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Optimization and Sensitivity Analysis of Trajectories for Autonomous Small Celestial Body Operations
Abstract
Within this paper, a method for on-board trajectory calculation in the vicinity of a small celestial body is introduced. Therefor, high precision methods of nonlinear optimization and optimal control are used. Additionally, a parametric sensitivity analysis is implemented. This tool allows to approximate a perturbed optimal solution in case of model parameter deviations from nominal values without noticeable computational effort. Parametric sensitivity analysis is a recent research area of great interest. Parameter perturbations that occur in the dynamic of the system as well as in boundary conditions or in state and control constraints can be analyzed. Thus, additional stability information is provided. Furthermore, the fast and reliable approximation of perturbed controls can be used for real-time control in time critical navigation phases.
... These requirements can be met by solving appropriate optimal control problems for the computation of high-quality trajectories. Based on the ESA solver WORHP [6] for nonlinear optimization problems and the corresponding transcription method TransWORHP [7], this has proven to be a very successful approach, for example within the simulated environments of KaNaRiA [8] and EnEx-CAUSE [9]. ...
... These problems are well studied and can be addressed with the ESA NLP solver WORHP [6]. This software is designed to efficiently solve huge problems (up to millions of variables) in a short amount of time and it is successfully applied to a wide range of (space) applications (including [8,20]). The underlying SQP algorithm takes advantage of the sparsity of the gradient, Jacobian and Hessian to accelerate the calculation. ...
Online optimization and trajectory planning are key aspects of autonomous deep space missions. Taking into account individual target criteria, such as time or energy optimality, any spacecraft maneuver can be traced back to a general problem definition of the form "move the spacecraft from its initial state to a desired final state, while considering a dynamic model and avoiding collisions". This corresponds to an optimal control problem (OCP) and can be solved using WORHP, the ESA solver for non-linear programming (NLP). More specifically, the OCP is transcribed into an NLP by time discretization and the corresponding optimal trajectory - including control variables - is calculated by sequential quadratic programming. In order to obtain a highly efficient solution algorithm, the naturally occurring sparsity of the Jacobian and the Hessian is exploited.
The effectiveness of this approach has already been demonstrated in several DLR projects, such as the deep space missions KaNaRiA and EnEx-CAUSE. In order to make such results immediately available for terrestrial applications, a transfer to current scientific questions is appropriate. Moreover, the transfer would provide a test platform and increase public acceptance. Conversely, the knowledge gained from terrestrial testing can help planning more detailed space missions.
In this work, the DLR project AO-Car for controlling an autonomous vehicle in road traffic is presented as such a transfer. The concept, originally developed in the context of KaNaRiA for trajectory planning and control, is successfully implemented on a research vehicle, a VW Passat. The vehicle is able to explore a parking area autonomously, to identify free parking spaces and to perform a parking maneuver. During the exploration, suddenly appearing objects are recognized. Depending on the scene, a collision avoidance trajectory is computed or an emergency stop is performed. The method presented is based on WORHP and offers a uniform framework for optimal driving maneuvers. It is highly flexible, as reaction speed and passenger comfort can be easily balanced by adaptive weighting of target criteria.
Andreas Folkers stellt in diesem Buch einen Regelalgorithmus zur Steuerung eines autonomen Fahrzeugs vor. Um eine nichtlineare Bewegungsdynamik bei gleichzeitig kurzer Rechenzeit zu berücksichtigen, wird der Regler als Neuronales Netz definiert und dessen Parameter im Vorfeld gelernt. Dieser Trainingsschritt wird im Setting des Deep Reinforcement Learning durch die Proximal-Policy-Optimierung innerhalb einer Simulation durchgeführt. Die Qualität des resultierenden Reglers wird schließlich sowohl durch die Simulation als auch bei der Anwendung auf ein reales Fahrzeug in einem Parkplatz-Szenario evaluiert.
Der Inhalt
• Grundlagen des Deep Learning und kontinuierliches Deep Reinforcement Learning
• Definition eines Deep Controller für Autonomes Fahren
• Training und Evaluierung des Deep Controller innerhalb einer Simulation
• Anwendung des Deep Controller auf einem Forschungsfahrzeug
Die Zielgruppen
• Dozierende und Studierende der Bereiche Angewandte Mathematik und Informatik
• Fach- und Führungskräfte auf den Gebieten Autonomes Fahren und Künstliche Intelligenz
Der Autor
Andreas Folkers studierte Technomathematik an der Universität Bremen und arbeitet dort als Wissenschaftlicher Mitarbeiter an angewandten Forschungsfragen im Kontext von Autonomen Fahrzeugen. Im Zentrum seiner Forschung stehen Algorithmen zur Fahrzeugsteuerung und Künstliche Intelligenz.
In this paper, we propose an active real-time capable 3D graph based simultaneous localization and mapping (Graph SLAM) approach, which actively estimates the state of an autonomous spacecraft relative to a simultaneously established map estimate. The graph is constructed in a tightly-coupled fashion, where an Extended Kalman Filter estimates the relative offset between two of its vertices. An additional relative measurement is derived by matching point clouds obtained by a light detection and ranging (LiDAR) system. In order to yield a significant speed-up, scan matching is implemented on the GPU. To reduce the uncertainty of either the state or the map estimate, we present an approach to actively control the system resting on an extended representation of uncertainty in the map. Furthermore, it adapts its behavior depending on the current uncertainty distribution in order to find a dynamic trade-off between exploitation (improve localization performance) and exploration (improve knowledge about the environment). Finally, we present a post-processing approach to discover landing sites in the map estimate without prior knowledge. The evaluation is conducted in a numerical simulation, where the spacecraft explores the real 3D model of Itokawa in its actual dynamic environment. Within that simulation, we use a shader-based GPU implementation for simulating LiDAR measurements. We evaluate the performance of the active SLAM approach and demonstrate that the use of the adaptive approach improves navigation and exploration performance at the same time.
With the rapid evolution of space technologies and increasin
g thirst for knowledge about the origin of life and the
universe, the need for deep space missions as well as for autonomous solutions for complex, time
-
critical mission
operations becomes urgent. Within this context, the project KaNaRiA aims at technology
development tailored to the
ambitious task of space resource mining on small planetary bodies using increased autonomy for
on
-
board mission
planning,
navigation and guidance.
This paper focuses on the specific challenges as well as first solutions and re
sults corresponding to the KaNaRiA
mission phases (1) interplanetary cruise, (2) target identification and characterization and (3) proximity operations.
Based on the KaNaRiA asteroid mining mission objectives, initially, a mission reference scenario as we
ll as a
reference mission architecture are described in this paper. KaNaRiA has been proposed as a multi
-
spacecraft mission
to the asteroid main belt. Composed of a flock of prospective scout spacecraft, a mother ship carrying the mining
payload and severa
l service modules placed on a 2.8 AU parking orbit around the Sun, KaNaRiA intends to
characterize main belt asteroid properties, identify targets for mining and perform a soft
-
landing
for in
-
situ
characterization and mining
.
Subsequently, the autonomous n
avigation system design of KaNaRiA for the interplanetary cruise is presented.
The navigation challenges
,
which arise in phases (1)
to
(3)
,
are discussed. Particular attention is given to the sensor
-
technology readiness
-
level, accuracy, applicability range
, mass and power budgets.
In order to navigate in the
vicinity of an asteroid,
an
information fusion algorithm is required that aggregates multi
-
sensor data as well as a
-
priori knowledge and solves the task known as simultaneous localization and mapping (S
LAM). In order to deal with
uncertain and inconsistent information and to explicitly represent different dimensions of uncertainty, a belief
-
function
-
based SLAM
approach
is used, which is a generalization of the popular FastSLAM
algorithm
.
T
he objective of
the guidance task is the autonomous planning of optimal
transfer
trajectories according to
mission driving criteria, e.g. transfer time and fuel consumption.
Optimal control problems and the calculation of
trajectory sensitivities for on
-
board stability a
nalysis as well as real
-
time optimal control are explained.
Bringing cognitive autonomy to a spacecraft requires an on
-
board
computational
module as a central spacecraft
component. This module is responsible for
state evaluation, mission planning
and deci
sion
-
making regarding
selection of potential targets, trajectory selection and FDIR. A knowledge
-
base serves as a
database
for decision
making
processes
.
With the aim to validate and test our methods, we create a virtual environment in which humans can in
teract with
the simulation of the mission. In order to achieve real
-
time performance, we propose a massively
-
parallel software
system architecture, which enables very efficient and easily adaptable communication between concurrent software
modules within K
aNaRiA.
We Optimize Really Huge Problems (WORHP) is a solver for large-scale, sparse, nonlinear optimization problems with millions of variables and constraints. Convexity is not required, but some smoothness and regularity assumptions are necessary for the underlying theory and the algorithms based on it. WORHP has been designed from its core foundations as a sparse sequential quadratic programming (SQP) / interior-point (IP) method; it includes efficient routines for computing sparse derivatives by applying graph-coloring methods to finite differences, structure-preserving sparse named after Broyden, Fletcher, Goldfarb and Shanno (BFGS) update techniques for Hessian approximations, and sparse linear algebra. Furthermore it is based on reverse communication, which offers an unprecedented level of interaction between user and nonlinear programming (NLP) solver. It was chosen by ESA as the European NLP solver on the basis of its high robustness and its application-driven design and development philosophy. Two large-scale optimization problems from space applications that demonstrate the robustness of the solver complement the cursory description of general NLP methods and some WORHP implementation details.
Solving an optimal control or estimation problem is not easy. Pieces of the puzzle are found scattered throughout many different disciplines. Furthermore, the focus of this book is on practical methods, that is, methods that I have found actually work! In fact everything described in this book has been implemented in production software and used to solve real optimal control problems. Although the reader should be proficient in advanced mathematics, no theorems are presented.
Traditionally, there are two major parts of a successful optimal control or optimal estimation solution technique. The first part is the “optimization” method. The second part is the “differential equation” method. When faced with an optimal control or estimation problem it is tempting to simply “paste” together packages for optimization and numerical integration. While naive approaches such as this may be moderately successful, the goal of this book is to suggest that there is a better way! The methods used to solve the differential equations and optimize the functions are intimately related.
The first two chapters of this book focus on the optimization part of the problem. In Chapter 1 the important concepts of nonlinear programming for small dense applications are introduced. Chapter 2 extends the presentation to problems which are both large and sparse. Chapters 3 and 4 address the differential equation part of the problem. Chapter 3 introduces relevant material in the numerical solution of differential (and differentialalgebraic) equations. Methods for solving the optimal control problem are treated in some detail in Chapter 4. Throughout the book the interaction between optimization and integration is emphasized. Chapter 5 describes how to solve optimal estimation problems. Chapter 6 presents a collection of examples that illustrate the various concepts and techniques. Real world problems often require solving a sequence of optimal control and/or optimization problems, and Chapter 7 describes a collection of these “advanced applications.”
While the book incorporates a great deal of new material not covered in Practical Methods for Optimal Control Using Nonlinear Programming [21], it does not cover everything. Many important topics are simply not discussed in order to keep the overall presentation concise and focused. The discussion is general and presents a unified approach to solving optimal estimation and control problems. Most of the examples are drawn from my experience in the aerospace industry. Examples have been solved using a particular implementation called SOCS . I have tried to adhere to notational conventions from both optimization and control theory whenever possible. Also, I have attempted to use consistent notation throughout the book.
In support of the MarcoPolo-R mission, we have carried out numerical
simulations of spacecraft trajectories about the binary asteroid 175706
(1996 FG3) under the influence of solar radiation pressure. We study the
effects of (1) the asteroid's mass, shape, and rotational parameters,
(2) the secondary's mass, shape, and orbit parameters, (3) the
spacecraft's mass, surface area, and reflectivity, and (4) the time of
arrival, and therefore the relative position to the sun and planets. We
have considered distance regimes between 5 and 20 km, the typical range
for a detailed characterization of the asteroids - primary and secondary
- with imaging systems, spectrometers and by laser altimetry. With solar
radiation pressure and gravity forces of the small asteroid competing,
orbits are found to be unstable, in general. However, limited orbital
stability can be found in the so-called Self-Stabilized Terminator
Orbits (SSTO), where initial orbits are circular, orbital planes are
oriented approximately perpendicular to the solar radiation pressure,
and where the orbital plane of the spacecraft is shifted slightly
(between 0.2 and 1 km) from the asteroid in the direction away from the
sun. Under the effect of radiation pressure, the vector perpendicular to
the orbit plane is observed to follow the sun direction. Shape and
rotation parameters of the asteroid as well as gravitational
perturbations by the secondary (not to mention sun and planets) were
found not to affect the results. Such stable orbits may be suited for
long radio tracking runs, which will allow for studying the gravity
field. As the effect of the solar radiation pressure depends on the
spacecraft mass, shape, and albedo, good knowledge of the spacecraft
model and persistent monitoring of the spacecraft orientation are
required.
The small gravitational forces associated with a minor celestial body, such as an asteroid or comet nucleus, allow visiting spacecraft to implement active control strategies to improve maneuverability about the body. One form of active control is hovering, where the nominal accelerations on the spacecraft are canceled by a nearly continuous control thrust. We investigate the stability of realistic hovering control laws in the body-fixed and inertial reference frames. Two implementations of the body-fixed hovering thrust solution are numerically tested and compared with an analytical stability result presented in the previous literature. The perturbation equations for inertial frame hovering are presented and numerically simulated to identify regions of stability. We find that body-fixed hovering can be made stable inside a region roughly approximated by the body's resonance radius and inertial frame hovering is stable in all regions outside the resonance radius. A case study of hovering above Asteroid (25143) Itokawa, target of the Hayabusa mission, is also presented.
We give a brief overview of important results in several areas of sensitivity and stability analysis for nonlinear programming, focusing initially on qualitative characterizations (e.g., continuity, differentiability and convexity) of the optimal value function. Subsequent results concern quantitative measures, in particular optimal value and solution point parameter derivative calculations, algorithmic approximations, and bounds. Our treatment is far from exhaustive and concentrates on results that hold for smooth well-structured problems.
Parametric nonlinear optimal control problems subject to control and state constraints are studied. Two discretization methods are discussed that transcribe optimal control problems into nonlinear programming problems for which SQP-methods provide efficient solution methods. It is shown that SQP-methods can be used also for a check of second-order sufficient conditions and for a postoptimal calculation of adjoint variables. In addition, SQP-methods lead to a robust computation of sensitivity differentials of optimal solutions with respect to perturbation parameters. Numerical sensitivity analysis is the basis for real-time control approximations of perturbed solutions which are obtained by evaluating a first-order Taylor expansion with respect to the parameter. The proposed numerical methods are illustrated by the optimal control of a low-thrust satellite transfer to geosynchronous orbit and a complex control problem from aquanautics. The examples illustrate the robustness, accuracy and efficiency of the proposed numerical algorithms.
From WORHP to TransWORHP
- M Knauer
- C Büskens