# Optimal Adaptive Control and Backstepping Control Method with Sliding Mode Differentiator

Authors: Shengxin Sun; Yang Zhao; Hao Wu
Journal: Complexity
 (2021)

---

## Abstract

In order to improve the success rate of space debris object capture, how to increase the resistance to interference in the space robot arm has become an issue of interest. In addition, since the space operation time is always limited, finite-time control has become another urgent requirement needed to be addressed. Considering external disturbances, two control methods are proposed in this paper to solve the control problem of space robot arm. Firstly, a linear sliding mode control method is proposed considering the model uncertainties and external disturbances. The robot arm can track the desired trajectory, while a trade-off between optimality and robustness of the solved system can be achieved. Then, in order to reduce conservativeness and relax restrictions on external disturbances, a novel backstepping control method based on a finite-time integral sliding mode disturbance observer is developed, which compensates for the effects of both model uncertainties and infinite energy-based disturbance inputs. Finally, simulation examples are given to illustrate the effectiveness of the proposed control method.


---

## Body

## 1. Introduction

With the rapid growth of space projects in the last several decades, the increasing space debris residues from satellites scrapped in space bring huge threat to the existing on-orbit spacecraft [1]. Therefore, how to reduce the amount of space debris and effectively lower their risk level becomes more and more urgent. Under this background, Active Debris Removal (ADR) has become a worldwide research hotspot [2–4].Space debris is mostly discarded space scrap, which is out of control and eventually moves freely due to complex nutation. Since these high-speed tumbling targets are very difficult to catch directly, it is necessary to reduce the relative speed between the chaser and the target before the next on-orbit capture [5–8]. Thus, the final capture of the target can be achieved when the relative speed is slow enough [9, 10].In terms of control for system with uncertainties and disturbances, numerous methods are proposed. Focused on solving the problem of asynchronous phenomena with different solutions, Cheng et al. [11–13] proposed a finite-time backstepping control method by incorporating a hidden Markov model, and the finite-time asynchronous control is achieved in the end. Considering time-varying full-state constraints and uncertainties, an adaptive fuzzy backstepping control was proposed for nonlinear state-constrained systems by Zhou et al. [14–16], and the parameter updating law is different compared with existing adaptive updating methods. By using the integral sliding mode design method for nonlinear stochastic systems, Wang et al. [17, 18] presented a new integral sliding mode control for fuzzy stochastic systems subjected to matched/mismatched uncertainties. The asymptotic stability of sliding mode dynamics is guaranteed while a simple search algorithm is provided to find the stability bound. Liu et al. [19, 20] proposed universal adaptive control to solve the universal control problem of a class of uncertain nonlinear systems. Yang and Tan [21, 22] designed an adaptive neural network for sliding mode control of flexible manipulators. However, the conservativeness and strict restrictions of the disturbances of the above controller are still needed to be addressed.Taking into account the phenomenon of disturbing moments in real systems, robust control with disturbance observer is an efficient control scheme to address it. The essence of robust control is to maintain the robustness of a closed-loop system. Robustness refers to the ability of a control system to maintain certain properties of the system under certain parameter regimes. Robust control theory is the theory of robustness in space by optimising certain performance indicators with infinite parameters. So, in this paper, a robust control system for a robotic arm will be designed based on the backstepping method and the model will be compensated with a finite-time integral sliding mode disturbance observer in that it can approximate the disturbance moment vector well. Thus, the system can achieve accurate tracking of the robotic arm attitude command.Considering all the above practical challenges, in this paper, we study the stabilization problem of the flexible deceleration brush detumbling mechanism attached to a space robot arm. Two sliding mode control methods are developed to solve this problem. First, an optimalH∞sliding mode control law is proposed for space multijoint robotic manipulator with consideration of both optimality and robustness of the detumbling system. The proposed control law can stabilize the overall closed-loop system with a prescribed H∞ performance level. Moreover, in this design, by using the weighting matrix method, the balance between the optimality and robustness of the detumbling system is achieved. Second, considering the fact that in practical space, the external disturbance always refers to the mismatched type, a novel backstepping control law with the finite-time integral sliding mode disturbance observer is developed, which can compensate the effects of model uncertainty and mismatched disturbance input simultaneously. Finally, simulation examples are given to verify the accuracy and effectiveness of the controller.


## 2. Materials and Methods

### 2.1. Design of Robust Sliding Mode Controller

With the consideration of microgravity environment, the potential energy of the system can be ignored. Then, the Lagrange function of the total kinetic energy of the space robot system can be expressed as(1)L=12q˙togTHq˙tog,where(2)q˙tog=vI0ωI0q˙T,H=ME3−Mr0g×∑i=17miJTiMr0g×∑i=07Ii−mir0i×r0i×∑i=17IiJRi+mir0i×JTi∑i=17miJTiT∑i=17JRiTIi−miJTiTr0i×∑i=17JRiTIiJRi+miJTiTJTi,JRi=k1⋯ki0⋯0∈R3×7,JTi=k1×ri−p1⋯ki×ri−pi0⋯0∈R3×7,where r0g× is r0g×, vI0 is the linear velocity of the base, ωI0 is the angular velocity of the base, mi is the mass of the ith rod of the space robot, q=q1,q2,q3,q4,q5,q6,q7∈R7 are the variables of each joint of the manipulator, Ji is the joint that connects the i−1th link and the ith link, rIi∈R3 is the position vector of the centre of mass of the ith lever of the manipulator in an inertial coordinate system, pIi∈R3 is the position vector of the ith joint in an inertial frame, mi is the mass of the ith bar of the space robot, Ii is the inertial of ith link, and M is a given constant.By using the Lagrange equation(3)ddt∂L∂q˙tog−∂L∂qtog=τ,the dynamic equation of the space robot system can be obtained as follows:(4)Hqtogq¨tog+Cqtog,q˙togq˙tog=τ,where Hqtog is the symmetric positive definite inertial matrix, Cqtog,q˙tog is a nonlinear term satisfying Cqtog,q˙togq˙tog=H˙qtogq˙tog−∂/∂qtog1/2q˙togTHq˙tog, and τ is the external control force and torque.Since the space racemization robot works in a complex microgravity environment, considering uncertainties such as external disturbance, friction, and parameter error, equation (4) can be further expressed as follows:(5)Hqtogq¨tog+Cqtog,q˙togq˙tog=τ,Hqtog=H0qtog+ΔHqtog,Cqtog,q˙tog=C0qtog,q˙tog+ΔCqtog,q˙tog,where H0qtog and C0qtog,q˙tog are nominal matrices and ΔHqtog and ΔCqtog,q˙tog are the corresponding uncertain matrices. Then, the above equation can be further rewritten as(6)H0qtogq¨tog+C0qtog,q˙togq˙tog=τ−ΔHqtogq¨tog−ΔCqtog,q˙togq˙tog.Define the dynamic compensation as(7)τ=H0qtogu+C0qtog,q˙togq˙tog,q¨tog=u−H0−1ΔHq¨tog−H0−1ΔCq˙tog,where u is control input vector, and then define the external disturbances τd as(8)δqtog,q˙tog,q¨tog=−ΔHq¨tog+ΔCq˙tog−τd.In order to make the space robot end track the time-varying desired trajectory, the state tracking errore∈R13 is defined as(9)e=q˙tog−q˙togdqtog−qtogd=q˜˙togq˜tog,where qtogd is the desired joint angle and q˙togd is the desired joint angular velocity.The trajectory tracking error equation of the space robot can then be obtained as follows:(10)e˙=Aqtog,q˙toge+Bu+Bw,where each parameter is specifically defined as(11)Aqtog,q˙tog=−H0−1C00I130,B=I130T,w=−H0−1δqtog,q˙tog,q¨tog,u=H0−1τ−H0q¨togd−C0q˙togd.Therefore, the force and torque exerted on the space robot can be solved as(12)τ=H0q¨togd+u+C0q˙togd.In order to facilitate the tracking control of the desired trajectory and reach the desired racemate point at the end, an auxiliary equation is designed as(13)z=De=D11D120Iq˜˙togq˜tog,where D11 and D12 are constant matrices.Substituting equation (13) into equation (10) yields(14)e˙=ANe+BNu+BNw,where(15)AN=D−1−H0−1C00D11−1−D11−1D12D,BN=D−1H0−10,D1=D11D12,u=H0D1e˙+C0D1e,w=H0D11H0−1δqtog,q˙tog,q¨tog.Then, the control force and torque of the space robot can be expressed as(16)τ=H0q¨tog+C0q˙togd,q¨tog=q¨togd+−D11−1H0−1C0D11−D11−1D12e˙−D11−1H0−1C0D12e+D11−1H0−1u=q¨togd−D11−1D12e˙−D11−1H0−1C0D11D12q˜˙togq˜togT−u=q¨togd−D11−1D12q˜˙tog−D11−1H0−1C0D1e˙−u.By designingu=−Ket, the external disturbance w in the system can be reduced. Given any positive real number γ, we have(17)J=minu∈L2max0≠w∈L2∫0∞1/2eTQe+1/2uTRudt∫0∞1/2wTwdt≤γ2.Design the Lyapunov functionVx=1/2xTPx, and then the Riccati equation can be solved as follows:(18)P˙+PAN+ANP+PBNR−1−1γ2IBNP+Q=0.WithC0−1/2M0 being skew symmetric, P can be designed as(19)P=DTM000ND,where N is a positive definite symmetric constant matrix.Substituting equation (19) into equation (18), one can obtain(20)0NN0−DTBNR−1−1γ2IBNTD+Q=0.By utilizing Cholesky decomposition method, we can have(21)R1TR1=R−1−1γ2I−1.At the same time:(22)Q=Q1TQ1Q12Q12TQ2TQ2,(23)D=R1TQ1R1TQ20I,(24)N=12Q1TQ2+Q2TQ1−12Q21T+Q12.Therefore, the robust optimalH∞ state feedback controller can be designed as(25)u=−R−1BTDe.


### 2.2. Design of Backstepping Controller with Finite-Time Observer

In the previous section, we design anH∞ sliding mode control law for space multijoint robotic manipulator, which can stabilize the overall closed-loop system with a prescribed H∞ performance level. It should be pointed out that in the previous control approach, the model uncertainty ΔAx is dealt with the robust control framework via the LMI technique, which may bring conservativeness in practical application when finding a feasible solution for spacecraft or space robotic manipulator system. Moreover, the considered external disturbance input wt refers to a matched type, which is a conservative assumption for practical space multijoint robotic manipulator, forthe orbit disturbance is always unmatched. To this end, in this section, we will revisit the control design problem for the space multijoint robotic manipulator, where the model uncertainty ΔAx is treated as a matched nonlinearity of the system instead of a model uncertainty, and the considered disturbance input wt is an unmatched term. A novel backstepping control law with the finite-time integral sliding mode disturbance observer is developed, which can compensate the effects of model uncertainty ΔAx and infinite energy type disturbance input wt.We first recall the following dynamic equation for space multijoint robotic manipulator:(26)q˙q¨=q˙M∘−1ΔMqq¨−M∘−1q△cq,q˙q˙.Then, we define the state vector as(27)x=qq˙.Thus, equation (26) can be rewritten as(28)x˙t=Axt+△Ax+But+△Hx˙,x,where(29)A=0I00,B=0I,△Ax=000M∘−1△Mqxt.As the previous discussion, in the practical space environment, there always exist unknown model nonlinearities and unmatched external disturbances, which are denoted asfx,t∈R6 and wt∈R6, respectively. Then, considering these effects in system (23), the system equation can be rewritten as(30)x˙t=Axt+△Ax+But+△Hx˙,x+ϕx,t,where(31)ϕx,t=fx,t+wt.In the following discussion, we will employ the finite-time integral sliding mode disturbance observer method to estimate the total nonlinearityϕx,t, based on which a backstepping control law will be designed to stabilize system (24).Before proceeding the subsequent design work, we decompose the system state vectorxt as xt=x1Ttx2TtT with x1t=qt and x2t=q˙t and decompose ϕx,t as ϕx,t=ϕ1Tx,tϕ2Tx,tT. Then, system (26) can be decomposed as(32)x˙1tx˙2t=0I00x1tx2t+000M∘−1△Mqx1tx2t+000M∘−1△Mqx˙1tx˙2t+ϕ1x,tϕ2x,t+0Iut.We now rewrite equation (32) as the following two subsystems:(33)x˙1t=x2t+ϕ1x,t,x˙2t=−M∘−1△Mqx2t−M∘−1△Mqx˙2t+ϕ2x,t+ut.Next, we define the following two backstepping variables as(34)z1t=x1t,z2t=x2t−rt,where rt∈R3×1 is the virtual input vector to be designed, which is constructed as(35)rt=−k1z1t+ϕ∧x,t,where k1∈R3∗3 is a positive definite diagonal matrix to be designed and ϕ∧x,t∈R3∗1 is the estimation of the nonlinearity ϕx,t. With all the information in hand, we present the following finite-time integral sliding mode disturbance observer (FTISMDO) for system (32) as(36)s0=z1t−pt,p⋅=z2t+rt+ϕ∧x,t,s1jt=s0j⋅t+∫0tl1jsiga1js0j+l2jsiga2js0j⋅dt,ϕ^˙jx,t=l1jsiga1js0j+l2jsiga2js0j⋅+b1jsigc1js1j+b2js1+L1jsgns1j,where s0=s01s02s03s04T, for j=1,2,3,4; L1j≥supt≥0ϕ˙jx,t denotes the norm bound of the unknown derivative of ϕjx,t; and l1j,l2j,a1j,a2j,b1j,b2j,L1j are the positive parameters to be designed. In particular, these parameters should satisfy 0<a1j<1, a2j=a1j/1+a1j, siga1js0j=s0ja1jsgns0j, and siga2js˙0j=s˙0ja2jsgns˙0j where sgns0j and sgns˙0j are two sign functions.In observer (36), get(37)s˙0=z˙1t−pt⋅=x2t+ϕ1x,t−z2t+rt+ϕ∧x,t=ϕ1x,t−ϕ∧x,t,(38)s¨0=ϕ˙1x,t−ϕ˙^x,t.Combining equations (36)–(38) yields(39)s˙1jt=s¨0jt+l1jsiga1js0j+l2jsiga2js˙0j=ϕ^1jx,t−b1jsigc1js1j−b2js1−L1jsgns1j.Design the Lyapunov functionV1j=1/2s1j2:(40)V˙1j=s1js˙1j=s1jϕ˙1jx,t−b1jsigcjs1j−b2js1j−L1jsgns1jℤ≤−b1jsigcj+1s1j−b2js1j2.So,V˙1j+b1jsigcj+1s1j+b2js1j2≤0.(41)s0j⋅t+∫0tl1jsiga1js0j+l2jsiga2js0j⋅dt=0.Because of (41), s0jt, s0j⋅t, s¨0jt can converge in finite time.So,(42)s¨0jt=−l1jsiga1js0j−l2jsiga2js˙0j.Based on equation (41) s0jt, s0j⋅t,s¨0jt can converge in finite time T0j and so does s0 in finite time Ts0=maxjT0j. It should be noted that Hurwitz condition should be met for s¨0jt+l1jsiga1js0j+l2jsiga2js˙0j=0, and also a2j∈0,1, a1j=a2j/2−a2j or a2j∈0,1, a1j=a1j/1+a1j. Therefore, after finite time Ts0+Ts1,s˙0=0=ϕ1x,t−ϕ∧x,t, namely, the estimate error is able to converge in finite time Ts0+Ts1.In observer (36), note that the information of s0j⋅ is required, which cannot be measured and obtained directly due to physical constraints. To this end, a high-order sliding mode differentiator (HOSMD) is employed here to estimate s0j⋅. The HOSMD is presented as follows:(43)y0⋅=h0=−αky0−ftk/k+1sgny0−ft+y1,y1⋅=h1=−αk−1y0−h0k−1/ksgny0−h0+y2,yk−1⋅=hk=−α1yk−1−hk−21/2sgnyk−1−hk−2+yk,yk⋅=−α0y1−h0,where α0,α1,…,αk>0 are positive constants to be selected. According to [23], the following conclusion holds after a finite-time transient process:(44)y0=ft,yr=hr−1=fr−1t,r=1,2,…,k.In HOSMDs (37) and (43), ft denotes s0it, and s˙0it can be calculated by y1t.Based on the FTISMDO (32) and HOSMDs (37)–(43), we now reconsider the backstepping variables z1t and z2t. For z1t, based on the virtual input vector rt defined in equation (35), it is derived that(45)z1⋅⋅t=x˙1t=x2t+ϕx,t=z2t+rt+ϕx,t=z2t−k1z1t+ϕ∧x,t+ϕx,t=z2t−k1z1t−ϕ∼x,t,where(46)ϕ∼x,t=ϕ∧x,t−ϕx,t.On the other hand, forz2t, it is derived that(47)z˙2t=x˙2t−r˙t=−M∘−1△Mqx2t−M∘−1△Mqx˙2t+ϕ2x,t+ut−k1z˙1t+ϕ∧⋅x,t=−M∘−1△Mqx2t−M∘−1△Mqx˙2t+ϕ2x,t+ut−k1x2t+k1ϕx,t+ϕ∧⋅x,t=−M∘−1△Mqz2t+M∘−1△Mqk1z1t+M∘−1△Mqϕ∧1x,t−M∘−1△Mqx˙2t+ϕ2x,t+ut−k1z2t+rt+k1ϕx,t=ut−ϕ∧⋅x,t+ϕ2x,t−k1z2t+rt+k1ϕx,t+dx,t,where the lumped disturbance vector dx,t is defined as(48)dx,t=M∘−1△Mqk1z1t+M∘−1△Mqϕ∧1x,t−M∘−1△Mqx˙2t.As a result, we obtain the following system which is equivalent to equation (33):(49)z˙1t=z2t−k1z1t−ϕ∼x,t,z˙2t=ut−ϕ∧⋅x,t+ϕ2x,t−k1z2t+rt+k1ϕx,t+dx,t.Theorem 1.
Considering system (49), suppose that the model uncertainty △Mq satisfies △Mq≤β1, where β1>0 is a known constant; under the following adaptive control law:(50)ut=ϕ∧⋅x,t−I3−k1ϕ∧1x,t+k1z2t+rt−z1t−β1M∘−1k1z1t+β1M∘−1ϕ∧1x,t+β1M∘−1x˙2tsgnz2t,the overall closed-loop system (49) is asymptotically stable.Proof.
Considering system (49), we define the Lyapunov function for the first subsystem of (49) as follows:(51)V1t=12z1Ttz1t.
Calculating the time derivative yields(52)V˙1t=12z1Ttz2t−k1z1t−ϕ∼x,t=−k1z1t+12z1Ttz2t−12z1Ttϕ∼x,t.
Next, we design the Lyapunov function for the second subsystem of (49) as(53)V2t=12z2Ttz˙2t+V1t,and then we have(54)V˙2t=z2Ttz˙2t+V˙1t=z2Ttut−ϕ∧⋅x,t+ϕ2x,t−k1z2t+rt+k1ϕx,t+dx,t−k1z1t+12z1Ttz2t−12z1Ttϕ∼x,t.
Considering the lumped disturbancedx,t in equation (50), notice that the following inequality holds:(55)dx,t≤β1M∘−1k1z1t+β1M∘−1ϕ∧1x,t+β1M∘−1x˙2t.
If we substitute the control law (50) into equation (54), it is shown that(56)V˙2t≤z2Ttut−ϕ∧⋅x,t+ϕ2x,t−k1z2t+rt+k1ϕx,t+β1M∘−1k1z1t+β1M∘−1ϕ∧1x,t+β1M∘−1x˙2t−k1z1t+12z1Ttz2t−12z1Ttϕ∼x,t,≤z2Ttϕ∧⋅x,t−I3−k1ϕ∧1x,t+k1z2t+rt−z1t−β1M∘−1k1z1t+β1M∘−1ϕ∧1x,t+β1M∘−1x˙2tsgnz2t+z2Tt−ϕ∧⋅x,t+ϕ2x,t−k1z2t+rt+k1ϕx,t+β1M∘−1k1z1t+β1M∘−1ϕ∧1x,t+β1M∘−1x˙2t−k1z1t+12z1Ttz2t−12z1Ttϕ∼x,t.
By some calculation, we have that(57)V˙2t≤−ϕ∼x,t−k1z1t−z1Ttϕ∼x,t≤−k1z1t<0holds for ∀zt∈R6,zt≠0, which means that the overall closed-loop system (49) is asymptotically stable. Thus, we complete the proof.


## 2.1. Design of Robust Sliding Mode Controller

With the consideration of microgravity environment, the potential energy of the system can be ignored. Then, the Lagrange function of the total kinetic energy of the space robot system can be expressed as(1)L=12q˙togTHq˙tog,where(2)q˙tog=vI0ωI0q˙T,H=ME3−Mr0g×∑i=17miJTiMr0g×∑i=07Ii−mir0i×r0i×∑i=17IiJRi+mir0i×JTi∑i=17miJTiT∑i=17JRiTIi−miJTiTr0i×∑i=17JRiTIiJRi+miJTiTJTi,JRi=k1⋯ki0⋯0∈R3×7,JTi=k1×ri−p1⋯ki×ri−pi0⋯0∈R3×7,where r0g× is r0g×, vI0 is the linear velocity of the base, ωI0 is the angular velocity of the base, mi is the mass of the ith rod of the space robot, q=q1,q2,q3,q4,q5,q6,q7∈R7 are the variables of each joint of the manipulator, Ji is the joint that connects the i−1th link and the ith link, rIi∈R3 is the position vector of the centre of mass of the ith lever of the manipulator in an inertial coordinate system, pIi∈R3 is the position vector of the ith joint in an inertial frame, mi is the mass of the ith bar of the space robot, Ii is the inertial of ith link, and M is a given constant.By using the Lagrange equation(3)ddt∂L∂q˙tog−∂L∂qtog=τ,the dynamic equation of the space robot system can be obtained as follows:(4)Hqtogq¨tog+Cqtog,q˙togq˙tog=τ,where Hqtog is the symmetric positive definite inertial matrix, Cqtog,q˙tog is a nonlinear term satisfying Cqtog,q˙togq˙tog=H˙qtogq˙tog−∂/∂qtog1/2q˙togTHq˙tog, and τ is the external control force and torque.Since the space racemization robot works in a complex microgravity environment, considering uncertainties such as external disturbance, friction, and parameter error, equation (4) can be further expressed as follows:(5)Hqtogq¨tog+Cqtog,q˙togq˙tog=τ,Hqtog=H0qtog+ΔHqtog,Cqtog,q˙tog=C0qtog,q˙tog+ΔCqtog,q˙tog,where H0qtog and C0qtog,q˙tog are nominal matrices and ΔHqtog and ΔCqtog,q˙tog are the corresponding uncertain matrices. Then, the above equation can be further rewritten as(6)H0qtogq¨tog+C0qtog,q˙togq˙tog=τ−ΔHqtogq¨tog−ΔCqtog,q˙togq˙tog.Define the dynamic compensation as(7)τ=H0qtogu+C0qtog,q˙togq˙tog,q¨tog=u−H0−1ΔHq¨tog−H0−1ΔCq˙tog,where u is control input vector, and then define the external disturbances τd as(8)δqtog,q˙tog,q¨tog=−ΔHq¨tog+ΔCq˙tog−τd.In order to make the space robot end track the time-varying desired trajectory, the state tracking errore∈R13 is defined as(9)e=q˙tog−q˙togdqtog−qtogd=q˜˙togq˜tog,where qtogd is the desired joint angle and q˙togd is the desired joint angular velocity.The trajectory tracking error equation of the space robot can then be obtained as follows:(10)e˙=Aqtog,q˙toge+Bu+Bw,where each parameter is specifically defined as(11)Aqtog,q˙tog=−H0−1C00I130,B=I130T,w=−H0−1δqtog,q˙tog,q¨tog,u=H0−1τ−H0q¨togd−C0q˙togd.Therefore, the force and torque exerted on the space robot can be solved as(12)τ=H0q¨togd+u+C0q˙togd.In order to facilitate the tracking control of the desired trajectory and reach the desired racemate point at the end, an auxiliary equation is designed as(13)z=De=D11D120Iq˜˙togq˜tog,where D11 and D12 are constant matrices.Substituting equation (13) into equation (10) yields(14)e˙=ANe+BNu+BNw,where(15)AN=D−1−H0−1C00D11−1−D11−1D12D,BN=D−1H0−10,D1=D11D12,u=H0D1e˙+C0D1e,w=H0D11H0−1δqtog,q˙tog,q¨tog.Then, the control force and torque of the space robot can be expressed as(16)τ=H0q¨tog+C0q˙togd,q¨tog=q¨togd+−D11−1H0−1C0D11−D11−1D12e˙−D11−1H0−1C0D12e+D11−1H0−1u=q¨togd−D11−1D12e˙−D11−1H0−1C0D11D12q˜˙togq˜togT−u=q¨togd−D11−1D12q˜˙tog−D11−1H0−1C0D1e˙−u.By designingu=−Ket, the external disturbance w in the system can be reduced. Given any positive real number γ, we have(17)J=minu∈L2max0≠w∈L2∫0∞1/2eTQe+1/2uTRudt∫0∞1/2wTwdt≤γ2.Design the Lyapunov functionVx=1/2xTPx, and then the Riccati equation can be solved as follows:(18)P˙+PAN+ANP+PBNR−1−1γ2IBNP+Q=0.WithC0−1/2M0 being skew symmetric, P can be designed as(19)P=DTM000ND,where N is a positive definite symmetric constant matrix.Substituting equation (19) into equation (18), one can obtain(20)0NN0−DTBNR−1−1γ2IBNTD+Q=0.By utilizing Cholesky decomposition method, we can have(21)R1TR1=R−1−1γ2I−1.At the same time:(22)Q=Q1TQ1Q12Q12TQ2TQ2,(23)D=R1TQ1R1TQ20I,(24)N=12Q1TQ2+Q2TQ1−12Q21T+Q12.Therefore, the robust optimalH∞ state feedback controller can be designed as(25)u=−R−1BTDe.


## 2.2. Design of Backstepping Controller with Finite-Time Observer

In the previous section, we design anH∞ sliding mode control law for space multijoint robotic manipulator, which can stabilize the overall closed-loop system with a prescribed H∞ performance level. It should be pointed out that in the previous control approach, the model uncertainty ΔAx is dealt with the robust control framework via the LMI technique, which may bring conservativeness in practical application when finding a feasible solution for spacecraft or space robotic manipulator system. Moreover, the considered external disturbance input wt refers to a matched type, which is a conservative assumption for practical space multijoint robotic manipulator, forthe orbit disturbance is always unmatched. To this end, in this section, we will revisit the control design problem for the space multijoint robotic manipulator, where the model uncertainty ΔAx is treated as a matched nonlinearity of the system instead of a model uncertainty, and the considered disturbance input wt is an unmatched term. A novel backstepping control law with the finite-time integral sliding mode disturbance observer is developed, which can compensate the effects of model uncertainty ΔAx and infinite energy type disturbance input wt.We first recall the following dynamic equation for space multijoint robotic manipulator:(26)q˙q¨=q˙M∘−1ΔMqq¨−M∘−1q△cq,q˙q˙.Then, we define the state vector as(27)x=qq˙.Thus, equation (26) can be rewritten as(28)x˙t=Axt+△Ax+But+△Hx˙,x,where(29)A=0I00,B=0I,△Ax=000M∘−1△Mqxt.As the previous discussion, in the practical space environment, there always exist unknown model nonlinearities and unmatched external disturbances, which are denoted asfx,t∈R6 and wt∈R6, respectively. Then, considering these effects in system (23), the system equation can be rewritten as(30)x˙t=Axt+△Ax+But+△Hx˙,x+ϕx,t,where(31)ϕx,t=fx,t+wt.In the following discussion, we will employ the finite-time integral sliding mode disturbance observer method to estimate the total nonlinearityϕx,t, based on which a backstepping control law will be designed to stabilize system (24).Before proceeding the subsequent design work, we decompose the system state vectorxt as xt=x1Ttx2TtT with x1t=qt and x2t=q˙t and decompose ϕx,t as ϕx,t=ϕ1Tx,tϕ2Tx,tT. Then, system (26) can be decomposed as(32)x˙1tx˙2t=0I00x1tx2t+000M∘−1△Mqx1tx2t+000M∘−1△Mqx˙1tx˙2t+ϕ1x,tϕ2x,t+0Iut.We now rewrite equation (32) as the following two subsystems:(33)x˙1t=x2t+ϕ1x,t,x˙2t=−M∘−1△Mqx2t−M∘−1△Mqx˙2t+ϕ2x,t+ut.Next, we define the following two backstepping variables as(34)z1t=x1t,z2t=x2t−rt,where rt∈R3×1 is the virtual input vector to be designed, which is constructed as(35)rt=−k1z1t+ϕ∧x,t,where k1∈R3∗3 is a positive definite diagonal matrix to be designed and ϕ∧x,t∈R3∗1 is the estimation of the nonlinearity ϕx,t. With all the information in hand, we present the following finite-time integral sliding mode disturbance observer (FTISMDO) for system (32) as(36)s0=z1t−pt,p⋅=z2t+rt+ϕ∧x,t,s1jt=s0j⋅t+∫0tl1jsiga1js0j+l2jsiga2js0j⋅dt,ϕ^˙jx,t=l1jsiga1js0j+l2jsiga2js0j⋅+b1jsigc1js1j+b2js1+L1jsgns1j,where s0=s01s02s03s04T, for j=1,2,3,4; L1j≥supt≥0ϕ˙jx,t denotes the norm bound of the unknown derivative of ϕjx,t; and l1j,l2j,a1j,a2j,b1j,b2j,L1j are the positive parameters to be designed. In particular, these parameters should satisfy 0<a1j<1, a2j=a1j/1+a1j, siga1js0j=s0ja1jsgns0j, and siga2js˙0j=s˙0ja2jsgns˙0j where sgns0j and sgns˙0j are two sign functions.In observer (36), get(37)s˙0=z˙1t−pt⋅=x2t+ϕ1x,t−z2t+rt+ϕ∧x,t=ϕ1x,t−ϕ∧x,t,(38)s¨0=ϕ˙1x,t−ϕ˙^x,t.Combining equations (36)–(38) yields(39)s˙1jt=s¨0jt+l1jsiga1js0j+l2jsiga2js˙0j=ϕ^1jx,t−b1jsigc1js1j−b2js1−L1jsgns1j.Design the Lyapunov functionV1j=1/2s1j2:(40)V˙1j=s1js˙1j=s1jϕ˙1jx,t−b1jsigcjs1j−b2js1j−L1jsgns1jℤ≤−b1jsigcj+1s1j−b2js1j2.So,V˙1j+b1jsigcj+1s1j+b2js1j2≤0.(41)s0j⋅t+∫0tl1jsiga1js0j+l2jsiga2js0j⋅dt=0.Because of (41), s0jt, s0j⋅t, s¨0jt can converge in finite time.So,(42)s¨0jt=−l1jsiga1js0j−l2jsiga2js˙0j.Based on equation (41) s0jt, s0j⋅t,s¨0jt can converge in finite time T0j and so does s0 in finite time Ts0=maxjT0j. It should be noted that Hurwitz condition should be met for s¨0jt+l1jsiga1js0j+l2jsiga2js˙0j=0, and also a2j∈0,1, a1j=a2j/2−a2j or a2j∈0,1, a1j=a1j/1+a1j. Therefore, after finite time Ts0+Ts1,s˙0=0=ϕ1x,t−ϕ∧x,t, namely, the estimate error is able to converge in finite time Ts0+Ts1.In observer (36), note that the information of s0j⋅ is required, which cannot be measured and obtained directly due to physical constraints. To this end, a high-order sliding mode differentiator (HOSMD) is employed here to estimate s0j⋅. The HOSMD is presented as follows:(43)y0⋅=h0=−αky0−ftk/k+1sgny0−ft+y1,y1⋅=h1=−αk−1y0−h0k−1/ksgny0−h0+y2,yk−1⋅=hk=−α1yk−1−hk−21/2sgnyk−1−hk−2+yk,yk⋅=−α0y1−h0,where α0,α1,…,αk>0 are positive constants to be selected. According to [23], the following conclusion holds after a finite-time transient process:(44)y0=ft,yr=hr−1=fr−1t,r=1,2,…,k.In HOSMDs (37) and (43), ft denotes s0it, and s˙0it can be calculated by y1t.Based on the FTISMDO (32) and HOSMDs (37)–(43), we now reconsider the backstepping variables z1t and z2t. For z1t, based on the virtual input vector rt defined in equation (35), it is derived that(45)z1⋅⋅t=x˙1t=x2t+ϕx,t=z2t+rt+ϕx,t=z2t−k1z1t+ϕ∧x,t+ϕx,t=z2t−k1z1t−ϕ∼x,t,where(46)ϕ∼x,t=ϕ∧x,t−ϕx,t.On the other hand, forz2t, it is derived that(47)z˙2t=x˙2t−r˙t=−M∘−1△Mqx2t−M∘−1△Mqx˙2t+ϕ2x,t+ut−k1z˙1t+ϕ∧⋅x,t=−M∘−1△Mqx2t−M∘−1△Mqx˙2t+ϕ2x,t+ut−k1x2t+k1ϕx,t+ϕ∧⋅x,t=−M∘−1△Mqz2t+M∘−1△Mqk1z1t+M∘−1△Mqϕ∧1x,t−M∘−1△Mqx˙2t+ϕ2x,t+ut−k1z2t+rt+k1ϕx,t=ut−ϕ∧⋅x,t+ϕ2x,t−k1z2t+rt+k1ϕx,t+dx,t,where the lumped disturbance vector dx,t is defined as(48)dx,t=M∘−1△Mqk1z1t+M∘−1△Mqϕ∧1x,t−M∘−1△Mqx˙2t.As a result, we obtain the following system which is equivalent to equation (33):(49)z˙1t=z2t−k1z1t−ϕ∼x,t,z˙2t=ut−ϕ∧⋅x,t+ϕ2x,t−k1z2t+rt+k1ϕx,t+dx,t.Theorem 1.
Considering system (49), suppose that the model uncertainty △Mq satisfies △Mq≤β1, where β1>0 is a known constant; under the following adaptive control law:(50)ut=ϕ∧⋅x,t−I3−k1ϕ∧1x,t+k1z2t+rt−z1t−β1M∘−1k1z1t+β1M∘−1ϕ∧1x,t+β1M∘−1x˙2tsgnz2t,the overall closed-loop system (49) is asymptotically stable.Proof.
Considering system (49), we define the Lyapunov function for the first subsystem of (49) as follows:(51)V1t=12z1Ttz1t.
Calculating the time derivative yields(52)V˙1t=12z1Ttz2t−k1z1t−ϕ∼x,t=−k1z1t+12z1Ttz2t−12z1Ttϕ∼x,t.
Next, we design the Lyapunov function for the second subsystem of (49) as(53)V2t=12z2Ttz˙2t+V1t,and then we have(54)V˙2t=z2Ttz˙2t+V˙1t=z2Ttut−ϕ∧⋅x,t+ϕ2x,t−k1z2t+rt+k1ϕx,t+dx,t−k1z1t+12z1Ttz2t−12z1Ttϕ∼x,t.
Considering the lumped disturbancedx,t in equation (50), notice that the following inequality holds:(55)dx,t≤β1M∘−1k1z1t+β1M∘−1ϕ∧1x,t+β1M∘−1x˙2t.
If we substitute the control law (50) into equation (54), it is shown that(56)V˙2t≤z2Ttut−ϕ∧⋅x,t+ϕ2x,t−k1z2t+rt+k1ϕx,t+β1M∘−1k1z1t+β1M∘−1ϕ∧1x,t+β1M∘−1x˙2t−k1z1t+12z1Ttz2t−12z1Ttϕ∼x,t,≤z2Ttϕ∧⋅x,t−I3−k1ϕ∧1x,t+k1z2t+rt−z1t−β1M∘−1k1z1t+β1M∘−1ϕ∧1x,t+β1M∘−1x˙2tsgnz2t+z2Tt−ϕ∧⋅x,t+ϕ2x,t−k1z2t+rt+k1ϕx,t+β1M∘−1k1z1t+β1M∘−1ϕ∧1x,t+β1M∘−1x˙2t−k1z1t+12z1Ttz2t−12z1Ttϕ∼x,t.
By some calculation, we have that(57)V˙2t≤−ϕ∼x,t−k1z1t−z1Ttϕ∼x,t≤−k1z1t<0holds for ∀zt∈R6,zt≠0, which means that the overall closed-loop system (49) is asymptotically stable. Thus, we complete the proof.


## 3. Simulation Experiment and Result Analysis

### 3.1. Robust Optimal Controller Simulation Results

The redundant manipulator has seven degrees of freedom, and the floating base has six degrees of freedom. The structure of the space robot is established by the DH method, as shown in Figure1, the dynamic parameters are shown in Table 1, and the total detumbling chaser is illustrated in Figure 2.Figure 1
Structure of space robot.Table 1
Kinetic parameters.
Parameter nameMatrixJoint1Joint2Joint3Joint4Joint5Joint6Joint7Lengtha (m)—0011000Angleα (°)—90900090900Joint distanced (m)—00.1000.30.10Joint angleθ (°)—θ1θ2θ3θ4θ5θ6θ7Parameter nameMatrixLink1Link2Link3Link4Link5Link6Link7m (kg)28112.82.825.3925.392.82.810.6Ix (kg·m2)4730.0050.0052.762.760.0050.0051.33Iy (kg·m2)4730.0050.0052.762.760.0050.0051Iz (kg·m2)4670.0050.0050.030.030.0050.0051.5Figure 2
Space debris detumbling chaser.Because optimal control, for a system ofx˙=A+Bu to design a suitable state feedback control rate u=−Kxt , make the performance index(58)J=∫0∞xTtQtxt+uTtRtutdt,where Qt is n×n of real symmetric semipositive definite weighted matrices and Rt is m×m of a real symmetric positive definite matrix.(59)Q1=eye13,R1=30eye13,gamma=1.9.From equation (25), we obtain the corresponding parameters as(60)B=eye13zeros13;A1=sqrtinvinvR−1gama^2eye13,T11=A1′Q1,T12=A1′∗R1,T0=T11T12;zeros13eye13;u=−pinvRBT0De.In this simulation, the end of the manipulator moves from pointArA=5.8300000T to point BrB=5−21000T. At the moment of t = 1 s, the interference force of 200 N is applied to the terminal flexible reducer brush, with the action time being 0.01 s, which can be regarded as an impulse force during the racemization process.The initial pose and final pose of the space racemization robot are known, and the trajectory planning is carried out within a specified time. After planning, the position changes of the end points are shown in Figure3. The desired trajectory of end-line velocity can be obtained by differentiation, as shown in Figure 4. Among them, the terminal attitude does not change, so it is unnecessary to repeat the terminal attitude planning.Figure 3
Terminal location planning (m).Figure 4
Terminal line velocity planning (m/s).The end planning trajectory in the above Cartesian space is transformed into the joint space. The joint angle, joint angular velocity, and joint angular acceleration after planning are shown in Figures5–7 , respectively. The first 10 s is the planned motion mode, and the last 5 s is the reserved stabilization time of the manipulator.Figure 5
Planning joint angles (rad).Figure 6
Planning joint angular velocity (rad/s).Figure 7
Plan for joint angular acceleration (rad/s−2).For the planned trajectory of the space robot, theH∞ robust optimal controller proposed in this paper is adopted, and the switching frequency of joint control torque is high. In order to simulate the actual working effect, a low-pass filter is incorporated. The joint torque is shown in Figure 8, while the joint angle error is shown in Figure 9. The upper and lower bounds of the control torque are small, and with the gradual convergence of the tracking error, the system is asymptotically stable, and the control torque gradually decreases and ultimately converges.Figure 8
Joint control torque (Nm).Figure 9
Joint angle tracking error (rad).H∞ robust optimal controller designed in this paper can deal with disturbances with stronger robustness. As for the joint angle error, H∞ robust optimal controller can maintain stability and suppress the tremor within only 0.5 s. If disturbance occurs, the H∞ robust optimal controller can generate larger control torques to suppress vibration, which is conducive to the racemization process. Moreover, the H∞ robust optimal controller has smaller tracking error and higher tracking accuracy and can stabilize the system within planned time, and thus its advantage of accuracy and effectiveness is clearly verified.


### 3.2. Backstepping Control Simulation Results

In this section, a simulation of the stabilization problem of a space multijoint robotic manipulator is implemented to illustrate the effectiveness of the proposed backstepping control law. Considering the double-joint robotic manipulator, the mechanical model of the manipulator is shown in Figure10. For manipulator model (1) with angular state variables q=q1q2T and angular velocity variables q˙=q˙1q˙2T, the initial attitude orientation and attitude angular velocity of the manipulator are set as(61)q0=π3−π6Trad,q˙0=00Trad/s.Figure 10
Mechanical model of the double-joint robotic manipulator.In addition, the inertia matrix, Coriolis force, and centrifugal force matrix are set as follows:(62)M0q=M1M2M2M3,C0q,q˙=0C1C20,where(63)M1=J1+m1r12+m2l12,M2=m1r1l1cosq2−q1,M3=J2+m2r22,C1=−m2r2l1q˙2sinq2−q1,C2=−m2r2l1q˙1sinq2−q1,and the parameters above are selected as(64)J1=0.1169kg⋅m2,J2=0.0042kg⋅m2,m1=6.1643kg,m2=1.3212kg,l1=0.27m,l2=0.23m,r1=0.1235m,r2=0.1133m.The uncertainties of the system are described as(65)ΔM=ΔM1ΔM2ΔM2ΔM3,ΔC=0ΔC1ΔC20,with(66)ΔM1=0.01+0.004ε+0.014ε2,ΔM2=0.018εcosq2−q1,ΔM3=0.008+0.007ε+0.015ε2,C1=−0.015εq˙2sinq2−q1,C2=−0.015εq˙1sinq2−q1,where ε∈0,1 describes the size of the uncertainty and it is designed as ε=0.5. The measurement errors of state variables are given as(67)Δx1=10−510sin5+0.5t8sin3+0.7t,Δx2=10−53sin5+0.5t2sin3+0.7t.The attitude angular and angular velocity trajectories are depicted in Figures11 and 12 , respectively, in which ideal performances are achieved.Figure 11
The trajectories of attitude angles (rad).Figure 12
The trajectories of attitude angular velocities (rad/s).Thesign⋅ function in control law (41) may lead to undesirable chattering when the sliding manifold crosses the sliding mode surface St=0. To avoid this phenomenon, the sign⋅ function can be replaced by(69)νt=StnormSt+ε1,where ε1>0 refers to the size of the bounder layer, which offers a continuous approximation to the sliding mode controller inside the boundary layer. The time response of the controller is presented in Figure 13, of which the upper bound is limited to 2 Nm.Figure 13
The control torque (Nm).For the designed finite-time integral sliding mode disturbance observer, the initial estimated value is set asϕ^10=00T, and the gain parameters in observer (36) are set as follows:(70)l1=0.018008,l2=0.017007,a1=0.014004,a2=0.014004;c1=0.014004,b1=0.017007,b2=0.018008,L1=0.018008.The estimation performance is given in Figure14, in which ideal estimation is achieved in finite time. Thus, the improved control performance of the backstepping controller with finite-time observer is demonstrated fully.Figure 14
The estimated errors of FTISMDO (rad/s).(68)k1=0.5000.5.The uncertainties of the system are described as follows: at 0 s, 10 s, and 30 s, moments of 15 Nm lasting for 0.02 s are entered in each dimension as a pulse. For the first dimension, a moment of 15 Nm acting for 0.03 s, with an action period of 10 s, is entered as a pulse. For the second dimension, a moment of 10 Nm acting for 0.02 s, with a period of 10 s is entered as a pulse. A set of random numbers between−4∗10−6,4∗10−6 is generated as a random disturbance.The attitude and angular velocity trajectories are depicted in Figures15 and 16 , respectively, in which ideal performances are achieved under the condition that a pulse acts on the target at 0 s, 10 s, and 30 s.Figure 15
The trajectories of attitude angles (rad).Figure 16
The trajectories of attitude angular velocities (rad/s).The time response of the controller is presented in Figure17, of which the upper bound is limited to 2 Nm.Figure 17
The control torque (Nm).The estimated performance is given in Figure18, where the ideal estimate is achieved in finite time. Thus, given multiple external disturbances, periodic disturbances, and random disturbances, the improved control performance of an inverse stepper controller with a finite-time observer can fulfil the requirements.Figure 18
The estimated errors of FTISMDO (rad/s).


## 3.1. Robust Optimal Controller Simulation Results

The redundant manipulator has seven degrees of freedom, and the floating base has six degrees of freedom. The structure of the space robot is established by the DH method, as shown in Figure1, the dynamic parameters are shown in Table 1, and the total detumbling chaser is illustrated in Figure 2.Figure 1
Structure of space robot.Table 1
Kinetic parameters.
Parameter nameMatrixJoint1Joint2Joint3Joint4Joint5Joint6Joint7Lengtha (m)—0011000Angleα (°)—90900090900Joint distanced (m)—00.1000.30.10Joint angleθ (°)—θ1θ2θ3θ4θ5θ6θ7Parameter nameMatrixLink1Link2Link3Link4Link5Link6Link7m (kg)28112.82.825.3925.392.82.810.6Ix (kg·m2)4730.0050.0052.762.760.0050.0051.33Iy (kg·m2)4730.0050.0052.762.760.0050.0051Iz (kg·m2)4670.0050.0050.030.030.0050.0051.5Figure 2
Space debris detumbling chaser.Because optimal control, for a system ofx˙=A+Bu to design a suitable state feedback control rate u=−Kxt , make the performance index(58)J=∫0∞xTtQtxt+uTtRtutdt,where Qt is n×n of real symmetric semipositive definite weighted matrices and Rt is m×m of a real symmetric positive definite matrix.(59)Q1=eye13,R1=30eye13,gamma=1.9.From equation (25), we obtain the corresponding parameters as(60)B=eye13zeros13;A1=sqrtinvinvR−1gama^2eye13,T11=A1′Q1,T12=A1′∗R1,T0=T11T12;zeros13eye13;u=−pinvRBT0De.In this simulation, the end of the manipulator moves from pointArA=5.8300000T to point BrB=5−21000T. At the moment of t = 1 s, the interference force of 200 N is applied to the terminal flexible reducer brush, with the action time being 0.01 s, which can be regarded as an impulse force during the racemization process.The initial pose and final pose of the space racemization robot are known, and the trajectory planning is carried out within a specified time. After planning, the position changes of the end points are shown in Figure3. The desired trajectory of end-line velocity can be obtained by differentiation, as shown in Figure 4. Among them, the terminal attitude does not change, so it is unnecessary to repeat the terminal attitude planning.Figure 3
Terminal location planning (m).Figure 4
Terminal line velocity planning (m/s).The end planning trajectory in the above Cartesian space is transformed into the joint space. The joint angle, joint angular velocity, and joint angular acceleration after planning are shown in Figures5–7 , respectively. The first 10 s is the planned motion mode, and the last 5 s is the reserved stabilization time of the manipulator.Figure 5
Planning joint angles (rad).Figure 6
Planning joint angular velocity (rad/s).Figure 7
Plan for joint angular acceleration (rad/s−2).For the planned trajectory of the space robot, theH∞ robust optimal controller proposed in this paper is adopted, and the switching frequency of joint control torque is high. In order to simulate the actual working effect, a low-pass filter is incorporated. The joint torque is shown in Figure 8, while the joint angle error is shown in Figure 9. The upper and lower bounds of the control torque are small, and with the gradual convergence of the tracking error, the system is asymptotically stable, and the control torque gradually decreases and ultimately converges.Figure 8
Joint control torque (Nm).Figure 9
Joint angle tracking error (rad).H∞ robust optimal controller designed in this paper can deal with disturbances with stronger robustness. As for the joint angle error, H∞ robust optimal controller can maintain stability and suppress the tremor within only 0.5 s. If disturbance occurs, the H∞ robust optimal controller can generate larger control torques to suppress vibration, which is conducive to the racemization process. Moreover, the H∞ robust optimal controller has smaller tracking error and higher tracking accuracy and can stabilize the system within planned time, and thus its advantage of accuracy and effectiveness is clearly verified.


## 3.2. Backstepping Control Simulation Results

In this section, a simulation of the stabilization problem of a space multijoint robotic manipulator is implemented to illustrate the effectiveness of the proposed backstepping control law. Considering the double-joint robotic manipulator, the mechanical model of the manipulator is shown in Figure10. For manipulator model (1) with angular state variables q=q1q2T and angular velocity variables q˙=q˙1q˙2T, the initial attitude orientation and attitude angular velocity of the manipulator are set as(61)q0=π3−π6Trad,q˙0=00Trad/s.Figure 10
Mechanical model of the double-joint robotic manipulator.In addition, the inertia matrix, Coriolis force, and centrifugal force matrix are set as follows:(62)M0q=M1M2M2M3,C0q,q˙=0C1C20,where(63)M1=J1+m1r12+m2l12,M2=m1r1l1cosq2−q1,M3=J2+m2r22,C1=−m2r2l1q˙2sinq2−q1,C2=−m2r2l1q˙1sinq2−q1,and the parameters above are selected as(64)J1=0.1169kg⋅m2,J2=0.0042kg⋅m2,m1=6.1643kg,m2=1.3212kg,l1=0.27m,l2=0.23m,r1=0.1235m,r2=0.1133m.The uncertainties of the system are described as(65)ΔM=ΔM1ΔM2ΔM2ΔM3,ΔC=0ΔC1ΔC20,with(66)ΔM1=0.01+0.004ε+0.014ε2,ΔM2=0.018εcosq2−q1,ΔM3=0.008+0.007ε+0.015ε2,C1=−0.015εq˙2sinq2−q1,C2=−0.015εq˙1sinq2−q1,where ε∈0,1 describes the size of the uncertainty and it is designed as ε=0.5. The measurement errors of state variables are given as(67)Δx1=10−510sin5+0.5t8sin3+0.7t,Δx2=10−53sin5+0.5t2sin3+0.7t.The attitude angular and angular velocity trajectories are depicted in Figures11 and 12 , respectively, in which ideal performances are achieved.Figure 11
The trajectories of attitude angles (rad).Figure 12
The trajectories of attitude angular velocities (rad/s).Thesign⋅ function in control law (41) may lead to undesirable chattering when the sliding manifold crosses the sliding mode surface St=0. To avoid this phenomenon, the sign⋅ function can be replaced by(69)νt=StnormSt+ε1,where ε1>0 refers to the size of the bounder layer, which offers a continuous approximation to the sliding mode controller inside the boundary layer. The time response of the controller is presented in Figure 13, of which the upper bound is limited to 2 Nm.Figure 13
The control torque (Nm).For the designed finite-time integral sliding mode disturbance observer, the initial estimated value is set asϕ^10=00T, and the gain parameters in observer (36) are set as follows:(70)l1=0.018008,l2=0.017007,a1=0.014004,a2=0.014004;c1=0.014004,b1=0.017007,b2=0.018008,L1=0.018008.The estimation performance is given in Figure14, in which ideal estimation is achieved in finite time. Thus, the improved control performance of the backstepping controller with finite-time observer is demonstrated fully.Figure 14
The estimated errors of FTISMDO (rad/s).(68)k1=0.5000.5.The uncertainties of the system are described as follows: at 0 s, 10 s, and 30 s, moments of 15 Nm lasting for 0.02 s are entered in each dimension as a pulse. For the first dimension, a moment of 15 Nm acting for 0.03 s, with an action period of 10 s, is entered as a pulse. For the second dimension, a moment of 10 Nm acting for 0.02 s, with a period of 10 s is entered as a pulse. A set of random numbers between−4∗10−6,4∗10−6 is generated as a random disturbance.The attitude and angular velocity trajectories are depicted in Figures15 and 16 , respectively, in which ideal performances are achieved under the condition that a pulse acts on the target at 0 s, 10 s, and 30 s.Figure 15
The trajectories of attitude angles (rad).Figure 16
The trajectories of attitude angular velocities (rad/s).The time response of the controller is presented in Figure17, of which the upper bound is limited to 2 Nm.Figure 17
The control torque (Nm).The estimated performance is given in Figure18, where the ideal estimate is achieved in finite time. Thus, given multiple external disturbances, periodic disturbances, and random disturbances, the improved control performance of an inverse stepper controller with a finite-time observer can fulfil the requirements.Figure 18
The estimated errors of FTISMDO (rad/s).


## 4. Conclusion

In this paper, the dynamic model of a space debris detumbling system is established. For the manipulator with uncertainty and external disturbance, we first design aH∞ robust optimal controller based on a linear quadratic performance index. The essence of H∞ control is to minimize the H∞ norm of the error transfer function when the interference is bounded and the maximum interference is considered. At the same time, the optimal control makes the system have robust optimal performance under a specified performance index. It can be proved that the robust state feedback controller can effectively compensate the uncertainties and bounded external disturbances, which means the manipulator can accurately track the desired trajectory, and the quadratic performance index reaches the optimal. Then, assuming that the disturbance is unmatched, a backstepping controller with a finite-time integral sliding mode disturbance observer is designed to further reduce the conservativeness existing in first robust controller and improve the control accuracy. The stability analysis shows that the finite-time integral sliding mode disturbance observer can efficiently compensate the unmatched lumped uncertainty in finite time..As this paper is a finite-time backstepping control of an observer, further research on finite-time backstepping control of an controlleron redundant robotic arms will be done in the future.



---

*Source: 9936224-2021-07-15.xml*