# Adaptive Backstepping Sliding Mode Control of Trajectory Tracking for Robotic Manipulators

Authors: Zhu Dachang; Du Baolin; Zhu Puchen; Wenqiang Wu
Journal: Complexity
 (2020)

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## Abstract

To achieve precise trajectory tracking of robotic manipulators in complex environment, the precise dynamic model, parameters identification, nonlinear characteristics, and disturbances are the factors that should be solved. Although parameters identification and adaptive estimate method were proposed for robotic control in many literature studies, the essential factors, such as coupling and friction, are rarely mentioned as it is difficult to build the precise dynamic model of the robotic manipulator. An adaptive backstepping sliding mode control is proposed to solve the precise trajectory tracking under external disturbances with complex environment, and the dynamic response characteristics of a two-link robotic manipulator are described in this paper. First, the Lagrange kinetic method is used to derive the precise dynamic model which includes the nonlinear factor with friction and coupling. Moreover, the dynamic model of two-link robotic manipulator is built. Second, the estimate function for the nonlinear part is selected, and backstepping algorithm is used for analyzing the stabilities of the sliding mode controller by using Lyapunov theory. Furthermore, the convergence of the proposed controller is verified subject to the external disturbance. At last, numerical simulation results are reported to demonstrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Nowadays, with the development of modern industrial technology, robotic manipulators are widely used in automobile manufacturing, aerospace, electronic assembly, precision medical operation, and other fields. To obtain steady state accuracy and fast dynamical response, it is necessary for high precision of the trajectory tracking ability of robotic manipulator. Unfortunately, it is difficult to be satisfied on account of the nonlinear characteristics in complex environment like clearance of joints, friction, external disturbance, strong couple, and so on. The first stage of trajectory tracking is to establish the precise mathematical model of the robotic manipulator. However, the nonlinear part of the dynamic model of the robotic manipulator is ignored in many literatures [1–5] or parameter identification by many approaches [6–8]; even the torque in the joint space and the moment of inertia were ignored in [9]. By calculating kinetic energy, potential energy, and generalized force, the Lagrange equation was utilized to build the dynamic equation for robotic manipulator [10, 11]. As the recurrence relationship is not established between multiple links, it cannot be applied easily to the whole dynamic modeling for robotic manipulators. To overcome inaccuracy of the dynamical model of robotic manipulator, friction, clearance, and external disturbance were considered, and intelligent control strategies have been developed by many researchers for the uncertain manipulator [12–14], for example. In this paper, the dynamic equation with recurrence method by using Lagrange energy method is provided as an accurate mathematical model for precision trajectory tracking.Furthermore, the design of intelligence controllers for nonlinear systems affected by disturbances is a topic that has been studied by several authors, and many different approaches have been proposed for this problem [15–17]. Backstepping is a systematic and recursive design method for nonlinear control applied to the feedback linearization system, which can guarantee global regulation and tracking performance [18, 19]. It is a regression design method which combines the selection of Lyapunov function with the design of controller, and the virtual control is designed with the requirements step by step, and then, control law of complex uncertain system is finally designed. The basic design idea is to decompose the complex nonlinear system into subsystems of no more than systematic order and then design the Lyapunov function of each subsystem separately. On the basis of ensuring that the subsystem has guaranteed convergence, the control law of the subsystem is obtained. In the design of the next subsystem, the control law of the preceding subsystem is taken as the tracking target of this subsystem, and finally the control law of the last subsystem is obtained. By analogy, the control law of the entire closed-loop system can be obtained, and the convergence of the closed-loop system is guaranteed by Lyapunov stability analysis method. Backstepping with intelligent algorithm can increase the quality of the transition process, reduce or even eliminate the uncertainty of the matching constraints, and provide a structured and systematic design method for the Lyapunov function design for complex nonlinear systems [20–23]. Sliding mode control (SMC) is another robust control method, which produces a switching control law (equivalent control law) to force the system to converge to the sliding surface within a boundary layer near the sliding surface under the convergence of the Lyapunov stability theory [24, 25]. Trajectory tracking control of a 6-DOF pneumatically actuated Gough-Stewart parallel robot was investigated by Lafmejani [26], and position control of the pneumatic actuator was performed based on backstepping sliding mode controller according to the dynamic model of system. A methodology of dynamic analysis and control for a hybrid humanoid robot arm was presented in [27], and an adaptive backstepping sliding mode controller was developed for the parameters uncertainties and disturbances of the hybrid humanoid robot arm.For the multi-input-multi-output (MIMO) nonlinear system with uncertainties and disturbances, sliding mode PI control with backstepping approach [28], presence of bounded uncertainties from unmodeled dynamic, parameters variations, disturbances, and visual fusion technology [29, 30], and asynchronous control with fuzzy approach [31] were proposed. To make the dynamic model of the robotic manipulator more accurate, an impedance-control strategy with dynamic compensation for interactive control of robot manipulators was presented in [32]. The kinematics and dynamics of multiple cooperative welding robot manipulators were studied on the basis of the Denavit–Hartenberg and Lagrange method, and adaptive neural control and dynamic movement primitives were considered in [33]. An adaptive backstepping sliding mode control of robotic manipulators is proposed in this paper to achieve the precision trajectory tracking with external disturbances. The main contributions of the current paper are summarized as follows:(1)
By using Lagrange energy function, the precise dynamic model of the robotic manipulator is built, and the nonlinear characteristics and uncertainties are analyzed. Furthermore, dynamic model of a two-link robotic manipulator is derived.(2)
According to the precision dynamic model of the two-link robotic manipulator, an estimate function of the nonlinear and coupling parts is proposed. Backstepping algorithm is used to construct the equivalent control law of sliding mode control through three steps, and the stability of the proposed controller is convergent by using Lyapunov theory.This paper is organized in the following manner: Section2 presents the precise dynamic modeling method and the precise dynamic model of the robotic manipulator. According to the previous method, the precise dynamic model of a two-link robotic manipulator is derived. Section 3 describes the sliding mode control with backstepping algorithm. Through the estimate function of nonlinear and coupling parts, the controller is proposed for the trajectory tracking under external disturbances, and its stability is discussed. Numerical simulation results are reported to demonstrate the effectiveness of the proposed method in Section 4. Finally, conclusions are provided in Section 5.


## 2. Description of Robotic Manipulator and Dynamic Modeling

To make the dynamic model recursive, the Newton–Euler method is used to establish all force balance between the links of the robotic manipulator, and the dynamic equation can be derived. Forward recursion is used for speed and acceleration transfer between all of the links and backward recursion is used for force transfer from the end-effector to each link of the robotic manipulator.


### 2.1. Description of Link Parameters of Robotic Manipulator

The dynamic parameters which describe the dynamic model are important for the control algorithms, effective simulation results, and accurate trajectory tracking algorithms. Dynamic equation of the robotic manipulator withn-DOF has been characterized in many literature studies [1–11] as follows:(1)Dqq¨+Cq,q˙q˙+Gq+Fq,q˙+τd=τ,where q,q˙,q¨∈Rn are the link position, velocity, and acceleration vectors, respectively, Dq∈Rn×n is the symmetric positive definite inertia matrix, Cq,q˙∈Rn×n is the Coriolis or centrifugal forces, Gq∈Rn×1 consolidates the gravitational force, Fq,q˙∈Rn×1 incorporates the friction terms, and τd represents external disturbance.However, quadratic velocity terms and dynamic coupling terms are not taken into account. So, the problem of model accuracy cannot be solved essentially only through parameter identification and compensation methods. The kinematic description of theith link is shown in Figure 1.Figure 1
Kinematic description of theith link.An infinitesimal elementei is selected which lies on the ith link, and the position of the center of mass ci is represented by vector pi, the position vector of ei is presented by vector pi∗, and the position vector from the center of mass to the infinitesimal element is ri. When integrating in the whole area of the ith link, the infinitesimal element can represent the motion performance of the whole link.The kinetic energy componentEkli of the ith link can be given by(2)Ekli=12∫Vlip˙i∗Tp˙i∗ρdV,where p˙i∗ is the linear velocity vector of the infinitesimal element, ρ is material density, and dV is the volume of the infinitesimal element.The position vector relationship betweenci and ei is satisfied with(3)ri=rixriyrizT=pi∗−pli,where pli is calculated by centroid theorem as(4)pli=1mli∫Vlipi∗ρdV,where mli is the mass of the ith link.Differentiating equation (3) with respect to time, one can get(5)p˙i∗=p˙li+ωi×ri=p˙li+Sωiri,where p˙li is the linear velocity vector of the center of mass, ωi is angular velocity of the infinitesimal element around the center of mass, and Sωi is the antisymmetric matrix of three-dimensional vector ωi=ωixωiyωizT and defined as follows:(6)Sωi=0−ωizωiyωiz0−ωix−ωiyωix0.Substituting equation (5) into equation (2), one can obtain(7)Ekli=12∫Vlip˙li+SωiriTp˙li+SωiriρdV.Equation (7) shows that the kinetic energy of each link consists of three parts, described as follows:(1)
Translational kinetic energy is expressed as(8)12∫Vlip˙liTp˙liρdV=12mlip˙liTp˙li.(2)
Implicated motion kinetic energy is expressed as(9)212∫Vlip˙liSωiriρdV=212p˙liTSωi∫Vlipi∗−pliρdV=0.(3)
Rotation kinetic energy, combining with equation (6), is expressed as(10)12∫VliriTSTωiSωiriρdV=12ωiT∫VliSTriSriρdVωi=12ωiTIliωi,whereSωiri=−Sriωi; Ili is defined as(11)Ili=∫riy2+riz2ρdV−∫rixriyρdV−∫rixrizρdV−∫rixriyρdV∫rix2+riz2ρdV−∫rizriyρdV−∫rixrizρdV−∫rizriyρdV∫riy2+rix2ρdV,(1)
whereIli is symmetric matrix and denotes inertia tensor related to the center of mass of the ith link in base coordinate system.Combining equation (8) with equation (10), the kinetic energy of the ith link can be given by(12)Ekli=12mliq˙TJpliTJpliq˙+12q˙TJoliTIliJoliq˙,where q=q1q2⋯qiT is joint variable vector.The kinetic energy component of the motor ofi# joint can be calculated in a similar way. Under the assumption of rigid transmission, there is(13)ktiq˙i=θ˙mi,where kti is the transmission ratio of gear speed reducer and θmi is the rotor angular position.With the law of angular velocity synthesis, the total angular velocity is derived as follows:(14)ωmi=ωi−1+ktiq˙izmi,where ωi−1 is the angular velocity of the i−1# link and zmi is the unit vector of rotor axis.The linear and angular velocities of the rotor center of mass can be expressed as(15)p˙mi=Jpmiq˙,ωmi=Jomiq˙,where Jpmi=Jp1mi⋯Jpi−1mi0⋯0, Jomi=Jo1mi⋯Joimi0⋯0, and the elements of matrix are given by(16)Jpimi=zi−1×pmi−pi−1,Jojmi=Jojli,j=1,2,…,i−1,ktizmi,j=i.If the rotor turns around its center, then,Jpimi=0.So, the kinetic energy componentEkmi of the ith rotor can be given by(17)Ekmi=12mmiq˙TJpmiTJpmiq˙+12q˙TJomiTImiJomiq˙,where mmi is the mass of the rotor and Imi is the inertia matrix of rotor.Summing different components of a single link in (12) and a single motor in (17), the total kinetic energy of the robotic manipulator can be calculated and yields (18)Ek=Ekli+Ekmi=12∑i=1n∑j=1nbijqq˙iq˙j=12q˙TBqq˙,where Bq is the inertia matrix, satisfied with(19)Bq=∑i=1nmliJpliTJpli+JoliTIliJoli+mmiJpmiTJpmi+JomiTImiJomi.The potential energy of theith link can be calculated as follows:(20)Uli=−∫Vlig0Tpi∗ρdV=−mlig0Tpli,where g0 is acceleration vector of gravity in the base coordinate system.So, the total potential energyU of the robotic manipulator can be expressed as(21)U=∑i=1nUli+Umi=−∑i=1nmlig0Tpli+mmig0Tpmi,where pmi is the position vector of rotor center of mass.According to (18) and (21), the Lagrange function is constructed as follows:(22)Lq,q˙=Ekq,q˙−Uq.The dynamic equation is derived by using Lagrange function, yielding(23)τi=ddt∂L∂q˙i−∂U∂qi,where(24)ddt∂L∂q˙i=ddt∂Ek∂q˙i=∑j=1nbijqq¨j+∑j=1n∑k=1n∂bijq∂qkq˙kq˙j,∂Ek∂qi=12∑j=1n∑k=1n∂bijq∂qiq˙kq˙j,∂U∂qi=−∑j=1nmljg0TJpiliq+mmjg0TJpimiq=giq.The dynamic equation of robotic manipulator withn-DOF can be derived as(25)∑j=1nbijqq¨j+∑j=1n∑k=1nhijkqq˙kq˙j+giq=τi,i=1,2,…,n,where hijk=∂b˙ij/∂qk−1/2∂bij/∂qi.Comparing (25) with (1), the elements cij of Cq,q˙ should be satisfied with(26)∑j=1ncijqj=∑j=1n∑k=1nhijkq˙kq˙j.Considering the viscous friction and Coulomb friction, equation (25) can also be rewritten as follows:(27)Bqq¨+Cq,q˙q˙+Fvq˙+Fssgnq˙+gq=τ−JTqhe,where he is contact force of end-effector, Jq is velocity Jacobi matrix, Fvq˙ is the torque of viscous friction, and Fssgnq˙ is the torque of Coulomb friction.


### 2.2. Parameters and Dynamic Model of Two-Link Robotic Manipulator

The dynamic mathematical model for a rigid planar robotic manipulator having two links and a contact surface with the external force acting on the surface is shown in Figure2. According to the coordinate system o−xy, it consists of two links having link lengths l1 and l2 with their centers of mass m1 and m2 lying at the middle of links, respectively. The length of the center of mass is p1 and p2, respectively.Figure 2
Two-link robotic manipulator plant with contact force at tip.The Lagrange method is used to build the precise dynamic model of a two-link robotic manipulator with their nominal values as listed in Table1.Table 1
Variables description of the two-link robotic manipulator.
DescriptionNominal valueCenter of mass link 1:m1 (kg)0.1Length of link 1:l1 (m)0.8Length of center of mass link 1:p1 (m)0.4Centroid inertia of link 1:I1 (kgm2)0.064Center of mass link 2:m2 (kg)0.1Length of link 2:l2 (m)0.4Length of center of mass link 2:p2 (m)0.2Centroid inertia of link 2:I2 (kgm2)0.016The total kinetic energy of the two-link robotic manipulator is(28)Ek=Ek1+Ek2=12m1p12θ˙12+12m2l12θ˙12+12m2p22θ˙1+θ˙22+m2l1p2θ˙12+θ˙1θ˙2cosθ2,where θ1 and θ2 are joint variables of the two-link robotic manipulator.The total potential energy of the two-link robotic manipulator is(29)Ep=Ep1+Ep2=m1gp1cosθ1+m2gl1cosθ1+p2cosθ1+θ2.By using (23), the dynamic equation of two-link robotic manipulator can be derived as(30)τ1=ddt∂L∂θ˙1−∂L∂θ1=D11θ¨1+D12θ¨2+D112θ˙1θ˙2+D122θ˙22+D1,where s1=sinθ1, c1=cosθ1s12=sinθ1+θ2, and(31)D11=m1p12+m2p22+m2l12+2m2l1p2c2,D12=m2p22+m2l1p2c2,D112=−2m2l1p2s2,D122=−m2l1p2s2,D1=m1p1+m2l1gs1+m2p2gs12.Meanwhile, the torque of joint 2τ2 can be also derived as(32)τ2=ddt∂L∂θ˙2−∂L∂θ2=D21θ¨1+D22θ¨2+D212θ˙1θ˙2+D211θ˙22+D2,where(33)D21=m1p22+m2l1p2c2,D22=m2p22,D212=0,D211=m2l1p2s2,D2=m2p2gs12.Combining (30) with (32), the robotic plant can be rewritten with following mathematical model:(34)D11D12D21D22θ¨1θ¨2+D112D212θ˙1θ˙2θ˙2θ˙1+D211D122θ˙22θ˙12+D1D2=τ1τ2.Equation (14) completely represents the relationship between actuated torque and displacement, velocity, and acceleration in the joint space. The issues with D11 and D22 represent the moment of inertia caused by the acceleration of joint 1 and joint 2, respectively. The issues with D12 and D21 represent the moment of inertia of the acceleration coupling between two joints. The issues with D122 and D211 represent the coupling moment term of the centripetal force caused by the velocity between two joints. The issues with D112 and D212 represent the coupling moment term of the Coriolis force between two joints. D1 and D2 represent the gravity moment term. Considering the effect of centroid inertia, (35) can be modified as(35)D11′=I1+I2+D11,D12′=I2+D12,D21′=I2+D21,D22′=I2+D22.Furthermore, to consider the effect of the contact force acting on the end-effector, the right side of (34) can be replaced by(36)τ′=τ−JTqhe,where he is the force and moment vector applied by the end-effector in the working environment and JTq is the velocity Jacobian matrix, yielding(37)Jq=−l1s1−l2s12−l2s12l1c1+l2c12l2c12.


## 2.1. Description of Link Parameters of Robotic Manipulator

The dynamic parameters which describe the dynamic model are important for the control algorithms, effective simulation results, and accurate trajectory tracking algorithms. Dynamic equation of the robotic manipulator withn-DOF has been characterized in many literature studies [1–11] as follows:(1)Dqq¨+Cq,q˙q˙+Gq+Fq,q˙+τd=τ,where q,q˙,q¨∈Rn are the link position, velocity, and acceleration vectors, respectively, Dq∈Rn×n is the symmetric positive definite inertia matrix, Cq,q˙∈Rn×n is the Coriolis or centrifugal forces, Gq∈Rn×1 consolidates the gravitational force, Fq,q˙∈Rn×1 incorporates the friction terms, and τd represents external disturbance.However, quadratic velocity terms and dynamic coupling terms are not taken into account. So, the problem of model accuracy cannot be solved essentially only through parameter identification and compensation methods. The kinematic description of theith link is shown in Figure 1.Figure 1
Kinematic description of theith link.An infinitesimal elementei is selected which lies on the ith link, and the position of the center of mass ci is represented by vector pi, the position vector of ei is presented by vector pi∗, and the position vector from the center of mass to the infinitesimal element is ri. When integrating in the whole area of the ith link, the infinitesimal element can represent the motion performance of the whole link.The kinetic energy componentEkli of the ith link can be given by(2)Ekli=12∫Vlip˙i∗Tp˙i∗ρdV,where p˙i∗ is the linear velocity vector of the infinitesimal element, ρ is material density, and dV is the volume of the infinitesimal element.The position vector relationship betweenci and ei is satisfied with(3)ri=rixriyrizT=pi∗−pli,where pli is calculated by centroid theorem as(4)pli=1mli∫Vlipi∗ρdV,where mli is the mass of the ith link.Differentiating equation (3) with respect to time, one can get(5)p˙i∗=p˙li+ωi×ri=p˙li+Sωiri,where p˙li is the linear velocity vector of the center of mass, ωi is angular velocity of the infinitesimal element around the center of mass, and Sωi is the antisymmetric matrix of three-dimensional vector ωi=ωixωiyωizT and defined as follows:(6)Sωi=0−ωizωiyωiz0−ωix−ωiyωix0.Substituting equation (5) into equation (2), one can obtain(7)Ekli=12∫Vlip˙li+SωiriTp˙li+SωiriρdV.Equation (7) shows that the kinetic energy of each link consists of three parts, described as follows:(1)
Translational kinetic energy is expressed as(8)12∫Vlip˙liTp˙liρdV=12mlip˙liTp˙li.(2)
Implicated motion kinetic energy is expressed as(9)212∫Vlip˙liSωiriρdV=212p˙liTSωi∫Vlipi∗−pliρdV=0.(3)
Rotation kinetic energy, combining with equation (6), is expressed as(10)12∫VliriTSTωiSωiriρdV=12ωiT∫VliSTriSriρdVωi=12ωiTIliωi,whereSωiri=−Sriωi; Ili is defined as(11)Ili=∫riy2+riz2ρdV−∫rixriyρdV−∫rixrizρdV−∫rixriyρdV∫rix2+riz2ρdV−∫rizriyρdV−∫rixrizρdV−∫rizriyρdV∫riy2+rix2ρdV,(1)
whereIli is symmetric matrix and denotes inertia tensor related to the center of mass of the ith link in base coordinate system.Combining equation (8) with equation (10), the kinetic energy of the ith link can be given by(12)Ekli=12mliq˙TJpliTJpliq˙+12q˙TJoliTIliJoliq˙,where q=q1q2⋯qiT is joint variable vector.The kinetic energy component of the motor ofi# joint can be calculated in a similar way. Under the assumption of rigid transmission, there is(13)ktiq˙i=θ˙mi,where kti is the transmission ratio of gear speed reducer and θmi is the rotor angular position.With the law of angular velocity synthesis, the total angular velocity is derived as follows:(14)ωmi=ωi−1+ktiq˙izmi,where ωi−1 is the angular velocity of the i−1# link and zmi is the unit vector of rotor axis.The linear and angular velocities of the rotor center of mass can be expressed as(15)p˙mi=Jpmiq˙,ωmi=Jomiq˙,where Jpmi=Jp1mi⋯Jpi−1mi0⋯0, Jomi=Jo1mi⋯Joimi0⋯0, and the elements of matrix are given by(16)Jpimi=zi−1×pmi−pi−1,Jojmi=Jojli,j=1,2,…,i−1,ktizmi,j=i.If the rotor turns around its center, then,Jpimi=0.So, the kinetic energy componentEkmi of the ith rotor can be given by(17)Ekmi=12mmiq˙TJpmiTJpmiq˙+12q˙TJomiTImiJomiq˙,where mmi is the mass of the rotor and Imi is the inertia matrix of rotor.Summing different components of a single link in (12) and a single motor in (17), the total kinetic energy of the robotic manipulator can be calculated and yields (18)Ek=Ekli+Ekmi=12∑i=1n∑j=1nbijqq˙iq˙j=12q˙TBqq˙,where Bq is the inertia matrix, satisfied with(19)Bq=∑i=1nmliJpliTJpli+JoliTIliJoli+mmiJpmiTJpmi+JomiTImiJomi.The potential energy of theith link can be calculated as follows:(20)Uli=−∫Vlig0Tpi∗ρdV=−mlig0Tpli,where g0 is acceleration vector of gravity in the base coordinate system.So, the total potential energyU of the robotic manipulator can be expressed as(21)U=∑i=1nUli+Umi=−∑i=1nmlig0Tpli+mmig0Tpmi,where pmi is the position vector of rotor center of mass.According to (18) and (21), the Lagrange function is constructed as follows:(22)Lq,q˙=Ekq,q˙−Uq.The dynamic equation is derived by using Lagrange function, yielding(23)τi=ddt∂L∂q˙i−∂U∂qi,where(24)ddt∂L∂q˙i=ddt∂Ek∂q˙i=∑j=1nbijqq¨j+∑j=1n∑k=1n∂bijq∂qkq˙kq˙j,∂Ek∂qi=12∑j=1n∑k=1n∂bijq∂qiq˙kq˙j,∂U∂qi=−∑j=1nmljg0TJpiliq+mmjg0TJpimiq=giq.The dynamic equation of robotic manipulator withn-DOF can be derived as(25)∑j=1nbijqq¨j+∑j=1n∑k=1nhijkqq˙kq˙j+giq=τi,i=1,2,…,n,where hijk=∂b˙ij/∂qk−1/2∂bij/∂qi.Comparing (25) with (1), the elements cij of Cq,q˙ should be satisfied with(26)∑j=1ncijqj=∑j=1n∑k=1nhijkq˙kq˙j.Considering the viscous friction and Coulomb friction, equation (25) can also be rewritten as follows:(27)Bqq¨+Cq,q˙q˙+Fvq˙+Fssgnq˙+gq=τ−JTqhe,where he is contact force of end-effector, Jq is velocity Jacobi matrix, Fvq˙ is the torque of viscous friction, and Fssgnq˙ is the torque of Coulomb friction.


## 2.2. Parameters and Dynamic Model of Two-Link Robotic Manipulator

The dynamic mathematical model for a rigid planar robotic manipulator having two links and a contact surface with the external force acting on the surface is shown in Figure2. According to the coordinate system o−xy, it consists of two links having link lengths l1 and l2 with their centers of mass m1 and m2 lying at the middle of links, respectively. The length of the center of mass is p1 and p2, respectively.Figure 2
Two-link robotic manipulator plant with contact force at tip.The Lagrange method is used to build the precise dynamic model of a two-link robotic manipulator with their nominal values as listed in Table1.Table 1
Variables description of the two-link robotic manipulator.
DescriptionNominal valueCenter of mass link 1:m1 (kg)0.1Length of link 1:l1 (m)0.8Length of center of mass link 1:p1 (m)0.4Centroid inertia of link 1:I1 (kgm2)0.064Center of mass link 2:m2 (kg)0.1Length of link 2:l2 (m)0.4Length of center of mass link 2:p2 (m)0.2Centroid inertia of link 2:I2 (kgm2)0.016The total kinetic energy of the two-link robotic manipulator is(28)Ek=Ek1+Ek2=12m1p12θ˙12+12m2l12θ˙12+12m2p22θ˙1+θ˙22+m2l1p2θ˙12+θ˙1θ˙2cosθ2,where θ1 and θ2 are joint variables of the two-link robotic manipulator.The total potential energy of the two-link robotic manipulator is(29)Ep=Ep1+Ep2=m1gp1cosθ1+m2gl1cosθ1+p2cosθ1+θ2.By using (23), the dynamic equation of two-link robotic manipulator can be derived as(30)τ1=ddt∂L∂θ˙1−∂L∂θ1=D11θ¨1+D12θ¨2+D112θ˙1θ˙2+D122θ˙22+D1,where s1=sinθ1, c1=cosθ1s12=sinθ1+θ2, and(31)D11=m1p12+m2p22+m2l12+2m2l1p2c2,D12=m2p22+m2l1p2c2,D112=−2m2l1p2s2,D122=−m2l1p2s2,D1=m1p1+m2l1gs1+m2p2gs12.Meanwhile, the torque of joint 2τ2 can be also derived as(32)τ2=ddt∂L∂θ˙2−∂L∂θ2=D21θ¨1+D22θ¨2+D212θ˙1θ˙2+D211θ˙22+D2,where(33)D21=m1p22+m2l1p2c2,D22=m2p22,D212=0,D211=m2l1p2s2,D2=m2p2gs12.Combining (30) with (32), the robotic plant can be rewritten with following mathematical model:(34)D11D12D21D22θ¨1θ¨2+D112D212θ˙1θ˙2θ˙2θ˙1+D211D122θ˙22θ˙12+D1D2=τ1τ2.Equation (14) completely represents the relationship between actuated torque and displacement, velocity, and acceleration in the joint space. The issues with D11 and D22 represent the moment of inertia caused by the acceleration of joint 1 and joint 2, respectively. The issues with D12 and D21 represent the moment of inertia of the acceleration coupling between two joints. The issues with D122 and D211 represent the coupling moment term of the centripetal force caused by the velocity between two joints. The issues with D112 and D212 represent the coupling moment term of the Coriolis force between two joints. D1 and D2 represent the gravity moment term. Considering the effect of centroid inertia, (35) can be modified as(35)D11′=I1+I2+D11,D12′=I2+D12,D21′=I2+D21,D22′=I2+D22.Furthermore, to consider the effect of the contact force acting on the end-effector, the right side of (34) can be replaced by(36)τ′=τ−JTqhe,where he is the force and moment vector applied by the end-effector in the working environment and JTq is the velocity Jacobian matrix, yielding(37)Jq=−l1s1−l2s12−l2s12l1c1+l2c12l2c12.


## 3. Adaptive Backstepping Sliding Mode Control Algorithm

In this section, an adaptive backstepping sliding mode controller is presented which achieves precision trajectory tracking property by guaranteeing the robustness and stability of the closed-loop system of robotic manipulator. The uncertainties included in the system are required for compensating the external disturbances and nonlinear dynamics in terms described as (27). The adaptive backstepping sliding mode control is constructed at the final step.The inverse dynamic equation can be expressed as(38)q¨=−B−1qCq,q˙q˙+Fvq˙+Fssgnq˙+gq+B−1qτ−JTqhe.Property 1.
Bq is the n×n symmetric inertial matrix and bounded:(39)c1A2≤ATBqA≤c2A2,∀A∈Rn×1,where c1 and c2 are positive constants.Property 2.
B˙q−2Cq,q˙ is skew symmetric matrix and satisfies(40)ATB˙q−2Cq,q˙A=0.Property 3.
Cq,q˙, gq, and Fvq˙+Fssgnq˙ are bounded as follows:(41)Cq,q˙q˙≤ckq˙2,gq≤gk,Fvq˙+Fssgnq˙≤fkq˙,where ck, gk, and fk are positive constants.Property 4.
JTqhe∈Rn is unknown disturbance and bounded as(42)JTqhe≤τk,τk>0.
Defining vectorsx1=q, x2=q˙, x=x1x2T, the desired track is defined as xd=x1dx2dT, so the state equation of the robotic manipulator can be given by(43)x˙1=x2,x˙2=B−1qτ−Cq,q˙q˙−gq−φq,q˙,where φq,q˙ is defined as the disturbance nonlinear function.
Assuming thatqd is the desired trajectory of the robotic manipulator in joint space, select e1 as the trajectory tracking error; it is defined as(44)e1=q−qd=x1−x1d.
Differentiating (44) with respect to time, one can get(45)e˙1=q˙−q˙d=x˙1−x˙1d=x2−x˙1d.
By using backstepping algorithm, lete2 be a virtual input; then, the feedback control law can be expressed as(46)e2=e˙1+k1e1,where k1∈Rn×n is a positive definite diagonal coefficient matrix according to the DOF of robotic manipulator.Theorem 1.
Consider a robotic manipulator with n-DOF with the dynamic in (28), by designing the following sliding mode controller as(47)τeq=−hλe2−k1e1+B−1qτ−Cq,q˙q˙−gq−φq,q˙−q¨d+k1e˙1=−hs,where h∈Rn×n is a positive definite diagonal coefficient matrix.Select the Lyapunov function of the first step as(48)V1=12e1Te1.Differentiating (48) with respect to time, one can get(49)V˙1=e1Te˙1=e1Te2−k1e1=e1Te2−e1Tk1e1.Ife2=0, then V˙1=−e1Tk1e1≤0. So, the sliding surface function is defined as s=λe1+e2, and the Lyapunov function of the second step is selected as(50)V2=V1+12sTs,where λ∈Rn×n is a positive definite diagonal coefficient matrix according to the DOF of robotic manipulator.Differentiating (50) with respect to time, one can get(51)V˙2=e1Te2−e1Tk1e1+sTλe˙1+e˙2=e1Te2−e1Tk1e1+sTλe2−k1e1+e¨1+k1e˙1=e1Te2−e1Tk1e1+sTλe2−k1e1+B−1qτ−Cq,q˙q˙−gq−φq,q˙−q¨d+k1e˙1.An adaptive algorithm can be assigned for nonlinear functionφq,q˙, yielding(52)eφq,q˙=φq,q˙−φ^q,q˙,e˙φq,q˙=−φ^⋅q,q˙,where eφq,q˙ is the estimation error and φ^q,q˙ is the estimated value of the nonlinear function φq,q˙.The Lyapunov function of the third step is selected as(53)V3=V2+12k2eφq,q˙Teφq,q˙.Differentiating (53) with respect to time, one can get(54)V˙3=V˙2−k2eφq,q˙Tφ^⋅q,q˙+1k2M−1qTs.So, the adaptive control law is derived asφ^⋅q,q˙=−1/k2M−1qTs.Considering the boundary ofφq,q˙ and Property 3, (54) can be rewritten as follows:(55)V˙3≤e1Te2−e1Tk1e1−λe1+e2Thλe1+e2=−e1e2TMe1e2,where(56)M=k1+λThλhλ−12Ihλ−12Ih.By giving the proper values ofk1, h, and λ, the matrix M can be positive, and V˙3≤0, and the stability of the robotic manipulator is guaranteed.


## 4. Simulation and Discussion

In this section, a two-link robotic manipulator is utilized to verify the effectiveness of the proposed control strategy. The structural parameters are described in Table1, and the dynamic equation is given by (35). The desired trajectory tracking of two joints is θ1d=θ2d=0.5sint.The parameter values used in the adaptive backstepping sliding mode control system areλ=diag5,5, k1=diag50,50, h=diag3,3, and the external disturbance is added by a pulse signal ϕ=0.5δt in 6 s, which is used to verify the robustness and stability of the two-link robotic manipulator system. The schematic for backstepping algorithm by using Matlab@ software is shown in Figure 3, and the schematic for the proposed control strategy is shown in Figure 4.Figure 3
Schematic for backstepping algorithm by using Matlab@ software.Figure 4
Schematic of the proposed control strategy.Figure5 shows the trajectory tracking performance of the proposed control with the actuated torque in joint space τ1 and τ2, shown in Figure 6, respectively.Figure 5
Trajectory tracking of two-link robotic manipulator. (a)1#link with q1. (b) 2#link with q2.
(a)(b)Figure 6
Actuated torque in the joint space as control signal. (a)τ1. (b) τ2.
(a)(b)The trajectory tracking errors of the two joints are shown in Figure7, respectively. The maximum tracking error of the 1# joint is equal to 0.0296 mm and in 1.6 s is 5.92%. The maximum tracking error of the 2# joint is equal to 0.0896 mm and in 1.2 s is 17.92% and in 2.2 s tends to convergence. However, the convergence time of 1# joint is in 2.8 s, which is affected by the coupling of 2# joint.Figure 7
Trajectory tracking errors of the two-link robotic manipulator in joint space.The trajectory tracking performance in end-effector space (operating space) is shown in Figure8. Furthermore, the mean square error of the estimation function φ^q,q˙ is derived, shown in Figure 9. Figure 9 shows that the maximum estimation error is 1.8×10−3, and the minimum error is 1.68×10−8, meaning that the proposed controller has a good effect in nonlinear estimation and tracking abilities.Figure 8
Trajectory tracking performance in the operating space of the end-effector.Figure 9
Mean square error of the estimation functionφ^q,q˙.To verify the robustness of the proposed control system, an external disturbance is added to the system in 6 s. The trajectory tracking of the two-link robotic manipulator under external disturbances is shown in Figure10. Furthermore, the trajectory tracking errors in joint space are shown in Figure 11. It shows that the maximum tracking error of the 1# joint is equal to 0.0316 mm, and the maximum tracking error of the 2# joint is equal to 0.076 mm. According to the given external disturbance, the percentages of the tracking errors are 6.32% and 15.2%, respectively.Figure 10
Trajectory tracking of the two-link robotic manipulator under external disturbances.
(a)(b)Figure 11
Tracking errors of the two-link robotic manipulator under external disturbances.


## 5. Conclusions

In this article, an adaptive backstepping sliding mode control subject to external disturbance is proposed. The dynamic model of the robotic manipulator is built by considering the coupling and nonlinear characteristics, and the estimate function of these nonlinear factors is proposed and used for the equivalent control law of sliding mode control. The control system is designed by the backstepping algorithm, and the stability and robustness of the two-link robotic manipulator are analyzed. Simulation results show that the proposed control system has good tracking performance and strong robustness for the external disturbance.



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*Source: 3156787-2020-08-14.xml*