Trans. JSASS Aerospace Tech. JapanVol. 8, No. ists27, pp. Pc_35-Pc_40, 2010

Original Paper

Copyright© 2010 by the Japan Society for Aeronautical and Space Sciences and ISTS. All rights reserved.

Pc_35

1

Empirical Model Reduction of Spinning Solar Sail

By Masahiko YAMAZAKI1) and Yasuyuki MIYAZAKI2)

1) Department of Aerospace Engineering, Graduate School of Science and Technology, Nihon University, Funabashi, Japan2) Department of Aerospace Engineering, College of Science and Technology, Nihon University, Funabashi, Japan

The Spinning solar sail is expected to be a future space exploration system. Considering the dynamic deformation of the sail membrane is an indispensable factor in designing the spacecraft. But the mathematical model of the sail membrane is complex. Computational analysis is difficult and time consuming. Therefore this has a negative impact on simulation, design and control problems. The model reduction technique is required to shorten the design period. It is a necessary step in order to put the gossamer structure to practical use. In this presentation, the dynamic property of the square type spinningsolar sail is revealed, and the requirements for constructing a reduction model are revealed. Then, empirical model reduction techniques are applied to gossamer structures, and the issue of constructing a low-order model is summarized.

Key Words: Solar Sail, Spin-Deploy, Membrane, Empirical Model Reduction

1. Introduction

In recent years, the use of membranes for space craft applications was attracted a great deal of attention, as so called gossamer structures. Very large and ultra-light structures have recognized as the most viable space deployable structures. But gossamer structures do not have practical applications at present.

Currently, JAXA has investigated a spin type solar power sail spacecraft 1). A solar sail is a space craft which converts light from the sun into thrust by reflecting the light. Spin type sail may be more advantageous than the sail deployed by extendible masts. Spin deployment technology is most important technology for a future deep space exploration.

From this development and past space experimentations, we discovered several problems concerning (1) the prediction of structural and dynamic characteristics in the micro-gravity and vacuum environment, (2) the control of maintenance including accuracy, (3) the control of the shape and the attitude of the sail, and (4) the computational cost of complex mathematical models.

As mentioned above, it is significant to construct a design process of gossamer structure.

We should do that starting with a small, low cost gossamer structure, and gradually develop large structures through different stages of experiments/trials with increasingly bigger structures. In order to resolve the second and third problems, we should construct a control system. A reduction model is necessary for problems two, three and four. We have finally managed to simulate nonlinear dynamics of gossamer structures. We do not have proven methods of constructing reduction models. We worked on simplified rough models without considering flexibility.

In this paper, the dynamic property of the square type spinning solar sail and the requirements for constructing a reduction model are revealed. Then we apply a model

reduction to gossamer structure. If we construct the model reduction technique, we can obtain a computational saving in problems of both structural and control design. Our aim is to develop a general methodology which is applicable to a wide range of gossamer multibody systems.

2. Square Type Solar Sail

2.1. Sail configuration

Fig. 1. Square solar sail. Fig. 2. Configuration of center tether.

Fig. 3. Deployment sequence.

Probe Vehicle

Center tether Membrane

(Received July 16th, 2009)

Trans. JSASS Aerospace Tech. Japan Vol. 8, No. ists27 (2010)

Pc_36

2

In this paper, the square type solar sail is considered (Figure 1). The sail craft is consists of a square membrane and a probe vehicle. The membrane is made up of four trapezium membranes or “petals” connected by rectangular membranes called “bridge” which is colored grey in the Figure 1. The square membrane is connected to the probe vehicle by the centre tether (Figure 2). In addition, tip of the square membrane is connected to the tip mass by the tip tether.

This model has mainly four features. The first one is the square opening between the probe vehicle and the square membrane. Tangling of the membrane and the probe vehicle is prevented by the presence of that opening. The second is the square shape and combination of four petals. On a practical level, it is easier to construct a square shape than circular one, due to the thickness and flexibility of the polyimide membrane. The division into four petals also facilitates the construction. The circular shape may seem to be suitable adapted for rotating in according to rotation dynamics, but construction is difficult. The third one is the tip masses. They assist in membrane deployment. In addition, they contribute to concentrating tensile force on a line connecting the tip mass with the probe vehicle in steady state. The fourth feature is that the wrinkle or slack is easily induced by the bridges. The advantage of the bridges is in the facility of construction. The disadvantage is the resulting flexibility and ensuing difficulty in predicting the motion of the sail. As mentioned above, the sail model that is addressed in this paper comes with features to cope with production error and geometric nonlinearity.

2.2. Sail fold & deploymentThe deployment method of the sail consists of two stages,

as shown in Figure 3. In stored configuration, each petal is folded up and rolled up around the probe vehicle. In the first stage, the petals are extracted and form a cross shape. The cross shape is maintained because the stoppers are constrained. In the second stage, the stopper is released and each petal is deployed to form a square shape.

3. Dynamic Property of Square Type Spinning Solar Sail

3.1. Mathematical model It is not adequate to evaluate the performance of the sail

only with ground experiment. In fact, there has not been developed the experimental method of the large membrane to simulate behavior under the micro-gravity and the vacuum environment. In such case, the numerical simulation is necessary to complement the evaluation. We may design the sail that the ambiguous effect of any factor will be minimized. The mathematical model may neglect the minor effect of such factors to reduce the computation cost. In case of solar sail, the local wrinkle of the membrane may decrease the performance of the propulsion. The spring effect and the hysteresis of the fold line may also give undesirable influence on the deployment dynamics and the equilibrium shape after the second stage. The effects of the wrinkle and the fold line depend on the product error which distributes whole of the membrane. We can hardly measure all of the error. Therefore, such effects can’t be estimated precisely. In the case of our square sail, such effects should be reduced by linking the

appropriate amount of tip mass so that the performance of the deployment is improved. Then we can evaluate the performance of the sail without programming the local buckling and bending of the membrane.

Such a model reduction is required to shorten the design period. The geometrical relations also have to be simplified in the finite element analysis of the deployment of large membrane for this requirement. For example, the number of the fold lines can be modified to be smaller than that of the flight model. There may be disagreement about the validity of such a simplification. The control model must be simpler than the detailed dynamics model in order to realize the on-board processing as well as to shorten the development period of the control law. The model reduction technique should be applied to construct the control model. Above model reduction is not theoretic. In section 4, we show the theoretical approach to construct a low-order mode.

Figure 4 illustrates an example of the deformed shape of a 20m-sized square sail with 300kg weight during the second stage. In this example, the membrane and the bridges are modeled by membrane element that is a plane stress model based on the tension field theory. The tethers are modeled by cable elements. In addition, the mass characteristic of the probe vehicle is asymmetric to the spin axis. And the model is assumed to have an initial nutation and the center tether damping.

Fig. 4. Deformed shape during the second stage.

Fig. 5. Strain energy density distribution.

The numerical simulation shows that the sail deploys within the several seconds, but the in-plane oscillation continues for a while. The oscillation gets small around one thousand seconds after the second stage starts because of the damping of the center tethers. The residual vibration of the membrane is not going to matter as much. The time to steady state may indicate the time to proceed to the next phase, e.g. the attitude maneuver.

( ) 2.0sect (b) 1000sect ( ) 2000sect

HighLow

M. YAMAZAKI and Y. MIYAZAKI: Empirical Model Reduction of Spinning Solar Sail

Pc_37

3

Figure 5 shows the distribution of the strain energy density. The red area represents higher density, and the blue one represents lower density. At the time, the energy concentrates at some specific area where the objects such as solar cells are attached to the membrane. But after a while, the almost all area of the petal has low energy and repeats the taut state and the slack state. The energy concentrates along the diagonal line of the square sail because the centrifugal force of the tip mass is much larger than that of the petal membrane. This figure suggests that some model reduction is possible in the deployed state when enough time, e.g. 2000sec, elapses after the second stage starts.

Figure 6 shows the change of the spin rate. The solid line represents a case where the number of fold lines is 18 (the same as the flight model), while the dotted line represents a case with 8 fold lines. Indeed there is a difference, but the difference is small and the deployment behaviors are similar. Thus the simplification of the fold pattern is a possibility for a rough understanding of deployment motion.

Fig. 6. Fold lines comparison by spin rate.

Even though we have omitted certain result, we should take into account the fact that the membrane doesn’t bear a compressive load. But we do not have to consider the bending rigidity of the fold line and the out-of-plane deformation at the wrinkle. If you need to analyze the detail of the deformed shape, especially the out-of-plane deformation, you should employ the shell element to model the membrane. You can use a simpler element, i.e. the tension field element if you don’t need the detailed shape and you want to reduce the computation time.

Through numerical simulation, we have found that in the case near steady state, the sail craft has low geometric nonlinearity. From this, we have concluded that it may be possible to express sail craft motion in a low-order model. However, because of the particularities of the sail (Buckling occurs and subsides repeatedly on various location on the sail, so it is not possible to express sail movement with the mode method, used with the linear system), it is important to consider dynamic nonlinearity in the construction of low-order models. In order to put gossamer structures into practical use, as stated above model reduction is indispensable. We have to construct a model reduction method suitable for gossamer structures.

4. Model Reduction

Linear input-output systems implement several model reduction techniques, including modal truncation, balanced

truncation, and optimal Hankel methods2). It is difficult to apply directly to the complex nonlinear systems. We have to change these methods to apply for complex nonlinear systems.

The principal idea of dimensional model reduction is to find a small number of generalized coordinates in which to express the dynamics, ideally with some bounds on the truncation error. In this section we show two reduction methods and some issue of model reduction of gossamer structure. The first one is focus on the input-output behavior of the state systems. We can implement the model reduction using by empirical gramians. The second one implement a model reduction using by an empirical eigenvector from increment of the system. These methods of model reduction are based on data. It requires only simple matrix computations, and may be applied directly to nonlinear systems.

4.1. Empirical model reduction -empirical gramian3)-In this section, we use the Karhunen-Loève decomposition.

The Karhunen-Loève decomposition for model reduction is useful for searching a low dimensional affine subspace of the state space, in which the dynamics of interest of the original system are contained. Once such a subspace is found, the Galerkin projection can be applied to project the dynamics onto it, so that the high-dimensional system is approximated by a small number of nonlinear ordinary differential equations (ODE).

4.1.1. Empirical gramiansEmpirical controllability and observability gramian are

constructed from a set of data. We tackle model reduction of the nonlinear systems of the

form ,t f x t t

t h x t

x w

z(1)

Here nx t is the state of the system, ptw , and qz t . The function tw is regarded as an input signal to

the system, and the function z t as an output signal. Define the following sets:

*1

1

1

, , ; , , 1, ,, , ; , 0, 1, ,, , ;standard unit vectors in

n n nr i i i

s i in n

n

T T T T T i rc c c c i se e

I(2)

Here is an arbitrary set of r orthogonal matrices, and is a set of s positive constants. r is different perturbation

direction, s is different perturbation magnitude, n is input number/state number of the full order system. Definition1: Empirical controllability gramian

For the system (1), the empirical controllability gramian Yis define by

02

1 11

1ˆpr s

ilm

m il m

t dtrsc

Y = (3)

where ilm t is given by*ilm ilm ilm ilm ilmt x t x x t x (4)

And ilmx t is the state of the system (1) corresponding

to the impulsive input m ilc T e tw t . Here t is dirac

delta function. Definition2: Empirical observability gramian

For the system (1), the empirical observability gramian X

Time [sec]

Spin

rate

[rpm

]

Trans. JSASS Aerospace Tech. Japan Vol. 8, No. ists27 (2010)

Pc_38

4

is define by

0

*2

11

1ˆr s

lml l

ml m

T t T dtrsc

X (5)

where lm n nt is given by *

: jlm jlmlm ilm ilmij t z t z z t z (6)

and ilmz t is the output of the system (1) corresponding to the initial condition m ilc T e0x t .

4.1.2. Empirical balanced truncation The empirical gramians indicate the importance of

particular subspaces of the state space, with respect to the inputs and outputs of the system. We constructed a low-order model as explained below; find a linear change of coordinates such that the empirical gramians are balanced, and perform the Galerkin projection onto the states corresponding to the largest eigenvalues of X , Y . Let T be the coordinate transformation such that the system is balanced; that is

-1* -1ˆ ˆ*Y XT T T T is diagonal matrix of Hankel values 2 n , and let 0[ ]P I be the k n projection

matrix( k dimensional subspace approximation). Then we can construct a k dimensional subspace approximant reduced order ODE model given by

-1 *

-1 *

,PP

PT T

T

y t f y t w t

z h y t(7)

Where ky t , and k n . The relation of full-order model to low-order model is given by -1 *PTx t y t .

4.1.3. Numerical exampleIn this section we give an example for the nonlinear

mechanical system and the gossamer structure system (Fig.7). First we show the procedure and effectivity of the empirical model reduction on the nonlinear mechanical model. Next we show the results of constructing a low-order model for the gossamer structure. The mechanical system consists of five uniform rigid rods in two dimensions, connected via torsional springs and dampers. The lowest rod is pinned to ground with a torsional spring, so that the system has a stable equilibrium in the upright vertical position. There is no gravity. The system has a single input, a torque about the lowest pin joint, and a single output, the horizontal displacement of the end of the last rod from the vertical symmetry axis. Gossamer system consists of rigid, truss and cable elements in two dimensions. The system has a single input, a torque about the z axis of rigid body, and five outputs, the displacement and velocity of node 8.

(a) Mechanical system (b) Gossamer system Fig. 7. Example for empirical model reduction.

Table 1. Parameter of the mechanical system.

Item ValueSpring constant [N/m] 3.0

Damping coefficient [Nms/rad] 0.5Rod mass [kg] 1.0Rod length [m] 1.0

Table 2. Parameter of the gossamer system.

Item ValueRod length lv [m] 1.0Rod length lh [m] 0.1 Rigid mass [kg] 1.0Rigid radius [m] 0.2

Angular velocity [rad/sec] 0.0

4.1.4. Nonlinear mechanical system One has to choose the region of operation. Then data should

be collected within this region and empirical gramians calculated. In this numerical calculation, we chose

0.4 2r and , since this corresponds to using both positive and negative inputs (or initial states) on each input separately.

Fig. 8. Horizontal displacement of the end of the last rod from the vertical symmetry axis.

We apply the balanced realization at the linearization of the system about its stable equilibrium and the empirical balanced realization. Figure 8 shows a comparison of the full-order model response with the low-order model response. In this calculation we give a impulsive input at 200[sec]t . Figure 9 shows the configuration part of the first four of the ten mode shapes and the corresponding singular values of the balanced realization. The singular values have been normalized so that they sum to one. The full-order model response is good approximated by only two large modes.

Fig. 9. Empirical balanced modes.

4.1.5. Gossamer structure system In the case of gossamer structures, we have to construct an

Hor

izon

tal d

ispl

acem

ent [

m]

-0.4

-0.2

0

0.2

0.4

100 150 200 250 300 350 400 450 500Time [sec]

2 3 40.502 0.410 0.019 0.008

M. YAMAZAKI and Y. MIYAZAKI: Empirical Model Reduction of Spinning Solar Sail

Pc_39

5

ODE model from a mathematical model of gossamer structures. We constructed an equation of motion as explained below, considering the finite element method, used widely for analyzing gossamer structures.

First, we take out a mass matrix and internal force vector from the subroutine of the finite element method. Next, we model the constraint condition on the penalty method. In addition, we considered the different magnitude of each node type, such as constraint node. Therefore we change a perturbation magnitude at each node. In the case of full-order model we calculate a dynamics using the finite element method formulated by the Energy Momentum Method (EMM) 4). In the case of low-order model we use the Runge-Kutta method. Following two numerical calculation we chose a perturbation direction . In addition we displayed the result in the rotating frame on a rigid-body. Case1

This system consists of rigid, truss and cable elements in two dimensions. Figure 10 shows a comparison of the full-order model response with the low-order model response. In this calculation we give an impulsive input at 0[sec]t .The result begins to diverge when the total number of modes is reduced by 3/4.

Fig. 10. The velocity of the node 8.

Case2This system consists of rigid and truss elements in two

dimensions. Figure 11 shows a comparison of the full-order model response with the low-order model response. In this calculation we give an impulsive input at 0[sec]t . Even if the total number of modes is reduced by 9/10, it is possible to approximate.

Fig. 11. The velocity of the node 8.

In the Case 1 model, there is high-frequency motion because of cable elements. If the time step of divided minutely,

the high order terms will cause divergence of the results. In future research, I would like to investigate the cause of

numerical instability and adapt the ideas proposed in this paper to the EMM (steady numerical integration of motion equation). The model-reduction method requires too much time for preliminary calculations, but it is necessary to examine how input affects output. Since gossamer structures have a large degree of freedom, this time consuming aspect is an important problem. However once we construct the low-order model, we can use structure design and control design, so building it is important for putting gossamer structures into practical use.

4.2. Empirical model reduction -empirical eigenvector5)-The empirical eigenvectors are used as the basis in which to

expand the incremental problem. We can get the basis from a simulation data for the response of the full-order model in time to construct a reduced basis which reproduces the full system in a statistical sense. Because of this “global” time viewpoint, the basis need not be updated. In addition we can adapt the method to the EMM.

4.2.1. Empirical eigenvectorConsider a solid body subjected to some dynamic loads

during a time interval I . Mark arbitrary N points on the surface of the body and/or in its interior, and record the positions of these N points at S instants during the interval I . The observations are collected into a set of 3N-dimensional vectors, 3 , 1, ,j N j Su (three displacement components for each of the N points). The set of vectors is centered by subtracting its average,

1

1 Si i

iSu u u (8)

and collect the centered vectors as columns of the matrix, 1 2

3

S

N SU = , , ,u u u u u u (9)

If an approximation to the vector ensemble ju is sought in the form

1

MT i i

i

u u u -u= M N (10)

Where u is the three displacement for reduced order model. The expectation

2E u -u (11)

is minimized, when i are the orthonormal eigenvectors of the covariance matrix dC . The covariance matrix calculated from U with a desired reduced order of M by

1 T

d MC UU (12)

corresponding to the eigenvalues 1 2 N .

4.2.2. Numerical exampleIn order to assess the efectivity of the model reduction, we

give an example for the gossamer structure. As shown in Figure 12, simple square membrane model with one side of the edge fixed is constructed using mass spring network model. The square membrane is divided into 10 10 meshes for the analyses in this section.

Vel

ocity

[m/s]

Vel

ocity

[m/s

]

Time [sec]

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0 0.1 0.2 0.3 0.4 0.5Time [sec]

Trans. JSASS Aerospace Tech. Japan Vol. 8, No. ists27 (2010)

Pc_40

6

Fig. 12. Square membrane model.

Table 3. Material parameter of membrane.

Item ValueYoung's modulus [GPa] 3.5

Poisson's ratio [-] 0.3Thickness [ m] 5.0Density [kg/m3] 1.4 103

I calculated the motion created by displacing Node A (see Figure 12) by 0.5mm (towards the x and y axis) and the letting go.

Fig. 13. Horizontal displacement at Node B.

Figure 13 displays the horizontal displacement at the Node B (see Figure 12). The solid line represents a full-order model, the dashed line represents a half modes of the full-order model, while the solid line with square marker represents a quarter modes of the full-order model. The general movements are similar, but there are small differences. The motion of the membrane in “tension state” is very different from that is the “compression state”. Because of the error of the low-order model, a bifurcation occurs that was not in the full-order model.

5. Conclusion

1. The overview of the deployment motion of the spinning square solar sail and requirements for constructing a reduction model are presented. It may be possible to express sail motion in a low-order model in the near steady state.

2. We recommended adapting the two empirical model reduction methods using the empirical gramian and the empirical eigenvector. For the use of empirical gramian, we constructed an equation of motion as explained below;

We took out a mass matrix and internal force

vector from the subroutine of the finite element analysis to construct an equation of motion. In this way, we can easily construct an ODE model from complex gossamer structures. We modeled the constraint condition on the penalty method. We considered the different magnitude of each node type. Therefore we varied perturbation magnitude at each node.

These two methods were applied directly to nonlinear systems. In addition these bases (the basis from empirical balanced transformation matrix and covariance matrix) need not be updated in the calculation. In the case of the empirical eigenvector method, we can adapt the method to the EMM.

Through some numerical examples, we have concluded that it may be possible to apply empirical model reduction to gossamer structures. For the future, we want to compare these two methods and apply real system.

Acknowledgments

The authors thank for IKAROS project team, especially, Dr. Osamu Mori, Dr. Nobukatsu Okuizumi and Dr. Hirotaka Sawada at JAXA, and Mr. Yoji Shirasawa at the University of Tokyo, Japan, for their valuable discussion on the dynamics of the square type solar sail. This study is partially supported by Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research (B).

References

1) Mori, O., et al.: Development of Deployment System for Small Size Solar Sail Mission, 26th ISTS, (2008), 2008-d-57, pp.1-6.

2) Moore, B. C.: Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Reduction, IEEE Transactions on Automatic Control, 26(1), (1981), pp. 17-32.

3) Lall, S., Marsden, J. E. and Glavaski, S.: Empirical Model Reduction of Controlled Nonlinear Systems, Proceedings of the IFAC World Congress, (1999), pp.473-478.

4) Miyazaki, Y. and Kodama, T.: Formulation and Interpretation of the Equation of Motion on the Basis of the Energy Momentum Method, Journal of Multi-body Dynamics, 218 (2004), No.1, pp.1-7.

5) Krysl, P., Lall, S. and Marsden, J. E.: Dimensional Model Reduction in Nonlinear Finite-Element Dynamics of Solids and Structures, International Journal for Numerical Methods in Engineering,(2001), pp. 479-504.

6) Miyazaki, Y.: Dynamic Behavior of Spinning Square Solar Sail, 18th Workshop on JAXA Astrodynamics and Flight Mechanics,(2008), pp.38-42.

7) Lall, S., Krysl, P., and Marsden, J. E.: Structure-Preserving Model Reduction of Mechanical Systems, Physica D: Nonlinear Phenomena, 184 Issue 1-4, (2003), pp. 304-318.

8) Miyazaki, Y.: Wrinkle/Slack Model and Finite Element Dynamics of Membrane, International Journal for Numerical Methods in Engineering, 66 No.7, (2006), pp.1179-1209.

9) Hahn, J. and Edgar, T.F.: An Improved Approach to the Reduction of Nonlinear Models Using Balancing of Empirical Gramians, Journal of Computers and Chemical Engineering, 26 No.10, (1999), pp.1379-1397.

xy

A

1 m

1 m

B

Disp

lace

men

t [m

]

0.95

0.96

0.97

0.98

0.99

1

1.01

0 0.02 0.04 0.06 0.08 0.1Time [sec]Time [sec]