A Study of Dynamic Modeling and Reaction Wheel Controller Design for Satellite AOCS with Flexible appendages Byoungsam Woo, Jangsoo Chae System Engineer, Satellite-Precision Sensor Division, Daewoo Heavy Industries Ltd.
Abstract In this study, a few of the modeling methods for flexible spacecraft were introduced and adopted to the modeling of a 3-axes stabilization satellite. The generated model was put into pre-built rigid body attitude control loop. A Lumped Parameter Model-GMM(Global Mode Model) here-was recommended for the case of the FEM model absence. Finally, GMM was compared with FEM in terms of designing a control filter. A 1st-order filter was designed to meet requirements of the controller since the new flexible model was applied, and that filter was added to motor controller and axis controller. MATLAB/Simulink was used as a tool for designing and simulation of the control loop and filter.
1. Introduction Satellites which were built in past had a small size and a relatively fewer requirements of electrical power since they had simple missions in space. But for the current satellites, the missions are very complex then the size of satellite and power need are getting bigger and bigger. As a result, a solar panel which has large size and light weight becomes most common part for many satellites. The satellite bus is generally considered as a rigid body but the large solar panel or antenna has flexibilities, so the flexible bus model is required for designing attitude and orbit control system(AOCS) especially for precise one[2][9][12]. In general, a flexible appendage of satellite has very small damping ratio in low frequency region and it can be considered as a sum of infinite number of flexible modes. The natural frequencies of each mode are widely distributed and some of them may locate near the attitude control bandwidth, so they would be stimulated by control torque. That's the reason why the flexibilities should be considered in designing AOCS. For designing conveniences, two or three flexible modes which have the biggest effect on AOCS are generally chosen in modeling. In this study, the modeling process is presented first and the resultant flexible model is applied to pre-built rigid body attitude controller[1]. After analysis of the flexible effect on control performance, the controller is reinforced with the 1st-order filters and the improved performance of controller is shown by simulation[3][4]. The goal of controller design is to satisfy its requirements. In this point of view, if the flexible modes are included, an additional design will be necessary though the requirements are satisfied with rigid body model by previous design. For the additional design, the GMM was used first before the FEM model.
2. Modeling methods for flexible satellite Although the FEM is generally used and most precise one, yet it doesn't give physical intuition and is not available at early phase of development. Then other methods can be used
for early phase of AOCS design or for nearly rigid satellite. Table 1 shows several methods for satellite's flexibility modeling which are used currently[6]. (Table 1. Modeling Methods for Satellite's Flexibility) Modeling Method
Characteristics Used for almost rigid satellite or at first step of controller design in the lack of Rigid Body flexibility data Lumped Parameter It is a kind of Assumed Parameter Method. It simply Models real flexibility Model with the couple of some rigid bodies Finite Elements Quite a precise model with utilizing NASTRAN etc. But it is not available for Method the first phase of controller design. It is totally analytical model for very simple structure. This method can be used Analytical PDE as a rough model for many satellite. Assumed Mode It represents real flexibility with some known mode. This method is good for a Method satellite with simple geometry, small flexibility.
3. Analysis of flexible solar panel The FEM model has a few defects; (1) large matrix(related with degree of freedom) in state-space representation, (2) provide no physical insight, and (3) available after all structural features are decided. In this study, so two kinds of Lumped parameter model-Cantilever mode model and GMM-are considered in modeling flexible solar panel of satellite[13].(Generally, a solar panel is most flexible part of current satellite.) To do this, we only considered 1-axis rotational motion of satellite with the assumption of no cross coupling effects between axes (See Fig. 1). The equation of motion of satellite in Fig. 1 is ∂ 2u EI ∂ 4u ∂ 2η EI ∂ 4η a = θ r̈( ξ + ) : deflection motion 2 + ( ρA ) 4 = θ r ̈x = 2 +( 3 ) l ∂t ∂x ∂t ml ∂ξ 4
I r θ r̈ - 2 ρ A ⌠ ⌡
a
a+l
x(
l ∂ 2u a ∂ 2η 2⌠ ) d ξ = M : rotational motion 2 ) d x = I r θ r ̈ - 2 m l ⌡ ( ξ + l )( 0 ∂t ∂t 2
(1) (2)
where E : Young's modulus, I : Moment of Inertia of cross section, I r : Moment of Inertia of total rigid body, M : Control torque, l : Length of flexible beam, a : Distance from center of mass to the root of beam, ρ : beam density, A : cross section area, u : relative deflection, θ r : Rigid Rotation Angle, x = l ξ + a , u = η ( ξ , t ) l , m = ρ A l .
(Fig. 1. Flexible Solar Array and Rotation Motion)
3.1 Cantilever Mode Analysis Assume the solar panel as a cantilever and with fixed satellite bus, one can easily get the solution of Eq. 1 and Eq. 2 with some series of well-known cantilever modes. The final model is generated by cantilever deformation function, Laplace transformation and some algebraic calculation. θ r( s ) =
∞ kn M (s) 2 2 + s θ r( s ) ∑ 2 2 n=1 s + w n I rs 1
kn=
2m l
2
[⌠ ⌡ Ψ n( ξ ) ( ξ + 0
Ir
(3)
a )d ξ ] 2 l
(4)
⌠1 Ψ 2 ( ξ ) d ξ ⌡0 n
where Ψ n ( ξ ) : n-th mode shape The modal gain k n can be calculated from flexible characteristics and n-th mode shape, Ψ n ( ξ ) . The block diagram of Eq. 3 is shown in Fig. 2(a). At Fig. 2(a), the first and second mode is appeared.
3.2 Global Mode Analysis The GMM analysis is a different method from cantilever modal analysis in the point that GMM analysis generates total flexible system directly. For current controller design, the GMM is more practical one than cantilever mode. By eliminating θ r ̈ between Eq. 1 and Eq. 2, Eq. 5 can be derived. ∂ 4ψ n ∂ 2η 2m l 2 a ⌠1 a ∂ 2η EI a M ( )( ξ + ) ( ξ + ) d ξ + ( ) = (ξ+ ) 2 2 3 4 ⌡ l l l Ir I 0 ∂t ∂t ml ∂ξ r
(5)
∞
η ( ξ, t ) = ∑ ψ n ( ξ) q n ( t )
(6)
n=1
where ψ n (ξ) : n-th mode shape, q n ( t ) : generalized coordinate of n-th mode Put Eq. 6 into Eq. 5 with q n ( t )= cos ( Ω n t ) and solve resulted differential equation, Eq. 7 and Eq. 8 are derived. 4
(
2 1 ∂ ψn EI a a 2 2 2 ml ) = Ω ψ Ω ( )( ξ + ) ⌠ (ξ + )ψ n d ξ n n n 3 4 ⌡ l l I 0 ∂ξ ml r
{
}
∞ 1 2m l 2 a a a M (ξ + )ψ n d ξ ]( q n ̈ + Ω n 2 q n ) = ( ξ + ) ∑ [ ψ n - ( I )( ξ + l ) ⌠ ⌡ l l Ir 0 n=1 r
(7) (8)
Replace θ r ̈ of Eq. 2 with Eq. 9 and solve Eq. 2 in homogeneous condition, then Eq. 10 can be derived. θ r = b n cos (Ω nt ) 1
2m l 2 ⌠ ⌡ (ξ+ 0
(9)
a )ψ n d ξ = I r b n l
(10)
Using Eq. 10 in Eq. 8, it yields Eq. 11. And finally, Eq. 11 has the common form of differential equation like Eq. 12 ∞ a a M ∑ [ ψ n - b n ( ξ + l )]( q n ̈ + Ω n 2 q n ) = ( ξ + l ) I n=1 r
{
}
(11)
A nM Ir
q n̈+ Ω n 2 q n =
(12)
where A n is coefficient parameter. On the other hand, take Laplace transform Eq. 2 and rewrite it in terms of θ r , θ r( s) =
M ( s) Ir
{ s1 + 2ml
2
2
∞ 1 q (s) a (ξ + )ψ n d ξ n ∑⌠ l M ( s) 0 n=1 ⌡
}
(13)
By Laplace transform Eq. 12 and put into Eq. 13, it yields, θ r(s) =
M( s) I rs 2
{
2m l 2
kn
∞
1+s2 ∑ [ n=1
2
s + Ω n2
]
}
1
a )ψ n d ξ ] 2 l kn= 1 2 ⌠ ψ 2 d ξ - 2 m l [ ⌠ 1 ( ξ + a )ψ d ξ ] 2 n ⌡0 n ⌡0 l Ir Ir
(14)
[⌠ ⌡ (ξ + 0
(15)
The block diagram of Eq. 14 is shown in Fig. 2(b). Fig. 2(b) shows only the first and second mode, but it can be extended easily by just adding another modes.
(a) Cantilever Model (b) Global Mode Model (Fig. 2 Block Diagram of Lumped parameter models)
3.3 FEM Model analysis The FEM model which is used in this study is derived from NASTRAN analysis. And it takes a state-space representation in order to be fit into control loop. To take a state-space form, it is necessary to be decided that inputs and outputs of satellite bus besides of bus plant itself. In this study, the control input comes from four reaction wheels and the control output is detected by gyros. In this state-space model, the modal factor of reaction wheels and gyros are shown as inputs and outputs(see Eq. 16).
A =
[ - w0
I 2
n
- 2 ζw n
C = [ Φ g y ro 0 ]
]
B =
[Φ0 ] wheel
(16)
D = [0]
where w n is natural frequency(diagonal matrix), ζ is damping ratio, Φ is modal matrix of wheels and gyros and I is unit matrix. In this study, the order of matrix was determined along with NASTRAN results. To put this model into pre-designed control loop, the State-Space block of MATLAB was used. It is
shown in Table 2 that the important results of NASTRAN analysis for controller design. (Table 2. Flexible Modes and Frequencies) Mode
Frequency(Hz)
1-6 7 8
0.0 0.49 0.73
Modal description Rigid node modes Symmetric array flapping Asymmetric array flapping
4. Modeling Results 4.1 Result of Global mode model analysis Table 3 shows variables and their values for GMM analysis. In this study, first two flexible modes(mode 7 & 8 in Table 2) are modeled. (Table 3. Global Mode Coefficients) Variable a
Value 0.435 m
l
3.00 m
EI m
97.60 N - m
2
Variable
Value
Ir
265kg - m 2
w1
3.0787 rad/sec
w2
4.5867 rad/sec
10.00 kg
To get the gain k 1 , k 2 , calculate Eq. 15 with the values of Table 3 and with these gain, the transfer function of dynamic model of flexible body can be built. The final values of gain are
k 1 = 0.4312
(17)
k 2 = 0.1003 The Bode plot of dynamic model with two flexible mode is shown in Fig. 3(a). Two surges appear clearly in the vicinity of flexible mode frequency-0.49 Hz and 0.73 Hz.
(a) GMM-two modes included (b) Roll axis controller : No filters (Fig. 3 Bode plot of Flexible Body(a) and Axis controller(b))
4.2 Results of FEM model analysis With FEM results, a Bode plot like Fig. 3(a) is easily acquired. At Fig. 3(b), the Bode plot of roll axis controller with FEM model is shown. The roll axis control loop (which shown in Fig. 4) has FEM model in the form of state-space block. For the detail of axis controller, reference [1] will be helpful. The same phenomenon which two surges caused by two flexible modes can be observed in Fig. 3(b). Of course, GMM assumes solar panel as one-end fixed cantilever and take many assumed variables then the characteristics of Fig. 3(a) and Fig. 3(b) are not exactly identical. But in terms of designing axis controller, GMM provide enough resemblance with real(FEM) model.
(Fig. 4 Open loop Block Diagram of Roll axis controller[1])
5. Filter Design The reaction wheel controller consists of motor and axis controller to control satellite attitude. The top level requirements imposed on the controller design are shown below. - Bandwidth of axis controller is 0.02 Hz and for motor controller, 0.2 Hz. - Overshoot should be less than 20 % and provide short settling time. - Gain margin over 10 dB, phase margin over 30 degrees. These requirements were satisfied with rigid body model[1]. But according to the Fig. 3(b) and the left column of Table 4, it is clear that the requirements isn't satisfied with flexible model. Then two 1st-order filters are designed to improve control characteristics and appended to the controller. Those results are shown in the right column of Table 4. (Table 4. Characteristics of Axis Controller with Flexible model) 1st order filter absence Roll Pitch Yaw
Gain Margin : Phase Margin Gain Margin : Phase Margin Gain Margin : Phase Margin
3.547 dB@ 0.7233 :
[email protected] 21.04 dB@ 0.2032 :
[email protected] 20.27 dB@ 0.1701 :
[email protected]
1st order filter exist Hz Gain Margin : 16.12 dB@ 0.1302 Hz Phase Margin : 76.07
[email protected] Hz Gain Margin : 14.21 dB@ 0.1300 Hz Phase Margin : 76.79
[email protected] Hz Gain Margin : 16.38 dB@ 0.1305 Hz Phase Margin : 77.48
[email protected]
Hz Hz Hz Hz Hz Hz
Motor Gain Margin : 12 dB@ 0.3303 Hz Gain Margin : 10.53 dB@ 0.3285 Hz Controller Phase Margin : 40 deg@ 0.1130 Hz Phase Margin : 38.84
[email protected] Hz
Generally, a thruster control is closely related to flexibility but reaction wheel control isn't[5]. In Table 4, there are no needs to add filters on pitch and yaw controller. But for roll axis, the gain margin is too small to satisfy requirements so 1st-order filter is designed for. 1st-order filter is added to each of motor and axis controller. The outer filter block of Fig. 4 is filter for axis controller and the filter for motor controller is included in nonlinear wheel model block of Fig. 4. According to Table 4, the gain margin of roll axis controller is increased from 3.547 dB to 16.12 dB by virtue of filters. In the case of pitch or yaw axis controller doesn't meet the requirements, then design of filters will be needed for those axis. The characteristics of roll controller is shown in Fig. 5.
(Fig. 5 Bode plot of Roll axis controller with 1st-order filters)
6. 3-Axes Simulation To perform the time domain analysis and 3-axes simulation, pre-built 3-axes simulation loop is used[1]. The purposes of simulation are to check a cross coupling between axes and to show the filter's ability that eliminates the unfavorable effects. Fig. 6 shows step response of 30 degree attitude command input on roll axis. In this figure, there are cross coupling effects in pitch and yaw response and those are caused by flexibility of satellite. For rigid satellite, there were no cross coupling effects like this[1]. The magnitude of cross coupling effects is about 0.05 degree in pitch and yaw axis.
7. Conclusion To include flexible mode in attitude controller design, a GMM that is useful before the completion of FEM model was derived and it was shown that the GMM is effective model for attitude controller design. The performance of controller was slightly changed by adding flexibility so 1st-order filter was adopted to improve the controller performance.
(Fig. 6 Step response of 30 deg input to roll axis)
8. Reference 1. Byoungsam Woo et al, "Study on the three-axes attitude controller design of Satellite using reaction wheel", Journal of Korea Society for aeronautical & Space Sciences, Vol. 25, No. 3, 1997, pp. 148-156. 2. Chetty, P. R. K., “Satellite Technology and its Application”, TAB books, PA, 1991, pp.205-233. 3. Chovotov, V. A., “Spacecraft attitude dynamics and control”, Krieger publishing company, Malabar, Florida, 1991, pp.1-13, 41-43, 65-90. 4. Hughes, P. C., “Spacecraft Attitude Dynamics”, John Wiley & Sons, New York, 1986, pp.93-128. 5. Iwens, R. P., “Basic spacecraft attitude control tools”, 1981. 6. Junkins J. L., Kim Y., "Introduction to Dynamics and Control of Flexible Structures", AIAA, 1993, pp. 140-217. 7. Kaplan, M. H., “Modern Spacecraft Dynamics & Control”, John Wiley & Sons, 1976, pp. 37-61. 8. Kuo, B. C., “Automatic Control System”, Prentice-Hall Int. Inc., 1995. 9. Larson, W. J. and Wertz, J. R., “Space Mission Analysis and Design 2nd Ed.”, Microcosm Inc., Torrance, 1992, pp.340-359. 10. Ogata, K., “Modern Control Engineering”, Prentice-Hall Int. Inc., 1990, pp. 118-122. 11. Shahian, B. and Hassul, M., “Control system design using MATLAB”, Prentice-Hall Int. Inc., 1993. 12. Wertz, J. R., “Spacecraft Attitude Determination and Control”, Reidel Publishing Com., 1986. 13. Xiao Y., "Elements of Flexible Spacecraft Dynamics", Lecture note, 1995, pp32-40.
