Three-axis antenna tracking system

Abstract

Techniques for controlling an antenna positioning system having a 3-axis antenna positioner with azimuth and elevation axes and a third axis orthogonal to the azimuth and elevation axes. An example method comprises estimating a time at which the distance between the 3-axis antenna positioner and a target satellite will be at a minimum, defining a 3-axis tracking interval, based on the estimated time, and determining starting and ending azimuth positions for the positioner, corresponding to the beginning and end of the interval. The example method further comprises determining minimum and maximum third-axis positions for the positioner within the 3-axis tracking interval, based on the starting and ending azimuth positions and based on positions of the target satellite within the interval, and controlling the positioner to track the satellite within the 3-axis tracking interval based on calculating elevation positions and third axis positions for the positioner from azimuth positions for the positioner.

Description

[0001] VS2611-WO-1 MBH 1096-0178 THREE-AXIS ANTENNA TRACKING SYSTEM RELATED APPLICATIONS This application claims the benefit of / priority to U.S. Provisional Patent Application Serial No. 63 / 694,701,filed 13 September 2024, the entire contents of which are incorporated herein by reference. TECHNICAL FIELD This disclosure is generally related to satellite communication systems and is more particularly related to the control of antenna positioners in such systems. BACKGROUND Communications between a ground station located on the Earth and a satellite orbiting the Earth in a non-geosynchronous orbit may rely on antenna positioning systems that can track the satellite as it traverses the visible sky and keep the antenna (or antennas) pointed at the satellite. This ensures that the primary lobe of the ground station antenna is pointed as directly pointed as possible towards the antenna system of the satellite, maximizing the signal strengths received at each end of the communications link. Some antenna positioning systems utilize a two-axis positioner for adjusting the antenna’s orientation. These two-axis positioners provide for motion around an azimuth axis as well as motion around an elevation axis, orthogonal to the azimuth. Thus, the two-axis positioner may assume a range of azimuth positions, around the azimuth axis, as well as a range of elevation positions (or simply “elevations”) around the elevation axis. Typically, the azimuth axis is perpendicular to the earth’s surface, and thus the azimuth positions are directions more-or-less tangent to the earth’s surface, such as North, East, South, or West. The elevation axis is orthogonal to the azimuth axis, and thus elevation positions might range from zero degrees, pointing to the horizon, to somewhere close to 90 degrees, pointing close to straight up. It will be appreciated, however, that the azimuth axis and elevation axis for the positioner might diDer from true Earth azimuth and elevation, due to the location and arrangement of the antenna positioning system. But, it is straightforward to convert from the two-axis coordinate system of the two-axis positioner to true Earth azimuth and elevation. A well-known problem with two-axis positioners arises when the satellite being tracked passes through the zenith, with respect to the positioner, i.e., directly overhead or substantially overhead from the perspective of the positioner. As the satellite passes through the zenith, the VS2611-WO-1 MBH 1096-0178 azimuth position of the typical 2-axis positioner needs to change rapidly by 180 degrees. The speed with which the positioner can change its azimuth position is limited by mechanical considerations, and thus the 2-axis positioner typically has a blind spot, or “keyhole,” directly or substantially overhead, where it is unable to continuously track the satellite passing through. This phenomenon, which is known as the “keyhole problem” or as “gimbal lock,” can cause degraded or lost communication. One way to address this problem is to use a three-axis positioner. These positioners have a third axis, e.g., orthogonal to azimuth and elevation axes. This provides a third degree of freedom for the positioner movement, allowing it to be adjusted within a range of positions around the third axis, as well as around the azimuth and elevation axes. This range of motion around the third axis may be limited, e.g., to a range substantially less than 90 degrees in width, while still allowing the three-axis positioner to follow the tracked satellite in such a way that discontinuities in the positioner’s velocity around the azimuth axis as the satellite passes overhead are avoided, thereby avoiding the keyhole problem discussed above. Control of a three-axis positioner, however, is substantially more complicated than that of a two-axis positioner. This is at least partly because for a given satellite position in the sky and a given position of the positioner with respect to one of the three axes, there will often be many combinations of positions with respect to the other two axes that will aim the antenna properly. Improved techniques for eDiciently and accurately controlling a three-axis positioner so as to avoid the keyhole problem while accounting for the physical dynamics of the antenna positioning system are needed. SUMMARY The techniques and apparatuses described herein address these problems, and include techniques for controlling an antenna positioning system having a 3-axis antenna positioner with azimuth and elevation axes and a third axis orthogonal to the azimuth and elevation axes. An example method comprises estimating a time at which the distance between the 3-axis antenna positioner and a target satellite will be at a minimum (and therefore elevation angle at a maximum), defining a 3-axis tracking interval, based on the estimated time, and determining starting and ending azimuth positions for the positioner, corresponding to the beginning and end of the interval. The example method further comprises determining minimum and maximum third-axis positions for the positioner within the 3-axis tracking interval, based on the starting and ending azimuth positions and based on positions of the target satellite within the VS2611-WO-1 MBH 1096-0178 interval, and controlling the positioner to track the satellite within the 3-axis tracking interval based on calculating elevation positions and third-axis positions for the positioner from azimuth positions for the positioner. The example method further comprises generating a cubic function to transition from a linear preposition motion of the 3-axis antenna positioner prior to entering the 3-axis tracking interval to motion of the 3-axis antenna positioner at a time of the maximum third-axis position, within the 3-axis tracking interval, and controlling the 3-axis antenna positioner from immediately prior to the 3-axis tracking interval to the time of the maximum third-axis position, based on the cubic function. Apparatuses and systems corresponding to the techniques described in detail below are also described. The techniques and apparatuses described herein provide improved control of a three-axis antenna positioner, particularly with respect to tracking high-elevation passes of the satellite being tracked. These techniques and apparatuses resolve the well-known keyhole problem that is faced by two-axis positioners, utilizing a smoothing out of the control function to avoid high accelerations. Other benefits and advantages of these techniques and apparatuses will be apparent upon reviewing the detailed description below and the attachedfigures. BRIEF DESCRIPTION OF THE FIGURES Figure 1 compares azimuth position and rates near the keyhole position for a 2-axis positioner and a 3-axis positioner according to some of the embodiments disclosed herein. Figure 2 illustrates two cross-azimuth profiles for a transition between a pre-position for the positioner and motion within a limited-azimuth-motion interval: a sinusoidal profile and a Z- shaped profile. Figure 3 is a processflow diagram illustrating an example method for controlling a 3-axis positioner according to some embodiments of the presently disclosed techniques. Figure 4 graphically illustrates an example adjustment of a cubic function to match velocity and acceleration for a transition between a pre-position for the positioner and motion within a limited-azimuth-motion interval. VS2611-WO-1 MBH 1096-0178 Figure 5 illustrates positioner dynamics of an 89-degree elevation pass for an example embodiment of the presently disclosed techniques, for azimuth, elevation, and cross-azimuth motion. Figure 6 illustrates a tradeoD between maximum azimuth velocity and maximum cross-azimuth range for a 3-axis positioner configured according to the techniques disclosed herein. Figure 7 is a processflow diagram illustrating a generalized method for controlling a 3-axis antenna positioning system having a 3-axis antenna positioner with an azimuth axis, an elevation axis, and a third axis orthogonal to the azimuth axis and elevation axis, according to some of the embodiments described herein. Figure 8 is a block diagram illustrating an example satellite-tracking antenna system configured to operate according to the techniques described herein. DETAILED DESCRIPTION The detailed discussion below describes the program tracking of a 3-axis positioner in an antenna positioning system. For the purposes of discussion, control techniques and algorithms are described for what might be referred to as an elevation-over-cross-azimuth-over-azimuth 3- axis positioner, i.e., a 3-axis positioner that has an azimuth axis, an elevation axis, and a third axis, referred to herein as a “cross-azimuth” axis, that is orthogonal to the azimuth and elevation axes, and positioned above the azimuth axis and below the elevation axis, with respect to the ground, which is assumed to be below the positioner. The cross-azimuth axis thus provides for controlled motion around the cross-azimuth axis, orthogonal to motions around the azimuth and elevation axes. It can be a limited range of motion axis and, looking from the ground up, it is above the azimuth axis and below the elevation axis. It should be noted that there is at least one other type of 3-axis positioner, where the limited motion third axis is above the elevation axis. The third axis in this type of positioner might be referred to as a “cross-elevation axis.” The principles discussed below for the elevation-over- cross-azimuth-over-azimuth 3-axis positioner can be applied to the at least one other type of 3- axis positioner, including this second type of 3-axis position, with straightforward adjustments to the mathematical relationships described below and to the transformations between the respective positioner 3-axis coordinate system and the Earth 3-axis coordinate system. The techniques described herein provide for elimination of the keyhole eDect, using control algorithms that provide smooth, non-jerky movement of the positioner in a manner that VS2611-WO-1 MBH 1096-0178 accounts for the limited speed around each of the three axis and the limited range in one or more of the axes. Note that the terms “speed,” “velocity,” and “rate,” as used herein with respect to motion of a positioner relative to one of its axes, are used interchangeably, and refer to angular velocities (e.g., degrees / second) unless the context clearly indicates otherwise. At low elevations, away from the azimuth keyhole, it is simplest for the positioner to act as a two-axis azimuth / elevation (Az / El) positioner, with the cross-azimuth position (i.e., the positioner’s orientation around the cross-azimuth axis)fixed at a reference position, e.g., at a “zero” position. Above a chosen elevation point, the system operates as a 3-axis positioner, employing motion around all three axes. To reduce the maximum azimuth velocity near the keyhole, the azimuth pedestal axis no longer needs to follow the normal azimuth earth angle but can move at a steady limited azimuth speed. Figure 1 illustrates examples of azimuth position and azimuth position velocity versus time (in seconds) for a near overhead pass with a 2-axis Az / El positioner and with a 3-axis positioner according to embodiments described herein. The 2-axis positioner, which exhibits the curve profiles in each part of Figure 1, has a high azimuth rate near the keyhole, which is at time=0 in Figure 1. With a well-designed tracking algorithm, motion around the cross-azimuth axis can be used to eliminate the need for this high azimuth rate, as seen in the linear profiles corresponding to the 3-axis positioner in Figure 1. Examples of such well-defined tracking algorithms are described in detail, below. Unlike with 2-axis positioners, where for every earth-centered position of the satellite there is one and only one correct combination of positions with respect to the two axes, with a 3-axis positioner there are multiple combinations of positions with respect to the three axes that will point the antenna to the same earth-centered coordinates, and multiple tracking algorithms that would track the correct positions. These multiple tracking algorithms, however, will have a wide range of trade-oDs with respect to the dynamics of the positioner during a tracking operation. Before describing details of tracking techniques and algorithms that eliminate the keyhole problem while accounting for the limited dynamics of the positioner, it is worth considering alternative approaches. One alternative would be to preposition the positioner and thenfix the azimuth position during the satellite pass. This is like an X / Y positioner, with the azimuth angle preselected. While this will eliminate the azimuth keyhole, this approach requires a high cross-azimuth travel range to VS2611-WO-1 MBH 1096-0178 cover low elevation positions. For example, with afixed azimuth with a 250km altitude at 90˚ inclination and with a maximum elevation of 75°, the cross-azimuth would need to reach 30° just to cover >5° elevation. To cover all the way to zero elevation the cross-azimuth range would need to reach 90°. So, this approach does not work with a limited-travel-range cross-azimuth axis. Another approach is to preposition the positioner andfix the cross-azimuth position during the satellite pass. However, a prepositioned,fixed cross-azimuth axis makes the azimuth velocity near the keyhole worse, other than >89.5° where the keyhole is still awful, just not a singularity (i.e., a discontinuity in the needed positioner orientation). This provides no practical benefits over an Az / El positioner. An X / Y positioner avoids the keyhole overhead, but has another singularity at low elevation at two azimuth positions, therefore it cannot handle dynamics at low elevation near those azimuth positions. Still another possibility is to use an Az / El positioner with afixed tilt axis. Afixed tilt axis does mitigate the keyhole by rotating the keyhole location. But it also requires three axes of motion and a 9.5˚ tilt would only limit the azimuth velocity to 10.7 deg / s for a 250 km altitude, direct overhead pass, whereas the same rotation in the cross-azimuth axis can limit the azimuth velocity to 6 deg / s. In another approach, the 3-axis positioner can start at a cross-azimuth position of 0 and move in a sinusoidal pattern. An example of this sinusoidal pattern is seen in Figure 2. This algorithm is simple because it tracks as an Az / El without any changes until it reaches near the keyhole. This is a viable alternative tracking algorithm. In yet another approach, the positioner can be prepositioned with respect to the cross-azimuth position and move in a Z-pattern, this is the linear pattern shown in Figure 2. Prepositioning reduces changes in directions in the cross-azimuth and possible stiction points where the velocity goes to zero while the positioner reverses directions. But, the cross-azimuth trajectory needs to be determined at the beginning of the pass so that there is no reversal of direction or profile discontinuities, leading to high dynamics. This is the selected tracking mode for the description that follows. To set a baseline for the detailed description of the 3-axis tracking algorithms disclosed herein, transformations from an Earth-based coordinate system to a pedestal-based coordinate VS2611-WO-1 MBH 1096-0178 systems are described below. By “pedestal-based” is meant that the coordinate system is referenced to the two or three-axes of the positioner. As noted above, the azimuth and elevation axes of the positioner may not be aligned with Earth azimuth or elevation with respect to the Earth’s surface, due to the mounting and orientation of the positioner hardware; these transformations allow directions referenced to the Earth-based coordinate system to be converted to the pedestal-based coordinate system. First is an earth-to-pedestal coordinate conversion given a known azimuth. Together, the expressions below may be referred to as “Equation 1”: ^^ = arccos(cos ^^^^^^^ cos (^^^^^^^ − ^^))^^^ ^^^(^^^^^^^ ^^)^^^Next is cross-azimuth. Together, the expressions below may be referred to as “Equation 2”: ^^ = arcsin sin ^^& ^^^^^cos ^^^ ' ^^ = ^^^^^^^ − ^^()^^^ − arcsin (tan ^^^ tan ^^^^^^^)The derivation of these transformations, which are for an elevation-over-cross-azimuth-over- azimuth positioner, is provided in Appendix A, below. Transformations for the cross-elevation- over-elevation-over-azimuth positioner mentioned above may be derived in a similar manner. The object of the tracking trajectory algorithms described herein is to produce a program tracking profile that: • Reduces maximum azimuth velocity at the keyhole while keeping cross-azimuth within limits • Has a smooth profile in all 3 axes - no discontinuities in position, velocity and acceleration • Moves in a single direction motion in cross-azimuth program track The algorithm should also be: • Computationally eDicient as a brute force method may not calculate during pre-pass time. • Robust for all low-earth orbit (LEO) and medium-earth-orbit (MEO) satellite orbits and positioner site locations. VS2611-WO-1 MBH 1096-0178 Basic steps for this algorithm may include: • Determine time, *+,-, at which the satellite to site location distance is minimum across the satellite pass. At that time the elevation position and azimuth velocity will also be at their maximum. Thus, determining the time *+,-at which the satellite to site location distance is minimized is equivalent to determining the time at which the elevation is at its highest and / or the time at which azimuth velocity is maximized. • Limit the azimuth velocity around *+,-and calculate the needed cross-azimuth position. • Determine the times of the minimum and maximum cross-azimuth position in that azimuth limited interval. • Create a cubic function to smoothly transition between the pre-pass position for the positioner and the positioner’s movement during the azimuth limited interval. These steps are illustrated in Figure 3 and discussed in more detail below. Thefirst step in the processflow diagram of Figure 3, shown at block 310, is estimating the peak elevation for the satellite’s trajectory and the corresponding time, *+,-. There is nothing diDerent about the satellite rise and set time for a 3-axis positioner, compared to 2-axis positioners, and there are no required changes to existing algorithms for determining the satellite rise and set times and the peak elevations, from existing models for the satellite’s orbit. As an example, for a given model of the satellite’s orbit, a golden-search algorithm across the interval of time between the rise and set times for the satellite may be used tofind an estimate of the peak elevation. Next, as shown at block 320, the azimuth velocity is limited, to a value at or below the maximum velocity at which the positioner can reliably and safely operate. In other words, a maximum azimuth velocity ^^′+^ / is determined. This maximum azimuth velocity is related to a time interval *#,+,^, which can be defined as the time needed to transition a full 180 degrees in maximum azimuth velocity: ^^′ 012°+^ / = ^4565^ .As an example, *#,+,^may be selected to be 30 seconds. Based on *+,-and *#,+,^, an interval for limited azimuth velocity operation of the positioner can be defined. The start and end points of this interval are: VS2611-WO-1 MBH 1096-0178 *= * − ^4565^ ^#,+,^_ ^ ^ +,- 9 , *#,+,^_8^): = *+,- + 4565^8 ^ ^ 9 .The during this interval, ^^_^<=<*(*), is then: ^^_^<=<*(*) = ^^^^^^^>^4565^_?^@AB ^^^^^^^>^4565^_?^^^^B^4565^ (* − *#,+,^_8^^^^) + ^^^^^^^(*#,+,^_8^^^^) . So, during the interval (*#,+,^_8^^^^, *#,+,^_^-C), the azimuth position for the positioner is given byEquation 3. The corresponding elevation and cross-azimuth at any given time during that interval can be from the azimuth position, using Equation 1. The minimum and maximum cross-azimuth positions will be within the interval (*#,+,^_8^^^^ , *#,+,^_^-C). These positions and their corresponding times can be found, using1 3 and a numerical method such as the golden-section search method, which can a minimum and maximum of a unimodal function and is order log(N) in complexity. This is illustrated in Figure 3 at block 330. Properly controlling the positioner motion to provide smooth and reliable operation requires a well-controlled transition between the positioner’s pre-position, i.e., it’s movement and position prior to the azimuth-limited interval, to the motion within the interval defined by Equation 3. Generation of this transition profile is shown at block 340. A corresponding transition at the end of the azimuth-motion-limited interval is also necessary. For the transition from the positioner’s pre-position motion to the motion within the azimuth-motion-limited interval, a linear preposition motion (constant velocity, zero acceleration) transitions to the azimuth limited motion right after the peak cross-azimuth distance, with a cubic function (constant jerk, D). The transition profile is: ^(*) = E 0, + F8(* − *8^^^^) + G D(* − *8^^^^)H .The starting a small value that is At a time, *J),-, the transition profile must be equal to the azimuth-limited motion of the positioner in position, velocity and acceleration. To solve, equate the velocity and accelerations and solve for *J),- then calculate the required E, and *8^^^^:^KK(*J),-) = D>*J),- − *8^^^^B = LKK(*J),-)) VS2611-WO-1 MBH 1096-0178 Solve for *8^^^^in terms of *J),-at the point in time at which the acceleration is eual for the transition profile and the azimuth-limited motion: *8^^^^ = *J),- − LKK(*J),-) / DSubstitute into velocity equation and simplify: 1^K>* 9J),-B = F8 +2 D>*J),- − (*J),- − LKK(*J),-) / D)B = LK(*J),-)The equation for may be referred to as “Equation 4”: LKK>*J),-B − R2D>LK>*J),-B − F8B = 0 .Equation 4 can be determined with a rootfinding method, like the bisection method. When numerically taking the derivative of the function y, the time step, T*, size should not be too small, or it will introduce noise, because the numerical precision of the propagator at millisecond intervals may not be suDicient for a double derivative. A time delta of 1.0 seconds has been found to work well, though others may also work. K( ) (( ) L * + T* / 2 − L * − T* / 2)L * ≈T*LKK(*) ≈ V(^WX^) 9V(^)WV(^ X^)X^Y .Graphically the *J),-point is where both the acceleration and the integral of the acceleration are equal, shown visually in Figure 4. Figure 4 includes graphs for the cross-azimuth position, cross-azimuth velocity, and cross-azimuth acceleration graphs for an initial sinusoidal profile and a Z-profile. The initial sinusoidal profile uses the azimuth motion in Equation 3, and the cross-azimuth is solved for using Equation 1. The second profile is the Z-profile that transitions from a preposition in cross-azimuth to the sinusoidal profile at t_join. The 2ndprofile is a cubic function in position, a quadratic function in velocity and a linear function in acceleration. The times t_start and t_join are solved for numerically, as there is no readily apparent closed- form solution. But, there is a unique solution, which is illustrated by the graphs at the bottom of Figure 4. The time between t_start and t_join is determined by the acceleration at t_join. If t_join is set too early, as shown in the bottom left graph in Figure 4, the acceleration will match VS2611-WO-1 MBH 1096-0178 but the z-profile will have too high of a velocity, since the area under the acceleration curve is greater. Likewise, if t_join is set too late the z-profile will have too low of a velocity to match the sinuosoidal profile and will lead to a discontinuity in velocity. t_start and t_join are thus adjusted numerically to avoid this mismatch in velocity. After *J),-,0is calculated, *8^^^^and E,can be calculated: *8^^^^ = *J),-,0 − R2(LK>*J),-,0B − F8) / D,0BThe limited interval is similar. Equation 4 is solved for the end of the profile,finding *J),-,9. Then, *^-Cand E[can be calculated: *^-C = *J),-,9 + R2(LK>*J),-,9B − F8) / D^>* = HJ),-,9B E[ + F8>*J),-,9 − ^-CB = L>*J),-,9B .Once the 2 can be used to calculate the azimuth and elevation positions. Between the minimum and maximum cross-azimuth positions, the limited azimuth equation will determine the position using Equation 1, everywhere else the preposition and transition equations will determine the cross- azimuth position. The full profile is given by the equations below.^^^(*) = Coordinate TransformE, + F8(* − *8^^^^) * < *8^^^^ Equation 2E, + F8(* − * 08^^^^) + ] G D(* − *8^^^^)H *8^^^^ < * < *J),-,0 Equation 2 , , *J),-,0 < * < *J),-,9 Equation 1^^#,+,^(*): Equation 3 E[+F − ^ + ] 08 -C G D(* − *^-C)H *J),-,9 < * < *^-C Equation 2E[+F8(* − *^-C) * > *^-C Equation 2 VS2611-WO-1 MBH 1096-0178 d= +1 if *+^ / > *+,- else -1Figure 5 illustrates position dynamics for an 89-degree elevation pass for an example profile as determined above, versus the earth coordinates and a trapezoidal profile, for azimuth, elevation, and cross-azimuth. Left to right, the graphs illustrate position, velocity and acceleration. From top to bottom, the graphs are for the azimuth, elevation and cross-azimuth axes. The “Earth” profile is the profile of a 2 axis Az / El system, there is no cross-azimuth for that system. The “trap” is a trapezoidal profile in the azimuth velocity, which is what results if the azimuth velocity and acceleration are simply limited to some value. The “single” profile is the profile described in detail herein. These graphs illustrate the motion in all 3 axes; the highest dynamics are all in azimuth. This shows that the trapezoidal profile and the single profile both limit the high azimuth velocity near the keyhole. The single profile has a lower acceleration than either of the other two. They have similar dynamics in elevation and cross-azimuth. This algorithm was designed for eDicient single thread computing (non-parallel). Each of the steps is either afixed set of calculations or a log(N) order of search algorithm. The table below details the number of orbit propagation calculations for an example implementation of the algorithm, based on a search tolerance of .01 seconds and a *#,+,^of 30 seconds. Note that xaz(t) refers to cross-azimuth position as a function of time, and xaz’’(t) is the second derivative of this function, i.e., the acceleration of cross-azimuth movement as a function of time. Step Method Number of Orbit Calculations VS2611-WO-1 MBH 1096-0178 ^^^′′(*) tolerance) x (5 for 1st& 2nd=a2 ∙ D^cd#,+,^ ∙ ^^^′(*)derivative) The total additional calculations in the table above are 167. When 3-axis positioning is based on a time interval, not a trigger elevation angle, then the maximum azimuth velocity is consistent across altitude. A direct overhead pass moves 180˚ in azimuth, so it will have the same azimuth velocity with an equal time interval. In the algorithm described above, the azimuth limited time interval and the satellite altitude determine the range of cross-azimuth motion. Therefore, using the lowest altitude at diDerent time intervals will determine the trade space between maximum azimuth velocity and maximum cross-azimuth range, absent any other limitations. This trade-oD can be seen in Figure 6. In an example implementation of the algorithm described above, the decision was made to keep the cross-azimuth time constant, set at a constant value of 60 seconds, so long as the calculated cross-azimuth position was within the travel limits of the positioner (+1 deg buDer for mount model). If the cross-azimuth position is not within the travel limits, the cross-azimuth time constant is reduced by three seconds and the cross-azimuth position is recalculated. This continues until the cross-azimuth position is within the travel limits or until the cross-azimuth time constant reaches 12 seconds (maximum azimuth velocity of 15 deg / s). High altitude satellite passes will therefore have a maximum azimuth velocity of 3° / s, in this implementation, and the minimum required cross-azimuth range to meet that 60 second cross- azimuth motion time. Low altitude satellites passes will operate at the lowest azimuth speed that the cross-azimuth travel range and altitude permit. Positioner requirements for an implementation like that described in detail above include that the positioner can move from any starting point to any ending point in 30 seconds, and that the positioner will track all circular orbits from 250 km to 8,000 km. The dynamic requirements are shown in Table2, below. Dynamic Requirements VS2611-WO-1 MBH 1096-0178 Range ±180˚ 0 to 180˚ ±15˚ The equations and detailed description of the tracking algorithm above are generally based on the simplifying assumption of ideal positioner movement. In practical implementations, mount model corrections specific to the positioner design should be applied correctly to determine the correct solution, otherwise strange oDsets can occur, especially near the keyhole where the small-angle approximation breaks down. Figure 7 is a processflow illustrating a method for controlling a 3-axis antenna positioning system having a 3-axis antenna positioner with an azimuth axis, an elevation axis, and a third axis orthogonal to the azimuth axis and elevation axis. The illustrated method is a generalization of the specific examples given above – accordingly, where there are minor diDerences in terminology between the following description of Figure 7 and the specific examples given above, the terminology used below should be understood as at least encompassing the similar terminology used above. Figure 7 and the description below refer to a “third axis” – in the detailed examples described above, this third axis is a cross-azimuth axis. It will be appreciated, however, that with other 3-axis positioners, this third axis may diDer. For instance, with a cross- elevation-over-elevation-over azimuth positioner, the third axis may be the cross-elevation axis. As shown at block 710, the method comprises the step of estimating a time at which the distance between a location of the 3-axis antenna positioner and a target satellite will be at a minimum. This may comprise, for example, using a golden search algorithm across the interval of time between the rise and set times for the target satellite, as was discussed above. VS2611-WO-1 MBH 1096-0178 As shown at block 720, the method further comprises defining a 3-axis tracking interval, based on the estimated time, and determining starting and ending azimuth positions for the 3-axis antenna positioner, corresponding to the beginning and end of the 3-axis tracking interval. An example of this was also described above, with the interval being referred to there as an interval for limited azimuth velocity operation of the positioner, or azimuth-motion-limited interval. As shown at block 730, the method comprises determining minimum and maximum third-axis positions for the 3-axis antenna positioner within the 3-axis tracking interval, based on the starting and ending azimuth positions and based on positions of the target satellite within the 3- axis tracking interval. In the detailed examples described above, the minimum and maximum third-axis positions are minimum and maximum cross-azimuth positions for the positioner. Skipping forward to block 750, the illustrated method still further comprises controlling the 3- axis antenna positioner to track the target satellite within the 3-axis tracking interval based on calculating elevation positions and third-axis positions for the 3-axis antenna positioner from azimuth positions for the 3-axis antenna positioner. In the detailed examples described above, the third-axis positions are cross-azimuth positions, and the elevation and cross-azimuth positions are calculated from azimuth positions according to Equation 1. The example method shown in Figure 7 further comprises the step of generating a cubic function to transition from a linear preposition motion of the 3-axis antenna positioner prior to entering the 3-axis tracking interval to motion of the 3-axis antenna positioner at a time of the maximum third-axis position, within the 3-axis tracking interval, and controlling the 3-axis antenna positioner from immediately prior to the 3-axis tracking interval to the time of the maximum third-axis position, based on the cubic function. This is shown at block 740 in Figure 7. A similar step may be carried out for generating a cubic function to transition from motion of the positioner from a time of the minimum third-axis position, within the 3-axis tracking interval, to linear post-position motion of the 3-axis antenna positioner following the 3-axis tracking interval. In some embodiments or instances, the cubic function has no discontinuity in position, velocity, or acceleration within the interval of time when the cubic function is used to control the 3-axis positioner. In various embodiments or instances of the method shown in Figure 7, the method may further comprise controlling the 3-axis antenna positioner as a 2-axis positioner for an interval immediately prior to controlling the 3-axis antenna positioner based on the cubic function. VS2611-WO-1 MBH 1096-0178 In some embodiments, controlling the 3-axis antenna positioner to track the target satellite within the 3-axis tracking interval comprises controlling the azimuth position of the 3-axis positioner at an at least approximately steady velocity across the 3-axis tracking interval. By “approximately steady velocity” is meant that the azimuth motion is controlled to an approximately constant velocity, allowing for tolerances in the ability of the physical apparatus and control system to maintain constant velocity and other insubstantial deviations. As noted above, in some embodiments, the third axis is a cross-azimuth axis that is above the azimuth axis and below the elevation axis, with respect to a ground position below the 3-axis antenna positioner. In other embodiments, the third axis is a cross-elevation axis above both the azimuth axis and elevation axis, with respect to a ground position below the 3-axis antenna positioner. Note that in any of the embodiments or instances described above, “controlling” the 3-axis antenna positioner according to any of the steps of the method may include incorporating adjustments into the control to account for mount model corrections, e.g., to account for imperfections in the dynamics of the positioner. Figure 8 is a block diagram illustrating an example satellite-tracking antenna system configured to operate according to the techniques described above. It will be appreciated that this system, which comprises a control system 810 and a 3-axis positioner 830, may resemble a conventional system with the exception of the details of the control system, which is configured, in this example with processing circuitry 812 and memory 818, to carry out a tracking algorithm according to the techniques described above. More particularly, the control system 810, which is for controlling a 3-axis antenna positioning system having a 3-axis antenna positioner with an azimuth axis, an elevation axis, and a third axis orthogonal to the azimuth axis and elevation axis, comprises processing circuitry 812 and memory 818 operatively coupled to the processing circuitry 812. The memory 818 comprises program instructions 820 and program data 822, and is configured, by way of the program instructions 820 and program data 822, to cause the processing circuitry 812 to: estimate a time at which the distance between a location of the 3-axis antenna positioner and a target satellite will be at a minimum; define a 3-axis tracking interval, based on the estimated time, and determine starting and ending azimuth positions for the 3-axis antenna positioner, corresponding to the beginning and end of the 3-axis tracking interval; determine minimum and maximum third-axis positions for the 3-axis antenna positioner within the 3-axis tracking VS2611-WO-1 MBH 1096-0178 interval, based on the starting and ending azimuth positions and based on positions of the target satellite within the 3-axis tracking interval; and control the 3-axis antenna positioner to track the target satellite within the 3-axis tracking interval based on calculating elevation positions and third-axis positions for the 3-axis antenna positioner from azimuth positions for the 3-axis antenna positioner. Power source 816 provides power for the processing circuitry 812, memory 818, and other components of the control system 810. The processing circuitry 812 controls the 3-axis positioner 830 via input / output interface circuitry 814, which provides drive and / or control signals for controlling actuators 832 in the 3-axis positioner 830; these actuators, which may comprise any of linear motors, torque motors, piezo motors, voice-coil motors, servo and stepper stages, etc., control the position and movement of the positioner according to its three degrees of motion. As noted above, thefirst and second degrees of motion are around an azimuth axis and elevation axis. In some embodiments, the third axis of the 3-axis positioner 830 is a cross- azimuth axis that is above the azimuth axis and below the elevation axis, with respect to a ground position below the 3-axis antenna positioner 830. In other embodiments, the third axis may be a cross-elevation axis above both the azimuth axis and elevation axis, with respect to a ground position below the 3-axis antenna positioner 830. In some embodiments of the control system 810, the program instructions 820 are further configured to cause the processing circuitry 812 to generate a cubic function to transition from a linear preposition motion of the 3-axis antenna positioner 830 prior to entering the 3-axis tracking interval to motion of the 3-axis antenna positioner 830 at a time of the maximum third- axis position, within the 3-axis tracking interval. The program instructions 820 in these embodiments are further configured to cause the processing circuitry 812 to control the 3-axis antenna positioner 830 from immediately prior to the 3-axis tracking interval to the time of the maximum third-axis position, based on the cubic function. In some embodiments, the program instructions 820 are further configured to cause the processing circuitry 812 to control the 3-axis antenna positioner 830 as a 2-axis positioner for an interval immediately prior to controlling the 3-axis antenna positioner 830 based on the cubic function. In some embodiments or instances, the cubic function has no discontinuity in position, velocity, or acceleration within the interval of time when the cubic function is used to control the 3-axis positioner 830. In some embodiments or instances, controlling the 3-axis VS2611-WO-1 MBH 1096-0178 antenna positioner 830 to track the target satellite within the 3-axis tracking interval comprises controlling the azimuth position of the 3-axis positioner 830 at an at least approximately steady velocity across the 3-axis tracking interval. The techniques and apparatuses above provide improved control of a three-axis antenna positioner, particularly with respect to tracking high-elevation passes of the satellite being tracked. These techniques and apparatuses resolve the well-known keyhole problem that is faced by two-axis positioners, utilizing a smoothing out of the control function to avoid high accelerations. Another benefit of various embodiments of the presently disclosed techniques and apparatuses is that they eliminate the two turnaround points of cross-azimuth motion that are seen in some control algorithms. These turnaround points can be areas of high error, because of the high static friction attendant to the turnaround. The algorithms described above include only one turnaround, among all three axes, at the elevation peak.

[0002] VS2611-WO-1 MBH 1096-0178 APPENDIX TO THE DESCRIPTION Following is a derivation of coordinate conversion between azimuth / elevation (AZ / EL) and azimuth / cross-azimuth / elevation (AZ / XAZ / EL) systems. The elevation-over-cross-azimuth-over-azimuth (AZ / XAZ / EL) pedestal is a three-axis positioner that can orient its RF beam away from the azimuth keyhole using a tilt axis mounted between the azimuth and elevation axis and orthogonal to both azimuth and elevation axes. This arrangement is a good choice for tracking targets that pass directly over the antenna. The rotation direction of the AZ and EL angles are the same as the standard AZ / EL pedestal. For purposes of the present discussion, the XAZ angle is defined so that a positive value of XAZ tilts the antenna beam towards the east when azimuth is at 0°. Using the East, North, Up coordinate frame (ENU), this model has a unit vector in the y-axis rotated by elevation, then cross-azimuth, then azimuth. The earth vector is the same, but it has no XAZ rotation: e^>−fg:BeV(hfg)e / >ij:BL⃗ = e^(−f^ + f^()^^^)e / (ij)L⃗m= This reduces to: sin fg: cos ij: + cos fg: sin ij: sin hfg− o= AZP / ELP / XAZ to AZ / EL Coordinate Conversion: Solve for ij using the third row of Equation 5, solve for fg by dividing thefirst and second rows of Equation 5: sin fg ∙ cos ij + cos fg sin ij sin hfgfg = tan 0 : : : : + f^()^^^ VS2611-WO-1 MBH 1096-0178 AZ / EL to AZP / ELP / XAZ Coordinate Conversion (vwxyw z{|): To convert from earth coordinates to pedestal coordinates. Eliminate the azimuth rotation on the left side by rotating both sides by e^>fg:B:= (−f^ + f^ ^^^) L⃗Apply sin ij: sin hfg sin>fg − fg:B cos ijn cos ij: o = ncos>fg − fg:B cos ij o Solve for ij:using by the third rows: ij: = cos 0(cos ij cos (fg − fg:))i f f / o / / Coor nae Converson (vwxyw }z{): For fg, rearrange solution from: 0sin>fg − fg − f^hfg = tan : ()^^^Btan ijtan hfg tan ij = sin>fg − fg: − f^()^^^Barcsin(tan hfg tan ij) = fg − fg: − f^()^^^f^: = fg − f^()^^^ − arcsin(tan ^^^ tan ij)Solve for ij:using the third row of5. sin ijij: = arcsin &hf '

Claims

AMENDED CLAIMS received by the International Bureau on 12 March 2026 (12.03.2026)What is claimed is:1 . A method for controlling a 3-axis antenna positioning system having a 3-axis antenna positioner with an azimuth axis, an elevation axis, and a third axis orthogonal to the azimuth axis and elevation axis, the method comprising: estimating a time at which the distance between a location of the 3-axis antenna positioner and a target satellite will be at a minimum; defining a 3-axis tracking interval, based on the estimated time, and determining starting and ending azimuth positions for the 3-axis antenna positioner, corresponding to the beginning and end of the 3-axis tracking interval; determining minimum and maximum third-axis positions for the 3-axis antenna positioner within the 3-axis tracking interval, based on the starting and ending azimuth positions and based on positions of the target satellite within the 3-axis tracking interval; generating a cubic function to transition from a linear preposition motion of the 3-axis antenna positioner prior to entering the 3-axis tracking interval to motion of the 3-axis antenna positioner at a time of the maximum third-axis position, within the 3-axis tracking interval; controlling the 3-axis antenna positioner to track the target satellite within the 3-axis tracking interval based on calculating elevation positions and third-axis positions for the 3-axis antenna positioner from azimuth positions for the 3-axis antenna positioner; and controlling the 3-axis antenna positioner from immediately prior to the 3-axis tracking interval to the time of the maximum third-axis position, based on the cubic function.

2. (Canceled).

3. The method of claim 1 , further comprising controlling the 3-axis antenna positioner as a 2- axis positioner for an interval immediately prior to controlling the 3-axis antenna positioner based on the cubic function.

244. The method of claim 1 or 3, wherein the cubic function has no discontinuity in position, velocity, or acceleration within the interval of time when the cubic function is used to control the 3-axis positioner.

5. The method of any one of claims 1 , 3, or 4, wherein controlling the 3-axis antenna positioner to track the target satellite within the 3-axis tracking interval comprises controlling the azimuth position of the 3-axis positioner at an at least approximately steady velocity across the 3-axis tracking interval.

6. The method of any one of claims 1 and 3-5, wherein the third axis is a cross-azimuth axis that is above the azimuth axis and below the elevation axis, with respect to a ground position below the 3-axis antenna positioner.

7. The method of any one of claims 1 and 3-5, wherein the third axis is a cross-elevation axis above both the azimuth axis and elevation axis, with respect to a ground position below the 3- axis antenna positioner.

8. A control system for controlling a 3-axis antenna positioning system having a 3-axis antenna positioner with an azimuth axis, an elevation axis, and a third axis orthogonal to the azimuth axis and elevation axis, the control system comprising: processing circuitry; and memory operatively coupled to the processing circuitry, the memory comprising program instructions configured to cause the processing circuitry to: estimate a time at which the distance between a location of the 3-axis antenna positioner and a target satellite will be at a minimum; define a 3-axis tracking interval, based on the estimated time, and determine starting and ending azimuth positions for the 3-axis antenna positioner, corresponding to the beginning and end of the 3-axis tracking interval; determine minimum and maximum third-axis positions for the 3-axis antenna positioner within the 3-axis tracking interval, based on the starting and ending azimuth positions and based on positions of the target satellite within the 3-axis tracking interval; generate a cubic function to transition from a linear preposition motion of the 3-axis antenna positioner prior to entering the 3-axis tracking interval to motion of the3-axis antenna positioner at a time of the maximum third-axis position, within the 3-axis tracking interval; control the 3-axis antenna positioner to track the target satellite within the 3-axis tracking interval based on calculating elevation positions and third-axis positions for the 3-axis antenna positioner from azimuth positions for the 3-axis antenna positioner; and control the 3-axis antenna positioner from immediately prior to the 3-axis tracking interval to the time of the maximum third-axis position, based on the cubic function.

9. (Canceled).

10. The control system of claim 8, wherein the program instructions are further configured to cause the processing circuitry to control the 3-axis antenna positioner as a 2-axis positioner for an interval immediately prior to controlling the 3-axis antenna positioner based on the cubic function.

11. The control system of claim 8 or 10, wherein the cubic function has no discontinuity in position, velocity, or acceleration within the interval of time when the cubic function is used to control the 3-axis positioner.

12. The control system of any one of claims 8, 10, and 11 , wherein controlling the 3-axis antenna positioner to track the target satellite within the 3-axis tracking interval comprises controlling the azimuth position of the 3-axis positioner at an at least approximately steady velocity across the 3-axis tracking interval.

13. The control system of any one of claims 8 and 10-12, wherein the third axis is a crossazimuth axis that is above the azimuth axis and below the elevation axis, with respect to a ground position below the 3-axis antenna positioner.

14. The control system of any one of claims 8 and 10-12, wherein the third axis is a crosselevation axis above both the azimuth axis and elevation axis, with respect to a ground position below the 3-axis antenna positioner.

15. A satellite-tracking antenna system comprising the control system of any one of claims 8 and 10-14, and further comprising the 3-axis positioner.27

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