Motion Control of Functionally Related Systems

Tarik Uzunović, Asif Šabanović

• Publisher: CRC Press
• Year: 2020
• Print ISBN:
  9780367208806
• eBook ISBN:
  9780429558955
• Topic: Technology & Engineering
• Subtopic: Electrical Engineering & Telecommunications
• Index: Technology & Engineering

This book is concerned with the development of design techniques for controlling motion of mechanical systems which are employed to execute certain tasks acting collaboratively. The book introduces unified control design procedure for functionally related systems. The controllers for many different tasks in motion control can be successfully designed by applying the proposed simple procedure. The book gives an overview of the control methods appearing in the motion control area and the detailed design procedures for the class of systems that are required to execute certain task together. Tasks can generally be divided in their components, denoted as functions in the book. It is shown how dynamics of those tasks can be described. Based on the presented description, several control methods were discussed. Applicability of the introduced control design approach was demonstrated in subsequent chapters for various tasks.

1

 

Introduction

 

 

1.1  Motivation

One can often encounter a task which has to be executed by multiple mechanical systems. Therefore, the systems need to operate cooperatively to realize the specified task. Control system design for such a situation is a very challenging mission. It is clear that the nature of the task creates functional relations between the systems and that imposes functional dependencies between the states and outputs of the systems (in one word, coordinates). It can be considered that these functional dependencies make the systems ’virtually’ interconnected, even though they may be physically separated. Thus, the systems can be treated as sub-systems of an overall complex system. The component sub-systems within the overall system are called functionally related systems, since the nature of the task is defining functional relations between them.

The conventional approach in motion control systems implies that a task execution in a system that consists of multiple physical units (sub-systems) is typically based on the generation of references for separate sub-systems and control of those subsystems using some of the known control strategies. The definition of these unit tasks to be executed by the sub-systems can be significantly demanding. This makes control of the overall system very complex, as it is sometimes very complicated to find appropriate references for the sub-systems that have to cooperate in order to accomplish a certain task. There is another important disadvantage with this approach. Basically, the task is being decomposed not according to the functions that form the task, but according to the sub-systems that will execute parts of the task. Problems arising with the conventional approach are becoming even more obvious when some function in the task needs to be executed by several sub-systems synchronously, or when a sub-system has to execute multiple functions in the same time. In that situation, it becomes very demanding to generate a reference for a single sub-system’s motion. Thus, some new strategy is needed. The aim of this book is to contribute to proposing control of complex motion tasks based on functions.

The following fact can be noticed; it would be reasonable to describe the system task using the functions that have to be controlled. The functions would establish certain functional relationships between coordinates that describe system units (sub-systems). Then, the task would be executed if those functional relationships track their references, where the references are actually references for the functions to be completed. This function-based description makes it possible to control task components, which are identified as the functions. Of course, at the end it is necessary to generate control signals for the independent system units, but the main goal is to find a systematic way to accomplish that mission. If the goal is achieved, a control system designer would only need to describe the desired task by specifying functions that form the task, and the systematic approach could be applied. Systems interconnected with certain functional relationships formed in order to accomplish certain tasks are called functionally related systems. For these systems, a task can be projected to a new coordinate space, denoted as the function space, in which the control design procedure is easier and more intuitive. After a control strategy is designed, one can move back to the original space and calculate control signals that have to be applied to the systems.

Several examples of functionally related systems involved in different tasks are discussed in the subsequent chapters. The examples include: (i) motion synchronization control, (ii) object manipulation control, (iii) formation control of mobile robots, (iv) control of a walking piezoelectric motor, and (v) control in a bilateral system. To explain the motivation for this book, a short discussion about example (ii) will be given here.

Let us assume that two robotic manipulators have to manipulate an object, i.e., to transport it to a specified destination. In order to perform this task, one needs to control grasping force on the object and motion of the object. First, the robots should grasp the object and then move it to the desired destination. Their motion has to be synchronized during the task, and the grasping force needs to controlled at the same time. The task can be very adequately described specifying two functions that have to be controlled: grasping force and motion. Therefore, execution of the whole task can be based on control of these functions.

From the listed examples, and many similar ones that can be created, one can understand that various motion control systems can be designed in the framework based on functionally related systems. Thus, a step towards generalized treatment of such systems is highly desirable. This would simplify control task specification and control strategy design in the motion control systems that can be described within the framework.

 

1.2  Objectives of the Book

From the short introduction given in the previous subsection, it can be observed that many different technical systems can be categorized as functionally related systems. Nevertheless, this claim will be strongly illustrated throughout this book. The book intends to offer a new perspective for motion control design for such systems. The basic idea is control strategy synthesis based on functions that have to be executed in the controlled system. Therefore, the main objective of the book is to give a systematic mathematical procedure for motion control design for the functionally related systems.

In order to accomplish the main objective, other objectives of this book can be listed as follows;

• Definition of an appropriate form to describe function space dynamics of the system with a task containing multiple functions, and dynamics of the system which has to execute several tasks with different priority, or the system including constraints to be satisfied in combination with tasks to be executed.
• Proposal of a systematic and unified method for control design in the function space. The aim is to obtain a unit control distribution matrix in the function space and enforce desired dynamics for each of the identified functions.
• Investigation of two-layer control in two different forms. In the first form, a high-level controller calculates references for low-level controllers, which enforce tracking of these references. In the second form, low-level compensators are compensating disturbances in the configuration space, which is then making the homework for high-level controller easier, since this controller is enforcing desired dynamics of the controlled functions for the system with compensated disturbances.
• Identification of constraints that exist in the selection of functions. The book will examine constraints that have to be taken into account when the function-based approach in control synthesis is used. The constraints will determine which functions can or cannot be accomplished at the same time in the control system, and what are the criteria that have to be taken into consideration when functions are being defined.
• Validation of the proposed approach for control synthesis for several systems. Successful application to different systems will prove the generality of the proposed approach.

2

Methods in Motion Control

This chapter presents an overview about methods in motion control and control of functionally related systems. The chapter starts with a general overview about control methods used in motion control. Then, several examples that can be found in the literature related to the control of functionally related systems are introduced.

2.1 General Structure of Control Systems

The dynamics of a fully actuated mechanical system having n degrees of freedom (n-DOF) can be described in the configuration space as [54]

A(q)q¨+b(q,q˙)+g(q)+Text=T	(2.1)

where

• q∈Rn×1 is the configuration vector,
• A(q)∈Rn×n stands for the symmetric positive definite kinetic energy matrix (sometimes referred to as inertia matrix) which has bounded strictly positive elements 0<aij−≤aij(q)≤aij+ (hence 0<A−≤∥A(q)∥≤A+, where A⁻, A⁺ are two known scalars satisfying 0 < A⁻ ≤ A⁺),
• b(q,q˙)∈Rn×1 represents the vector of Coriolis forces, viscous friction forces, and centripetal forces, and it is bounded by ∥b(q,q˙)∥≤b+,
• g(q)∈Rn×1 is the vector of gravity terms bounded by ∥g(q)∥≤g+,
• Text∈Rn×1 represents the vector of external forces acting on the system, and
• T∈Rn×1 denotes the vector of generalized joint forces, and it will be usually called control vector or input force vector.

It is important to note that (2.1) written in the given form may also be representing the dynamics of several physically separated systems, which can be treated as subsystems of an augmented system having in total n degrees of freedom.

If one desires to control the system described in (2.1), an appropriate control vector T needs to be determined. For example, the goal can be that configuration vector q tracks the reference configuration vector q^ref. To achieve this goal, a control system designer has to choose the control vector in the following form

T=A(q)q¨des+b(q,q˙)+g(q)+Text	(2.2)

where q¨=q¨des enforces a desired closed-loop dynamics of the controlled system. Different approaches to the design of control (2.2) differ in how to obtain the defined control vector (input force vector).

The first approach is to define the system disturbance (or just disturbance in shorter form) as Td=b(q,q˙)+g(q)+Text, which consists of the forces that are mostly unknown, and then use a disturbance observer to estimate the disturbance and apply it as a part of the control vector. The other part of the control vector is A(q)q¨des and it is selected based on a desired closed-loop behavior of the system. Having the disturbance as a component in the control vector makes the system behave in the ideal case as a nominal system A(q)q¨=A(q)q¨des whose dynamics is known. However, in reality disturbance estimation T^d is not equal to the real disturbance, and disturbance estimation error Td−T^d is reflected in the resulting dynamics, which is in reality A(q)q¨=A(q)q¨des−(Td−T^d). With a good disturbance compensation, estimation error can be sufficiently small. Then, only desired acceleration q¨des needs to be selected and the second component of the input force vector A(q)q¨des can be calculated. If the inertia matrix is unknown, then the matrix can be represented as A(q) = A_n(q) + ΔA(q), where A_n(q) is th...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface
• Authors
• Symbols
• 1. Introduction
• 2. Methods in Motion Control
• 3. Design of a Motion Control System for Functionally Related Systems
• 4. Functions in Motion Control Systems
• 5. Motion Synchronization and Object Manipulation in 2-D Space
• 6. Formation Control
• 7. A New Driving Principle for Piezoelectric Walkers
• 8. Illustrative Simulation Examples
• 9. Conclusion
• Bibliography
• Index