Nonlinear Model Predictive Control for Motion Generation of Humanoids

Das Ziel dieser Arbeit ist die Untersuchung und Entwicklung numerischer Methoden zur Bewegungserzeugung von humanoiden Robotern basierend auf nichtlinearer modell-prädiktiver Regelung. Ausgehend von der Modellierung der Humanoiden als komplexe Mehrkörpermodelle, die sowohl durch unilaterale Kontaktbedingungen beschränkt als auch durch die Formulierung unteraktuiert sind, wird die Bewegungserzeugung als Optimalsteuerungsproblem formuliert. In dieser Arbeit werden numerische Erweiterungen basierend auf den Prinzipien der Automatischen Differentiation für rekursive Algorithmen, die eine effiziente Auswertung der dynamischen Größen der oben genannten Mehrkörperformulierung erlauben, hergeleitet, sodass sowohl die nominellen Größen als auch deren ersten Ableitungen effizient ausgewertet werden können. Basierend auf diesen Ideen werden Erweiterungen für die Auswertung der Kontaktdynamik und der Berechnung des Kontaktimpulses vorgeschlagen. Die Echtzeitfähigkeit der Berechnung von Regelantworten hängt stark von der Komplexität der für die Bewegungerzeugung gewählten Mehrkörperformulierung und der zur Verfügung stehenden Rechenleistung ab. Um einen optimalen Trade-Off zu ermöglichen, untersucht diese Arbeit einerseits die mögliche Reduktion der Mehrkörperdynamik und andererseits werden maßgeschneiderte numerische Methoden entwickelt, um die Echtzeitfähigkeit der Regelung zu realisieren. Im Rahmen dieser Arbeit werden hierfür zwei reduzierte Modelle hergeleitet: eine nichtlineare Erweiterung des linearen inversen Pendelmodells sowie eine reduzierte Modellvariante basierend auf der centroidalen Mehrkörperdynamik. Ferner wird ein Regelaufbau zur GanzkörperBewegungserzeugung vorgestellt, deren Hauptbestandteil jeweils aus einem speziell diskretisierten Problem der nichtlinearen modell-prädiktiven Regelung sowie einer maßgeschneiderter Optimierungsmethode besteht. Die Echtzeitfähigkeit des Ansatzes wird durch Experimente mit den Robotern HRP-2 und HeiCub verifiziert. Diese Arbeit schlägt eine Methode der nichtlinear modell-prädiktiven Regelung vor, die trotz der Komplexität der vollen Mehrkörperformulierung eine Berechnung der Regelungsantwort in Echtzeit ermöglicht. Dies wird durch die geschickte Kombination von linearer und nichtlinearer modell-prädiktiver Regelung auf der aktuellen beziehungsweise der letzten Linearisierung des Problems in einer parallelen Regelstrategie realisiert. Experimente mit dem humanoiden Roboter Leo zeigen, dass, im Vergleich zur nominellen Strategie, erst durch den Einsatz dieser Methode eine Bewegungserzeugung auf dem Roboter möglich ist. Neben Methoden der modell-basierten Optimalsteuerung werden auch modell-freie Methoden des verstärkenden Lernens (Reinforcement Learning) für die Bewegungserzeugung untersucht, mit dem Fokus auf den schwierig zu modellierenden Modellunsicherheiten der Roboter. Im Rahmen dieser Arbeit werden eine allgemeine vergleichende Studie sowie Leistungskennzahlen entwickelt, die es erlauben, modell-basierte und -freie Methoden quantitativ bezüglich ihres Lösungsverhaltens zu vergleichen. Die Anwendung der Studie auf ein akademisches Beispiel zeigt Unterschiede und Kompromisse sowie Break-Even-Punkte zwischen den Problemformulierungen. Diese Arbeit schlägt basierend auf dieser Grundlage zwei mögliche Kombinationen vor, deren Eigenschaften bewiesen und in Simulation untersucht werden. Außerdem wird die besser abschneidende Variante auf dem humanoiden Roboter Leo implementiert und mit einem nominellen modell-basierten Regler verglichen

High performance implementation of MPC schemes for fast systems

In recent years, the number of applications of model predictive control (MPC) is rapidly increasing due to the better control performance that it provides in comparison to traditional control methods. However, the main limitation of MPC is the computational e ort required for the online solution of an optimization problem. This shortcoming restricts the use of MPC for real-time control of dynamic systems with high sampling rates. This thesis aims to overcome this limitation by implementing high-performance MPC solvers for real-time control of fast systems. Hence, one of the objectives of this work is to take the advantage of the particular mathematical structures that MPC schemes exhibit and use parallel computing to improve the computational e ciency. Firstly, this thesis focuses on implementing e cient parallel solvers for linear MPC (LMPC) problems, which are described by block-structured quadratic programming (QP) problems. Speci cally, three parallel solvers are implemented: a primal-dual interior-point method with Schur-complement decomposition, a quasi-Newton method for solving the dual problem, and the operator splitting method based on the alternating direction method of multipliers (ADMM). The implementation of all these solvers is based on C++. The software package Eigen is used to implement the linear algebra operations. The Open Message Passing Interface (Open MPI) library is used for the communication between processors. Four case-studies are presented to demonstrate the potential of the implementation. Hence, the implemented solvers have shown high performance for tackling large-scale LMPC problems by providing the solutions in computation times below milliseconds. Secondly, the thesis addresses the solution of nonlinear MPC (NMPC) problems, which are described by general optimal control problems (OCPs). More precisely, implementations are done for the combined multiple-shooting and collocation (CMSC) method using a parallelization scheme. The CMSC method transforms the OCP into a nonlinear optimization problem (NLP) and de nes a set of underlying sub-problems for computing the sensitivities and discretized state values within the NLP solver. These underlying sub-problems are decoupled on the variables and thus, are solved in parallel. For the implementation, the software package IPOPT is used to solve the resulting NLP problems. The parallel solution of the sub-problems is performed based on MPI and Eigen. The computational performance of the parallel CMSC solver is tested using case studies for both OCPs and NMPC showing very promising results. Finally, applications to autonomous navigation for the SUMMIT robot are presented. Specially, reference tracking and obstacle avoidance problems are addressed using an NMPC approach. Both simulation and experimental results are presented and compared to a previous work on the SUMMIT, showing a much better computational e ciency and control performance.Tesi

Constrained-Differential-Kinematics-Decomposition-Based NMPC for Online Manipulator Control with Low Computational Costs

Flexibility combined with the ability to consider external constraints comprises the main advantages of nonlinear model predictive control (NMPC). Applied as a motion controller, NMPC enables applications in varying and disturbed environments, but requires time-consuming computations. Hence, given the full nonlinear multi-DOF robot model, a delay-free execution providing short control horizons at appropriate prediction horizons for accurate motions is not applicable in common use. This contribution introduces an approach that analyzes and decomposes the differential kinematics similar to the inverse kinematics method to assign Cartesian boundary conditions to specific systems of equations during the model building, reducing the online computational costs. The resulting fully constrained NMPC realizes the translational obstacle avoidance during trajectory tracking using a reduced model considering both joint and Cartesian constraints coupled with a Jacobian transposed controller performing the end-effector’s orientation correction. Apart from a safe distance from the obstacles, the presented approach does not lead to any limitations of the reachable workspace, and all degrees of freedom (DOFs) of the robot are used. The simulative evaluation in Gazebo using the Stäubli TX2-90 commanded of ROS on a standard computer emphasizes the significantly lower online computational costs, accuracy analysis, and extended adaptability in obstacle avoidance, providing additional flexibility. An interpretation of the new concept is discussed for further use and extensions

Visual Servoing NMPC Applied to UAVs for Photovoltaic Array Inspection

The photovoltaic (PV) industry is seeing a significant shift toward large-scale solar plants, where traditional inspection methods have proven to be time-consuming and costly. Currently, the predominant approach to PV inspection using unmanned aerial vehicles (UAVs) is based on photogrammetry. However, the photogrammetry approach presents limitations, such as an increased amount of useless data during flights, potential issues related to image resolution, and the detection process during high-altitude flights. In this work, we develop a visual servoing control system applied to a UAV with dynamic compensation using a nonlinear model predictive control (NMPC) capable of accurately tracking the middle of the underlying PV array at different frontal velocities and height constraints, ensuring the acquisition of detailed images during low-altitude flights. The visual servoing controller is based on the extraction of features using RGB-D images and the Kalman filter to estimate the edges of the PV arrays. Furthermore, this work demonstrates the proposal in both simulated and real-world environments using the commercial aerial vehicle (DJI Matrice 100), with the purpose of showcasing the results of the architecture. Our approach is available for the scientific community in: https://github.com/EPVelasco/VisualServoing_NMPCComment: This paper is under review at the journal "IEEE Robotics and Automation Letters

Systems and control : 21th Benelux meeting, 2002, March 19-21, Veldhoven, The Netherlands

Book of abstract

Novel Formulation and Application of Model Predictive Control.

Model predictive control (MPC) has been extensively studied in academia and widely accepted in industry. This research has focused on the novel formulation of model predictive controllers for systems that can be decomposed according to their nonlinearity properties and several novel MPC applications including bioreactors modeled by population balance equations (PBE), gas pipeline networks, and cryogenic distillation columns. Two applications from air separation industries are studied. A representative gas pipeline network is modeled based on first principles. The full-order model is ill-conditioned, and reduced-order models are constructed using time-scale decomposition arguments. A linear model predictive control (LMPC) strategy is then developed based on the reduced-order model. The second application is a cryogenic distillation column. A low-order dynamic model based on nonlinear wave theory is developed by tracking the movement of the wave front. The low-order model is compared to a first-principles model developed with the commercial simulator HYSYS.Plant. On-line model adaptation is proposed to overcome the most restrictive modeling assumption. Extensions for multiple column modeling and nonlinear model predictive control (NMPC) also are discussed. The third application is a continuous yeast bioreactor. The autonomous oscillations phenomenon is modeled by coupling PBE model of the cell mass distribution to the rate limiting substrate mass balance. A controller design model is obtained by linearizing and temporally discretizing the ODES derived from spatial discretization of the PBE model. The MPC controller regulate the discretized cell number distribution by manipulating the dilution rate and the feed substrate concentration. A novel plant-wide control strategy is developed based on integration of LMPC and NMPC. It is motivated by the fact that most plants that can be decomposed into approximately linear subsystems and highly nonlinear subsystems. LMPCs and NMPCs are applied to the respective subsystems. A sequential solution algorithm is developed to minimize the amount of unknown information in the MPC design. Three coordination approaches are developed to reduce the amount of information unavailable due to the sequential MPC solution of the coupled subsystems and applied to a reaction/separation process. Furthermore, a multi-rate approach is developed to exploit time-scale differences in the subsystems

Efficient solution approach to nonlinear optimal control problems and application to autonomous driving

Diese Arbeit beschäftigt sich mit der numerischen Lösung von dynamischen nichtlinearen Optimierungsaufgaben und der Entwicklung neuer Methoden für deren Analyse, um die Effizienz der Berechnungen zu erhöhen. Der Betrieb vieler natürlicher und technischer Prozesse kann als nichtlineares Optimierungsproblem mit Beschränkungen formuliert werden. Aufgrund der steigenden Komplexität wird die Lösung eines solchen Problems zu einer Herausforderung, insbesondere wenn das Problem in Echtzeit gelöst werden muss. Der Ansatz des kombinierten Mehrfachschießverfahren mit Kollokation ist effizient, um solche Probleme zu lösen, auch wenn sie eine schnelle Dynamik aufweisen. So ist das erste Ziel dieser Arbeit die weitere Verbesserung der Rechenleistung durch die Bereitstellung einer analytischen Hesse-Matrix und die Realisierung eines Parallelberechnungs-Schemas. Zunächst wurden die Formeln zur Berechnung der Sensitivitäten zweiter Ordnung für den kombinierten Ansatz abgeleitet. Mit Hilfe des Mehrfachschießverfahrens können die Lösungen von Modellgleichungen und Auswertungen von Sensitivitäten erster und zweiter Ordnung für jedes Zeitintervall unabhängig voneinander berechnet werden. Der zweite Beitrag widmet sich daher der Realisierung eines parallelen Rechenschemas. Dadurch wird ein hoher Beschleunigungsfaktor durch Parallelisierung erreicht, der zu einer Reduzierung des Rechenaufwands führt. Als dritter Beitrag wurde eine neuartige Korrelationsanalyse der Steuergrößen eingeführt, die auf die Notwendigkeit hinweist, die analytische Hesse-Matrix anstelle seiner Approximation einzusetzen, um ein Optimierungsproblem effizient zu lösen. Die numerische Leistung dieser drei Beiträge wurde mit Hilfe von herausfordernden dynamischen Optimierungsproblemen einschließlich der optimalen Steuerung eines großen Problems mit mehr als tausend dynamischen Variablen demonstriert. Die kombinierte Methode wandelt das Problem der kontinuierlichen dynamischen Optimierung in ein nichtlineares Programmierungsproblem mit einer vorgegebenen Anzahl der Zeitintervalle um. Es gibt jedoch keine umfassenden Regeln, um diese Anzahl der Zeitintervalle passend zu wählen. Daher widmet sich das vierte Ziel dieser Arbeit der Analyse der zugrunde liegenden Optimierungsprobleme mit dem besonderen Fokus auf der Anzahl der diskreten Zeitintervalle. Aus Anwendungssicht sollte die Anzahl der Zeitintervalle so gewählt werden, dass gleichzeitig die Bilanz zwischen der numerischen Genauigkeit und der Rechenlast zur Lösung des diskreten Optimierungsproblems erreicht wird. Darüber hinaus ist es unerlässlich, die Mindestanzahl an Zeitintervallen zu finden, um diese Genauigkeit zu gewährleisten. So wurde im Rahmen der Kollokation auf finiten Elementen ein neuartiger Bilevel-Ansatz vorgeschlagen, bei dem die äußere Schleife für die Ermittlung der minimalen Anzahl von Zeitintervallen zuständig ist und die innere Schleife eine obere Grenze des Approximationsfehlers auswertet, indem sie ein Fehlermaximierungsproblem durch Manipulation der Steuergrößen löst. Auf diese Weise kann eine Mindestanzahl von Zeitintervallen festgelegt werden, die eine benutzerdefinierte Fehlertoleranz gewährleistet. Außerdem wird der Einfluss der Anfangsbedingungen auf den maximalen Approximationsfehler berücksichtigt, so dass die ermittelte Anzahl von Intervallen für unterschiedliche Anfangsbedingungen gilt und somit für die nichtlineare modellprädiktive Regelung (engl.: nonlinear model predictive control (NMPC)) angewendet werden kann. Mehrere Fallstudien wurden verwendet, um die Wirksamkeit des vorgeschlagenen Ansatzes zu demonstrieren. Sowohl die theoretisch entwickelten Methoden als auch der kombinierte Ansatz wurden mit Hilfe von Open-Source-Software als allgemeines Framework für Testzwecke implementiert. Schließlich wurden die entwickelten Methoden auf das autonome Fahren im NMPC-Framework angewendet. Autonomes Fahren ist der aktuelle Trend in der Automobilindustrie mit dem Ziel, vollautomatisierte oder selbstfahrende Fahrzeuge zu entwickeln und zu produzieren. Reglerentwurf und -betrieb von autonomen Fahrzeugen stellen mehrere Herausforderungen dar, weshalb umfangreiche und intensive Forschungsarbeiten notwendig sind, um den wachsenden industriellen Bedarf abzudecken. Die Fahrzeugbewegung wurde als ein dynamisches Optimierungsproblem dargestellt, das online effizient gelöst wird. Der erfolgreiche Test der NMPC mit zwei Modellfahrzeugen (im Maßstab 1:5 und 1:8 im Vergleich zum realen Fahrzeug) zeigte die Effizienz des entwickelten Ansatzes.This thesis deals with the numerical solution of dynamic nonlinear optimization problems and the development of new methods for their analysis in order to increase the efficiency of calculations. The operation of many natural and technical processes can be formulated as a nonlinear optimal control problem with constraints. Because of the increasing complexity, the solution of such a problem becomes challenging, in particular if it has to be obtained in real-time. The approach of combined multiple-shooting with collocation is efficient for solving such problems even if they contain fast dynamics. Thus, the first target of this work is to further improve its computational performance by providing an analytical Hessian and realizing a parallel-computing scheme. First, the formulas for computing the second-order sensitivities for the combined approach were derived. Using multiple-shooting, the solutions of model equations and evaluations of both first-order and second-order sensitivities can be provided independently for each time interval. Therefore, the second contribution is dedicated to the realization of a parallel computing scheme. As a result, a high speedup factor is attained through parallelization leading to reduction of computational expenses. As a third contribution, a novel control-variable correlation analysis was introduced, which indicates the necessity of employing the analytical Hessian instead of its approximation to efficiently solve an optimization problem. The numerical performance of these three contributions was demonstrated through challenging dynamic optimization problems including optimal control of a large-scale problem containing more than one thousand dynamic variables. The combined method converts the continuous dynamic optimization problem into a nonlinear programming problem using a given number of time intervals. However, there have been no comprehensive rules to properly choose this number. Therefore, the fourth target of this work is devoted to the analysis of the underlying optimization problem with the special focus on the number of discrete time intervals. From the application point of view, the number of time intervals should be selected to simultaneously achieve the balance between the numerical accuracy and the computation load for solving the discretized optimization problem. Moreover, it is imperative to find the minimum number of time intervals to guarantee this balance. Thus, in the context of collocation on finite elements, a novel bilevel approach was proposed, where the outer loop is responsible for finding the minimum number of time intervals and the inner loop evaluates an upper limit of the approximation error by solving an error maximization problem by manipulating the control variables. In this way, a minimum number of time intervals can be determined guaranteeing a user defined error tolerance. Moreover, the impact of the initial conditions on the maximum approximation error is taken into account so that the determined number of intervals is valid for varying initial conditions and thus can be applied to nonlinear model predictive control (NMPC). Several case studies were conducted to demonstrate the efficacy of the proposed approach. Both theoretically developed methods as well as the combined approach were implemented using open-source software as a generalized framework for testing purposes. Finally, the developed methods were applied to autonomous driving in the NMPC framework. Autonomous driving is the current trend in the automotive industry with the aim of designing and producing fully automated or self-driving vehicles. Control design and field operation of autonomous vehicles impose several challenges and thus extensive as well as intensive research studies need to be made to cover the growing industrial demand. In this work, the vehicle motion was modeled as a dynamic optimization problem which is efficiently solved on-line. The successful test of the NMPC with two model vehicles (with scale of 1:5 and 1:8 to real vehicles) demonstrated the effectiveness of the developed approach

Optimization-based methods for real-time generation of safe motions in mobile robots

Having robots operating in unstructured and dynamically changing environments is a challenging task that requires advanced motion generation approaches that are able to perform in real-time while maintaining the robot and environment safety. The progress in the field of numerical optimization, as well as the development of tailored algorithms, made Nonlinear Model Predictive Control (NMPC) an appealing candidate for real-time motion generation. By considering the robot model as prediction model and through appropriate constraints on the robot states and control inputs, NMPC can enforce safety to the resulting motion in a straightforward way. This thesis addresses the problem of real-time generation of safe motions for mobile robots and mobile manipulators. The different structure of the considered robots introduces different safety risks during the robot motion and so the motion generation problem for each robot is addressed in separate parts of this thesis. In the first part, the problem of motion generation for mobile robots navigating in environments populated by static and/or moving obstacles is considered. For the generation of the desired motion, real-time NMPC is used. We argue that, in order to tackle the risk of collision with the environment, traditional distance-based approaches are incapable of maintaining the robot safety when the NMPC uses relatively short prediction horizons. Instead, we propose two NMPC approaches that employ two alternative collision avoidance constraints. The first proposed NMPC approach is applied to a scenario of safe robot navigation in a human crowd. The NMPC serves as a motion generation module in a safe motion generation framework, complete with a crowd prediction module. The considered collision avoidance constraint is built upon an appropriate Control Barrier Function (CBF). The second NMPC approach is applied to a scenario of robot navigation among moving obstacles, where the dynamics of the considered robot are significant. The proposed collision avoidance constraint is built upon the notion of avoidable collision state, which considers not only the robot-obstacle distance but also their velocity as well as the robot actuation capabilities. The simulation results indicate that both methods are effective and able to maintain the robot safety even in cases where their purely distance-based counterparts fail. The second part of the thesis addresses the problem of safe motion generation for mobile manipulators, called to execute tasks that may require aggressive motions. Here, in addition to the risk of collision with its environment, the robot, consisting of multiple articulated bodies, is also susceptible to self-collisions. Moreover, fast motions can always result to loss of balance. To solve the problem, we propose a real-time NMPC scheme that uses the robot full dynamics, in order to enforce kinodynamic feasibility, while it also considers appropriate collision and self-collision avoidance constraints. To maintain the robot balance we enforce a constraint that restricts the feasible set of robot motions to those generating non-negative moments around the edges of the support polygon. This balance constraint, inherently nonlinear, is linearized using the NMPC solution of the previous iteration. In this way, we facilitate the solution of the NMPC in real-time, without compromising the robot safety. Although the proposed NMPC is effective when applied to MM with low degrees of freedom, when the robot becomes more complex the use of its full dynamic model as a prediction model in an NMPC can lead to unacceptably large computational times that are not compatible with the real-time requirement. However, the use of a simplified model of the robot in an NMPC can compromise the robot safety. For this reason, we propose an optimization-based controller equipped with balance constraints as well as CBF-based collision avoidance constraints. The proposed controller can serve as an intermediate between a motion generation module that does not consider the robot full dynamics and the robot itself in order to ensure that the resulting motion will be at least safe. Simulation results indicate the effectiveness of the proposed method

Advanced control designs for output tracking of hydrostatic transmissions

The work addresses simple but efficient model descriptions in a combination with advanced control and estimation approaches to achieve an accurate tracking of the desired trajectories. The proposed control designs are capable of fully exploiting the wide operation range of HSTs within the system configuration limits. A new trajectory planning scheme for the output tracking that uses both the primary and secondary control inputs was developed. Simple models or even purely data-driven models are envisaged and deployed to develop several advanced control approaches for HST systems