In contrast, a spacecraft uses reactive collision avoidance (RCA) to avoid any ... RCA brings increased formation safety, and if also used in nominal scenarios,.
Three-Dimensional Reactive Collision Avoidance with Multiple Colliding Spacecraft for Deep-Space and Earth-Orbiting Formations 1
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D.P. Scharf, B. Acikmese, S.R. Ploen, F.Y. Hadaegh NASA Jet Propulsion Laboratory, California Institute of Technology, Pasadena, USA
Introduction Spacecraft in formation must naturally avoid colliding. In nominal operations, relative trajectories are often planned for the entire formation with collision-avoidance constraints [1,2]. We refer to this approach as predictive collision avoidance. This path-planning problem, however, is NP-hard, and so does not scale well with number of spacecraft [3,4]. Further, if spacecraft fault and, for example, begin drifting, collision avoidance is no longer assured. Fault-tolerant predictive collision avoidance has only been addressed for the two-spacecraft case of autonomous rendezvous and docking [5], and suffers the same scalability issue. In contrast, a spacecraft uses reactive collision avoidance (RCA) to avoid any incoming objects, whether a nominal, maneuvering formation member or a drifting, faulted object. To elucidate the difference, consider fault-tolerant predictive collision avoidance for rendezvous. In this case, the target spacecraft does not maneuver, and chase spacecraft’s trajectory is planned so that if it begins to drift, it does not drift into the target. For RCA, both the chase and target would have the ability to avoid collision, and the target would take action if an imminent collision with a faulted chase was detected. RCA brings increased formation safety, and if also used in nominal scenarios, eliminates the scalability issue of formation path-planning. That is, spacecraft can plan collision-avoidance unconstrained paths and then monitor for imminent collisions, executing RCA maneuvers as necessary. While this path-planning architecture is technically sub-optimal, we argue that in most practical formations, collisions during maneuvers and reconfigurations are exceedingly rare. Then collision-avoidance unconstrained path-planning with RCA is optimal. That is, this path-planning architecture is generally optimal in practice. Moreover, it allows path-planning to be decentralized, since each spacecraft can plan its own trajectories and execute its own RCA maneuvers. RCA has its own challenges. An RCA algorithm must (i) run in real-time on flight processors, (ii) account for limited acceleration, and so recognize initial conditions for which avoidance cannot be guaranteed, (iii) avoid multiple colliding spacecraft, (iv) account for erratically maneuvering spacecraft, and (v) ideally, minimize the fuel used to avoid collision. While there has been extensive RCA research in robotics, it focuses on two-dimensional, wheeled (i.e., non-holonomic) robots or robotic arms and static obstacles. Other methods, such as potential fields or pursuit/evasion games, rely on unlimited acceleration or are overly conservative. Instead, we extend an RCA algorithm previously developed by the authors for two-dimensional deep-space formations [6] to three dimensions and Earth-orbiting formations. We first show that through a Monte Carlo analysis that collisions during realistic formation maneuvers are rare. Next, we develop the full three-dimensional, fuel-optimal solution, and then a relaxation, which has is shown to be both effective and run easily in real-time. The relaxation is basically to only avoid the most imminent collision. The RCA algorithm uses double-integrator dynamics. However, based on research that shows this th approximation is accurate over approximately 1/7 of an orbit [7], we apply the algorithm to low Earth orbit (LEO) scenarios, and show its feasibility. Note that while reconfigurations generally take half an orbit to several orbits, RCA maneuvers are executed in tens of minutes or less. Finally, as a stressing case, three spacecraft are commanded to the same relative orbit location in LEO, and all repeatedly avoid collision.
Methods To show that collision avoidance constraints are rarely active in practice, we perform a Monte Carlo analysis. First, formations in LEO are randomly generated based on passive relative orbits (PROs) [8]. Then spacecraft assignments are randomly permuted and the spacecraft reconfigure using linearized Lambert targeting (i.e., two-impulse transfer via the Hill-Clohessy-Wiltshire (HCW) Equations). All independent inter-spacecraft distances are recorded, and the minimum is taken over the inter-spacecraft distances to give a minimum distance envelope for that formation and that reconfiguration. The minimum of this envelope gives the closest any two spacecraft approached over the entire reconfiguration. To give a measure of the probability of collision, these closest approach distances are then plotted in a histogram. Figures 1 and 2 illustrate the process.
Figure 1. Example of Formation and Reconfiguration Trajectories in HCW Frame from Monte Carlo Study for 7-Spacecraft Formation with 1 km Spacing. PROs are blue dotted lines.
Figure 2. Inter-Spacecraft Distances During Reconfiguration. All 21 distances for 7-SC formation versus time during 1-orbit. Minimum-distance envelope is shown in black.
Turning to the generalization of the RCA algorithm, from [6], the architecture of the algorithm is a receding-horizon model-predictive control with a parameterized RCA trajectory. The parameterization allows the fuel-optimal RCA problem to be solved off-line and stored as a look-up table for on-board implementation. Regarding the receding horizon, each time-step, the optimal trajectory is re-solved, and the first timestep of the acceleration command is executed. This approach handles spacecraft maneuvering onto and off of collision courses, and does not require any information from the other spacecraft regarding their intentions. The RCA trajectory is parameterized as a constant direction and acceleration. To generalize to three-dimensions requires generalizing the trajectory parameterization of the RCA algorithm. Summarizing, the trajectory parameterization is a constant direction at maximum acceleration. Three dimensions requires two direction angles instead of one. This new parameterization is substituted into the polynomial expressions from [6] that determine fuel consumption. Given a colliding spacecraft, the fuel cost for evading in any direction can be plotted on the surface of the sphere. Again, a look-up table can be generated that is searched over exhaustively for the optimal. The fidelity of the table can be adjusted to improve run time. However, optimality is sacrificed for speed for reasons discussed in the Results. As a computationally simpler approach, the 2D RCA algorithm was also extended to 3D via a heuristic: only avoid the most imminent collision. Assuming colliding spacecraft are traveling at constant velocity, the times of collision are calculated. For the spacecraft with the shortest time to collision, the relative position and relative velocity define a “collision plane.” In this 2D plane, the previous RCA algorithm can be used. This approach was motivated by the observation that even in 3D, the optimal avoidance direction lies in the collision plane. A spacecraft would only choose to go out-of-plane to avoid multiple spacecraft simultaneously.
Results First, we show the results of the Monte Carlo formation configuration/reconfiguration analysis. Figure 3 shows an example histogram of the closest approaches for 50 000 random formation configurations/reconfigurations.
Figure 3. Histogram and Close-Up of Closest Approaches for 50 000 Random Formation Configurations and Reconfigurations. Closest approach was 25 m for 1 km initial separations. Simple extrapolation gives two cases less than 10 m. Figure 4 shows a plot of the fuel required to avoid collision as a function of direction for an example two-spacecraft collision scenario. The black area represents directions in which collision cannot be avoided. The equator is the collision plane, and note that the minimum fuel direction lays on it (the center of the blue depression). To find the globally optimal avoidance direction, a min-max problem is solved on the surface of a sphere. For each direction, the maximum fuel cost is found over the Figure 4. Example of 3D RCA Fuel Optimality “cost spheres” of all colliding Calculation. Colors represent the amount of fuel spacecraft. Then the overall required to avoid collision in that direction. Blue is minimum is found. least. Black means avoidance is impossible. The heuristic three-dimensionalization method, namely, avoid only the most imminent collision, was implemented in software and tested in several scenarios. This approach can be considered a greedy algorithm that optimally avoids the nearest-in-time colliding spacecraft. It is not globally optimal, but can be significantly faster. Figure 5 shows a visualization of one example deep space simulation. One spacecraft is running the greedy 3D RCA algorithm, avoiding three others. The bright orange line shows the path of the avoidance. The red, green, and blue lines indicate the collision avoidance spheres, the radii of which are selectable. Finally, Figure 6 shows that RCA can keep a formation safe even with operator error. The example consists of a 4-spacecraft string-of-pearls in LEO with 1 km-separations. SC 2 and 4 are commanded to the position of SC 3. All are running greedy 3D RCA with collision avoidance spheres of 50 m in radius. As the spacecraft approach one another, RCA activates and avoids collision (the troughs of the inter-spacecraft distances in Figure 6). After avoidance, the spacecraft plan paths back to the commanded position, which induces a looping trajectory that temporarily increases separation. Then RCA activates again as they approach the same location, and the pattern repeats.
Discussion As shown by Figure 3, collision avoidance constraints are active only 1-in-25 000 times on average for the formation configurations considered. The full paper will include more scenarios, such as all spacecraft on one PRO. The full paper will also include a more extensive literature review, more examples of cost spheres, and a step-by-step comparison of the globally-optimal and greedy algorithms. The stress case of Figure 6 will be extended to 10 spacecraft to show scalability. The 3D aspects of the stress case will also be discussed. Conclusion A scalable, real-time implementable reactive collision avoidance algorithm has been developed and demonstrated for three-dimensional formations in deep space and Earth orbit.
Figure 5. Example of Greedy 3D RCA Algorithm in Simulation. The orange line shows the trail of the SC running the RCA algorithm avoiding three SC. Even though RCA trajectory parameterization is a straight line, complex avoidance motions can be realized. Kinks in the orange line show when another colliding SC became the most imminent.
References [1] G. Singh and F. Hadaegh, “Collision avoidance guidance for formation-flying applications,” AIAA Guid., Nav.,& Contr. Conf., 2001. [2] G. Yang et al., “Fuel optimal manoeuvres for multiple spacecraft formation reconfiguration using multiagent optimization,” Int. J. Robust & Nonlinear Contr., Vol. 12, no. 2,3, pp. 243–283, 2002. [3] J. Reif and M. Sharir, “Motion Planning in the Presence of Moving Obstacles,” J. Assoc. Computing Machinery, Vol. 41(4), pp. 764-790, 1994. [4] A. Richards et al., “Spacecraft trajectory planning with avoidance Figure 6. Inter-Spacecraft Distances for Example constraints using mixed-integer linear Stress Case of Greedy 3D RCA Algorithm. The programming,” J. Guid., Contr., & separations approach the 100 m limit before RCA Dyn., Vol. 25(4), pp. 755–764, 2002. avoids collision. After avoidance, SC plan paths [5] L. Breger and J. How, “Safe back to same position, re-triggering RCA. trajectories for autonomous rendezvous of spacecraft,” J. Guid., Contr., & Dyn., Vol. 31(5), pp. 1478–1489, 2008. [6] D.P. Scharf et al., “A Direct Solution for Fuel-Optimal Reactive Collision Avoidance of Collaborating Spacecraft,” Amer. Control Conf., 2006. [7] D.P. Scharf et al., “On the Validity of the Double Integrator Approximation in Deep Space Formation Flying,” 1st Intl. Symp. Formation Flying Missions and Tech., 2002. [8] D.P. Scharf et al., “A Survey of Spacecraft Formation Flying Guidance and Control (Part I): Guidance,” Amer. Control Conf., 2003.
