# Control of Flexible Joint Robot Based on Motor State Feedback and Dynamic Surface Approach

Authors: Pengxiao Jia
Journal: Journal of Control Science and Engineering
 (2019)

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## Abstract

A controller based on dynamic surface control and observer is proposed by using motor state feedback for trajectory tracking of flexible joint robot with uncertain link dynamic model. Considering the link state information cannot be obtained, an observer is designed to estimate the link state information, and a dynamic surface controller is proposed based on link state observer. The controller based on the observer compared to backstepping controller avoids the repeated differentiation problem. At the same time, the dynamic surface method avoids the measurement of high order signal. The simulation results show that the designed controller has a good trajectory tracking effect, which effectively suppresses the residual vibration of the flexible joint robot. Moreover, the proposed controller and observer are robust to the uncertainty and external disturbance of the link dynamic model. The proposed controller can be directly applied in the flexible joint robot without installing additional sensors, which is very important for industrial applications.


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## Body

## 1. Introduction

In recent years, not only are the requirements for robots in modern manufacturing limited to repetitive tasks, but also the robots can be quickly transformed in multiple tasks with the market demand changing from large batches, single mode to small batches and diversified directions[1]. Modern manufacturing thus calls for a flexible approach to manage production process, while the most flexible factor in a manufacturing process is human operators. In this instance, human and robot working collaboratively will be more efficient [2]. According to ISO 10218-2, collaborative robots are now allowed to work hand in hand with humans while being required to safely perform physical interactions in the dynamically changing and unstructured working environments [3]. Therefore, flexible joints are used extensively on collaborative robots, because when the flexible joint robot encounters obstacles during the operation, contact force between robot and obstacles may be relatively slight, and the robot can stop immediately. In [4], a robot with a variable stiffness joint is proposed, and the leaf spring is used to generate compliance. A servo-controller to asymptotically regulate the driving torque with unknown parameters of flexible robot is derived in [5].Recently, many control methods of manipulator have been proposed such as adaptive neural network control [6–8], robust control [9], vibration control [10], and fuzzy control [11]. For the complex nonlinear systems whose model is uncertain, backstepping is a widely used method. But it suffers from the curse of dimension in the process of controller design. In order to avoid differentiating virtual control signals, dynamic surface control method is proposed in [12] by using first-order filters within the backstepping controller design. Currently, dynamic surface control method has been widely used in manipulators, marine vehicles, robots with flexible joints, and spacecrafts [13]. However, most studies on dynamic surface control focus on single-input single-output system [14]. Dynamic surface control is first applied to the controller design for nonlinear multi-input multioutput system like the flexible joint robot in [15]; however, the research is based on full-state feedback control, which needs to measure the link position, while most manipulators only have motor sensors installed, and the link state information cannot be acquired; therefore, this method cannot be used in industry directly. In practice, installing additional sensors can lead to a high cost design for the motion control system for industrial robots [16]. In order to obtain the link state information of the flexible joint manipulator, observer which can estimate the state variable unmeasurable is necessary. In [17], a neural network observer is designed to estimate the position and velocity of the link, but it is worth mentioning that this work is only used for single-input single-output system.Generally, the dynamic model of a flexible joint manipulator can be divided into two parts: the link dynamic and the motor dynamic. Considering the parameters of motor are easy to obtain, it can be assumed that the motor dynamic model is precisely known. Since the parameters such as the quality of link are quite different from the actual values, and the quality of payload changes with the operation of manipulator, and the manipulator is sometimes subject to external disturbance, the observer, and the controller need to be robust. Therefore, in this paper, a controller based on dynamic surface control and observer is proposed by using motor state feedback for trajectory tracking of flexible joint robot with uncertain link dynamic model. Compared with existing results, the contribution of this paper is listed as follows. First, because the states of the link are nonmeasurable in industrial robot, a high-gain observer is proposed to estimate link positions and velocities. Since there is no need to install additional sensors, the proposed method will further facilitate its practical applications. Second, the trajectory tracking controller is designed for the multiple-input multiple-output system using the dynamic surface method considering the uncertain link dynamic model. It can be seen that the dynamic surface method ends the complexity arising due to the explosion of terms by introducing a first-order filter into the virtual control input at each backstepping design procedure, simplifies the controller design steps, and avoids the measurement of high-order signals. In addition, the designed controller can be adjusted by a limited number of parameters, which facilitates the debugging of the controller performance. Third, the system stability is analyzed by using Lyapunov stability analysis method, and simulation studies are performed to illustrate the theoretical results. The system stability analysis proves that the dynamic surface controller based on the state observer makes all signals in the closed-loop system be UUB, and the first-order filter is introduced so that the tracking error no longer converges to zero, but can converge to small enough neighborhoods around the design parameters. It also proves that the designed controller based on observer is robust to the parameter uncertainty of the mass of the links and payload.The paper is organized as follows: Section2 states the problem formulation. A link observer is designed for the flexible joint robot considering the uncertainty and disturbance in Section 3. Section 4 presents the designed controller along with stability analysis. Section 5 provides simulation results to illustrate the theoretical results. Section 6 concludes this paper.


## 2. Problem Formulation

Consider a flexible joint robot, whose dynamics can be described as [18](1)Dqlq¨l+Cql,q˙lq˙l+Gql+kql-qr=0(2)Jq¨r-kql-qr=τwhere joints 1,2, and 3 are flexible joints, qr∈R3×1 and ql∈R3×1 represent, respectively, the motor angles and link positions, D(ql)∈R3×3 is the link inertia matrix, C(ql,q˙l)q˙l∈R3×1 represents the Coriolis and centrifugal forces, G(ql)∈R3×1 is the gravitational force vector, k represents joint stiffness, τ∈R3×1 represents torque, and J∈R3×3 is the matrix of the moments of the inertia of the motors.Equations (1) and (2) have several fundamental properties which can be exploited to facilitate control system design. These properties are as follows.Property 1.
The link inertia matrixD(ql) is symmetric and positive definite, and both D(ql) and D-1(ql) are uniformly bounded as follows: D(ql)≤MD and D(ql)-1≤MID. Since k is a constant matrix, D(ql)-1k≤MDk, where MD, MID, and MDk are positive constants.Property 2.
C ( q l , q ˙ l ) is uniformly bounded as follows: C(ql,q˙l)≤MC, where MC is a positive constant.Property 3.
The gravitational termG(ql) is uniformly bounded as follows: G(ql)≤MG, where MG is a positive constant.Property 4.
The matrixD˙(ql)-2C(ql,q˙l) is skew-symmetric.Property 5.
J is symmetric, positive definite, J≤MJ, where MJ is a positive constant.


## 3. Link State Observer

For the flexible joint robot described in (1) and (2), a state observer for estimating the position and velocity of the link is designed by using the method in [19].If we definex1=ql, x2=q˙l, x3=qr, x4=q˙r, the dynamic system (1) and (2) are described as(3)x˙1=x2x˙2=-D-1x1Cx1,x2x2+Gx1+kx1-x3x˙3=x4Jx˙4=τ+kx1-x3It is assumed that the parameters in (3) are nominal parameters. When the parameters of the link dynamic model are uncertain and there is external disturbance, the real state equation can be expressed as follows:(4)x˙1=x2x˙2=-D-1x1Cx1,x2x2+Gx1+kx1-x3+Eudx˙3=x4Jx˙4=τ+kx1-x3where Eud is the uncertainty and external disturbance, the specific form of which is as follows:(5)Eudx1,x2,x3=-D--1x1C-x1,x2x2+G-x1+kx1-x3+τdl+D-1x1Cx1,x2x2+Gx1+kx1-x3where D-(x1),C-(x1,x2)x2,G-(x1) are real parameters, τdl is the external disturbance, and Eud is bounded as follows: Eud≤Eud∗,Eud∗ is a constant.It is assumed thatqr and q˙r are measurable. The two outputs are defined as follows:(6)y1=x3y2=x4Now consider the following change of coordinates:(7)z1=k-1Jx4z2=x1z3=x2The reduced order system obtained via the above change of coordinates satisfies(8)z˙1=k-1Jx4=z2+k-1τ-y1(9)z˙2=x˙1=x2=z3(10)z˙3=x˙2=-D-1z2Cz2,z3z3+Gz2+kz2-y1Define ϕ(z2,z3,y1)=-D-1(z2)[C(z2,z3)z3+G(z2)+k(z2-y1)].Equations (8), (9), and (10) can be expressed as follows:(11)z˙=Az+Φz,y1,τy2=J-1kCzwhere(12)A∈R9×9=0I000I000C∈R3×9=I00Φz,y1,τ=k-1τ-y10ϕz2,z3,y1 Define H-=Hk-1J; (11) can be expressed as follows:(13)z^˙=Az^+Φz^,y1,τ+ΔH-y2-Cz^=Az^+Φz^,y1,τ+ΔHCz-z^Choose a matrix H that places all the eigenvalues of (A-HC) in the left half of the complex plane. H∈R9×3=H1T∣H2TH3TT and Hi∈R3×3, i=1,2,3The gain matrix can be expressed asΔ∈R9×9=ΓI000Γ2I000Γ3I, Γ∈R3×3=diag{Γ1,Γ2,Γ3}, and Γ1≥1,Γ2≥1,Γ3≥1.Consider the real dynamic system (4), ϕ-(z^2,z^3,y1)=ϕ(z^2,z^3,y1)+Eud; Φ can be expressed as follows:(14)Φz^,y1,τ=k-1τ-y10ϕz^2,z^3,y1=k-1τ-y10ϕ-z^2,z^3,y1-EudSubstituting (7) into (13) yields the equations of observer.(15)x^˙1=x^2+Γ2H2k-1Jx4-x^4(16)x^˙2=-D-1x^1[Cx^1,x^2x2+Gx^1+kx1-x3-Γ3H3k-1Jx4-x^4(17)k-1Jx^˙4=x^1+k-1τ-x3+ΓH1k-1Jx4-x^4Define x~i=xi-x^i; the observer state error can be expressed as follows:(18)x~˙1=x~2-Γ2H2k-1Jx~4(19)x~˙2=-D-1x1Cx1,x2x2+Gx1+kx1-x3+Eud+D-1x^1Cx^1,x^2x2+Gx^1+kx1-x3-Γ3H3k-1Jx~4=ϕ-x1,x2,x3-ϕx^1,x^2,x3-Γ3H3k-1Jx~4(20)k-1Jx~˙4=x~1-ΓH1k-1Jx~4


## 4. Dynamic Surface Controller Based on Observer

A visualization of the proposed controller structure is shown in Figure1. In this section, we assume we cannot measure the position x1 and the velocity x2 of the link. We use the observer proposed in the previous section to estimate the position x1 and the velocity x2.Figure 1
Controller structure.


### 4.1. Controller Design

First, we letS1=x^1-qld and compute its derivative; we have(21)S˙1=x^˙1-q˙ld=x^2-q˙ld+Γ2H2k-1Jx~4Define the virtual control variable as follows:(22)x-2=-η1S1+q˙ld-Γ2H2k-1Jx~4where η1>0 is the parameter of controller.Then we letx-2 pass through a first-order filter, we obtain a new variable x2f, and the filter is expressed as follows:(23)α2x˙2f+x2f=x-2,x2f0=x-20where α2 is the time constant.Second, we letS2=x^2-x2f and compute its derivative; we have(24)S˙2=x^˙2-x˙2f=-D-1x^1Cx^1,x^2x2+Gx^1+kx^1-x3+Γ3H3k-1Jx~4-x-2-x2fα2Define the virtual control variable as follows:(25)x-3=k-1kx^1+Cx^1,x^2x^2+Gx^1+Dx^1-η2S2-Γ3H3k-1Jx~4+x-2-x2fα2where η2>0 is the parameter of controller.Then we letx-3 pass through a first-order filter, we obtain a new variable x3f, and we have(26)α3x˙3f+x3f=x-3,x3f0=x-30where α3 is the time constant.Third, we letS3=x3-x3f and compute its derivative; we have(27)S˙3=x˙3-x˙3f=x4-x-3-x3fα3Define the virtual control variable as follows:(28)x-4=-η3S3+x-3-x3fα3where η3>0 is the parameter of controller.Then we letx-4 pass through a first-order filter, we obtain a new varialbe x4f, and we have(29)α4x˙4f+x4f=x-4,x4f0=x-40where α4 is the time constant.Fourth, we letS4=x4-x4f and compute its derivative; we have(30)S˙4=x˙4-x˙4f=J-1τ+kx1-x3-x-4-x4fα4=J-1τ+kx^1-kx3+kx~1-x-4-x4fα4Choose the control input as follows:(31)τ=kx3-kx^1+Jx-4-x4fα4-η4S4where η4>0 is the parameter of controller.


### 4.2. Stability Analysis

Assumption 6.
The position and velocity of the link are bounded.Assumption 7.
The uncertainty and external disturbance of the model are bounded.Assumption 8.
x ^ 1 , x ^ 2 are bounded.Property 9.
Δ, Γ are positive definite diagonal matrix and Γ2≤λmax(Γ),-Γ2≤-λmin(Γ).Property 10.
The eigenvalues ofΓ satisfy these conditions: λmax(Γm)=λmaxm(Γ), λmin(Γm)=λminm(Γ) where m is an integer, λmin(·) denotes the minimum eigenvalue, and λmax(·) denotes the maximum eigenvalue.Define the boundary layer errorri=xif-x-i and we have(32)r2=x2f-x-2=x2f+η1S1-q˙ld+Γ2H2k-1Jx~4(33)r3=x3f-x-3=x3f-k-1kx^1+Cx^1,x^2x^2+Gx^1+Dx^1-η2S2+Γ3H3k-1Jx~4+x-2-x2fα2=x3f-k-1kx^1+Cx^1,x^2x^2+Gx^1+Dx^1-η2S2-r2α2(34)r4=x4f-x-4=x4f+η3S3-x-3-x3fα3=x4f+η3S3+r3α3Now, S˙1can be expressed as(35)S˙1=S2+x2f-q˙ld+Γ2H2k-1Jx~4=S2+r2-η1S1+Γ2H2k-1Jx~4S˙2 can be expressed as(36)S˙2=-D-1x^1Cx^1,x^2x^2+Gx^1+kx^1-x3+Γ3H3k-1Jx~4-x-2-x2fα2=D-1x^1kr3+S3-η2S2+Γ3H3k-1Jx~4S˙3 can be expressed as(37)S˙3=x4-x-3-x3fα3=S4+x4f-x-3-x3fα3=S4+r4-η3S3S˙4 can be expressed as(38)S˙4=-η4S4+J-1kx~1The derivatives of ri are(39)r˙2=-r2α2-Ξ2S1,S2,r2,Qld(40)r˙3=-r3α3-Ξ3S1,S2,S3,r2,r3,Qld(41)r˙4=-r4α4-Ξ4S1,S2,S3,S4,r2,r3,r4,Qldwhere(42)Qld=qldTq˙ldTq¨ldTTΞ2S1,S2,r2,x~˙4,Qld=-η1S˙1+q¨ld+Γ2H2k-1Jx~˙4Ξ3S1,S2,S3,r2,r3,x~1,x~2,x~4,Qld=k-1kx^˙1+x^˙1T∂Cx^1+x^˙2T∂Cx^2x^2+Cx^1,x^2x^˙2+∂Cx^1x^˙1+x^˙1T∂Cx^1-η2S2-r2α2+Eud+Dx^1-η2S˙2-r˙2α2+EudΞ4S1,S2,S3,S4,r2,r3,r4,x~1,x~2,x~4,Qld=-η3S˙3-r˙3α3.Now, we choose a Lyapunov function:(43)V=Vc+Vowhere Vc is the Lyapunov function of controller and Vo is the Lyapunov function of observer.(44)Vc=12∑i=14SiTSi+∑i=13ri+1Tri+1Define x~=Δ-1k-1Jx~4x~1x~2T. Given a positive definite diagonal matrix Λ=diag{Γ,Γ,Γ} the derivative of x~ can be expressed as(45)x~˙=ΛA-HCx~+Δ-100ϕ-x1,x2,x3-ϕx^1,x^2,x3TLet Ω be the solution to the equation (A-HC)TΩ+Ω(A-HC)=-I.The Lyapunov function of observer can be expressed as(46)Vo=12x~TΛ3Ωx~Using (35) to (41), the time derivative of Vc is(47)V˙c=S1TS2+r2-η1S1+S2TD-1x^1kr3+S3-η2S2+S3TS4+r4-η3S3+S4T-η4S4+J-1kx~1+∑i=13ri+1T-1αi+1ri+1+Ξi+1≤S1S2+S1r2-η1S12+D-1x^1k2S2r3+S3-η2S22+S3S4+S3r4-η3S32-η4S42+J-1k2S4x~1+∑i=13-1αi+1ri+1+ri+1Ξi+1According toProperty1, applying Young’s inequality to (47), we obtain(48)V˙c≤S12+S224+S12+r224+MDkS22+r324+MDkS22+S324+S32+S424+S32+r424+λmaxJ-1kS42+x~124-∑i=14ηiSi2+∑i=13-1αi+1ri+12+ri+12+Ξi+124=2S12+14+2MDkS22+2+MDk4S32+14+λmaxJ-1kS42-∑i=14η1S42+5r224+4+MDk4r32+5r424+∑i=13-1αi+1ri+12+Ξi+124+λmaxJ-1k4x~12 where λmax(·) denotes the maximum eigenvalue.Let(49)η1∗=η1-2,η2∗=η2-14-2MDk,η3∗=η3-2-MDk4,η4∗=η4-14-λmaxJ-1k,1α2∗=1α2-54,1α3∗=1α3-4+MDk4,1α4∗=1α4-54,and we have(50)V˙c≤-∑i=14ηi∗Si2-∑i=231αi+1∗ri+12+∑i=23Ξi+124+λmaxJ-1k4x~12According toProperty9, it follows that(51)V˙c≤-∑i=14ηi∗Si2-∑i=231αi+1∗ri+12+∑i=23Ξi+124+λmaxJ-1k4Γ22Γ-2x~12≤-∑i=14ηi∗Si2-∑i=231αi+1∗ri+12+∑i=23Ξi+124+λmaxJ-1k4λmaxΓ2x~2 Using (14) to (20), the time derivative of Vo is(52)V˙o=12x~˙TΛ3Ωx~+x~TΛ3Ωx~˙=12ΛA-HCx~+Δ-100ϕ-x1,x2,x3-ϕx^1,x^2,x3TTΛ3Ωx~+12Λ3x~TΩΛA-HCx~+Δ-100ϕ-x1,x2,x3-ϕx^1,x^2,x3TTx~=12Λ4x~Tx~+12Δ-100ϕx1,x2,x3-ϕx^1,x^2,x3+EudTΛ3Ωx~+12Λ3x~TΩΔ-100ϕx1,x2,x3-ϕx^1,x^2,x3+EudTwhere ϕ(x1,x2,x3)-ϕx^1,x^2,x3 can be expressed as(53)ϕx1,x2,x3-ϕx^1,x^2,x3≤D-1x1kx3-x1-D-1x^1kx3-x^1+D-1x^1Cx^1,x^2-D-1x1Cx1,x2x2+D-1x^1Gx^1-D-1x1Gx1According toProperty1 and Assumptions6,7, and8, we have(54)D-1x1kx3-x1-D-1x^1kx3-x^1≤D-1x1kx3-x1+D-1x1kx1-D-1x1kx^1-D-1r^2kx3-x^1≤D-1x1kx1-x^1+D-1x1-D-1x^12kx3-x^1≤D-1x1k2x1-x^1+ς1k2D-1x1-D-1x^1≤MDKx1-x^1+ς1λmaxkD-1x1-D-1x^1≤ς2x1-x^1=ς2x~1where λmax(k) denotes the maximum eigenvalues of k, ς1, and ς2 are all positive constant, and x3-x^1≤ς1.In the same way, we can get(55)D-1x^1Cx^1,x^2x^˙2-D-1x1Cx1,x2x2≤ς3x~2+ς4x~1(56)D-1x^1Gx^1-D-1x1Gx1≤ς5x~1where ς3, ς4, ς5 are all positive constants.And then, we have(57)ϕx1,x2,x3-ϕx^1,x^2,x3≤ς6x~1+ς7x~2where ς6, ς7 are positive constants.Applying (57) to (52), we can get(58)V˙o≤-12λminΓ4x~2+λmaxΓ3Ω2Γ-32x~ς6x~1+ς7x~2+Eud≤-12λminΓ4x~2+λmaxΩς6+ς7Γ32x~2+λmaxΩEud∗≤-12λminΓ4x~2+λmaxΩς6+ς7λmaxΓ3x~2+λmaxΩEud∗Therefore, we have(59)V˙=V˙c+V˙o≤-∑i=14ηi∗Si2-∑i=231αi+1∗ri+12+∑i=23Ξi+124+λmaxJ-1kλmaxΓ24x~2-12λminΓ4x~2+λmaxΩς6+ς7λmaxΓ3x~2+λmaxΩEud∗Let(60)β∗=12λminΓ4-λmaxJ-1kλmaxΓ24-λmaxΩς6+ς7λmaxΓ3,and we have(61)V˙≤-∑i=14ηi∗Si2-∑i=131αi+1∗ri+12-β∗x~2+λmaxΩEud∗+∑i=23Ξi+124≤-2ρV+Υwhere(62)ρ∈0,min⁡ηi∗,1αi∗,β∗λmaxΛ3Ω,Υ=∑i=23Ξi+124+λmaxΩEud∗.If we choose ηi, Γ, and Η, and we let ρ≥Υ/2μ, when (t)=μ, V˙≤0.Therefore, V≤μ is invariant set and we have(63)0≤V≤Υ2ρ+V0-Υ2ρe-2ρtIf we choose the value appropriately, we can make Υ/2ρ small enough; the tracking error and observation error converge to a small neighborhood of the origin; that is, the tracking error of the closed-loop system and the observation error are semiglobally consistent and ultimately bounded and can be arbitrarily small by choosing the appropriate parameters.


## 4.1. Controller Design

First, we letS1=x^1-qld and compute its derivative; we have(21)S˙1=x^˙1-q˙ld=x^2-q˙ld+Γ2H2k-1Jx~4Define the virtual control variable as follows:(22)x-2=-η1S1+q˙ld-Γ2H2k-1Jx~4where η1>0 is the parameter of controller.Then we letx-2 pass through a first-order filter, we obtain a new variable x2f, and the filter is expressed as follows:(23)α2x˙2f+x2f=x-2,x2f0=x-20where α2 is the time constant.Second, we letS2=x^2-x2f and compute its derivative; we have(24)S˙2=x^˙2-x˙2f=-D-1x^1Cx^1,x^2x2+Gx^1+kx^1-x3+Γ3H3k-1Jx~4-x-2-x2fα2Define the virtual control variable as follows:(25)x-3=k-1kx^1+Cx^1,x^2x^2+Gx^1+Dx^1-η2S2-Γ3H3k-1Jx~4+x-2-x2fα2where η2>0 is the parameter of controller.Then we letx-3 pass through a first-order filter, we obtain a new variable x3f, and we have(26)α3x˙3f+x3f=x-3,x3f0=x-30where α3 is the time constant.Third, we letS3=x3-x3f and compute its derivative; we have(27)S˙3=x˙3-x˙3f=x4-x-3-x3fα3Define the virtual control variable as follows:(28)x-4=-η3S3+x-3-x3fα3where η3>0 is the parameter of controller.Then we letx-4 pass through a first-order filter, we obtain a new varialbe x4f, and we have(29)α4x˙4f+x4f=x-4,x4f0=x-40where α4 is the time constant.Fourth, we letS4=x4-x4f and compute its derivative; we have(30)S˙4=x˙4-x˙4f=J-1τ+kx1-x3-x-4-x4fα4=J-1τ+kx^1-kx3+kx~1-x-4-x4fα4Choose the control input as follows:(31)τ=kx3-kx^1+Jx-4-x4fα4-η4S4where η4>0 is the parameter of controller.


## 4.2. Stability Analysis

Assumption 6.
The position and velocity of the link are bounded.Assumption 7.
The uncertainty and external disturbance of the model are bounded.Assumption 8.
x ^ 1 , x ^ 2 are bounded.Property 9.
Δ, Γ are positive definite diagonal matrix and Γ2≤λmax(Γ),-Γ2≤-λmin(Γ).Property 10.
The eigenvalues ofΓ satisfy these conditions: λmax(Γm)=λmaxm(Γ), λmin(Γm)=λminm(Γ) where m is an integer, λmin(·) denotes the minimum eigenvalue, and λmax(·) denotes the maximum eigenvalue.Define the boundary layer errorri=xif-x-i and we have(32)r2=x2f-x-2=x2f+η1S1-q˙ld+Γ2H2k-1Jx~4(33)r3=x3f-x-3=x3f-k-1kx^1+Cx^1,x^2x^2+Gx^1+Dx^1-η2S2+Γ3H3k-1Jx~4+x-2-x2fα2=x3f-k-1kx^1+Cx^1,x^2x^2+Gx^1+Dx^1-η2S2-r2α2(34)r4=x4f-x-4=x4f+η3S3-x-3-x3fα3=x4f+η3S3+r3α3Now, S˙1can be expressed as(35)S˙1=S2+x2f-q˙ld+Γ2H2k-1Jx~4=S2+r2-η1S1+Γ2H2k-1Jx~4S˙2 can be expressed as(36)S˙2=-D-1x^1Cx^1,x^2x^2+Gx^1+kx^1-x3+Γ3H3k-1Jx~4-x-2-x2fα2=D-1x^1kr3+S3-η2S2+Γ3H3k-1Jx~4S˙3 can be expressed as(37)S˙3=x4-x-3-x3fα3=S4+x4f-x-3-x3fα3=S4+r4-η3S3S˙4 can be expressed as(38)S˙4=-η4S4+J-1kx~1The derivatives of ri are(39)r˙2=-r2α2-Ξ2S1,S2,r2,Qld(40)r˙3=-r3α3-Ξ3S1,S2,S3,r2,r3,Qld(41)r˙4=-r4α4-Ξ4S1,S2,S3,S4,r2,r3,r4,Qldwhere(42)Qld=qldTq˙ldTq¨ldTTΞ2S1,S2,r2,x~˙4,Qld=-η1S˙1+q¨ld+Γ2H2k-1Jx~˙4Ξ3S1,S2,S3,r2,r3,x~1,x~2,x~4,Qld=k-1kx^˙1+x^˙1T∂Cx^1+x^˙2T∂Cx^2x^2+Cx^1,x^2x^˙2+∂Cx^1x^˙1+x^˙1T∂Cx^1-η2S2-r2α2+Eud+Dx^1-η2S˙2-r˙2α2+EudΞ4S1,S2,S3,S4,r2,r3,r4,x~1,x~2,x~4,Qld=-η3S˙3-r˙3α3.Now, we choose a Lyapunov function:(43)V=Vc+Vowhere Vc is the Lyapunov function of controller and Vo is the Lyapunov function of observer.(44)Vc=12∑i=14SiTSi+∑i=13ri+1Tri+1Define x~=Δ-1k-1Jx~4x~1x~2T. Given a positive definite diagonal matrix Λ=diag{Γ,Γ,Γ} the derivative of x~ can be expressed as(45)x~˙=ΛA-HCx~+Δ-100ϕ-x1,x2,x3-ϕx^1,x^2,x3TLet Ω be the solution to the equation (A-HC)TΩ+Ω(A-HC)=-I.The Lyapunov function of observer can be expressed as(46)Vo=12x~TΛ3Ωx~Using (35) to (41), the time derivative of Vc is(47)V˙c=S1TS2+r2-η1S1+S2TD-1x^1kr3+S3-η2S2+S3TS4+r4-η3S3+S4T-η4S4+J-1kx~1+∑i=13ri+1T-1αi+1ri+1+Ξi+1≤S1S2+S1r2-η1S12+D-1x^1k2S2r3+S3-η2S22+S3S4+S3r4-η3S32-η4S42+J-1k2S4x~1+∑i=13-1αi+1ri+1+ri+1Ξi+1According toProperty1, applying Young’s inequality to (47), we obtain(48)V˙c≤S12+S224+S12+r224+MDkS22+r324+MDkS22+S324+S32+S424+S32+r424+λmaxJ-1kS42+x~124-∑i=14ηiSi2+∑i=13-1αi+1ri+12+ri+12+Ξi+124=2S12+14+2MDkS22+2+MDk4S32+14+λmaxJ-1kS42-∑i=14η1S42+5r224+4+MDk4r32+5r424+∑i=13-1αi+1ri+12+Ξi+124+λmaxJ-1k4x~12 where λmax(·) denotes the maximum eigenvalue.Let(49)η1∗=η1-2,η2∗=η2-14-2MDk,η3∗=η3-2-MDk4,η4∗=η4-14-λmaxJ-1k,1α2∗=1α2-54,1α3∗=1α3-4+MDk4,1α4∗=1α4-54,and we have(50)V˙c≤-∑i=14ηi∗Si2-∑i=231αi+1∗ri+12+∑i=23Ξi+124+λmaxJ-1k4x~12According toProperty9, it follows that(51)V˙c≤-∑i=14ηi∗Si2-∑i=231αi+1∗ri+12+∑i=23Ξi+124+λmaxJ-1k4Γ22Γ-2x~12≤-∑i=14ηi∗Si2-∑i=231αi+1∗ri+12+∑i=23Ξi+124+λmaxJ-1k4λmaxΓ2x~2 Using (14) to (20), the time derivative of Vo is(52)V˙o=12x~˙TΛ3Ωx~+x~TΛ3Ωx~˙=12ΛA-HCx~+Δ-100ϕ-x1,x2,x3-ϕx^1,x^2,x3TTΛ3Ωx~+12Λ3x~TΩΛA-HCx~+Δ-100ϕ-x1,x2,x3-ϕx^1,x^2,x3TTx~=12Λ4x~Tx~+12Δ-100ϕx1,x2,x3-ϕx^1,x^2,x3+EudTΛ3Ωx~+12Λ3x~TΩΔ-100ϕx1,x2,x3-ϕx^1,x^2,x3+EudTwhere ϕ(x1,x2,x3)-ϕx^1,x^2,x3 can be expressed as(53)ϕx1,x2,x3-ϕx^1,x^2,x3≤D-1x1kx3-x1-D-1x^1kx3-x^1+D-1x^1Cx^1,x^2-D-1x1Cx1,x2x2+D-1x^1Gx^1-D-1x1Gx1According toProperty1 and Assumptions6,7, and8, we have(54)D-1x1kx3-x1-D-1x^1kx3-x^1≤D-1x1kx3-x1+D-1x1kx1-D-1x1kx^1-D-1r^2kx3-x^1≤D-1x1kx1-x^1+D-1x1-D-1x^12kx3-x^1≤D-1x1k2x1-x^1+ς1k2D-1x1-D-1x^1≤MDKx1-x^1+ς1λmaxkD-1x1-D-1x^1≤ς2x1-x^1=ς2x~1where λmax(k) denotes the maximum eigenvalues of k, ς1, and ς2 are all positive constant, and x3-x^1≤ς1.In the same way, we can get(55)D-1x^1Cx^1,x^2x^˙2-D-1x1Cx1,x2x2≤ς3x~2+ς4x~1(56)D-1x^1Gx^1-D-1x1Gx1≤ς5x~1where ς3, ς4, ς5 are all positive constants.And then, we have(57)ϕx1,x2,x3-ϕx^1,x^2,x3≤ς6x~1+ς7x~2where ς6, ς7 are positive constants.Applying (57) to (52), we can get(58)V˙o≤-12λminΓ4x~2+λmaxΓ3Ω2Γ-32x~ς6x~1+ς7x~2+Eud≤-12λminΓ4x~2+λmaxΩς6+ς7Γ32x~2+λmaxΩEud∗≤-12λminΓ4x~2+λmaxΩς6+ς7λmaxΓ3x~2+λmaxΩEud∗Therefore, we have(59)V˙=V˙c+V˙o≤-∑i=14ηi∗Si2-∑i=231αi+1∗ri+12+∑i=23Ξi+124+λmaxJ-1kλmaxΓ24x~2-12λminΓ4x~2+λmaxΩς6+ς7λmaxΓ3x~2+λmaxΩEud∗Let(60)β∗=12λminΓ4-λmaxJ-1kλmaxΓ24-λmaxΩς6+ς7λmaxΓ3,and we have(61)V˙≤-∑i=14ηi∗Si2-∑i=131αi+1∗ri+12-β∗x~2+λmaxΩEud∗+∑i=23Ξi+124≤-2ρV+Υwhere(62)ρ∈0,min⁡ηi∗,1αi∗,β∗λmaxΛ3Ω,Υ=∑i=23Ξi+124+λmaxΩEud∗.If we choose ηi, Γ, and Η, and we let ρ≥Υ/2μ, when (t)=μ, V˙≤0.Therefore, V≤μ is invariant set and we have(63)0≤V≤Υ2ρ+V0-Υ2ρe-2ρtIf we choose the value appropriately, we can make Υ/2ρ small enough; the tracking error and observation error converge to a small neighborhood of the origin; that is, the tracking error of the closed-loop system and the observation error are semiglobally consistent and ultimately bounded and can be arbitrarily small by choosing the appropriate parameters.


## 5. Simulation

To demonstrate the effectiveness of the proposed controller, we perform simulation studies with the model of flexible joint robot (1) and (2). We choose the parameters of the robot in [17]. The reference trajectories of the three joints are(64)qld=πt232t≤2sπ4-πt-42322s<t≤4sπ4t>4sA simple PID controller has been developed in simulations for motor control for the three flexible joints. For simplicity, ‘Ref’ represents the reference trajectories; as shown in Figure 2, the position of three links connected by three flexible joints show an obvious vibration. Therefore, effective control must be taken to achieve vibration suppression.Figure 2
Response of three links.Besides the proposed controller, we also perform the comparative studies using the conventional dynamic surface controller, and both controllers have the same parameters. It is assumed that the position and the velocity of link are measurable in the method using the conventional dynamic surface controller. In the following simulation studies, we assume the estimated mass parameters of link dynamic model deviate from the actual values, and the link dynamic model is subject to a bounded external disturbance. The bounded external disturbance isτdl=50sin⁡(π/1.25). The actual values of three links and payload are half of the nominal values in the simulation. In the following simulation, ‘DSC’ represents the trajectories using the conventional dynamic surface controller that requires link information, and ‘ODSC’ represents trajectories using the controller proposed in this paper that does not require link information. Results of simulation are shown in Figures 3–9.Figure 3
The position and tracking error of link1.Figure 4
The position and tracking error of link2.Figure 5
The position and tracking error of link3.Figure 6
Control input of ODSC.Figure 7
The estimation error of position and velocity of link1.Figure 8
The estimation error of position and velocity of link2.Figure 9
The estimation error of position and velocity of link3.It can be observed from Figures3, 4, and 5 that even if the estimated parameters have large deviations and the robot is subjected to large external disturbance, the state observer can still accurately estimate the positions and velocities of the links. The controller designed by using dynamic surface method based on state observer proposed in this paper can effectively achieve accurate trajectory tracking of link and suppress vibration of the flexible joints. Compared to the conventional dynamic surface controller using actual values, the control effect of the proposed controller is almost the same. The simulation results show that the proposed controller is robustness to external disturbance and the parameter uncertainty of the mass of the mass of the links and payload. Figure 6 shows the control signals of three joints using the proposed controller.Figures3, 4, and 5 show that the controller proposed in this paper can achieve good trajectory tracking effect. Although the tracking error no longer converges to zero, it can converge to any small area associated with the design parameters, which is consistent with that the introduction of the first-order filter can still guarantee the uniform asymptotic stability of the whole system proved in Section 4.2. Figures 7, 8, and 9 show that the observer can approach the actual values with any small precision when the dynamic model has uncertain parameters, which means that the observer is robust to the uncertainty.Figures3, 4, and 5 show that the proposed controller achieves the same response as the conventional dynamic surface controller requiring the link information. Therefore, the proposed controller based on motor state feedback can still achieve the accurate trajectory tracking control for the flexible joint robot in the absence of the link state information. The proposed method does not need to modify the physical structure of existing industrial robot and will further facilitate its practical applications.


## 6. Conclusions

A controller for flexible joint robot with uncertain model is proposed based on motor state feedback. Considering the link state information cannot be obtained, an observer is designed to estimate the link state information. Considering the uncertain model and the external disturbance, a dynamic surface controller is proposed based on link state observer. The controller based on the observer compared to backstepping controller avoids the repeated differentiation problem. At the same time, the dynamic surface method avoids the measurement of high order signal. The Lyapunov stability proves that the controller based on observer makes the closed-loop system error signal consistent and ultimately bounded. The introduced first-order filter no longer makes the tracking error converge to zero, but can converge to an arbitrarily small area related to design parameters. The simulation results show that the designed controller has a good trajectory tracking effect, which effectively suppresses the residual vibration of the flexible joint robot. Moreover, the proposed controller and observer are robust to the uncertainty and disturbance of the link dynamic. The proposed controller can be directly applied in the industrial application and do not need install additional sensors, which is very important for industrial applications.



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*Source: 5431636-2019-07-01.xml*