# The Art of Satellite Constellation Design: What You Need to Know

There is no defined common process for constellation design since it is a process that varies significantly with the mission objectives. During constellation design, the preferred solutions are those that satisfy the mission requirements while minimizing the overall cost of realizing the mission. As a consequence, it is desired that a minimum number of satellites be employed to accomplish the mission objectives. Another important cost factor is the number of orbital planes utilized. Placing satellites on significantly differing orbital planes require multiple launches that increases launch cost and complicates the launch sequence. The primary input for constellation design is the geographical areas that need to be covered and how frequently they needs to be covered. Quite often a mission requires global coverage and sometimes continuous global coverage.

This article describes key parameters that a constellation designer needs to consider and their trade-offs. Also, we will describe the steps involved in designing a frequently used constellation geometry: The Walker Delta Constellation.

### How it began

The Space Age began with the launch of Sputnik-1, world’s first Artificial Satellite, on October 1957. The commencement of Space Age created a rapid demand for space-based applications, mainly in areas of navigation, communication and observation. The idea of satellites functioning in a coordinated manner came up in 1958. The synchronized behaviour by a group of satellites, known as a satellite constellation, provides significant improvement in temporal and spatial coverage. The importance of satellite constellations cannot be overstated.

The first constellation called the TRANSIT was developed by the US Navy in 1960 to provide navigational assistance to their ballistic missiles. Since then, the developments in satellite constellations was propelled by its wide applicability. Satellite constellation design is often misrepresented as a mere act of replicating multiple copies of a single satellite in modified orbits. The satellite constellation design process is somewhat akin to developing a multicellular organism with each cell representing a satellite. The infinite number of choices for the six Keplerian orbital parameters make the constellation design extraordinarily difficult. Various constellation geometries were proposed to reduce this complexity. The most notable constellation geometry is the Walker-Delta constellation proposed by John Walker in 1970. Walker’s geometry made the orbital parameters dependant on one another in a particular way, thereby reducing the complexity. Walker-Delta technique provides the most symmetric geometry among all the constellation design techniques. Thus, it is most suitable for global coverage for several applications related to earth-observation. Off late another technique, called the flower constellation’ technique developed at Texas A&M University, is becoming popular which can address cases where only local coverage is desirable. In fact, flower constellations can provide “repeating ground track” orbits which are not restricted to an inertial plane, thus widening the varieties of satellite constellation. However, detailed discussions about flower constellations will be deferred for a later article.

### Designing a Satellite Constellation

There are no definite rules for designing a satellite constellation. The parameters defining a satellite constellation are ‘mission dependant’. Generally all the satellites in the constellation have similar altitude profiles, eccentricity and inclination so that perturbations affect the satellites in the same way and the geometry can be preserved without much station-keeping. The principal factors to be defined while designing a Satellite Constellation are listed below.

Table: Parameters to be considered during Constellation Design

 Parameters Mission Impacts Number of Satellites Affects the coverage and the principal cost Number of Orbital Planes Varies based on coverage needs. Highly advantageous to have minimum number of orbital planes as transfer between the orbits increases the launch, and transfer costs. Minimum Elevation Angle Must be consistent with all satellites. Determines the coverage of single satellite. Altitude Increasing the Altitude increases the coverage and the launch, transfer cost. Decreases the number of Satellites. For communication applications, increase/decrease in altitude can correspondingly change latency. Inclination Determines the latitude distribution of coverage and selected based on coverage needs. Plane Spacing Uniform plane spacing results in continuous ground coverage. Eccentricity Circular orbits are popular, because then the satellite is at a constant altitude requiring a constant strength signal to communicate. For some cases, elliptic orbits are chosen where we need satellites to stay over a particular region for longer duration. Tundra and Molniya orbits are two such examples.

Apart from the six orbital parameters and the ones mentioned above, one important design consideration is collision avoidance. Apart from loss of mission, collision between satellites in the constellation or between other existing satellites will result in space debris which might have a devastating effect on the other satellites, like it was depicted in a recent movie Gravity’. The most unfortunate example is the collision between Iridium 33 and Kosmos 2251 on February 2009. This resulted in millions of small debris, most of which still orbit the Earth. To prevent unnecessary space debris, we also require a well defined ‘end of life strategy’. Typically, at end of life,  satellites are either de-orbited or transferred to graveyard orbits (suitable for satellites in geostationary orbits).

Figure above shows ground trace for a typical communication satellite. The ground trace (shaded area) is circular with radius $\lambda_{max}$ and is subtended by a cone with half angle $\theta$. The continuous coverage often called the street of coverage is represented by considering a chordal range of $\lambda_{street}$ on both sides of the ground trace (assumed circular), as shown in figure below. The adjacent orbits should be decided such that the bulges of one orbital plane fills the dips of the other orbital plane. Hence to guarantee continuous coverage the maximum distance between adjacent orbit planes $D_{max}$ can be selected as

$D_{max} = \lambda_{street} + \lambda_{max}.$

As one can guess, coverage increases with the increase in number of satellites or with the increase in altitude. However this also increases the principal and launch costs. Hence there exists a trade-off between coverage and mission cost.

Let us consider a satellite constellation at altitude 1000 km with an inclination of 45° and in circular orbits. Let it have 2 orbital planes with 4 satellites in each plane. The graph below shows the variation in coverage with the variation in number of satellites. Here the ground terminal elevation mask is set at 10°, which means that the ground terminal can see the full sky except 10° from the horizon.

The variation in coverage vs altitude, with the total number of satellites fixed at 8 is shown below. The values of coverage is estimated using the software SaVi.

We can clearly see that the coverage increases significantly with the increase in Altitude. The downside of using satellites at higher altitude apart from the launch cost is the increase in power needs for data transmission and longer signal propagation periods (higher latency).

### Coverage by a LEO Constellation

In contrast to Geostationary satellites , many LEO satellites are needed to provide continuous coverage over an area. However satellites in Low Earth Orbits enjoy the benefits of shorter distance to the Earth’s surface. The key advantages of using LEO constellation are listed below.

• Shorter signal propagation periods (low latency). The minimum theoretical latency for LEO satellite is 1-4 milliseconds whereas the latency for GEO satellite is 125 milliseconds.
• Lower power needed for data transmission and instrumentation
• Better resolution for imaging applications, and also for other earth-observation applications.

The high velocities of LEO satellites relative to the surface imply short contact periods to ground stations and short observation periods of specific surface areas by a single satellite. Hence several satellites in appropriate complementary orbits are necessary to provide continuous coverage.

### Walker-Delta Constellation

A frequently used design technique is the Walker-Delta pattern constellation for a global coverage of the Earth’s surface by a minimum number of satellites in circular orbits. The Walker constellation is denoted by a notation

i: t/p/f

where

• i : inclination
• t : total number of satellites
• p : number of equally spaced orbit planes
• f : relative phase difference between satellites in adjacent planes

A Walker-Delta pattern contains of total of ‘t’ satellites in ‘p’ orbital planes with $s=\frac{t}{p}$ satellites in each orbital plane. All orbital planes are assumed to be in same inclination ‘i’ with reference to the equator. The phase difference between satellites in adjacent plane is defined as the angle in the direction of motion from the ascending node to the nearest satellite at a time when a satellite in the next most westerly plane is at its ascending node. This is illustrated in figure below. In order for all of the orbit planes to have the same phase difference with each other, the phase difference between adjacent satellites must be a multiple ‘f’ of $\frac{360^\circ}{t}$, where ‘f’ can be an integer between 0 to p – 1.

### Designing a Walker-Delta Constellation

After defining the number of satellites, number of orbital planes, semi major axis and inclination, specific to the mission, the true anomaly and the right ascension of ascending node can be calculated using the spacing rule defined by John Walker. The eccentricity and argument of perigee can be ignored as most Walker constellation orbits are circular. The steps involved in designing a Walker constellation are simplified and listed below.

1. Calculate the number of satellites needed to satisfy the mission requirements , ‘t’.
2. Select the number of orbital planes that provide maximum coverage and at the same time obeys the specified cost constraint, ‘p’.
3. The ascending node of the ‘p’ orbital planes should be equally distributed around the equator at intervals of $\frac{360^\circ}{p}$.
4. Define the number of satellites per plane, $s = \frac{t}{p}$.
5. Within each orbit plane, ‘s’ satellites should be equally distributed at interval $\frac{360^\circ}{s}$.
6. The spacing between the satellites in adjacent planes should be ‘f’ multiplied by spacing between the satellites in a orbit plane [(i.e) $\frac{360^\circ}{s}$.] divided by the number of orbital planes.
7. Spacing (angular) between satellites in adjacent planes = $f \times \frac{360^\circ}{s \times p}$.

Walker-Delta constellation design is a milestone in constellation design process but it should be noted that it is one among the various options available and does not necessarily provide the best characteristics for a given mission. For further reading about Walker-Delta constellation, see here and here.

### Galileo Constellation

A famous example of Walker-Delta Constellation is the Galileo constellation. The satellites are placed as a 56°: 27/3/1 constellation, having 27 satellites in orbit, placed in 3 orbit planes separated by 120°. The altitude of the constellation is 23,222 km. The orbital planes are at an inclination i = 56° and hosts 9 satellites at an angular distance of 40° in a plane. The phase shift between adjacent orbits is $f \times \frac{40^\circ}{3} = 13.33^\circ$.  The Galileo constellation has been optimised and its orbital parameters are chosen in such a way that it provides continuous global coverage.

The Ground traces (green and pink in color) of two Galileo satellites (marked as two dots) are highlighted in the figure below.  The figure also shows the orbit of the two satellites along with their directions.

### Conclusion (A New Beginning)

Satellite constellations provides effective solutions for the skyrocketing demands in numerous fields. A breakdown of various parameters influencing a satellite constellation has been presented in this article. The choice of appropriate values to these parameter are limited to the mission needs and to the constellation designer. As a consequence of complexity in the design process, even after 50 years, the constellation design process is still considered to be in it’s infant stage. The time proven simplified constellation geometries and design processes have reached a plateau and no longer satisfy the modern mission needs. This has created a need for new benchmarking processes. Numerous researches are being carried out in this field and one could expect cosmic advancements in the near future.

Study carried out by Raja P, Intern at Astrome Technologies.