Use of thermal data to estimate satellite rotation rate in the UPMSat-2

Attitude determination is a fundamental task of the Attitude Determination and Control Subsystem (ADCS). It involves determining the spacecraft’s orientation with respect to a reference system and calculating its angular velocity, which is essential for understanding where the payload is pointing and the spacecraft’s stability. The determination of the attitude is obtained using sensors such as Sun sensors, Nadir sensors, or Star trackers. Regarding the spacecraft angular velocity, gyroscopes or MEMs can be used to measure the angular velocity of the satellite, but in their absence the velocity can be derived from the attitude data of the other sensors. However, this method requires a sampling rate to be twice the frequency of the movement, otherwise the angular velocity cannot be calculated with traditional methods. To address this issue, this study proposes a thermal analysis of the external temperatures of a rotating satellite to obtain its rotation rate. This method is especially useful for satellites with low or limited data sample rates. The proposed methodology is used in the UPMSat-2 case to demonstrate the functionality of its experimental ADCS.


Introduction
Attitude determination represents a fundamental task for most of the spacecrafts, as it is essential to know the spacecraft's orientation in order to control the pointing direction of payloads and antennas, increase the power generation of the satellite by pointing the solar panels towards the Sun, or provide additional data for sensors, cameras, or scientific payloads in order to interpret the received data.
Typically, attitude determination is achieved by utilizing sensors capable of computing reference directions, such as the direction of the Sun, nadir, or known stars.These reference directions, measured in the body frame of the satellite, are then compared with the modeled reference directions to determine the attitude of the spacecraft with respect to a reference coordinate system [9,1].Using the information provided by these sensors, the attitude of the satellite at the time of measurement can be obtained.However, when calculating the angular velocity of the spacecraft (in the absence of MEMs or gyros), consecutive measurements are required to derive its angular velocity.According to the Nyquist-Shannon theorem, to obtain the true value of the angular velocity, the sampling rate of the measurements should be at least twice the frequency of the spacecraft's movement.Failing to meet this condition will result in an apparent angular velocity instead of the true value, necessitating a non-traditional approach to calculate the angular velocity accurately.
Recently, a novel approach to spacecraft attitude determination has been proposed, which uses the temperature and absorbed heat fluxes of the external surfaces [5,6].This approach considers the existing correlation between thermal data and the orientation of the spacecraft.Other studies, such as [7], have also explored the use of thermal data to determine the attitude of satellites.Furthermore, Gadalla Mohamed et al. [3] demonstrated that the temperature difference between surfaces of a moving spacecraft can be significantly affected by its rotation, enabling the determination of the spacecraft's rotation rate.
The UPMSat-2 is a 50 kg university microsatellite that serves as a technological demonstrator.One of the payloads on board the UPMSat-2 is its magnetic control law [2].This control law, derived from the B-dot, enables the satellite to orient one of its faces perpendicular to the orbital plane while setting a desired rotation rate along that axis.To validate the performance of the control law, knowledge of the satellite's attitude and angular velocity is necessary.However, during the commissioning phase, the satellite's communication sample rate only allows for the reception of one data package per minute, coinciding with the expected angular velocity of the spacecraft.This limitation renders traditional methods ineffective in deriving the angular velocity of the spacecraft.Therefore, thermal data must be used to calculate the angular velocity of the UPMSat-2.

Methodology
In order to compute the angular velocity of the UPMSat-2, it is necessary to establish a relationship between the spacecraft's dynamics and the evolution of its temperature.First, a model that describes the dynamics of the spacecraft within the orbit must be developed.The results obtained from this model will be fed into a thermal model that will compute the temperature of external faces of the satellite.Thus, a comprehensive description of both the dynamic and thermal models is essential to solve this problem.This approach resembles the one proposed by Labibian et al. [6].However, due to the limited number of data packages received during each pass of the satellite above the ground station (only 6-10), the convergence of a Kalman filter is not guaranteed with such a small dataset.Therefore, an alternative approach is required.The spacecraft's attitude dynamics can be defined using the following equations: Equation ( 1) represents the Euler equation for rigid body dynamics, where ω denotes the angular velocity of the spacecraft, I is the inertia matrix of the spacecraft, m coil represents the control torque and m dist is the disturbance torque, which is often ignored except for the easily computable gravity torque.Equation (2) describes the change in the attitude of the satellite expressed in terms of the quaternion q.Since the objective of this work is to demonstrate the effectiveness of this control law on board the UPMSat-2, and considering that each pass typically lasts around 5-10 minutes, the impact of the control law on spacecraft angular velocity, once stabilized, is not significant within this time frame.This means that an initial value of the angular velocity can be set at the beginning of each pass and kept constant.
Regarding the thermal model, the evolution of the temperature for each node can be described by the following expression: On the left side of the equation, the symbol m i denotes the mass of the i th node, c i represents its thermal capacity, and T i corresponds to the temperature of the node.On the right side of the equation, Ẇdiss denotes the heat generated by the power dissipated by various components of the satellite.The term Qcond refers to the heat transmitted through conduction between different parts of the spacecraft, while Qrad represents the heat emitted by the spacecraft in the form of infrared radiation.Lastly, Q in represents the different thermal loads.In the near-Earth space environment, the term Q in can be further decomposed into the following terms: where Qi,sun represents the heat flux due to solar radiation, Qi,albedo is the heat flux from albedo radiation, Qi,IR accounts for the heat flux from the infrared emission of the Earth.The heat fluxes Qi,sun , Qi,albedo , Qi,IR , and Qrad involve the exchange of heat between the space environment and the satellite (see Figure 2).On the other hand, Qi,cond and Ẇdiss represent the heat fluxes exchanged between different components or parts of the satellite.The following paragraphs provide detailed descriptions of the models used for each of these heat sources.The heat absorbed by the satellite that comes from the Sun radiation can be expressed as follows: Qi,sun = where α i represents the absorptance of the considered node, A i is the area of the node, G denotes the Sun radiation (1360 W/m 2 , in near Earth space environment), and θ is the angle between the Sun direction and the normal to the surface.In the case that there are solar panels, it is important to account for the portion of energy that is converted into electrical power.This leads to the following equation: where η represents the conversion efficiency of the solar cells, and F pf is the solar panel package factor, which considers that not the entire surface area of the solar panel is capable of generating electrical power.The heat flux resulting from albedo radiation is the portion of solar radiation reflected by the Earth and absorbed by the spacecraft.Since albedo radiation originates from the Sun, the general expression for modeling the heat flux from albedo radiation shares similarities with the model for solar radiation.A simplified function to model the heat flux from albedo radiation can be represented as follows: where ρ p represents the reflectivity coefficient of the Earth surface, whose mean value increases with the latitude (ϕ), F i,p denotes the view factor of the surface under consideration with respect to the Earth and Φ is the angle between the Sun direction and the sub-satellite point.In a Sunsynchronous orbit, as is the case of the UPMSat-2, the factor cos Φ can be computed in the following way: where β, represents the angle between the ascending node of the spacecraft orbit and the right ascension angle of the Sun, δ is the declination of the Sun (which depends on the epoch), and ν is the true anomaly measured form the equator plane to the spacecraft position in the illuminated zone.It should be noted that in the case cos Φ ≤ 0 means that the satellite is above a part of the Earth's surface that is shadowed, which means that there is no albedo radiation.In such cases, cos Φ is set to 0. Another heat source that needs to be considered is the infrared radiation emitted from the Earth towards the spacecraft.The heat flux from this source can be described by the following equation: In the above equation, ε i represents the emissivity of the external surface of the node, while ε p is the emissivity of the planet.The symbol σ corresponds to the Stefan-Boltzmann constant (5.6703•10 −8 W/(m 2 K 4 ), and T p denotes the equivalent temperature of the planet as if it emitted as a black body (typically taken as T p =278 K for the Earth).The term Qrad represents the heat infrared radiation emitted by the spacecraft to the deep space and other bodies.All the radiative terms present in the model can be grouped into a single variable, Qi,∞ whose value can be obtained using the following equation: It should be mentioned that the infrared emission of the deep space to the spacecraft does not appear in this model, as its contribution is negligible compared to the other heat sources.
The heat exchange resulting from conduction occurs as a result of temperature gradients within different parts that are in contact.Heat exchange between two different nodes through conduction can be modeled as follows.
where k i,j represents the thermal conductivity coefficient, and L is the distance between the center of the nodes.The term is sometimes replaced by the variable G L .This new variable is particularly useful for simplified models as it represents a global value of conductivity EASN-2023 Journal of Physics: Conference Series 2716 (2024) 012099 between conductively coupled nodes.However, for more detailed models, the exact coupling points and properties need to be considered, leading to the creation of a more complex model.The view factor of the different external areas can be calculated analytically [8].Calculating the view factor of an external surface of a satellite resembles the case of a small planar surface tilted to a sphere of radius R, at a distance H with h = H/R; the tilt angle ψ is between the normal and the line of centers.Two different cases shall be studied.
If |ψ| ≤ arccos(1/h) (plane not cutting the sphere): If |ψ| ≥ arccos(1/h) (plane cutting the sphere), with x = √ h 2 − 1 and y = −x cot ψ In the above expressions, some terms that relate the heat exchange with the attitude of the spacecraft appear, specifically θ the angle of the Sun direction to the normal and ψ the angle between the normal direction of the node and the nadir direction.Calculating this angle requires the use of a transformation matrix that relates the reference system to the body coordinate system C br , as the normal direction of the node n and the nadir or Sun directions r ⊙ are expressed in different coordinate systems.The calculation can be performed using the following equation: whereC br (q) is expressed in terms of the quaternion as: 2 (q 1 q 2 + q 3 q 4 ) 2 (q 1 q 3 − q 2 q 4 ) 2 (q 1 q 2 − q 3 q 4 ) 1 − 2q 2  1 − 2q 2 3 2 (q 2 q 3 + q 1 q 4 ) 2 (q 1 q 3 + q 2 q 4 ) 2 (q 2 q 3 − q Once the thermal and dynamic models are established, the problem to solve becomes an optimization problem where the objective is to find the angular velocity that minimizes the following cost function: In the above expression, T m j,i is the modelled temperature of node i, T data j,i represents the actual temperature measured for that node, and N is the total number of data points, which is calculated as the number of nodes, N n , times the number of different data packages, N d .

Simulation, results and discussion
In order to study the expected thermal behaviour of the UPMSat-2 at different spin rates, a reduced thermal model of the satellite was made based on the detailed thermal model created by IDR/UPM in ESATAN-TMS [4].Although the detailed thermal model contains much more information, its complexity makes the calculation of the transient temperatures along the orbit unnecessarily complicated.Therefore, a reduced model will be used.It consists of 7 nodes, 6 of them representing each panel of the satellite.The last node represents the inner part (trays, payloads, etc.) and is conductively coupled to the external panels.The parameters of the reduced model include the thermal capacities, conductive couplings, emissivity, and absorptivity of the panels.A good initial estimation of the reduced model parameters can be obtained from the detailed model itself.In Table 1 the properties considered for the external faces are shown.
The Sun vector, orbital position, and nadir direction are required.These variables can be obtained from the TLE data of the satellite and the Sun ephemeris model.Also, an initial value of the attitude and the angular velocity direction is also required, but they can be estimated through the onboard magnetometers and Sun sensors.
Implementing this approach for every initial condition during each pass of the satellite above the ground station and for each spin rate would be inefficient.Therefore, conducting simulations to obtain lower and upper bounds for the optimal angular velocity is beneficial.are presented.From this simulation, it is evident that a slow spin rate of 0.01 rad/s cannot fit the flight data because of the amplitude of the oscillations.Furthermore, it is observed that the flight data exhibit small oscillations that cannot be accurately captured by high spin rates (0.2 rad/s or higher).These observations provide insight into setting lower and upper bounds for the rotation rates.If this analysis is done for other passes, the plots are similar and the same conclusions can be extracted.For spin rates in between, it is not easy to see directly which one fits better with the data, is in this range of value where the optimal value of the spin rate shall be found.In Figure 3 the value of the cost function is plotted, representing the temperature deviation between the flight data and the model as a function of the spin rate.As expected, when the spin rate increases, the error decreases, reaching a stationary value for high spin rates.In particular, three significant local minimum points can be identified in the graph, specifically at 0.052 rad/s, 0.10 rad/s and 0.21 rad/s.These values correspond to multiples of the most probable satellite spin rate.However, the global minimum, around 0.1 rad/s (1 rpm), is the most plausible spin rate for the UPMSat-2.The local minimum at 0.052 rad/s seems to only be caused by the fact that the simulated spin is half the spin rate.However, the differences between maximum and minimum in the model oscillations (see the dotted blue curve in Figure 3) are higher than those in the data, as it clearly states the value of the cost function.Something similar can be said about the 0.21 rad/s spin rate.These results show that an angular velocity speed in the vicinity of 0.10 rad/s is the most likely for the satellite and coincides with the theoretical angular velocity programmed in the control law.

Conclusions
In this work, the in-orbit performance of the UPMSat-2 attitude control law was studied.The control law studied governing the UPMSat-2 can stabilize an axis in an orientation almost perpendicular to the orbital plane and then control the angular velocity around this axis to the desired target velocity.In order to measure the performance of the control law, the angular velocity and attitude of the satellite must be calculated.
The analysis of the satellite data posed challenges due to its low sampling rate, which is approximately one data package per minute.This sampling rate is comparable to the expected period of the main motion of the satellite, which is approximately one revolution per minute.Consequently, a stroboscopic effect arises, rendering traditional analysis methods ineffective for data interpretation.
In this study, a thermal analysis was conducted to determine the actual angular velocity of the satellite.The thermal data analysis enabled the exclusion of rotation rates below 0.052 rad/s, as they did not align with the overall temperature trend observed in the satellite.Furthermore, the analysis revealed that a spin rate close to 0.1 rad/s exhibited the best agreement with the flight data.In particular, this finding coincides with the target angular velocity set for the control law.

Figure 1 .
Figure 1.Main heat sources in the space environment.

Figure 2 .
Figure 2. Model temperatures for different spin rates (lines) and flight temperature measurements (dots) for one pass.

Figure 3 .
Figure 3. Variation of the temperature deviation between the flight data and the model as a function of the spin rate.

Table 1 .
Thermal properties considered for the external nodes.Parameter ± X & ± Y faces +Z face − Z face