Method of high precision pointing control for spacecraft system based on the active-passive integrated orthogonal micro-vibration

isolation platform

Zhizhou Chen 1,2 , Feng Li 1,2 , Xueping Hu 1,2 , Xiaolong Ma* 1,2 , Longyu Ma 1,2 , Junwei Shi 1,2 1 Shanghai Institute of Aerospace System Engineering, Shanghai, P. R. China 2 Shanghai Key Laboratory of spacecraft Mechanism, Shanghai, P. R. China

ABSTRACT To acquire the high precision pointing control for spacecraft system under the disturbance of micro-vibration, and resolve conflicts between the vibration suppression and pointing control, a simultaneous-distributed control method was proposed in this paper. Based on this method and the Stewart platform, an active-passive integrated orthogonal micro-vibration isolation platform with multiple dimensions was developed. Six line-actuators according to the configuration of Gough- Stewart parallel mechanism were selected to build the platform. Specially, the displacement of the line-actuator was converted from the end deflection of two groups of active and passive integrated cantilever beams, which contained viscoelastic damping plate, aluminum material and macro fiber composite (MFC). It should be noted that the dynamic model of the platform system was strongly coupled and it was decoupled by employing the orthogonal configuration. By employing the simultaneous-distributed control strategy, the integrated control algorithm of vibration isolation and pointing was deeply studied. Furthermore, the dynamic behavior and control effects for the 6 degree of freedom (6-DOF) Stewart platform with different control strategies were compared and analyzed. Results showed the method had a good vibration isolation effect and reliable pointing accuracy, which could be applied for the spacecraft requiring high precision pointing.

Keywords : Spacecraft, micro-vibration, precision pointing, Stewart platform

1. INTRODUCTION

With the continuous development of space applications, observation satellites, remote sensing and communication spacecraft are increasingly demanding observation resolution and pointing accuracy [1-3]. On the other hand, small vibrations caused by control Moment Gyro (CMG), thermal pumps and other equipment from satellite platforms can easily reduce the pointing accuracy and stability of precision loads (such as space telescopes). In order to reduce the effect of the micro- vibration for the satellite platform to the precision load, a vibration isolation system can be introduced [4,5]. For precise loads with pointing control, it is necessary to have high precision pointing control capability with micro-vibration isolation. However, the balance of precise loads and pointing control in the vibration isolation system is still an essential problem to be solved in the design of spacecraft. As the typical characteristics of high precision pointing and high stability requirements of spacecraft system, the vibration isolation system should have the functions of low frequency

*Corresponding author: mxl1905@163.com

following, medium frequency suppression and high frequency isolation [6]. Therefore, a vibration isolation system must be designed between the precision load and the disturbance source. In fact, the stiffness of the vibration isolator is used to ensure the following of low frequency and the attenuation of high frequency, and the damping is used to suppress the resonance peak near the resonance frequency [7].

In this paper, an active and passive integrated vibration isolation scheme is designed based on the 6-bar Gough-Stewart platform. In Section 1, a dynamic model based on Stewart platform is introduced and the corresponding dynamic equation of load under the action of Stewart platform is developed. Next, based on the dynamic decoupling of Stewart platform with orthogonal configuration, the active and passive single-bar vibration isolation system is studied. The active and passive vibration isolation device is designed by combining MFC material as the actuator and viscoelastic damping material. Furthermore, pointing control is conduced on the basis of vibration isolation system, and the integrated control strategy of vibration isolation and pointing is discussed. Then, a single rod and its Stewart platform are respectively taken as an example to analyze and verify the effectiveness of the method by numerical simulation.

2. DYNAMIC MODEL BASED ON STEWART PLATFORM

The schematic diagram of the classic 6-legged Gough-Stewart platform and its hinge point layout are shown in Fig. 1. The upper platform including object load is regarded as a rigid body with 6 degrees of freedom. And the lower platform is fixed on the earth. It should be noted the symbol " ⊗ " represents the center of mass of the upper platform whose distance from the plane is denoted as h. The upper and lower platforms are respectively provided with 6 hinge points, which are connected by 6 support rods. Notably, each support rod is simplified into a rod with certain stiffness, damping and actuator. Cartesian coordinate systems M(O u , x u , y u , z u ) and B (O d , x d , y d , z d ) is established to the upper platform and the lower platform, respectively.

Figure 1: Schematic diagram of (a) The classic Gough-Stewart platform (b) layout of the hinge

point of the upper and lower platform The lower platform coordinate system is regarded as the reference coordinate system, and the generalized coordinate column vector composed of the 6 DOF motion variables of the upper platform is denoted as:

q(t) = { x p y p z p θ ϕ ψ} (1)

Where, x p , y p , z p is the coordinate position of the center of the circle in the coordinate system; θ ϕ ψ respectively are roll Angle, yaw Angle and pitch Angle. Axial expansion of support rods between hinge points of the platform is denoted as I(t) = {I i (t)}, i = 1, ⋯,6 . In the

oh

control process, the relationship between variables and can be established by Jacobian matrix, which are given as:

I ̇ = Jq̇ (2)

If it is denoted as the unit vector in which the fourth supporting rod is located, the vector is the vector from the center of the circle of the upper platform to the hinge points of the upper platform. Then the Jacobian matrix is:

J = [ e 1 e 2 ⋯ e 6 r u1 × e 1 r u2 × e 2 ⋯ r u6 × e 6 ] (3)

According to the analytical dynamics theory, the motion of the non-conservative system obeys Lagrange equation:

d dt

∂L ∂q̇ + ∂ϕ

∂q̇ −∂L

∂q = Q (4)

Where, Q is the generalized force corresponding to the operating force of the strut actuator and the coordinate q; L, ϕ respectively are the Lagrange function and Rayleigh dissipation function of the system:

L = 1

2 q̇ T Mq̇ −1

2 q T Kq, ϕ = 1

2 q̇ T cq̇ (5)

Where M, K, C are the mass matrix, stiffness matrix and damping matrix of the system respectively. If the column vector composed of six support rods as the driving force is written as:

f = {f 1 f 2 ⋯ f 6 } T (6)

When the load on the upper platform has virtual displacement δq , it can be known from the formula (2) that the virtual work done by the six supports is:

δw = f T δI = f T Jδq = Q T δq (7)

Therefore, the generalized force corresponding to the force acting on the coordinate is:

Q = J T f (8)

By substituting equations (5) and equations (8) into the equation(4), the dynamics equation of the system is:

Mq̈ + Cq̇ + Kq = J T f(t) (9)

Ignoring the kinetic energy of the support, the mass matrix is determined by the mass characteristics of the upper platform. In addition, according to the diagonal matrix of damping and stiffness of the six struts, the virtual work principle also shows that the equivalent damping matrix and stiffness matrix are:

C = J T diag(c i )J, K = J T diag(k i )J (10)

For the differential equation, its characteristic equation can be written as:

Mq̈ + Kq = 0 (11)

By modal analysis, the natural frequency and modal matrix of the system can be obtained.And the modal matrix is denoted as:

ϕ = [ϕ 1 ϕ 2 ⋯ ϕ 6 ] (12)

Where is the n k -order modal vector of the system ϕ nk = (n k = 1,2, ⋯6) . 3. DESIGN OF VIBRATION ISOLATION SYSTEM FOR ORTHOGONAL STEWART PLATFORM

3.1. Orthogonal Configuration of Stewart Platform

In order to facilitate the design of vibration controller, Stewart platform is usually required to meet the characteristics of dynamic decoupling. Actually, decoupling is for the sake of achieving the consistency of modal coordinates and physical coordinates.

In this paper, a decoupled orthogonal configuration is adopted that the plane of the two adjacent struts is perpendicular to the surface of the lower platform, which could achieve the decoupling of the stiffness matrix. Then, the dynamic decoupling is realized by adjusting the pair. Also, the vertical relationship between adjacent struts can be realized through the integrated design of the upper hinge. Meanwhile, the surface of the struts and the surface of the lower platform are perpendicular to each other, which can decouple and reduce the length of the rod.

3.2. Single Pole Active and Passive Integrated Vibration Isolator Design

Based on dynamic decoupling, a Stewart platform configuration design is carried out to decompose the dynamic parameters of the single rod including natural frequency and damping.

The block diagram of different measurement feedback control vibration isolation schemes of the system is shown in Figure 2. It can be seen the control strategies are centralized and distributed. Centralized mode refers to the control mode in which measurement is carried out on Gough-Stewart platform and single rod is operated. This control mode belongs to multi-input multi-output control (MIMO), and independent modal space control (IMSC) is generally adopted. Since the motion amplitude of the support rod is very small, the modal matrix can be regarded as a constant matrix.

Distributed control is measured and actuated within the range of each single rod. This control mode is single input single output control (SISO), which is relatively simple and has high reliability. Based on the dynamic decoupling design, SISO control can achieve the same effect compared to MIMO. Therefore, a distributed control strategy is selected to measure and operate within the range of each single rod.

Figure 2: Block diagram of the different control strategy The single rod is consisted of vibration isolator, connecting rod, flexible hinge 1, flexible hinge 2 parts, which is shown in Figure 3. It can be found the damper adopts cantilever beam with double constraint damping layer. Besides, the damping layer adopts viscoelastic material (VEM). The actuator adopts piezoelectric actuated cantilever beam design. Flexible hinge adopts flexible ball joint. Viscoelastic material and piezoelectric material constitute the main body of active and passive integration.

Figure 3: The composition of the single rod system The configuration of the active and passive integrated vibration isolator is shown in Fig. 4, which consists of two layers respectively composed of four end-closed piezoelectric and viscoelastic laminated cantilever beams with a 90° distribution and their corresponding lug plates .The cantilever beam is a vibration isolator and the lug is used to ensure that the end of the beam meets the boundary condition of fixed support.

Figure 4: Active-passive integrated vibration isolator

4. VIBRATION ISOLATION/DIRECTIONAL INTEGRATED CONTROL SCHEME DESIGN

4.1. The Control Strategy

Vibration isolation direction integration has two modes, which are time-sharing and simultaneous control. The simultaneous control mode is adopted in this study. When pointing control is introduced, corresponding strategy and settlement procedures must be added. Based on active vibration isolation in Section 2.2, simultaneous distributed control strategy is adopted. Therefore, vibration isolation and pointing integration are the weighted synthesis of the two controls. Among them, tracking is achieved by measuring feedback of low bandwidth pointing sensor. High frequency jitter is isolated by passive vibration isolation. At this time, the actuator control signal is the weight of vibration isolation and pointing. Due to the adoption of position closed-loop, the loop rigidity increases. Then the system bandwidth will move slightly to the right.

It should be pointed out that the smaller the ratio of the load to the inertia of the spacecraft could result to the more suitable for the pointing control of the load end. If the pointing control bandwidth is lower than the attitude control bandwidth, the pointing control will has no effect. However, if the pointing control bandwidth is higher than the attitude control bandwidth, it may be coupled with the low frequency system such as the star and the sail. So the comprehensive dynamic model of the star attitude control, flexible accessories and vibration isolation system can be evaluated. If the bandwidth of directional control exceeds the frequency of passive vibration isolation, it is inevitable to introduce a notch filter to suppress mechanical resonance, which is irreconcilable with the application of large active damping in this frequency band of active vibration isolation.

4.2 Control Law Design

The single-DOF system can be used for mechanical equivalence without considering the force and hinge stiffness of the strut actuator. Thus, the transfer function from the bottom of the single rod disturbance to the load is as follows:

2

L[X C ] L[X d ] ≜X C (s)

X d (s) = 2ζw n s + w n

s 2 + 2ζw n s + w n 2 (13)

Where L[⋯] i s Laplace transform. s is Laplace variable. ζ, w n are respectively the damping ratio and natural frequency of the equivalent system.

Skyhook control algorithm is used to increase the damping control effect of vibration isolation system. Skyhook control of acceleration feedback is essentially integral control of acceleration feedback:

u 1 (t) = T d ∫x c ̈ dt = T d ẋ c (14)

Where T d is differential gain. Or written in the Laplace domain:

U 1 (s) = L[u 1 ] = T d sX c (s) (15)

Because the pointing instruction is generally in low frequency band and changes slowly, it can also be regarded as differential control of position x c feedback. Therefore, position tracking control is designed as position feedback control, and the gain is denoted as k p :

2

L[X C ] L[X d ] ≜X C (s)

X d (s) = 2ζw n s + w n

s 2 + 2ζw n s + w n 2 (16)

According to the formula (15) and (16) formula, vibration isolation and position tracking control are their weights, and the transfer function of feedback path becomes PD control of position feedback, and the turning frequency is k p T d ⁄ :

U(s) = U 1 (s) + U 2 (s) = (T d s + k p )X c (s) = k p (1 + T d

s) X c (s) (17)

k p

In order to improve the steady-state accuracy of tracking control, the integral part 1 (T i s) ⁄ is added. In this way, the control law for vibration isolation and pointing control becomes the traditional PROPORTIONAL control (PID) :

U(s) = (k p + 1

T i s + T d s) X c (s) (18)

4. NUMERICAL RESULTS

In this section, the proposed control method will be verified by numerical simulation. The payload of the upper platform and the base platform of the spacecraft are 500kg and 6500kg respectively, and the inertia ratio is about 2.5% and the mass ratio is about 8%. Therefore, the base platform can be regarded as fixed. The rotational inertia at the center of mass of the payload on the upper platform with respect to the shaft is respectively I xx = 188kgm 2 , I yy = I zz = 261kgm 2 . The vibration isolation frequency of the vibration isolation system is required to be 1Hz~3Hz, and the pointing control accuracy is 0.008°.

What is more, the performance of single rod system vibration isolation and pointing integration control of Stewart platform is analyzed. The disturbance at the bottom of the rod is presented in the form of harmonic with random, which is made of a frequency range of 0~200Hz, as shown in Fig. 5. And the pointing control instruction is set as a sine function with a frequency of 0.1Hz. Fig. 6 shows the tracking curve without disturbance at the bottom. with the bottom disturbance added, Figure 7 and Fig. 8 are the tracking curves before and after active control is applied respectively. It can be seen the control error curve within the first 10s. Conclusions can be obtained that the pointing control accuracy is significantly improved by an order of magnitude under the condition of active and passive vibration isolation.

Figure 5: Disturbance characteristic from

Figure 6: Tracking curve without bottom

bottom of single rod

disturbance of single rod

Figure 7: Passive vibration isolation/pointing control effect of single rod with disturbance

Figure 8: Active-passive vibration isolation-

pointing control effect of single rod with

disturbance In order to analysis the point-tracking control effects on the Stewart model, the tracking plan of payload is given as:

ρ = αe γ (19)

When only the PI control link in the control law is used for tracking control (i.e. T d = 0 ), the turning frequency of PI controller is 8 times of the instruction frequency. The tracking control results of yaw ( R y ) and pitch ( R z ) directions of the payload are shown in Fig. 9. Figure 10 shows the disturbance in the basic yaw and pitch directions, where the tracking control parameters in Fig. 10 are still adopted. It can be seen the tracking curve in Fig. 11 has an obviously jitter. Namely, the control effect is poor. On the basis of the control parameters in Fig. 10, the active vibration isolation link in vibration control is added, which means the PID control link in the equation is adopted. The tracking results are shown in Fig. 12. It can be seen from Figure 12 that the tracking jitter is obviously suppressed. Therefore, conclusions can be obtained the PID control strategy has a good effect on the vibration suppression of the platform.

Figure 9: Pointing control effect of the Stewart platform without disturbance

Figure 10: Disturbance characteristics from

the bottom of the Stewart platform

Figure 11: Passive vibration isolation/pointing control effect of the Stewart

Figure 12: Active-passive vibration isolation/pointing control effect of the Stewart

platform with disturbance

platform with disturbance 5. CONCLUSION

In this paper, a single-bar vibration isolation system is designed by combining viscoelastic damping material and piezoelectric film material. And a multi-dimensional vibration isolation system with active and passive integration is constructed by combining the single-bar system according to the configuration of Gough-Stewart parallel mechanism. The control strategy is designed for vibration isolation-pointing control with high pointing accuracy and high stability. Numerical results show that the system can meet the vibration isolation and pointing requirements of load platform. Based upon these results, conclusions can be obtained:

(1) The design method and rig of the micro-vibration isolation platform can work well to achieve the required vibration suppression and pointing control for spacecraft system.

(2) The control scheme presented in this paper has a good effect on the vibration suppression and pointing control of the micro-vibration isolation platform.

6. ACKNOWLEDGEMENTS

This research was supported by the National Natural Science Foundation of China with Grant No. U21B6002. 7. REFERENCES

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