Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400...

Real-Time Optimal Control and Trajectory Planning with Applications to Industrial Robots

Christof Büskens

INRIA, Optimal Control: Algorithms and Applications (May/June 2007)

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C. BüskensC. Büskens

Optimal Control: Industrial Robots

Foces:

-centrifugal

-Coriolis

-gravity

-frictional (dry)

-restoring

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Parcel Robots and minimal Robot Trajectory Planning

Unloading of ContainersParameter Identification/

Trajectory PlanningRobot Dynamics- kinematics- (minimal) dynamic- collision prevention

with: Deutsche Post, DHL Express, BIBA, EADS

with: SEF–Robotics, ADC, Lüneburg

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Classification of Optimal Control Solutions

open-loop:

closed-loop:

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Mixed Strategy

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Open-Loop vs. Closed-Loop

nonlinearlinearModel

Method

open-loop

closed-loop

knowledge

Part I

Part II

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Part I

nonlinear open-loop nonlinear closed-loop

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• Part I: (Motivation)– Example: Industrial Robot

• Part II: (Optimal Control)- Perturbed Optimal Control Problems- Perturbed NLP Problems- Numerical Solution of NLP-Problems

• Part III: (Real-Time Optimal Control)- Parametric Sensitivity Analysis / Solution Differentiability- Real–Time Solution

• Open–Loop Strategy• Mixed Strategies (Constraints)

- Examples: - Industrial Robot ABB IRB 6400 - Emergency Landing

Overview

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Example: ABB IRB 6400 (Optimal Control Problem)

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Optimal Control Problems with Perturbations: ODE

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Methods for solving Optimal Control Problems

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Overview

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Direct approaches for OCP I: ODE

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Direct approaches for OCP II: ODE

MACH1: [B./Gerdts]WORHP: [B./Gerdts]

NUDOCCCS: [B.]

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Sparse NLP Solver

WORHP

We Optimize Really Huge Problems

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WORHP Born for Space Applications

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Principle Method

QP

SQP

Optimization Process

Objective function

Constraints

Sparsity

User

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SQP Algorithm

Evaluation ofInitial Estimate

Main Iteration

2nd order Information(Hessian)

Compute new iteration point

Preparation of QPinput data

Check Optimality Criteria

QP Solver

Solution of QuadraticSubproblem

Provides search direction

Hessian Update

Find a suitable stepsise

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Requirements

What is a good NLP Solver? • fast• robust• precise• large-scale• user-friendly• flexible• …

fast

robust

precise

large-scale

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Main Features of WORHP

• Sparse Structures– efficient storage and computation

• Automatic Hessian structure– simpler usability for large-scale sparse problems

• Reverse Communication– flexible interface for demanding problem formulations

• Modularity– modules for different intentions and purposes

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Sparse NLP Solver: Sparse Matrices

Sparse Matrix(here: )

Higher efficiency by storing only nonzero entries in column-

compressed format

Lower storage

requirement(here: -87.5%)

Reduced computational time

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Reverse Communication

Advantages

Simple implementationNo worries after start

Caller

Evaluation

ObjectiveGradient

Evaluation

ConstraintsJacobian

Disadvantages

Inflexible user-interfaceNo influence after start

Traditional Calling Sequence:

SQP

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Reverse Communication

Caller

New Calling Sequence:

Evaluation

ObjectiveGradient

Evaluation

ConstraintsJacobian

Advantages

Flexible problem evaluationFull control on optimization

Disadvantages

Harder to implementWORHP

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Features of WORHP: Sentinel

User Solution WORHP

Sentinel

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Group Strategy for Gradient Computation

Challenges of large-scale problems:– How to determine the minimum number of groups?– Efficient algorithm?– Suboptimal groups acceptable?– Extendable to Hessian computation?

Example:Gradient computation for the Rayleigh optimal control problem with

100.000 variables

Evaluations

Standard: 100.000 Groups: 7

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Modularity

Modules to adapt to the User‘s requirements:

Speed, Precision, Robustness, Simplicity, …

GradientFixed Stepsize

GradientAdaptive Stepsize

GradientGroup Strategy

Advanced ErrorReporting

Matrix structureCreation/Analysis

Pre-Iteration-Analysis

WORHP

QPSOL

QP-Matrixdiagonal-BFGS

QP-MatrixHessian

QP-MatrixIdentity

QP-Matrixdiag.-Hessian

QP-MethodNonsmooth Newton

QP-MethodInterior Point

direct SolversMA47, SuperLU

iterative SolversCGN, CGS, BiCG

InterfaceTranscriptor

InterfaceReverse Comm.

InterfaceTraditional

InterfaceAMPL

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Example: ABB IRB 6400 (optimal control)

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Hierarchical and Pareto Optimization

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Offline

Real-Time Capable?

UnperturbedProblem

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Methods for solving Optimal Control Problems

• The result of any performance process is limited by the most scarcely available ressource: the Time. (Drucker)

• We have enough Time, if only we use it wisely. (Goethe)

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Overview

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Perturbed NLP problems

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Parametric Sensitivity Analysis / Solution Differentiability

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Sensitivity Analysis of OCP

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Example: ABB IRB 6400 (parametric sensitivity analysis)

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Overview

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offline

Real-Time Optimization (General Idea)

UnperturbedProblem

SensitivityAnalysis

onlineReal-Time Optimization

Real-Time Optimal Control

SensitivitiesSolution

Solution

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Real-Time Optimization

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Small Perturbation?

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Primary Goals of Real-Time Optimization

Hierarchical AAO-order:

• real-time ability

• admissibility (feasibility)

• optimality

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Idea 1: Adaptive optimal closed-loop control

Adaptive optimal control (closed-loop)

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Idea 2: Real-Time model predictive control

Trajectory error estimation:

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Real-Time model predictive control II

• iterative process (Newton Type, no gradient calculation)• self-correcting• any-time property

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Real-Time model predictive control III

Potential for speedup:

• dynamically moving horizon

• iterative process abortable anytime

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offline

Idea 3: Mixed Strategy

UnperturbedProblem

SensitivityAnalysis

onlineReal-Time Optimization

Real-Time Optimal Control

SensitivitiesSolution

Solution

onlineMathematical

Model

InformationAdvanced

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Real-Time Optimization (Mixed Strategy)

• iterative process (Newton Type, no gradient calculation)• self-correcting• any-time property

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Iterative Process

Convergence ?

• Order of convergence ?

• Existence of a fixed point ?

• Uniqueness of a fixed point ?

• Order of optimality:

worse, unchanged, improved?

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Convergence of the Mixed Strategy

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Convergence of the Mixed Strategy

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Real-Time Optimal Control: (Sensitivity Derivatives)

Indirect Approaches (Minimumprinciple of Pontryagin)• Existence of Derivatives (Still Research)

[ Malanovski, Maurer, Pesch, ...]• Linear Perturbations (in the State) (Feedback Control Laws)

[ Kelley, Breakwell, Speyer, Bryson, Ho, Pesch, Bock, Krämer–Eis, ...]• General Perturbations (Linear Approximations)

[ Maurer, Augustin, Pesch, Kugelmann, ...]

Direct Approaches (NLP Problems)• Existence of Derivatives (NLP)

[ Fiacco, Robinson, ...]• Convergence (Discretized OCP OCP)

[ Alt, B., Dontchev, Felgenhauer, Malanowski, Maurer, ...]• Real–Time Optimal Control (Linear Approximations)

[ B., Maurer, ...]• Real–Time Optimal Control (Nonlinear Feedback Approx.)

[ B. ]

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Overview

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Example: ABB IRB 6400 (real-time optimal control)

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Example: ABB IRB 6400 (real-time optimal control)

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Example: Emergency Landing

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Thank You!

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Real-Time Optimization (Advanced Iteration Methods)

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Aims and Cooperations

Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400...

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Transcript of Suboptimal Improvement of the Classical Riccati Controllerbonnans/...- Industrial Robot ABB IRB 6400...