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Trajectory Design Employing Convex Optimization for Landing on Irregularly Shaped Asteroids

Mission proposals that land on asteroids are becoming popular. However, in order to have a successful mission the spacecraft must reliably and softly land at the intended landing site. The problem under investigation is how to design a fuel-optimal powered descent trajectory that can be quickly computed on- board the spacecraft, without interaction from ground control. An optimal trajectory designed immediately prior to the descent burn has many advantages. These advantages include the ability to use the actual vehicle starting state as the initial condition in the trajectory design and the ease of updating the landing target site if the original landing site is no longer viable. For long trajectories, the trajectory can be updated periodically by a redesign of the optimal trajectory based on current vehicle conditions to improve the guidance performance. One of the key drivers for being completely autonomous is the infrequent and delayed communication between ground control and the vehicle. Challenges that arise from designing an asteroid powered descent trajectory include complicated nonlinear gravity fields, small rotating bodies and low thrust vehicles. There are two previous studies that form the background to the current investigation. The first set looked in-depth at applying convex optimization to a powered descent trajectory on Mars with promising results.1, 2 This showed that the powered descent equations of motion can be relaxed and formed into a convex optimization problem and that the optimal solution of the relaxed problem is indeed a feasible solution to the original problem. This analysis used a constant gravity field. The second area applied a successive solution process to formulate a second order cone program that designs rendezvous and proximity operations trajectories.3, 4 These trajectories included a Newtonian gravity model. The equivalence of the solutions between the relaxed and the original problem is theoretically established. The proposed solution for designing the asteroid powered descent trajectory is to use convex optimization, a gravity model with higher fidelity than Newtonian, and an iterative solution process to design the fuel optimal trajectory. The solution to the convex optimization problem is the thrust profile, magnitude and direction, that will yield the minimum fuel trajectory for a soft landing at the target site, subject to various mission and operational constraints. The equations of motion are formulated in a rotating coordinate system and includes a high fidelity gravity model. The vehicle's thrust magnitude can vary between maximum and minimum bounds during the burn. Also, constraints are included to ensure that the vehicle does not run out of propellant, or go below the asteroid's surface, and any vehicle pointing requirements. The equations of motion are discretized and propagated with the trapezoidal rule in order to produce equality constraints for the optimization problem. These equality constraints allow the optimization algorithm to solve the entire problem, without including a propagator inside the optimization algorithm.

Pinson, Robin M.

The Lunar Orbiter program

The Luna and Zond series of unmanned U.S.S.R. spacecraft were designed to investigate the moon and its vicinity. Sixteen Luna spacecraft and, six Zond spacecraft have obtained lunar data. These series have included flyby, lunar-orbiting, and soft-landing missions. A variety of experiments were carried out by these spacecraft including studies of magnetism, X-ray and gamma emissions, gravitational anomalies, and chemical composition. Soil samples, near- and farside photography (both color and black and white), and earth-cloud photography were also acquired. Luna 17 and 23, carried automatic roving vehicles (Lunokhod 1 and 2) that traversed portions of the lunar surface. Lunokhod 1 roamed in Mare Imbrium near Sinus Iridum, and Lunokhod 2 roamed in the Crater Le Monnier at the eastern edge of Mare Serenitatis. The Luna 16, 20, and 24 missions soft-landed on the lunar surface, scooped up lunar material, and returned these samples to earth. The photographic samples received are in the form of paper prints. Some publications containing photographs are described.

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The future impact of computation on planetary exploration

Informed speculations are advanced on plausible candidates for space exploration missions in decades ahead; while none are approved NASA projects, they are deemed accessible to current technology. A Venus radar mapper, a minirover system (several rovers) for Mars surface exploration (geology, areochemistry, meteorology, detection of biota), investigation of Halley's comet (1986 apparition) with the aid of a solar sail vehicle, flybys of Jovian (Galilean) satellites and a landing on Ganymede, and a soft landing on the Saturnian large satellite Titan for examination of its atmosphere and surface are sketched. Anticipated data rates, software, mission reliability, and spacecraft autonomy are discussed, along with anticipated improvements in information transmission hardware and software, and some conjectures beyond the turn of the century.

Whitney, W. M.

Surveyor terminal guidance.

Terminal guidance system instrumentation for Surveyor project lunar soft-landing spacecraft

SPACECRAFT INSTRUMENTATION

Spacecraft control

Spacecraft control studies on antenna pointing, capsule sterilization, propulsive lander, optical sensing for soft landing spacecraft, and gyro data reduction computer programs

SPACECRAFT CONTROL

Lossless Convexification of Control Constraints for a Class of Nonlinear Optimal Control Problems

In this paper we consider a class of optimal control problems that have continuous-time nonlinear dynamics and nonconvex control constraints. We propose a convex relaxation of the nonconvex control constraints, and prove that the optimal solution to the relaxed problem is the globally optimal solution to the original problem with nonconvex control constraints. This lossless convexification enables a computationally simpler problem to be solved instead of the original problem. We demonstrate the approach in simulation with a planetary soft landing problem involving a nonlinear gravity field.

planetary soft landing