# Stabilization of a Class of Nonlinear Underactuated Robotic Systems through Nonsingular Fast Terminal Sliding Mode Control

Authors: Yaobin Song; Hui Li; Xiaoling Shi
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

To facilitate the stabilization of nonlinear underactuated robotic systems under perturbation, a novel nonsingular fast terminal sliding mode control method is proposed. Based on the system transformation into an integrator chain, the combination of twisting-like algorithm and a nonsingular fast terminal sliding mode control technique is employed to achieve the stabilization of the studied systems, which can drive the robot states (joint positions and velocities) to the desired region and then maintain the system at the equilibrium point in finite time. The robustness of the proposed method is validated by the Lyapunov direct method. Finally, numerical simulation results further demonstrate that the proposed method has better performance on the convergent speed of the system state (robot joint positions and velocities) than state-of-the-art methods, especially for the underactuated joints.


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## Body

## 1. Introduction

Robotics technologies have been widely applied to various industrial applications, such as agriculture [1, 2], mining [3], and manufacturing [4–6]. To enable the robots to successfully operate in these fields, the design methodology of robot control is a necessary issue that practitioners need to consider. Among various robot designs, the control of underactuated robots has become an important and attractive research issue [7–9]. Such kind of robots has many advantages, such as low costs, compactness of design, and lightweight. Meanwhile, the difficulties of controller design are also accompanied due to the existence of the robot joints that cannot be driven directly [9]. Besides, the underactuated robots are often highly nonlinear in their dynamics, making the control even challenging when facing uncertainties [10].To overcome these control difficulties, multiple studies on the control of underactuated robotic systems have been proposed, such as fuzzy control [11–14], optimal control [15–17], partial feedback linearization [18, 19], energy-based approach [20–22], and sliding mode control [23–27]. In particular, the sliding mode control has been a popular control method due to its variable structure controller design that can endow nonlinear systems with good robustness under the uncertainties which are commonly met in real plants [28].For the control of practical underactuated systems, many researchers have put in lots of efforts [29–31]. One of the most classical systems is the cart-pole system (CPS), which has been taken as the benchmark for control in many studies [32–35]. Based on the opinion of [36], although many researchers have proposed various control schemes for dealing with the upward swing problem [37–39], the control issue for the CPS that has the initial pose of the pendulum at upper-half plan has not been well solved. In addition, the introduction of a nonlinear damping perturbation on the underactuated joint will further worsen the system stability at the equilibrium point, making the control more difficult [40]. Therefore, many studies do not consider such a perturbation. Nevertheless, some studies show that fast terminal sliding mode control can effectively cope with such an issue [41, 42].Inspired by the aforementioned issue, this work attempts to propose a control method that can not only manage the CPS under the damping perturbation but also achieve better performance on the convergent speed of the system state. The proposed method firstly requires a coordinate transform to govern a new integration system in accordance with the concept of differential parametrization. Next, a sliding mode control method based on a reaching controller (PD) and a variation of twisting-like controller [43] is proposed, where the sliding mode variables are designed based on the inspiration of the nonsingular fast terminal sliding mode control method [44]. There are two sliding mode variables designed in this control method: one can drive the underactuated joint to the desired position and velocity, while another one can stabilize the overall system state. The stability analysis is also provided via the Lyapunov direct method for proving that the proposed controller can enable the closed-loop system to converge in finite time. Finally, a comparative study is made with state-of-the-art methods based on numerical simulations for demonstrating that the proposed method has better performance on the settling time (convergent speed) of system state.The contributions of this work are illustrated as follows:(1)
This work uses a method that combines the coordinate transformation and twisting-like methods to model a CPS system, making the dynamics of the CPS system simpler to be controlled(2)
A nonsingular terminal sliding-mode control method is proposed on the simplified underactuated robotic system, where the robustness of the designed method is proved via the Lyapunov method(3)
The simulation results show that the proposed method has better control performance on the convergent speed, than other state-of-the-art methodsThe remainder of this paper is organized as follows: Section2 describes the problem statement of this work; Section 3 illustrates the detailed methodologies on the system treatments and controller design, as well as the stability analysis; Section 4 shows the comparative study based on numerical simulations; Section 5 briefly summarizes this paper.


## 2. Problem Statement

Consider the following dynamics of ann-link underactuated robotic system:(1)Mqq¨+Cq,q˙q˙=τgq+τdq+Bτc,where q∈ℝn refers to the robot joint position; Mq∈ℝn×n refers to the inertia matrix (symmetric positive-definite); Cq,q˙ refers to the Coriolis matrix; τgq∈ℝn refers to the torques generated from the gravity; τdq∈ℝn refers to the torques generated from the viscous friction; τc∈ℝn refers to the control inputs; and B∈ℝn×n refers to a coefficient matrix.Let the initial robot joint positions and velocities beq0∈ℝn and q˙0∈ℝn, respectively, and the robot joint positions and velocities be qd∈ℝn and q˙d∈ℝn, respectively. The objective of the designed controller τc is to drive the studied robotic system from the initial state (q0 and q˙0) to the goal state (qd and q˙d) in finite time. Note that the robot states are known and all parameters in (1) are determined and the goal state is selected as an unstable equilibrium point.Remark 1.
The torques generated from the viscous frictionτdq are nonlinear disturbances and only apply to the underactuated joints, making the stability of the studied system to be easily destroyed.


## 3. Methodologies

This section provides the detailed methodologies on the design of the nonsingular fast terminal sliding mode control. Besides, the corresponding stability is proved through the Lyapunov direct method. The overall schematic diagram is illustrated in Figure1.Figure 1
Overall schematic diagram of the proposed method.


### 3.1. System Transformation

This studied underactuated robotic system has the following parameters that satisfy the aforementioned nonlinear model (1):(2)q=q1,q2T∈ℝ2,Mq=m1+m2m2lcosq2m2lcosq2m2l2∈ℝ2×2,Cq,q˙=0−m2lsinq200∈ℝ2×2,τg=0m2glsinq2∈ℝ2×2,τd=0dT∈ℝ2×2,B=10T∈ℝ2×2,τc=u∈ℝ.Define(3)q¯1=q1l,q¯2=q2,u¯=um2g,m¯=m1m2.A normalized system of (2) can be derived:(4)1+m¯cosq¯2cosq¯21q¯¨+−q¯˙22sinq¯2−sinq¯2=u¯−dq¯˙2,where q¯=q¯1,q¯2T∈ℝ2.To form a closed control loop, let(5)u¯=m¯η+ηsin2q¯2−q¯˙22sinq¯2+sinq¯2cosq¯2−dq¯˙2cosq¯2.Then, system (4) becomes(6)q¯¨=−cosq¯21η+sinq¯2−dq¯˙20.Next, constructing a new system with new state variablesζ=ζ1,ζ2,ζ3,ζ4T∈ℝ4 and a new control input ξ∈ℝ, we obtain(7)ζ˙1=ζ2,ζ˙2=ζ3+ϕζ3,ζ4,ζ˙3=ζ4,ζ˙4=ξ.First, define two transforms(8)ζ1=q¯1+lnφq¯2,ζ3=tanq¯2,where(9)φq¯2=1+tanq¯2/21−tanq¯2/2.Then, the correspondingζ2 and ζ4 can be obtained:(10)ζ2=q¯˙1+q¯˙2cosq¯2,ζ4=q¯˙2cos2q¯2.Besides, to satisfy (7),(11)ϕζ3,ζ4=ζ41+ζ323/2ζ3ζ4−d1+ζ32.Remark 2.
ϕζ3,ζ4 is a nonlinear term to represent the perturbation that causes the plant to be unstable. Besides, with the convergence of the control input ζ4, this term also converges, indicating that the perturbation vanishes in the steady state. For the detailed proof, readers can refer to Appendix.
Based on the concept of the differential parameterizations, introduce a normalized timet¯, where dt¯=εdt. Correspondingly, a change of coordinate on (7) is formulated:(12)ζ¯=diagε2,ε1,ε0,ε−1ζ,ξ¯=ε−2ξ,where ζ¯=ζ¯1,ζ¯2,ζ¯3,ζ¯4T∈ℝ4 refers to the new state variables under the normalized time t¯; ξ¯∈ℝ refers to the new control input under the normalized time; and ε=g/l∈ℝ+ refers to the scale of time.Remark 3.
The normalized timet¯ is dimensionless, and the differentiation of ζ¯ is thus performed based on the normalized time t¯.
Finally, combining (12) which is equivalent to system (2), the plant can be represented by(13)ζ¯˙=Γζ¯,ξ¯,where Γ:ℝ4×ℝ⟶ℝ4.


### 3.2. Controller Design

This section reveals the controller design, and the following definition will be used throughout this paper.Definition 1.
(14)ρab=basignb,signb=1,b>0,0,b=0,−1,b<0.
Based on the core idea of sliding mode control, the control input is generated from two modes: the reaching mode and the sliding mode, where the reaching mode drives the state variables to the sliding mode manifold and then the sliding mode keeps the state variables to move along the sliding mode manifold until they converge to the goal state.
Herein, the control lawξ¯ζ¯ is therefore composed of two controllers:(15)ξ¯ζ¯=ξ¯rζ¯+ξ¯sζ¯,where ξ¯rζ¯ refers to the controller for the reaching mode and ξ¯sζ¯ refers to the controller for the sliding mode.
The controller for the reaching mode is realized in a PD form:(16)ξ¯ζ¯=−λ1ζ¯3+λ2ζ¯4λ3,where λ1,λ2,λ3∈ℝ+.
The controller for the sliding mode is composed of two sliding mode manifolds (s1=0 and s2=0):(17)ξ¯sζ¯=−λ4signs1+λ5signs2λ3,where λ4,λ5∈ℝ+ and λ4>λ5; the two sliding mode manifolds are constructed as(18)s1=λ1ρv1ζ¯1+λ1+λ2ρv1+v2ζ¯2+λ2+λ3ρv2+v3ζ¯3+λ3ρv4ζ¯4,s2=λ1ρv2+v3ζ¯2+λ2ρv3+v4ζ¯3+λ3ρv4ζ¯4,with v1,v2,v3,v4∈ℝ+.


### 3.3. Stability Analysis

Theorem 1.
Consider plant (13) with the following closed-loop control law:(19)ξ¯ζ¯=−λ1ζ¯3+λ2ζ¯4λ3+−λ4signs1+λ5signs2λ3,where(20)s1=λ1ρv1ζ¯1+λ1+λ2ρv1+v2ζ¯2+λ2+λ3ρv2+v3ζ¯3+λ3ρv4ζ¯4,s2=λ1ρv2+v3ζ¯2+λ2ρv3+v4ζ¯3+λ3ρv4ζ¯4,with λi∈ℝ+i=1,2,3,4,5, λ4>λ5, and vi∈ℝ+,i=1,2,3,4, and the two sliding mode variables (s1 and s2) can converge to zero in finite time, making the overall closed-loop system asymptotically stable.Proof.
This proof uses the Lyapunov direct method to prove the stability of the proposed control method. Select a Lyapunov function as(21)Vζ¯=λ12λ3ζ¯32+12ζ¯42+12λ3v4s12+12λ3v4s22.
Then, the derivative of this Lyapunov function is(22)V˙ζ¯=−λ2λ3ζ¯42+1λ3ζ¯4λ4signs1+λ5signs2+s1s˙12λ2v4+s2s˙22λ2v4=−λ2λ3ζ¯42+1λ3ζ¯4λ4signs1+λ5signs2+β1s1,s˙1+β2s2,s˙2,where(23)β1s1,s˙1=−ζ¯4v4−1μ1s12+μ2s1+μ3ϕζ¯3,ζ¯4,β2s2,s˙2=−ζ¯4v4−1μ4s22+μ5s2+μ6ϕζ¯3,ζ¯4,with μi∈ℝ+,i=1,2,3,4,5,6.
By Young’s inequality, the following inequality can be obtained:(24)1λ3ζ¯4λ4signs1+λ5signs2≤12λ3ζ¯42+λ4signs1+λ5signs22.
Therefore, an upper bound exists in (22):(25)V˙ζ¯≤−2λ2−12λ3ζ¯42−12λ3λ4signs1+λ5signs22−ζ¯4v4−1μ1s12+μ2s1+μ3ϕζ¯3,ζ¯4−ζ¯4v4−1μ4s22+μ5s2+μ6ϕζ¯3,ζ¯4.
Sinceϕζ¯3,ζ¯4 is bounded (as proved in the Appendix), a conclusion that the overall system is asymptotically stable can be made.


## 3.1. System Transformation

This studied underactuated robotic system has the following parameters that satisfy the aforementioned nonlinear model (1):(2)q=q1,q2T∈ℝ2,Mq=m1+m2m2lcosq2m2lcosq2m2l2∈ℝ2×2,Cq,q˙=0−m2lsinq200∈ℝ2×2,τg=0m2glsinq2∈ℝ2×2,τd=0dT∈ℝ2×2,B=10T∈ℝ2×2,τc=u∈ℝ.Define(3)q¯1=q1l,q¯2=q2,u¯=um2g,m¯=m1m2.A normalized system of (2) can be derived:(4)1+m¯cosq¯2cosq¯21q¯¨+−q¯˙22sinq¯2−sinq¯2=u¯−dq¯˙2,where q¯=q¯1,q¯2T∈ℝ2.To form a closed control loop, let(5)u¯=m¯η+ηsin2q¯2−q¯˙22sinq¯2+sinq¯2cosq¯2−dq¯˙2cosq¯2.Then, system (4) becomes(6)q¯¨=−cosq¯21η+sinq¯2−dq¯˙20.Next, constructing a new system with new state variablesζ=ζ1,ζ2,ζ3,ζ4T∈ℝ4 and a new control input ξ∈ℝ, we obtain(7)ζ˙1=ζ2,ζ˙2=ζ3+ϕζ3,ζ4,ζ˙3=ζ4,ζ˙4=ξ.First, define two transforms(8)ζ1=q¯1+lnφq¯2,ζ3=tanq¯2,where(9)φq¯2=1+tanq¯2/21−tanq¯2/2.Then, the correspondingζ2 and ζ4 can be obtained:(10)ζ2=q¯˙1+q¯˙2cosq¯2,ζ4=q¯˙2cos2q¯2.Besides, to satisfy (7),(11)ϕζ3,ζ4=ζ41+ζ323/2ζ3ζ4−d1+ζ32.Remark 2.
ϕζ3,ζ4 is a nonlinear term to represent the perturbation that causes the plant to be unstable. Besides, with the convergence of the control input ζ4, this term also converges, indicating that the perturbation vanishes in the steady state. For the detailed proof, readers can refer to Appendix.
Based on the concept of the differential parameterizations, introduce a normalized timet¯, where dt¯=εdt. Correspondingly, a change of coordinate on (7) is formulated:(12)ζ¯=diagε2,ε1,ε0,ε−1ζ,ξ¯=ε−2ξ,where ζ¯=ζ¯1,ζ¯2,ζ¯3,ζ¯4T∈ℝ4 refers to the new state variables under the normalized time t¯; ξ¯∈ℝ refers to the new control input under the normalized time; and ε=g/l∈ℝ+ refers to the scale of time.Remark 3.
The normalized timet¯ is dimensionless, and the differentiation of ζ¯ is thus performed based on the normalized time t¯.
Finally, combining (12) which is equivalent to system (2), the plant can be represented by(13)ζ¯˙=Γζ¯,ξ¯,where Γ:ℝ4×ℝ⟶ℝ4.


## 3.2. Controller Design

This section reveals the controller design, and the following definition will be used throughout this paper.Definition 1.
(14)ρab=basignb,signb=1,b>0,0,b=0,−1,b<0.
Based on the core idea of sliding mode control, the control input is generated from two modes: the reaching mode and the sliding mode, where the reaching mode drives the state variables to the sliding mode manifold and then the sliding mode keeps the state variables to move along the sliding mode manifold until they converge to the goal state.
Herein, the control lawξ¯ζ¯ is therefore composed of two controllers:(15)ξ¯ζ¯=ξ¯rζ¯+ξ¯sζ¯,where ξ¯rζ¯ refers to the controller for the reaching mode and ξ¯sζ¯ refers to the controller for the sliding mode.
The controller for the reaching mode is realized in a PD form:(16)ξ¯ζ¯=−λ1ζ¯3+λ2ζ¯4λ3,where λ1,λ2,λ3∈ℝ+.
The controller for the sliding mode is composed of two sliding mode manifolds (s1=0 and s2=0):(17)ξ¯sζ¯=−λ4signs1+λ5signs2λ3,where λ4,λ5∈ℝ+ and λ4>λ5; the two sliding mode manifolds are constructed as(18)s1=λ1ρv1ζ¯1+λ1+λ2ρv1+v2ζ¯2+λ2+λ3ρv2+v3ζ¯3+λ3ρv4ζ¯4,s2=λ1ρv2+v3ζ¯2+λ2ρv3+v4ζ¯3+λ3ρv4ζ¯4,with v1,v2,v3,v4∈ℝ+.


## 3.3. Stability Analysis

Theorem 1.
Consider plant (13) with the following closed-loop control law:(19)ξ¯ζ¯=−λ1ζ¯3+λ2ζ¯4λ3+−λ4signs1+λ5signs2λ3,where(20)s1=λ1ρv1ζ¯1+λ1+λ2ρv1+v2ζ¯2+λ2+λ3ρv2+v3ζ¯3+λ3ρv4ζ¯4,s2=λ1ρv2+v3ζ¯2+λ2ρv3+v4ζ¯3+λ3ρv4ζ¯4,with λi∈ℝ+i=1,2,3,4,5, λ4>λ5, and vi∈ℝ+,i=1,2,3,4, and the two sliding mode variables (s1 and s2) can converge to zero in finite time, making the overall closed-loop system asymptotically stable.Proof.
This proof uses the Lyapunov direct method to prove the stability of the proposed control method. Select a Lyapunov function as(21)Vζ¯=λ12λ3ζ¯32+12ζ¯42+12λ3v4s12+12λ3v4s22.
Then, the derivative of this Lyapunov function is(22)V˙ζ¯=−λ2λ3ζ¯42+1λ3ζ¯4λ4signs1+λ5signs2+s1s˙12λ2v4+s2s˙22λ2v4=−λ2λ3ζ¯42+1λ3ζ¯4λ4signs1+λ5signs2+β1s1,s˙1+β2s2,s˙2,where(23)β1s1,s˙1=−ζ¯4v4−1μ1s12+μ2s1+μ3ϕζ¯3,ζ¯4,β2s2,s˙2=−ζ¯4v4−1μ4s22+μ5s2+μ6ϕζ¯3,ζ¯4,with μi∈ℝ+,i=1,2,3,4,5,6.
By Young’s inequality, the following inequality can be obtained:(24)1λ3ζ¯4λ4signs1+λ5signs2≤12λ3ζ¯42+λ4signs1+λ5signs22.
Therefore, an upper bound exists in (22):(25)V˙ζ¯≤−2λ2−12λ3ζ¯42−12λ3λ4signs1+λ5signs22−ζ¯4v4−1μ1s12+μ2s1+μ3ϕζ¯3,ζ¯4−ζ¯4v4−1μ4s22+μ5s2+μ6ϕζ¯3,ζ¯4.
Sinceϕζ¯3,ζ¯4 is bounded (as proved in the Appendix), a conclusion that the overall system is asymptotically stable can be made.


## 4. Simulations

To validate the performance of the proposed controller, a comparative study is made based on numerical simulations in MATLAB. In this comparative study, the control method proposed in [36] is used for the comparison, where it has been shown that the proposed method has better performance than the control method [45]. Therefore, this paper only presents the comparison between the proposed method and the method in [36]. The used parameters for the numerical simulations are depicted in Table 1. The initial state is selected as q0=−0.5,0.3T and q˙0=0.1,−0.2T. The goal state is selected as qd=0,0T and q˙d=0,0T. Note that the used units are radians and radians/second.Table 1
Parameters of the controller.
ParametersValuesε0.2d0.8815λ11.5λ28λ34λ46λ50.8v10.8v20.2v30.8v41Figure2 and Table 2 present the performances of the proposed control method and the control method in [36]. These results demonstrate that the proposed method has better performance on the settling time, especially for the underactuated joint q1.Figure 2
Performance of system state under the proposed method and the method in [36].
(a)(b)(c)(d)Table 2
Performance of settling time.
Settling time (s)q1q˙1q2q˙2Proposed method1.65340.94572.59972.2618Method in [36]2.14121.18782.56402.1761Figure3 demonstrates the stability of the designed sliding mode variables s1 and s2, where these two sliding mode variables converge to zero in finite time. Based on these two sliding mode variables, the designed controller can stabilize the system to the desired state.Figure 3
Convergence of sliding mode variables.To validate the robustness of the proposed control method, the testing of the studied robotic system under different damping perturbations is conducted. Herein,d is changed from 0 to 2, as shown in Figure 4. The results show that the system control becomes difficult with the increase of the damping perturbation d. Nevertheless, the proposed control method can manage such perturbation within a range.Figure 4
Performance of the proposed method under different damping perturbations.
(a)(b)(c)(d)


## 5. Conclusions

This paper presents a novel control method for controlling a class of nonlinear underactuated robotic systems under nonlinear damping perturbation. The devised method is based on the concept of nonsingular fast terminal sliding mode controls which can obviously improve the settling time (response time). Especially for the underactuated robot joints, the method can not only reduce the settling time but also achieve stability. In addition, this control method is robust to a range of damping perturbations and can ensure finite-time convergence. The simulation results show that the studied underactuated system under the proposed method has better performance on the settling time (or convergent speed) than the state-of-the-art methods and validate that the proposed method can adapt to various damping perturbations. In the future, the trajectory tracking problem under such a perturbation will be an interesting direction for further research.



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