We present a convex programming algorithm for the numerical solution of the minimum fuel powered descent guidance problem associated with Mars pinpoint landing. Our main contribution is the formulation of the trajectory optimization problem, which has nonconvex control constraints, as a finite-dimensional convex optimization problem, specifically as a second-order cone programming problem. Second-order cone programming is a subclass of convex programming, and there are efficient second-order cone programming solvers with deterministic convergence properties. Consequently, the resulting guidance algorithm can potentially be implemented onboard a spacecraft for real-time applications.

Convex programming approach to powered descent guidance for mars landing

To increase the science return of future missions to Mars and to enable sample return missions, the accuracy with which a lander can be deliverer to the Martian surface must be improved by orders of magnitude. The prior work developed a convex-optimization-based minimum-fuel powered-descent guidance algorithm. In this paper, this convex-optimization-based approach is extended to handle the case when no feasible trajectory to the target exists. In this case, the objective is to generate the minimum-landing-error trajectory, which is the trajectory that minimizes the distance to the prescribed target while using the available fuel optimally. This problem is inherently a nonconvex optimal control problem due to a nonzero lower bound on the magnitude of the feasible thrust vector. It is first proven that an optimal solution of a convex relaxation of the problem is also optimal for the original nonconvex problem, which is referred to as a lossless convexification of the original nonconvex problem. Then it is shown that the minimum-landing-error trajectory generation problem can be posed as a convex optimization problem and solved to global optimality with known bounds on convergence time. This makes the approach amenable to onboard implementation for real-time applications.

Minimum-Landing-Error Powered-Descent Guidance for Mars Landing Using Convex Optimization

Planetary soft landing is one of the benchmark problems of optimal control theory and is gaining renewed interest due to the increased focus on the exploration of planets in the solar system, such as Mars. The soft landing problem with all relevant constraints can be posed as a finite-horizon optimal control problem with state and control constraints. The real-time generation of fuel-optimal paths to a prescribed location on a planet's surface is a challenging problem due to the constraints on the fuel, the control inputs, and the states. The main difficulty in solving this constrained problem is the existence of nonconvex constraints on the control input, which are due to a nonzero lower bound on the control input magnitude and a nonconvex constraint on its direction. This paper introduces a convexification of the control constraints that is proven to be lossless; i.e., an optimal solution of the soft landing problem can be obtained via solution of the proposed convex relaxation of the problem. The lossless convexification enables the use of interior point methods of convex optimization to obtain optimal solutions of the original nonconvex optimal control problem.

Lossless Convexification of Nonconvex Control Bound and Pointing Constraints of the Soft Landing Optimal Control Problem

This paper presents a new onboard-implementable, real-time convex optimization-based powered-descent guidance algorithm for planetary pinpoint landing. Earlier work provided the theoretical basis of convexification, the equivalent representation of the fuel-optimal pinpoint landing trajectory optimization problem with nonconvex control constraints as a convex optimization problem. Once the trajectory optimization problem is convexified, interior-point method algorithms can be used to solve the problem to global optimality. Though having this guarantee of convergence motivated earlier convexification results, there were no real-time interior point method algorithms available for the computation of optimal trajectories on flight computers. This paper presents the first such algorithm developed for onboard use and flight-tested on a terrestrial rocket with the NASA Jet Propulsion Laboratory and the NASA Flight Opportunities Program in 2013. First, earlier convexification results are summarized and the result...

Customized Real-Time Interior-Point Methods for Onboard Powered-Descent Guidance

The problem of powered descent guidance and control for autonomous precision landing for next-generation planetary missions is addressed. The precision landing algorithm aims to trace a fuel-optimal trajectory while keeping geometrical constraints such as the line of sight to the target site. The design of an autonomous control algorithm managing such mission scenarios is challenging due to fact that critical geometrical constraints are coupled with the translational and rotational motions of the lander spacecraft, leading to a complex motion-planning problem. This problem is approached within the model predictive control framework by representing the dynamics of the rigid body in a uniform gravity field via a piecewise affine system taking advantage of the unit dual-quaternion parameterization. Such a parameterization in turn enables a six-degree-of-freedom motion planning in a unified framework while also admitting a quadratic cost on the required control commands to minimize propellant consumption. A n...

Constrained Autonomous Precision Landing via Dual Quaternions and Model Predictive Control

I. Abstract In this paper, we formulate and compare a number of powered terminal descent guidance algorithms for Mars pinpoint landing (PPL). The PPL guidance problem involves finding a trajectory that transfers the spacecraft from any g iven state at engine ignition to a desired terminal state (usually within 100m of a desired target) without violating fuel limits or any state constraints and control constraints. Sp ecifically, we first formulate the fuel-optimal guidance problem and show that a direct method can be used to reduce it to a finite-dimensional convex program. Modern interior point methods can then be used to find the global solution to any desired level of accuracy. Nex t, we discuss a class of suboptimal guidance laws based on simple polynomial basis functions. The performance of the sub-optimal guidance laws under a variety of realistic mission constraints are compared to the global fuel-optimal solution.

https://behcet.ae.utexas.edu/sites/default/files/comparison_aiaa.pdf

A Comparison of Powered Descent Guidance Laws for Mars Pinpoint Landing.

We present a powered descent guidance algorithm for Mars pinpoint landing that solves the minimum fuel trajectory optimization problem via a direct numerical method. Our main contribution is the formulation of the trajectory optimization problem, which has nonconvex control constraints, as a finite dimensional convex optimization problem, specifically as a finite dimensional semidefinite program (SDP). Since efficient SDP solvers with deterministic convergence properties exist, the resulting guidance algorithm can potentially be implemented onboard. In this paper, we present a powered descent guidance algorithm for Mars pinpoint landing. This problem is gaining importance due to the renewed interest in the manned and robotic exploration of Mars. The pinpoint landing problem can be defined as guiding a lander spacecraft to a given target on Mars’ surface with an accuracy of less than several hundred meters. This involves an entry phase through the Mars atmosphere, a phase of descent with a parachute, and then the final phase of powered descent, which is initiated with the parachute cutoff. The powered descent guidance problem for pinpoint landing is defined as finding the fuel optimal trajectory that takes a lander with a given initial state (position and velocity) to a prescribed final state in a uniform gravity field, with magnitude constraints on the available net thrust, and various state constraints. A version of this problem is also known in the optimal control literature as the soft landing problem, 1–3 and its solutions have well known characterizations. One important characterization is that the net thrust magnitude must be either at the minimum or maximum value at any given time during the maneuver. A closed form solution of the problem for the one dimensional case (with vertical motion only) is given by. 1 However, a closed form solution of the thrust profile is not available for the general three dimensional case with additional state and control constraints. Direct numerical methods for trajectory optimization are attractive because explicit consideration of the necessary conditions (adjoint equations, transversality conditions, maximum principle) are not required. 4 The infinite dimensional optimal control problem is directly converted into a finite dimensional parameter optimization problem 5–7 which is then solved via a nonlinear programming method. However, real-time onboard solution of a nonlinear program via a general iterative algorithm may not be desirable without explicit knowledge of the convergence properties of the algorithm. As Mars pinpoint landing requires realtime onboard computation of the optimal trajectory, it is essential to exploit the structure of the soft landing problem in order to design algorithms with guaranteed convergence to the global optimum. This leads us to formulate the problem in a convex optimization 8,9 framework, specifically to formulate the resulting parameter optimization problem as a semidefinite program (SDP). 10,11 SDP problems have low complexity, and they can be solved in polynomial time. There exist algorithms, such as interior-point methods, 12–14 that compute the global optimum with a deterministic stopping criteria, and with prescribed level of accuracy. Therefore, they are very well-suited for real-time onboard computations.

A Powered Descent Guidance Algorithm for Mars Pinpoint Landing

In this paper we develop a robust nonlinear algorithm for the attitude control of a solar sailcraft with M single degree-of-freedom articulated control vanes. A general attitude controller that tracks an admissible trajectory while rejecting disturbances such as torques due to center-of-mass to center-of-pressure offsets is applied to this problem. We then describe a methodology based on nonlinear programming to allocate the required control torques among the control vanes. A simplified allocation strategy is then applied to a solar sail with four articulated control vanes, and simulation results are given. The performance of the control algorithm and possible limitations of vane-only control are then discussed.

Attitude Dynamics and Control of Solar Sails with Articulated Vanes

This paper is a product of research supported by NASA under RASC (the Revolutionary Aerospace Systems Concepts) program. It presents an overall system architecture, and covers issues of deployment, navigation, and control related to a formation of two spacecraft in the neighborhood of the Sun-Earth L2 Lagrange point (on the Sun-Earth line), that serves as an observatory of Earth's atmosphere. The observatory concept definition study was a multi-center NASA effort conducted in 2003, and covered a much wider scope than is presented in this focused paper.The Earth observatory at L2 is a unique design concept that can improve the knowledge and understanding of dynamic, chemical and radiative mechanisms that cause changes in the atmosphere, and can lead to the development of models and techniques to predict short and long-term climate changes.

Earth Atmosphere Observatory Formation at L2

In this paper the translational equations of motion of a formation of n spacecraft in Earth orbit, n(sub f) of which are drag-free spacecraft, are derived in a coordinate-free manner using the balance of linear momentum and direct tensor notation. A drag-free spacecraft consists of a spacecraft bus and a proof mass shielded from external disturbances in an internal cavity. By controlling the spacecraft so that the proof mass remains centered in the cavity, the spacecraft follows a purely gravitational orbit. The results described in this paper provide a first step toward coupling drag-free control technology with formation flying in order to mitigate the effect of differential aerodynamic drag on formation flying missions (e.g., Earth imaging applications) in low Earth orbit.

A. Behcet Acikmese

Papers

A Comparison of Powered Descent Guidance Laws for Mars Pinpoint Landing.

A Powered Descent Guidance Algorithm for Mars Pinpoint Landing

Attitude Dynamics and Control of Solar Sails with Articulated Vanes

Earth Atmosphere Observatory Formation at L2

Dynamics of drag free formations in Earth orbit