Pfadverfolgung –
Von Normalformen zu prädiktiven Reglern
Timm Faulwasser
Laboratoire d’Automatique, Ecole Polytechnique Fédérale de Lausanne
Institut für Angewandte Informatik, Karlsruher Institut für Technologie
Joint work with:
Tobias Weber, Janine Matschek, Juan Pablo Zometa, Rolf Findeisen (OvGU MD)
Sven Lorenz, Johann Dauer (DLR Braunschweig)
Elgersburg Workshop | 11.02.2016
Stabilization, Trajectory Tracking and Path Following?
• Reference set-point
• Control error
time invariant problem
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Set-point Stabilization
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Stabilization, Trajectory Tracking and Path Following?
Set-point Stabilization
• Reference trajectory
• Reference set-point
• Tracking error
• Control error
time varying problem
performance limits?
time invariant problem
Qui, L. & Davison, E. Performance limitations of non-minimum phase systems in the servomechanism problem. Automatica,
1993, 29, 337-349
Seron, M.; Braslavsky, J.; Kokotovic, P. V. & Mayne, D. Q. Feedback limitations in nonlinear systems: from Bode integrals to
cheap control. IEEE Trans. Automat. Contr., 1999, 44, 829-833
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Trajectory Tracking
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Stabilization, Trajectory Tracking and Path Following?
Path Following
• Reference trajectory
• Reference path
• Tracking error
• Path-following error
time varying problem
performance limits?
time invariant problem
additional design parameters
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Trajectory Tracking
Applications? Controller design?
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Typical Applications
• Autonomous vehicles
• Machine tools
• Motion problems
•
…
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Path Following
Problem definition? Controller design?
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Path Following
• Set-point stabilization, trajectory tracking and path following
• Problem analysis: tailored normal forms
• Path followability results
• Examples
• Feedforward path following and closed-loop path following
Model Predictive Path Following Control
• Convergence properties
• MPFC for a robotic manipulator
• Feedforward path following for an autonomous helicopter
Outlook and Summary
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Outline
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Output Path-following Problem
• Output path
regular 1d curve
Control task
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Convergence to path
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Convergence on path
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Satisfaction of constraints
Questions
• Problem structure
• Constraints
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Suitable formulation?
Controller design?
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
• Dynamic system
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state space
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= path manifold
output space
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Analysis of Path-following Problems
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Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Vector Relative Degree (I)
[Isidori `95; Nijmeijer & van der Schaft `90]
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Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Vector Relative Degree (II)
[Isidori `95; Nijmeijer & van der Schaft `90]
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Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Vector Relative Degree (III)
[Isidori `95; Nijmeijer & van der Schaft `90]
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Analysis of Path-following Problems
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Output space
= path manifold
Timing law
Augmented system
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
State space
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Transverse Normal Form
Idea: map to suitable coordinates
Transverse normal form
[Nielsen & Maggiore `08; Banaszuk & Hauser `95; F. & Findeisen `16]
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Augmented system
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Transverse Normal Form
Idea: map to suitable coordinates
Example? Path followability?
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Transverse normal form
Augmented system
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Augmented system
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Example – Fully Actuated Robot
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Example – Fully Actuated Robot
Transverse normal form
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Augmented system
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Path Followability and Transverse Normal Forms
Transverse normal form
[Faulwasser `13]
Necessary conditions? Constraints? Feedback design?
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Augmented system
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[Faulwasser `13]
Feedback control?
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Path Followability in the Presence of Constraints
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Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Feedforward Path Following
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Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Closed-loop Path Following
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Path Following
• Set-point stabilization, trajectory tracking and path following
• Problem analysis: tailored normal forms
• Path followability results
• Examples
• Feedforward path following and closed-loop path following
Model Predictive Path Following Control
• Convergence properties
• MPFC for a robotic manipulator
• Feedforward path following for an autonomous helicopter
Outlook and Summary
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Outline
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Principle of Predictive Control
Model Predictive Control (MPC) = repeated optimal control
at
2. Solve
3. Apply
for
Advantages:
constraints, MIMO systems, optimization of transients, dist. implementation
NMPC for path-following problems?
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
1. State observation
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Principle of Predictive Path Following
Optimal Control Problem
Stability/convergence? Example?
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Idea
• Prediction based on augmented dynamics.
• Costs penalize path-following error (and its time derivative).
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Benchmark Problem
• Robot KUKA LBR IV  up to 7 joints.
• Make the robot write on a blackboard.
• Allow interaction between user and robot.
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Application to a Robotic Manipulator
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Implementation of Predictive Path-Following on KUKA LWR IV
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Dynamics of the robot
– Path parameter dynamics
– Error outputs
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Cost function, terminal penalty
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Prediction horizon
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Optimization solved with ACADO Toolkit
, sampling time
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
– State space representation
Results?
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Joint work with: Juan P. Zometa, Janine Matschek, Tobias Weber, Rolf Findeisen
(all OvG Magdeburg)
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Application to a Robotic Manipulator
[Faulwasser et al. `15]
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Joint work with: Juan P. Zometa, Janine Matschek, Tobias Weber, Rolf Findeisen
(all OvG Magdeburg)
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Application to a Robotic Manipulator
[Faulwasser et al. `15]
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Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Application to a Robotic Manipulator
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Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Application to a Robotic Manipulator
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References
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Aguiar et al. (2005). Path-following for non minimumphase systems removes performanc e limitations. IEEE Transactions on
Automatic Control 50, 234-239.
Dacic & Kokotovic (2006). Path-following for linear systems with unstable zero dynamics. Automatica 42, 1673-1683.
Aguiar et al. (2008). Performance limitations in reference tracking and path-following for nonlinear systems. Automatica 44, 598610.
Skjetne et al. (2005). Robust output maneuvering for a class of nonlinear systems. Automatica 40, 373-383.
Do & Pan (2006). Global robust adaptive path followng of underactuated ships. Automatica 42, 1713-1722.
...
Differential geometric reformulation
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•
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Banaszuk & Hauser (1995). Feedback linearization of transverse dynamics for periodic orbits
Sys. Contr. Lett., 26, 95-105.
Nielsen & Maggiore (2008). On local transverse feedback linearization. SIAM Journal on Control and Optimization, 47, 2227-2250.
Faulwasser (2013). Optimization-based Solutions to Constrained Trajectory-tracking and Path-following Problems. Shaker, Aachen,
Germany.
…
Feedforward path following
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•
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Shin & McKay (1985). Minimum-time control of robotic manipulators with geometric path constraints
IEEE Trans. Automat. Contr., 30, 531 – 541.
Verscheure et al. (2009). Time-Optimal Path Tracking for Robots: A Convex Optimization Approach.
IEEE Trans. Autom. Contr., 54, 2318-2327.
Faulwasser, Hagenmeyer & Findeisen (2011). Optimal Exact Path-Following for Constrained Differentially Flat Systems. Proc. of
18th IFAC World Congress, pp. 9875–9880.
Faulwasser, Hagenmeyer & Findeisen (2014). Constrained Reachability and Trajectory Generation for Flat Systems. Automatica,
vol. 50, pp. 1151-1159.
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Lyapunov and back-stepping approaches to path following
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Predictive control for path following (theory)
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Faulwasser & Findeisen (2008). Nonlinear model predictive path-following control. Proc. of Workshop Future Directions of
Nonlinear Model Predictive Control, Pavia, Italy.
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Faulwasser, Kern & Findeisen (2009). Model predictive path-following for constrained nonlinear systems. Proc. of 48th IEEE
Conference on Decision and Control, pp. 8642-8647.
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Faulwasser & Findeisen (2016). Nonlinear Model Predictive Control for Constrained Output Path Following. IEEE Trans. Automatic
Control.
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Faulwasser (2013). Optimization-based Solutions to Constrained Trajectory-tracking and Path-following Problems. Shaker, Aachen,
Germany.
Predictive control for path following (applications)
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Faulwasser et al. (2013). Predictive Path-following Control: Concept and Implementation for an Industrial Robot. Proc. of IEEE
Conference on Control Applications (CCA), pp. 128-133.
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Dauer et al. (2013). Optimization-based Feedforward Path Following for Model Reference Adaptive Control of an Unmanned
Helicopter. Proc. of AIAA Guidance, Navigation and Control Conference.
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Lam et al. (2013). Model predictive contouring control for biaxial systems. IEEE Trans. Contr. Syst. Techn., 21, 552-559.
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Böck & Kugi (2014). Real-time Nonlinear Model Predictive Path-Following Control of a Laboratory Tower Crane
IEEE Trans. Contr. Syst. Techn., 22, 1461-1473.
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Faulwasser et al. (2015). Implementation of Constrained Nonlinear Model Predictive Path-Following Control for an Industrial
Robot. Submitted to IEEE Trans. Contr. Syst. Techn.
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
References
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Summary and Conclusion
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Path following: combines trajectory generation and tracking
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Typical applications of path following: autonomous vehicles & robots, machine tools, …
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Tailored Byrnes-Isidori Normal Form is the key step for analysis
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MPC can be used to tackle path-following problems
Open questions
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Combination of path following and force control?
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Model uncertainty and repetitive motions?
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Implication of exo-systems with inputs for regulator problems?
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…
Thanks to:
• Juan Pablo Zometa, Tobias Weber, Janine Matschek, Rolf Findeisen (OvG University Magdeburg)
• Johann Dauer, Sven Lorenz (DLR Braunschweig)
Timm Faulwasser | Pfadverfolgung - Von Normalformen zu MPC
Summary
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