The Fire Satellite model¶

The fire satellite model is a multidisciplinary test case that involves 3 disciplines: Power, Orbit and Attitude & control. This usecase has been firstly proposed by Wertz et al. [wertz1999].

The model deals with the design of a realistic satellite, the goal of which is to detect and monitor forest fires from Earth’s orbit, through the use of optical sensors. Three disciplines are needed to design this satellite. The orbit discipline is responsible of computing the orbit period, the satellite velocity, the maximal slewing angle and the eclipse period. The power discipline is used to estimate the total power of the system and the area of the solar arrays. Finally, the attitude & control discipline computes the total torque of the satellite and the power required for the attitude control system. These disciplines exchange several coupling variables. The multidisciplinary analysis that is used to compute the converged values of the coupling variables is performed through a Fixed Point Iteration algorithm.

This test case is composed of nine random variables. These variables are distributed according to truncated Normal distribution $\mathcal{TN}(\mu,\sigma,a,b)$ . The hyperparameters are $\mu$ the mean, $\sigma$ the standard deviation, $a$ the lower bound and $b$ the upper bound.

$H \sim \mathcal{TN}(\num{18e6}, \num{1e6}, \num{15e6}, \num{21e6})$ : the altitude ( $\unit{\m}$ )
$P_{other} \sim \mathcal{TN}(1000, 50, 850, 1150)$ : the power other than attitude control system ( $\unit{\W}$ )
$F_{s} \sim \mathcal{TN}(1400, 20, 1340, 1460)$ : the average solar flux ( $\unit{\W\per\m^2}$ )
$\theta \sim \mathcal{TN}(15, 1, 12, 18)$ : the deviation of moment axis ( $\unit[angle-symbol-degree=deg]{\degree}$ )
$L_{sp} \sim \mathcal{TN}(2, 0.4, 0.8, 3.2)$ : the moment arm for radiation torque ( $\unit{\m}$ )
$q \sim \mathcal{TN}(0.5, 0.1, 0.2, 0.8)$ : the reflectance factor (-)
$R_D \sim \mathcal{TN}(5, 1, 2, 8)$ : the residual dipole of spacecraft ( $\unit{\A\m^2}$ )
$L_\alpha \sim \mathcal{TN}(2, 0.4, 0.8, 3.2)$ : the moment arm for aerodynamic torque ( $\unit{\m}$ )
$C_d \sim \mathcal{TN}(1, 0.3, 0.1, 1.9)$ : the drag coefficient (-)

The three outputs of interest are:

$\tau_{torque}$ : the total torque of the satellite ( $\unit{\N\m}$ )
$P_{tot}$ : the total power of the satellite ( $\unit{\W}$ )
$A_{sa}$ : the area of the solar array ( $\unit{\m^2}$ )

Different deterministic quantities are also present:

$c$ : the speed of light, $\qty{2.9979e8}{\m\per\s}$
$\omega_{max}$ : the maximum rotational velocity of reaction wheel, $\qty{6000}{\rpm}$
$n$ : the number of reaction wheels that could be active, 3
$\delta_{\theta_{slew}}$ : the slewing time period, $\qty{760}{\s}$
$A_s$ : the area reflecting radiation, $\qty{13.85}{\m^2}$
$i$ : the sun incidence angle, $\qty{0}{\deg}$
$M$ : the magnetic moment of earth, $\qty{7.96e15}{\A\m^2}$
$\rho$ : the atmospheric density, $\qty{5.1480e-11}{\kg\per\m^3}$
$A$ : the cross-sectional in flight direction, $\qty{13.85}{\m^2}$
$P_{hold}$ : the holding power, $\qty{20}{\W}$
$\mu$ : the Earth gravity constant, $\qty{398600.4418e9}{\m^3\per\s^2}$
$Id$ : the inherent degradation of array, 0.77
$t$ : the thickness of solar panels, $\qty{0.005}{\m}$
$n_{sa}$ : the number of solar arrays, 3
$\epsilon_{deg}$ : the degradation in power production capability, $\qty{0.0375}{\percent\per\year}$
$LT$ : the lifetime of spacecraft, $\qty{15}{\year}$
$r_{lw}$ : the length to width ratio of solar array, 3
$D$ : the distance between panels, $\qty{2}{\m}$
$I_{bodyX}$ : the inertia of body, X axis, $\qty{6200}{\kg\m^2}$
$I_{bodyY}$ : the inertia of body, Y axis, $\qty{6200}{\kg\m^2}$
$I_{bodyZ}$ : the inertia of body, Z axis, $\qty{4700}{\kg\m^2}$
$\rho_{sa}$ : the average mass density to arrays, $\qty{700}{\kg\m^3}$
$\eta$ : the power efficiency, 0.22
$\phi_{target}$ : the target diameter, $\qty{235000}{\m}$
$R_E$ : the Earth radius, $\qty{6378140}{\m}$

We assume that the input variables are independent.

The following figure depicts the interaction between the disciplines.

The orbit discipline is defined as follows. First, the satellite velocity $v$ is computed from the Earth radius $R_E$ and the altitude $H$ .

$v = \sqrt{\frac{\mu}{R_E+H}}$

with $\mu$ the Earth gravity constant. Then, the orbit period $\Delta_{t_{orbit}}$ is calculated,

$\Delta_{t_{orbit}} = \frac{2\pi(R_E+H)}{v}$

The eclipse period $\Delta_{t_{ecplise}}$ and maximum slewing angle $\theta_{slew}$ are then computed,

$\Delta_{t_{ecplise}} = \frac{\Delta_{t_{orbit}}}{\pi}\arcsin\left(\frac{R_E}{R_E+H}\right)$

$\theta_{slew} = \arctan\left(\frac{\sin\left(\frac{\phi_{target}}{R_E}\right)}{1-\cos\left(\frac{\phi_{target}}{R_E}\right)+\frac{H}{R_E}} \right)$

with $\phi_{target}$ the target diameter.

The attitude and control discipline is governed by the following equations.

$\tau_{tot} = \max(\tau_{slew},\tau_{dist})$

with

$\tau_{slew} = \frac{4\theta_{slew}}{\Delta t_{slew}^2} I_{max}$

and

$\tau_{dist} = \sqrt{\tau_g^2 + \tau_{sp}^2 + \tau_m^2 + \tau_a^2}$

$\tau_{g} = \frac{3\mu}{2(R_E+H)^3}|I_{max}-I_{min}|\sin(2\theta)$

$\tau_{sp} = L_{sp}\frac{F_s}{C}A_s(1+q)\cos(i)$

$\tau_{m} = \frac{2 M R_D}{(R_E+H)^3}$

$\tau_{a} = \frac{1}{2} L_{alpha} \rho C_d A v^2$

with $\tau_{tot}$ the total torque, $\tau_{slew}$ the slewing torque, $\tau_{dist}$ the disturbance torque, $\tau_{g}$ the gravity gradient torque, $\tau_{sp}$ the solar radiation torque, $\tau_{m}$ the magnetic field interaction torque, $\tau_{a}$ the aerodynamic torque.

The attitude control power $P_{ACS}$ is finally defined by

$P_{ACS} = \tau_{tot}\omega_{max} + n P_{hold}$

The power discipline has 16 inputs and computes the total solar array size and total power by,

$P_{tot} = P_{ACS} + P_{other}$

$A_{sa} = \frac{P_{sa}}{P_{EOL}}$

with,

$P_{EOL} = P_{BOL}(1-\epsilon_{deg})^{LT},$

the power production capability at the end of life, defined by

$P_{BOL} = \eta F_s I_d \cos(i)$

the power production capability at the beginning of life, and

$P_{sa} = \frac{\left( \frac{P_e T_e}{0.6} + \frac{P_d}{T_d}{0.8} \right)}{T_d}$

is the required power output. $P_e$ and $P_d$ are the satellite requirements during eclipse and daylight (here $P_e = P_d = P_{tot}$ ). $T_e$ and $T_d$ are the time per orbit spent in eclipse and daylight.

Finally, the inertia can be derived as follows,

$I_{min} = \min(I_{totX},I_{totY},I_{totZ})$

$I_{max} = \max(I_{totX},I_{totY},I_{totZ})$

with $I_{tot} = I_{sa} + I_{body}$ the total moment of inertia in the three dimensions, that depends on,

$I_{saX} = m_{sa} \left( \frac{1}{12} (L^2+t^2) + (D+\frac{L}{2})^2 \right)$

$I_{saY} = m_{sa} (t^2+W^2)$

$I_{saZ} = m_{sa} \left( \frac{1}{12} (L^2+W^2) + (D+\frac{L}{2})^2 \right)$

with $L = \sqrt{\frac{A_{sa} r_{tw}}{n_{sa}}}$ the length of the solar array, $W = \sqrt{\frac{A_{sa}}{r_{tw}n_{sa}}}$ the width of the solar array, and $m_{sa} = 2\rho_{sa}L W t$ the mass of the solar array.

Two tunings parameters are present :

$tolFPI$ : the tolerance on the fixed point iteration algorithm used in the multidisciplinary analysis, 1e-3
$maxFPIIter$ : the maximum number of iterations of the fixed point iteration algorithm used in the multidisciplinary analysis, 50

References¶

Load the use case¶

We can load this model from the use cases module as follows :

>>> from openturns.usecases import fireSatellite_function
>>> m = fireSatellite_function.FireSatelliteModel()
>>> # Load the Fire satellite use case (with 3 outputs: total torque, total power and solar array area)
>>> model = m.model()
>>> # Load the Fire satellite use case with total torque as output
>>> modelTotalTorque = m.modelTotalTorque()
>>> # Load the Fire satellite use case with total power as output
>>> modelTotalPower = m.modelTotalPower()
>>> # Load the Fire satellite use case with solar array area as output
>>> modelSolarArrayArea = m.modelSolarArrayArea()

API documentation¶

class FireSatelliteModel

Data class for the Fire Satellite.

Examples

>>> from openturns.usecases import fireSatellitefunction
>>> # Load the FireSatellite model model
>>> m = fireSatellitefunction.FireSatelliteModel()

Attributes:

dimDimension of the problem: dim = 9
HAltitude (m), TruncatedNormal distribution: First marginal, ot.TruncatedNormal(18e6,1e6,18e6-3e6,18e6+3e6)
PotherPower other than ACS (W), TruncatedNormal distribution: Second marginal, ot.TruncatedNormal(1000.0,50.0,1000.0-150.0,1000.0+150.0)
FsAverage solar flux (W/m^2), TruncatedNormal distribution: Third marginal, ot.TruncatedNormal(1400.0,20.0,1400.0-60.0,1400.0+60.0)
thetaDeviation of moment axis (deg), TruncatedNormal distribution: Fourth marginal, ot.TruncatedNormal(15.0,1.0,15.0-3.0,15.0+3.0)
LspMoment arm for radiation torque (m), TruncatedNormal distribution: Fifth marginal, ot.TruncatedNormal(2.0,0.4,2.0-1.2,2.0+1.2)
qReflectance factor (-), TruncatedNormal distribution: Sixth marginal, ot.TruncatedNormal(0.5,0.1,0.5-0.3,0.5+0.3)
RDResidual dipole of spacecraft (A.m^2), TruncatedNormal distribution: Seventh marginal, ot.TruncatedNormal(5.0,1.0,5.0-3.0,5.0+3.0)
LalphaMoment arm for aerodynamic torque (m), TruncatedNormal distribution: Eighth marginal, ot.TruncatedNormal(2.0,0.4,2.0-1.2,2.0+1.2)
CdDrag coefficient (-), TruncatedNormal distribution: Nineth marginal, ot.TruncatedNormal(1.0,0.3,1.0-0.9,1.0+0.9)
distributionXJointDistribution: The joint distribution of the input parameters.
modelPythonFunction: The Fire Satellite model with H, Pother, Fs, theta, Lsp, q, RD, Lalpha and Cd as variables. This function retrieves three outputs : the total torque, the total power and the area of solar array
modelTotalTorquePythonFunction: The Fire Satellite model retrieving only the Total Torque as output, with H, Pother, Fs, theta, Lsp, q, RD, Lalpha and Cd as variables.
modelTotalPowerPythonFunction: The Fire Satellite model retrieving only the Total Power as output, with H, Pother, Fs, theta, Lsp, q, RD, Lalpha and Cd as variables. This function retrieves three outputs : the total torque, the total power and the area of solar array
modelSolarArrayAreaPythonFunction: The Fire Satellite model retrieving only the Solar Array Area as output, with H, Pother, Fs, theta, Lsp, q, RD, Lalpha and Cd as variables. This function retrieves three outputs : the total torque, the total power and the area of solar array
cSpeed of light, constant: c = 2.9979e8 m/s
omega_maxMaximum rotational velocity of reaction wheel, constant: omega_max = 6000 rpm
nNumber of reaction wheels that could be active, constant: n = 3
delta_theta_slewSlewing time period, constant: delta_theta_slew = 760 s
AsArea reflecting radiation, constant: As = 13.85 m^2
iSun incidence angle, constant: i = 0 deg
MMagnetic moment of earth, constant: M = 7.96e15 A.m^2
rhoAtmospheric density, constant: rho = 5.1480e-11 kg/m^3
ACross-sectional in flight direction, constant: A = 13.85 m^2
PholdHolding power, constant: Phold = 20 W
muEarth gravity constant: mu = 398600.4418e9 m^3/s^2
IdInherent degradation of array, constant: Id = 0.77
tThickness of solar panels, constant: t = 0.005 m
n_saNumber of solar arrays, constant: n_sa = 3
epsilon_degDegradation in power production capability, constant: epsilon_deg = 0.0375 percent per year
LTLifetime of spacecraft, constant: LT = 15 years
r_lwLength to width ratio of solar array, constant: r_lw = 3
DDistance between panels, constant: D = 2 m
I_bodyXInertia of body, X axis, constant: I_bodyX = 6200 kg.m^2
I_bodyYInertia of body, Y axis, constant: I_bodyY = 6200 kg.m^2
I_bodyZInertia of body, Z axis, constant: I_bodyZ = 4700 kg.m^2
rho_saAverage mass density to arrays, constant: rho_sa = 700 kg.m^3
etaPower efficiency, constant: eta = 0.22
phi_targetTarget diameter, constant: phi_target = 235000 m
REEarth radius, constant: RE = 6378140 m
tolFPITolerance on Fixed Point Iteration used in the multidisciplinary analysis: tolFPI = 1e-3 (deterministic)
maxFPIIterMaximum number of iterations of Fixed Point Iteration used in the multidisciplinary analysis: maxFPIIter = 50 (deterministic)

Methods

`attitudeControl`(inputs)	Function computing the attitude and control discipline outputs to retrieve the power of ACS and total torque
`multidisciplinaryAnalysis`(x)	Function computing the multidisciplinary analysis to retrieve the total torque, the total power and the area of solar array
`orbit`(inputs)	Function computing the orbit discipline outputs and retrieve the slewing angle, the velocity, the orbit duration and the eclipse duration
`power`(inputs)	Function computing the power discipline outputs to retrieve the inertia, the total power and the area of solar array

attitudeControl(inputs)

Function computing the attitude and control discipline outputs to retrieve the power of ACS and total torque

Inputs:: dictionary of inputs of the Attitude and Control discipline

multidisciplinaryAnalysis(x)

Function computing the multidisciplinary analysis to retrieve the total torque, the total power and the area of solar array

X:: list of inputs

orbit(inputs)

Function computing the orbit discipline outputs and retrieve the slewing angle, the velocity, the orbit duration and the eclipse duration

Inputs:: dictionary of inputs of the Orbit discipline

power(inputs)

Function computing the power discipline outputs to retrieve the inertia, the total power and the area of solar array

Inputs:: dictionary of inputs of the Power discipline

Examples based on this use case¶

Example of multi output Kriging on the fire satellite model

OpenTURNS

An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

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The Fire Satellite model¶

References¶

Load the use case¶

API documentation¶

Examples based on this use case¶