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? Set-point Stabilization Reference set-point Control error time invariant problem 2
Stabilization, Trajectory Tracking and Path Following? Trajectory Tracking Set-point Stabilization Reference trajectory Tracking error Reference set-point 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 3
Stabilization, Trajectory Tracking and Path Following? Trajectory Tracking Path Following Reference trajectory Tracking error Reference path Path-following error time varying problem performance limits? time invariant problem additional design parameters Applications? Controller design? 4
Path Following Typical Applications Autonomous vehicles Machine tools Motion problems Problem definition? Controller design? 5
Outline 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 6
Output Path-following Problem Dynamic system Output path regular 1d curve Control task Convergence to path Convergence on path Satisfaction of constraints Questions Problem structure Suitable formulation? Constraints Controller design? 7
Analysis of Path-following Problems state space output space = path manifold 8
Vector Relative Degree (I) [Isidori `95; Nijmeijer & van der Schaft `90] 9
Vector Relative Degree (II) [Isidori `95; Nijmeijer & van der Schaft `90] 10
Vector Relative Degree (III) [Isidori `95; Nijmeijer & van der Schaft `90] 11
Analysis of Path-following Problems State space Output space = path manifold Timing law Augmented system 12
Transverse Normal Form Idea: map to suitable coordinates Augmented system Transverse normal form [Nielsen & Maggiore `08; Banaszuk & Hauser `95; F. & Findeisen `16] 13
Transverse Normal Form Idea: map to suitable coordinates Augmented system Transverse normal form Example? Path followability? 14
Example Fully Actuated Robot Augmented system 15
Example Fully Actuated Robot Augmented system Transverse normal form 16
Path Followability and Transverse Normal Forms Augmented system Transverse normal form [Faulwasser `13] Necessary conditions? Constraints? Feedback design? 17
Path Followability in the Presence of Constraints [Faulwasser `13] Feedback control? 18
Feedforward Path Following 19
Closed-loop Path Following 20
Outline 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 21
Principle of Predictive Control Model Predictive Control (MPC) = repeated optimal control 1. State observation at 2. Solve 3. Apply for Advantages: constraints, MIMO systems, optimization of transients, dist. implementation NMPC for path-following problems? 22
Principle of Predictive Path Following Idea Prediction based on augmented dynamics. Costs penalize path-following error (and its time derivative). Optimal Control Problem Stability/convergence? Example? 23
Application to a Robotic Manipulator Benchmark Problem Robot KUKA LBR IV up to 7 joints. Make the robot write on a blackboard. Allow interaction between user and robot. 24
Implementation of Predictive Path-Following on KUKA LWR IV Dynamics of the robot State space representation Path parameter dynamics Error outputs Cost function, terminal penalty Prediction horizon, sampling time Optimization solved with ACADO Toolkit Results? 26
Application to a Robotic Manipulator Joint work with: Juan P. Zometa, Janine Matschek, Tobias Weber, Rolf Findeisen (all OvG Magdeburg) [Faulwasser et al. `15] 27
Application to a Robotic Manipulator Joint work with: Juan P. Zometa, Janine Matschek, Tobias Weber, Rolf Findeisen (all OvG Magdeburg) [Faulwasser et al. `15] 28
Application to a Robotic Manipulator 29
Application to a Robotic Manipulator 30
References Lyapunov and back-stepping approaches to path following 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, 598-610. 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 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 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. 31
References Timm Faulwasser Pfadverfolgung - Von Normalformen zu MPC Predictive control for path following (theory) Faulwasser & Findeisen (2008). Nonlinear model predictive path-following control. Proc. of Workshop Future Directions of Nonlinear Model Predictive Control, Pavia, Italy. Faulwasser, Kern & Findeisen (2009). Model predictive path-following for constrained nonlinear systems. Proc. of 48th IEEE Conference on Decision and Control, pp. 8642-8647. Faulwasser & Findeisen (2016). Nonlinear Model Predictive Control for Constrained Output Path Following. IEEE Trans. Automatic Control. Faulwasser (2013). Optimization-based Solutions to Constrained Trajectory-tracking and Path-following Problems. Shaker, Aachen, Germany. Predictive control for path following (applications) 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. 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. Lam et al. (2013). Model predictive contouring control for biaxial systems. IEEE Trans. Contr. Syst. Techn., 21, 552-559. Böck & Kugi (2014). Real-time Nonlinear Model Predictive Path-Following Control of a Laboratory Tower Crane IEEE Trans. Contr. Syst. Techn., 22, 1461-1473. Faulwasser et al. (2015). Implementation of Constrained Nonlinear Model Predictive Path-Following Control for an Industrial Robot. Submitted to IEEE Trans. Contr. Syst. Techn. 32
Summary and Conclusion Summary Path following: combines trajectory generation and tracking Typical applications of path following: autonomous vehicles & robots, machine tools, Tailored Byrnes-Isidori Normal Form is the key step for analysis MPC can be used to tackle path-following problems Open questions Combination of path following and force control? Model uncertainty and repetitive motions? Implication of exo-systems with inputs for regulator problems? Thanks to: Juan Pablo Zometa, Tobias Weber, Janine Matschek, Rolf Findeisen (OvG University Magdeburg) Johann Dauer, Sven Lorenz (DLR Braunschweig) 33