Table of Contents

Cover

Introduction

1 Three-dimensional Modeling

1.1. Rotation matrices
1.2. Euler angles
1.3. Inertial unit
1.4. Dynamic modeling
1.5. Exercises
1.6. Corrections

2 Feedback Linearization

2.1. Controlling an integrator chain
2.2. Introductory example
2.3. Principle of the method
2.4. Cart
2.5. Controlling a tricycle
2.6. Sailboat
2.7. Sliding mode
2.8. Kinematic model and dynamic model
2.9. Exercises
2.10. Corrections

3 Model-free Control

3.1. Model-free control of a robot cart
3.2. Skate car
3.3. Sailboat
3.4. Exercises
3.5. Corrections

4 Guidance

4.1. Guidance on a sphere
4.2. Path planning
4.3. Voronoi diagram
4.4. Artificial potential field method
4.5. Exercises
4.6. Corrections

5 Instantaneous Localization

5.1. Sensors
5.2. Goniometric localization
5.3. Multilateration
5.4. Exercises
5.5. Corrections

6 Identification

6.1. Quadratic functions
6.2. The least squares method
6.3. Exercises
6.4. Corrections

7 Kalman Filter

7.1. Covariance matrices
7.2. Unbiased orthogonal estimator
7.3. Application to linear estimation
7.4. Kalman filter
7.5. Kalman–Bucy
7.6. Extended Kalman filter
7.7. Exercises
7.8. Corrections

8 Bayes Filter

8.1. Introduction
8.2. Basic notions of probabilities
8.3. Bayes filter
8.4. Bayes smoother
8.5. Kalman smoother
8.6. Exercises
8.7. Corrections

References

Index

End User License Agreement

List of Illustrations

Chapter 1

Figure 1.1. Rotation of angle θ in a plane
Figure 1.2. Rotations in ℝ³ following various viewing angles
Figure 1.3. Changing the coordinate system to the system
Figure 1.4. Composition of rotations
Figure 1.5. The coordinate system ) attached to the robot
Figure 1.6. Illustration of the formula when . Left: the robot follows a circ...
Figure 1.7. Navigation mechanization equations
Figure 1.8. Quadrotor. For a color version of this figure, see www.iste.co.uk/ja...
Figure 1.9. Rotation Of the vector u around the vector n; Left: perspective view...
Figure 1.10. Left: a view of the rotation of angle φ around n; right: a visualiz...
Figure 1.11. Inclinometer pendulum moving over the surface of the Earth
Figure 1.12. Car trying to detect whether the point m is braking
Figure 1.13. Redermor built by GESMA (Groupe d’Etude Sous-Marine de l’Atlantique...
Figure 1.14. 3D representation of the robot together with its shadow in the hori...
Figure 1.15. Parametrization for the direct geometric model of a manipulator rob...
Figure 1.16. Floating wheel. For a color version of this figure, see www.iste.co...
Figure 1.17. The inertial unit inside the robot is static
Figure 1.18. Earth with the different frames. The origin of the frames have been...
Figure 1.19. Simulation of the underwater robot. For a color version of this fig...
Figure 1.20. Simulation the manipulator robot
Figure 1.21. Motion of the wheel with a torque-free precession. For a color vers...
Figure 1.22. Jacobian matrix of the evolution function at point x_o. For a color ...
Figure 1.23. Graph of the differential delays for the floating wheel. For a colo...
Figure 1.24. The robot with the trajectory painted blue and the robot fixed ...
Figure 1.25. Fake periodic trajectory returned by the inertial unit which feels ...
Figure 1.26. Left: simulation of the controller based on the Lie bracket techniq...
Figure 1.27. Left: The car goes from 0 toward all cardinal directions. The frame...
Figure 1.28. The robot moves like a satellite around the Earth. For a color vers...
Figure 1.29. Dynamic model
Figure 1.30. The robot follows the equator toward the east. For a color version ...

Chapter 2

Figure 2.1. Simple pendulum with state vector x = (x₁, x₂)
Figure 2.2. The nonlinear system, once transformed, becomes linear and decoupled...
Figure 2.3. Robot cart propelled by a ventilator
Figure 2.4. Cart (also called Dubin’s car)
Figure 2.5. Dynamic feedback linearization
Figure 2.6. Tricycle robot to be controlled
Figure 2.7. Sailing robot to be controlled
Figure 2.8. Internal frame (in light grey) and external frame (in dark grey) of ...
Figure 2.9. Graph of the differential delays for the sailing robot
Figure 2.10. By using a linearizing controller, the robot docks in its place in ...
Figure 2.11. Inverted rod pendulum to be modeled and controlled
Figure 2.12. The inverted rod pendulum, looped by a high gain K, behaves like a ...
Figure 2.13. Manipulator robot whose end effector must follow a circle
Figure 2.14. System composed of three pools containing water and connected by tw...
Figure 2.15. Our robot (with eyes) following a vehicle (here a car) whose state ...
Figure 2.16. Robot B (dotted) has to follow robot A
Figure 2.17. Reaction wheel pendulum
Figure 2.18. Robot 1 is in pursuit of robot 2
Figure 2.19. The SAUCISSE robot in a pool
Figure 2.20. Underwater float controlled by a ballast
Figure 2.21. Simple pendulum with state vector x = (x₁, x₂)
Figure 2.22. Illustration of a singularity, when both arms are stretched out. Th...
Figure 2.23. Illustration of the execution of the program; cart A is in bold gre...
Figure 2.24. Differential dependency graph of the underwater robot
Figure 2.25. Differential dependency graph after adding the integrators
Figure 2.26. During the transient phase, all robots go toward their setpoints in...
Figure 2.27. After the transient phase, all robots follow the leader with a cons...
Figure 2.28. Representation of the position of the float with respect to the tim...
Figure 2.29. Precise line following
Figure 2.30. Robot accurately describing the Van der Pol cycle
Figure 2.31. Left: vector field of the pendulum for u = 0. Right: system control...
Figure 2.32. Left: Vector field for the pendulum controlled with a sliding mode ...
Figure 2.33. Sets for K = 1, 5, 10 in red. When K is large, the probability to...

Chapter 3

Figure 3.1. Hyperbolic tangent function used as saturation function
Figure 3.2. Sawtooth function used to avoid the jumps in the heading control
Figure 3.3. Skating robot moving like a snake
Figure 3.4. Various simulations illustrate the control law for the skating robot
Figure 3.5. Thrust supplied by the engine u₂
Figure 3.6. Evolution of the front skate angle β in cruising regime
Figure 3.7. Feedback control of the sailing robot
Figure 3.8. Left: the VAIMOS sailing robot of IFREMER (in the background) and th...
Figure 3.9. Variables used in the state equations of the robot
Figure 3.10. Nominal vector field that the robot tries to track, when possible
Figure 3.11. a) The nominal vector field may be incompatible with the wind (here...
Figure 3.12. Some directions are not possible for the sailboat. These unfeasible...
Figure 3.13. Keep close hauled strategy by remaining within the strip centered o...
Figure 3.14. Rudder control for the sailing robot
Figure 3.15. Adjusting the maximum sail angle (or the sheet length); left: Carte...
Figure 3.16. Petri net supervising the navigation of the robot
Figure 3.17. Experiment of the spiral composed of five stages: a) the robot begi...
Figure 3.18. Journey from Brest to Douarnenez made by VAIMOS: a) the robot leave...
Figure 3.19. Car moving on a plane (view from above)
Figure 3.20. Car describing the Van der Pol cycle
Figure 3.21. Coordinate system change allowing us to take advantage of the rotat...
Figure 3.22. Path of the controlled system in order to remain within the disk
Figure 3.23. Vector field associated with our system
Figure 3.24. a) μ-STIC plane made by the ENSTA Bretagne; b) Faser Ultra Stick pl...
Figure 3.25. The quadrotor to be controlled. For a color version of this figure,...
Figure 3.26. Dynamic of the quadrotor represented by a causal chain which links ...
Figure 3.27. Simulation of the quadrotor which follows the Van der Pol cycle. Fo...
Figure 3.28. Underwater robot that has to follow an isobath. For a color version...
Figure 3.29. Simulation of an underwater robot (blue) following an isobath. The ...
Figure 3.30. The planet (blue) at coordinates (0, 0) and the satellite (red) at ...
Figure 3.31. Using cylindrical symmetry to switch from an order 3 system to an o...
Figure 3.32. Discretization of the system’s dynamics by a partitioning of the st...
Figure 3.33. Simulation for the line following of a sailboat. For a color versio...
Figure 3.34. Set where the strategy θ ≔ π + ψ − qζ is chosen in the (e, ψ) -sp...
Figure 3.35. Set of singularities of our controller in the -space. For a colo...
Figure 3.36. a) Initial phase: the robot stands up and turns to its left; b) the...
Figure 3.37. Sequence of back-stepping inversions to simplify the system
Figure 3.38. Left: u = 0. Center: u = −e₁ (x) − e₃ (x). Right: u = −e₁ (x)− e₂ (...

Chapter 4

Figure 4.1. a) Geographical coordinate system; b) we would like to express a in ...
Figure 4.2. a) The map gives a local Cartesian viewpoint around the robot; b) a ...
Figure 4.3. Getting Cartesian coordinates in a local frame centred in m
Figure 4.4. Illustration of second-order B-splines
Figure 4.5. Voronoi diagram
Figure 4.6. Delaunay triangulation
Figure 4.7. Artificial potential fields
Figure 4.8. On the sphere, the robot follows the robot
Figure 4.9. The robot has to follow a path without hitting the obstacles
Figure 4.10. Ten points for which we want to build a Voronoi diagram
Figure 4.11. In bold: an RSL-type (Right-Straight-Left) Dubins path leading from...
Figure 4.12. Controller (dotted) obtained by the potential method
Figure 4.13. Illustration of the flocking behavior from a random initialization
Figure 4.14. Platooning on the circle
Figure 4.15. The car is attracted counterclockwise by the circle
Figure 4.16. The robot following an eight circuit
Figure 4.17. The blue robot follows the red one . For a color version of this...
Figure 4.18. Left: control points with the associated Bézier polynomial; right: ...
Figure 4.19. Voronoi diagram
Figure 4.20. Circles circumscribed to the Delaunay triangles
Figure 4.21. Sawtooth functions for performing the angular difference and rejoin...
Figure 4.22. Top: the four superimposed Dubins paths of type RSR, LSR, RSL and L...
Figure 4.23. Calculating the length of the RSL Dubins path based on the fact tha...
Figure 4.24. Left: the four Dubins paths possible; right: the best of the four p...
Figure 4.25. Another situation in which the robot has to perform quite a complex...
Figure 4.26. (a) initialization at time t = 0;(b)–(f) the controller yields a un...
Figure 4.27. Generation of a vector field with a stable limit cycle correspondin...
Figure 4.28. Automaton switching from one field to the other. For a color versio...
Figure 4.29. Top: the two vector fields; bottom left: A superposition of the two...

Chapter 5

Figure 5.1. Operating principle of an accelerometer
Figure 5.2. Coriolis vibratory gyroscope and gimbal gyroscope
Figure 5.3. Principle of the Sagnac effect for optical gyroscopes
Figure 5.4. A robot moving on a plane, measures the angles in order to locate it...
Figure 5.5. Illustration of the Inscribed Angle Theorem
Figure 5.6. The three cars perceive the landmarks with the same angle
Figure 5.7. The emission time to and the offsets between the arrival times τ₂, τ...
Figure 5.8. Goniometric localization
Figure 5.9. The robots ℛ_a and ℛ_a can see each other, which will aid them in the ...
Figure 5.10. The robot measures the distance to the landmark
Figure 5.11. Left: raw lidar data; middle: data associated with the small-residu...

Chapter 6

Figure 6.1. Illustration of the least squares method in the nonlinear case
Figure 6.2. The Newton method to minimize ‖f (p) − y‖². For a color version of t...
Figure 6.3. Robot equipped with eight telemeters trying to localize itself
Figure 6.4. Representation of the function x · y on the paving [−1, 1]² left: ve...
Figure 6.5. Left: graph of the function g (x, y); right: contour lines. The sear...
Figure 6.6. Monte Carlo method for estimating the parameters a and b
Figure 6.7. The line cuts the segment [ab
Figure 6.8. If the point is on the segment [a, b] then α corresponds to the di...
Figure 6.9. The position of the robot as found from the eight distances given by...

Chapter 7

Figure 7.1. Cloud of points that represents a pair of two random variables
Figure 7.2. Left: dependent but non-correlated variables (x, y); Right: independ...
Figure 7.3. A confidence ellipse is the image of the unit circle by an affine fu...
Figure 7.4. The Gaussian random vector y :(ӯ, Γ_y) is the image by an affine appl...
Figure 7.5. Nonlinear estimator E(x|y)
Figure 7.6. Biased linear estimator
Figure 7.7. Among these four linear estimators, estimator (b), which is unbiased...
Figure 7.8. Unbiased orthogonal linear estimator
Figure 7.9. The Kalman filter is composed of a corrector followed by a predictor
Figure 7.10. Confidence ellipses associated with the six covariance matrices
Figure 7.11. The robot must locate itself by measuring its distance to the three...
Figure 7.12. Kalman filter used to estimate the state of the inverted rod pendul...
Figure 7.13. Dead reckoning for a tricycle robot
Figure 7.14. Goniometric localization
Figure 7.15. Goniometric localization for two communicating robots
Figure 7.16. Telemetric measurements collected by the robot
Figure 7.17. Probability density of a non-centered unit Gaussian random vector
Figure 7.18. Density of the random vector y after transformation of x by a linea...
Figure 7.19. Confidence ellipses associated to Γ₁,…, Γ₆, from thinnest to thicke...
Figure 7.20. Left Cloud of points, Right: confidences ellipses
Figure 7.21. Cloud of points generated by simulation with t ∈ {0, 1, 2, 3; 4, 5}...
Figure 7.22. Confidence ellipsoids for t ∈ {0, 1, 2, 3; 4, 5}.
Figure 7.23. The two lines in bold represent the two estimators and
Figure 7.24. When δ approaches 0, the Brownian noise (integral of the white nois...
Figure 7.25. White noise with its corresponding Brownian noise for different val...
Figure 7.26. Estimated path for the mass
Figure 7.27. Graphical illustration of the Kalman filter on a simple example
Figure 7.28. Confidence ellipse (in bold) associated with the localization of th...
Figure 7.29. Top: confidence ellipses for k from 0 to 19; bottom: standard devia...
Figure 7.30. Localization of the underwater robot using the Kalman filter

Chapter 8

Figure 8.1. Left: a Bayes network. Right: a graph that has a cycle and is thus n...
Figure 8.2. A Bayes network associated with the state equation
Figure 8.3. Markov chain describing the evolution of the weather in Brest
Figure 8.4. Graph representing the evolution and the observation of the door rob...
Figure 8.5. Illustration of the Bayes filter
Figure 8.6. The Redermor, built by GESMA (Groupe d’Etude Sous-Marine de l’Atlant...
Figure 8.7. The robot uses the information of the non-visibility of a tree to im...
Figure 8.8. A virtual step is added to allow us to use the Kalman equations
Figure 8.9. Confidence tube obtained with the predictor
Figure 8.10. Confidence tube obtained with the Kalman filter together with the e...
Figure 8.11. Confidence tube obtained with the Kalman smoother together with the...

First edition published 2015 in Great Britain and the United States by ISTE Press and Elsevier © ISTE Press 2015.

First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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The rights of Luc Jaulin to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2019943737

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ISBN 978-1-78630-523-7

Introduction

A mobile robot can be defined as a mechanical system capable of moving in its environment in an autonomous manner. For that purpose, it must be equipped with:

– sensors, which will collect knowledge of its surroundings (which it is more or less aware of) and determine its location;
– actuators, which will allow it to move;
– an intelligence (or algorithm, regulator), which will allow it to compute, based on the data gathered by the sensors, the commands to send to the actuators in order to perform a given task.

Finally, to this we must add the surroundings of the robot, which correspond to the world in which it evolves, and its mission, which is the task it has to accomplish. Mobile robots are constantly evolving, mainly from the beginning of the 2000s, in military domains (airborne drones [BEA 12], underwater robots [CRE 14], etc.), and even in medical and agricultural fields. They are in particularly high demand for performing tasks considered to be painful or dangerous to humans. This is the case for instance in mine-clearing operations, the search for black boxes of damaged aircraft on the seabed and planetary exploration. Artificial satellites, launchers (such as Ariane V), driverless subways and elevators are examples of mobile robots. Airliners, trains and cars evolve in a continuous fashion towards more and more autonomous systems and will very probably become mobile robots in the following decades.

Mobile robotics is the discipline which looks at the design of mobile robots [LAU 01]. It is based on other disciplines such as automatic control, signal processing, mechanics, computing and electronics. The aim of this book is to give an overview of the tools and methods of robotics which will aid the design of mobile robots. The robots will be modeled by state equations, i.e. first-order (mostly nonlinear) differential equations. These state equations can be obtained by using the laws of mechanics. It is not in our objectives to teach, in detail, the methods of robot modeling (refer to [JAU 05] and [JAU 15] for more information on the subject), but merely to recall its principles. By modeling, we mean obtaining the state equations. This step is essential for simulating robots as well as designing controllers. We will however illustrate the principle of modeling in Chapter 1 on deliberately three-dimensional examples. This choice was made in order to introduce important concepts in robotics such as Euler angles and rotation matrices. For instance, we will be looking at the dynamics of a wheel and the kinematics of an underwater robot. Mobile robots are strongly nonlinear systems and only a nonlinear approach allows the construction of efficient controllers. This construction is the subject of Chapters 2 and 3. Chapter 2 is mainly based on control methods that rely on the utilization of the robot model. This approach will make use of the concept of feedback linearization, which will be introduced and illustrated through numerous examples. Chapter 3 presents more pragmatic methods which do not use the state model of the robot and which will be referred to as without model or mimetic. This approach uses a more intuitive representation of the robot and is adapted to situations in which the robots are relatively simple to control remotely, such as in the case of cars, sailing boats or airplanes. Chapter 4 looks at guidance, which is placed at a higher level than control. In other words, it focuses on guiding and supervising the system which is already under control by the tools presented in Chapters 2 and 3. There will therefore be an emphasis on finding the instruction to give to the controller in order for the robot to accomplish its given task. The guidance will then have to take into account knowledge of the surroundings, the presence of obstacles and the roundness of the Earth. The nonlinear control and guidance methods require good knowledge of the state variables of the system, such as those which define the position of the robot. These position variables are the most difficult to find and Chapter 5 focuses on the problem of positioning. It introduces the classical nonlinear approaches that have been used for a very long time by humans for positioning, such as observing beacons and stars, using a compass or counting steps. Although positioning can be viewed as a particular case of state observation, the specific methods derived from it warrant a separate chapter. Chapter 6 on identification focuses on estimating, with a certain precision, non-measured quantities (parameters, position) from other, measured ones. In order to perform this identification, we will mainly be looking at the so-called least squares approach, which consists of finding the vector of variables that minimizes the sum of the squares of the errors. Chapter 7 presents the Kalman filter. This filter can be seen as a state observer for dynamic linear systems with coefficients that vary in time. A generalization of the Kalman filter to the case where the functions are nonlinear and the noise is non-Gaussian is provided in Chapter 8. The resulting observer, which is called the Bayes filter, computes the probability density function of the state vector at specific times.

The MATLAB^® and PYTHON code related to some the exercises of this book, together with explanatory videos, can be found at the following address:


www.ensta-bretagne.fr/jaulin/isterob.html