# Robust Control for a Two DOF Robot Manipulator

Authors: Fatma Massaoudi; Dorsaf Elleuch; Tarak Damak
Journal: Journal of Electrical and Computer Engineering
 (2019)

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## Abstract

In this paper, we present robust control techniques applied on a manipulator robot system: modified sliding mode control (MSMC) and backstepping control (BSC). The purpose is to evaluate SMC and BSC performances, taking into account the model uncertainties. Then, the obtained results of MSMC technique are compared with those of the adaptive sliding mode. Both methods have comparable simulation results which show a good quality of robustness. However, simulation results prove that the modified SMC is more robust, mostly under the effect of external variations and uncertainties.


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## Body

## 1. Introduction

Different control methodologies and approaches have been used for nonlinear systems (NLSs). Control architectures depend on the system model or structure. These controllers aim to guarantee system robustness, fast convergence, and stability. They can solve many problems such as external disturbance invariance to system uncertainties [1].In the literature, many controllers have been developed for NLS such as backstepping control [2] and linearizing feedback [3], robust control, adaptive control, and sliding mode technique [4, 5]. This theory was proposed by Emelyanov in 1950 and has been widespread thanks to its high performance and its ability to solve the problem of stability and robustness [1, 6]. SMC is one of the powerful methods to control uncertain nonlinear systems as robot manipulators. In [7–9], authors have proved that modified SMC is robust than tracking control. Yet, many researchers have faced some problems like “chattering” which is a high-frequency oscillation of the input and can lead to the saturation and destruction of actuators [10].Hence, in order to solve this problem, many researchers used boundary layer saturation method, estimated uncertainties strategy [1, 11] and high-order sliding mode control [12]. Others have combined sliding mode with another control technique such as adaptive control [13] or backstepping control for NLS such as robotic manipulators [14], in order to reach high performance and ameliorate robustness.Backstepping approach presents a systematic method for designing a control structure to track a reference signal, by selecting an appropriate Lyapunov function [15].It is applied to the globally stable and asymptotically adaptive tracking control of a strict feedback nonlinear system.Backstepping control offers many advantages over the conventional approaches like improving the system performance [16]. It is also known by its robustness and effectiveness [17].This technique has been very efficiently applied thanks to its systematic and recursive design methodology for nonlinear feedback control [18]. Unlike the feedback linearization method problems such as the precise model requirement, the backstepping approach offers a choice of design tools for nonlinearities accommodation. Moreover, backstepping is known by its flexibility, thanks to its use of recursive Lyapunov functions. For this, the final designed control law can guarantee the stability of the total control system. In aviation technology, the backstepping technique has been used to solve the stabilization and trajectory tracking problems of quadrotor UAVs [19, 20]. Yet, it is not easy to get satisfactory performance because the controller parameters obtained by the backstepping method are arbitrarily chosen. It is necessary to select proper parameters to obtain a good response. In fact, an improper selection of these parameters leads to inappropriate responses or may even lead to the instability of the system.Many researchers have studied adaptive sliding backstepping control of semistrict feedback systems like Koshkouei and Zinober [21].In this work, we present a modified SM approach for robust manipulators developed in [4]. We also propose a backstepping controller and we compare the results of both control types.This paper is organized as follows. In Section2, we present the system modeling. In Section 3, sliding mode method is developed; then backstepping control is developed in Section 4. Section 5 is devoted to present all results of sliding mode simulation and backstepping simulation followed by a comparative study. Finally, Section 6 contains conclusions.


## 2. System Modeling

In this paper, a two-degree-of-freedom manipulator DOF is considered (Figure1) [1, 4].Figure 1
A two DOF serial robot manipulator.The two DOF manipulator dynamic model can be given as follows [1, 4]:(1)τ1τ2=M11M12M21M22q¨1q¨2+−cq2−cq˙1+q˙2cq˙10q˙1q˙2+Fc100Fc2sgnq˙1sgnq˙2+v100v2q˙1q˙2+d1d2,where q=q1,q2T: the joints position vector; q˙=q˙1,q˙2T: the joints velocity vector; q¨=q¨1,q¨2T: the joints acceleration vector; d=d1,d2T: the joints external disturbance vector; u=τ1,τ2T: the joints control vector; Fci: the Coulomb friction forces; vi: the viscose friction coefficients. M11=a1+2a3cosq2−2a4sinq2, M12=M21=a2+a3cosq2−a4sinq2, and M22=a2, where c=a3sinq2+a4cosq2, a1=l1+m1lc12+Ie+melce2+meI12, a2=Ie+melce2, a3=cos30melcel1, and a4=sin30melcel1.The system equation (1) can be written as follows:(2)q¨=M−1u−Cq˙−Fcisgnq˙−viq˙−d,where fq,q˙=M−1−Cq˙−Fcisgnq˙−viq˙.Then, the second form of the dynamic model has the following expression:(3)q¨=Bqu+fq,q˙,where q=q1,q2T, u=τ1,τ2T, Bq=M−1.In the presence of uncertainties, functionsfq,q˙ and Bq have the following form [4]:(4)fq,q˙=f⌢q,q˙+Δfq,q˙,Bq=B⌢q+ΔBq,where Δf and ΔB are the unknown-parts of f and B, respectively, and f⌢ and B⌢ are the known parts.


## 3. Sliding Mode Controller

### 3.1. Definition

Sliding mode control depends on the choice of SM surface and how to guarantee the stability and convergence using Lyapunov function. Then, a suitable sliding mode control law is proposed.Under the sliding conditionsSxS˙x<0, the sliding surface should verify Sx=0 and S˙x=0.Sliding mode controllers are robust and rapid. But, classical SMC has some disadvantages, especially the chattering phenomenon, which can lead to some troubles such as the heat and saturation of the robotic mechanical parts [1, 6].As a remedy to this problem, many solutions can be used like boundary layer technique, combination of other types of control with sliding mode and higher-order sliding mode [1, 12].An adaptive SMC is used for the two DOF manipulator in [4].The sliding surface function is chosen as follows [1, 4]:(5)St=ddt+Λ2∫edt=e˙+2Λe+Λ2∫edt,where Λ is an n × n diagonal definite positive constant matrix and e is the vector of tracking error, e=q−qd.The error rate vector ise˙=q˙−q˙d.In [4], the authors use an adaptive sliding mode control in which the SMC law is given by the following expression:(6)ut=B^−1q¨r−f^−KS−Γ∫Sdt,where Γ is the diagonal definite constant positive matrix and q¨r is the reference acceleration. K, B⌢−1, and f⌢ are defined in [4].Considerueq=B^−1q¨r−f⌢; u1=−B^−1KS and u⌢2=−B^−1Γ∫Sdt.So,(7)ut=ueq+u1+u⌢2,where u⌢2 is an adaptive term used to compensate system uncertainties.In this paper,u⌢2 is replaced by another term according to sliding technique.(8)u⌢2=−B⌢−1KηsatS.So, the modified SMC expression is given by(9)ut=B⌢−1q¨r−f⌢−KS+ηsatS.The SMC block diagram is shown in Figure2 [4].(10)R=ηsatS.Figure 2
SMC block diagram [4].


### 3.2. Stability Analyzes

The Lyapunov function has the following expression:(11)v=12STS.The derivative of Lyapunov function isv˙=STS˙ and S˙=q¨−q¨r.The SMC law is described by(12)ut=B⌢−1q¨r−f⌢−KS+ηsatS.Sinceq¨=B⌢qu+f⌢q,q˙, v˙=STq¨−q¨r=STB⌢qB⌢−1q¨r−f⌢−KS+ηsatS+f⌢−q¨r.Thenv˙=−STKS+ηsatS.The stability is proved whenv˙≤0. So, η≥−S/satS.


## 3.1. Definition

Sliding mode control depends on the choice of SM surface and how to guarantee the stability and convergence using Lyapunov function. Then, a suitable sliding mode control law is proposed.Under the sliding conditionsSxS˙x<0, the sliding surface should verify Sx=0 and S˙x=0.Sliding mode controllers are robust and rapid. But, classical SMC has some disadvantages, especially the chattering phenomenon, which can lead to some troubles such as the heat and saturation of the robotic mechanical parts [1, 6].As a remedy to this problem, many solutions can be used like boundary layer technique, combination of other types of control with sliding mode and higher-order sliding mode [1, 12].An adaptive SMC is used for the two DOF manipulator in [4].The sliding surface function is chosen as follows [1, 4]:(5)St=ddt+Λ2∫edt=e˙+2Λe+Λ2∫edt,where Λ is an n × n diagonal definite positive constant matrix and e is the vector of tracking error, e=q−qd.The error rate vector ise˙=q˙−q˙d.In [4], the authors use an adaptive sliding mode control in which the SMC law is given by the following expression:(6)ut=B^−1q¨r−f^−KS−Γ∫Sdt,where Γ is the diagonal definite constant positive matrix and q¨r is the reference acceleration. K, B⌢−1, and f⌢ are defined in [4].Considerueq=B^−1q¨r−f⌢; u1=−B^−1KS and u⌢2=−B^−1Γ∫Sdt.So,(7)ut=ueq+u1+u⌢2,where u⌢2 is an adaptive term used to compensate system uncertainties.In this paper,u⌢2 is replaced by another term according to sliding technique.(8)u⌢2=−B⌢−1KηsatS.So, the modified SMC expression is given by(9)ut=B⌢−1q¨r−f⌢−KS+ηsatS.The SMC block diagram is shown in Figure2 [4].(10)R=ηsatS.Figure 2
SMC block diagram [4].


## 3.2. Stability Analyzes

The Lyapunov function has the following expression:(11)v=12STS.The derivative of Lyapunov function isv˙=STS˙ and S˙=q¨−q¨r.The SMC law is described by(12)ut=B⌢−1q¨r−f⌢−KS+ηsatS.Sinceq¨=B⌢qu+f⌢q,q˙, v˙=STq¨−q¨r=STB⌢qB⌢−1q¨r−f⌢−KS+ηsatS+f⌢−q¨r.Thenv˙=−STKS+ηsatS.The stability is proved whenv˙≤0. So, η≥−S/satS.


## 4. Backstepping Controller

### 4.1. Definition

The backstepping control scheme is a nonlinear control method based on the Lyapunov theorem control. It is a systematic design technique applied to strict feedback systems. BSC has a systematic way of constructing the Lyapunov function along the control input design.This strategy is used especially in robotic field and it is based on the following points:(i)
Defining the error variable(ii)
Defining Lyapunov function(iii)
Derivating Lyapunov function(iv)
Determining a virtual variable(v)
Defining the next integrator until the apparition of the real control variablesThe most important object of backstepping control is to make loop systems equivalent to one-order system that verifies the Lyapunov stability.


### 4.2. Stability Analyzes

Consider the following state vector:(13)xi=x1,x2,x3,x4T=q1,q˙1,q2,q˙2.Then, the system model is subdivided into two subsystems. The first subsystem is described by(14)x˙1=x2,x˙2=τ1M11−M12M11x˙4+2cx2x4M11+cM11x42−Fc1M11sgnx2−v1M11x2−d1M11.And the second subsystem is described by(15)x˙3=x4,x˙4=τ2M22−M21M22x˙2−cx22M22−Fc2M22sgnx4−v2M22x4−d2M22.The error variable is given as(16)zi=xi−xid,i=1,…,n, where n = 2 and xid is the desired state vector.The derivative ofzi is(17)z˙i=x˙i−x˙id,z˙i=x˙i−x˙id.The Lyapunov function can be written as(18)vi=12zi2.Then its derivative is(19)v˙i=ziz˙i,v˙i=−cizi2+zicizi+xi+1−x˙id,where ci are the backstepping variables.In order to guarantee the system stability, the following condition should be verified:(20)v˙i≤0,the virtual controlαi=−cizi=−cixi−xid.The Lyapunov function can be written as(21)vi+1=12zi+12+vi.The stability is proved when(22)v˙i+1≤0.


#### 4.2.1. For the First Subsystem

The first error variable isz1=x1−x1d and its derivative is z˙1=x˙1−x˙1d=x2−x˙1d.The first Lyapunov function isv1=1/2z12 and its derivative is v˙1=z1z˙1. So, v˙1=−c1z12+z1c1z1+x2−x˙1d.The virtual control is(23)α1=−c1z1=−c1x1−x1d.The first error variable isz2=c1z1+x2−x˙1d, z2=−α1+x2−x˙1d and its derivative is z˙2=c1z˙1+x˙2−x¨1d, z˙2=c1x˙1−x˙1d+x˙2−x¨1d.Sincex˙2=f1+g1τ1, with f1=−M12/M11x˙4+2cx2x4/M11+c/M11x42−Fc1/M11sgnx2−v1/M11x2−d1/M11 and g1=1/M11.Then,z˙2=f1+g1τ1−x¨1d+c1z2+α1.The second Lyapunov function isv2=1/2z22+v1 and its derivative is v˙2=z2z˙2+v˙1.The system stability can be obtained whenv˙2≤0, v˙2=−c1z12+z1z2+z2−c2z2−z1, v˙2≤−c1z12+z1z2+z2z1−z1+c2z2−c2z2+g1τ1+f1−x¨1d+c1z2+c1α1, z1+c2z2+g1τ1+f1−x¨1d+c1z2+c1α1=0.Then,τ1=1/g1x¨1d−c1x2+c1x˙1d−f1−x1+x1d−c2x2+c2x˙1d−c1c2x1+c1c2x1d.Thus, the expression of the first subsystem BSC law is(24)τ1=M11x¨1d−c1x2+c1x˙1d+M12M11x˙4−2cM11x2x4−cM11x42+Fc1M11sgnx2+v1M11x2+d1M11−x1+x1d−c2x2+c2x˙1d−c1c2x1+c1c2x1d.


#### 4.2.2. For the Second Subsystem

The same steps used for the first subsystem are followed to obtain the expression of the second subsystem BSC which is given by(25)τ2=M22x¨2d−c3x4+c3x˙2d+M21M22x˙2+cM22x22+Fc2M22sgnx4+v2M22x4+d2M22−x3+x2d−c4x4+c4x˙2d−c3c4x3+c3c4x2d.


## 4.1. Definition

The backstepping control scheme is a nonlinear control method based on the Lyapunov theorem control. It is a systematic design technique applied to strict feedback systems. BSC has a systematic way of constructing the Lyapunov function along the control input design.This strategy is used especially in robotic field and it is based on the following points:(i)
Defining the error variable(ii)
Defining Lyapunov function(iii)
Derivating Lyapunov function(iv)
Determining a virtual variable(v)
Defining the next integrator until the apparition of the real control variablesThe most important object of backstepping control is to make loop systems equivalent to one-order system that verifies the Lyapunov stability.


## 4.2. Stability Analyzes

Consider the following state vector:(13)xi=x1,x2,x3,x4T=q1,q˙1,q2,q˙2.Then, the system model is subdivided into two subsystems. The first subsystem is described by(14)x˙1=x2,x˙2=τ1M11−M12M11x˙4+2cx2x4M11+cM11x42−Fc1M11sgnx2−v1M11x2−d1M11.And the second subsystem is described by(15)x˙3=x4,x˙4=τ2M22−M21M22x˙2−cx22M22−Fc2M22sgnx4−v2M22x4−d2M22.The error variable is given as(16)zi=xi−xid,i=1,…,n, where n = 2 and xid is the desired state vector.The derivative ofzi is(17)z˙i=x˙i−x˙id,z˙i=x˙i−x˙id.The Lyapunov function can be written as(18)vi=12zi2.Then its derivative is(19)v˙i=ziz˙i,v˙i=−cizi2+zicizi+xi+1−x˙id,where ci are the backstepping variables.In order to guarantee the system stability, the following condition should be verified:(20)v˙i≤0,the virtual controlαi=−cizi=−cixi−xid.The Lyapunov function can be written as(21)vi+1=12zi+12+vi.The stability is proved when(22)v˙i+1≤0.


### 4.2.1. For the First Subsystem

The first error variable isz1=x1−x1d and its derivative is z˙1=x˙1−x˙1d=x2−x˙1d.The first Lyapunov function isv1=1/2z12 and its derivative is v˙1=z1z˙1. So, v˙1=−c1z12+z1c1z1+x2−x˙1d.The virtual control is(23)α1=−c1z1=−c1x1−x1d.The first error variable isz2=c1z1+x2−x˙1d, z2=−α1+x2−x˙1d and its derivative is z˙2=c1z˙1+x˙2−x¨1d, z˙2=c1x˙1−x˙1d+x˙2−x¨1d.Sincex˙2=f1+g1τ1, with f1=−M12/M11x˙4+2cx2x4/M11+c/M11x42−Fc1/M11sgnx2−v1/M11x2−d1/M11 and g1=1/M11.Then,z˙2=f1+g1τ1−x¨1d+c1z2+α1.The second Lyapunov function isv2=1/2z22+v1 and its derivative is v˙2=z2z˙2+v˙1.The system stability can be obtained whenv˙2≤0, v˙2=−c1z12+z1z2+z2−c2z2−z1, v˙2≤−c1z12+z1z2+z2z1−z1+c2z2−c2z2+g1τ1+f1−x¨1d+c1z2+c1α1, z1+c2z2+g1τ1+f1−x¨1d+c1z2+c1α1=0.Then,τ1=1/g1x¨1d−c1x2+c1x˙1d−f1−x1+x1d−c2x2+c2x˙1d−c1c2x1+c1c2x1d.Thus, the expression of the first subsystem BSC law is(24)τ1=M11x¨1d−c1x2+c1x˙1d+M12M11x˙4−2cM11x2x4−cM11x42+Fc1M11sgnx2+v1M11x2+d1M11−x1+x1d−c2x2+c2x˙1d−c1c2x1+c1c2x1d.


### 4.2.2. For the Second Subsystem

The same steps used for the first subsystem are followed to obtain the expression of the second subsystem BSC which is given by(25)τ2=M22x¨2d−c3x4+c3x˙2d+M21M22x˙2+cM22x22+Fc2M22sgnx4+v2M22x4+d2M22−x3+x2d−c4x4+c4x˙2d−c3c4x3+c3c4x2d.


## 4.2.1. For the First Subsystem

The first error variable isz1=x1−x1d and its derivative is z˙1=x˙1−x˙1d=x2−x˙1d.The first Lyapunov function isv1=1/2z12 and its derivative is v˙1=z1z˙1. So, v˙1=−c1z12+z1c1z1+x2−x˙1d.The virtual control is(23)α1=−c1z1=−c1x1−x1d.The first error variable isz2=c1z1+x2−x˙1d, z2=−α1+x2−x˙1d and its derivative is z˙2=c1z˙1+x˙2−x¨1d, z˙2=c1x˙1−x˙1d+x˙2−x¨1d.Sincex˙2=f1+g1τ1, with f1=−M12/M11x˙4+2cx2x4/M11+c/M11x42−Fc1/M11sgnx2−v1/M11x2−d1/M11 and g1=1/M11.Then,z˙2=f1+g1τ1−x¨1d+c1z2+α1.The second Lyapunov function isv2=1/2z22+v1 and its derivative is v˙2=z2z˙2+v˙1.The system stability can be obtained whenv˙2≤0, v˙2=−c1z12+z1z2+z2−c2z2−z1, v˙2≤−c1z12+z1z2+z2z1−z1+c2z2−c2z2+g1τ1+f1−x¨1d+c1z2+c1α1, z1+c2z2+g1τ1+f1−x¨1d+c1z2+c1α1=0.Then,τ1=1/g1x¨1d−c1x2+c1x˙1d−f1−x1+x1d−c2x2+c2x˙1d−c1c2x1+c1c2x1d.Thus, the expression of the first subsystem BSC law is(24)τ1=M11x¨1d−c1x2+c1x˙1d+M12M11x˙4−2cM11x2x4−cM11x42+Fc1M11sgnx2+v1M11x2+d1M11−x1+x1d−c2x2+c2x˙1d−c1c2x1+c1c2x1d.


## 4.2.2. For the Second Subsystem

The same steps used for the first subsystem are followed to obtain the expression of the second subsystem BSC which is given by(25)τ2=M22x¨2d−c3x4+c3x˙2d+M21M22x˙2+cM22x22+Fc2M22sgnx4+v2M22x4+d2M22−x3+x2d−c4x4+c4x˙2d−c3c4x3+c3c4x2d.


## 5. Simulation Results

In this section, tracking results of SMC and backstepping control are shown.The desired position vectors are chosen as follows [4]:(26)qd1=π61−cos1.5πt+sinπt,qd2=π61−cos2πt+sin1.5πt.In order to reduce the chattering phenomenon, in equation (14), “sign” is replaced by saturation function “sat,” where(27)satS=signS,ifS≥ϕ,satS=Sϕ,otherwise.The design parameters of simulation areη=1, K=1000, and ϕ=0.8.The BSC parameters are chosen as follows:(28)c1=30,c2=40,c3=80,c4=300.The end-effect or mass isme=2kg.


### 5.1. Results with the External Perturbation

The external perturbations are applied att = 3 s, d1=200, and d2=400.Figure3 represents the tracking manipulator trajectory of joints J1 and J2, respectively, by sliding mode controller and backstepping controller.Figure 3
Joints trajectories. (a) J1; (b) J2.
(a)
(b)The simulation results prove that both SMC and BSC are robust under external uncertainties. In fact, by using these techniques, both joint trajectories follow the desired trajectory with a high performance. However, SMC is more stable and robust especially via the joint J2 tracking trajectory.Figure4 represents torques variations by SMC and BSC for Joint J1 and Joint J2, respectively.Figure 4
Joints torques.
(a)
(b)We admit that both SMC and BSC are efficient and robust techniques. But, the control signal with BSC is smoother than SMC technique.Figure5 represents sliding surfaces and the trajectories error tracking of the two joints J1 and J2. Simulation results prove the good performances of the SMC technique.Figure 5
Sliding surfaces and trajectory errors. (a) s1, (b) s2, (c) e1, and (d) e2.
(a)
(b)
(c)
(d)


### 5.2. Results with Sinusoidal Perturbation

Att = 3 s(29)d1=300cos1.7πt+12+270sinπt,d2=250sin1.35πt+12+300cos2t.Figure6 represents the trajectory tracking manipulator of joints J1 and J2, respectively, by using SMC and BSC controllers.Figure 6
Joints trajectories. (a) J1; (b) J2.
(a)
(b)The obtained results show the performance of both methods. Yet, BSC is more rapid while SMC is more robust against external perturbations.Figure7 represents torque variations by SMC and BSC for Joint J1 and Joint J2, respectively.Figure 7
Joints torques.
(a)
(b)The following results are comparable for both controllers. This concludes their effectiveness and robustness.Figure8 represents sliding surfaces and trajectories error tracking of joints J1 and J2. The convergence of sliding surfaces is verified. Good performances of SMC technique are then proved.Figure 8
Sliding surfaces and trajectory errors. (a) s1, (b) s2, (c) e1, and (d) e2.
(a)
(b)
(c)
(d)


### 5.3. Modified SMC with External and Internal Perturbations

Att = 1.2 s,(30)d1=200,d2=400,and at t = 2.6 s, me=5kg.Figure9 represents the trajectories tracking of the manipulator and torque variations, using SMC technique for joints J1 and J2, respectively. In spite of the existence of external and internal uncertainties, the actual trajectory keeps on following the desired one. This method has been successfully performed. Robustness under uncertainties has been deduced.Figure 9
Joints trajectories and torques.
(a)
(b)
(c)
(d)Figure10 represents sliding surfaces and trajectories error tracking of joints J1 and J2. Convergence of sliding surfaces has been proved via simulation results.Figure 10
Sliding surfaces and trajectory errors. (a) s1, (b) s2, (c) e1, and (d) e2.
(a)
(b)
(c)
(d)Att = 1.2 s,(31)d1=300cos1.7πt+12+270sinπt,d2=250sin1.35πt+12+300cos2t,and at t = 2.6 s, me=5kg.Figure11 represents the trajectories tracking of the manipulator and torques variations using SMC for joints J1 and J2, respectively. Simulation results are performing. In spite of the existence of internal perturbations added to external sinusoidal variations, the system is stable and robust.Figure 11
Joints trajectories and torques.
(a)
(b)
(c)
(d)Figure12 represents sliding surfaces and the trajectories error tracking of joints J1 and J2. Convergence and performance of the system controlled by sliding mode have been reached.Figure 12
Sliding surfaces and trajectory errors. (a) s1, (b) s2, (c) e1, and (d) e2.
(a)
(b)
(c)
(d)


## 5.1. Results with the External Perturbation

The external perturbations are applied att = 3 s, d1=200, and d2=400.Figure3 represents the tracking manipulator trajectory of joints J1 and J2, respectively, by sliding mode controller and backstepping controller.Figure 3
Joints trajectories. (a) J1; (b) J2.
(a)
(b)The simulation results prove that both SMC and BSC are robust under external uncertainties. In fact, by using these techniques, both joint trajectories follow the desired trajectory with a high performance. However, SMC is more stable and robust especially via the joint J2 tracking trajectory.Figure4 represents torques variations by SMC and BSC for Joint J1 and Joint J2, respectively.Figure 4
Joints torques.
(a)
(b)We admit that both SMC and BSC are efficient and robust techniques. But, the control signal with BSC is smoother than SMC technique.Figure5 represents sliding surfaces and the trajectories error tracking of the two joints J1 and J2. Simulation results prove the good performances of the SMC technique.Figure 5
Sliding surfaces and trajectory errors. (a) s1, (b) s2, (c) e1, and (d) e2.
(a)
(b)
(c)
(d)


## 5.2. Results with Sinusoidal Perturbation

Att = 3 s(29)d1=300cos1.7πt+12+270sinπt,d2=250sin1.35πt+12+300cos2t.Figure6 represents the trajectory tracking manipulator of joints J1 and J2, respectively, by using SMC and BSC controllers.Figure 6
Joints trajectories. (a) J1; (b) J2.
(a)
(b)The obtained results show the performance of both methods. Yet, BSC is more rapid while SMC is more robust against external perturbations.Figure7 represents torque variations by SMC and BSC for Joint J1 and Joint J2, respectively.Figure 7
Joints torques.
(a)
(b)The following results are comparable for both controllers. This concludes their effectiveness and robustness.Figure8 represents sliding surfaces and trajectories error tracking of joints J1 and J2. The convergence of sliding surfaces is verified. Good performances of SMC technique are then proved.Figure 8
Sliding surfaces and trajectory errors. (a) s1, (b) s2, (c) e1, and (d) e2.
(a)
(b)
(c)
(d)


## 5.3. Modified SMC with External and Internal Perturbations

Att = 1.2 s,(30)d1=200,d2=400,and at t = 2.6 s, me=5kg.Figure9 represents the trajectories tracking of the manipulator and torque variations, using SMC technique for joints J1 and J2, respectively. In spite of the existence of external and internal uncertainties, the actual trajectory keeps on following the desired one. This method has been successfully performed. Robustness under uncertainties has been deduced.Figure 9
Joints trajectories and torques.
(a)
(b)
(c)
(d)Figure10 represents sliding surfaces and trajectories error tracking of joints J1 and J2. Convergence of sliding surfaces has been proved via simulation results.Figure 10
Sliding surfaces and trajectory errors. (a) s1, (b) s2, (c) e1, and (d) e2.
(a)
(b)
(c)
(d)Att = 1.2 s,(31)d1=300cos1.7πt+12+270sinπt,d2=250sin1.35πt+12+300cos2t,and at t = 2.6 s, me=5kg.Figure11 represents the trajectories tracking of the manipulator and torques variations using SMC for joints J1 and J2, respectively. Simulation results are performing. In spite of the existence of internal perturbations added to external sinusoidal variations, the system is stable and robust.Figure 11
Joints trajectories and torques.
(a)
(b)
(c)
(d)Figure12 represents sliding surfaces and the trajectories error tracking of joints J1 and J2. Convergence and performance of the system controlled by sliding mode have been reached.Figure 12
Sliding surfaces and trajectory errors. (a) s1, (b) s2, (c) e1, and (d) e2.
(a)
(b)
(c)
(d)


## 6. Conclusion

In this work, two robust techniques have been applied to control a two-degree-of-freedom robot. A modified sliding mode control has been developed and a backstepping control has been studied for this uncertain nonlinear system. Simulation results prove the performance and effectiveness of both controllers. Sliding mode control is more robust, taking into account uncertainties, while backstepping is more rapid. Modified SMC has reached the same performance and robustness as adaptive sliding mode control. In order to ameliorate robustness and performance of uncertain nonlinear systems, a combination of both SMC and BSC may be proposed and high-order sliding mode control may also be developed.



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*Source: 3919864-2019-04-17.xml*