Derivation of a Sun-tracking law for payloads with pointing restrictions in the UPMSat-3 mission

Satellites dedicated to remote sensing either for the Earth or space, typically require their instruments to be pointing toward the location of their object of study. These satellites can face significant challenges as different components also have its own pointing requirements to be operative or work optimally. For instance, we have the case of solar panels for electrical power generation, where perfect pointing of the panels toward the Sun can affect negatively to payload operation. Common solutions for maximizing power generation include the usage of orientable solar panels. However, this approach increases the complexity of the satellite, raising the cost of the solar panels and their associated mechanism. In this study, we propose a new approach that enables high-consuming remote sensing payloads to operate for extended periods without using orientable solar panels. To ensure maximum power generation without compromising the satellite’s pointing constraints, an optimal tracking law is derived. This law maximizes the projected solar array area at each instant, resulting in maximum electrical power generation. The proposed method is validated against an actual mission scenario. This work offers significant benefits for satellite operators, reducing the need for costly orientable solar panels and enhancing the overall efficiency of satellite missions.


Introduction
All satellites require energy to operate their subsystems and payloads.While there are various energy sources available, such as nuclear or chemical energy, the majority of Earth orbiting satellites primarily rely on solar energy.In small satellites, due to their limited size, the power generated by panels mounted on the satellite faces may be insufficient to meet the power requirements.One of the most commonly employed methods to increase the electrical power generated by satellites is incorporating deployable solar panels, which ultimately increase the total surface area of the solar array.However, the integration of deployable structures has not only financial implications, but also various structural and operational considerations, especially when dealing with large deployable panels.Consequently, it becomes crucial to employ strategies that encompass both design and operational modifications, specifically tailored to the unique characteristics of each satellite, as the solar array area can only be expanded up to a certain point before encountering increased complexity and design-related issues.
In addition to deployable solar panels, another common strategy to maximize power generation is to point the solar panels toward the Sun, since photovoltaic cells generate power proportionally to the incoming solar radiation.By continuously adjusting the orientation of the panels, satellites can significantly reduce the required panel area, leading to cost savings and mitigating potential structural and operational challenges.Typically, solar panels are oriented in a way that aligns their normal direction with the direction of the Sun.This process, known as Sun-tracking, has been extensively studied in the literature for ground-mounted solar panels [1][2][3].However, compared to ground-mounted solar panels, there are many more factors that are relevant for optimizing satellite's power generation.Among them, the satellite's orbit, its position within the orbit, and its attitude stand out.
A satellite equipped with a 3-axis stabilization control can rotate freely to achieve a favorable orientation of the solar panels toward the Sun.However, during normal operations, the satellite often needs to point its instruments, sensors, or antennas toward a specific geographical point or space location.In these cases, the maximum available solar power cannot always be extracted, as the perfect alignment between the solar array and the Sun is rarely compatible with the pointing constraints.This leads to a trade-off between satellite operations, performance and power generation, resulting in a non-trivial optimization problem.To address this challenge, two primary solutions have been explored: the use of orientable panels or the implementation of optimal Sun-tracking laws that account for pointing restrictions.Orientable solar panels offer significant advantages over the non-orientable ones, as they can rotate around one axis to track the Sun.While this simplifies the guidance law, its use also increases the overall complexity, cost and risk of system malfunction.Therefore, for low-cost missions the use of Sun-tracking laws is preferred over orientable panels to avoid these drawbacks.
The literature lacks an extensive exploration of the problem of optimizing power production in satellites while meeting pointing constraints.Some authors have focused on nadir-pointing satellites [4,5] and have developed mathematical models that consider the satellite's rotation around the nadir direction and, if applicable, the orientation of solar panels.Other strategies propose to determine which solar panel could generate more power and point it toward the Sun, without disturbing the nadir-pointing direction [6].More advanced studies also consider the dynamics of the vehicle.An example can be found in reference [7] which develops a technique to prevent sudden uncontrolled maneuvers while keeping the nadir pointing direction.Other authors such as Kristiansen and Gravdahl [8] have focused on the net power gain of solar panels.In their work, they propose a balance between the power generated by the panels and the additional energy consumption by the attitude control subsystem required to maintain the optimal Sun-tracking attitude.An important drawback of this approach is that the optimization process involves considering the total power generated over an entire orbit, making it unsuitable for real-time systems unless the guidance law was pre-calculated and transmitted to the satellite in advance.
In this paper, a pointing law is derived to achieve the optimal satellite attitude to maximize solar power production while adhering to payload pointing constraints.The goal is to maximize the projected area of the satellite perpendicular to the Sun direction, which is proportional to the generated power, at each instant in the orbit.The proposed method is applicable to satellites with non-orientable solar panels of any shape.This contribution offers a valuable tool for space mission planning and optimization, particularly in scenarios where power generation is critical.

Mathematical formulation of the problem
The power generated by a solar panel can be approximated by the following expression: where A is the solar panel area, f pf refers to the packing factor of the solar panel, which indicates the percentage of the solar panel area that is covered by solar cells, η denotes the efficiency conversion between solar energy to electrical energy, G is the irradiance of the Sun, and θ is the angle between the normal of the solar panel and the Sun direction.The cosine of this angle can also be expressed as the scalar product of these two directions: where Â denotes the normalized area vector of the panel and r⊙ is the normalized Sun direction vector.According to equation ( 1), one way to increase the power generation of the solar panel is aligning its normal direction toward the Sun (i.e., maximizing cos θ).
The optimization problem to maximize the generated power while adhering to the pointing constraints, can be formulated as follows: where the inequality represents the pointing constraint.In this expression, the normalized vector of the restricted direction is denoted by ẑ1 , ẑ denotes the normalized direction of the satellite that should point towards the target direction, and α is the maximum allowable deviation of this axis from the target direction.To tackle this optimization problem, it is convenient to refer the area Â to the body axes and the Sun vector in the general reference frame.These vectors are expressed in two different sets of coordinate systems, but the result of the dot product only makes sense if both vectors are expressed in the same coordinate system.Therefore, it is necessary to introduce the transformation matrix C br between these two sets of axes to express one of the vectors in the other coordinate system.Thus, the optimization problem is formulated as follows: where [•] b , [•] r denotes a vector expressed with respect to the body and the general reference frame respectively.

Analysis of the optimization problem
The problem in equation ( 4) constitutes an optimization problem with inequality constraints.The solution of this problem can lie in the interior of the region defined by the constraint or on the surface of the restriction cone.Therefore, the first step is the computation of the maximum value without considering any constraints.Second, this solution is evaluated in order to check whether the constraint is satisfied.In the case that this solution does not satisfy the restriction, it is discarded.Then, it would be necessary to solve the constrained optimization problem.This last solution will be located on the cone surface.To achieve this, a common strategy is to use the Lagrange multiplier method.By applying this method, the optimization problem is reduced to maximizing the following functional: where the restriction of the problem is introduced in the cost function as an equality restriction The complexity of finding a solution to this problem depends on the variables chosen to express the transformation matrix between the coordinate systems.For example, one can express the solution in terms of the components of the direction cosine matrix.However, this approach requires solving a system of 12 equations: 9 equations derived from the nine values of the direction cosine matrix, 1 equation representing the pointing constraint, and 2 additional equations from the definition of an orthogonal matrix.Specifically, these additional equations are C br • C br T = I and |C br | = 1.Alternatively, other formulations, such as a predefined sequence of Euler angles, reduce the size of the system of equations to four: one equation for each Euler angle and one for the pointing restriction.However, the complexity of the expression increases significantly as sinusoidal functions emerge in every term of the equation.Possibly, the most balanced representation arises from the use of quaternions, which yields a system of 6 polynomial equations: 4 equations derived from the four components of a quaternion, 1 equation from the pointing restriction, and 1 equation from an additional restriction that must satisfy the resulting quaternion, q T q = 1, which is analogous to the properties of the rotation matrices.
In order to understand the solutions of the optimization problem, an analytical solution for one of the simplest situations is analyzed here.In this scenario, we are going to assume the following solar array area vector Â = [0, 0, 1], with a pointing restriction to the nadir direction.Also, the normal of the solar panel and the restriction axis are aligned.These considerations also apply to the university satellite UPMSat-3, which is being developed by the IDR/UPM Institute and the STRAST group.Taking into account these considerations, the following system of equations is derived: Here, as the restriction axis is aligned with the normal of the solar panel, any rotation around this axis does not change the angle between the Sun direction and the solar panel.Therefore, the rotation around this axis can be considered as a parameter, having no influence on the optimal solution.In terms of the quaternion, since the restriction axis is aligned with the z-axis, this EASN-2023 Journal of Physics: Conference Series 2716 (2024) 012098 means that q 3 is the aforementioned parameter, by selecting its value to zero, q 3 = 0, the system of equations can be simplified.Now, from equation (7c) the following relation is obtained: By replacing q 2 from expression (8) into equation (7e), the following analytical expression for q 1 is obtained: Note that there are two solutions for q 1 , but since the positive and negative quaternions represent the same rotation, the sign can be freely chosen.In this case, we adopt the positive value.Substituting the expression for q 1 into equation ( 8), the expression for q 2 is derived.Finally, q 4 is obtained from expression (7f), resulting in the expressions listed in Table 1.
Table 1.Solutions for the constrained problem.
Solution 1 Solution 2 There are two possible solutions for the optimization problem, one choosing the positive value of q 4 and the other employing its negative value.One of these results will correspond to the maximum of the cost function, while the other will represent the minimum.
Since the optimization problem involves an inequality constraint, it is also necessary to solve the unconstrained problem.Therefore, now the problem without the pointing restriction is solved.For this undertaking, λ is set to zero and equation (7e) is removed from the system of equations.
The process of obtaining a solution is similar to the previous case, producing the same expression as equation.( 8) from formula (7c).Now, using equation (7a), an expression for q 4 in terms of q 1 can be derived: Substituting q 2 and q 4 in terms of q 1 in expression (7d) it is possible to obtain µ: Once more, two solutions arise, one corresponding to the maximum and the other to the minimum of the cost function, depending on the selected expression of µ.Finally, by substituting µ into equation (10) and using expression (7f), an analytical expression for q 1 can be derived, Table 2. Solutions for the unconstrained problem.Solution 1 Solution 2 thereby solving the unconstrained optimization problem.The two possible solutions are listed in Table 2.
It should remember that it is necessary to evaluate the cost function for each solution obtained in Tables 1 and 2 and check if it satisfies the problem constraints in order to identify the optimal solution of this problem.
Typically, in these optimization problems, it is interesting to examine the structure of the obtained solution as relevant information can be inferred.Quaternions offer a convenient mathematical representation for spatial orientations and rotations.Furthermore, a quaternion can be parameterized in terms of a rotation axis ⃗ e and the rotation angle θ as: Interestingly, the obtained solutions resemble this parameterization, providing a geometrical interpretation of the solutions.In both solutions q 1 is associated with the term and q 3 = 0. Meaning that the in both solutions the rotation axis is the same: The rotation axis direction is perpendicular to the projection of the Sun direction onto the xy-plane (the plane perpendicular to the restriction axis).
The differences arise in the rotation angle.In the constrained problem, the rotation angle is always α.However, one possible solution rotates in the direction of the Sun (maximum), while in the other solution, it rotates the same angle but in the opposite direction (minimum).As for the unconstrained solution, since r ⊙,z directly represents the cosine of the angle between the Sun direction and the z-axis, the rotation angle corresponds to aligning the solar panel with the Sun direction.Once again, two solutions are possible: perfect alignment of the solar panel with the Sun (maximum) or alignment in the opposite direction (minimum).
By analyzing this, we can gain insight into the behavior of the actual optimization process.Furthermore, understanding the result of this simple case demonstrates that an analytical solution to the problem can be derived without relying on an optimization algorithm in more general scenarios.This could be achieved by determining the corresponding axis of rotation and the required rotation angle.

Results
To validate this methodology, a case study from the development of the UPMSat-3 satellite has been carried out.The orbit of the UPMSat-3 is Sun-synchronous, with Local Time at the Descending Node (LTDN) of 10:30 am at an altitude of 500 km.The nominal attitude of the satellite requires the −z face, where a photonic radiometer is installed, pointing to Nadir.A maximum deviation of 20 deg from the nadir position is allowed for the Sun-tracking law.The solar array is composed by three solar panels (one fixed and two deployable) pointing to the +z direction of the satellite.The time evolution of the dimensionless electrical power generated for both the nadir and the proposed optimal Sun-tracking law is shown in Fig. 1.The reference power corresponds to the one obtained in the unconstrained Sun-tracking strategy at each instant (i.e., the maximum theoretical power).According to the results, the power generated with the optimal Sun-tracking law is, as expected, higher than the nadir fixed attitude.Additionally, it should be noted that a perfect alignment between the solar array and the Sun is achieved when the dimensionless power profile reaches the value of 1.Despite this situation appearing only once and lasting for a short amount of time, it is clear that the proposed optimal Sun-tracking law is a major improvement to increase the electrical power generation in satellites with pointing restrictions.In terms of mean power generation, an increase in approximately 41% with respect to the fixed nadir attitude is reported.Of course, this result changes depending on the maximum allowed deviation from the target direction.If the same simulation is done for a maximum deviation angle of 5 and 10 deg, the increase in the mean power generation corresponds to 10.2 % and 20.4 % respectively, which is less than the latter case but still a significant increase in the power generation.An important advantage of increasing power generation along the orbit is that it decreases the electrical current demanded from the battery, since the effective peak power to be supplied would be lower.This allows to extend the operational life of the power system and could potentially lead to reduction in the satellite mass by employing smaller batteries.Furthermore, the lesser current demanded improves the overall efficiency of the power system by reducing the conversion and energy losses due to the internal resistance of the battery during the charge and discharge cycles.

Conclusions
In this work, an optimal Sun-tracking law for a satellite with pointing constraint has been derived.This tracking law is the result of solving an optimization problem where the projected solar array area is maximized.A solution was derived analytically for a particular case, although a general solution is not yet available.This scenario, the same as in the UPMSat-3 mission, considers the normal vector of the solar array aligned with the direction of the satellite that has to fulfill the pointing requirement.The solution was analyzed to give a geometric interpretation of the quaternion components that were obtained.Finally, this law was applied for the UPMSat-3 mission and the effectiveness of the proposed Sun-tracking law was compared with pure nadir pointing.The results from this simulation showed that the optimal Sun-tracking law was able to generate more considerably power than the nadir pointing attitude, specifically an increase of 41% was observed for the maximum allowed deviation angle.

Figure 1 .
Figure1.Time evolution of the dimensionless electrical power generated using the nadir pointing strategy (gray) and the proposed optimal Sun-tracking law (black).