Post on 10-Mar-2021
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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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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• 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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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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• 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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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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• 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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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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• 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 (real-time optimal control)
click me
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Example: ABB IRB 6400 (real-time optimal control)
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Example: Emergency Landing
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Example: Emergency Landing
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Example: Emergency Landing
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Example: Emergency Landing
click me
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Thank You!
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Real-Time Optimization (Advanced Iteration Methods)
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Aims and Cooperations