2.2.1. Payload Subsystem
The payload subsystem model can be built with two granularities: ideal optical visible granularity and resolution-constrained granularity.
The former is characterized by the angle of visibility, which is determined by the target and the light shaft and can be calculated based on the location of the camera and the angle of connection between the two. If the angle is smaller than the corner of the field, it is considered to be optical visibility, otherwise, it is not visible. The latter granularity is constrained by the resolution of the payload and is determined by the smallest resolvable element in the image.
where
is the angle of the target and the light shaft and
is the visual angle.
Resolution-constrained granularity needs to consider data generation, data compression, and data storage of the payload camera. The rectangular field of view generated by the optical camera is shown in
Figure 2. S is a satellite camera, and C1C2C3C4 is the rectangular field of view.
Based on the camera’s field of view, the positions of the four points in a rectangle can be calculated. Images can be converted to pixel information and data after the fit resolution is selected. If the pixel data is empty, the amount of data generated is constant, otherwise, the amount of data generated is as follows,
where
is the resolution of the rectangle image’s long side,
is the resolution of the rectangle image’s short side,
is image depth,
is the data volume of the picture, the unit is the byte.
If the sampling frequency is
, the total data volume generated in the period
is
Data are stored by compression; compressibility is
and the compressed data volume is
When data are transmitted to the Earth station, link transmission must be calculated. This calculation process is managed by the track telemetry and control subsystem and is shown in the telemetry and control subsystem.
2.2.2. Attitude and Orbit Control Subsystem
The attitude and orbit control subsystem model can be described as the payload and orbital kinematics constraints granularity, capability constraints of attitude and orbit control subsystem granularity, and capability constraints of attitude and orbit control devices granularity.
The payload and orbital kinematics constraints granularity is a mathematical representation that accounts for the changes in orbit and attitude kinematics parameters over time, without taking into consideration the control process. Orbit is calculated as an elliptical orbit, and the kinematics formula is
The first line in the formula represents the elliptical track motion equation, E is the eccentric anomaly, M is the mean anomaly, e is eccentricity, n is the average orbital angular velocity, and is perigee time.
Setting attitude mode as gaze mode, the expected attitude angle of the satellite can be calculated based on the projection
of the target.
where
,
, and
are roll angle, pitch angle, and yaw angle, where
is the expected yaw angle.
Capability constraints of the attitude and orbit control subsystem granularity require the closed-loop control process of sensors, controllers, and actuators, as well as the consideration of the influence of force and torque on the attitude and orbit control state parameters.
Sensor model [
32] is
where
is measured value,
is real value,
is measured error. The measured error can be calculated using a normal distribution.
is the theoretical model of the sensor, and
is the error model of the sensor.
is install information of sensor,
is a celestial state, i.e., solar vector, Earth vector.
The output command of the satellite controller [
32]
is
where
is a controller algorithm including an exclusion algorithm, filtering algorithm, attitude determination algorithm, and control algorithm.
is the state parameters of the satellite controller,
is the expected satellite state, and
is related to the measurement parameters of the sensor and telecommand,
where
is telecommand.
The expected satellite state includes default parameters for the controller and parameters set through ground telecommand,
is a relation function,
where
is the controller default parameters.
The real output value of the actuator [
32] is as follows
where
is the ideal output value of the actuator, and
is the output error of the actuator. The output value of the actuator
includes force, torque, and angular momentum.
is the theoretical model of the actuator, and
is the error model of the actuator.
is install information of actuator.
For fixed actuators, the install parameter
is invariable, only related to the initial install information
.
For moving actuators, the install parameter is related to the initial state and motion parameter.
where
is the install parameter of the moving actuator,
is the motion parameter of the moving actuator,
is the kinematics rule of the moving device, and
is the initial install information of the moving device. The changes in motion parameters are related to the initial state, control commands, or telecommand
Moving devices of satellites include solar arrays, antenna, etc. For solar arrays, motion parameters mainly include the speed of the solar array and the rotation angle of the solar array . The install parameter mainly includes the normal vector of the solar array . For the antenna, the motion parameter mainly includes the rotation rate of the antenna , , and the rotation angle of the antenna , . The install parameter is the normal vector of the antenna .
The dynamic equation is as follows
where
is environmental force,
is control force.
is environmental torque,
is control torque,
is the angular momentum of the satellite.
The capability constraints of attitude and orbit control devices granularity further considers the coupling effect between each device and the surrounding environment based on the previous granularity.
The error model of the sensor and actuator is
where
and
are coupling state variables of the sensor and actuator. There are mainly two types of state variables: one is transient state variables, which can be updated immediately after time, such as voltage and switch state. The other one is the steady-state state variable, which changes relatively slowly over time.
Taking a laser gyro as an example, considering the coupling with thermal field and radiation, the angular velocity measurement value
is
where
is the wavelength,
is the normalized refractive index of the optical path,
,
are area and perimeter enclosed by a closed optical path,
is a measurement of the frequency difference between the front and back beams of light,
where
is the real value of frequency difference,
is zero bias error,
is random walk error, and
is an error caused by the accumulation of radiation fluence. Temperature couples between
and thermal field calculation,
coupled with particle radiation. The error model is as follows
where
,
,
are zero bias compensation coefficient obtained by fitting measurement data,
is the temperature of a laser gyro,
is lock zone threshold,
is peak jitter rate,
is laser gyro scale factor,
is radiation flux along satellite orbit,
is the relationship function between error and radiation flux.
Taking the wheel as an example, considering the coupling between friction coefficient and thermal field, the torque output by the wheel in the body coordinates is
where
is output torque of the motor,
is bearing static friction torque,
is the frictional coefficient,
is the speed of the wheel, and
is the unit installation vector of the wheel. The output torque is as follows,
where
is torque voltage ratio coefficient,
is the input voltage.
Temperature coupling between
and thermal field calculation. The model is as follows
where
is the density of lube,
is the amount of lube,
is the specific heat of lube,
is the difference between temperature and nominal temperature of the wheel, and
is the bearing diameter of the wheel.
Taking into account the friction factor of bearings, the angular acceleration of the wheel is
where
is the inertia of the wheel.
Device-level performance indicators must not only account for the impact of inter-subsystem interactions on device performance but also consider the varying degrees of impact that these interactions have on different parts of the device. For instance, the measurement accuracy of gyroscopes is affected by shaft temperature, with higher temperatures leading to greater errors. Therefore, it is important to consider the calculation of the bearing temperature field. However, the temperature of other parts of the satellite has a smaller impact, and therefore a multi-scale approach with local refinement and overall coarseness should be adopted when dividing the mesh.
2.2.3. Power Subsystem
The power subsystem is characterized by a set of parameters including the output of the solar array, load power consumption, battery charging and discharging, and remaining capacity. The calculation model of each parameter is as follows
(1) Output model of the solar array
The output model of a solar array can be modeled with constant output power granularity and considering solar array state granularity.
Constant power granularity refers to the fact that the solar array generates zero electricity in the shadow area and the output power is calculated at a constant value in the illumination area.
The considering solar array state granularity refers to the relationship between the output power of the solar array and the satellite’s state, taking into account the material characteristics of the solar array and the incidence angle of solar optics.
where
is shadow area identification,
is the power generated by sunlight,
is an area of the solar array,
is the power temperature coefficient of the solar array,
is other parameter,
is the normal vector of the solar array and included angle of solar vector,
is difference between working temperature and standard temperature.
is not only related to the position and attitude of the satellite but also to the relative angle
of the sail relative to the body.
(2) Load power consumption
The load power can be built with three granularities: constant load granularity, considering device switch granularity, and considering device state granularity.
The power consumption in the constant load granularity is calculated using fixed values, which can be either the rated power or the average power consumption over a specified period.
The considering device switch granularity involves the real-time statistical analysis of the switch state of each device, as well as the power consumption of each device, which is constant and remains the same over time.
where
is the total number of electrical devices,
is the switch state of the device
, power on is 1, power off is 0 and
is the power of the device
.
The considered device state granularity refers to the relationship between the actual power consumption and the device’s working state. For example, this includes the power consumption and speed of wheels. The formula is as follows
where
is the inertia of the wheel, and
is the angular velocity of the wheel,
is the angular acceleration of the wheel.
(3) Battery charging and discharging
The battery charging and discharging model includes ideal battery granularity and nonideal battery granularity.
In the ideal battery granularity model, when the power generated by the solar array exceeds the electrical power of the equipment and the battery is not fully charged, the battery is charged at constant power, with any excess electricity dissipated through the charging regulator. Conversely, when the Earth shadow or power supply of the solar array is insufficient, the battery is discharged at constant power.
The consumption power of the charging regulator is as follows
where
is the maximum charging power of the battery.
where
is load power,
is minimum battery capacity, and
is maximum battery capacity.
Battery remaining capacity is as follows
where
is the simulation step.
The power relationship in nonideal battery granularity is unaltered, but varying charging and discharging coefficients must be taken into account when calculating power consumption.
where
is charging coefficients,
is discharging coefficients.
2.2.4. Thermal Control Subsystem
The thermal balance expression of satellites in space is
where
is the direct solar radiation heat absorbed by satellites,
is the Earth infrared radiation heat absorbed by satellites,
is the infrared radiation heat absorbed by the satellite,
is space background heating amount,
is the heat generated by the satellite,
is a change in internal energy of satellites,
is the heat emitted by satellites into space. Due to the low temperature of the spatial background and the small heating heat, the heating amount of the spatial background
can be ignored.
The temperature field of the satellite is solved using the finite element method, and the thermal balance equation of each finite element node is [
33]
where
is node temperature,
is node mass,
is node specific heat capacity,
is solar radiation heat,
is Earth radiation heat,
is Earth reflection radiation heat,
is heat radiation,
is heat conduction,
is internal thermal power,
is external surface heat dissipation,
is internal surface heat dissipation,
is node initial temperature.
The thermal control subsystem model includes three parts: finite element partitioning calculation, external heat flow calculation, and internal heat flow calculation.
(1) Finite element partition calculation model
The heat balance equation for each node remains constant regardless of the specific research object, but the number of nodes distributed throughout the satellite may vary depending on the research object. The thermal balance equations of each node remain unchanged, but the division of nodes in different parts of satellites can differ based on the research object. Taking
Figure 3 as an example, the satellite structure mainly includes the main body, solar array, and various devices. When temperature is not the primary factor affecting certain indicators, the temperature of each part can be considered at a coarser level of detail. Typically, all structures and devices can be divided into meshes with the same level of accuracy. However, when calculating indicators that involve the temperature characteristics of a specific part, a finer mesh should be used for that part, while other parts can still use a coarser granularity to increase computational efficiency. All meshing is performed through Ansys software.
(2) External heat flow calculation [
33]
The external heat flow calculation model can be described as constant external heat flow granularity and node state external heat flow granularity.
The constant external heat flow granularity does not account for local temperature differences, and the radiation parameters are calculated using average values in the local film area and the light area, respectively.
The node state external heat flow granularity is a model that considers the orbit, attitude, and other states that need to be accounted for in the calculation of external heat flow, as well as material attributes.
Node
solar radiation heat is
where
is solar absorption rate on the outer surface of satellites,
is solar constant,
is the surface area of node
, and
is the direct solar radiation angle coefficient of the node
.
where
is the dot product of the vector
and vector
,
is the shadow zone identification,
is a solar vector, and
is an illuminated surface normal vector.
Earth radiation heat of node
is as follows
where
is the outer surface emission rate of the node
,
is the average of surface infrared radiation density, and
is the Earth’s infrared radiation angle coefficient of the node
. Define
, where
is satellite orbital altitude,
is the average radius of the Earth. When 0 ≤
≤
,
. When
≤
≤ π,
. When
≤
≤
Earth reflection radiation heat of node
is
where
is the average reflection density of the Earth’s surface facing the solar radiation, and
is the Earth’s infrared radiation factor of the node
.
where
is the Earth vector.
(3) Internal heat flow calculation [
33]
The internal heat flow calculation model can be described as constant internal heat flow granularity and node state internal heat flow granularity.
The constant internal heat flow granularity is consistent with the constant external heat flow granularity, and all devices operate at their rated thermal power.
The node state internal heat flow granularity is consistent with node state external heat flow granularity.
Thermal radiation from other thermal nodes to node
is
where
is the absorption factor of node
to node
,
is the emissivity of node
, and
is Boltzmann constant.
Other thermal nodes to the thermal conduction of node
are
where
is the conduction factor between node
and node
.
The internal thermal power of the node
is equal to the sum of the thermal power of all devices in the node
range
where
is the total number of all devices belonging to the node
,
is the thermal power of the component
where
is the work status of devices, The relationship function between
and thermal power is
.
External surface heat dissipation of node
is
Internal surface heat dissipation of the node
is as follows
where
is the internal surface emissivity of the node
.
2.2.5. Propulsion Subsystem
The model granularity of the propulsion subsystem includes constant granularity, constant pressure model granularity, and depressurization model granularity. The core state parameters include specific impulse and fuel consumption rates.
The relationship between the mass of the remaining propellant in the storage tank and time is
where
is initial propellant mass in the tank,
is mass consumption of propellant,
is propellant consumption rate,
is ignition command.
Constant granularity refers to the calculation of specific impulse and fuel consumption rates based on constant values.
The constant pressure model granularity is concerned with the relationship between specific impulse, fuel quality, and the working status of temperature, pressure, and valve switching duration.
where
is flow coefficient,
is the total area of the thruster injection hole,
is the gravitational acceleration, and
is the relative density of the propellant liquid.
The storage tank maintains constant gas pressure, that is
. The gas mass in the tank can be obtained from the ideal gas equation.
where
is the average molar mass of gas, and
is the universal gas constant.
In the depressurization model granularity, the working status of the specific impulse, fuel quality consumption and temperature, pressure, and valve switch duration is related. The mass of gas in the storage tank is constant, that is
. The gas pressure in the storage tank can be calculated through the ideal gas equation
2.2.6. TTC Subsystem
TTC subsystem includes communication visible constraint granularity, Earth occlusion granularity, and SNR transmission granularity.
The communication visible constraint granularity focuses on the connection state of the satellite and earth radar station and does not consider the specific transmission process.
The Earth occlusion granularity focuses on geometric constraints. When the satellite and the Earth station are not obstructed by the Earth, their transmission ability is considered connecting, otherwise disconnected. The elevation between the satellite and the Earth station can be calculated. The elevation angle is the angle formed by the horizon horizontal line where the ground station is located at the center line of the antenna. When the actual elevation angle
is greater than the minimum elevation angle
, it is determined that the two can communicate. Assuming that the position vector of satellites and Earth stations is
and
, the included angle
and
is
.
In the SNR transmission granularity, the transmission capacity is influenced by factors such as the relative position of the receiver and transmitter, performance, data compression, and transmission loss, the basic transfer equation is as follows
where
is the receiving antenna power,
is the transmitter power,
is the transmitter antenna gain,
is receive antenna gain,
is the path loss,
is other various losses.
The unit of operating frequency f is MHz, and The distance d between two interfaces is measured in kilometers.
mainly includes the influence of antenna pointing. Loss of transmitter antenna pointing is
, the loss of receiver antenna pointing
. The loss is related to the directional pattern of the antenna, A computational model for a point beam antenna is as follows
where
is the angle between the
Z-axis of the transmitting antenna instrument coordinate system and the vector between the transmitting node and the receiving node.
The current carrier-to-noise ratio from sender to receiver
is
when
, the sender from the receiver is normal, otherwise it will not be connected.