Behcet Acikmese

Convex Programming Approach to Powered Descent Guidance for Mars Landing

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.

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

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.

Brief paper: Lossless convexification of a class of optimal control problems with non-convex control constraints

We consider a class of finite time horizon optimal control problems for continuous time linear systems with a convex cost, convex state constraints and non-convex control constraints. We propose a convex relaxation of the non-convex control constraints, and prove that the optimal solution of the relaxed problem is also an optimal solution for the original problem, which is referred to as the lossless convexification of the optimal control problem. The lossless convexification enables the use of interior point methods of convex optimization to obtain globally optimal solutions of the original non-convex optimal control problem. The solution approach is demonstrated on a number of planetary soft landing optimal control problems.

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

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 planets 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.

Swarm Keeping Strategies for Spacecraft under J2 and Atmospheric Drag Perturbations

This paper presents several new open-loop guidance methods for spacecraft swarms comprised of hundreds to thousands of agents with each spacecraft having modest capabilities. These methods have three main goals: preventing relative drift of the swarm, preventing collisions within the swarm, and minimizing the fuel used throughout the mission. The development of these methods progresses by eliminating drift using the Hill-ClohessyWiltshire equations, removing drift due to nonlinearity, and minimizing the J2 drift. In order to verify these guidance methods, a new dynamic model for the relative motion of spacecraft is developed. These dynamics are exact and include the two main disturbances for spacecraft in Low Earth Orbit (LEO), J2 and atmospheric drag. Using this dynamic model, numerical simulations are provided at each step to show the eectiveness of each method and to see where improvements can be made. The main result is a set of initial conditions for each spacecraft in the swarm which provides hundreds of collision-free orbits in the presence of J2. Finally, a multi-burn strategy is developed in order to provide hundreds of collision free orbits under the inuence of atmospheric drag. This last method works by enforcing the initial conditions multiple times throughout the mission thereby providing collision free motion for the duration of the mission.

Observers for systems with nonlinearities satisfying incremental quadratic constraints

We consider the problem of designing observers to asymptotically estimate the state of a system whose nonlinear time-varying terms satisfy an incremental quadratic inequality that is parameterized by a set of multiplier matrices. Observer design is reduced to solving linear matrix inequalities for the observer gain matrices. The proposed observers guarantee exponential convergence of the state estimation error to zero. In addition to considering a larger class of nonlinearities than previously considered, this paper unifies earlier related results in the literature. The results are illustrated by application to several examples.

A Markov chain approach to probabilistic swarm guidance

This paper introduces a probabilistic guidance approach for the coordination of swarms of autonomous agents. The main idea is to drive the swarm to a prescribed density distribution in a prescribed region of the configuration space. In its simplest form, the probabilistic approach is completely decentralized and does not require communication or collaboration between agents. Agents make statistically independent probabilistic decisions based solely on their own state, that ultimately guides the swarm to the desired density distribution in the configuration space. In addition to being completely decentralized, the probabilistic guidance approach has a novel autonomous self-repair property: Once the desired swarm density distribution is attained, the agents automatically repair any damage to the distribution without collaborating and without any knowledge about the damage.

Decentralized observer with a consensus filter for distributed discrete-time linear systems

Abstract This paper presents a decentralized observer with a consensus filter for the state observation of discrete-time linear distributed systems. Each agent in the distributed system has an observer with a model of the plant that utilizes the set of locally available measurements, which may not make the full plant state detectable. This lack of detectability is overcome by utilizing a consensus filter that blends the state estimate of each agent with its neighbors’ estimates. It is proven that the state estimates of the proposed observer exponentially converge to the actual plant states under arbitrarily changing, but connected, communication and pseudo-connected sensing graph topologies. Except these connectivity properties, full knowledge of the sensing and communication graphs is not needed at the design time. As a byproduct, we obtained a result on the location of eigenvalues, i.e., the spectrum, of the Laplacian for a family of graphs with self-loops.

Stability analysis with quadratic Lyapunov functions: Some necessary and sufficient multiplier conditions

In this paper, we present several conditions which are both necessary and sufficient for quadratic stability of an uncertain/nonlinear system. These conditions involve multiplier matrices which characterize the uncertain/nonlinear terms in the system description. It is known that some of these conditions are sufficient for quadratic stability. One of the main contributions of this paper is to demonstrate that these conditions are also necessary conditions, hence, they are not conservative conditions for quadratic stability. By presenting multiplier matrices for many common types of uncertain/nonlinear terms, the paper also demonstrates the usefulness of the multiplier matrix approach in the analysis and control of nonlinear/time-varying/uncertain systems.

Automated Custom Code Generation for Embedded, Real-Time Second Order Cone Programming

Abstract In this paper, we discuss the development of an Interior Point Method (IPM) solver for Second Order Cone Programming optimization problems that is capable of producing customized ANSI-C code for embedded, real-time applications. The customized code is generated for a given problem structure and makes use of no dynamic memory allocation, minimizes branching, wastes no mathematical or logical computations, and has minimal dependencies to standard libraries. The resulting software is designed to be easy to implement on embedded hardware with limited computing capabilities, while still providing accurate results rapidly enough for real-time use. The core IPM algorithm is a fairly standard primal-dual IPM, which makes use of Mehrotra predictor-corrector method with Nesterov-Todd scalings and Newton search directions. We make use of the Approximate Minimum Degree heuristic to maximize the sparsity of the Cholesky factorizations that are ultimately used to solve for the search directions. We conclude the paper by presenting the computational performance results from two example problems: a Mars landing optimal control problem and a reaction wheel allocation problem. The code generated for the Mars landing problem was successfully validated in three flights onboard a NASA test rocket, and was used in real-time to generate the optimal landing trajectories that guided the rocket. To the best of our knowledge, this was the first time that a real-time embedded convex optimization algorithm was used to control such a large vehicle, where mission success and safety critically relied on the real-time optimization algorithm.