KinematicsJens Wittenburg

Theory and Applications

Kinematics

Jens Wittenburg

Kinematics Theory and Applications

ISBN 978-3-662-48486-9 ISBN 978-3-662-48487-6 (eBook) DOI 10.1007/978-3-662-48487-6 Library of Congress Control Number: Springer Heidelberg New York Dordrecht London This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer-Verlag GmbH Berlin Heidelberg is part of Springer Science+Business Media (www.springer.

com

2015951089

Jens Wittenburg Institut für Technische Mechanik Karlsruhe Institute of Technology (KIT) Karlsruhe, Germany

)

© Springer Verlag Berlin Heidelberg 2016

For Rosemarie

Preface

This book is devoted to the kinematics of the single rigid body and of sys-tems of inter-connected rigid bodies. Engineers are confronted with an endlessvariety of systems ranging from simple planar mechanisms to robots, walk-ing machines, prothetic devices, vehicles, stewart platforms, shaft couplings,gears etc. Subjects of kinematics are relationships between two and morepositions, finite and infinitesimal displacements and continuous motions.

The book is intended for use as textbook in advanced courses on kine-matics of mechanisms. It focuses on a solid theoretical foundation and onmathematical methods applicable to the solution of problems of very diversenature. Applications are demonstrated in a large number of fully worked-outproblems. In kinematics a wide variety of mathematical tools is applicable.The most important tools are vectors, tensors, matrices, complex numbers,quaternions, dual numbers, dual vectors, dual quaternions and elements ofline geometry. Whereever possible vector equations are formulated instead oflengthy scalar coordinate equations. The principle of transference is appliedto problems of very diverse nature.

The book has 19 chapters. Chapters 1 – 13, 16 and 18 are devoted tospatial kinematics and Chapts. 14, 15 and 17 to planar kinematics. In Chapt.19 nonlinear dynamics equations of motion are formulated for general spatialmechanisms. Nearly one half of the book is dealing with position theory andthe other half with motion.

Chapter 1 on Finite rotation about a fixed point introduces the directioncosine matrix, the similarity transformation, Euler and Bryan angles, Euler-Rodrigues parameters, quaternions, Cayley-Klein parameters, Rodrigues-,Euler- and Wiener vectors.

Subjects of Chapt. 2 on Line geometry are Plucker vectors with applica-tions to the line of intersection of two planes, to lines intersecting four givenlines, to the linear complex, to linear congruences and to ruled surfaces.

Chapter 3 on Finite screw displacement introduces the (4 × 4) transfor-mation matrix for general rigid-body displacements, Chasles’ and Halphen’s

vii

viii Preface

theorems and the screw triangle. Dual numbers and dual vectors lead to theprinciple of transference. This principle is applied to the composition anddecomposition of finite and infinitesimal screw displacements and to deter-mining the manifold of screw displacements effecting a prescribed line dis-placement.

In Chapt. 4 entitled Degree of freedom of a mechanism Grublers formula isdeveloped and illustrated by applications to various spatial mechanisms withor without overconstraint.

Chapter 5 is devoted to Spatial closed kinematic chains with a singleindependent and with six dependent joint variables. Such chains are basicelements of single-degree-of-freedom mechanisms. Efficient methods are de-veloped for expressing the dependent variables in terms of the independentvariable and of constant Denavit-Hartenberg parameters. As closure condi-tions Woernle-Lee equations and Lee’s half-angle equations are formulated.For products of vectors in these equations a novel technique is developed. Itis applied to ten mechanisms ranging from the RCCC mechanism with nineto the general 7R mechanism with twenty-one parameters. The final sectionis devoted to the inverse kinematics of serial 6-d.o.f. robots.

Chapter 6 entitled Overconstrained mechanisms starts with Bricard’s the-orem on single-loop-mechanisms with revolute joints. The technique devel-oped in Chapt. 5 is applied to Bennett’s 4R-mechanism, Goldberg’s 5R-mechanism, to Bricard’s 6R-mechanisms known as line-symmetric, plane-symmetric and trihedral mechanisms and to Dietmaier’s 6R-mechanism. Inaddition, Steffen’s mobile polyhedra, the Bricard-Borel mechanism, Bricard’shyperbolic mechanism, a cam mechanism and the HEUREKA octahedron areanalyzed.

Chapter 7 is devoted to the position theory for the terminal body ofspatial three-body chains with two revolute, prismatic or cylindrical joints.

Subject of Chapt. 8 is the direct kinematics of the Stewart platform ofgeneral geometry. Coordinate-free vector equations are formulated for thedual quaternion specifying the position of a platform The same problem issolved by elementary means for a Stuart platform with triangular geometry.

Chapter 9 entitled Angular velocity, angular acceleration is the first chap-ter devoted not to position theory, but to continuous motion. Key words areinstantaneous screw axis, velocity screw, velocity and acceleration distribu-tion in a rigid body, angular velocity of a body expressed in terms of positionsand velocities of three points, novel formulas for striction point and distribu-tion parameter of raccording axodes, strapdown inertial navigation, motionon curved surfaces, mecanum wheel.

In Chapt. 10 Kinematic differential equations are developed relating an-gular velocity to the time derivatives of direction cosines, Euler- and Bryanangles, Euler-Rodrigues parameters, Cayley-Klein parameters, Rodrigues pa-rameters, Wiener parameters and the Euler vector.

Preface ix

Chapter 11 is devoted to Kinematics of tree-structured systems with jointsof arbitrary nature. Positions, velocities and accelerations of bodies are ex-pressed as functions of generalized joint variables and of their time deriva-tives. The structure of the tree is described by its path matrix with elements+1 , −1 and zero.

Chapter 12 on Screw systems begins with the resultant of two generalvelocity screws, with the raccording hyperboloids of revolution associatedwith the relative motion of two bodies rotating about skew axes and withthe analogy between force screw and velocity screw. The principle of virtualpower leads to the concept of reciprocal screws. Screw systems reciprocal tofirst-order, second-order and third-order screw systems are investigated.

Chapter 13 on Shaft couplings starts with an analysis of Hooke’s jointincluding the polhode and herpolhode cones of its cross-shaped central body.From the principle of transference results are obtained for a joint couplingskew axes. A simple formula is developed for the transmission ratio of a chainof series-connected Hooke’s joints. Final sections are devoted to three classesof homokinetic shaft couplings and to an elementary analysis of the tripodjoint.

In Chapt. 14 on Displacements in a plane complex numbers are used fordescribing translation, rotation about a point and reflection in a line andresultants of these three elementary displacements. Relationships betweenthree and between four positions of a plane including Burmester’s pole curveare studied. The last section is devoted to Heesch’s work on tilings.

In Chapt. 15 on Plane motion the first part is devoted to centrodes andto the theorems of Burmester and Kennedy/Aronhold with a series of il-lustrative examples. Instantaneous centers of rotation and of acceleration,Bresse circles and normal poles are expressed in complex form. Curvaturetheory of plane trajectories leads to the Euler-Savary equation, the cubic ofstationary curvature and to Ball’s point. A short section on Holditch’s the-orem with applications is followed by the theory of trochoids in general andof cycloids in particular. An application of cycloids is demonstrated by theanalysis of optimal dwell mechanisms. The final section is devoted to theproblem of maneuvering a rectangle of maximum size in the space betweentwo nonorthogonal straight lines and a point.

In Chapt. 16 entitled Theory of gearing the first part on gears with par-allel axes has the key words analytical meshing conditions for calculatingthe tooth flank conjugate to a given flank, external and internal pin gears,curvature relationship for meshing tooth flanks, Camus’ theorem, cycloidalgears, involute gearing, addendum modification, helicoidal gears. Subject ofthe final section is Giovanozzi’s theory of general spatial involute gearing.

Chapter 17 on the Planar four-bar begins with sections on Grashof’s con-dition, the transfer function relating input and output angles, classical andnew formulas for stationary values of the transmission ratio, on four-barsfor the transmission of forces (shears, prongs etc.) and on coupler curves

x Preface

(Roberts-Tschebychev theorem, double points, cusps, symmetrical couplercurves). Two sections are devoted to applications of four-bars in planar robotsand in Jeantaud’s steering mechanism. For the former a simple position andvelocity analysis is formulated. Final sections are devoted to Tschebychev’soptimal straight-line approximations by coupler curves, to Peaucellier’s in-versor generating an exact straight line, to Burmester’s four-position theoryand to the trajectory of the composite system center of mass of a four-bar.

Chapter 18 on the Spherical four-bar has sections on the transfer function,on conditions analogous to Grashof’s condition for the planar four-bar and oncoupler curves (symmetry conditions, double points, cusps, number of pointsof intersection with a circle, stereographic projection). The chapter ends withthe investigation of a spherical parallel robot.

In Chapt. 19 entitled Dynamics of mechanisms nonlinear differential equa-tions of motion for spatial mechanisms are developed from the principle ofvirtual power. Equations for tree-structured systems are based on the kine-matics formulation in Chapt. 11 . Equations for mechanisms with closed kine-matic chains are obtained by incorporating additional kinematical constraintequations.

The author expresses his gratefulness to the following colleagues, researchassistants and students who supported the development of this manuscript:Prof. Ljubomir Lilov / Kliment-Ochridsky Univ. Sofia (for his critical readingof parts of the manuscript and for contributing Eq.(1.181) and Sect. 9.10 ),Gunther Stelzner (for fruitful discussions and contributions to Sect. 15.6 ),Andrey Shutovich (for contributing ideas to Chapt. 18 and for his criti-cal reading of Chapt. 8 ) and Xu Tongsheng, Benjamin Rutschke, AndreasFunkhanel and Simon Fritz (for converting pencil-on-paper figures into datafiles). Last, but not least, I thank the Springer team for technical advice andfor the patience in waiting for completion of the manuscript

Karlsruhe,August 2015 Jens Wittenburg

Preface xi

Notation

Vectors are printed boldface: r , ω .

Tensors are printed serif: Unit tensor I , dyad D = rr , inertia tensorJ =

∫m(r2I− rr) dm .

Products: ω · r , ω × r , J · ω .

Formulas: I ·ω ≡ ω , (n× r)×n = r−nn · r = (I−nn) · r (unit vector n ) ,∫mr× (ω × r) dm =

∫m(r2I− rr) dm · ω = J · ω (angular momentum) ,∫

m(ω × r)2 dm = ω · ∫

m(r2I− rr) dm · ω = ω · J · ω (kinetic energy) ,

ω × r + ω × (ω × r) = ω × r + ωω · r − ω2r = (ω × I + ωω − ω2 I ) · r(acceleration) .

Matrices are underscored: A , unit matrix I . The transpose of A is AT .

A right-handed cartesian basis with unit basis vectors e1 , e2 , e3 is calledbasis e . This symbol e denotes also the column matrix [ e1 e2 e3 ]

T of thethree unit vectors.

The equation r = r1e1 + r2e2 + r3e3 defines the coordinates r1 , r2 , r3 ofa vector r in basis e and the column matrix r = [ r1 r2 r3 ]

T of these coor-dinates. This matrix is not the vector since it is different in different bases.The relationship between vector and coordinate matrix is r = eT r = rT e .

The coordinates ω1 , ω2 , ω3 of a vector ω define the skew-symmetric ma-

trix ω =

[0 −ω3 ω2

ω3 0 −ω1

−ω2 ω1 0

]. If r is the column matrix of the coordinates of

another vector r in the same reference basis, ω r is the column matrix ofthe coordinates of ω × r . Also ω r = −r ω .

Right-handed cartesian bases fixed on bodies i = 0, 1, 2, . . . are denoted e0 ,e1 , e2 etc. The coordinates of a vector r in basis ek are denoted rk1 , r

k2 , r

k3 ,

and the column matrix of these coordinates is denoted rk . Whether r21 isthe square of some scalar r1 or the first component of a vector r in basise2 is seen from the context.

Matrices with vectors as elements are boldface underscored: Basis e =[ e1 e2 e3 ]

T , column matrix ω = [ω1 . . .ωn ]T of vectors ω1 , . . . , ωn .

Matrices with tensors as elements are serif underscored: Diagonal matrix Jof inertia tensors J1 . . . Jn .

The rule of ordinary matrix multiplication (AB )ij =∑

k AikBkj is gener-alized for matrices with vectors and tensors as elements. Examples:

1. r = eT r ; 2. e·eT = I ; 3. e1 ·e2T =[e1i ·e2j

](direction cosine ma-

trix) ; 4. r·e×eT =

[0 r3 −r2

−r3 0 r1r2 −r1 0

]= −r ; 5. J·ω = [ J1 ·ω1 . . . Jn ·ωn ]

T

(diagonal matrix J of inertia tensors, column matrix ω ) .

Contents

1 Rotation about a Fixed Point. Reflection in a Plane . . . . . 11.1 Direction Cosine Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Similarity Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Bryan Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Rotation Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Euler-Rodrigues Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.7 Relationships Between Euler-Rodrigues Parameters and

Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.8 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.9 Relationships Between Three Positions of a Body . . . . . . . . . . . 241.10 Relationships Between four Positions . . . . . . . . . . . . . . . . . . . . . . 261.11 Cayley-Klein Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.12 Euler Vector. Exponential Form of the Direction Cosine Matrix 331.13 Rodrigues Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.14 Wiener Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.15 Illustrative Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.15.1 Generalized Coordinates Associated with a GivenDirection Cosine Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.15.2 Resultant of two 180◦-Rotations . . . . . . . . . . . . . . . . . . . . 38

Critical Bryan Angle φ2 = ±π/2 . . . . . . . . . . . . . . . . . . . 401.15.4 Determine all Direction Cosine Matrices Having three

Prescribed Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.15.5 Rear Axle of a Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421.15.6 Rotation Determined from Three Positions of a

Body-Fixed Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.15.7 Rodrigues Vector Determined from Prescribed Point

Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441.15.8 Spherical Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

xiii

1.15.3 Rotations (n, φ) Resulting in Positions with the

xiv Contents

2 Line Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.1 Normal Vector of a Plane. Equation of a Plane . . . . . . . . . . . . . 632.2 Plucker Vectors. Plucker Coordinates of a Line . . . . . . . . . . . . . 642.3 Reflection in a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.4 Plucker Vectors of the Line of Intersection of two Planes . . . . . 662.5 Condition for two Lines to Intersect . . . . . . . . . . . . . . . . . . . . . . . 662.6 Plucker Vectors of the Common Perpendicular of two Lines . . 692.7 Linear Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2.7.1 Null Point. Null Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.7.2 Axis. Pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.7.3 Determine the Null Point if the Null Plane is Given

and Vice Versa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.7.4 Determine a Linear Complex from Given Complex

Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.7.5 Reciprocal Polars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2.8 Linear Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772.9 Ruled Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2.9.1 Intersection of three Linear Complexes . . . . . . . . . . . . . . 782.9.2 Striction Point. Distribution Parameter . . . . . . . . . . . . . 78

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3 Finite Screw Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.1 (4× 4) Transformation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.2 Chasles’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.3 Scalar Measures of a Screw Displacement . . . . . . . . . . . . . . . . . . 903.4 Roth’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.5 Screw Displacement Determined from Displacements of

Three Body Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.6 Halphen’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.7 Resultant of two Screw Displacements. Screw Triangle . . . . . . 953.8 Dual Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.9 Dual Vectors. Dual Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.10 Principle of Transference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.10.1 Dual Basis. Dual Direction Cosine Matrix . . . . . . . . . . . 1023.10.2 Screw Axis, Screw Angle and Translation Determined

from Dual Direction Cosines . . . . . . . . . . . . . . . . . . . . . . . 1043.10.3 Dual Euler Angles. Dual Bryan Angles . . . . . . . . . . . . . . 108

1.16 Reflection in a Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

1.15.9 Rotations Effecting a Prescribed Line Displacement . . . 461.15.10 Sensor Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481.15.11 Decomposition of a Rotation into three Rotations . . . 511.15.12 Decomposition of a Rotation into three Rotations.

Quaternion Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 55

Contents xv

3.10.4 Dual Rodrigues Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103.10.5 Dual Euler-Rodrigues Parameters. Dual Quaternions . . 110

3.11 Resultant of two Screw Displacements. Dual-QuaternionFormulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3.12 Equations for the Screw Triangle . . . . . . . . . . . . . . . . . . . . . . . . . 1183.13 Resultant of two Infinitesimal Screw Displacements.

Cylindroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4 Degree of Freedom of a Mechanism . . . . . . . . . . . . . . . . . . . . . . . 1374.1 Grubler’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1384.2 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

4.2.1 Five-Point-Contact Joint . . . . . . . . . . . . . . . . . . . . . . . . . . 1414.2.2 Shaky Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1434.2.3 Closed Chain Formed by Four Planar Four-Bars . . . . . . 1444.2.4 Trihedral Plane-Symmetric Bricard Mechanism. . . . . . . 1454.2.5 Line-Symmetric Bricard Mechanism . . . . . . . . . . . . . . . . 1494.2.6 Homokinetic Shaft Coupling . . . . . . . . . . . . . . . . . . . . . . . 1534.2.7 Mobile Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

5 Spatial Simple Closed Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1595.1 Joint Variables. Denavit-Hartenberg Parameters . . . . . . . . . . . . 1615.2 Screw Displacements. Coordinate Transformations . . . . . . . . . . 1625.3 Closure Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

5.3.1 Woernle-Lee Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1665.3.2 Half-Angle Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

5.4 Systematic Analysis of Mechanisms . . . . . . . . . . . . . . . . . . . . . . . 1775.4.1 RCCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1775.4.2 RCRCR and CRRRC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1815.4.3 RCPRC , CCPRR and RCPCR . Independent

Variable in the Prismatic Joint . . . . . . . . . . . . . . . . . . . . . 1845.4.4 Mechanisms in Rows 6 and 7 of Table 5.1 .

Independent Variable is an Angle . . . . . . . . . . . . . . . . . . . 1845.4.5 5R-C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1855.4.6 RRCRPR , RRCPRR , RRCRRP . Independent

Variable in the Prismatic Joint . . . . . . . . . . . . . . . . . . . . . 1885.4.7 Mechanism 7R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1885.4.8 4R-3P . Independent Variable is an Angle . . . . . . . . . . . . 1925.4.9 6R-P . Independent Variable is an Angle . . . . . . . . . . . . . 1925.4.10 6R-P . Independent Variable in the Prismatic Joint . . . 194

5.5 Mechanisms with Special Parameter Values . . . . . . . . . . . . . . . . 1945.5.1 7R with Three Parallel Joint Axes in Series . . . . . . . . . 1945.5.2 RRSRR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

3.14 Screw Displacements Effecting a Prescribed Line Displacement 127

xvi Contents

5.6 Generalized Velocities. Generalized Accelerations . . . . . . . . . . . 1975.6.1 RCCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1985.6.2 Mechanism 7R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

5.7 Spatial Serial Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

6 Overconstrained Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2056.1 Bricard’s Theorem on Closed Chains with Revolute Joints . . . 2066.2 Bennett Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2076.3 Kinematical Chains with Five Revolute Joints . . . . . . . . . . . . . . 210

6.3.1 Goldberg Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2126.4 Kinematical Chains with Six Revolute Joints . . . . . . . . . . . . . . . 217

6.4.1 Line-Symmetric Bricard Mechanism . . . . . . . . . . . . . . . . 2176.4.2 Plane-Symmetric Bricard Mechanism . . . . . . . . . . . . . . . 2216.4.3 Trihedral Bricard Mechanism . . . . . . . . . . . . . . . . . . . . . . 2236.4.4 Dietmaier’s Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

6.5 Mobile Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2316.6 RRCRP Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2336.7 4R-P Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2346.8 Bricard-Borel Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2356.9 Hyperboloid and Paraboloid Mechanisms . . . . . . . . . . . . . . . . . . 2366.10 Cam Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2406.11 Heureka Octahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

7 Two-Joint Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2577.1 Work Space of Points of the Terminal Body . . . . . . . . . . . . . . . . 258

7.1.1 Chains RR Defining Tori . . . . . . . . . . . . . . . . . . . . . . . . . . 2597.1.2 Chains RP Defining Hyperboloids of Revolution . . . . . . 2627.1.3 Chains PR Defining Elliptic Cylinders . . . . . . . . . . . . . . 2627.1.4 Chains PP Defining Planes . . . . . . . . . . . . . . . . . . . . . . . . 263

7.2 Chains RR Leading the Terminal Body Through PrescribedPositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

7.3 Chains CC Leading the Terminal Body Through PrescribedPositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

8 Stewart Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2778.1 Direct Kinematics of the General Stewart Platform . . . . . . . . . 2788.2 Triangle-Configuration of the Stewart Platform . . . . . . . . . . . . . 284References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

199

Contents xvii

9 Angular Velocity. Angular Acceleration . . . . . . . . . . . . . . . . . . . 2899.1 Definitions. Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2899.2 Inverse Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2929.3 Instantaneous Screw Axis. Pitch. Velocity Screw. Linear

Complex of Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2939.4 Angular Velocity of a Body in Terms of Positions and

Velocities of Three Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2999.5 Raccording Axodes. Striction Point. Distribution Parameter . 3019.6 Spatial Rotation About a Fixed Point . . . . . . . . . . . . . . . . . . . . . 3059.7 The Ancient Chinese Southpointing Chariot . . . . . . . . . . . . . . . 3079.8 Acceleration Distribution. Instantaneous Center of

Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3099.9 Angular Acceleration of a Body in Terms of Positions,

Velocities and Accelerations of Three Points . . . . . . . . . . . . . . . 3169.10 Strapdown Inertial Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3189.11 Motion on a Curved Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3219.12 Mecanum Wheel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

10 Kinematic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 32910.1 Direction Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32910.2 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33310.3 Bryan Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33310.4 Euler-Rodrigues Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33410.5 Cayley-Klein Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33710.6 Rodrigues Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33810.7 Wiener Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33910.8 Euler Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33910.9 Correction Formulas for Euler-Rodrigues Parameters . . . . . . . . 345References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

11 Direct Kinematics of Tree-Structured Systems . . . . . . . . . . . . 34911.1 Kinematics of Individual Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . 35011.2 Kinematics of Entire Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

11.2.1 Recursive Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35411.2.2 Explicit Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

12 Screw Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35912.1 Resultant of two Velocity Screws. Cylindroid . . . . . . . . . . . . . . . 35912.2 Relative Velocity Screw. Raccording Hyperboloids . . . . . . . . . . 36212.3 Rotary Piercing of Tubes over Plug . . . . . . . . . . . . . . . . . . . . . . . 36512.4 Analogy Between Force Screw and Velocity Screw . . . . . . . . . . 36612.5 Virtual Power of a Force Screw. Reciprocal Screws . . . . . . . . . . 36812.6 Reciprocal Screw Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

xviii Contents

12.6.1 First-Order Screw System and Reciprocal Fifth-OrderSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

12.6.2 Second-Order Screw System and ReciprocalFourth-Order System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

12.6.3 Third-Order Screw System and ReciprocalThird-Order System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

13 Shaft Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38713.1 Hooke’s Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

13.1.1 Polhode and Herpolhode Cones of the Central Cross . 39013.2 Fenyi’s Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

13.2.1 Raccording Axodes of the Central Ring . . . . . . . . . . . . . 39513.3 Series-Connected Hooke’s Joints . . . . . . . . . . . . . . . . . . . . . . . . . . 39713.4 Homokinetic Shaft Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

13.4.1 Couplings With a Spherical Joint . . . . . . . . . . . . . . . . . . . 40113.4.2 Couplings With Three Parallel Serial Chains . . . . . . . . 40313.4.3 Ball-in-Track Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40413.4.4 Tripod Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410

14 Displacements in a Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41114.1 Complex Numbers in Planar Kinematics . . . . . . . . . . . . . . . . . . 411

14.1.1 Curvature of a Plane Curve . . . . . . . . . . . . . . . . . . . . . . . . 41414.2 Elementary Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41514.3 Resultant Displacements. Commutativity Conditions . . . . . . . . 41814.4 Relationships Between Three Positions . . . . . . . . . . . . . . . . . . . . 42514.5 Relationships Between Four Positions. Pole Curve . . . . . . . . . . 43214.6 Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448

15 Plane Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45115.1 Instantaneous Center of Rotation. Centrodes . . . . . . . . . . . . . . . 451

15.1.1 Theorems of Burmester and Kennedy/Aronhold . . . . . . 45315.1.2 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

15.2 Velocity and Acceleration in Complex Formulation . . . . . . . . . 47015.2.1 Instantaneous Center of Rotation . . . . . . . . . . . . . . . . . . . 47115.2.2 Instantaneous Center of Acceleration . . . . . . . . . . . . . . . 47215.2.3 Inflection Circle. Bresse Circles . . . . . . . . . . . . . . . . . . . . . 47315.2.4 Center of Acceleration and Bresse Circles of the

Inverse Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47515.3 Curvature of Plane Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . 476

15.3.1 Normal Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47615.3.2 Normal Poles of the Inverse Motion . . . . . . . . . . . . . . . . . 47815.3.3 Euler-Savary Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

Contents xix

15.3.4 Radii of Curvatures of Centrodes . . . . . . . . . . . . . . . . . . . 48515.3.5 Cubic of Stationary Curvature. Directrix . . . . . . . . . . . . 48715.3.6 Ball’s Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

15.4 Holditch’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49315.5 Trochoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

15.5.1 Basic Equations in Complex Notation . . . . . . . . . . . . . . . 49715.5.2 Double Generation of Trochoids . . . . . . . . . . . . . . . . . . . . 49915.5.3 Cycloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50115.5.4 Ordinary Cycloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50615.5.5 Involute of a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50715.5.6 Dwell Mechanisms Based on Cycloids . . . . . . . . . . . . . . . 509

15.6 Rectangle Moving Between two Lines and a Point . . . . . . . . . . 51315.6.1 Obtuse-Angled Corner . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51515.6.2 Right-Angled Corner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52015.6.3 Acute-Angled Corner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

16 Theory of Gearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52916.1 Parallel Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530

16.1.1 Curvature Relationship of Meshing Tooth Flanks . . . . . 53216.1.2 Camus’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53416.1.3 Cycloidal Gearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53516.1.4 Construction of Conjugate Flanks . . . . . . . . . . . . . . . . . . 53716.1.5 Pin Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54016.1.6 External Involute Spur Gears . . . . . . . . . . . . . . . . . . . . . . 54316.1.7 Internal Involute Spur Gears . . . . . . . . . . . . . . . . . . . . . . . 55116.1.8 Involute Helical Gearing . . . . . . . . . . . . . . . . . . . . . . . . . . . 552

16.2 Skew Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55816.2.1 Construction of Conjugate Flanks . . . . . . . . . . . . . . . . . . 55816.2.2 General Spatial Involute Gearing . . . . . . . . . . . . . . . . . . . 560

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564

17 Planar Four-Bar Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56717.1 Grashof Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56817.2 Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57317.3 Interchange of Input Link and Fixed Link . . . . . . . . . . . . . . . . . 57617.4 Inclination Angle of the Coupler. Transmission Angle . . . . . . . 57717.5 Transmission Ratio. Angular Acceleration of Output Link . . . 57817.6 Stationary Values of the Transmission Ratio . . . . . . . . . . . . . . . 58117.7 Transmission of Forces and Torques . . . . . . . . . . . . . . . . . . . . . . . 58717.8 Coupler Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590

17.8.1 Roberts/Tschebychev Theorem. Cognate Four-Bars . . . 59017.8.2 Parameter Equations for Coupler Curves . . . . . . . . . . . . 59417.8.3 Implicit Equation for Coupler Curves . . . . . . . . . . . . . . . 59517.8.4 Symmetrical Coupler Curves . . . . . . . . . . . . . . . . . . . . . . . 603

xx Contents

18 Spherical Four-Bar Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 63918.1 Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64018.2 Grashof Type Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64118.3 Coupler Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643

18.3.1 Implicit Equation for Coupler Curves . . . . . . . . . . . . . . . 64418.3.2 Symmetrical Coupler Curves . . . . . . . . . . . . . . . . . . . . . . . 64618.3.3 Geometrical Locus of Double Points . . . . . . . . . . . . . . . . 64818.3.4 Stereographic Projection . . . . . . . . . . . . . . . . . . . . . . . . . . 65118.3.5 Cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65418.3.6 Parameter Equations for Coupler Curves . . . . . . . . . . . . 656

18.4 Spherical Parallel Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662

19 Dynamics of Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66319.1 Conservative Single-Degree-of-Freedom Mechanisms . . . . . . . . . 66319.2 The General Problem of Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 66419.3 Principle of Virtual Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66519.4 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669

19.4.1 Systems with Tree Structure . . . . . . . . . . . . . . . . . . . . . . . 66919.4.2 Constraint Forces and Torques in Joints . . . . . . . . . . . . . 67119.4.3 Systems with Closed Kinematical Chains . . . . . . . . . . . . 672

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636

17.9 Slider-Crank. Inverted Slider-Crank . . . . . . . . . . . . . . . . . . . . . . . 60617.10 Planar Parallel Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60917.11 Four-Bars with Prescribed Transmission Characteristics . . . . 611

17.11.1 Prescribed Pairs of Input-Output Angles . . . . . . . . . . 61117.11.2 Prescribed Transmission Ratios . . . . . . . . . . . . . . . . . . 61317.11.3 Jeantaud’s Steering Mechanism . . . . . . . . . . . . . . . . . . 614

17.12 Coupler Curves with Prescribed Properties . . . . . . . . . . . . . . . 61617.12.1 Coupler Curves Passing Through Prescribed Points . 61617.12.2 Straight-Line Approximations . . . . . . . . . . . . . . . . . . . . 61717.12.3 Tschebychev’s Straight-Line Approximations . . . . . . . 619

17.13 Peaucellier Inversor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62617.14 Four-Bars Producing Prescribed Positions of the Coupler

Plane. Burmester Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62917.14.1 Three Prescribed Positions . . . . . . . . . . . . . . . . . . . . . . 62917.14.2 Four Prescribed Positions. Center Point Curve. Circle

Point Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63017.14.3 Five Prescribed Positions . . . . . . . . . . . . . . . . . . . . . . . . 63217.14.4 Crank-Rockers Producing Four Prescribed Positions

in Prescribed Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63217.15 Trajectory of the Center of Mass of a Four-Bar . . . . . . . . . . . . 634

Contents xxi

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679

References to Additional Literature . . . . . . . . . . . . . . . . . . . . . . . 677. . . .

Chapter 1

Rotation about a Fixed Point.Reflection in a Plane

Subject of this chapter are relationships between two positions of a rigid bodywith a fixed point. Note: Motions leading from one position to the other arenot investigated. Consequently, terms such as velocity or angular velocity donot occur.Literature: Rodrigues [25], Schoenflies [28], Klein/Muller [11], Meyer/Mohrmann [17], Mises [19], Kuipers [14], Mayer [16], Murnaghan [23], Muller[20, 21, 22], Rooney [26], Altmann [1], Geradin/Park/Cardona [7], Angeles[2], Shuster [29], Kolve [13], Bronstein/Semendjajev/Musiol/Muhlig [5].

For describing positions two right-handed cartesian bases are defined. Theorigins of both bases coincide with the fixed point of the body. The basis withunit basis vectors e1i (i = 1, 2, 3) serves as reference basis. The other basiswith unit basis vectors e2i (i = 1, 2, 3) is fixed on the body. It representsthe body. The bases are denoted e1 and e2 , respectively. The unit basisvectors of any right-handed cartesian basis e satisfy the six orthonormalityconditions

ei · ej = δij (i, j = 1, 2, 3) (1.1)

as well as the right-handedness condition

e1 · e2 × e3 = +1 . (1.2)

The positions between which relationships are to be established are the so-called initial position in which e2 coincides with e1 and an arbitrary finalposition.

1.1 Direction Cosine Matrix

The unit basis vectors e1i and e2j (i, j = 1, 2, 3) of the two bases define the

altogether nine direction cosines e1i · e2j = cos� (e1i , e2j ) (i, j = 1, 2, 3). They

1J. Wittenburg, Kinematics, DOI 10.1007/978-3-662-48487-6_1 © Springer-Verlag Berlin Heidelberg 2016

2 1 Rotation about a Fixed Point. Reflection in a Plane

are abbreviated

a12ij = a21ji = e1i · e2j (i, j = 1, 2, 3) . (1.3)

These direction cosines are the coordinates of e1i (i = 1, 2, 3) in e2 as wellas the coordinates of e2j (j = 1, 2, 3) in e1 . Definition: The (3× 3) direction

cosine matrix A12 has in row i the coordinates of e1i (i = 1, 2, 3) in e2 and,consequently, in column j the coordinates of e2j (j = 1, 2, 3) in e1 . Thethree equations

e1i =

3∑j=1

a12ij e2j (i = 1, 2, 3) (1.4)

are combined in matrix form in the equation

e1 = A12e2 . (1.5)

The symbols e1 and e2 , until now simply the names of bases, denote the col-umn matrices of the unit basis vectors: e1 = [ e11 e12 e13 ]

T , e2 = [ e21 e22 e23 ]T .

The exponent T denotes transposition. The use of boldface letters indicatesthat the elements of e1 and e2 are vectors. The matrix product A12e2 isevaluated following the rule of ordinary matrix algebra, although one of thematrices has vectors as elements and the other scalars. For two matrices eachhaving vectors as elements both the inner product (dot product) and theouter product (cross product) exist. Example: Scalar multiplication of (1.5)

from the right by e2T

produces for the direction cosine matrix the explicitexpression

A12 = e1 · e2T . (1.6)

This equation is the matrix form of the nine Eqs.(1.3).The direction cosine matrix is the first mathematical quantity used for

specifying the relationship between two positions. Other quantities are in-troduced later. In what follows, properties of the matrix are discussed. Sinceeach row contains the coordinates of one of the unit basis vectors of e1 , thedeterminant of the matrix is the scalar triple product e11 ·e12×e13 . Accordingto (1.2) this equals +1 . Hence

detA12 = +1 . (1.7)

The six orthonormality conditions (1.1) express the fact that the scalar prod-uct of any two rows i and j and also of any two columns i and j of A12

equals the Kronecker delta:

3∑k=1

a12ika12jk = δij ,

3∑k=1

a12kia12kj = δij (i, j = 1, 2, 3). (1.8)

1.1 Direction Cosine Matrix 3

A matrix having these properties is called orthogonal matrix. Because of

(1.8) the product A12A12T equals the unit matrix. Hence the matrix has theimportant property that its inverse equals its transpose:

(A12)−1

= A12T . (1.9)

Definition: The co-factor of the element a12ij is (−1)i+j times the deter-

minant of the (2× 2)-submatrix of A12 left after deleting row i and columnj . Because of the orthogonality of the matrix every element equals it ownco-factor1. Omitting the upper indices 12 these identities can be written inthe forms

aii = ajjakk −ajkakj ,ajk = aijaki −aiiakj ,akj = ajiaik −aiiajk

⎫⎬⎭ (i, j, k = 1, 2, 3 cyclic) . (1.10)

With (1.9) the inverse of (1.5) is

e2 = A21e1 with A21 = A12T . (1.11)

Let now v be an arbitrary vector (not necessarily the position vector ofa point). In the bases e1 and e2 it has different coordinates v1i and v2i ,respectively (i = 1, 2, 3) :

v =

3∑i=1

v1i e1i =

3∑i=1

v2i e2i . (1.12)

The sums are written as matrix products. For this purpose the column ma-trices v1 = [ v11 v

12 v

13 ]

T and v2 = [ v21 v22 v

23 ]

T of the coordinates of v in thetwo bases are defined. They are called coordinate matrices of v in e1 and ine2 , respectively. It should be noted that the term vector is used for v andnot as abbreviation for coordinate matrix. We also distinguish between thecoordinate vi and the component viei of a vector. A component is itself avector, whereas a coordinate is a scalar.

In terms of coordinate matrices (1.12) has the form

e1Tv1 = e2

Tv2 . (1.13)

From (1.11) it follows that e2T

= e1TA12 . Substitution of this expression

produces the equation e1Tv1 = e1

TA12v2 and, consequently,

v1 = A12v2 . (1.14)

1 Special case of the general formula (A−1)ij = cji/detA valid for an arbitrary (n× n)-

matrix A ( cji co-factor of the matrix element aji )

4 1 Rotation about a Fixed Point. Reflection in a Plane

More directly, this equation follows from the fact that the components ofv are linear combinations of unit basis vectors. The equation is the rule bywhich vector coordinates are transformed from one basis into another. Be-cause of this equation the direction cosine matrix is also called transformationmatrix. The absolute value of a vector does not change under a transforma-tion. Indeed, with both sets of coordinates the scalar product v · v is the

same: v1Tv1 = v2

TA12TA12v2 = v2

Tv2 .

Example: Let r and r∗ be the position vectors of an arbitrary body-fixedpoint in the initial position and in the final position of basis e2 , respectively.The coordinate matrices r∗1 in e1 and r∗2 in e2 satisfy the equation r∗1 =A12r∗2 . However, since r∗ coincides with r , when e2 coincides with e1 ,also the identity r1 = r∗2 is true. From this it follows that the coordinatesof r∗ and r in e1 are related through the equation

r∗1 = A12r1 . (1.15)

End of example.

Imagine in addition to the bases e1 and e2 a third basis e3 , i.e., a thirdposition of the rigid body. Then three direction cosine matrices are definedby the equations

e1 = A13e3 , e1 = A12e2 , e2 = A23e3 . (1.16)

The matrices are not independent. The last two equations establish the rela-tionship e1 = A12A23e3 . Comparison with the first equation yields

A13 = A12A23 . (1.17)

This equation shows that orthogonal matrices constitute a group with respectto multiplication. They satisfy the four conditions: 1) The product of anytwo elements of the group is itself an element (A13 is an orthogonal matrix).2) The product is associative. 3) There is a unit element, namely, the unitmatrix. 4) For every element of the group the inverse element exists, namely,the transposed matrix. The group is denoted SO(3) with O for orthogonal.

Next, the eigenvalue problem A12v = λv or (A12 − λI)v = 0 is inves-tigated where A12 is an arbitrary direction cosine matrix. The equation isthe transformation rule A12v2 = v1 in the special case v1 = λv2 . Sincethe absolute value of a vector does not change under a transformation, itcan be predicted that all three eigenvalues have the absolute value one. Theeigenvalues are the roots of the characteristic equation det (A12 − λI) = 0 .Without the superscript of A12 this is the cubic equation

−λ3 + λ2 trA − λ[(a11a22 − a12a21) + (a22a33 − a23a32)

+(a33a11 − a31a13)] + detA = 0 . (1.18)

1.1 Direction Cosine Matrix 5

According to (1.7) the free term is +1 . Every expression in parentheses is theco-factor of one diagonal element of A12 . According to the first Eq.(1.10) theco-factor is identical with the diagonal element. Consequently, the expressionin square brackets represents the trace of the matrix. Thus, the equationreads

−λ3 + λ2 trA12 − λ trA12 + 1 = 0 . (1.19)

It shows that every direction cosine matrix has the eigenvalue λ = +1 .Division by (λ−1) produces for the other eigenvalues the quadratic equationλ2 − (trA12 − 1)λ+ 1 = 0 . It has the roots

λ2,3 =trA12 − 1

2± i

√1−

( trA12 − 1

2

)2= cosϕ± i sinϕ = e±iϕ (1.20)

with

cosϕ =trA12 − 1

2. (1.21)

If A12 is the unit matrix, it has the triple eigenvalue +1 . In the casetr A12 = −1 it has the double eigenvalue λ2,3 = −1 .

Let n be the normalized eigenvector associated with the eigenvalue λ =+1 . It is calculated from the equations

(A12 − I)n = 0 , n21 + n2

2 + n23 = 1 . (1.22)

This eigenvector n represents the coordinate matrix of a unit vector n whichhas identical coordinate matrices in the bases e1 and e2 . Also this vectorn is called eigenvector of A12 .

Imagine that, starting from the initial position, the body-fixed basis e2

is rotated about the eigenvector n . The final position depends upon therotation angle. Independent of the angle the vector n has identical coordinatematrices n1 = n2 = n in e1 and in e2 . The existence of the eigenvectorguarantees the existence of an angle which carries the basis from its initialposition to the final position given by the matrix A12 . Hence the

Theorem 1.1. (Euler) The displacement of a body-fixed basis from an initialposition e1 to an arbitrary final position e2 is achieved by a rotation througha certain angle about an axis which is fixed in both bases. The axis has thedirection of the eigenvector associated with the eigenvalue λ = +1 of thedirection cosine matrix A12 .

In Sect. 1.5 it is shown that the rotation angle in Euler’s theorem is theangle ϕ in Eq.(1.20) for the eigenvalues λ2 and λ3 . Euler’s theorem guar-antees that the direction cosine matrix A12 can be expressed in terms ofthe coordinates of its eigenvector n and of the rotation angle ϕ . This is thesubject of Sect. 1.5 .

6 1 Rotation about a Fixed Point. Reflection in a Plane

The complex conjugate eigenvectors associated with the eigenvalues cosϕ±i sinϕ are determined at the end of Sect. 1.5 .

1.2 Similarity Transformation

Let A and B be arbitrary (n × n)-matrices, A nonsingular. The transfor-mation of B into the matrix

B∗ = ABA−1 (1.23)

is called similarity transformation of B .

Theorem 1.2. The matrices B and ABA−1 have identical characteristicpolynomials and, therefore, identical eigenvalues and identical traces.

Proof: The characteristic polynomial of ABA−1 is det (ABA−1 − λI) =

det (ABA−1 −AλI A−1) = det [A (B − λI)A−1] = det (B − λI) = 0

(the determinant of a product of matrices equals the product of the determi-nants of the factors). Since the trace is the sum of all eigenvalues, also thetraces are identical. End of proof.

Theorem 1.3. Let n and nB be the eigenvectors of ABA−1 and of B ,respectively, associated with one and the same eigenvalue λ (arbitrary). Theyare related by the equation

n = AnB . (1.24)

Proof: By definition, ABA−1 n = λn and B nB = λnB . With (1.24) thefirst equation is ABA−1AnB = λAnB or B nB = λnB . This is the secondequation. End of proof.

Example: Let A12 be the direction cosine matrix relating the coordinatesof vectors in two bases e1 and e2 . Let, in particular, a1 , a2 and b1 , b2

be the coordinate matrices of two vectors a and b , respectively, so that,for example, b1 = A12b2 . The cross-product a× b has in the two bases thecoordinate matrices a1b1 and a2b2 with the skew-symmetric matrices ai =⎡⎢⎣

0 −ai3 ai

2

ai3 0 −ai

1

−ai2 ai

1 0

⎤⎥⎦ (i = 1, 2) . Hence a1 b1 = A12 a2 b2 . On the right-hand side

b2 = A12T b1 is substituted. This results in the transformation formula

a1 = A12 a2 A12T . (1.25)

It is a similarity transformation. End of example.

1.3 Euler Angles 7

1.3 Euler Angles

Calculations with nine direction cosines subject to six constraint equationsare cumbersome. In the present section and in following sections the elementsof the direction cosine matrix are formulated in various ways in terms of eitherthree coordinates without constraint equations or in terms of four coordinateswith one constraint equation. In the present section so-called Euler anglesare introduced. The final position e2 of the body-fixed basis is the result ofthree successive rotations (Fig. 1.1a ). In the initial position prior to the firstrotation the body-fixed basis coincides with e1 . The first rotation is carriedout through an angle ψ about the axis e13 (in the usual right-handed sense).It carries the body-fixed basis into an intermediate position e2

′′. The second

rotation is carried out through an angle θ about the axis e2′′

1 . It carries thebody-fixed basis into a new intermediate position e2

′. The third rotation is

carried out through an angle φ about the axis e2′

3 . It carries the body-fixedbasis into the final position e2 . A characteristic feature of Euler angles is thatthe second and the third rotation are carried out about axes which are theresult of the previous rotation (or rotations). Another characteristic featureis the sequence (3,1,3) of rotation axes.

The desired expression for the matrix A12 in terms of the three anglesis obtained from the transformation Eqs.(1.5) for the individual rotations.Figure 1.1a yields the equations

e1 = Aψe2′′ , e2

′′= Aθe

2′ , e2′= Aφe

2 (1.26)

with

Fig. 1.1 Euler angles. Definition (a) and application in a two-gimbal suspension (b)

8 1 Rotation about a Fixed Point. Reflection in a Plane

Aψ =

⎡⎢⎣ cosψ − sinψ 0

sinψ cosψ 0

0 0 1

⎤⎥⎦ , Aθ =

⎡⎢⎣ 1 0 0

0 cos θ − sin θ

0 sin θ cos θ

⎤⎥⎦ ,

Aφ =

⎡⎢⎣ cosφ − sinφ 0

sinφ cosφ 0

0 0 1

⎤⎥⎦ . (1.27)

From (1.26) it follows that A12 = AψAθAφ . When this is multiplied out anduse is made of the abbreviations cψ , cθ , cφ for cosψ , cos θ , cosφ andsψ , sθ , sφ for sinψ , sin θ , sinφ , the matrix is obtained in the form

A12 =

⎡⎢⎣ cψcφ − sψcθsφ −cψsφ − sψcθcφ sψsθsψcφ + cψcθsφ −sψsφ + cψcθcφ −cψsθsθsφ sθcφ cθ

⎤⎥⎦ . (1.28)

The advantage of having only three coordinates and no constraint equationis paid for by the disadvantage that the direction cosines are complicatedfunctions of the three coordinates. There is still another problem. Figure 1.1ashows that in the case θ = nπ (n = 0,±1, . . .) the axis of the third rotationcoincides with the axis of the first rotation. This has the consequence that ψand φ cannot be distinguished.

Euler angles can be illustrated by means of a rigid body in a two-gimbalsuspension system (Fig. 1.1b ). The bases e1 and e2 are attached to thematerial base and to the suspended body, respectively. The angles ψ , θand φ are, in this order, the rotation angle of the outer gimbal relative tothe material base, of the inner gimbal relative to the outer gimbal and ofthe body relative to the inner gimbal. With this device all three angles canbe adjusted independently since the intermediate bases e2

′′and e2

′are

materially realized by the gimbals. For θ = nπ (n = 0, 1, . . .) the planes ofthe gimbals coincide (gimbal lock).

Euler angles are ideally suited as position variables for the study of motionsin which θ(t) is either exactly or approximately constant, whereas ψ andφ are (exactly or approximately) proportional to time, i.e., ψ ≈ const andφ ≈ const . Euler angles are advantageous also whenever there exist twophysically significant directions, one fixed in the reference basis e1 and theother fixed in the body-fixed basis e2 . In such cases, e13 and e23 are giventhese directions so that θ is the angle between the two (as examples seeEqs.(10.10) and (10.83)). However, the use of Euler angles is not restrictedto such special cases.

If the matrix A12 is given, the corresponding Euler angles are calculatedfrom the equations