Automation for Robotics

Luc Jaulin

• Publisher: Wiley-ISTE
• Year: 2015
• ISBN:
  9781119081425
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Robotics
• Index: Technology & Engineering

In order to enable a better understanding of the key concepts of automation, this book develops the fundamental aspects of the field while also proposing numerous concrete exercises and their solutions. The theoretical approach that it presents fundamentally uses the state space and makes it possible to process general and complex systems in a simple way, involving several switches and sensors of different types. This approach requires the use of developed theoretical tools such as linear algebra, analysis and physics, generally taught in preparatory classes for specialist engineering courses.

1

Modeling

We will call modeling the step that consists of finding a more or less accurate state representation of the system we are looking at. In general, constant parameters appear in the state equations (such as the mass or the inertial moment of a body, the coefficient of viscous friction, the capacitance of a capacitor, etc.). In these cases, an identification step may prove to be necessary. In this book, we will assume that all the parameters are known, otherwise we invite the reader to consult Eric Walter’s book [WAL 14] for a broad range of identification methods. Of course, no systematic methodology exists that can be used to model a system. The goal of this chapter and of the following exercises is to present, using several varied examples, how to obtain a state representation.

1.1. Linear systems

In the continuous-time case, linear systems can be described by the following state equations:

Linear systems are rather rare in nature. However, they are relatively easy to manipulate using linear algebra techniques and often approximate in an acceptable manner the nonlinear systems around their operating point.

1.2. Mechanical systems

The fundamental principle of dynamics allows us to easily find the state equations of mechanical systems (such as robots). The resulting calculations are relatively complicated for complex systems and the use of computer algebra systems may prove to be useful. In order to obtain the state equations of a mechanical system composed of several subsystems S₁, S₂, . . . , S_m, assumed to be rigid, we follow three steps:

1) Obtaining the differential equations. For each subsystem S_k, with mass m and inertial matrix J, the following relations must be applied:

where the f_i are the forces acting on the subsystem S_k, M_fi represents the torque created by the force f_i on S_k, with respect to its center. The vector a represents the tangential acceleration of S_k and the vector

represents the angular acceleration of S_k. After decomposing these 2m vectorial equations according to their components, we obtain 6m scalar differential equations such that some of them might be degenerate.

2) Removing the components of the internal forces. In differential equations there are the so-called bonding forces, which are internal to the whole mechanical system, even though they are external to each subsystem composing it. They represent the action of a subsystem S_k on another subsystem S_ℓ. Following the action–reaction principle, the existence of such a force, denoted by f^_k,_ℓ , implies the existence of another force f^ℓ,k, representing the action of S_ℓ on S_k, such ...

Table of contents

• Cover
• Contents
• Title Page
• Copyright
• Introduction
• 1: Modeling
• 2: Simulation
• 3: Linear Systems
• 4: Linear Control
• 5: Linearized Control
• Bibliography
• Index