - 1} .

(39)

{\dot{\hat{b}}}_{02} = \frac{γ_{02}}{α_{02}} |s_{2}| {|ω_{c} - ω|}^{γ_{02} - 1} .

(40)

{\dot{\hat{b}}}_{11} = \frac{γ_{11}}{α_{11}} |s_{1}| {|υ_{c} - υ|}^{γ_{11} - 1} |υ| .

(41)

{\dot{\hat{b}}}_{12} = \frac{γ_{12}}{α_{12}} |s_{2}| {|ω_{c} - ω|}^{γ_{12} - 1} |ω| .

(42)

A deviation of the

X_{ICR}

location from its nominal value leads to a change in the model parameters of

Δ \tilde{M}

Δ \tilde{C}

, and

Δ \tilde{R}

. Additionally, changes in friction forces are also observed in

Δ \tilde{R}

, which constitute lumped uncertainties, as indicated earlier in Equation (22). Hence, any change in modeling error and external disturbances is reflected in velocity errors and sliding error dynamics. Consequently, the estimation rule of Equations (39)–(42) regularly monitors the changes and updates the estimation of the lumped uncertainties that occur during operations.

Now, Equation (38) is simplified as

\begin{matrix} \dot{V} \leq & - S^{T} γ {|{\dot{e}}_{z}|}^{γ - 1} k_{1} S - 2 α_{0} {\dot{\hat{b}}}_{0}^{T} |{\bar{b}}_{0}| - 2 α_{1} {\dot{\hat{b}}}_{1}^{T} |{\bar{b}}_{1}| \\ = - (S^{T} γ {|{\dot{e}}_{z}|}^{γ - 1} k_{1} S + 2 α_{0} {\dot{\hat{b}}}_{0}^{T} |{\bar{b}}_{0}| + 2 α_{1} {\dot{\hat{b}}}_{1}^{T} |{\bar{b}}_{1}|) . \end{matrix}

(43)

Substituting Equations (39)–(42) into Equation (43) yields

\begin{matrix} \dot{V} \leq & - (γ_{1} k_{11} {|υ_{c} - υ|}^{γ_{1} - 1} |s_{1}| |s_{1}| + γ_{2} k_{12} {|ω_{c} - ω|}^{γ_{2} - 1} |s_{2}| |s_{2}| \\ + 2 \frac{γ_{01}}{α_{01}} α_{01} |s_{1}| {|υ_{c} - υ|}^{γ_{01} - 1} |{\bar{b}}_{01}| + 2 \frac{γ_{02}}{α_{02}} α_{02} |s_{2}| {|ω_{c} - ω|}^{γ_{02} - 1} |{\bar{b}}_{02}| \\ + 2 \frac{γ_{11}}{α_{11}} α_{11} |s_{1}| {|υ_{c} - υ|}^{γ_{11} - 1} |υ| |{\bar{b}}_{11}| + 2 \frac{γ_{12}}{α_{12}} α_{12} |s_{2}| {|ω_{c} - ω|}^{γ_{12} - 1} |ω| |{\bar{b}}_{12}|) . \end{matrix}

(44)

To achieve a more concise expression, we define the following:

\begin{matrix} Γ_{1} = γ_{1} k_{01} {|υ_{c} - υ|}^{γ_{1} - 1} |s_{1}|, \\ Γ_{2} = γ_{2} k_{02} {|ω_{c} - ω|}^{γ_{2} - 1} |s_{2}|, \\ Γ_{3} = 2 \frac{γ 01}{α 01} {|υ_{c} - υ|}^{γ_{01} - 1} |s_{1}|, \\ Γ_{4} = 2 \frac{γ 02}{α 02} {|ω_{c} - ω|}^{γ_{02} - 1} |s_{2}|, \\ Γ_{5} = 2 \frac{γ 11}{α 11} {|υ_{c} - υ|}^{γ_{12} - 1} |s_{1}| |υ|, \\ Γ_{6} = 2 \frac{γ 12}{α 12} {|ω_{c} - ω|}^{γ_{12} - 1} |s_{2}| |ω|, \end{matrix}

Now,

\dot{V}

is rewritten as

\begin{matrix} \dot{V} \leq & - (Γ_{1} |s_{1}| + Γ_{2} |s_{2}| + Γ_{3} α_{01} |{\bar{b}}_{01}| + Γ_{4} α_{02} |{\bar{b}}_{02}| + Γ_{5} α_{11} |{\bar{b}}_{11}| + Γ_{6} α_{12} |{\bar{b}}_{12}|) \\ \leq - η (|s_{1}| + |s_{2}| + |α_{01} {\bar{b}}_{01}| + |α_{02} {\bar{b}}_{02}| + |α_{11} {\bar{b}}_{11}| + |α_{12} {\bar{b}}_{12}|), \end{matrix}

(45)

where

η =

min

\{Γ_{1}, Γ_{2}, Γ_{3}, Γ_{4}, Γ_{5}, Γ_{6}\}

It is noteworthy that the subsequent inequality is valid [41] for

X, Y, Z \in ℜ

{(X^{2} + Y^{2} + Z^{2})}^{\frac{1}{2}} \leq |X| + |Y| + |Z| .

(46)

Equation (33) can be rewritten as

V (ψ) = {(\frac{1}{\sqrt{2}})}^{2} [S_{1}^{2} + S_{2}^{2} + {(α_{01} {\bar{b}}_{01})}^{2} + {(α_{02} {\bar{b}}_{02})}^{2} + {(α_{11} {\bar{b}}_{11})}^{2} + {(α_{12} {\bar{b}}_{12})}^{2}]

(47)

Hence, comparing Equation (45), Equation (46), and Equation (47), Equation (45) can be simplified as

\dot{V} \leq - \sqrt{2} η V^{\frac{1}{2}} .

(48)

Lemma 1

([42,43]). Consider the nonlinear system described by

\dot{ξ} = f (ξ)

, whose equilibrium is

ξ = 0

, suppose there is a

C^{1}

(continuously differentiable) function (

V (x)

) defined in a neighborhood (

D \subset R^{n}

) of the origin and there are real numbers (

Υ > 0

and

0 < φ < 1

) such that

V (x) > 0

on D and

\dot{V} (x) + Υ V^{φ} (x) \leq 0

(along the trajectory) on D. Then, the origin of the system is finite-time stable. Moreover, the settling time, depending on the initial slate (

x (0) = x_{o}

), is given by

T (x_{o}) \leq \frac{1}{Υ (1 - φ)} V^{1 - φ} (x_{o}),

(49)

for

x_{o}

in some open neighborhood of the origin.

Hence, the finite-time stability criterion of the system is satisfied based on Lemma 1, implying that V converges from the initial

V (0)

to the origin according to the inequality given by Equation (48).

T (0) \leq \frac{\sqrt{2}}{2 η} V^{\frac{1}{2}} (0) .

(50)

Despite uncertainties and disturbances, the sliding surface (

S

{\bar{b}}_{0}

, and

{\bar{b}}_{1}

converge to the origin within a finite time (

T (0)

5. Numerical Results

A numerical example is used to implement the proposed control strategy. The SSMR is required to follow a desired circular trajectory path starting from an initial pose of

q (0) = {[2.0 1.0 p i / 4]}^{T}

. The reference circular trajectory is defined in Equation (13), where the reference velocities are given by

υ_{r e f} = 0.5 m / s

and

ω_{r e f} = 0.2 rad / s

. As noted in [8], a fixed coordinate of the

ICR

should be selected as

X_{ICR} \in (- a, b)

. Hence, the nominal value of

X_{ICR}

is assigned to

X_{ICR 0}

. Referring to Figure 1, the dimensions of the SSMR and nominal values of model parameters are given in Table 1.

The complete control-block diagram for the wheeled mobile robot is shown in Figure 2. The block diagram consists of an outer-loop control and an inner-loop control. The design of the outer-loop control aims to minimize the robot’s pose errors and to generate a command velocity input that the inner-loop control can utilize. On the other hand, the purpose of the inner-loop control is to minimize the error between the command and true velocities. It is noted that the parametric change in

X_{ICR}

introduces parametric uncertainties in

\tilde{M}

\tilde{C}

, and

\tilde{R}

. Moreover, the parametric change in the coefficient of friction introduces parametric uncertainties in

\tilde{R}

. In order to validate the robustness performance of against modeling uncertainties, the following uncertainties are considered:

δ μ_{x} = 0.15 |sin (π / 4)|

for

t \geq 10 s e c

δ μ_{y} = 0.2 |sin (π / 4)|

for

15 \leq t \leq 20 s

δ X_{ICR} = - 0.09 |sin (π / 4)|

for

25 \leq t \leq 30 s e c

, and

δ τ_{d} = 35 |sin (π / 4)|

for

35 \leq t \leq 45 s e c

Generally, validating the proposed control involves trade-offs between robustness and the desired tracking performance. In the following, we outline the criteria for the parameter selection guideline for the proposed control law and their values for the SSMR under study. Increases in the

k_{x}

k_{y}

, and

k_{θ}

values can lead to faster convergence of the velocity tracking error (

{\dot{e}}_{z}

) to zero but may also increase overshoots. The

α

parameter is used to adjust the dynamic behaviors of the estimation update rules, whereas

γ

affects both the dynamic behaviors of estimation update rules and sliding surfaces. By fixing the

γ

value, the smaller

α

value enhances the fast convergence of the estimation update rules to the true value of modeling error and external disturbances but may invoke unwanted overshoot. An increase in

β

enables the attainment of a quicker convergence rate. However, the selection of the

β

value should compromise between enhancing the convergence rate and controlling the transient response. The

k_{1}

k_{2}

, and

σ

parameters in the reaching control law are crucial in determining the system’s robustness. Increases in the values of

k_{1}

and

k_{2}

enhance the system’s robustness at the expense of the smoothness of the control signal. Conversely, the

σ

parameter effectively manages the trade-off between control-signal chattering and robustness. The constant gains underwent tuning in various simulation tests, and the final values are provided in Table 2.

The robustness of NTSMC against model uncertainties and external disturbances is examined by considering the time-varying parametric perturbation of

X_{ICR}

, as well as friction coefficients along the x and y directions during turning maneuvers. Figure 3 and Figure 4 illustrate the time history of the velocity and angular velocity of the SSMR, respectively. It is clear that the velocity does not converge near the initial time, which denotes unstable motion; however, it is stabilized for the rest of the simulation time. Compared to other types of uncertainties, the external disturbance has a notable effect on velocity, as shown in Figure 3. On the other hand, disturbance in lateral friction force has a notable influence on the angular velocity of the robot, as shown in Figure 4. However, as shown in the time history of the figures, the designed NTSMC eliminates both disturbances and performs better trajectory tracking. The robot poses are plotted in Figure 5, and the RMS values for pose errors are presented in Table 3. These confirm that the control enables the SSMR to track the target pose despite the disturbances. As shown in Figure 6, the robot follows the desired circular path precisely in spite of the disturbances. Figure 7 shows that the proposed torque input is a continuous control and does not exhibit a noticeable chattering problem. It is observed from the simulation results that time-varying parametric perturbation of

X_{I C R}

during turning maneuvers is successfully compensated for by a robust NTSMC. Finally, the simulation results shown in Figure 8 clearly reveal the asymptotically convergent property of NTSMC.

6. Conclusions

Accurately determining uncertainties caused by changes in

ICR

location, friction force, and external disturbances poses a significant challenge. The proposed control strategy generally comprises the position control loop using the backstepping technique and the velocity control loop using NTSMC to eliminate the position and velocity errors, respectively. The Lyapunov function was used to obtain the lumped parametric uncertainties, and the finite-time stability of the overall system was also verified. Circular trajectory path simulations for a typical SSMR were conducted to verify the proposed approach. The proposed sliding surfaces converge to the origin despite the uncertainties and disturbances, which demonstrates that the control strategy is robust against modeling uncertainties and disturbances. In addition, the simulation result validates that NTSMC provides chattering- and singularity-free control. Notably, the structure and parameters of lumped uncertainties are predefined in the proposed method. In the future, a more flexible, non-parametric, learning-based uncertainty prediction will be investigated to achieve adjustable parameters for lumped uncertainties by conducting experiments on SSMRs.

Author Contributions

M.D.T. is responsible for the basic concepts, discussing the results, and writing the article. T.Z. is the corresponding author, responsible for the basic concepts, discussing the results, proofreading, and revising the manuscript. D.S.Z. is responsible for proofreading and revising the manuscript. All authors have read and agreed to the final version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The original data presented in the study are openly available on GitHub at https://github.com/tzouCA/NTSMC.git accessed on 12 December 2024.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. SSMR with an ICR of

X_{ICR}

; kinematic parameters of a, b, and c; a robot heading angle of

θ

; velocities at the center of mass (COM) of

υ

υ_{y}

, and

ω

; and velocity components of

υ_{i x}

and

υ_{i y}

with a velocity vector of

υ_{i}

for the

i th

wheel for

i = 1, \dots, 4

Figure 1. SSMR with an ICR of

X_{ICR}

; kinematic parameters of a, b, and c; a robot heading angle of

θ

; velocities at the center of mass (COM) of

υ

υ_{y}

, and

ω

; and velocity components of

υ_{i x}

and

υ_{i y}

with a velocity vector of

υ_{i}

for the

i th

wheel for

i = 1, \dots, 4

Figure 2. The complete proposed control with reference velocities (

q_{ref}

) of

υ_{r e f}

and

ω_{r e f}

, an actual robot pose of

q

, a transformation matrix of

T (θ)

, an outer-loop position error of

e

with a backstepping control of

{\dot{z}}_{c}

, an inner-loop velocity error of

{\dot{e}}_{z}

with a NTSMC of

τ

, and a torque disturbance of

τ_{d}

Figure 2. The complete proposed control with reference velocities (

q_{ref}

) of

υ_{r e f}

and

ω_{r e f}

, an actual robot pose of

q

, a transformation matrix of

T (θ)

, an outer-loop position error of

e

with a backstepping control of

{\dot{z}}_{c}

, an inner-loop velocity error of

{\dot{e}}_{z}

with a NTSMC of

τ

, and a torque disturbance of

τ_{d}

Figure 3. The time histories of the command velocity (

υ_{c}

) and the true velocity (

υ

) during simulation.

Figure 3. The time histories of the command velocity (

υ_{c}

) and the true velocity (

υ

) during simulation.

Figure 4. The time histories of the angular velocity command (

ω_{c}

) and the true angular velocity (

ω

) during simulation.

Figure 4. The time histories of the angular velocity command (

ω_{c}

) and the true angular velocity (

ω

) during simulation.

Figure 5. A comparison between the desired robot pose (

q_{ref}

) and the actual pose (

q

) in the simulation.

Figure 5. A comparison between the desired robot pose (

q_{ref}

) and the actual pose (

q

) in the simulation.

Figure 6. Robot’s trajectory following a desired circular path.

Figure 7. Input torque produced by the left side of the wheel (

τ_{L}

) and the right side of the wheel (

τ_{R}

Figure 7. Input torque produced by the left side of the wheel (

τ_{L}

) and the right side of the wheel (

τ_{R}

Figure 8. The time history of sliding surfaces

S_{1}

and

S_{2}

Figure 8. The time history of sliding surfaces

S_{1}

and

S_{2}

Table 1. SSMR dimensions and nominal values of model parameters [13].

Parameter	Value
a	0.55 m
b	0.37 m
$2 c$	0.63 m
r	0.2 m
m	116 kg
I	20 kgm²
$X_{ICR 0}$	$- 0.18$ m
$μ_{x 0}$	$0.15$
$μ_{y 0}$	$0.35$

Table 2. Control and sliding surface design constants.

Constant	Value
$k_{x}$	$1.5$
$k_{y}$	$20.5$
$k_{θ}$	$2.5$
$α_{0} = α_{1}$	22
$β_{1}$ = $β_{2}$	$2.4$
$γ_{1}$ = $γ_{2}$	$1.5$
$k_{11}$ = $k_{12}$	$4.5$
$k_{21}$ = $k_{22}$	20
$σ_{1}$ = $σ_{2}$	$0.5$

Table 3. Root mean square (RMS) of tracking error.

RMS Error in X-Direction	RMS Error in Y-Direction	RMS Error in $θ$
$0.0131 m$	$0.0042 m$	$0.0214 rad$

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MDPI and ACS Style

Teji, M.D.; Zou, T.; Zeleke, D.S. Backstepping-Based Nonsingular Terminal Sliding Mode Control for Finite-Time Trajectory Tracking of a Skid-Steer Mobile Robot. Robotics 2024, 13, 180. https://doi.org/10.3390/robotics13120180

AMA Style

Teji MD, Zou T, Zeleke DS. Backstepping-Based Nonsingular Terminal Sliding Mode Control for Finite-Time Trajectory Tracking of a Skid-Steer Mobile Robot. Robotics. 2024; 13(12):180. https://doi.org/10.3390/robotics13120180

Chicago/Turabian Style

Teji, Mulugeta Debebe, Ting Zou, and Dinku Seyoum Zeleke. 2024. "Backstepping-Based Nonsingular Terminal Sliding Mode Control for Finite-Time Trajectory Tracking of a Skid-Steer Mobile Robot" Robotics 13, no. 12: 180. https://doi.org/10.3390/robotics13120180

APA Style

Teji, M. D., Zou, T., & Zeleke, D. S. (2024). Backstepping-Based Nonsingular Terminal Sliding Mode Control for Finite-Time Trajectory Tracking of a Skid-Steer Mobile Robot. Robotics, 13(12), 180. https://doi.org/10.3390/robotics13120180

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Backstepping-Based Nonsingular Terminal Sliding Mode Control for Finite-Time Trajectory Tracking of a Skid-Steer Mobile Robot

Abstract

1. Introduction

2. Kinematic Model

3. Dynamic Model

4. Control Problem Formulation

5. Numerical Results

6. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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