trajectory optimization from euler … to lawden … to today
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Trajectory Optimization From Euler … to Lawden … to Today. Christopher D’Souza The Charles Stark Draper Laboratory Houston, TX. Why Optimize?. Engineers are always interested in finding the ‘best’ solution to the problem at hand Fastest Fuel Efficient - PowerPoint PPT PresentationTRANSCRIPT
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
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AIAA Lunch and Learn
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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?
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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
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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
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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
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The Optimality Conditions and Pontryagin’s Minimum The Optimality Conditions and Pontryagin’s Minimum PrinciplePrinciple
These are also called the Euler-Lagrange equations
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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)
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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
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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
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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
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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
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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
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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
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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
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Pseudospectral Point Distribution (N = Pseudospectral Point Distribution (N = 10)10)
} }
Quadratic clustering at ends
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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
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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
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Three Stage to Orbit Thrust Three Stage to Orbit Thrust ProfileProfile
Maximum Thrust
Minimum ThrustCoast
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Three Stage to Orbit Thrust Direction Three Stage to Orbit Thrust Direction ProfileProfile
First Stage Separation
Second Stage Separation
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Three Stage to Orbit Mass Three Stage to Orbit Mass ProfileProfile
First Stage Separation
Second Stage Separation
Coast
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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
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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
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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
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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
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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
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Multi-Agent Hybrid Trajectory Optimization Example: Position Multi-Agent Hybrid Trajectory Optimization Example: Position ProfileProfile
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Multi-Agent Multi-Agent Hybrid Trajectory Optimization Hybrid Trajectory Optimization ExampleExample
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Hybrid Trajectory Optimization Example – Control Hybrid Trajectory Optimization Example – Control HistoryHistory
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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
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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
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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
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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.
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QuestionsQuestions??