# Adaptive Tracking Control for a Class of Manipulator Systems with State Constraints and Stochastic Disturbances

Authors: Wei Sun; Wenxing Yuan; Jing Zhang; Qun Sun
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

An adaptive controller is constructed for a class of stochastic manipulator nonlinear systems in this paper. The states are constrained in the compact set. A tan-type Barrier Lyapunov Function (BLF) is employed to deal with state constraints. The proposed control scheme guarantees the output error convergence to a small neighbourhood of zero. All the signals in the closed-loop system are bounded. The simulation results illustrate the validity of the proposed method.


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## Body

## 1. Introduction

Adaptive control of strict-feedback nonlinear systems has received a lot of attention since the appearance of recursive backstepping design [1] and feedforward design [2, 3]; a great deal of work has been done for this class of systems in the past decades; for example, see [4–9]. Constraints are widespread in many real systems such as robotic manipulators and physical engineering systems. Therefore robotic manipulators have received increasing attention over the last few years. For this reason, many methods have been used to handle the issue of constraints, for example, [10–16]. In detail, in [10], the author studies the neural network adaptive control design for robotic systems with the velocity constraints and input saturation. An adaptive finite-time controller is considered in [11] for a class of strict-feedback nonlinear systems with parametric uncertainties and full state constraints. In [12], the problem of position control of manipulators operating in the task space under state constraints has been addressed. Literatures [15, 16] discussed the trajectory and tracking control problems of the mobile manipulator with constrained end-effector and adaptive controllers are proposed.The practical systems are inevitable to contain the stochastic disturbance and it can cause instability of system. So the stability of stochastic nonlinear systems has attracted great attention [17–19]. In [20], the author has proposed the state equations of the stochastic dynamics of an open-chain manipulator in a fluid environment. This paper has given an algorithm for the discretization of the state equations and explained how the interaction between a fluid and a manipulator can be taken into account in the control of manipulators. The work [21] studies the problem of output feedback stabilization for a class of stochastic feedforward nonlinear systems with state and input delays and the unknown output function. In [22], the stochastic response of a mobile robotic manipulator has been investigated. This paper has studied the sensitivities of the joints responses to base velocity, the surface roughness coefficients, manipulator configuration, and damping in detail.In this paper, an adaptive tracking controller will be designed for stochastic manipulator nonlinear systems with full state constraints. A backstepping technique with a tan-type Barrier Lyapunov Function (BLF) will be constructed to address the state constraints problem and all the states in stochastic nonlinear systems are not violated. The error signals have converged to an arbitrarily small neighbourhood of zero and all the signals in the closed-loop system are bounded.


## 2. System Description

### 2.1. Problem Statement and Preliminary Results

Consider the one-link manipulator which contains motor dynamics and stochastic disturbances. The model is described as [9](1)Dq¨+θq˙-Nsin⁡q=τr+τd,Mτ˙r+Hτr=u-kmq˙,whereq,q˙,q¨ are the link position, velocity, and acceleration, respectively; τr denotes the torque produced by the electrical system; τd is the known torque stochastic disturbance; u is the electromechanical torque control input; D is a known mechanical inertia; θ is unknown coefficient of viscous friction at the joint; N is a known constant related to the mass of the load and the coefficient of gravity; M is the known armature inductance; H is the known armature resistance; km is the known back electromotive force coefficient; m is the link mass; and l is the length of the link which is known. Then, following change of coordinates ξ1=q,ξ2=q˙,ξ3=τr, we can get(2)dξ1=ξ2dt,dξ2=1Dξ3-θξ2+10sin⁡ξ1dt+1Dsin⁡ξ1ξ1dω,dξ3=1Mu-kmξ2-Hξ3dt,y=ξ1.All the states are constrained in the compact set asΩξ≔ξit∈R,ξit≤kci, where i=1,2,3, kci are positive constants.Given a reference trajectoryqd, the control objective is to design an adaptive control algorithm such that q tracks the desired trajectory qd as much as possible; all the signals in closed-loop systems are bounded; the state constraint requirements are not violated. To facilitate control system design, the following assumption and lemma are proposed.Assumption 1.
The reference trajectoryydt and its derivatives up to the n-th ones are continuous and bounded. That is, for any constant kci>0, there exist positive constants Yi,i=0,…,n, such that ydt≤Y0<kc1, ydit≤Yi,i=1,…,n.Remark 2.
This assumption is reasonable. Assumption1 is the worst case one. The requirement on derivatives is widely fixed in backstepping control [23]. This is because the standard backstepping technique requires the reference signal to be continuous and derivable to design the desired controller. y0<kci in assumption is always true in practice for the requirement of output tracking control. A similar assumption is also considered in [24, 25].Next, consider the following stochastic nonlinear system:(3)dx=fxdt+gxdω,wherex∈Rn is the system state vector; fx∈Rn and gx∈Rn×r satisfy the locally Lipschitz functions and the linear growth condition and f0=0,g0=0; ω is an r-dimensional standard Wiener process.Definition 3 (see [2]).
For any Lyapunov functionVx,t∈C2,1, we define the differential operator L as follows: (4)LVx,t=∂V∂t+∂V∂xf+12TrgT∂2V∂x2g,whereTr· is the matrix trace.Lemma 4 (see [26]).
There exist aC2,1 function V, two constants γ>0,ρ>0, and K∞ class functions α1,α2 satisfying α1x≤Vx≤α2x and LVx≤-γVx+ρ; then, there is a unique strong solution which satisfies (5)EVx≤Vx0e-γt+ργ.


## 2.1. Problem Statement and Preliminary Results

Consider the one-link manipulator which contains motor dynamics and stochastic disturbances. The model is described as [9](1)Dq¨+θq˙-Nsin⁡q=τr+τd,Mτ˙r+Hτr=u-kmq˙,whereq,q˙,q¨ are the link position, velocity, and acceleration, respectively; τr denotes the torque produced by the electrical system; τd is the known torque stochastic disturbance; u is the electromechanical torque control input; D is a known mechanical inertia; θ is unknown coefficient of viscous friction at the joint; N is a known constant related to the mass of the load and the coefficient of gravity; M is the known armature inductance; H is the known armature resistance; km is the known back electromotive force coefficient; m is the link mass; and l is the length of the link which is known. Then, following change of coordinates ξ1=q,ξ2=q˙,ξ3=τr, we can get(2)dξ1=ξ2dt,dξ2=1Dξ3-θξ2+10sin⁡ξ1dt+1Dsin⁡ξ1ξ1dω,dξ3=1Mu-kmξ2-Hξ3dt,y=ξ1.All the states are constrained in the compact set asΩξ≔ξit∈R,ξit≤kci, where i=1,2,3, kci are positive constants.Given a reference trajectoryqd, the control objective is to design an adaptive control algorithm such that q tracks the desired trajectory qd as much as possible; all the signals in closed-loop systems are bounded; the state constraint requirements are not violated. To facilitate control system design, the following assumption and lemma are proposed.Assumption 1.
The reference trajectoryydt and its derivatives up to the n-th ones are continuous and bounded. That is, for any constant kci>0, there exist positive constants Yi,i=0,…,n, such that ydt≤Y0<kc1, ydit≤Yi,i=1,…,n.Remark 2.
This assumption is reasonable. Assumption1 is the worst case one. The requirement on derivatives is widely fixed in backstepping control [23]. This is because the standard backstepping technique requires the reference signal to be continuous and derivable to design the desired controller. y0<kci in assumption is always true in practice for the requirement of output tracking control. A similar assumption is also considered in [24, 25].Next, consider the following stochastic nonlinear system:(3)dx=fxdt+gxdω,wherex∈Rn is the system state vector; fx∈Rn and gx∈Rn×r satisfy the locally Lipschitz functions and the linear growth condition and f0=0,g0=0; ω is an r-dimensional standard Wiener process.Definition 3 (see [2]).
For any Lyapunov functionVx,t∈C2,1, we define the differential operator L as follows: (4)LVx,t=∂V∂t+∂V∂xf+12TrgT∂2V∂x2g,whereTr· is the matrix trace.Lemma 4 (see [26]).
There exist aC2,1 function V, two constants γ>0,ρ>0, and K∞ class functions α1,α2 satisfying α1x≤Vx≤α2x and LVx≤-γVx+ρ; then, there is a unique strong solution which satisfies (5)EVx≤Vx0e-γt+ργ.


## 3. Control Design and Stability Analysis

Step 1.
Define the tracking errore1=ξ1-yd,e2=ξ2-α1, where αi are the virtual controllers and αi<α¯i with α¯i are positive constants, and ei∈Ωe≔ei∈R,ei<kbi,i=1,…,n with kb1=kc1-Y0>0,kbi=kci-α¯i-1>0. We can get (6)de1=dξ1-dyd=e2+α1-y˙ddt.Consider a candidate BLF as follows: (7)V1=kb14πtan⁡πe144kb14.Based on Definition 3, one has(8)LV1=e13cos2πe14/4kb14e2+α1-y˙d.Design the virtual controller α1 as(9)α1=-K1sin⁡πe14/4kb14cos⁡πe14/4kb14e13+y˙d,where K1>0 is a design parameter. Substituting (9) into (8), we can obtain(10)LV1≤-K1tan⁡πe144kb14+e13cos2πe14/4kb14e2.Step 2.
Define the tracking errore3=ξ3-α2; the following can be obtained: (11)de2=dξ2-dα1=1De3+α2-θξ2+10sin⁡ξ1dt+1Dsin⁡ξ1ξ1dω-∂α1∂ξ1ξ2+∑i=12∂α1∂ydi-1ydidt.Choose a candidate BLF as follows: (12)V2=V1+kb24πtan⁡πe244kb24+12θ~2.Here θ~=θ^-θ, θ^ denotes the estimation of θ. Based on Definition 3, we can obtain(13)LV2=LV1+e23cos2πe24/4kb241De3+α2-θξ2+10sin⁡ξ1-∂α1∂ξ1ξ2-∑i=12∂α1∂ydi-1ydi+3kb24e24cos⁡πe24/4kb24+2πe26sin⁡πe24/4kb242kb24cos3πe24/4kb241Dsin⁡ξ1ξ12+θ~θ^˙≤LV1+θ~θ^˙+e23cos2πe24/4kb241Dα2-θ^ξ2+10sin⁡ξ1-∂α1∂ξ1ξ2-∑i=12∂α1∂ydi-1ydi+3kb24e2cos⁡πe24/4kb24+2πe23sin⁡πe23/4kb242kb24cos⁡πe24/4kb24ξ1Dsin⁡ξ12+e23e3Dcos2πe24/4kb24-θ~e23ξ2Dcos2πe24/4kb24.Design the virtual controller α2 as(14)α2=-K2Dsinπe24/4kb24cos⁡πe24/4kb24e23+θ^ξ2-10sin⁡ξ1+D∂α1∂ξ1ξ2+D∑i=12∂α1∂ydi-1ydi-3kb24e2cos⁡πe24/4kb24+2πe23sin⁡πe23/4kb242Dkb24cos⁡πe24/4kb24ξ12sin2ξ1,where K2>0 is a design parameter. An tuning function is chosen as τ1=e23ξ2/Dcos2πe24/4kb24. Substituting (14) into (13) yields(15)LV2≤-∑i=12Kitan⁡πei44kbi4+θ~θ^˙-τ1+e23e3Dcos2πe24/4kb24.Step 3.
Defining the tracking errore3=ξ3-α2, we have (16)de3=dξ3-dα2=1Mu-kmξ2-Hξ3dt-∂α2∂ξ1ξ2+∂α2∂ξ21Dξ3-θξ2+10sin⁡ξ1+∑i=13∂α2∂ydi-1ydi+∂α2∂θ^θ^˙+12∂2α2∂ξ221Dsinξ1ξ12dt.Construct a candidate BLF as follows: (17)V3=V2+kb34πtan⁡πe344kb34.Based on Definition 3, computing LV3, we can figure out(18)LV3=LV2+e33cos2πe34/4kb341Mu-kmξ2-Hξ3-∂α2∂ξ1ξ2+∂α2∂ξ21Dξ3-θξ2+10sin⁡ξ1+∑i=13∂α2∂ydi-1ydi+∂α2∂θ^θ^˙+12∂2α2∂e221Dsinξ1ξ12≤e23e3Dcos2πe24/4kb24+e33cos2πe34/4kb341Mu-ξ2-0.5ξ3-∂α2∂ξ1ξ2+∂α2∂ξ21Dξ3-θξ2+10sin⁡ξ1+∑i=13∂α2∂ydi-1ydi+∂α2∂θ^θ^˙+12∂2α2∂e221gmsinξ1ξ12+∑i=12Kitan⁡πei44kbi4+θ~θ^˙-τ1.Design the controller u as follows:(19)u=M-K3sin⁡πe34/4kb34cos⁡πe34/4kb34e33+∑i=13∂α2∂ydi-1ydi+∂α2∂θ^θ^˙+12∂2α2∂e221Dsin⁡ξ1ξ12-θ^T1D∂α2∂ξ2ξ2+∂α2∂ξ1ξ2+∂α2∂ξ21Dξ3-ϑe2g2e3ϑe3,where K3>0 is a design parameter. Choosing τ2=τ1+1/D∂α2/∂ξ2ξ2 and substituting (18) and (19) into (15) result in(20)LV3≤-∑i=13Kitan⁡πei44kbi4+θ~θ^˙-τ2.The adaptive law is given as θ^˙=τ2-σθ^ with a design parameter σ. Then, we can obtain(21)LV3≤-∑i=13Kitan⁡πei44kbi4-σθ~θ^≤-∑i=13Kitan⁡πei44kbi4+σθ22-σθ~22.Choose η1=minK1π/kb12,K2π/kb22,K3π/kb32,σ; then (21) can be rewritten as(22)LV3≤-η1∑i=13kbi2πtan⁡πei44kbi4+12θ~2+C1,where C1=σθ2/2; that is, (23)LV3≤-η1V3+C1.Theorem 5.
Consider system (1) under Assumption 1; the controller u is given in (19), and the adaption law θ^˙=τ2-σθ^. Then, the following are guaranteed:(1)
The full state constraints are not violated.(2)
All the signals in the closed-loop system are bounded.(3)
The error signalsei(t)will converge toΞ=ei:Eei4≤4kbi4/πtan-1V30+C1/η1π/kbi4.Proof.
Define a candidate BLF as follows:(24)V3=∑i=13kbi4πtan⁡πei44kbi4+12θ~2.From (23), we can get that (25)LV3≤-η1V3+C1.Based on Lemma 4, we know that (26)0≤EV3t≤V30e-η1t+C1η1.Then, the following inequality holds: (27)Ekbi4πtan⁡πei44kbi4≤EV3t≤V30e-η1t+C1η1.Hence, (28)Eei4≤4kbi4πtan-1V30e-η1t+C1η1πkbi4≤4kbi4πtan-1V30+C1η1πkbi4.
Hence, the size of4kbi4/πtan-1V30+C1/η1π/kbi4 can be made small enough by choosing appropriate parameters. On the other hand, from the above inequality, we can obtain that EV3t is limited by C/η1. Then, we know that V3t is bounded. So θ~ is bounded and θ^=θ~+θ is bounded. Then we know that ei, yd, αi, and u are bounded. From e1=ξ1-yd and yd≤Y0, we know that ξ1≤|e1|+yd<kb1+Y0=kc1. From e2=ξ2-α1 and α1≤α¯1, we can obtain that ξ2≤e2+α1<kb2+α¯1=kc2. In a similar way, we can obtain ξ3<kc3. Thus, the full state constraints are not violated.


## 4. Simulation

In this section, simulation is introduced to demonstrate the effectiveness of the proposed scheme.(29)dξ1=ξ2dt,dξ2=1Dξ3-θξ2+10sin⁡ξ1dt+1Dsin⁡ξ1ξ1dω,dξ3=1Mu-Kmξ2-Hξ3dt,y=ξ1,whereD=1, M=1, N=10, B=1, H=0.5, and km=1. All the states are constrained in the compact set as Ωξ≔ξit∈R,ξit≤1.5,. The reference signal is chosen as yd=0.5sin⁡t.In the simulation, the design parameters are chosen asξi0=0.1, Ki=1, i=1,2,3, θ^0=0.1, kb1=kb2=1, kb3=0.9, and kb4=1.1. The results of the simulation are shown in Figures 1–4. The output tracking q and qd is illustrated in Figure 1. It can be seen that the position state q can primely track the desired trajectory qd. It is shown in Figure 2 that all the states are strictly constrained in ξi∣-1.5≤ξit≤1.5,i=1,2,3. The parameter updating law θ^ and input u are all bounded as shown in Figures 3 and 4. The simulation results demonstrate the effectiveness of the proposed adaptive control scheme.Figure 1
The trajectories ofq and qd.Figure 2
The trajectories ofξ1, ξ2, and ξ3.Figure 3
The estimation ofθ^.Figure 4
The trajectory of control inputu.


## 5. Conclusions

This study carries out the adaptive tracking control for a class of stochastic manipulator nonlinear systems. An adaptive controller is proposed to ensure that the mean square of the tracking error can be made arbitrarily small. Simulation results are presented to illustrate the effectiveness of the proposed control strategy.



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*Source: 5080684-2018-06-20.xml*