An Exponentially Convergent Adaptive Robust Based Space Manipu- lator Noise Control Method Jinglong Liu 1,a,b , Ning Li 2,a,b , Xueping Hu 3,a,b , Xiaolong Ma 4,a,b a. Shanghai Institute of Aerospace System Engineering, No. 3888, Yuanjiang Road, Minhang Dis- trict, Shanghai, China b. Shanghai Key Laboratory of Spacecraft Mechanism, No. 3888, Yuanjiang Road, Minhang Dis- trict, Shanghai, China

ABSTRACT Because most of the traditional nonlinear control methods are model-based control methods, it will dramatically increase the parameter identification burden of multi-rigid-body system dynamic model. Meanwhile, Unmodeled dynamics, parameter drift, model uncertainty, complex motor hyste- resis characteristics, etc are unfortunately in the practical systems, which makes it very hard to ac- curately build the noise model and identify the model parameters. Among them, the accurate values of the viscous friction coefficient, elastic friction coefficient, the distance between the centroid and the rotation center of the connecting rod, the inertial moment and products of each joint, these ex- act values can hardly be obtained. In this paper, direct adaptive control and the robust model ref- erence adaptive control based on Lyapunov direct method are adopted, which can make the noise error close to zero under the condition of parameter uncertainty and unknown nonlinear friction characteristics.

1. INTRODUCTION

Due to the long arm and large joint flexibility of the space manipulator, there is a large residual noise in the space robot. Due to the existence of residual noise, the manipulator needs a long time to wait for the elimination of residual noise when performing space operation tasks. According to NASA's statistics, about 25% of the space manipulator's operation task time is spent waiting for the natural elimination of the manipulator's residual noise, which actually depends on the residual noise consumed by the natural damping characteristics of the joints and motor ends. This greatly reduces the working efficiency of the space manipulator and wastes the valuable on orbit working time of astronauts. Therefore, reasonable and effective noise suppression measures are inevitable problems in the task of space manipulator.

1 ljlong@mail.nwpu.edu.cn

2 lining_20080513@126.com

3 hher805@163.com

4 mxl1905@163.com

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Noise suppression can be divided into passive suppression [1], semi-active suppression [2] and ac- tive suppression [3]. Among them, passive suppression is a passive control method, which can sup- press the residual noise by increasing the physical inertia and damping of the system. This method is simple and reliable, and the suppression effect of high-frequency noise is obvious, but the flexi- bility is poor, and the suppression effect of low-frequency noise is not ideal. Increasing physical damping is the most commonly used method in passive residual noise, including using viscoelastic materials, setting friction devices, elastic devices, fluid dampers [4], controllable dampers [5] and so on. Active noise suppression is an active control. According to different control algorithms, it can also be divided into model-based control [6], non-model-based control and noise suppression based on modern control theory. PID control is a traditional and widely recognized control method. It has the advantages of simple structure, easy to use, high reliability and good robustness. For the manipulator, both position con- trol and force control can increase the damping of the closed-loop transfer function of the system and attenuate the system noise well. However, the PID controller requires that the system model parameters cannot change greatly. The torque sensor can directly sense the noise torque and load torque. The inverse dynamic calculation of the joint end can be avoided through torque feedback, which saves the hardware consumption of the computer. The PID control based on noise torque feedback can well realize the noise suppression [7] of the system, and is not sensitive to the change of load parameters. However, the measurement noise of the torque sensor is large, the thermal sta- bility is not ideal, and the interface circuit will produce a certain deviation. The basic controller design method has some limitations. Firstly, the design of the controller de- pends heavily on the experience of the designer. Secondly, because the conventional PID controller is essentially a linear controller, which is generally used in single input and single output system, the joint controller of multi degree of freedom manipulator is an unstable system, and the control effect is insufficient. For the pole system with single input and single output, the classical controller design methods based on pole assignment, root locus or Bode diagram are generally adopted, and the controller is generally in the form of PD or PID. For the nonlinear SISO system, the feedforward method [8] is generally used to compensate the nonlinearity of the system, and then the system is treated as a lin- ear system. The controller design methods using this idea mainly include pole placement method [9] and root locus method [10][11]. Double closed loop or multi closed loop control is a control idea often used in practical control. The inner loop ensures the stability and robustness of the system, and the outer loop is used for expected control. Based on this idea, on the basis of traditional PID, many scholars change the integer order of differential link to fractional order [12] to correct the influence of load change on the phase mar- gin of closed-loop system, but fractional order differential controller is only suitable for linear steady-state system. In this paper, direct adaptive control and robust model reference adaptive control based on Lyapun- ov direct method are adopted, which can make the noise error close to zero under the condition of parameter uncertainty and unknown nonlinear friction characteristics.

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2. PROBLEM FORMULATION

2.1. General Manipulator Dynamics Equation The dynamic equation of a single joint general manipulator with uncertain dynamic parameters is described by the following equation

(1)

1 0 ( ) cos ( ) I d mgl u f           

 2 4 3 I ml  mg Where refers to the manipulator joint angle, refers to the moment of inertia, refers

l to the gravity, refers to the control input, refers to the unknown nonlinear friction, refers

u ( , ) f u 

d to the distance between the center of mass and the rotation center of the connecting rod, refers to

1  the viscous friction coefficient of connecting rod motion, refers to the uncertain value of viscous

0  friction coefficient, refers to the elastic friction coefficient.

2.2. Space Manipulator Dynamics Equation By ignoring the gravity term, the dynamic equation of a single joint space manipulator with uncer- tain dynamic parameters is described by the following equation

0 1 1 ( ( )) d u f I I I           

(2)

The eq(2) can be rewritten as

(3)

1 0 ( ( )) u f         

I I I          1 0 ,   Where , refer to the nonnegative bounded real number, refers to a

1 , , d

0 1 1 0

positive real number.

max ( ) f F  

Assumption 1: Boundedness hypothesis .

For eq(3), we can introduce the reference model as follow

(4)

1 0 m m m a a br      

m  r 1 0 , , a a b Where refers to the reference model output, refers to the command input, refer to the positive real number. We can define the error signal as

m e     (5)

2.3. Control Objectives According to the previous sections, the control objectives of this problem can reduce to

1 0 , ,  u When are unknown, we can design the control law , that is, there exists an arbitrary initial

t  state , such that for . 3. CONTROLLER DESIGN AND STABILITY ANALYSIS

0  ( ) 0, ( ) 0 e t e t  

According to the eq(3) and eq(4), we can derive the following equation

(6) That is

1 0 1 0 ( ( )) m m m a a br u f                

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(7)

1 0 1 1 0 0 ( ) ( ) ( ) e a e a e br u f a a                

If we define the state vector , the state space expression of eq(7) is

[ , ] T x e e 

0 0 u              x Ax Ax Z

(8)

Where

1 1 0 0 ( ) ( ) ( ) br f a a            

0 1 0 0 , u a a                    A Z

0 1

A P Since the eigenvalue of matrix has a negative real part, there are positive definite matrices Q and , so the following formula holds

T   A P PA Q (9) ˆ e If we define the auxiliary signal as follow

2 3 2 3 ˆ 0 1 0 1 p p e e p e p e p p e            Px

   

(10)

1 2

p p p p       P Where .

1 2

2 3

If the control law is designed as this

(11)

0 1 2 u k r k k v      

0 1 2 , , k k k v Where are the gain coefficients to be adjusted respectively, is the robust term. Theory 1: 0 1 2 , , k k k For eq(8), if we choose the control law eq(11), and the gain coefficients for the adaptive law and robust compensation term are chosen as follow

(12)

0 0 1 1 2 2 max ˆ ˆ ˆ ˆ , , , sgn k er k e k e v F e       

0 1 2 , ,  0 1 , , ( ) f  

Where is a sign function, are the positive real number. For any , and arbi-

sgn( )

trary initial condition, will converge asymptotically to 0 and bounded. Proof: Step1: Define the candidate Lyapunov function

( ), ( ) e t e t

1 1 1 1 ( ) ( ) ( ) ( ) 2 2 2 2

T V t b k a k a k                  x Px (13)

2 2 2 0 0 0 1 1 1 2 0 1 2

By the reason of

1 1 1 1 1 ( ) ( ) ( ) 2 2 2 2 2 1 1 ( ) ( ) 2 2 1 1 ( ) ( ) 2 2 1 ( ) 2

   

T T T T T

x Px x Px x Px Ax + Z Px x P Ax + Z

  

T T T T

x A + Z Px x P Ax Z

(14)

  

T T T T

x A P + PA x Z Px x PZ

  

T T

x Q x x PZ

0 0 0 ( ) 1 u u                 Z (15)

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                    

T p p e e u e u p p

0 ˆ , ( ) ( ) 1

 

1 2

 

x PZ

2 3

(16)

       

ˆ( ( ) ( ) ( ))

e br f a a k r k k v

1 1 0 0 0 1 2

     

         

ˆ ˆ ˆ ˆ ( ) ( ) ( ) ( )

er b k e a k e a k e f v

0 0 0 1 1 1 2

That is

1 ˆ ˆ ˆ ˆ ( ) ( ) ( ) ( ) 2

     

          

T

V er b k e a k e a k e f v

x Qx

0 0 0 1 1 1 2

k k k b k a k a k

       

       

0 1 2 0 0 0 1 1 1 2 0 1 2

( ) ( ) ( )

(17)

k k er b k e a k

1 ˆ ˆ ( )( ) ( )( ) 2

     

       

T

0 1 0 0 0 1 0 1

x Qx

k e a k e f v

    

     

ˆ ˆ ( )( ) ( )

2

1 1 2 2

If we bring in the eq(12), it will be

1 ˆ ( ( ) ) 2

(18)

T V e f v      x Qx

Because

max max ˆ ˆ ˆ ˆ ˆ ( ) ( sgn ) 0 e f v e f F e e f F e           (19) That is

T V x     Q x Qx

2 max ( ) 1 0 2 2

(20)

max ( )  Q Q

Where are the maximum eigenvalues of matrix , is refer to the Euclidian 2-norm.

0  x When , . According to the LaSalle invariance principle, the closed-loop system is asymp-

0 V 

totically stable, i.e. there exists such that and will converge asymptotically to 0 and bounded.

, 0 t x   ( ) e t ( ) e t

0, 0 V V  

V t  0 1 2 , , k k k On account of , that means is bounded when . Therefore, are bounded. Q.E.D. 4. SIMULATIONS

In this section, a typical simulation case will be expounded to present the availability of the pro- posed exponentially convergent adaptive robust noise control method. The dynamic equation of this control plant is

(21)

1 0 ( ) u f          

0 1 0.5, 0.7, 70       Where .

The friction model is expressed as . The reference model is described as

( ) 1.6 0.5sgn( ) f     

, i.e. reference model parameters are chosen as

30 50 150 , sin(0.05 ) m m m r r t        

1 0 30, 50, 150 a a b    . The initial states of the control plant and the reference model are taken as   0,0 .

40 0 0 40       Q If the weight matrix are defined as and solve the continuous-time algebraic Riccati

46 0.4 0.4 0.68       P equation eq(9), we will obtain . Then, the auxiliary signal can represent as

ˆ 0.4 0.68 e e e  

.

From the expression of and simulation test, we define the upper bound of robust compensation

( ) f 

max 3.0 F  term as in eq(12). The control law is eq(11), and the adaptive law is eq(12) with giving

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0 1 2 1.5, 1.5, 1.5       ˆ( ) sat e . The saturation function is used to replace the switching function ˆ sgn( ) e 0.3  , and the boundary thickness value is defined as . The simulation results are shown in Figure 1-3.

Angular Position Tracking

x d x

4

3

Angular Position / [deg]

2

1

0

-1

-2

0 50 100 150 -3

Figure 1: Angle Position Tracking.

Time / [s]

Angular Rate Tracking

0.5

xdot d xdot

Angular Rate / [deg/s]

0

0 50 100 150 -0.5

Figure 2: Angle Rate Tracking.

Time / [s]

Control Input

1.5

u

1

Control Input / [Nm]

0.5

0

-0.5

-1

0 50 100 150 -1.5

Figure 1: Angle Position Tracking. The simulation results show that the proposed exponentially convergent adaptive robust controller does not have to rely on the information of the controlled object. Meanwhile, it can adapt to the un- certainty of unknown friction characteristics and parameters, and ensure the high-precision tracking of objects and models. 5. Conclusion and future work

Time / [s]

This paper aims at the problems encountered in the complex noise control of space manipulator. The model perturbation and the fixed parameters of PID control will cause serious noise problem for tracking angle position command. The mathematical model of space manipulator considering viscous friction, elastic friction and unknown nonlinearity is established. By designing a robust ex- ponential convergence adaptive controller, the angular position tracking error and angular rate tracking error are asymptotically converge at zero simultaneously. Based on the scheme, the simula- tion results meet the anticipated effect. The future work is to extend this method to a method that does not rely on prior knowledge, such as the upper bound of friction. 6. ACKNOWLEDGEMENTS

We gratefully acknowledge the Shanghai Key Laboratory of Spacecraft Mechanism, National De- fense Science and Technology Key Laboratory of Space Structure and Mechanism Technology for their supports to this paper research. This research was also supported by the National Natural Sci- ence Foundation of China with Grant No. U21B6002. 7. REFERENCES

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