Historically speaking, the success rate of robotic planetary landers has been unsatisfyingly

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1 Guidance for Autonomous Precision Landing on Airless Bodies Ingo Gerth and Erwin Mooij Deft University of Technology, Delft, The Netherlands, 2629 HS EXTENDED ABSTRACT Historically speaking, the success rate of robotic planetary landers has been unsatisfyingly low. The cumulative probability of failure based on historical data was about 20 % by the beginning of the last decade. 1 This may in part be traced to the limited capabilities of past systems: all landers were blind, and were unable to evade any surface hazards encountered. Because such high landing risks are unacceptable for next-generation spacecraft, much research is going into Hazard Detection and Avoidance (HDA) technologies for landers at the moment. Such systems allow landers to see using vision-based technologies and thus mitigate the risk of partial and fatal mission loss. Research has shown, for example, that the risk of failure could have been reduced by a factor of four for the Mars Science Laboratory mission. 2 Applying HDA impacts the GNC-system design significantly, and poses special requirements to guidance as well. Hence, it is the purpose of this paper to investigate the suitability of three landing guidance methods for application in GNC frameworks that use HDA technologies. The focus is on approach-phase guidance, the interval between main braking and (vertical) terminal descent. Only landings on airless bodies are considered, as atmospheric entry trajectories are typically designed very differently, calling for different guidance requirements. A second feature that is of interest for future landers and affects guidance is the capability to land precisely (landing area with a diameter D 400 m). 3 One way of achieving this level of precision is to apply optical navigation methods. This, in combination with HDA, MSc student in Spaceflight, Department of Astrodynamics and Space Missions, Deft University of Technology, Kluyverweg 1, 2629 HS, The Netherlands. Intern at ESA ESTEC, D-HSO/IL (Lunar Lander Office). AIAA student member. Assistant professor. Associate Fellow, AIAA. 1 of 6

2 broadens the envelope of possible mission scenarios. It can enlarge the accessible area by up to a factor of three, 2 and allows for landings near previously landed surface assets. Renewed interest in robotic Lunar exploration can be observed around the major space agencies in the past few years. Despite recent setbacks, ESA is investigating a robotic Lunar lander as well. 3 The work of this paper is based on the requirements stipulated by this mission. The ESA Lunar Lander shall not only demonstrate HDA, but also precision landing technologies. Accordingly, the requirements for the approach-phase guidance laws discussed in this paper are as follows: Pinpoint-landing ability. The precise landing target must be specifiable in the guidance law. Adaptivity. Because the HDA system can command diverts to alternative landing site locations if the current site should be found to be unsafe and a safer one exists guidance shall be able to re-plan the trajectory to a new target on-line. Computational efficiency. Although comparably powerful space-grade computers have emerged, the image processing needed for HDA and optical navigation will put a heavy load on the CPU. Thus, guidance will be restricted in terms of computational resources, too, and shall work as efficiently as possible. Fuel optimality. Finally, the guidance law shall be fuel optimal as to maximize the feasible payload mass. This is of elevated importance when used in combination with HDA, because divert manoeuvres may require extra propellant. An extensive literature survey considering 24 guidance algorithms has been performed with the goal of selecting three candidates for further study. Out of the 24, 17 were considered in a detailed trade-off that considered the requirements stated above. The selection was based on data found in the literature (such as References 4 and 5) and expert judgment. The requirement fulfilling algorithms that were chosen for this study are E-Guidance, 6 D Souza s guidance mode, 7 and a convex optimization-based mode. 8 All of them feature pinpoint-landing terminal-state-vector guidance, adaptivity, and fuel optimality. However, complexity is increasing from the former to the latter. Thus, one goal of this research is to identify the gain from applying more complicated algorithms. The studied algorithms are briefly introduced next. Firstly, E-Guidance is based on the principle of constraining the spacecraft s total acceleration to a linear polynomial of the form: a c = a g = c 1 + c 2 t go g (1) where a c is the commanded thrust acceleration, a the total acceleration acting on the vehicle, g is the local gravity vector, c denotes coefficients, and t go denotes the time-to-go, a 2 of 6

3 quantity decreasing from the vehicle s current position along the trajectory to zero at landing. The coefficients are computed along the trajectory as follows: c 1 = c 2 2 t go 6 t 2 go 6 t 2 go 12 t 3 go v f v 0 (2) r f r 0 v 0 t go where v denotes the velocity vector, r the position vector, and the indices f and 0 the final and current conditions respectively. As this is an underconstrained system, the time-to-go is left as a free variable. One way to estimate t go is to derive it from the rocket equation, which was found to be close to the fuel-optimal solution. This method finds the time from: t go = τ (1 exp ( V/V e )) (3) where V e is the engine s exhaust velocity, and τ and V are given by: τ = τ 0 t V = (ẋ ẋ f ) 2 + (ẏ ẏ f ) 2 + (ż ż f ) 2 (4) The characteristic time τ needs to be initialised with τ 0 shown in the previous formula. This is done by using a reasonable estimate for t go in the first few steps and computing τ 0 from this as: τ 0 = ṁ m (5) After that, a switch to the procedure outlined above occurs. Because divisions by t go occur in Equation (2), the coefficients are simply not recomputed after a threshold value of 2 s. This chosen as a trade between sacrificing some landing precision and safety. Initial results have been obtained for a Mercury landing scenario using this E-Guidance scheme. In this scenario, the lander is initially located at 20 km downrange from the landing site and has a speed of 712 m/s. Figure 1 shows the results of the time-to-go estimation. The switch over from the initialisation to the primary method and the threshold are readily observable. The graph is also almost linear, indicating a stable estimation and a consistent prediction of the flight time already early in the simulation. The trajectory resulting from the estimated values for t go is shown in Figure 2. As the arrows indicate, in this scenario the thrust is increasing along the trajectory, but guides the vehicle safely to the landing site. The reduced distance in between ticks towards the end shows that the lander is slowing down. As far as HDA is concerned, the resulting trajectory is favorable because it resembles a straight line. This ensures that the landing site can stay 3 of 6

4 60 tgo [s t go switch-over t go threshold 0 t Figure 1. Results from the time-to-go estimation for the Mercury landing scenario. in the field of view of the cameras for HDA during the entire descent. In this preliminary 3-DOF simulation, the landing errors in position and velocity are negligibly small. y [km x Figure 2. The guided landing trajectory following from E-Guidance. No results have been obtained for the other two algorithms yet. Nevertheless they are briefly introduced next. The fuel mode proposed by D Souza optimizes the following performance index: 7 min J = Γt f tf t 0 a T a dt (6) where Γ is a weighting on the final time, which can be used to adjust the trajectory. Tuning Γ can be used for avoiding sub-terranean trajectories, for example. Using the calculus of variations, it can be shown that the optimal control accelerations are: a c = 4 v t go 6 r t 2 go g (7) where r = r r f, and v = v v f, and g = (0 0 g) T. The time-to-go can then be obtained by solving the following quartic equation, which is analytically shown in the corresponding paper: 7 ( ) Γ + g2 t 4 go 2v T vt 2 go 12v T r t go 18r T r = 0 (8) 2 4 of 6

5 The method is thus an explicit, three-dimensional guidance mode. Necessary and sufficient optimality conditions are fulfilled. Note the similarity to E-Guidance in terms of the commanded acceleration. Finally, the mode based on convex optimization theory 9 solves the following problem: 8 tf min a(t)t a(t) dt t f,a( ) 0 ṁ(t) = α T(t), z 0, β l β β u (9) m(0) = m 0, r(0) = r 0, ṙ(0) = ṙ 0, r(t f ) = r f, ṙ(t f ) = 0 where ṁ is the mass flow, z stands for the vertical component of the position vector with respect to the local surface horizontal, and β denotes the viewing angle between the camera and the landing site with the upper and lower limits u and l. Hence, the optimizer will solve an explicitly constrained problem that takes HDA limitations into account, which the other two algorithms cannot. Because this guidance mode is too complex to be briefly introduced here, the detailed discussion is deferred to later work. Further research will focus on the implementation of all three guidance modes in a highfidelity simulator that takes HDA into account. The algorithms will then be thoroughly tested and compared in a cardinal fashion as proposed by Steinfeldt, Braun, and Stephen. 10 This allows for a quantitative comparison of all guidance modes. Possible improvements of the algorithms will be identified and documented. The expected result is thus the identification of the best guidance law for autonomous robotic landings on airless bodies, where HDA is taken into account. Sensitivity and robustness testing will be performed for the chosen law. Initial results could be obtained for E-Guidance, showing promising suitability for HDA and precision landing. It is expected that even better results can be obtained for the other two guidance algorithms. References 1 Strandmoe, S. E., Jean-Marius, T., and Trinh, S., Toward a vision based autonomous planetary lander, AIAA Guidance, Navigation and Control Conference, AIAA , Huertas, A., Cheng, Y., and Matthies, L., Real-time Hazard Detection for Landers, NASA Science Technology Conference, Fisackerly, R., Pradier, A., Gardini, B., Houdou, B., Philippe, C., De Rosa, D., and Carpenter, J., The ESA Lunar Lander Mission, AIAA SPACE 2011 Conference & Exposition, AIAA , Steinfeldt, B. A., Grant, M. J., Matz, D. M., Braun, R. D., and Barton, G. H., Guidance, Navigation, and Control Technology System Trades for Mars Pinpoint Landing, AIAA Atmospheric Flight Mechanics Conference and Exhibit, AIAA , Atlanta, GA, of 6

6 5 Sostaric, R. R. and Rea, J. R., Powered Descent Guidance Methods For The Moon and Mars, AIAA Guidance, Navigation, and Control Conference and Exhibit, AIAA , Cherry, G. W., A general, explicit, optimizing guidance law for rocket-propelled spaceflight, AIAA/ION Astrodynamics Guidance and Control Conference, AIAA , D Souza, C. N., An optimal guidance law for planetary landing, AIAA/AAS Astrodynamics Specialist Conference and Exhibit, AIAA , Acikmense, B. and Ploen, S. R., Convex Programming Approach to Powered Descent Guidance for Mars Landing, Journal of Guidance, Control, and Dynamics, Vol. 30, No. 5, Sept. 2007, pp Boyd, S. and Vandenberghe, L., Convex Optimization, Cambridge University Press, Steinfeldt, B. A., Braun, R. D., and Paschall, S. C. I., Guidance and Control Algorithm Robustness Baseline Indexing, AIAA/AAS Astrodynamics Specialist Conference and Exhibit, AIAA , of 6