On discrete control of nonlinear systems with applications to robotics

Much progress has been reported in the areas of modeling and control of nonlinear dynamic systems in a continuous-time framework. From implementation point of view, however, it is essential to study these nonlinear systems directly in a discrete setting that is amenable for interfacing with digital computers. But to develop discrete models and discrete controllers for a nonlinear system such as robot is a nontrivial task. Robot is also inherently a variable-inertia dynamic system involving additional complications. Not only the computer-oriented models of these systems must satisfy the usual requirements for such models, but these must also be compatible with the inherent capabilities of computers and must preserve the fundamental physical characteristics of continuous-time systems such as the conservation of energy and/or momentum. Preliminary issues regarding discrete systems in general and discrete models of a typical industrial robot that is developed with full consideration of the principle of conservation of energy are presented. Some research on the pertinent tactile information processing is reviewed. Finally, system control methods and how to integrate these issues in order to complete the task of discrete control of a robot manipulator are also reviewed

Simplification of Manipulator Dynamic Model for Nonlinear Decoupled Control

This paper presents the development of simplified manipulator dynamic models which satisfy the desired steady-state error specification in the joint-variable space or in the Cartesian space under a nonlinear decoupled controller. The formulae which relate the tracking errors of joint variables in the joint-variable space or the manipulator hand in the Cartesian space to the dynamic modeling errors are first developed. Using these formulae, we derive the maximum error tolerance for each dynamic coefficient of the equations of motion. Then each simplified dynamic coefficient of the equations of motion can be expressed as a linear combination of the product terms of sinusoidal and polynomial basis functions. To illustrate the approach, a computer simulation has been carried out to obtain two simplified dynamic models of a Stanford robot arm which satisfy the specified error tolerances in the joint-variable space and in the Cartesian space under respective nonlinear decoupled controllers. Finally, to measure the time complexity of simplified models, the number of mathematical operations in terms of multiplication and addition for computing the joint torques is tabulated and discussed with the parallel computation result of Newton-Euler equations of motion

Robust nonlinear state feedback under structured uncertainty

This work follows the global input/output linearization approach for the design of control systems for nonlinear plants of Kravaris and Chung. A robust nonlinear state feedback is proposed for uncertainties considered as a class of bounded perturbations to the state model. A Liapunov-based approach is used to guarantee uniform ultimate boundedness.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/37402/1/690340708_ftp.pd

Nonlinear state feedback synthesis by global input/output linearization

This paper studies the design of feedback controllers for trajectory tracking in single-input/ single-output nonlinear systems x = f(x) + g(x) u, y = h(x) . A nonlinear transformation of the form v = k(x) + Λ(x) u that transforms this nonlinear input/output system into a linear system is first constructed. On the basis of this transformation, an approach for designing control laws for trajectory tracking is presented. The control law is robust in the sense that small changes in it do not produce large steady state errors or loss of stability. The theory provides a unified framework for treating control problems arising in nonlinear chemical processes; this is illustrated by a batch reactor control example.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/37396/1/690330408_ftp.pd

Robust decentralised variable structure control for rigid robotic manipulators

In this thesis, the problem of robust variable structure control for non-linear rigid robotic manipulators is investigated. Robustness and convergence results are presented for variable structure control systems of robotic manipulators with bounded unknown disturbances, nonlinearities, dynamical couplings and parameter uncertainties. The major outcomes of the work described in this thesis are summarised as given below. The basic variable structure theory is surveyed, and some basic ideas such as sliding mode designs, robustness analysis and control1er design methods for linear or non-linear systems are reviewed. Three recent variable structure control schemes for robotic manipulators are discussed and compared to highlight the research developments in this area. A decentralised variable structure model reference adaptive control scheme is proposed for a class of large scale systems. It is shown that, unlike previous decentralised variable structure control schemes, the local variable structure controller design in this scheme requires only three bounds of the subsystem matrices and dynamical interactions instead of the upper and the lower bounds of all unknown subsystem parameters. Using this scheme, not only asymptotic convergence of the output tracking error can be guaranteed, but also the controller design is greatly simplified. In order to eliminate chattering caused by the variable structure technique, local boundary layer controllers are presented. Furthermore, the scheme is applied to the tracking control of robotic manipulators with the result that strong robustness and asymptotic convergence of the output tracking error are obtained

Design of Adaptive Sliding Mode Fuzzy Control for Robot Manipulator Based on Extended Kalman Filter

In this work, a new adaptive motion control scheme for robust performance control of robot manipulators is presented. The proposed scheme is designed by combining the fuzzy logic control with the sliding mode control based on extended Kalman filter. Fuzzy logic controllers have been used successfully in many applications and were shown to be superior to the classical controllers for some nonlinear systems. Sliding mode control is a powerful approach for controlling nonlinear and uncertain systems. It is a robust control method and can be applied in the presence of model uncertainties and parameter disturbances, provided that the bounds of these uncertainties and disturbances are known. We have designed a new adaptive Sliding Mode Fuzzy Control (SMFC) method that requires only position measurements. These measurements and the input torques are used in an extended Kalman filter (EKF) to estimate the inertial parameters of the full nonlinear robot model as well as the joint positions and velocities. These estimates are used by the SMFC to generate the input torques. The combination of the EKF and the SMFC is shown to result in a stable adaptive control scheme called trajectory-tracking adaptive robot with extended Kalman (TAREK) method. The theory behind TAREK method provides clear guidelines on the selection of the design parameters for the controller. The proposed controller is applied to a two-link robot manipulator. Computer simulations show the robust performance of the proposed scheme

Prediction of the Response and Optimal Control of Stochastic Parametrically and Externally Excited Nonlinear Systems

Mechanical Engineerin

Aufbau von Modellen zur Lageregelung von Industrierobotern

Die vorliegende Arbeite soll zur Entwicklung von Modellen zur Lageregelung von Industrierobotern in drei Bereichen beitragen: der Kinematik, der Dynamik und der Lageregelung durch die Rückführungsentkopplung. Eingangs wurde das Hilfsmittel zur Kinematik und Dynamik von Industrierobotern, nämlich das körperfeste Koordinatensystem, erläutert. Die Theorie des in Kapitel 2 eingeführten Eingangsachsen-Koordinatensystems zeigt, daß das normale Eingangsachsen-Koordinatensystem vielseitige Vorteile hat. Es empfiehlt sich, bei den kinematischen und dynamischen Aufgaben von Industrierobotern das normale Eingangsachsen-Koordinatensystem zu verwenden. Die Lösungsverfahren der kinematischen Aufgaben wurden zunächst am Beispiel des Stanford-Arms ausführlich vorgestellt. Zusätzlich beschäftigte sich Kapitel 3 mit der kinematischen Singularität und der Methode zur Behandlung der Singularität. Während die Positionssingularität bei dem Entwurf des Arbeitsraums vermieden werden kann, kann die Orientierungssingularität an irgendeinem Punkt im Arbeitsraum auftreten. Da die Orientierungsgenauigkeit bei allen Fertigungsaufgaben eine untergeordnete Rolle spielt, wird eine gewisse Komponente der Winkelgeschwindigkeit des Endeffektors bei der Inversion der Jacobimatrix vernachlässigt, um die Positionsgenauigkeit zu gewährleisten. Im darauffolgenden Kapitel wurde die rekursive Newton-Eulersche Formulierung zum dynamischen inversen System auf der Basis des normalen Eingangsachsen-Koordinatensystems erneut hergeleitet. Dazu wurde ein Algorithmus entwickelt, der weniger rechenaufwendig ist als der auf der originalen Formulierung [42] basierende. Die Implementierung dieses Algorithmus wurde an einem Mikroprozessor des Multi-Mikrocomputersystems MMC 216 realisiert. Die Meßergebnisse zeigen, daß die auf der Basis dieses Algorithmus aufgebaute Mikroprozessorensoftware, die für das dynamische inverse System von heutigen Industrierobotern entwickelt wurde, das Abtastzeitkriterium des Industrieroboters befriedigt. Gegenüber der konventionellen Gelenk-Lageregelung hat die Kartesische Lageregelung den Vorteil, daß die Bahnabweichung des Endeffektors unmittelbar im Kartesischen Weltkoordinatensystem geregelt wird. Damit bleibt die Genauigkeit der Bewegung von Robotern im ganzen Arbeitsraum, ausschließlich in der Nähe von kinematischen singulären Punkten, gleich. Wenn in der Zukunft eine Einrichtung zur Messung der Position und Orientierung des Endeffektors eingesetzt werden könnte, würden störende Nichtlinearitäten und Elastizitäten in den Antriebsachsen und Getrieben durch die Kartesische Lageregelung umgangen. Die Schwierigkeit liegt aber darin, die Orientierungsabweichung zu definieren. Durch die Einführung der Eulerschen Parameter als Orientierungsabweichung wurde dieses Problem gelöst und in Kapitel 5 ein Modell der Kartesischen Lageregelung aufgebaut. Die Theorie ließ sich durch eine Simulation bestätigen. Das letzte Kapitel beschäftigte sich mit Robotern mit geschlossenen Ketten. Zunächst wurde eine dynamische Beziehung zwischen einem Roboter mit einer geschlossenen Kette und seiner entsprechenden geschnittenen offenen Kette hergeleitet. Basierend auf diesem Ansatz können alle für Roboter mit einer einfachen Kette entwickelten Modelle einfach auf Roboter mit geschlossenen Ketten erweitert werden. Alle in der vorliegenden Arbeit entwickelte Modelle sind von großer Bedeutung, wenn eine sehr leistungsfähige Lageregelung beim Einsatz von Industrierobotern bei Fertigungsaufgaben erforderlich ist.The present work should contribute to the development of models for the position control of industrial robots in three areas: kinematics, dynamics and position control through feedback decoupling. At the beginning, the aid for the kinematics and dynamics of industrial robots, namely the body-fixed coordinate system, was explained. The theory of the input axis coordinate system introduced in Chapter 2 shows that the normal input axis coordinate system has many advantages. It is advisable to use the normal input axis coordinate system for the kinematic and dynamic tasks of industrial robots. The methods of solving the kinematic problems were first presented in detail using the example of the Stanford arm. Chapter 3 also dealt with the kinematic singularity and the method for the treatment of the singularity. While the position singularity can be avoided in the design of the work space, the orientation singularity can occur at any point in the work space. Since the orientation accuracy plays a subordinate role in all manufacturing tasks, a certain component of the angular velocity of the end effector is neglected in the inversion of the Jacobian matrix in order to ensure the position accuracy. In the following chapter, the recursive Newton-Euler formulation for the dynamic inverse system was derived again on the basis of the normal input axis coordinate system. For this purpose, an algorithm was developed that is less computationally complex than that based on the original formulation [42]. The implementation of this algorithm was implemented on a microprocessor of the multi-microcomputer system MMC 216. The measurement results show that the microprocessor software built on the basis of this algorithm, which was developed for the dynamic inverse system of today's industrial robots, satisfies the scanning time criterion of the industrial robot. Compared to the conventional joint position control, the Cartesian position control has the advantage that the path deviation of the end effector is regulated directly in the Cartesian world coordinate system. This means that the accuracy of the movement of robots in the entire work area remains exclusive near kinematic singular points, the same. If a device for measuring the position and orientation of the end effector could be used in the future, annoying non-linearities and elasticities in the drive axles and gears would be avoided by the Cartesian position control. However, the difficulty lies in defining the orientation deviation. This problem was solved by introducing Euler's parameters as an orientation deviation and in Chapter 5 a model of the Cartesian position control was built. The theory could be confirmed by a simulation. The last chapter dealt with closed chain robots. First, a dynamic relationship was established between a robot with a closed chain and its corresponding cut open chain. Based on this approach, all models developed for robots with a simple chain can easily be extended to robots with closed chains. All models developed in the present work are of great importance if a very powerful position control is required when using industrial robots for manufacturing tasks