trajectory optimization from euler … to lawden … to today

AIAA Lunch and Learn

Christopher D’SouzaChristopher D’SouzaThe Charles Stark Draper LaboratoryThe Charles Stark Draper Laboratory

Houston, TXHouston, TX

Trajectory OptimizationTrajectory OptimizationFrom Euler … to Lawden … to TodayFrom Euler … to Lawden … to Today

2 - 9/04


Why Why Optimize?Optimize?

Engineers are always interested in finding the ‘best’ solution to the problem Engineers are always interested in finding the ‘best’ solution to the problem

at handat hand FastestFastest

Fuel EfficientFuel Efficient

Optimization theory allows engineers to accomplish thisOptimization theory allows engineers to accomplish this Often the solution may not be easily obtainedOften the solution may not be easily obtained

In the past, it has been surrounded by a certain mystiqueIn the past, it has been surrounded by a certain mystique

This seminar is aimed at demystifying trajectory optimizationThis seminar is aimed at demystifying trajectory optimization Practical trajectory optimization is now within reachPractical trajectory optimization is now within reach

State of the art computersState of the art computers

State of the art algorithmsState of the art algorithms

In order to fully appreciate trajectory optimization, however, one must In order to fully appreciate trajectory optimization, however, one must

understand something about it’s historyunderstand something about it’s history We need to understand where we’ve been in order to appreciate where we areWe need to understand where we’ve been in order to appreciate where we are

3 - 9/04


The Greeks started The Greeks started it!it!

Queen Dido of Carthage (7 century Queen Dido of Carthage (7 century

BC)BC)

Daughter of the king of TyreDaughter of the king of Tyre

Fled Tyre to TunisiaFled Tyre to Tunisia

Agreed to buy as much land as she could Agreed to buy as much land as she could

“enclose with one bull’s hide”“enclose with one bull’s hide”

Set out to choose the largest amount of land Set out to choose the largest amount of land

possible, with one border along the seapossible, with one border along the sea

A semi-circle with side touching the oceanA semi-circle with side touching the ocean

Founded CarthageFounded Carthage

Fell in love with Aeneas but committed Fell in love with Aeneas but committed

suicide when he leftsuicide when he left

Story immortalized in Homer’s AeneidStory immortalized in Homer’s Aeneid

4 - 9/04


The Italians The Italians CounteredCountered

Joseph Louis Lagrange (1736-Joseph Louis Lagrange (1736-

1813)1813) His work His work Mécanique Analytique Mécanique Analytique

(Analytical Mechanics)(Analytical Mechanics) (1788) was a (1788) was a

mathematical masterpiece mathematical masterpiece

Invented the method of ‘variations’ Invented the method of ‘variations’

which impressed Euler and became which impressed Euler and became

‘calculus of variations’‘calculus of variations’

Invented the method of multipliers Invented the method of multipliers

(Lagrange multipliers)(Lagrange multipliers)

Sensitivities of the performance index Sensitivities of the performance index

to changes in states/constraintsto changes in states/constraints

Became the ‘father’ of ‘Lagrangian’ Became the ‘father’ of ‘Lagrangian’

DynamicsDynamics

Euler-Lagrange EquationsEuler-Lagrange Equations

Obtained the equilibrium points of Obtained the equilibrium points of

the Earth-Moon and Earth-Sun the Earth-Moon and Earth-Sun

systemsystem

5 - 9/04


The Multi-Talented Mr. The Multi-Talented Mr. EulerEuler

Euler (1707-1783)Euler (1707-1783) Friend of LagrangeFriend of Lagrange

Published a treatise which Published a treatise which

became the de facto standard of became the de facto standard of

the ‘calculus of variations’the ‘calculus of variations’

The Method of Finding Curves The Method of Finding Curves

that Show Some Property of that Show Some Property of

Maximum or MinimumMaximum or Minimum

He solved the brachistachrone He solved the brachistachrone

((brachistosbrachistos = shortest, = shortest, chronoschronos

= time) problem very easily= time) problem very easily

Minimum time path for a bead Minimum time path for a bead

on a stringon a string

CycloidCycloid

6 - 9/04


The Plot Thickens: The Plot Thickens: HamiltonHamilton

William Hamilton (1805-1865)William Hamilton (1805-1865) Published work on least action in mechanical Published work on least action in mechanical

systems that involved two partial differential systems that involved two partial differential equationsequations

Inventor of the quaternionInventor of the quaternion

Karl Gustav Jacob Jacobi (1804-Karl Gustav Jacob Jacobi (1804-1851)1851)

Discovered ‘conjugate points’ in the fields of Discovered ‘conjugate points’ in the fields of extremalsextremals

Gave an insightful treatment to the second Gave an insightful treatment to the second variationvariation

Jacobi criticized Hamilton’s workJacobi criticized Hamilton’s work Only one PDE was requiredOnly one PDE was required

Hamilton-Jacobi equationHamilton-Jacobi equation

Became the basis of Bellman’s work 100 Became the basis of Bellman’s work 100 years lateryears later

and Jacobiand Jacobi

7 - 9/04


The ‘Chicago The ‘Chicago School’School’

At the beginning of the twentieth century At the beginning of the twentieth century Gilbert Bliss and Oskar Bolza gathered a Gilbert Bliss and Oskar Bolza gathered a number of mathematicians at the number of mathematicians at the University of Chicago University of Chicago

Made major advances in calculus of variations Made major advances in calculus of variations following on the work of Karl Wilhelm Theodor following on the work of Karl Wilhelm Theodor WeierstrassWeierstrass

Applied this to the field of ballistics during WW I Applied this to the field of ballistics during WW I Artillery firing tablesArtillery firing tables

Second Variation Conditions (conjugate point Second Variation Conditions (conjugate point conditions)conditions)

Built on the work of Legendre, Jacobi, and Built on the work of Legendre, Jacobi, and ClebschClebsch

Graduated many of the premiere applied Graduated many of the premiere applied mathematicians of the early/mid 20mathematicians of the early/mid 20thth century century

M. R. HestenesM. R. Hestenes

E. J. McShaneE. J. McShane

8 - 9/04


Derek and the PrimerDerek and the Primer

During the 1950s, Derek Lawden applied During the 1950s, Derek Lawden applied

the calculus of variations to exo-the calculus of variations to exo-

atmospheric rocket trajectoriesatmospheric rocket trajectories Published Published Optimal Space Trajectories for Optimal Space Trajectories for

NavigationNavigation

Concerned with thrusting and coasting arcsConcerned with thrusting and coasting arcs

‘‘Invented’ the Invented’ the primer vectorprimer vector Direction is along the thrust directionDirection is along the thrust direction

Directly related to the velocity Lagrange Directly related to the velocity Lagrange

multipliermultiplier

Provided a methodology for determining Provided a methodology for determining

optimal space trajectoriesoptimal space trajectories

9 - 9/04


The Russians are ComingThe Russians are Coming

In the mid 1950s a group of Russian Air Force In the mid 1950s a group of Russian Air Force

officers went to the Steklov Mathematical Institute officers went to the Steklov Mathematical Institute

outside of Moscow to find out whether the outside of Moscow to find out whether the

mathematicians could determine a particular set of mathematicians could determine a particular set of

optimal aircraft maneuversoptimal aircraft maneuvers

Pontryagin, the director of the Institute, accepted Pontryagin, the director of the Institute, accepted

the challenge and went on to invent a ‘new the challenge and went on to invent a ‘new

calculus of variations’calculus of variations’ The Maximum PrincipleThe Maximum Principle

Used the concept of control parameters, Used the concept of control parameters, upravlenieupravlenie, or , or uu

Solved the original problem and in the process Solved the original problem and in the process

revolutionized optimal control and trajectory revolutionized optimal control and trajectory

optimizationoptimization

– – PontryaginPontryagin

10 - 9/04


The American Response – Bryson The American Response – Bryson

Arthur Bryson, then at Harvard, an Arthur Bryson, then at Harvard, an aerodynamicist, came across the paper by aerodynamicist, came across the paper by Pontryagin and immediately recognized its Pontryagin and immediately recognized its valuevalue

He applied it to a problem of finding an He applied it to a problem of finding an minimum time to climb trajectory and minimum time to climb trajectory and presented it to the militarypresented it to the military

It was sent to Pax River and was demonstrated by Lt. John It was sent to Pax River and was demonstrated by Lt. John Young (using an altitude vs Mach number table at 1000 ft Young (using an altitude vs Mach number table at 1000 ft intervals)intervals)

338 seconds vs the predicted 332 seconds338 seconds vs the predicted 332 seconds

PathPath Accelerate to M = 0.84 at just about ground level where drag Accelerate to M = 0.84 at just about ground level where drag

rise beginsrise begins

Climb at constant Mach number to 30,000 ftClimb at constant Mach number to 30,000 ft

Shallow dive to 24,000 ft followed by a slow climb to 30000 ft, Shallow dive to 24,000 ft followed by a slow climb to 30000 ft,

increasing energy until the energy equals the final energyincreasing energy until the energy equals the final energy

Climb very rapidly to desired altitude (20 km)Climb very rapidly to desired altitude (20 km)

Applied this new ‘optimal control theory’ to Applied this new ‘optimal control theory’ to various aerospace engineering problems, various aerospace engineering problems, particularly those of interest to the US militaryparticularly those of interest to the US military

11 - 9/04


The Inescapable KalmanThe Inescapable Kalman

Rudolf Kalman first came on the Rudolf Kalman first came on the

scene in the late 50s leading the scene in the late 50s leading the

way to the way to the state spacestate space paradigm paradigm

of control theory along with the of control theory along with the

concepts of controllability and concepts of controllability and

observabilityobservability

He then introduced an integral He then introduced an integral

performance index that had performance index that had

quadratic penalties on the state quadratic penalties on the state

error and control magnitudeerror and control magnitude Demonstrated that the optimal Demonstrated that the optimal

controls were linear feedbacks of the controls were linear feedbacks of the

state variablesstate variables

Led to time varying linear systems Led to time varying linear systems

and MIMO systemsand MIMO systems

He later collaborated with Bucy He later collaborated with Bucy

to give us the Kalman-Bucy to give us the Kalman-Bucy

filterfilter

As some may know, these concepts were integral to the success of the guidance and navigation systems on the Apollo program

12 - 9/04


Other Trajectory Optimization Other Trajectory Optimization LegendsLegends

Richard BellmanRichard Bellman Introduced a new view and an extension of Introduced a new view and an extension of

Hamilton-Jacobi theory called Dynamic Hamilton-Jacobi theory called Dynamic

Programming and the Hamilton-Jacobi-Bellman Programming and the Hamilton-Jacobi-Bellman

equationequation

Led to a family of extremal pathsLed to a family of extremal paths

Provides Provides optimal nonlinear feedbackoptimal nonlinear feedback

Curse of dimensionalityCurse of dimensionality

John BreakwellJohn Breakwell Among the first to apply the calculus of variations Among the first to apply the calculus of variations

to optimal spacecraft and missile trajectoriesto optimal spacecraft and missile trajectories

Prof. Angelo MieleProf. Angelo Miele Among the first to develop numerical procedures Among the first to develop numerical procedures

for solving trajectory optimization problems for solving trajectory optimization problems

(SGRA)(SGRA)

Dr. Henry (Hank) KellyDr. Henry (Hank) Kelly Developed conditions for singular optimal control Developed conditions for singular optimal control

problems (called the Kelley Conditions in Russia)problems (called the Kelley Conditions in Russia)

13 - 9/04


So So What?What?

The brief reconnaissance into the history of trajectory The brief reconnaissance into the history of trajectory

optimization is intended to demonstrate the rich heritage optimization is intended to demonstrate the rich heritage

which we possesswhich we possess

It was also intended to prepare us for a discussion of It was also intended to prepare us for a discussion of

where we are and where we are goingwhere we are and where we are going

We began this seminar asking the question: Why We began this seminar asking the question: Why

optimize?optimize?

Because we are engineers and we want to find the ‘best’ solutionBecause we are engineers and we want to find the ‘best’ solution

So, how do we go about optimizing?So, how do we go about optimizing?

14 - 9/04


What to What to Optimize?Optimize?

Engineers intuitively know what they are interested in Engineers intuitively know what they are interested in optimizingoptimizing

Straightforward problemsStraightforward problems FuelFuel

TimeTime

PowerPower

EffortEffort

More complexMore complex Maximum marginMaximum margin

Minimum riskMinimum risk

The mathematical quantity we optimize is called a The mathematical quantity we optimize is called a cost cost functionfunction or or performance indexperformance index

15 - 9/04


The Trajectory Optimization The Trajectory Optimization NomenclatureNomenclature

Dynamical constraints Dynamical constraints Examples: equations of motion (Newton’s Laws)Examples: equations of motion (Newton’s Laws)

Controls (Controls (uu)) Exogenous (independent) variables which operate on the systemExogenous (independent) variables which operate on the system

Examples: Thrust, flight control surfacesExamples: Thrust, flight control surfaces

States (States (xx)) Dependent variables which define the ‘state’ of the systemDependent variables which define the ‘state’ of the system

Examples: position, velocity, massExamples: position, velocity, mass

Terminal constraintsTerminal constraints Conditions that the initial and final states must satisfyConditions that the initial and final states must satisfy

Example: circular orbit with a particular energy and inclinationExample: circular orbit with a particular energy and inclination

Path constraintsPath constraints Conditions which must be satisfied at all points of the trajectoryConditions which must be satisfied at all points of the trajectory

Example: Thrust boundsExample: Thrust bounds

Point constraintsPoint constraints Conditions at particular points along the trajectoryConditions at particular points along the trajectory

Examples: way points, maximum heatingExamples: way points, maximum heating

Trajectory optimization seeks to obtain both the states and the controls which Trajectory optimization seeks to obtain both the states and the controls which optimize the chosen performance index while satisfying the constraintsoptimize the chosen performance index while satisfying the constraints

16 - 9/04


The Optimal Control The Optimal Control ProblemProblem

The general trajectory optimization problem can be posed as: The general trajectory optimization problem can be posed as:

find the states and controls which find the states and controls which

subject to the dynamicssubject to the dynamics

which takes the system from to the terminal constraintswhich takes the system from to the terminal constraints

17 - 9/04


The Optimality Conditions and Pontryagin’s Minimum The Optimality Conditions and Pontryagin’s Minimum PrinciplePrinciple

These are also called the Euler-Lagrange equations

18 - 9/04


The Optimality Conditions and Pontryagin’s Minimum The Optimality Conditions and Pontryagin’s Minimum PrinciplePrinciple

The boundary conditions are

There is one additional condition (sometimes called the Weierstrass Condition) which must satisfy

for any (the set of controls that meet the constraints)

All of these conditions are collectively called the Pontryagin Minimum Principle (PMP)

19 - 9/04


Comments on the Pontryagin Minimum Comments on the Pontryagin Minimum ConditionsConditions

The Pontryagin conditions are very powerful tools to help find optimal The Pontryagin conditions are very powerful tools to help find optimal

trajectoriestrajectories Infinite Dimensional ConditionsInfinite Dimensional Conditions

It is a two-point boundary value problemIt is a two-point boundary value problem

States are specified at the initial timeStates are specified at the initial time

Costates (Lagrange multipliers) are specified at the final timeCostates (Lagrange multipliers) are specified at the final time

Some states (or combinations of states) are specified at the final timeSome states (or combinations of states) are specified at the final time

Equivalent to solving a PDEEquivalent to solving a PDE

Most problems cannot be solved in closed formMost problems cannot be solved in closed form Closed form solutions lend themselves to analysisClosed form solutions lend themselves to analysis

Need to use numerical methods to obtain solutions for real-world problemsNeed to use numerical methods to obtain solutions for real-world problems

No guarantee of a solutionNo guarantee of a solution

Convergence issuesConvergence issues

Stability issuesStability issues

In the process we convert an infinite dimensional problem into a finite dimensional In the process we convert an infinite dimensional problem into a finite dimensional

problemproblem

Implicit in numerical integrationImplicit in numerical integration

20 - 9/04


How to Optimize? How to Optimize?

Two general types of methods exist for solving optimal control Two general types of methods exist for solving optimal control problemsproblems

DirectDirect Methods Methods Discretize the states and controls at points in timeDiscretize the states and controls at points in time

NodesNodes

Convert the problem into a parameter optimization problemConvert the problem into a parameter optimization problem States and controls at the nodes become the optimizing parametersStates and controls at the nodes become the optimizing parameters

Use an NLP (Non-Linear Program) to solve the parameter optimization problemUse an NLP (Non-Linear Program) to solve the parameter optimization problem

Advantages: Fast SolutionAdvantages: Fast Solution

Disadvantages: Difficult to determine/prove optimalityDisadvantages: Difficult to determine/prove optimality

InIndirectdirect Methods Methods Operate on the Pontryagin Necessary ConditionsOperate on the Pontryagin Necessary Conditions

This is a two-point boundary value problemThis is a two-point boundary value problem Use Shooting methodsUse Shooting methods

Advantages: Easy to determine optimalityAdvantages: Easy to determine optimality

Disadvantages: (Very) difficult to convergeDisadvantages: (Very) difficult to converge

21 - 9/04


Direct MethodsDirect Methods

CollocationCollocation A method in which you choose states A method in which you choose states andand

controls at points in time along the trajectory controls at points in time along the trajectory These points are called These points are called nodesnodes

States and control values at the nodes become States and control values at the nodes become the optimizing variablesthe optimizing variables

Convert the infinite dimensional problem into a Convert the infinite dimensional problem into a finite dimensional, parameter optimization finite dimensional, parameter optimization problemproblem

Enforce the constraints at the nodesEnforce the constraints at the nodes DynamicDynamic

PathPath

Solved using a NonLinear Program (NLP)Solved using a NonLinear Program (NLP)

Types of Spacing Types of Spacing Uniform spacingUniform spacing

Nonuniform spacingNonuniform spacing

x

t

22 - 9/04


Numerical Optimization Numerical Optimization SolversSolvers

The general form of the nonlinear programming problem The general form of the nonlinear programming problem

(NLP) is(NLP) is

My favorite is SNOPT developed by Philip Gill My favorite is SNOPT developed by Philip Gill Sparse sequential quadratic programming (SQP)Sparse sequential quadratic programming (SQP)

Can be used for problems with thousands of constraints and variablesCan be used for problems with thousands of constraints and variables

State of the artState of the art

23 - 9/04


Trajectory Optimization Trajectory Optimization PackagesPackages

POST (POST (PProgram to rogram to OOptimize ptimize SSimulated imulated TTrajectories)rajectories) Direct/Multiple shooting FORTRAN program originally developed in 1970 for Space Shuttle Direct/Multiple shooting FORTRAN program originally developed in 1970 for Space Shuttle

Trajectory Optimization by NASA LangleyTrajectory Optimization by NASA Langley

Generalized point mass, discrete parameter targeting and optimization program. Generalized point mass, discrete parameter targeting and optimization program.

Provides the capability to target and optimize point mass trajectories for a powered or Provides the capability to target and optimize point mass trajectories for a powered or unpowered vehicle near an arbitrary rotating, oblate planet unpowered vehicle near an arbitrary rotating, oblate planet

SORT (SORT (SSimulation and imulation and OOptimization ptimization RRocket ocket TTrajectories)rajectories) FORTRAN program originally developed for ascent vehicle trajectoriesFORTRAN program originally developed for ascent vehicle trajectories

Used to generate Space Shuttle guidance targets and maintained by Lockheed-MartinUsed to generate Space Shuttle guidance targets and maintained by Lockheed-Martin

Can be used with a optimization package to optimize the trajectoryCan be used with a optimization package to optimize the trajectory Variable Metric MethodsVariable Metric Methods

NPSOLNPSOL

OTIS (OTIS (OOptimal ptimal TTrajectories through rajectories through IImplicit mplicit SSimulation)imulation) FORTRAN program for simulating and optimizing point mass trajectories of a wide variety of FORTRAN program for simulating and optimizing point mass trajectories of a wide variety of

aerospace vehicles from NASA Glenn supported by Boeing (Steve Paris) in Seattleaerospace vehicles from NASA Glenn supported by Boeing (Steve Paris) in Seattle Originally developed by Hargraves and ParisOriginally developed by Hargraves and Paris

Designed to simulate and optimize trajectories of launch vehicles, aircraft, missiles, satellites, Designed to simulate and optimize trajectories of launch vehicles, aircraft, missiles, satellites, and interplanetary vehiclesand interplanetary vehicles

Can be used to analyze a limited set of multi-vehicle problems, such as a multi-stage launch Can be used to analyze a limited set of multi-vehicle problems, such as a multi-stage launch system with a fly back boostersystem with a fly back booster

Hermite-Simpson collocation method which uses NZOPT as NLPHermite-Simpson collocation method which uses NZOPT as NLP

24 - 9/04


State of the Art Optimizers for Optimal State of the Art Optimizers for Optimal ControlControl

SOCS (SOCS (SSparse parse OOptimization for ptimization for CControl ontrol SSystems)ystems) General-purpose FORTRAN software for solving optimal control problems from Boeing (Seattle)General-purpose FORTRAN software for solving optimal control problems from Boeing (Seattle)

Trajectory optimizationTrajectory optimization

Chemical process control Chemical process control

Machine tool path definitionMachine tool path definition

Uses Trapezoid, Hermite-Simpson or Runge-Kutta integrationUses Trapezoid, Hermite-Simpson or Runge-Kutta integration

NLP is SPRNLP written by Betts and HuffmanNLP is SPRNLP written by Betts and Huffman

Uniform node spacing, but can have multiple intervals Uniform node spacing, but can have multiple intervals

Provides mesh refinement for complex problemsProvides mesh refinement for complex problems

DIDO (DIDO (DDirect and irect and IInnDDirect irect OOptimization)ptimization) Also named after Queen Dido of CarthageAlso named after Queen Dido of Carthage

General-purpose user-friendly MATLAB software for solving optimal control problems from NPSGeneral-purpose user-friendly MATLAB software for solving optimal control problems from NPS

Non-uniform node spacing with multiple intervalsNon-uniform node spacing with multiple intervals Legendre-Gauss-Lobatto pointsLegendre-Gauss-Lobatto points

Uses a sparse numerical optimization solver (SNOPT)Uses a sparse numerical optimization solver (SNOPT)

Can determine if the necessary conditions are satisfiedCan determine if the necessary conditions are satisfied

Has been used to solve a wide variety of missile and spacecraft problemsHas been used to solve a wide variety of missile and spacecraft problems

Very fast even for complex problemsVery fast even for complex problems

Current research is being directed toward real-time usesCurrent research is being directed toward real-time uses

25 - 9/04


The Wave of the Future – Pseudospectral The Wave of the Future – Pseudospectral MethodsMethods

Pseudospectral methods choose the collocation points in such Pseudospectral methods choose the collocation points in such a way as to minimize integration errora way as to minimize integration error

Number of nodes dependent on accuracy desiredNumber of nodes dependent on accuracy desired

The nodes are non-uniformly spaced in timeThe nodes are non-uniformly spaced in time Quadratic spacing at the endsQuadratic spacing at the ends

Number determines the spacingNumber determines the spacing

They use (global basis) functions which (optimally) They use (global basis) functions which (optimally) approximate the states and controls and enforce the (dynamic approximate the states and controls and enforce the (dynamic and path) constraints at the nodes over the interval [-1, 1]and path) constraints at the nodes over the interval [-1, 1]

Chebyshev-GaussChebyshev-Gauss

Legendre-GaussLegendre-Gauss

Chebyshev-Gauss-LobattoChebyshev-Gauss-Lobatto

Legendre-Gauss-LobattoLegendre-Gauss-Lobatto

Pseudospectral methods yield ‘spectral accuracy’Pseudospectral methods yield ‘spectral accuracy’ Optimal interpolationOptimal interpolation

Particularly well suited for trajectory optimization problems where much of Particularly well suited for trajectory optimization problems where much of the activity occurs at the ends of the intervalsthe activity occurs at the ends of the intervals

} Includes the end points

26 - 9/04


Pseudospectral Point Distribution (N = Pseudospectral Point Distribution (N = 10)10)

} }

Quadratic clustering at ends

27 - 9/04


Launch Vehicle Example: Three Stage to Launch Vehicle Example: Three Stage to OrbitOrbit

Suppose we wish to find the optimal trajectory for a three stage Suppose we wish to find the optimal trajectory for a three stage vehicle to get the maximum payload to orbitvehicle to get the maximum payload to orbit

Performance indexPerformance index

Differential constraints (equations of motion)Differential constraints (equations of motion)

Terminal constraintsTerminal constraints

Throttle capability (minimum, maximum specified)Throttle capability (minimum, maximum specified)

Coast of at least 5 seconds between second and third stageCoast of at least 5 seconds between second and third stage Maximum of 115 secondsMaximum of 115 seconds

28 - 9/04


Problem Specific Problem Specific IssuesIssues

Coordinate Systems Coordinate Systems

DynamicsDynamics

InertialInertial

SphericalSpherical

EquinoctialEquinoctial

Controls Controls

AnglesAngles

Thrust componentsThrust components

Direction cosinesDirection cosines

ScalingScaling

For good convergence properties, we need all the variables to be of ‘order 1’For good convergence properties, we need all the variables to be of ‘order 1’

So we scale the states, the controls and the time to achieve thisSo we scale the states, the controls and the time to achieve this

The ‘art’ of trajectory optimizationThe ‘art’ of trajectory optimization

Tuning knobsTuning knobs

29 - 9/04


Three Stage to Orbit Thrust Three Stage to Orbit Thrust ProfileProfile

Maximum Thrust

Minimum ThrustCoast

30 - 9/04


Three Stage to Orbit Thrust Direction Three Stage to Orbit Thrust Direction ProfileProfile

First Stage Separation

Second Stage Separation

31 - 9/04


Three Stage to Orbit Mass Three Stage to Orbit Mass ProfileProfile

First Stage Separation

Second Stage Separation

Coast

32 - 9/04


Orbit Orbit TransferTransfer

Optimal transfers between two orbits have been the subject of Optimal transfers between two orbits have been the subject of directed research for the past 40 yearsdirected research for the past 40 years

Much analytical and computational effort has been devoted to this taskMuch analytical and computational effort has been devoted to this task

Primer vector theory has been appliedPrimer vector theory has been applied

Numerical solutions are sometimes difficult to obtainNumerical solutions are sometimes difficult to obtain

The Legendre PseudoSpectral (LPS) method has been used to The Legendre PseudoSpectral (LPS) method has been used to extensively analyze this problemextensively analyze this problem

Impulsive burn approximationsImpulsive burn approximations

Finite burn effectsFinite burn effects

Types of coordinate systemsTypes of coordinate systems CartesianCartesian

EquinoctialEquinoctial Nonsingular orbital elements

33 - 9/04


Impulsive Orbit Impulsive Orbit TransferTransfer

Elliptical-Elliptical HohmannElliptical-Elliptical HohmannTransferTransfer

Analytic Solution:Analytic Solution:vv11= = 2076.72 m/s2076.72 m/s

vv2 2 = 87.46 m/s= 87.46 m/s

LPS Solution:LPS Solution:vv11= = 2076.71 m/s2076.71 m/s

vv2 2 = 87.49 m/s= 87.49 m/s

Elliptical-Elliptical Transfer with Elliptical-Elliptical Transfer with Inclination ChangeInclination Change

Analytic Solution:Analytic Solution:vv11= = 2106.13 m/s2106.13 m/s

vv2 2 = 239.69 m/s= 239.69 m/s

LPS Solution:LPS Solution:vv11= = 2106.17 m/s2106.17 m/s

vv2 2 = 239.65 m/s= 239.65 m/s

34 - 9/04


Finite Burn Orbit Transfer: LEO (ISS) to LEO (Sun Finite Burn Orbit Transfer: LEO (ISS) to LEO (Sun Synchronous)Synchronous)

Finite Burn Accumulated V V = 8027.5 m/s

Impulsive Burn Accumulated V V = 6548.6 m/s

Orbital Orbital

ElementsElements

InitialInitial Final OrbitFinal Orbit

aa 6772 km6772 km 7062 km7062 km

ee 7.08E-47.08E-4 1.115E-31.115E-3

ii 51.651.6oo 98.298.2oo

58.658.6oo 120.1120.1oo

238.3238.3oo 282.0282.0oo

35 - 9/04


Further Applications of Further Applications of LPSLPS

ISS Momentum DesaturationISS Momentum Desaturation

Constellation DesignConstellation Design

Libration point formation designsLibration point formation designs

Entry Trajectory DesignEntry Trajectory Design

Planetary Mission DesignPlanetary Mission Design

36 - 9/04


What is Next? -- MAHC What is Next? -- MAHC

Multi-Agent Hybrid Control (MAHC)Multi-Agent Hybrid Control (MAHC)

2121stst Century extension of 20 Century extension of 20thth Century optimal control Century optimal control

A general optimization framework for multiple vehiclesA general optimization framework for multiple vehicles

Multiple constraints on each vehicleMultiple constraints on each vehicle

Allow for discrete decision variables Allow for discrete decision variables

ExampleExample

Two stage vehicleTwo stage vehicle

Return vehicle must land at a particular point

Latitude: -28.25N § 1 km

Longitude: -70.1 E § 1 km

Ascent vehicle continues to a desired orbit while maximizing mass to

orbit

The discrete state space is as follows

37 - 9/04


Multi-Agent Hybrid Trajectory Optimization Example: Position Multi-Agent Hybrid Trajectory Optimization Example: Position ProfileProfile

38 - 9/04


Multi-Agent Multi-Agent Hybrid Trajectory Optimization Hybrid Trajectory Optimization ExampleExample

39 - 9/04


Hybrid Trajectory Optimization Example – Control Hybrid Trajectory Optimization Example – Control HistoryHistory

40 - 9/04


What is Next? -- Real-time Trajectory Optimization What is Next? -- Real-time Trajectory Optimization

‘‘Real-time’ trajectory optimizationReal-time’ trajectory optimization Computational capability is increasing with Moore’s lawComputational capability is increasing with Moore’s law

Time is approaching when these (direct) methods can be Time is approaching when these (direct) methods can be

implemented on board vehicles and optimized in ‘real-time’implemented on board vehicles and optimized in ‘real-time’

1 Hz1 Hz

Guidance cycles (outer loop) slower than control cycles (inner Guidance cycles (outer loop) slower than control cycles (inner

loop)loop)

Application to orbit (transfer) problemApplication to orbit (transfer) problem

IssuesIssues

ConvergenceConvergence

Stability of solutionsStability of solutions

41 - 9/04


What is Next? - What is Next? - NOGNOG

Neighboring Optimal Guidance (NOG)Neighboring Optimal Guidance (NOG)

A real-time guidance scheme which determines a new A real-time guidance scheme which determines a new

optimal path which is ‘close’ to the nominal (a priori) optimal path which is ‘close’ to the nominal (a priori)

optimal pathoptimal path

Neighboring optimalNeighboring optimal

Operates on deviations from the optimal trajectoryOperates on deviations from the optimal trajectory

Very robustVery robust

Based upon the second variation sufficient conditionsBased upon the second variation sufficient conditions

42 - 9/04


ConclusioConclusionn

Trajectory optimization has advanced greatly over the past 40 yearsTrajectory optimization has advanced greatly over the past 40 years

We are at the threshold of a new era for solving exciting complex We are at the threshold of a new era for solving exciting complex optimization problemsoptimization problems

New methods exist for solving (general) optimal control problems New methods exist for solving (general) optimal control problems Trajectory optimization problems are a subset of this classTrajectory optimization problems are a subset of this class

These methods give (reasonably) fast solutions even given poor guessesThese methods give (reasonably) fast solutions even given poor guesses Fast computersFast computers

Good algorithmsGood algorithms

Don’t need to know the details of the methods or devote your career to Don’t need to know the details of the methods or devote your career to optimizationoptimization

Just your problemJust your problem

Solution of complex trajectory optimization problems is Solution of complex trajectory optimization problems is within reach of the practicing engineerwithin reach of the practicing engineer

43 - 9/04


Selected ReferencesSelected References

Lietmann, G., Lietmann, G., Optimization TechniquesOptimization Techniques, Academic Press, 1962., Academic Press, 1962.

Lawden, D.F., Lawden, D.F., Optimal Trajectories for Space NavigationOptimal Trajectories for Space Navigation, , Butterworths, 1963.Butterworths, 1963.

Bryson, A.E. and Ho, Y-C., Bryson, A.E. and Ho, Y-C., Applied Optimal ControlApplied Optimal Control, Hemisphere , Hemisphere Publishing Company, 1975.Publishing Company, 1975.

Gill, P.E., Murray, W., and Wright, M.H., Gill, P.E., Murray, W., and Wright, M.H., Practical OptimizationPractical Optimization, , Academic Press, 1981.Academic Press, 1981.

Fletcher, R., Fletcher, R., Practical Methods of OptimizationPractical Methods of Optimization, Wiley Press, , Wiley Press, 1987.1987.

Betts, J.T., Betts, J.T., Practical Methods for Optimal Control Using Practical Methods for Optimal Control Using Nonlinear ProgrammingNonlinear Programming, SIAM: Advances in Control and Design , SIAM: Advances in Control and Design Series, 2001.Series, 2001.

44 - 9/04


QuestionsQuestions??

trajectory optimization from euler … to lawden … to today

Documents