# Lecture series in

# OPTIMAL CONTROL FOR SPACE MISSION DESIGN

# JAXA, Sagamihara Campus, Japan

This lecture series aims to help students and professionals alike to quickly get familiar with the state-of-the-art methods in optimal control for space mission design.

Each lecture is one-day long, and consists of two parts. The morning session ("Theory") introduces the mathematical principles underlying the method*. The afternoon session ("Hands-on") will guide the students through some simple examples they can actually code.

*Although some knowledge of optimal control theory is preferred, a quick overview will be given before each lecture.

For more information on the lecture series, contact me.

## Indirect Methods for Space Trajectory Optimization

### Prof. Casalino

When: April 20th, 2015

Where: JAXA/ISAS Sagamihara Campus, Room Nyusatsu 1st floor

Registration fee: No fee, but registration is required beforehand (simpy write your name and affiliation in the email).

Handouts: Download here!. -->

Abtract:

This lecture is focused on indirect optimization methods, which, with direct methods, are most widely used today in trajectory optimization (even though evolutionary algorithms are gaining space).

Part I - Indirect Methods for Space Trajectory Optimization: How to make them. The discussion starts with a brief review of the underlying mathematics with particular reference to calculus of variations. Some aspects concerning problem modeling and how these influence the position of the optimal problem will be discussed and example applications will be presented. Both impulsive and low-thrust trajectories will be considered; handling of complex constraints will also be discussed.

Part II - Indirect Methods for Space Trajectory Optimization: How to make them WORK. The most relevant issue when using indirect methods is finding a suitable tentative solution that allows for convergence. Numerical techniques that improve the method robustness will be described. Even though each problem presents unique characteristics, general considerations to favor convergence will be presented and their features and relevance discussed. Finally, new and original applications, subject of ongoing research, will be presented.The problem of designing trajectories that exploit low-thrust propulsion is significantly more challenging than the case of using only impulsive chemical propulsion. Where the gravity field can be well approximated as that of a point mass, many analytical results are available for the case of impulsive manoeuvres. However, this is not the case when low thrust is considered.

About the lecturer:

Lorenzo Casalino is an associate professor at Politecnico di Torino, after obtaining his PhD degree under Professor Colasurdo. His research interests lie mainly in space trajectory optimization. He has developed and applied indirect methods and evolutionary algorithms to various problems concerning space trajectories and also developed techniques for approximate solutions. He cooperated with ASI, CNES and ESA in various studies. He leads the Politecnico di Torino/Universita' di Roma “Sapienza” team in the Global Trajectory Optimisation Competitions (GTOC); the team took the first place in GTOC2 and GTOC6, and was responsible for hosting GTOC3 and GTOC7.

## An overview of the Q-Law and other analytical techniques

### Dr. Petropoulos

When: March 31st, 2014

Where: JAXA/ISAS Sagamihara Campus, Room #1432

Registration fee: No fee, but registration is required beforehand (simpy write your name and affiliation in the email).

Handouts: Download here!.

Abtract:

The problem of designing trajectories that exploit low-thrust propulsion is significantly more challenging than the case of using only impulsive chemical propulsion. Where the gravity field can be well approximated as that of a point mass, many analytical results are available for the case of impulsive manoeuvres. However, this is not the case when low thrust is considered.

This talk will start with a brief overview of historical and recent analytical results for special cases involving low thrust. The next part of the talk will briefly describe some semi-analytical work done by the speaker for the problem of spiral escape and capture involving orbits of arbitrary eccentricity.

The third part of the talk will focus on the development of the Q-law, a low-thrust Lyapunov feedback control law for orbit transfers between arbitrary orbits around a central body. This will include a brief discussion of other Lyapunov feedback control laws, and what makes for a good Lyapunov function. Also to be addressed are the performance improvements seen by optimising the weighting parameters in the Q-law using a genetic algorithm. Some of the common pitfalls that might be faced in implementing or understanding the Q-law will also be discussed. A comparison will also be made to optimal orbit transfers.

The talk will hopefully take more the form of a discussion, especially as regards the Q-law implementation, and so audience questions will be welcome throughout.

About the lecturer:

Anastassios Petropoulos is a Senior Engineer in Mission Design at the NASA/Caltech Jet Propulsion Laboratory, where he started in 2001, after obtaining his PhD degree under Professor Longuski at Purdue University in Indiana, USA. His current research interests lie in low-thrust and gravity-assist trajectory design algorithms. He is the developer of the Q-law algorithm, a Lyaponov feedback control law for low-thrust orbit transfers around a massive body. The Q-law allows the easy computation of transfers that lie close to the pareto front in flight time and propellant consumption. Dr. Petropoulos has worked on numerous NASA Discovery proposals in a mission design capacity, the Mars Science Laboratory mission (long-term trajectory propagation studies), various Mars mission studies, and mission studies to the outer planets. He was also the Mission Design Lead for the NASA Jupiter Europa Orbiter flagship mission study, and continues to work on the ongoing Europa Clipper mission study. Dr. Petropoulos also leads the JPL team in the Global Trajectory Optimisation Competitions (GTOC); the team took the first place in GTOC1 and GTOC5, and was responsible for hosting GTOC2 and GTOC6.

## The Pontryagin Maximum Principle: definition, proof, and examples

### Dr. Campagnola

When: February 13th, 2014

Where: JAXA/ISAS Sagamihara Campus, Room #1432

Registration fee: No fee, but registration is required beforehand (simpy write your name and affiliation in the email).

Abtract:

The lecture will focus on the Pontryagin Maximum Principle, on its proof (which is more geometric in nature than the calculus of variation), and on some examples.

## Pseudo-spectral methods and finite elements in time

### Prof. Vasile

When: August 1st, 2012

Where: JAXA/ISAS Sagamihara Campus, Room #7204, Building G

Registration fee: No fee, but registration is required beforehand (simpy write your name and affiliation in the email).

Social event: Dinner in Machida, starting 19:30, restaurant TBD. Please contact me if you want to participate. We'll take the ISAS shuttle for Fuchinobe station at 19:00.

Handouts: Download here!.

Abtract:

Over the last two decades, computational methods for the solution of optimal control problems have advanced substantially, relaying also on the parallel increase in computational power. Nowadays, extremely complex problems can be solved on a standard PC with only a few hours of computational time. A major advancement came from the development of robust and accurate direct approaches derived from other disciplines.

This lecture will present two techniques for the direct numerical solution of optimal control problems: pseudospectral methods and finite elements in time. Starting from an introduction to the concept of finite element in time, the evolution of this approach from the initial indirect solution of optimal control problems with low-order elements to the more recent use of high order elements on spectral basis for the direct transcription of optimal control problems will be discussed. A conceptual comparison will then be presented between pseudospectral methods and finite elements in time highlighting differences and commonalities, along with some space-related examples to illustrate the advantages and drawbacks of these methods and to indicate possible future directions.

The lecture will then introduce a special type of finite element based on the analytical solution of the equations of motions, expressed in non-singular equinoctial elements, under the effect of a low-thrust action. Some examples will show how to use these analytical solutions to quickly design low-thrust trajectories optimising multiple criteria.

The lecture is divided in two parts: an initial theoretical part followed by a hands-on part in which the attendees will be able to apply the concepts acquired in the first part to solve simple problems.

About the lecturer: Prof. Vasile .

## Differential dynamic programming

### Prof. Russell

When: tbd

## Solving Lambert problem using Optimal Control Theory

### Prof. Bando

When: tbd