An autonomous yaw planning method based on yaw angle dawn-dusk value optimization calculation

Abstract

The application discloses an autonomous yaw planning method based on yaw angle dawn-dusk value optimization calculation, proposes a method for improving satellite heat dissipation conditions by rotating the satellite yaw angle, and designs a method for calculating the dawn angle and the dusk angle of the yaw rotation according to the thought, so as to realize the constraint conditions of considering the calculation capacity, period and memory capacity of the on-board computer, appropriately simplifying the link involving complex operation in the yaw dawn-dusk angle calculation method by using an approximate expression, and improving the calculation efficiency and memory space occupation. The application solves the demand of small heat dissipation area and large heat dissipation power of the existing on-board computer of the remote sensing satellite.

Description

Technical Field

[0001] This invention belongs to the field of spacecraft guidance, navigation and control, and relates to a control method for autonomous yaw planning of satellites. Background Technology

[0002] Remote sensing satellite cameras generally require that sunlight not directly or indirectly enter the camera lens. Structurally, the satellite is designed with a sunshade to block sunlight. The orbit of a high-orbit remote sensing satellite determines that the positions of the sun and the satellite change with the sun's revolution around the sun and the Earth's rotation. In order for the sunshade to block sunlight in real time, the satellite's attitude also needs to change accordingly.

[0003] Existing high-orbit remote sensing satellites fly normally (yaw angle 0°) from 0:00 to 12:00 local time, perform a 180° yaw maneuver around 12:00 local time, and fly in reverse (yaw angle 180°) from 12:00 to 24:00 local time. Since the overall power consumption of existing remote sensing satellites is not high, and the current satellite structure design is sufficient to meet current heat dissipation requirements, existing technology does not consider the heat flow from sunlight on the satellite's heat dissipation surface.

[0004] Remote sensing satellites carry both remote sensing and communication payloads. The communication payloads consume significant power, resulting in a much higher overall power consumption compared to traditional remote sensing satellites. However, the satellite structure still adopts the traditional remote sensing satellite design, with a smaller heat dissipation surface area than traditional communication satellites. Therefore, this places higher demands on the satellite's heat dissipation. Consequently, a method is needed to improve the satellite's heat dissipation or reduce heat accumulation on its heat dissipation surface. Summary of the Invention

[0005] The technical problem solved by this invention is to overcome the shortcomings of the prior art and provide an autonomous yaw planning method based on the optimization calculation of yaw angle morning and evening values. This method can calculate the morning and evening values ​​of the yaw angle of the satellite rotation. By controlling the satellite to keep the yaw angle at the morning value from 0:00 to 12:00 local time and to keep the yaw angle at the evening value from 12:00 to 24:00 local time, the problem of satellite heat dissipation is solved.

[0006] The technical solution of this invention is: an autonomous yaw planning method based on yaw angle twilight value optimization calculation, comprising:

[0007] Define the yaw angle values ​​for dawn and dusk;

[0008] Based on the requirement of minimizing the cumulative amount of solar irradiation projection on the entire satellite's heat dissipation surface, i.e. minimizing the integral value of the solar unit vector projection on the north and south plates, an integral model of solar projection on the heat dissipation surface of the orbital period satellite is established.

[0009] Based on the morning and evening values ​​of the yaw angle, the established solar projection integral model of the satellite heat dissipation surface of the orbital period is split to obtain heat flow integral model A based on the morning value and heat flow integral model B based on the evening value;

[0010] Calculate the fixed yaw angle morning value Ψ`morning and evening value Ψ`evening in heat flux integral model A based on morning value and heat flux integral model B based on evening value;

[0011] Based on the obtained fixed yaw angle values ​​Ψ`morning and Ψ`evening, the autonomous yaw planning method is determined.

[0012] The definition of the yaw angle morning value Ψmorning and evening value Ψevening includes:

[0013] The satellite maintains a fixed yaw angle during each orbital period. This fixed yaw angle is set with two reference values:

[0014] Morning (Ψ): The yaw angle planned for the satellite between 0:00 and 12:00 local time, ranging from 0° to 90°;

[0015] Evening (Ψevening): The yaw angle planned for the satellite between 12:00 and 24:00 local time, ranging from 90° to 180°.

[0016] The establishment of the solar projection integral model for the heat dissipation surface of a periodic satellite is specifically as follows:

[0017]

[0018] In the formula, J represents the solar projection integral of the satellite's heat dissipation surface over one orbital period, and β S S is the angle between the solar vector and the XOZ plane of the satellite body. x S y S z R is the lower solar vector of the southeastern solar system. z (ψ * ), R y (θ) are based on ψ * , The direction cosine matrix of θ rotation, ψ * , θ represents the yaw angle, pitch angle, and roll angle based on the Southeastern 312 turn sequence, respectively; in the model, minJ represents the result of appropriately selecting ψ * , The three values ​​of θ minimize the solar projection integral of the satellite's heat dissipation surface over one orbital period.

[0019] The heat flux integral model A based on the morning value represents the selection of ψ * , The three values ​​of θ allow the satellite to accumulate min J over the period from 0:00 to 12:00 local time; that is:

[0020]

[0021] The heat flux integral model B based on the twilight value represents the selection of ψ... * , The three values ​​of θ allow the satellite to accumulate min J over the period from 12:00 to 24:00 local time; that is:

[0022]

[0023] The southeastern solar vector S x S y S z Specifically:

[0024] S x =cosδ s sinα

[0025] S y =-sinδ s cosΔρ-cosδ s cosαsinΔρ

[0026] S z =-sinδ s sinΔρ+cosδ s cosαcosΔρ

[0027] Where δ s ρ is the solar declination, α is the current local time angle of the satellite, and Δρ is the latitude drift of the satellite's nadir position.

[0028] The R z (ψ * ), R y The calculation method for (θ) is as follows:

[0029] Establish θ and ψ * Relational model;

[0030]

[0031] θ = arcsin(T) x cosψ+T y sinψ)

[0032] ψ=ψ *

[0033] Among them (T) x ,Ty ,T z () is the target pointing vector under the Southeastern Sector; according to θ and ψ * R can be obtained z (ψ * ), R y (θ).

[0034] The target pointing vector (T) under the Southeast system x ,T y ,T z The calculation process for ) is as follows:

[0035]

[0036]

[0037] Among them, (P) x ,P y ,P z (A,B) represents the target's position vector under the Southeastern Spherical Radio System, (A,B) represents the target's longitude and latitude, λ0 represents the satellite's fixed-point longitude, and R... e R is the Earth's radius. s For the radius of the synchronous orbit, This is due to latitude drift.

[0038] The calculation of the fixed yaw angle morning value Ψ`morning and evening value Ψ`evening in the heat flux integral model A based on morning value and the heat flux integral model B based on evening value includes:

[0039] Given the initial yaw angle value Ψm n Determine the corresponding parameter K n The parameter K n This includes the target pointing vector under the southeastern satellite system, the solar position vector, and the three-axis attitude angles; the heat flux J is calculated based on the heat flux integral model A based on the morning value. n The n = 1.2…N; where N is a uniformly spaced fraction from 0:00 to 12:00 local time; in J n Find min J; since min J has only one minimum point within the yaw angle range of 0° to 90°, the corresponding Ψm is... n The optimal yaw angle morning value Ψ`morning obtained through iteration;

[0040] Given an initial yaw angle Ψe n Determine the corresponding parameter K` n The parameter K` nThis includes the satellite's southeastern target pointing vector, solar position vector, and three-axis attitude angles; the heat flux J` is calculated based on the heat flux integral model B based on the twilight value. n The n = 1.2…N; where N is a uniformly spaced fraction from 0:00 to 12:00 local time; in J` n Find min J`; since min J` has only one minimum point within the yaw angle range of 90° to 180°, the corresponding Ψe is... n The optimal yaw angle value Ψ`evening is obtained through iteration.

[0041] Based on the obtained fixed yaw angles Ψ`morning and Ψ`evening, the autonomous yaw planning method is determined, including:

[0042] From 0:00 to 12:00, the satellite controls its yaw angle to maintain the morning value Ψ`morning. Around 12:00 local time, it performs yaw attitude maneuvers, moving the yaw angle from the morning value Ψ`morning to the evening value Ψ`evening. From 12:00 to 24:00 local time, the satellite controls its yaw angle to maintain the evening value Ψ`evening. Around 00:00 local time, it performs yaw attitude maneuvers, moving the yaw angle from the evening value Ψ`evening back to the morning value Ψ`morning.

[0043] The advantages of this invention compared to existing technologies are as follows: Existing remote sensing satellite yaw control methods only consider the prevention of direct or indirect sunlight from entering the camera lens during the control process, without addressing the satellite's heat dissipation problem. This invention addresses the heat dissipation problem of communication and remote sensing satellites and the current state of yaw planning control by proposing an autonomous yaw planning method based on optimized calculation of yaw angle and twilight values. This method is based on a solar cumulative minimum optimization index function. Through algorithmic optimization, it recursively calculates the optimal yaw rotation angle, thereby minimizing the cumulative heat flow on the satellite's heat dissipation surface through yaw rotation control per orbital cycle, thus improving the satellite's heat dissipation conditions. For the calculation methods of the solar vector and target vector used in this method, complex computational steps are appropriately optimized using approximate expressions to reduce computational load and simplify the process. Attached Figure Description

[0044] Figure 1 This is a flowchart of the method of the present invention. Detailed Implementation

[0045] This invention discloses an autonomous yaw planning method based on yaw angle twilight value optimization calculation, comprising:

[0046] Define the yaw angle values ​​for dawn and dusk;

[0047] Based on the requirement of minimizing the cumulative amount of solar irradiation projection on the entire satellite's heat dissipation surface, i.e. minimizing the integral value of the solar unit vector projection on the north and south plates, an integral model of solar projection on the heat dissipation surface of the orbital period satellite is established.

[0048] Based on the morning and evening values ​​of the yaw angle, the established solar projection integral model of the satellite heat dissipation surface of the orbital period is split to obtain heat flow integral model A based on the morning value and heat flow integral model B based on the evening value;

[0049] Calculate the fixed yaw angle morning value Ψ`morning and evening value Ψ`evening in heat flux integral model A based on morning value and heat flux integral model B based on evening value;

[0050] Based on the obtained fixed yaw angle values ​​Ψ`morning and Ψ`evening, the autonomous yaw planning method is determined.

[0051] The definition of the yaw angle morning value Ψmorning and evening value Ψevening includes:

[0052] The satellite maintains a fixed yaw angle during each orbital period. This fixed yaw angle is set with two reference values:

[0053] Morning (Ψ): The yaw angle planned for the satellite between 0:00 and 12:00 local time, ranging from 0° to 90°;

[0054] Evening (Ψevening): The yaw angle planned for the satellite between 12:00 and 24:00 local time, ranging from 90° to 180°.

[0055] The establishment of the solar projection integral model for the heat dissipation surface of a periodic satellite is specifically as follows:

[0056]

[0057] In the formula, J represents the solar projection integral of the satellite's heat dissipation surface over one orbital period, and β S S is the angle between the solar vector and the XOZ plane of the satellite body. x S y S z R is the lower solar vector of the southeastern solar system. z (ψ * ), R y (θ) are based on ψ * , The direction cosine matrix of θ rotation, ψ * , θ represents the yaw angle, pitch angle, and roll angle based on the Southeastern 312 turn sequence, respectively; in the model, minJ represents the result of appropriately selecting ψ * , The three values ​​of θ minimize the solar projection integral of the satellite's heat dissipation surface over one orbital period.

[0058] The heat flux integral model A based on the morning value represents the selection of ψ * , The three values ​​of θ allow the satellite to accumulate min J over the period from 0:00 to 12:00 local time; that is:

[0059]

[0060] The heat flux integral model B based on the twilight value represents the selection of ψ... * , The three values ​​of θ allow the satellite to accumulate min J over the period from 12:00 to 24:00 local time; that is:

[0061]

[0062] The southeastern solar vector S x S y S z Specifically:

[0063] S x =cosδ s sinα

[0064] S y =-sinδ s cosΔρ-cosδ s cosαsinΔρ

[0065] S z =-sinδ s sinΔρ+cosδ s cosαcosΔρ

[0066] Where δ s ρ is the solar declination, α is the current local time angle of the satellite, and Δρ is the latitude drift of the satellite's nadir position.

[0067] The R z (ψ * ), R y The calculation method for (θ) is as follows:

[0068] Establish θ and ψ * Relational model;

[0069]

[0070] θ = arcsin(T) x cosψ+T y sinψ)

[0071] ψ=ψ *

[0072] Among them (T) x ,T y ,T z () is the target pointing vector under the Southeastern Sector; according to θ and ψ * R can be obtained z (ψ * ), R y (θ).

[0073] The target pointing vector (T) under the Southeast system x ,T y ,T z The calculation process for ) is as follows:

[0074]

[0075]

[0076] Among them, (P) x ,P y ,P z (A,B) represents the target's position vector under the Southeastern Spherical Radio System, (A,B) represents the target's longitude and latitude, λ0 represents the satellite's fixed-point longitude, and R... e R is the Earth's radius. s For the radius of the synchronous orbit, This is due to latitude drift.

[0077] The calculation of the fixed yaw angle morning value Ψ`morning and evening value Ψ`evening in the heat flux integral model A based on morning value and the heat flux integral model B based on evening value includes:

[0078] Given the initial yaw angle value Ψm n Determine the corresponding parameter K n The parameter K n This includes the target pointing vector under the southeastern satellite system, the solar position vector, and the three-axis attitude angles; the heat flux J is calculated based on the heat flux integral model A based on the morning value. n The n = 1.2…N; where N is a uniformly spaced fraction from 0:00 to 12:00 local time; in J n Find min J; since min J has only one minimum point within the yaw angle range of 0° to 90°, the corresponding Ψm is... n The optimal yaw angle morning value Ψ`morning obtained through iteration;

[0079] Given an initial yaw angle Ψe nDetermine the corresponding parameter K` n The parameter K` n This includes the satellite's southeastern target pointing vector, solar position vector, and three-axis attitude angles; the heat flux J` is calculated based on the heat flux integral model B based on the twilight value. n The n = 1.2…N; where N is a uniformly spaced fraction from 0:00 to 12:00 local time; in J` n Find min J`; since min J` has only one minimum point within the yaw angle range of 90° to 180°, the corresponding Ψe is... n The optimal yaw angle value Ψ`evening is obtained through iteration.

[0080] Based on the obtained fixed yaw angles Ψ`morning and Ψ`evening, the autonomous yaw planning method is determined, including:

[0081] From 0:00 to 12:00, the satellite controls its yaw angle to maintain the morning value Ψ`morning. Around 12:00 local time, it performs yaw attitude maneuvers, moving the yaw angle from the morning value Ψ`morning to the evening value Ψ`evening. From 12:00 to 24:00 local time, the satellite controls its yaw angle to maintain the evening value Ψ`evening. Around 00:00 local time, it performs yaw attitude maneuvers, moving the yaw angle from the evening value Ψ`evening back to the morning value Ψ`morning.

[0082] like Figure 1 The diagram shown is a flowchart of an embodiment of the method of the present invention. The main steps are as follows:

[0083] (1) Calculate the satellite's current local time angle and latitude drift of the nadir position in real time.

[0084] First, define the local time angle of a satellite: the phase of the satellite's projection relative to the Sun in the equatorial plane. 0:00 (0°) corresponds to a 180° phase difference between the satellite and the Sun; 6:00 corresponds to a 90° phase difference; 12:00 (180°) corresponds to the satellite being in the same phase as the Sun; and 18:00 corresponds to a 270° phase difference. The satellite operates in a geostationary orbit, and its local time angle increases uniformly over time.

[0085] Local time angle of the satellite passing through the ascending node:

[0086] α G =Ω+π-λ S (a)

[0087] Where Ω is the right ascension of the ascending node of the satellite, and λ S This is the right ascension of the sun.

[0088] Due to the orbital inclination, the satellite's nadir position drift Δλ (longitude) and Δρ (latitude) are:

[0089]

[0090] Δρ=i sin(α-α G (c)

[0091] α is the current local time angle of the satellite, and i is the orbital inclination of the satellite.

[0092] As can be seen from equations (b) and (c), when operating at a small inclination angle of 5°, the daily longitude drift of the satellite's nadir point is much smaller than the latitude drift. Therefore, ignoring the small changes in the satellite's longitude, it can be approximated that the satellite only moves in the north-south direction along the meridian of its fixed position.

[0093] (2) Based on the analysis in step 1, the calculation of the solar vector under the southeastern system is simplified, and the formula for calculating the solar position vector under the southeastern system can be obtained, where δ s This refers to the solar declination.

[0094] S x =cosδ s sinα

[0095] S y =-sinδ s cosΔρ-cosδ s cosαsinΔρ

[0096] S z =-sinδ s sinΔρ+cosδ s cosαcosΔρ(d)

[0097] (3) Calculate the target position vector under the southeast system, where (P x ,P y ,P z (A,B) represents the target's position vector under the Southeastern Spherical Radio System, (A,B) represents the target's longitude and latitude, λ0 represents the satellite's fixed-point longitude, and R... e R is the Earth's radius. s For the radius of the synchronous orbit, This is due to latitude drift.

[0098]

[0099]

[0100] (4) Given the initial value of the yaw angle Ψ0, calculate the attitude angles of the three axes (Southeast series 312 rotation sequence).

[0101]

[0102] θ = arcsin(T) x cosψ+T y sinψ)

[0103] ψ=ψ * (f)

[0104] (5) According to θ and ψ * R can be obtained z (ψ * ), R y (θ).

[0105] (6) Calculate the cumulative heat flow of the satellite per orbital cycle based on the heat flow integral model A based on the morning value and the heat flow integral model B based on the evening value.

[0106] Model A:

[0107]

[0108] Model B:

[0109]

[0110] (6) Based on steps 1 to 6, the yaw angle morning and evening values ​​are iteratively optimized. Since the yaw angle morning value has only one minimum point in the range of 0° to 90° (and the evening value is in the range of 90° to 180°), the optimal yaw angle morning value Ψ`morning and the optimal yaw angle evening value Ψ`evening can be obtained iteratively.

[0111] (7) Based on the obtained fixed yaw angles Ψ`morning and Ψ`evening, determine the autonomous yaw planning method, including:

[0112] From 0:00 to 12:00, the satellite maintains a constant yaw angle at the morning value Ψ`morning. Around 12:00 local time, it performs yaw attitude maneuvers, moving the yaw angle from the morning value Ψ`morning to the evening value Ψ`evening. From 12:00 to 24:00 local time, the satellite maintains a constant yaw angle at the evening value Ψ`evening. Around 00:00 local time, it performs yaw attitude maneuvers, moving the yaw angle from the evening value Ψ`evening back to the morning value Ψ`morning.

[0113] Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make possible changes and modifications to the technical solutions of the present invention based on the above-disclosed technical content without departing from the spirit and scope of the present invention. Therefore, any simple modifications, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solutions of the present invention shall fall within the protection scope of the technical solutions of the present invention.

Claims

1. An autonomous yaw planning method based on yaw angle twilight value optimization calculation, characterized in that, include: Define the yaw angle values ​​for dawn and dusk; Based on the requirement of minimizing the cumulative amount of solar irradiation projection on the entire satellite's heat dissipation surface, i.e. minimizing the integral value of the solar unit vector projection on the north and south plates, an integral model of solar projection on the heat dissipation surface of the orbital period satellite is established. Based on the morning and evening values ​​of the yaw angle, the established solar projection integral model of the satellite heat dissipation surface of the orbital period is split to obtain heat flow integral model A based on the morning value and heat flow integral model B based on the evening value; Calculate the fixed yaw angle morning value Ψ`morning and evening value Ψ`evening in heat flux integral model A based on morning value and heat flux integral model B based on evening value; Based on the obtained fixed yaw angles, the morning value Ψ`morning and the evening value Ψ`evening, the autonomous yaw planning method is determined; The establishment of the solar projection integral model for the heat dissipation surface of a periodic satellite is specifically as follows: In the formula, J represents the solar projection integral of the satellite's heat dissipation surface over one orbital period, and β S S is the angle between the solar vector and the XOZ plane of the satellite body. x S y S z R is the lower solar vector of the southeastern solar system. z (ψ * ), R y (θ) are based on ψ * , The direction cosine matrix of θ rotation, ψ * , θ represents the yaw angle, pitch angle, and roll angle based on the Southeastern 312 turn sequence, respectively; in the model, minJ represents the result of appropriately selecting ψ * , The three values ​​of θ minimize the solar projection integral of the satellite's heat dissipation surface over one orbital period.

2. The autonomous yaw planning method based on yaw angle twilight value optimization calculation according to claim 1, characterized in that, The definition of the yaw angle morning value Ψmorning and evening value Ψevening includes: The satellite maintains a fixed yaw angle during each orbital period. This fixed yaw angle is set with two reference values: Morning (Ψ): The yaw angle planned for the satellite between 0:00 and 12:00 local time, ranging from 0° to 90°; Evening (Ψevening): The yaw angle planned for the satellite between 12:00 and 24:00 local time, ranging from 90° to 180°.

3. The autonomous yaw planning method based on yaw angle twilight value optimization calculation according to claim 1, characterized in that, The heat flux integral model A based on the morning value represents the selection of ψ * , The three values ​​of θ allow the satellite to accumulate minJ over local time from 0:00 to 12:00; that is:

4. The autonomous yaw planning method based on yaw angle twilight value optimization calculation according to claim 3, characterized in that, The heat flux integral model B based on the twilight value represents the selection of ψ... * , The three values ​​of θ allow the satellite to accumulate minJ over the period from 12:00 to 24:00 local time; that is:

5. The autonomous yaw planning method based on yaw angle twilight value optimization calculation according to claim 3, characterized in that, The southeastern solar vector S x S y S z Specifically: S x =cosδ s Sinai S y =-sinδ s cosΔρ-cosδ s cosαsinΔρ S z =-sinδ s sinΔρ+cosδ s cosαcosΔρ Where δ s ρ is the solar declination, α is the current local time angle of the satellite, and Δρ is the latitude drift of the satellite's nadir position.

6. The autonomous yaw planning method based on yaw angle twilight value optimization calculation according to claim 1, characterized in that, The R z (ψ * ), R y The calculation method for (θ) is as follows: Establish θ and ψ * Relational model; θ=arcsin(T x cosψ+T y sinψ) ψ=ψ* Among them (T) x ,T y ,T z () is the target pointing vector under the Southeastern Sector; according to θ and ψ * R can be obtained z (ψ * ), R y (θ).

7. The autonomous yaw planning method based on yaw angle twilight value optimization calculation according to claim 6, characterized in that, The target pointing vector (T) under the Southeast system x ,T y ,T z The calculation process for ) is as follows: Among them, (P) x ,P y ,P z (A,B) represents the target's position vector under the Southeastern Spherical Radio System, (A,B) represents the target's longitude and latitude, λ0 represents the satellite's fixed-point longitude, and R... e R is the Earth's radius. s For the radius of the synchronous orbit, This is due to latitude drift.

8. The autonomous yaw planning method based on yaw angle twilight value optimization calculation according to claim 3, characterized in that, The calculation of the fixed yaw angle morning value Ψ`morning and evening value Ψ`evening in the heat flux integral model A based on morning value and the heat flux integral model B based on evening value includes: Given the initial yaw angle value Ψm n Determine the corresponding parameter K n The parameter K n This includes the target pointing vector under the southeastern satellite system, the solar position vector, and the three-axis attitude angles; the heat flux J is calculated based on the heat flux integral model A based on the morning value. n The n = 1.2…N; where N is a uniformly spaced fraction from 0:00 to 12:00 local time; in J n Find minJ; since minJ has only one minimum point within the yaw angle range of 0° to 90°, the corresponding Ψm is... n The optimal yaw angle morning value Ψ`morning obtained through iteration; Given an initial yaw angle Ψe n Determine the corresponding parameter K` n The parameter K` n This includes the satellite's southeastern target pointing vector, solar position vector, and three-axis attitude angles; the heat flux J` is calculated based on the heat flux integral model B based on the twilight value. n The n = 1.2…N; where N is a uniformly spaced fraction from 0:00 to 12:00 local time; in J` n Find minJ`; since minJ` has only one minimum point within the yaw angle range of 90° to 180°, the corresponding Ψe is... n The optimal yaw angle value Ψ`evening is obtained through iteration.

9. The autonomous yaw planning method based on yaw angle twilight value optimization calculation according to claim 1, characterized in that, Based on the obtained fixed yaw angles Ψ`morning and Ψ`evening, the autonomous yaw planning method is determined, including: From 0:00 to 12:00, the satellite controls its yaw angle to maintain the morning value Ψ`morning. Around 12:00 local time, it performs yaw attitude maneuvers, moving the yaw angle from the morning value Ψ`morning to the evening value Ψ`evening. From 12:00 to 24:00 local time, the satellite controls its yaw angle to maintain the evening value Ψ`evening. Around 00:00 local time, it performs yaw attitude maneuvers, moving the yaw angle from the evening value Ψ`evening back to the morning value Ψ`morning.

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