Introduction

The objective of space surveillance is to create an inventory of objects above a certain size that are in orbit around Earth. This inventory (catalog) provides information about the origin of the objects (name, launch country) and their trajectory (orbital parameters), allowing them to be located at a later date. To fulfil this objective, different types of sensors must be used: first of all, detection systems with a wide field of view to see passing objects above a certain size and roughly calculate their orbit in order to locate them later on; then, tracking systems with a narrow field of view are used to follow a specific object, in order to take trajectographic measurements and improve knowledge of its trajectory. Basically, these detection and tracking systems are radars for low-orbiting objects or telescopes for objects in higher orbits. They can be located either on the ground or in orbit. The population of smaller debris below the size limit is discerned in a statistical and no longer deterministic way: this information is given by flux models such as Ordem (NASA) or Master (ESA). Some authors consider that space surveillance also includes functions relating to space weather and the monitoring of near-Earth asteroids whose trajectories could threaten Earth. In the rest of this article, we will only be considering the functions related to space traffic control, that is to say detecting and tracking artificial objects in orbit around the Earth.

The main source of information is the space surveillance system set up by the United States, which provides the most comprehensive information. Russia has a similar system for which very little information is available. On the other hand, the International Scientific Optical Network (ISON) telescope network provides a detailed catalog of objects in geostationary orbit. Finally, France has limited capability with the GRAVES system.

USA

(See NASA Handbook 8719.14, 2008; Chatters & Crothers, 2009.)

The US space surveillance network (SSN) is a combination of optical and radar sensors used to support the Joint Space Operations Center’s (JSpOC) mission to detect, track, identify, and catalog all manmade objects orbiting the Earth. The JSpOC, located at the Vandenberg Air Forces Base, is in charge of programming the sensors, and recovering and analysing the data to compile and manage the catalog. The American network carries out the following functions:

Several types of sensors are used to carry out these functions:

The SSN sensors are divided into three categories: dedicated, collateral, and contributing. A dedicated sensor is a US Strategic Command (USSTRATCOM) operationally controlled sensor with a primary mission of space surveillance support. A collateral sensor is a USSTRATCOM operationally controlled sensor with a primary mission other than space surveillance (usually, the site’s secondary mission is to provide surveillance support). Contributing sensors are those owned and operated by other agencies that provide space surveillance support upon request from the JSpOC.

The SSN is able to follow objects of around 10 cm in low orbit (possibly 5 cm for low-altitude objects and those on inclined orbits) and objects of around 1 m in geostationary orbit. On 1 April 2011, the public catalog contained 14,870 objects.

Combined, these types of sensors make up to 100,000 satellite observations each day. This enormous amount of data comes from SSN sites such as Maui, Hawaii; Eglin, Florida; Thule, Greenland; and Diego Garcia, Indian Ocean. The data is transmitted directly to JSpOC via satellite, ground wire, microwave and phone. Every available means of communication is used to ensure a backup is readily available if necessary (Figure 8.1.1).

To set up and manage the catalog, the SSN uses two different orbitography models: a model called General Perturbations (GP), an analytical model, which uses a simplified representation of the forces and a Special Perturbations (SP) model based on numerical integration with a more precise representation of the forces. Only less precise information produced with the GP model is available on the Space Track website. This information is given in Two-Line Element (TLE) format: the SSN and Committee on Space Research (COSPAR) number of the object, average orbital parameters on two lines (see detail and format on the Space Track website, https://www.space-track.org/).

The TLE only provides a rough knowledge of the orbit: the uncertainties can reach several kilometers from the creation of the TLE and then quickly deteriorate over time. In particular, this information is not precise enough to carry out a reliable collision risk prediction. Nevertheless, since 2010, the JSpOC has been setting up agreements with operators wishing to be provided with collision warnings based on precise data (SP) and messages called Conjunction Summary Messages (CSM) are sent to the operators. CSMs contain precise information on dangerously close objects and the associated uncertainties, which makes it possible to calculate the collision probability.

Russia

France

Since 2005, France has had a limited low-orbit space surveillance capability using the GRAVES system (Grand Réseau Adapté à la Veille Spatiale). The system consists of a single bistatic radar associated with significant calculation systems and a secure transmission network.

The transmitter site is located near Dijon: it consists of four transmission antennas, each covering a sector of 45° in azimuth and 20° in elevation: the total cover in azimuth is 180°. The radar continuously transmits in VHF.

The receiving site, located near Apt, consists of 100 antennas distributed over a metallic disc. Each antenna is linked up to an individual receiver and the detection beam is then digitized (Figure 8.1.3).

The system is operated by the French Air Force. It enables a catalog of around 2500 objects located below 1000 km in altitude to be independently compiled and maintained. Every object is seen at least once every 24 hours and the orbital parameters are determined using a single passage.

Transmitting antennas emit a continuous low-frequency signal towards a given angular section of space. The receiving site, located nearly 400 km away, houses a large number of omnidirectional antennas. Based on the elementary signals picked up by these antennas, a narrow-lobe beam is produced. The direction of this lobe provides an angular measurement of the object detected, while the frequency shift between the emitted signals and the received signals measures its radial velocity.

Based on this brand-new concept, the GRAVES radar provides angular and radial velocity measurements. These are fed into the orbital processing algorithms developed by ONERA (Office National d’Études et de Recherches Aérospatiales) to calculate the orbital parameters of the detected satellites (Figure 8.1.4).

ESA

In 2008, the ESA Member States decided to set up a space surveillance program (SSA – Space Situational Awareness), which aims to equip Europe with independent systems 10 years from now (Declaration of Preparatory Space Surveillance Programme, 2010). In the field of space surveillance, services cover the detection and tracking of objects over a given size and the establishment of a catalog of these objects, the launch of collision risk warnings, the recommendation of avoidance maneuvers, as well as the detection of explosions in orbit. Services will also cover the prediction of high-risk re-entries. In the field of space weather, services involve monitoring the Sun, solar wind, radiation belts, the magnetosphere and the ionosphere. Furthermore, preliminary activities concerning near-Earth objects and establishing a catalog of these objects will be carried out.

The preparatory phase of this programme covers the 2009–2012 timeframe and includes four elements:

The operational development phase will be decided at the end of the preparatory phase at an Agency Council meeting at ministerial level, decided an additional preparatory phase in order to consolidate the definition.

Prevention of Collision Risk

Following the increase in the number of objects in space and the heightened risk for satellites, preventive measures (mitigation) have been decided, first of all at the level of the main space agencies, then at international level within the framework of the Inter-Agency Space Debris Coordination Committee (IADC), which brings together 12 agencies: ASI (Italian Space Agency), CNES (French Space Agency), CNSA (China National Space Administration), CSA (Canadian Space Agency), DLR (German Space Agency), ESA (European Space Agency), ISRO (Indian Space Research Organisation), JAXA (Japan Aerospace Exploration Agency), NASA (US National Aeronautics and Space Administration), NSAU (State Space Agency of Ukraine), ROSCOSMOS (Russian Federal Space Agency), and UKSA (UK Space Agency). So, in 2002, the IADC published the document “IADC Space Debris Mitigation Guidelines”, which is still the reference in this field today. This subject was also discussed within the more general framework of the United Nations, at the Committee on Peaceful Uses of Outer Space (COPUOS). This committee was created in 1959 and brings together 70 countries: its work resulted in the publication in 2007 of the document UN-COPUOS Mitigation Guidelines, which defines the main principles of the preventive measures to apply in space regarding space debris. These documents recognize, in particular, the need to urgently protect two areas in space because of their congestion and their importance for space applications, namely the low Earth orbit (LEO), which corresponds to altitudes below 2000 km, and the GEO region, which corresponds to altitudes between ± 200 km around the geostationary orbit and to inclinations between ± 15 degrees (Figure 8.1.5).

At the same time, the International Organization for Standardization (ISO) is developing standards for space debris from these reference documents, which manufacturers and operators will be able to use.

These texts are not compulsory in nature. They are only recommendations. The applicable rules depend on each country and are presented in laws (e.g., Space Operations Act in France) or by licensing systems as in the United States or the United Kingdom. These different texts remain consistent for they are all derived from the IADC document.

Prediction of Collision Risk

• Space users must have access to reliable and precise information on the trajectories of all objects above a certain size: today this function is fulfilled by the JSpOC, whose policy of data dissemination was developed after the collision between Iridium 33 and Cosmos 2251. The other sources of information are not as comprehensive and their availability is not guaranteed.

• A code of conduct must exist: the work carried out within the framework of COPUOS and IADC resulted in recommendations that are applied voluntarily by most operators. These recommendations consist of preventive measures, particularly relating to end-of-life management, and operational measures aiming to predict objects coming dangerously close to each other in orbit or at launch. The application of these measures is now reinforced by their integration into national legal systems such as licensing systems or laws on space operations.

• Increased coordination is necessary between operators sharing neighboring orbital positions. This is particularly the case in geostationary orbits where a station longitude is allocated to every satellite by the ITU. Every operator therefore knows its “neighbors” and can inform them of maneuvers for positioning, longitude change or end-of-life exit from the area. This information is also particularly important during operational life, as a breakdown may lead to a satellite drifting towards positions occupied by others.

Passive Safe Operation Alternatives

If courteous behavior is unlikely and responsive avoidance is infeasible, the next alternative is to operate satellites in orbits that are unlikely to suffer crowding and collision risk. This might compromise essential mission capability. Either more satellites might be required for the same capability or additional propellant, propulsion, and guidance systems will be required.

This is the approach used in astronomy. Given comparable atmospheric clarity (seeing), telescopes must be located far from terrestrial sources of illumination (light pollution). Radio telescopes must be able to discriminate among extremely weak receptions from distant galaxies. They mitigate electromagnetic interference by being located remote from human electromagnetic activity. This makes these operations less efficient and more expensive.

We will explore several alternatives for safe operation, minimizing fratricidal risks and vulnerabilities to debris. For given mission requirements, we will characterize compromises to mission execution and the additional costs of more assured safety. We intend this to be the basis for incorporating safety as an architectural consideration.

Inherent collision risk is the most important safety discriminant. Our ability to know where satellites are continuously and precisely is another consideration. Orbits and the satellites themselves should also consider how well and how often they can be monitored. Space surveillance in any embodiment will not be able to see everything, everywhere, all of the time. More and better distributed sensors will only push the threshold of perception farther, not eliminate it. As sensors proliferate the value added by each new one will diminish.

We can mitigate the diminishing return of more sensors by designing satellites, orbits, and constellations with observability as a consideration. Useful highly eccentric, highly inclined orbits are least observable where their orbits are likely to change most, during rapid perigee passage in the far Southern Hemisphere.

Ideally, increasing observability could have no impact on mission capability or satellite control. Dilectis and Mortari pioneer analytic techniques in orbit design for observation from designated Earth sites (Dilectis & Mortari, 2011). Satellite signatures can also be augmented for observability. Many close approaches could be eliminated with better and more responsive trajectory data that would make possible very small mitigation maneuvers.

Observability is much more than line of sight to a satellite from a terrestrial sensor or even the duration of access. The number of independent observation opportunities, the geometry of the access, the frequency of revisits, and measurement quality matter greatly. Vallado and Griesbach have examined the effect on orbit determination precision of these satellite–sensor pair characteristics (Vallado & Griesbach, 2011). They have exquisitely cataloged hundreds of possible trajectory observation sites. Harrison and Finkleman used the same techniques to estimate the potential space situational awareness contributions of the numerous sensor systems (predominantly telescopes) that might participate word-wide (Finkleman, 2010).

Many dimensions of capability must be considered, for example whether a sensor is active (radar or laser) or passive (telescope), whether the satellite must be illuminated by the Sun or Moon, what on-site communications and data processing there are, and whether satellite signatures are augmented actively or passively. Vallado and Griesbach codify the constraints. The range of orbits that is feasible with adequate observability is potentially large enough that a greater variety of orbits could be employed for the sake of mitigating crowding and collision probability.

Figure 8.2.3 shows the extent of inclination of low Earth orbit satellites that are accessible with reasonable frequency and duration using modest telescopes with active laser illumination. These all contribute to precise orbits and ephemerides (POE) that others use to calibrate their instruments and orbit estimation techniques.

Characteristics of Unsafe Orbits

There are some obvious safety guidelines. First, an inclined orbit will cross the planes of all satellites with lower inclination twice on each revolution. Even satellites at the same altitude in different planes at different inclinations whose orbital locations (true anomalies) are out of phase will eventually come close together because of orbit perturbations. The higher the inclination, the more orbit planes of lower inclination satellites the subject satellite will cross. If the satellite belongs to a constellation all members of which are at the same altitude and inclination, the separation between orbits decreases with increasing latitude. The orbits will be clustered densely near Poles. The more satellites in each plane, the higher the frequency of close approaches.

Despite the unarguably greater encounter risk, highly inclined orbits are very desireable. Highly inclined orbits are a necessity for operators that must serve users at high latitudes. Critical inclination with minimal precession of line of apsides is a relatively high inclination.

Iridium is an example of a high risk constellation. On 30 August 2011, more than 1000 approaches within 5 km were estimated with rocket bodies, other debris, and even other Iridium satellites over the ensuing week. Much of the debris from the Cosmos 2251–Iridium 33 collision remains in Iridium-like orbits, increasing the encounter risk to other Iridium satellites. Iridium satellites themselves and the cross-linked communication network are extremely robust to on-orbit losses, but that might be a necessity.

Alternative Orbits for Surveillance

John Draim was principal inventor of the elliptical Ellipso, Virgo, and Cobra constellations and holds many patents in space technology. He also holds the only patents that provide nearly continuous global coverage with only four satellites (Draim, 1987). Circular orbit arrays for continuous global coverage require a minimum of five satellites. Using elliptical constellations, coverage can be concentrated geographically by latitude and/or longitude, and/or by time of day. Circular constellations generally waste much coverage on areas of little interest. Draim has received international recognition for his work on elliptical constellations that can provide users with more efficient system designs using the minimum number of satellites. His most recent “Droplet” designs allow the use of fixed ground antennas (similar to GEO systems) pointed toward mid- and high-latitudes and thus avoid any radio frequency (RF) interference with the GEO systems. In addition, they are designed to avoid major orbital debris fields, as well as the infamous van Allen radiation belt.

There are no general analytical techniques for constellation design. Selecting satellite orbits for a specific mission is a trade-off among: coverage, station-keeping effort, launch effort, onboard power requirements, communication access and energy requirements, persistence, and several other factors. There are no unique or universally optimal solutions. Numerous combinations of orbit characteristics, numbers of satellites, and onboard communication or surveillance capabilities will satisfy almost any specific mission criterion. We add avoiding close approaches with other satellites and minimizing the consequences of fragmentation to the trade-off criteria.

The first example is Draim’s eight satellite droplet constellation (Draim, 2009). Through numerical optimization, Draim arrived at constellation families for complete coverage of the Northern Hemisphere. The families include constellations of eight, 12, and 16 satellites, with 63.435 degree inclination, 16 hour periods, and two day repeating ground tracks. Figure 8.2.4 is the eight satellite configuration.

This constellation provides near stationary presence at high latitudes with very small antenna pointing excursions.

Figure 8.2.5 demonstrates that the satellite laser ranging network alone has complete visibility of the constellation at apogee. Darker areas are access by a single sensor, lightest areas are triple or more coverage. Figure 8.2.6 demonstrates complete coverage at perigee.

Figure 8.2.6 shows that the droplet orbits are visible to one or more single-lens reflex (SLR) sensors at perigee altitude. There is a small region off the tip of South America which enjoys only single or double coverage.

Conjunction searches against the complete, openly available satellite catalog confirm almost complete freedom from close approaches. Over weeks, there are only two approaches within 100 km of any of the Droplet Constellation satellites.

This is a representative example of an exquisitely safe constellation that fulfills a well defined mission. In some ways, this constellation has unique advantages. For example, such continuous coverage of a mid-latitude region anywhere in longitude would require many more satellites in more conventional orbits, particularly circular.

There are several additional practical and cost trade-off elements. In general, more satellites might be necessary than in more conventional architectures. These include capabilities of each satellite in terms of aperture required for desired resolution, motion required for reasonable relative collaborations such as frequency difference of arrival discrimination, and launch costs.

TLEs and State-Vector Propagation

The propagation of the state-vector of any satellite starts with the selection of the most appropriate set of information at a reference epoch. This information can consist of the 3-dimensional position and velocity in Cartesian components, or be given as Kepler elements; the exact form is unimportant for the purpose of this conjunction analysis. For obvious reasons, one should use a state-vector that is most reliable, and that is defined and available in a way that is common for all spacecraft being analyzed. The well-known sets of Two Line Elements (TLEs) generated by USSTRATCOM (2012) satisfy both requirements: they are typically given with frequencies ranging between several times per day for manned spacecraft, to several times per week for “average” active satellites, to once per (few) week(s) for extinct rocket boosters and such. The state-vector information in each TLE is given on two successive lines of ASCII code, which provide information on the Kepler elements, derivatives thereof, and a few other parameters related to the dynamic behavior of the vehicle under consideration, such as an atmospheric drag scaling parameter. The TLEs are made available by the US authorities (USSTRATCOM, 2012) and CelesTrak (Kelso, 2010). As mentioned before, the database encompasses about 16,000 objects at the time of writing, and it is the most complete and consistent set of data publicly available.

In principle, the propagation of a satellite state-vector can be done in a number of ways. The most common way is to numerically integrate the position and velocity of a satellite, while evaluating the instantaneous accelerations that act on the vehicle for each individual time-step. Such an integration can be done with Runge–Kutta (RK) integration schemes, where the step-size (fixed or variable) and the accuracy of the specific technique (RK4(5), RK7(8)) can be chosen at will (typically, this is a trade-off between accuracy and efficiency). Alternatives for RK methods are also possible, of course. Another option is to integrate the Kepler elements that describe the position and velocity of the vehicle in a geometric way. In that case, one uses the so-called Lagrange Planetary Equations, or the Gauss form thereof (e.g., Cornelisse et al., 1979). Here again, the integration type and specifics can be chosen at will.

However, all of this loses its relevance when one chooses to use the TLEs. Since these elements essentially provide the Kepler elements, the values of which were estimated in an orbit determination with an environment model with limited accuracy (central gravity, a few low-order terms of the gravity field, possible resonance, aerodynamic drag, solar radiation pressure, third-body acceleration due to Sun and Moon), one should stick to the conventions of this force model when making extrapolations of these state-vector solutions. Fortunately, such a propagation can easily be done, since USSTRATCOM and CelesTrak provide propagators ready for this task. In the past, and depending on the problem at hand, the user had to choose between the SGP4, SDP4, SGP8 and SDP8 propagators (SGP and SDP are abbreviations for simplified general perturbations and simplified deep-space perturbations, respectively) (Hoots & Roehrich, 1980). However, through time improvements have been made to these propagators, and currently the best version publicly available is the SGP4 propagator as described by Vallado et al. (2006); it inherently covers al possible orbital regimes (i.e., low and high orbits).

Brute Force

In the most simple conjunction analysis, one follows a so-called brute-force approach: propagate the pair of any two vehicles to a certain epoch, evaluate whether they approach one another too close or not, and then move on to the next epoch (e.g., 1 s further in time). Once the entire time interval of interest has been covered, the user switches to the next pair of vehicles. Ideally, the test “coming too close” suffices to deal with the volume physically taken by the (combination of) spacecraft, i.e. distances of several tens of meters. In reality, one has to incorporate uncertainties in the representation of the satellite orbit, which has repercussions for the techniques described here and the subsequent section.

It will be clear that this approach is not very efficient, if only for the fact that in this way the state-vector of a particular (user) satellite is integrated repeatedly (which can easily be solved, of course). Also, the relative velocity between an arbitrary pair of vehicles can be several tens of km/s, so when tested against a minimum distance of, e.g., 25 km at specific epochs, care must be taken not to miss any potential conjunction (or worse). To prevent this from happening, one typically increases the critical distance R_cr to a threshold value R_th, which allows for the relative motion in between evaluated steps Δt (Figure 8.3.3).

Knowing that the individual objects move with an absolute velocity, which cannot be larger than the local escape velocity, one can readily derive the following equation for this threshold radius):

(2)

One further refinement is to increase this distance with the effect of possible accelerations during each evaluation step. As a worst case, one can assume that the acceleration of an individual vehicle is the gravity acceleration at sea level, so the relative acceleration is twice this value. The new expression then becomes:

(3)

The expressions above show one of the main problems with this approach: in order to identify a reasonable amount of possible conjunctions, the testing criterion R_th cannot be set too high, i.e. the evaluation time-intervals of this brute-force method cannot be set too large. As an example, if R_cr and Δt are set at 25 km and 1 s, respectively, one would have to test against an R_th of 36 km. This implies the computation and pair-wise comparison of 86,400 times 16,000 objects for a single day (which will be very demanding on the computers involved). Instead, switching to an evaluation step-size of 10 s (60 s) would result in a value for R_th of 138 km (732 km). Although the number of evaluations decreases by factors 10 and 60, respectively, the larger value for R_th will significantly increase the number of possible conjunctions (which all need further in-depth analysis).

Clearly, this process can be optimized. Without going into detail, an optimal step-size appears to be about 20 s; optimal in the sense of reasonable number of conjunctions for a reasonable cpu-time. Yet, an analysis of a full catalog for a single day easily costs about a full day of cpu-time. To make this first-order assessment of possible conjunctions more efficient, filters and sieves are typically applied.

Filters and Sieves: Theory

Following the brute-force method, one would have to analyze all pairs of state-vectors and determine their mutual distance for all epochs in the dataset. As indicated, this would be an extremely laborious process. So, it would be very advantageous to have a tool to eliminate certain combinations of satellites directly, without going into the computation of distance, velocity, and such, for each individual evaluation epoch. Tools have been developed for this purpose: filters and sieves. In essence, they do the same thing: a quick yet reliable elimination of “impossible combinations”; filters do this for a complete interval of interest (e.g., a day), whereas sieves do this for an individual epoch.

Perturbed-Orbit Propagation

A simplified representation of the orbital state does not suffice for accurate orbit prediction. The unperturbed Kepler model may serve well as a first-order approximation or as a means to analytically analyze the orbit-propagation problem. It will not hold, though, when predictions of the satellite motion under the influence of not only the gravitational attraction of the main body but also a variety of perturbing accelerations are required. These perturbations may come from the Earth’s atmosphere or magnetic field, the third-body attraction of the Moon or the Sun, or the solar-radiation pressure, to name but a few.

In general-purpose astrodynamics tools, the state of an orbiting body and the related equations of motion, are often described in Cartesian coordinates. Its form is very simple, and if we separate the external acceleration into a main and perturbing component, the equations read:

(14)

where r_I is the position vector in the inertial frame, μ the standard gravitational parameter of the central body, and a_pert,I the sum of perturbing accelerations.

For numerical integration, eq. (14) can be written as a system of first-order differential equations:

(15)

with

(16)

These Cartesian coordinates have a physical meaning (position and velocity) and the orthogonality of the system results in simple equations of motion and transformations. It is therefore the preferred state model in many celestial-mechanics applications. However, an unperturbed two-body problem is an orbit, the shape and orientation of which can be described by five constants, and the position along the orbit by a single variable. The Cartesian model does not use this information, requiring a smaller step-size to reduce the integration errors.

The two-body orbit-shape and orientation description is the basis for the traditional Kepler elements. These elements define the shape and orientation of the satellite orbit and its position along this orbit (usually given by the true anomaly, θ). Such a state model, including the differential (or variational) equations to account for the change in orbit parameters due to the perturbations, is often more efficient and/or accurate in numerical integration. This is a result of the larger step-size that can be used without the loss of accuracy, because the main gravitational force has no or a linear effect on the variation of the orbital elements. This is in contrast with the Cartesian velocity components that are all driven by the main force with a quadratic dependency on the position. The major disadvantages of using the perturbed Kepler elements are the extra complexity of the model that make it harder to implement and verify, and singularities for both circular, equatorial and polar orbits.

To account for these singularities the modified equinoctial element (MEE) set, a mathematical combination of the Kepler elements, has been defined (Walker et al., 1985):

(17)

(18)

(19)

(20)

(21)

(22)

where the traditional Kepler elements are given by: a, the semi-major axis; e, the eccentricity; ω, the argument of periapsis; Ω, the right ascension of the ascending node; i, the inclination; and θ, the true anomaly. All but L are constant for an unperturbed two-body problem. When perturbation forces act on the body, the following differential equations describe the change of the MEE due to these forces (Betts & Erb, 2003):

(23)

(24)

(25)

(26)

(27)

(28)

In the above equations a_e1, a_e2 and a_e1 are the three components of the perturbing acceleration vector, expressed in the radial direction, e₁, perpendicular to the radius vector, e₂, and a component perpendicular to the orbital plane, e₃. As models for the perturbing accelerations are commonly available in an inertial, Cartesian frame, a_e can easily be obtained from a_pert,I through a frame transformation involving ω, Ω, i, and θ.

The MEE has three disadvantages over Kepler elements. First, it is obvious that the MEE lose the physical meaning that Kepler elements have. Second, the variational equations are more complex, resulting in a higher computational cost. Third, the rotation matrix in MEE is more complex than the Euler angles in the Kepler elements (this transformation matrix is required to transform the perturbing forces from the inertial to the orbital frame).

Despite these disadvantages, the major advantage of being singularity free is of prime importance. Of course, many more orbital-state models and related variational equations have been developed over the years. The MEE has been compared with many of them and came out as the best one in terms of accuracy and computational speed (Hintz, 2008).

A second orbital-state model that shows a potential of being a fast and accurate orbit propagator is the unified state model (Vittaldev et al., 2010, 2012). The unified state model (USM) uses seven state variables to describe the state of an orbiting body: three to define the shape of the orbit in the velocity phase space and four quaternions to define the orientation of the orbital plane with respect to the inertial reference frame.

C, R_f1 and R_f2 are functions of the radial and angular momentum that determine the energy of the satellite and the size and position of the orbit in velocity phase space (the velocity hodograph); see Figure 8.3.7. R_f1 and R_f2 are the components of R in a convenient intermediate reference frame that is rotated about the line of nodes. R has a constant direction, and points in the direction of the velocity at periapsis. C is always perpendicular to the radius vector. The satellite’s velocity, v, is the sum of both, i.e., v = R + C.

For an unperturbed two-body problem, the magnitude of both R and C is constant. Since the orbit in velocity phase space is constant for an unperturbed two-body problem, these variables are also constant for the unperturbed case. Where the MEE use the shape and size of an orbit (ellipse) in the position phase space, the USM uses C and R to define the orbit (circle) in the velocity phase space.

The USM uses a quaternion to keep track of the orientation of the orbital plane (Altman, 1972). q₄, q₁, q₂, q₃ are the four quaternion elements, , and are also called Euler parameters. Since four parameters are used to define three degrees of freedom, the following constraint on the quaternions needs to be introduced: .

Quaternions have two major advantages over Euler angles. No singular conditions occur in a rotation, the degrees of freedom are never lost, i.e., there is no gimbal-lock, and all angular functions are algebraic instead of trigonometric (Kuipers, 1999).

As mentioned, the unified state variables C, R_f1 and R_f2 are constant for the unperturbed two-body problem. If perturbing forces act on the body, the equations of motion in terms of the perturbing force components in the e₁, e₂ and e₃ direction read:

(29)

(30)

where:

(31)

which becomes singular for q₃ = q₄ = 0. This corresponds with retrograde equatorial orbits (i = 180°). This means that the USM breaks down for retrograde equatorial orbits. This orbit can be avoided by reformulating it with a negative orbital velocity and zero inclination.

The body-fixed velocity components v_e1 and v_e2 are given by:

(32)

The parameter γ in eq. (29) is used for compactness, and is defined as:

(33)

This term also becomes singular for q₃ = q₄ = 0, which are retrograde equatorial orbits as explained above.

The variable p is the ratio between C and the velocity perpendicular to the radius vector and ω_i is a body-fixed angular velocity component:

(34)

The angular velocity about the e₂-axis, ω₂, is zero, because there is no velocity component out of the instantaneous local orbital plane.

Orbital Stability and Monte-Carlo Analysis

Stability has many meanings. In the context of the current discussion we will look at the dynamical stability of satellite orbits, and in principle we would like to answer the following questions. What is the influence of small changes in initial state on future states? How does a small change in velocity affect the future states of the satellite? Is an Earth orbit stable and in what sense is it stable? A Monte-Carlo analysis may help us to answer these questions.

Small changes in initial values have small influence on a satellite’s orbit size and shape, i.e., a two-body orbit is orbital stable. However, small changes in initial values have large influences, increasing with time, on the position of a satellite in its orbit. The orbit is Lyapunov unstable (or dynamically stable) in the along-track direction, which can be shown as follows.

Imagine two satellites with unperturbed circular orbits around Earth. The orbits are almost equal, but have a small difference in radius, Δr. The angular velocity, n, of these satellites is given by:

(35)

and thus the difference in angular velocity is:

(36)

The relative angular position, Δθ, will thus increase with increasing time (Battin, 1999):

(37)

This shows that no matter how small the difference in radial position, the difference in angular position along the orbit will always increase with time. In other words, the orbit is unstable in the along-track direction. This along-track dynamical instability leads to a numerical instability for orbit computations. For conventional numerical integration this causes an unknown error in the along-track direction.

These analytical expressions will, of course, not hold for perturbed orbits although they may give a good first estimate. For perturbed orbits, though, one numerically propagates the trajectory for the two different initial states. One can easily extend this concept to the uncertain measurement of the initial satellite position and velocity. Suppose the initial position and velocity are known with an error e_r and e_v, respectively. This means that the initial state could be anywhere in the interval (r₀-e_r,r₀ + e_r) and (v₀-e_v,v₀ + e_v), where r₀ and v₀ are the actual initial position and velocity, respectively. Let us indicate this interval with (r) and (v).

To predict where the satellite will be after a certain period of time we need to take all possible initial conditions (r) and (v) into account. That means that if we assume a certain distribution of (r) and (v), e.g., a normal (= Gaussian) or uniform distribution, we could sample the respective intervals n times and do an orbit propagation for each of those samples. This is the basis of a Monte-Carlo analysis.

As an example we will show the results of a Monte-Carlo simulation, where 10,000 initial points have been randomly picked from a given 3-dimensional position interval with widths of 1 km. The resulting positions after 1.5 orbits are shown in Figure 8.3.8 (left). The red Monte-Carlo solution set is positioned along the blue nominal orbit; this shows the along-track instability. It grows from about 30 km after 1.5 orbits to over 400 km after 20.5 orbits. Note that the effect of position and velocity changes is largest after n + 0.5 orbits.

Figure 8.3.8 (right) shows a similar solution set, but now caused by a larger uncertainty in the velocity of 100 m/s, after 1.5 orbits. Again, the along-track instability is clear from the shape of the solution set that spreads out mostly in the direction along the nominal orbit. In fact, after 20.5 orbits the satellite can be located almost everywhere along its orbit. When the uncertainty in the velocity is 1 km/s in all components, the solution encloses almost the full orbit after only 1.5 revolutions.

Current collision-detection methods are based on estimates of the collision probability. As we have seen before, this collision probability depends on measurement errors in the state of the satellites at time t₀. These errors are estimated by propagating the (assumed) known error probability density function at a certain starting time t₀ to time t₁, the estimated time of closest approach.

This probability propagation is traditionally done using analytical or conventional numerical integration methods. These methods introduce errors in the predicted trajectory at every integration step. These errors, however small they may be, can grow into much larger errors, mainly due to the along-track instability of satellite orbits. Furthermore, the satellite position is only predicted at fixed-points in time and not over a continuous time interval. Thus, they do not cover every moment that collisions may occur.

Given the limitations of Monte-Carlo analysis coupled with the fact of discrete-time prediction this may not be the best approach to try and detect possible collisions between satellites. Therefore, in the next section we will introduce an alternative method.

Verified Interval Propagation

Verified integration methods form a novel type of integration methods that can both enclose integration errors and integrate for a continuous range of (uncertain) initial values and parameters (Römgens et al., 2011). Moreover, they provide solutions valid over a continuous time interval rather than for discrete time points. Verified integration has been developed by a small group of mathematicians over the last 30 years, but has not yet been applied to many real-world problems (Neher, 2005).

Verified interval orbit propagation is the application of verified integration to solve the equations of motion that model satellite trajectories. This group of integration methods can integrate for a range (interval) of initial positions and velocities, and for a range of perturbing forces (parameters) that are applied during the integration time. It creates a guaranteed satellite trajectory enclosure, i.e., a 3-dimensional trajectory enclosing all possible trajectories corresponding to a range of initial conditions and forces. Moreover, it guarantees to include all numerical integration and rounding errors, thereby making the result verified, and all this with a single integration. An example of such an orbit enclosure is shown in Figure 8.3.9, where adjacent cuboids enclose the satellite’s position during different time intervals.

The next logical step in the development of collision-detection software is the realization that if the interval orbit enclosures of two satellites intersect, it may indicate that the satellites are at the same position at the same time. However, this thought may appear to be more complex than it seems.

Figure 8.3.10 shows the interval position enclosure and Monte-Carlo fixed-point solution set computed using MEE, for a circular equatorial LEO at 400 km, after 10 hours. The cuboid represents the interval enclosure, valid during a time interval of 1 s. The Monte-Carlo solution set, created using 100,000 individual conventional integrations randomly picked (uniform distribution) from within the interval initial values and parameters, lies within the interval enclosure.

The figure also shows the overestimation that occurs when enclosing a non-convex 3-dimensional shape in a 3-dimensional interval. The Monte-Carlo solution set can be considered a good indication of the true solution set, although it does contain numerical integration errors. All other positions inside the solution enclosure are non-solutions. Although the true solution set will always be inside the enclosure, practical application of verified integration becomes useless when there are relatively many non-solutions inside the solution enclosure, e.g., knowing that a satellite is within a cuboid enclosing Earth and all LEO space around it is not very useful.

However, also the opposite is true: if the two enclosures do not intersect it is certain that the two satellites have not collided. And, since the true solution is mathematically guaranteed to lie inside the enclosure, the conclusion about not colliding is absolute.

In terms of orbital-state model, we have the three models as discussed previously in the section “Perturbed-Orbit Propagation.” Vittaldev et al. (2010, 2012) concluded that for perturbed orbits, the USM performs better than Cartesian coordinates for both fixed-step and variable-step integration, in some cases by more than an order of magnitude. Römgens et al. (2011) found that also in interval formulation the USM is superior to the Cartesian model. Moreover, in some cases the MEE perform even better. Therefore, a combination of USM and MEE is the best way to minimize the overestimation error mentioned earlier and to postpone solution explosion.

In Figure 8.3.11, the interval enclosures of different state models (MEE, USM, Cartesian and Kepler elements) are shown for an unperturbed orbit, as well as the intersection of the four models. As the theory of interval analysis guarantees that the actual satellite position lies within each individual enclosure, this means that the smallest possible enclosure will contain the satellite position. So if we use a hybrid state model we can use the intersection of the enclosures to reduce the overestimation. For the perturbed-orbit case we will use a combination of USM and MEE. In practise this means that we propagate the satellite orbit with both state models, determine the intersected enclosure and update the enclosures of the individual models.

To apply hybrid verified interval propagation to satellite-collision detection we will try to reproduce the largest and most notable collision in Earth orbit, i.e., the collision between two intact satellites, the operational communication satellite, Iridium-33, and the inactive navigation satellite, Cosmos-2251. This collision occurred at 16:56 UTC on February 10, 2009, at 789 km above the Taymyr Peninsula in Siberia (Iannotta, 2009).

For the sake of the example, the uncertainty in initial conditions has been taken small: the initial interval widths used are 10 m in all position components and 0.1 m/s in the velocity components. Drag and the J₂-effect are the perturbations that are explicitly modeled. The uncertainty in the density is enclosed in an interval with a width of 2 × 10⁻⁶ kg/m³, J₂ is taken as the interval (1.08262 × 10⁻³, 1.08264 × 10⁻³), and an uncertainty in all other perturbations is modeled by an interval with a width of 2 × 10⁻⁹ km/s². Both orbits are propagated using verified interval integration with an initial step-size of 10 s.

All interval trajectories are checked for intersections with other trajectories. Two possible collisions are found per revolution for the initial integration with a step-size of 10 s. One is at the location where the actual collision occurred and one is at the close approach area of both orbits at the opposite side of the globe (Figure 8.3.12).

When the step-size (time interval) is reduced to 1.25 s, possible collisions remain around the location of the actual collision, but all close approaches on the Southern Hemisphere are determined safe. The first possible collision is detected around 15:15 UTC, the second between 16:55:48.75 and 16:55:51.25. This second collision is the exact time of the real collision. Reducing the step-size to 0.3125 s removes the possible collision at 15:15. The real collision can still not be ruled out, which was expected since the satellites did actually collide.

The interval trajectories and the detected collision interval computed using a step-size of 5 s are shown in Figure 8.3.12. The solid and dashed lines indicate the nominal trajectory of Cosmos-2251 and Iridium-33, respectively. The time dimension of the interval trajectory is indicated by the opacity of the intervals, where a darker interval indicates a later moment in time. Projections of the interval trajectory and the black collision interval are shown on all three planes. This demonstrates that a true collision can successfully be detected by verified collision detection.

• The ISS is on orbit year-round, for perhaps more than two decades, making it much more vulnerable to collision than the Shuttle simply due to exposure time.

• The ISS requires complex interaction among six national and continental space agencies to operate, and notably requires coordination with Russia, who controls the propulsion system of ISS.

• It is the largest satellite ever put into orbit, making it uniquely exposed to the ambient debris flux.

• The mass of ISS itself means that over 100 kg of costly propellant are required just to execute a 1 m/s avoidance maneuver.

• About a dozen governmental and commercial visiting vehicles are on the manifest to arrive and depart ISS every year, all of which desire a stable target for rendezvous.

• “Red threshold”–1 × 10^–4: For any conjunction for which the probability of collision exceeds 1-in-10,000, the operations team must move the ISS out of the path of any conjuncting debris, unless such an avoidance maneuver in itself would cause substantial risk to the vehicle or crew. Two examples of waiving a red threshold maneuver include the case when such an operation might lead to the failure of a crewed visiting vehicle to rendezvous with ISS, or when the ISS propulsion system is incorrectly configured for a maneuver.

• “Yellow threshold”–1 × 10^–5: For a conjunction for which the probability of collision exceeds 1-in-100,000, the flight control team must maneuver ISS to safety unless it causes significant impacts to operations.

Notification Criteria

The first step of the operational flow in collision avoidance operations is the initial screening of the Space Station against the other objects in the debris catalog performed by USSTRATCOM, and the subsequent notification of a conjunction to the Flight Control Team in Houston. The thresholds at which USSTRATCOM warns NASA of a possible threat are variously called “notification criteria” or the “alert box”, and the concepts and thresholds for this part of the operation have evolved over time as NASA’s ISS analysts adjusted to operational realities and learned from experience.

The NASA flight dynamics team that initially established the processes for coming ISS operations utilized Shuttle notification criteria as a starting point in initial trials from 1997–2001, using the Shuttle-Mir “Phase I” program as a test bed. However, with the various considerations that made ISS different from Shuttle, the operations teams from NASA and Russia continually modified the original Shuttle notification thresholds into what have become the current criteria for the “kickoff” of an ISS conjunction scenario:

In most cases, this process guarantees that any object that could collide with ISS is identified early enough for the operations team to take action, while minimizing the likelihood of false alarms. However, there are exceptional situations in which even these safeguards allow an object to enter the alert boxes late in the process and disrupt the normal flow of operations, perhaps even precluding the capability of moving the vehicle to safety. Causes for such “late notification conjunctions” include:

In these late-notification scenarios, there typically is still enough time to assess the conjunction and execute an avoidance maneuver, even though it might require a compressed timeline. There are scenarios, however, where the notification comes with so little time before the time of closest approach that the only option available is to safe the crew in their Soyuz emergency escape vehicles and prepared to depart, in case the debris strikes ISS. This scenario has well-scripted procedures for the crew, and, fortunately, the ISS team has only rarely had to exercise such a “shelter-in-place.”

Planning and Executing a Debris Avoidance Maneuver

Once the ISS team has received notification that a piece of debris will conjunct with ISS within 72 hours, the flight dynamics team at NASA begins the ongoing process of computing the probability of collision. The team will frame those results within the maneuver thresholds that the flight rules define, whether the results are “red”, “yellow”, or neither, and begin the process for planning a debris avoidance maneuver if required. For conjunctions that show little risk of collision, where the Pc is extremely low, usually very little activity among the flight controllers is required other than monitoring by the flight dynamics team. However, for higher-risk situations, planning for a maneuver and evaluating the probability of collision occur simultaneously throughout the scenario.

After learning from some early mistakes in the program that included trying to execute ground procedures too quickly, Russian and NASA operations teams evolved to a standard, minimum lead time to execute a debris avoidance maneuver, and documented it in the multilateral Operations Interface Procedures. Russia has authority over the propulsion system of ISS, so NASA has to issue a formal Planning Process Change Request 28.5 hours before the time of closest approach in order to safely plan the maneuver. Subsequently, NASA has to develop, verify, and uplink the commands to the vehicle. This sequence, if fully executed, ends with a debris avoidance maneuver about 135 minutes (one orbit and a half) before the conjunction.

The avoidance maneuver is evaluated all along the 28.5 hour window. If the trajectory teams believe the risk is high, the ISS operations teams will proceed toward an avoidance maneuver. Between this point and the planned time of ignition, NASA continues to evaluate the collision risk, giving an opportunity to cancel the operation if the risk drops convincingly below the maneuver thresholds defined in the flight rules. There have been several cases in the history of the ISS where a red threshold violation raised 28 hours before the time of closest approach vanished right before the conjunction due to the elimination of large prediction errors as the conjunction approaches. In general, the later the operations team can wait to make a final decision on whether to maneuver, the less likely the maneuver itself will be required. For this reason, and in order to reduce the impact of late-notification conjunctions, the ISS program partners plan to develop new vehicle capabilities that permit a much shorter timeline for planning and executing a debris avoidance maneuver.

When a debris avoidance maneuver is required, two major considerations go into determining exactly the delta velocity: (a) what is required to ensure the maneuver itself keeps the ISS safe from the debris; (b) whether the maneuver negatively affects downstream operations because of the unexpected change in the orbit of the ISS. In order to keep the ISS safe from the debris, trajectory specialists generally aim to perform an impulse between 0.5 and 1.0 m/s in the posigrade direction as often as possible. A velocity of 0.5 m/s is considered a minimum to guarantee overcoming any trajectory uncertainties of the debris and the ISS. For other satellites operating in a lower-drag environment at a higher altitude, 0.5 m/s is often a vastly larger impulse than is required to guarantee safety.

In addition to the safety consideration of the maneuver, ISS specialists must also consider the substantial impacts to the wide array of vehicles due to launch, rendezvous, dock, undock, and deorbit with the station up to two months after the debris avoidance maneuver has been executed. Soyuz and Progress, for example, have certain restrictions on rendezvous altitude and phase angle. Even if a particular vehicle does not have a defined constraint, changes in the orbit of the ISS impact their flight profiles, and perhaps even the date on which they can launch. That is, these vehicles have limits on where the ISS can be in its orbit in order to guarantee a rendezvous and docking in, say, two days, given the program-determined launch date for that vehicle. For a crewed vehicle like Soyuz meeting these constraints is extremely important, while for unmanned cargo vehicles there is typically some flexibility in their rendezvous duration before arriving at the ISS.

Keeping the ISS safe from collision is of utmost importance when designing the impulse of a debris avoidance maneuver, but the ISS trajectory teams must also take into account substantial secondary requirements in order to maximize the efficiency of the program as a whole.

• assembly in space to obtain larger units;

• joining of a vehicle in orbit by an ascending one to exchange crew and goods;

• rescue, repair, retrieval of spacecraft.

• The American space program selected manual control of the approach by an astronaut for at least the final approach prior to contact.

• The Russian space program selected automatic control of the approach by an onboard control system, with the possibility of monitoring and of manual steering by a human operator (in the spacecraft or on ground) as backup.

• The first orbital RVD was performed by the United States on 16 March 1966, when Neil Armstrong and Dave Scott docked their Gemini VIII under manual control to an unmanned Agena target vehicle, a rocket upper stage specially equipped for this mission with a docking interface and visual target patterns.

• The first automatic RVD took place on 30 October 1967 between two Soviet vehicles: Cosmos 186 and Cosmos 188. Both vehicles were unmanned early Soyuz-type spacecraft, which were equipped with the IGLA rendezvous sensor system.

Safety Requirements for Rendezvous Missions

Safety requirements focus on protecting the highest value aspects in the type of mission concerned. For the rendezvous, capture and departure part of a mission it is critical to determine how these high value elements may potentially be compromised during those operations and how the general safety requirements translate into specific technical requirements for

In manned mission scenarios (at least one of the two vehicles is manned), the highest value to be protected is the life of the crew in orbit and the possibility of their safe return to Earth. Rendezvous, coupling and departure operations must comply with the failure tolerance requirements, which for manned missions typically are, as established in the ISS program (NASA 1998):

In this formulation of failure tolerance requirements, errors by human operators that will have repercussions on the ways of interaction by ground and onboard operators with the rendezvous and departure operations between chaser and target vehicles are also included. The above requirements can be translated for onboard systems and maneuvers into the more familiar fail operational–fail safe requirement. The resulting requirements for approach, capture and departure operations can be interpreted for the manned mission scenario as follows:

It has to be noted that failure tolerance requirements are not necessarily additive for each of the four phases discussed above. A first failure in the rendezvous control system, e.g. during the approach phase, would also be considered a first failure in the departure phase.

For unmanned missions (both vehicles are unmanned), the highest values to be protected will depend on the type of mission, as shown in the following examples. The highest value can be:

In Case 1 the failure tolerance requirements resulting for the rendezvous, capture and departure operations would be similar to the case of an unmanned chaser in a manned mission scenario, i.e. after the first failure the rendezvous control systems of the chaser must be able to continue/resume the mission and the trajectory must be safe.

After a second failure, in any combination with a first failure, the target must be safe and the chaser must still have the capabilities to be removed from orbit.

In Case 3 the requirement of mission continuation after failure would be extremely important for mission success. This could lead, if affordable, even to a double failure tolerance requirement for mission success. However, a requirement for a keeping the target safe and removing the chaser from the orbit after a second failure, would in this case not make any sense, as the mission would end anyway, if rendezvous and capture cannot be performed any longer.

In Case 4, where it is assumed that the chaser can be used also for the servicing of other targets, the situation is similar to Case 2, with the extension, however, that after a second failure the chaser must still have the capabilities to be removed from the orbit.

These examples show that safety and failure tolerance requirements for unmanned missions cannot be defined once for all, but depend on the mission objectives and will have to be defined for each mission accordingly.

Safety Risks during Rendezvous, Capture and Departure

As addressed in the previous section, there are some important safety risks in a rendezvous and coupling operation. The potential consequences of these risks in manned and unmanned mission scenarios are stated hereafter.

Safety requirements and safety risks of capture, coupling and release have been briefly discussed here for completeness. The present section will concentrate, however, on the first two issues. Capture and undocking are mechanical issues and, therefore, out of scope of this analysis; and the safety problems of departure are in essence not different from those of the approach.

• orbital disturbance,

• navigation error,

• thrust error,

• thruster failures,

• malfunction of the rendezvous control system or of one of its parts.

Orbital Disturbances

Drag of the residual atmosphere is in LEO, and solar pressure is in GEO, the largest orbital disturbance of rendezvous trajectories (Figures 8.5.1.4 and 8.5.1.5). Other effects, such as deviation of the gravitational field of the Earth from an ideal sphere play a role in the long range rendezvous operations, but do not create a safety danger.

Calculation of Trajectory Deviations

In a circular orbit, the relative motion between two bodies, e.g. chaser and target, are described (in a frame centred in, and moving with, an undisturbed target point in orbit) by the well known Hill equations:

(12)

where ω is the orbital rate and is the mass of the chaser vehicle. The disturbance accelerations are . This system of second order differential equations can be solved generally only by numerical integration.

For step changes of the velocity at start and for constant forces, the disturbed trajectory can be calculated by the Clohessy–Wiltshire (CW) equations, which are a particular solution of the Hill equations.

(13)

The CW equations give acceptable results as long as the altitude difference between the two vehicles is , the orbit radius. In orbit- (x)-direction, the curvature of the orbit will eventually lead to deviations in the navigation. Also, they provide sufficiently accurate results to describe the trajectory deviations caused by external disturbances, only as long as the disturbance forces/accelerations are constant over the time considered. For disturbance forces that are variable over the time considered, the Hill equations need to be integrated numerically.

Disturbances can be produced by external forces, such as residual air drag or solar pressure, and by forces produced by the spacecraft itself, such as thrust deviations and thruster failures. Trajectory deviations can further be caused by navigation errors, i.e. errors in the measurement of position and velocities. For the detailed equations applicable for each disturbance and error case the reader is referred to the reference by Fehse (2003). In the following only the results for some particular cases will be discussed.