Lunar Landing Problem Research Articles

On flexibility of constraints system under approximation of optimal control problems

For finite-dimensional mathematical programming problems (approximating problems) being obtained by a parametric approximation of control functions in lumped optimal control problems with functional equality constraints, we introduce concepts of rigidity and flexibility for a system of constraints. The rigidity in a given admissible point is treated in the sense that this point is isolated for the admissible set; otherwise, we call a system of constraints as flexible in this point. Under using a parametric approximation for a control function with the help of quadratic exponentials (Gaussian functions) and subject to some natural hypotheses, we establish that in order to guarantee the flexibility of constraints system in a given admissible point it suffices to increase the dimension of parameter space in the approximating problem. A test of our hypotheses is illustrated by an example of the soft lunar landing problem.

Smooth Interpolation-Based Fixed-Final-Time Command Generation

This paper proposes a command generation approach based on the smooth interpolation without twist constraints and the input–output relation developed by the generalized model predictive static programming. Commands are efficiently generated to achieve the desired terminal condition at a specified final time in an interpolative manner featuring a noniterative nature. The effectiveness of this approach is demonstrated through three different simulation studies including a double integrator benchmark, a soft lunar landing problem, and a cluster missile guidance problem.

Deep Neural Networks Based Real-time Optimal Control for Lunar Landing

Recent research on deep learning control, a new control algorithm based on machine learning able to learn deep architectures, has shown excellent performance on robots and drones. With the development of intelligent control like deep learning and reinforcement learning, accuracy, real-time, adaptability, robustness and autonomy of control algorithm have been achieved by the intelligent controls. Traditional control methods have difficulties to achieve nice performance in complex situations. Deep learning offers powerful algorithms to real-time search near-optimal controllers of lunar landing spacecraft with nonlinear dynamics. In terms of lunar landing control system, deep architectures offer the possibility to get an approximate solution of co-state equation without time-consuming iterative process. Furthermore, real-time optimal thrust during lunar landing may be derived directly through deep neural networks. As a single infrastructure for machine learning in both production and research, TensorFlow is chosen for training the deep artificial neural networks in this paper. Numerical simulations demonstrate the effectiveness of deep neural networks. The results of deep neural networks based optimal control are contrasted with traditional optimal algorithm, whose main idea is to track the pre-designed optimal trajectory by ground station. This research provides an effective approach to cope with the lunar landing problem.

On Application of Gaussian Functions for Numerical Solution of Optimal Control Problems

It is proved that the linear combinations of shifts and contractions of the Gaussian function can be used for an arbitrarily accurate approximation in the space of continuous functions of one variable on any fixed intervals. On the example of the soft lunar landing problem, a method for the numerical solution of optimal control problems based on this approximation procedure of the control function is described. Within the framework of the same example, the sensitivity of constraint functionals to the specification error of optimal parameters is investigated using three approaches as follows: 1) Pontryagin’s maximum principle (both numerically and theoretically); 2) the control parametrization technique in combination with the method of sliding nodes; 3) the newly proposed method. A comparative analysis is performed that confirms the effectiveness of the third method.

Desensitized Optimal Trajectory for Multi-phase Lunar Landing

Lunar landing problem has been formulated as desensitized optimal control problem and solved by Legendre Pseudospectral method. The problem has been split into three phases to account for mission constraints, namely the braking with rough navigation stage (from 18km to 7km), attitude hold stage (holding the attitude for 35sec) for a typical mission scenario. In presence of uncertainties, following the open loop reference trajectory using the closed loop linear quadratic regulator contribute greatly in trajectory dispersions. The goal is to desensitize this multi-phase optimal trajectory with reduced error in presence of initial state error, thrust error, Moon’s gravity uncertainty. To achieve this, the fuel minimization cost function is augmented with closed loop covariance to generate the open loop reference trajectory. Following this reference trajectory using closed loop linear quadratic regulator, shows significant reduction in landing error. The amount of extra fuel consumed by desensitizing optimal trajectory is less significant when compared to improvement in landing accuracy to meet mission constraints assuring the desensitized optimal control a viable technique for lunar landing trajectory design.

Optimal Trajectory Planning for Multiphase Lunar Landing

Lunar landing problem is formulated as an optimal control problem and has been solved by Legendre Pseudospectral method. The landing problem is split into various stages to take into account various mission constraints. Based on These constraints, the problem is split into three stages Hough Braking, Attitude Hold and Retargeting Legendre Pseudospectral method is used for solving this multiphase lunar landing problem. The idea of this method is to discretize the trajectory optimization as a nonlinear programming problem. Legendre Pseudospectral method was applied to approximate the state and the state differential equations as algebraic constraints. Optimal solutions of the above problem are those control variables which dynamically satisfy all mission constraints. The cost function considered is minimum fuel satisfying all mission constraints envisaged. The method guarantees a null (Mann fuel solution which satisfies terminal mission constraints in each stage as part of the formulation. The mission constraints considered in studying this problem are desired altitude, attitude and velocity while reaching the desired landing site from a specified perilune height. At touchdown, the lander orientation should be vertical favoring landing leg of the Lander to touch the Moons surface as desired. The commanded acceleration should be within the limit of the main engine thruster capability (saturation limit) which enforces maximum and minimum thrust constraints. (C) 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

Chance-constrained dynamic programming with application to risk-aware robotic space exploration

Existing approaches to constrained dynamic programming are limited to formulations where the constraints share the same additive structure of the objective function (that is, they can be represented as an expectation of the summation of one-stage costs). As such, these formulations cannot handle joint probabilistic (chance) constraints, whose structure is not additive. To bridge this gap, this paper presents a novel algorithmic approach for joint chance-constrained dynamic programming problems, where the probability of failure to satisfy given state constraints is explicitly bounded. Our approach is to (conservatively) reformulate a joint chance constraint as a constraint on the expectation of a summation of indicator random variables, which can be incorporated into the cost function by considering a dual formulation of the optimization problem. As a result, the primal variables can be optimized by standard dynamic programming, while the dual variable is optimized by a root-finding algorithm that converges exponentially. Error bounds on the primal and dual objective values are rigorously derived. We demonstrate algorithm effectiveness on three optimal control problems, namely a path planning problem, a Mars entry, descent and landing problem, and a Lunar landing problem. All Mars simulations are conducted using real terrain data of Mars, with four million discrete states at each time step. The numerical experiments are used to validate our theoretical and heuristic arguments that the proposed algorithm is both (i) computationally efficient, i.e., capable of handling real-world problems, and (ii) near-optimal, i.e., its degree of conservatism is very low.

Optimal Lunar Landing Trajectory Design for Hybrid Engine

The lunar landing stage is usually divided into two parts: deorbit burn and powered descent phases. The optimal lunar landing problem is likely to be transformed to the trajectory design problem on the powered descent phase by using continuous thrusters. The optimal lunar landing trajectories in general have variety in shape, and the lunar lander frequently increases its altitude at the initial time to obtain enough time to reduce the horizontal velocity. Due to the increment in the altitude, the lunar lander requires more fuel for lunar landing missions. In this work, a hybrid engine for the lunar landing mission is introduced, and an optimal lunar landing strategy for the hybrid engine is suggested. For this approach, it is assumed that the lunar lander retrofired the impulsive thruster to reduce the horizontal velocity rapidly at the initiated time on the powered descent phase. Then, the lunar lander reduced the total velocity and altitude for the lunar landing by using the continuous thruster. In contradistinction to other formal optimal lunar landing problems, the initial horizontal velocity and mass are not fixed at the start time. The initial free optimal control theory is applied, and the optimal initial value and lunar landing trajectory are obtained by simulation studies.

Three-dimensional trajectory optimization of soft lunar landings from the parking orbit with considerations of the landing site

Minimum fuel, three-dimensional trajectory optimization from a parking orbit considering the desired landing site is addressed for soft lunar landings. The landing site is determined by the final longitude and latitude; therefore, a two-dimensional approach is limited and a three-dimensional approach is required. In addition, the landing site is not usually considered when performing lunar landing trajectory optimizations, but should be considered in order to design more accurate and realistic lunar landing trajectories. A Legendre pseudospectral (PS) method is used to discretize the trajectory optimization problem as a nonlinear programming (NLP) problem. Because the lunar landing consists of three phases including a de-orbit burn, a transfer orbit phase, and a powered descent phase, the lunar landing problem is regarded as a multiphase problem. Thus, a PS knotting method is also used to manage the multiphase problem, and C code for Feasible Sequential Quadratic Programming (CFSQP) using a sequential quadratic programming (SQP) algorithm is employed as a numerical solver after formulating the problem as an NLP problem. The optimal solutions obtained satisfy all constraints as well as the desired landing site, and the solutions are verified through a feasibility check.

Linear State Feedback Control for Spacecraft with On-Off Type Reaction Control System

This paper considers design methods for controllers of linear time invariant systems that use on-off type actuators. While classical control theory uses phase-plane analysis, the describing function approach and pulse modulation technologies, these approaches do not permit the direct application of optimal controller design techniques that include non-linear effects. In this paper, we propose the application of input-state linearization theory to the general class of single on-off input plants. The resulting linearized space allows the application of any linear controller design tool. The method is further extended to be applicable to on-off controllers with a dead-band. Some numerical studies including the lunar landing problem are presented to demonstrate the effectiveness of the approach.

線形システムのオンオフ状態フィードバック制御系の設計

This paper studies the controller design problem for linear time-invariant systems that have on-off type actuators. Although phase-plane analysis, the describing function approach and pulse-modulation technologies have been studied in the classical literature, such approaches do not permit the direct application of optimal controller design techniques including non-linear effects. In this paper, we propose the application of input-state linearization theory to the general class of single on-off input plants. The linearized space resulting from this method allows the application of any linear controller design tool. The method is also extended to be applicable to on-off controllers with a dead-band. To demonstrate the effectiveness of this approach, some numerical studies are presented including the lunar landing problem, and the results are evaluated and compared with numerical optimal solutions.