![Inverse Kinematics Using a Converging
Paths Algorithm
Omid Heidari1(B)
and Alba Perez Gracia1,2(B)
1
Department of Mechanical Engineering, Idaho State University, Pocatello, USA
{heidomid,perealba}@isu.edu
2
Remy Robotics, Barcelona, Spain
Abstract. This article presents a numerical method for the inverse kine-
matics of serial chains by utilizing dual quaternion formulation of the
robot kinematics within a converging paths algorithm. The method is
inspired by iterative techniques such as FABRIK, however adding infor-
mation on the kinematics of the chain to be solved. The method has been
tested with 2R, planar 4R and spatial 4R robots. Future work includes
optimizing the method to compare with other fast numeric algorithms
for inverse kinematics.
Keywords: Inverse kinematics · Serial manipulators · FABRIK ·
Iterative methods
1 Introduction
Inverse kinematics (IK) methods are widely used in robotics and computer
graphics. Analytic and numerical techniques have been develop for solving the
IK problem. Closed-form or analytic methods were among the first to be devel-
oped and are successful for simple chains; however, they are developed in a
case-by-case strategy and they are hard to generalize. See [7] for a review. Some
of the analytic methods use the description of the geometric constraints of the
chain. Others use the matrix formulation to the forward kinematics equations
of the chain to solve the inverse problem when this is equated to the desired
position [4]. Analytic methods have the advantage of avoiding iterative solutions
that in principle are more time consuming; however many times they reduce to
finding the roots of a univariate polynomial [5], which carries its own numerical
problems. New algorithms such as IKFast [3] allow a more systematic approach
combining analytic and numerical solutions.
A good review for numerical methods, especially those applied to computer
graphics, can be found in [2]. Numerical methods are grouped into those based
on the Jacobian, Newton-based methods and heuristic methods. Jacobian-based
methods [8] are the most popular and a variety of implementations can be found,
see for instance [6]. Heuristic methods are developed as a computationally lean
alternative to other numerical methods. Most of them consist of a simple app-
roach that yields many fast iterations. Among them we highlight Forward And
c
The Editor(s) (if applicable) and The Author(s), under exclusive license
to Springer Nature Switzerland AG 2021
J. Lenarčič and B. Siciliano (Eds.): ARK 2020, SPAR 15, pp. 7–14, 2021.
https://doi.org/10.1007/978-3-030-50975-0_2](https://image.slidesharecdn.com/89071-250426042615-049bb3ed/85/Advances-in-Robot-Kinematics-2020-Jadran-Lenarcic-25-320.jpg)
![8 O. Heidari and A. P. Gracia
Backward Reaching Inverse Kinematics (FABRIK) [1]. This method is an itera-
tive approach where it tries to find the position of each joint by locating a point
on line. It is not costly in terms of computation neither it produces complex
solutions. However, the algorithm works best with spherical joints, mostly used
for IK in computer animations and gaming.
Desired properties of IK methods are speed, numerical stability and conver-
gence. In addition, being applicable to a broad number of kinematic chains is
desirable too. In this work we present a IK algorithm that combines some of the
advantages of the numerical methods with the insight of the analytic ones. The
method is tested for some simple open-chain robots consisting of revolute joints.
The core concept is to break the serial chain and create two legs for a given pose.
Then the end-effectors (ee) of these legs move towards each other smoothly till
they match and complete the chain. Moving towards a better solution at each
step is insured by incorporating pseudo-inverse in Cartesian space. At each step,
the solution in the trigonometric joint space is projected to the surface condition
for the joint variables to make sure that the constraints are satisfied.
2 Dual Quaternion Inverse Kinematics
In this section we describe a method to state the inverse kinematics for serial
chains with revolute joints. Let us consider a kinematic serial chain with n
degrees of freedom; let the joint space of the chain be defined by the vector
θ = {θ1, . . . , θn} and the joint axes be defined by the Plücker coordinates of the
lines Si, i = 1, . . . n.
2.1 Product of Exponentials in Dual Quaternion Form
In general, for an arbitrary k-degree of freedom robot, kR, given by k joint
axes S1 . . . Sk with rotation angles θ1 . . . θk, the forward kinematics of relative
displacements is given by the product of exponentials,
D = eS1
θ1
2 eS2
θ2
2 . . . eSk
θk
2 (1)
This can be done using any suitable algebra. In this case, we use Clifford
algebra of dual quaternions which can be seen as 2k
products of cosines and sines
of the angles. Select an order, for instance (c1c2 . . . ck, s1c2 . . . ck, . . . , s1s2 . . . sk).
The forward kinematics can be written as a linear combination of the joint axes
and Clifford products of joint axes by these coefficients,
I S1 S2 . . . S1S2 . . . Sk
⎧
⎪
⎨
⎪
⎩
cos θ1
2 cos θ2
2 . . . cosθk
2
.
.
.
sin θ1
2 sin θ2
2 . . . sin θk
2
⎫
⎪
⎬
⎪
⎭
= P, (2)](https://image.slidesharecdn.com/89071-250426042615-049bb3ed/85/Advances-in-Robot-Kinematics-2020-Jadran-Lenarcic-26-320.jpg)
![Inverse Kinematics Using a Converging Paths Algorithm 11
The procedure starts with setting the left leg to a random unit dual quater-
nion that satisfies Study’s quadric. Then, this pose is used to calculate the right
leg’s configuration that can be the closest. To this end, a pseudo-inverse matrix
is computed which may result in a solution that does not satisfy the constraints
in Eq. 5. To solve this problem, the algorithm incorporate a method of projection
where solutions out of workspace are projected based on the shortest distance
they can have to the constraint surface. The projected vector in the joint space
is, then, used to compute the Cartesian pose of the right leg and its transforma-
tion is set equal to the left leg equations to see if it is in the workspace of the
left leg.
Equation 1, for a 4R serial chain, can be converted to the following:
eS1
θ1
2 eS2
θ2
2 = De−S4
θ1
2 e−S3
θ3
2 (7)
Separating the left and right transformations of Eq. 7 by setting them to two
different dual quaternion, L = R, results in:
eS1
θ1
2 eS2
θ2
2 = L, eS3
θ3
2 eS4
θ4
2 = T (8)
where hat symbol is the dual quaternion conjugate operator and T = R̂D. As
it was stated in the previous sections, Eq. 8 can be written as a linear combination
of Clifford products of joint axes:
[L]v = l, [R]u = t (9)
where l and t are 8-element vectors representing dual quaternions on each
side, L and T respectively. [L] and [R] are corresponding matrices containing joint
axes and their Clifford products. It is noteworthy that elements of these matrices
are constant depending on a reference pose and do not change by changing the
configuration of the robot. Moreover, v and u are 4-element vectors including
the products of sines and cosines of angels expressed in Eq. 4.
3.1 Algorithm Description
The back and forth procedure explained in the previous section between left and
right leg is shown more in details by Algorithm 1. This iterative process repeats
till the ee of one leg matches the ee of the other leg in its workspace. To make
sure that there is a solution to the inverse kinematics problem, the forward kine-
matics is solved so that we have a D that is in the workspace.](https://image.slidesharecdn.com/89071-250426042615-049bb3ed/85/Advances-in-Robot-Kinematics-2020-Jadran-Lenarcic-29-320.jpg)
![12 O. Heidari and A. P. Gracia
Algorithm 1: Converging Paths Inverse Kinematics
Result: v and u that put the left and right leg in a configuration at
which their end-effectors reach each other.
initialization:
L = random unit dual quaternion;
δ = 0.0001;
while constraints are not satisfied do
l = Vector(L);
v = [L]−1
l;
vprojected = Project(v);
l = [L]vprojected;
L = DualQuaternion(l);
R = L;
r = Vector(R̂D);
u = [R]−1
r;
uprojected = Project(u);
t = [R]uprojected;
T = DualQuaternion(t);
R = DT̂;
L = R;
if CheckConstraints(v) δ CheckConstraints(u) δ then
Exit while loop;
end
end
There are some functions used in this algorithm that affect its performance
and speed. Vector() converts a dual quaternion to an 8-element vector and
DualQuaternion() does the opposite. CheckConstraint() verifies if the constraints
in Eq. 5 are satisfied, returning a value for each constraint that can be used to
evaluate the convergence. At last, Project() is the function that finds a solution
with the least distance between the given point and the constraint surface.
There are different approaches to accomplish this. One method uses vector
calculus on the parameterization to compute the normal vector to the surface
and then find the line normal to the surface and passing through the given point.
This is done by finding a line whose moment is equal to the moment of the point
in the direction of the line.
Another method that is used in the algorithm is through Lagrange Mul-
tipliers. The objective function to minimize is the squared Euclidean distance
between the given point and points lying on the constraint surface.
The proposed algorithm is in the early stage and the authors are exploring dif-
ferent approaches to do the projection. However, the Lagrangian method seems
to have a faster response at the current stage. Numerical results suggest con-
vergence by iteratively approaching each leg to the other using Moore–Penrose
pseudo inverse method, however convergence needs to be formally proved. At
each step the pose of one leg’s end effector is set equal to the other one to see if
it is in its workspace or not. If not, the corresponding vector in joint space gets
projected to the constraint surface to make sure that the ee is in the workspace.](https://image.slidesharecdn.com/89071-250426042615-049bb3ed/85/Advances-in-Robot-Kinematics-2020-Jadran-Lenarcic-30-320.jpg)
![Inverse Kinematics Using a Converging Paths Algorithm 13
3.2 Numerical Example
This section is dedicated to an example of solving the IK of a 4R serial manipu-
lator by the proposed method. The first step is to compute matrices [L] and [R]
for an arbitrary located joint axes.
L =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 −0.639867 0.0305296 0.732783
0 0.512895 −0.510064 0.567498
0 0.572284 0.859595 0.310715
1 0 0 −0.210788
0 −0.13248 1.10913 −0.322849
0 −0.912657 −0.523315 0.545166
0 0.66982 −0.349916 −0.138578
0 0 0 0.14111
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
R =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 −0.0689129 0.84182 0.263595
0 −0.994039 0.491349 −0.0865131
0 −0.0844797 −0.223418 0.802942
1 0 0. 0.527558
0 −1.12461 0.562694 −0.792586
0 0.040865 −0.732085 0.10385
0 0.43654 0.510156 0.0228117
0 0 0 0.378327
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(10)
After 9 iterations, the algorithm finds a solution with an error of 0.001.
Figure 3 shows the solution and the steps of convergence as well as the constraint
surface.
The convergence of the right and left leg to the
solution in joint space in terms of x1,x2 and x3.
Green: uprojected and vprojected, Blue: u and v,
Red: expected solution.
The convergence of the left (black) and right
(green) leg to the solution in Cartesian space.
Fig. 3. Convergence in joint and Cartesian spaces.
The computation time is relatively high in its current version, as the steps
have not been optimized for spped yet. The interest of the method is going to
depend on finding efficient processes for each of the steps of the algorithm.](https://image.slidesharecdn.com/89071-250426042615-049bb3ed/85/Advances-in-Robot-Kinematics-2020-Jadran-Lenarcic-31-320.jpg)
![Among the most noteworthy of living Swedish scholars are Adolph
Norén, Axel Koch and Esaias Tegnér, Junior, philologists; Hans
Hildebrand and Oscar Montelius, archæologists; P. Fahlbeck, Nils and
Magnus Hœjer, Martin Weibull, Ernest Carlson, historians; A. M.
Mittag-Leffler, mathematician; Hugo Hildebrandsson, meteorologist;
E. A. H. Key, E. O. T. Westerlund, Anton Wetterstrand, F. J.
Biornstrœm, T. F. Hartelius, Curt Wallis, prominent in various
branches of medical science.
King Oscar with fervent interest and unfailing liberality has
encouraged various scientific explorations, and has had the
satisfaction to see the greatest geographical discoveries of the
century successfully made by Swedes, the circumnavigation of Asia
and Europe, and the discovery of the Northeast Passage by Baron N.
A. E. Nordenskiold, and the exploration of Central Asia by Sven
Hedin, which has forever settled the learned disputes of ages. A
third expedition, the most daring of scientific exploits ever
attempted, still keeps the world in suspense as to its final outcome.
July 11, 1897, S. A. Andrée, a scientifically experienced aëronaut,
with two companions, Nils Strindberg and Knut Frænkel, started in a
balloon constructed for the purpose, and with provisions for three
years, from an island of Spitzbergen, with the purpose of reaching
the North Pole. The daring aëronauts have not been heard from
since their departure, but authorities like Baron Nordenskiold have
expressed the best of hopes that they may have reached Franz
Joseph’s Land in safety, whence they might regain settled regions.[6]
S. A. Andrée belongs to a class of men, the Swedish engineers, who
have won distinction for their ability, and on whom the examples set
by Christopher Polhem and John Ericsson have had a stimulating
influence. There are among them two inventors of the very first
rank, who belong to the reign of Oscar II., Alfred Nobel (d. 1896),
the inventor of dynamite, and Gustavus de Laval, the Swedish
Edison. The latter is world-famous for his separator and other
inventions, which have revolutionized the dairy industry. Alfred
Nobel, the disciple of John Ericsson, has not only the glory of having
invented one of the most useful helpers of mechanic and industrial](https://image.slidesharecdn.com/89071-250426042615-049bb3ed/85/Advances-in-Robot-Kinematics-2020-Jadran-Lenarcic-51-320.jpg)
Advances in Robot Kinematics 2020 Jadran Lenarčič
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- 4. Springer Proceedings in Advanced Robotics Series Editors: Bruno Siciliano · Oussama Khatib 15 Jadran Lenarčič Bruno Siciliano Editors Advances in Robot Kinematics 2020
- 5. Springer Proceedings in Advanced Robotics 15 Series Editors Bruno Siciliano Dipartimento di Ingegneria Elettrica e Tecnologie dell’Informazione Università degli Studi di Napoli Federico II Napoli, Napoli Italy Oussama Khatib Robotics Laboratory Department of Computer Science Stanford University Stanford, CA USA Advisory Editors Gianluca Antonelli, Department of Electrical and Information Engineering, University of Cassino and Southern Lazio, Cassino, Italy Dieter Fox, Department of Computer Science and Engineering, University of Washington, Seattle, WA, USA Kensuke Harada, Engineering Science, Osaka University Engineering Science, Toyonaka, Japan M. Ani Hsieh, GRASP Laboratory, University of Pennsylvania, Philadelphia, PA, USA Torsten Kröger, Karlsruhe Institute of Technology, Karlsruhe, Germany Dana Kulic, University of Waterloo, Waterloo, ON, Canada Jaeheung Park, Department of Transdisciplinary Studies, Seoul National University, Suwon, Korea (Republic of)
- 6. The Springer Proceedings in Advanced Robotics (SPAR) publishes new develop- ments and advances in the fields of robotics research, rapidly and informally but with a high quality. The intent is to cover all the technical contents, applications, and multidisci- plinary aspects of robotics, embedded in the fields of Mechanical Engineering, Computer Science, Electrical Engineering, Mechatronics, Control, and Life Sciences, as well as the methodologies behind them. The publications within the “Springer Proceedings in Advanced Robotics” are primarily proceedings and post-proceedings of important conferences, symposia and congresses. They cover significant recent developments in the field, both of a foundational and applicable character. Also considered for publication are edited monographs, contributed volumes and lecture notes of exceptionally high quality and interest. An important characteristic feature of the series is the short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. More information about this series at http://www.springer.com/series/15556
- 7. Jadran Lenarčič • Bruno Siciliano Editors Advances in Robot Kinematics 2020 123
- 8. Editors Jadran Lenarčič Jožef Stefan Institute Ljubljana, Slovenia Bruno Siciliano Department of Electrical Engineering and Information Technology University of Naples Federico II Naples, Italy ISSN 2511-1256 ISSN 2511-1264 (electronic) Springer Proceedings in Advanced Robotics ISBN 978-3-030-50974-3 ISBN 978-3-030-50975-0 (eBook) https://doi.org/10.1007/978-3-030-50975-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed 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, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
- 9. Foreword At the dawn of the century’s third decade, robotics is reaching an elevated level of maturity and continues to benefit from the advances and innovations in its enabling technologies. These all are contributing to an unprecedented effort to bringing robots to human environment in hospitals and homes, factories and schools; in the field for robots fighting fires, making goods and products, picking fruits and watering the farmland, saving time and lives. Robots today hold the promise for making a considerable impact in a wide range of real-world applications from industrial manufacturing to health care, transportation, and exploration of the deep space and sea. Tomorrow, robots will become pervasive and touch upon many aspects of modern life. The Springer Tracts in Advanced Robotics (STAR) was launched in 2002 with the goal of bringing to the research community the latest advances in the robotics field based on their significance and quality. During the latest fifteen years, the STAR series has featured publication of both monographs and edited collections. Among the latter, the proceedings of thematic symposia devoted to excellence in robotics research, such as ISRR, ISER, FSR, and WAFR, has been regularly included in STAR. The expansion of our field as well as the emergence of new research areas has motivated us to enlarge the pool of proceedings in the STAR series in the past few years. This has ultimately led to launching a sister series in parallel to STAR. The Springer Proceedings in Advanced Robotics (SPAR) is dedicated to the timely dissemination of the latest research results presented in selected symposia and workshops. This volume of the SPAR series is dedicated to the proceedings of a special edition of ARK on Advances in Robot Kinematics. Returning to Ljubljana, Slovenia, where it was first founded in September 1988, ARK marks this year an important milestone in reaching its seventeenth gathering, establishing itself as a major anchor of research advances in robot kinematics serving the global robotics community. v
- 10. The volume edited by Jadran Lenarčič and Bruno Siciliano contains 43 scientific contributions. This collection spans a wide range of research developments in robot mechanisms, kinematics, analysis, design, planning, and control. Rich by topics and authoritative contributors, ARK brings this unique reference on the current developments and new directions in the field of kinematics. A fine addition to the SPAR series and a genuine tribute to ARK contributors, organizers, and founder! May 2020 Bruno Siciliano Oussama Khatib SPAR Editors vi Foreword
- 11. Preface The series of international symposia Advances in Robot Kinematics (ARK) was organized for the first time in Ljubljana in 1988. Since then, they were organized every two years, in Slovenia, Austria, Italy, France, and Spain, under the patronage of the International Federation for the Promotion of Machine Sciences (IFToMM). The first edited book was published by Springer in 1991, one year after the con- ference in Linz. Since 1994, a new volume has been published every two years. Each edited book is linked to a corresponding symposium, in which the participants exchange their results and opinions in a meeting that brings together the best researchers and scientists in the field of robot kinematics. The current book is the 15th, and the last three were included in the SPAR series. The book contains 43 contributions and a large team of reviewers contributed their critical and insightful recommendations to the authors. In the 1980s, when we began organizing these symposia and publishing the books, we did not expect robot kinematics to remain at the forefront of robotics for so many years. However, in the current turbulent times of artificial intelligence, the analytical and in-depth work of kinematicians is even more important than it used to be. Kinematics remains an immense domain of topics that need to be explored if we are to continue the development of complex robot mechanisms. Industrial robots, and especially humanoid robots, open up many scientific intrigues that have not yet been answered. As an example, let us just mention that current humanoid robots are incapable of shrugging their shoulders. A movement that seems humanly childish is crucial for the reachability of the human arm, avoiding obstacles and, last but not least, for communications between people. High-performance computing and even artificial intelligence are undoubtedly tools that will help kinematicians solve the extremely difficult mathematical problems they have to deal with on a daily basis. The future of kinematics seems increasingly fascinating and unpredictable. After 32 years, ARK is now back to where it all started. This edition is not held in the traditional June dates of the symposium because of the pandemic. As we write this preface, we still hope to be able to safely gather in Ljubljana later this year. We are grateful to the contributors of this volume for their work and enthu- siasm. Some have been submitting their contributions for many years, and it is vii
- 12. because of them that the conference maintains its high quality. Special thanks go to Tadej Petrič, Conference Publishing Chair, who provided excellent technical sup- port, and Aleš Ude, Conference Organizing Committee Chair. We are also grateful to the Springer staff who have supported our work throughout these years. We hope that this new ARK book will again attract scholars and researchers specializing in robot kinematics and will outline the research guidelines for many years to come. Jadran Lenarčič May 2020 Bruno Siciliano viii Preface
- 13. Organization Conference Chairmen J. Lenarčič Jožef Stefan Institute, Ljubljana, Slovenia B. Siciliano University of Naples Federico II, Italy Honorary Chairman B. Roth Stanford University, USA Scientific Committee J. Angeles McGill University, Montreal, Canada O. Altuzarra University of the Basque Country, Spain M. Carricato University of Bologna, Italy M. Husty University of Innsbruck, Austria A. Kecskemethy University of Duisburg, Germany O. Khatib Stanford University, USA J. M. McCarthy University of California at Irvine, USA J.-P. Merlet Inria, Sopia-Antipolis, France V. Parenti-Castelli University of Bologna, Italy F. Park Seoul National University, Korea A. Perez Idaho State University, USA J. Selig London South Bank University, UK M. M. Stanisic University of Notre Dame, USA F. Thomas Institute of Industrial Robotics, Spain P. Wenger Ecole Centrale de Nantes, France ix
- 14. Organizing Committee Chair A. Ude Jožef Stefan Institute, Ljubljana, Slovenia Publications Chair T. Petrič Jožef Stefan Institute, Ljubljana, Slovenia x Organization
- 15. Contents Advances in Robot Kinematics Facts and Thoughts . . . . . . . . . . . . . . . . 1 Jadran Lenarčič Inverse Kinematics Using a Converging Paths Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Omid Heidari and Alba Perez Gracia Design Parameters Influence on the Static Workspace and the Stiffness Range of a Tensegrity Mechanism . . . . . . . . . . . . . . . . 15 G. M. Cruz-Martinez, J-C Avila Vilchis, A. Vilchis Gonzalez, S. Abdelaziz, and P. Poignet Bennett Based Balanced Butterfly Linkage, Deployable Linkage with Inherent Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Volkert van der Wijk A Compliant Linkage for Cooperative Object Manipulation Through a Heterogeneous Mobile Multi-robot System . . . . . . . . . . . . . . 33 Juan Mauricio Toro Ramos, Dhruvin Doshi, Daniel Schütz, and Franz Dietrich One Degree of Freedom 7-R Closed Loop Linkage as a Building Block of Nanorobots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Meysam T. Chorsi, Pouya Tavousi, Caitlyn Mundrane, Vitaliy Gorbatyuk, Horea Ilies, and Kazem Kazerounian Modeling and Control of a Redundant Tensegrity-Based Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Jérémy Begey, Marc Vedrines, Pierre Renaud, and Nicolas Andreff Motion Parameterization of Parallel Robots Used in Lower Limb Rehabilitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Iosif Birlescu, Manfred Husty, Calin Vaida, Bogdan Gherman, Ionut Ulinici, Remus Bogateanu, and Doina Pisla xi
- 16. Robust Trajectory Planning of Under-Actuated Cable-Driven Parallel Robot with 3 Cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Edoardo Idà, Sébastien Briot, and Marco Carricato On the Plane Symmetric Bricard Mechanism . . . . . . . . . . . . . . . . . . . . 73 J. M. Selig Exact Coupler-Curve Synthesis of Four-Bar Linkages with Fully Analytical Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Shaoping Bai, Rui Wu, and Ruiqin Li A General Method for Determining Algebraic Input-Output Equations for Planar and Spherical 4R Linkages . . . . . . . . . . . . . . . . . 90 Mirja Rotzoll, M. John D. Hayes, Manfred L. Husty, and Martin Pfurner The Forward Kinematics of the 4-1 Cable-Driven Parallel Robot with Non Elastic Sagging Cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 J-P. Merlet The Geometrical Arrangement of Joint Constraints that Makes Natural Motion Possible: Experimental Verification on the Ankle . . . . . 109 Michele Conconi, Nicola Sancisi, and Vincenzo Parenti-Castelli Development of a Vector Geometrical Model for PKM Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 J.-B. Guyon, B. Boudon, H. Chanal, and B. Blaysat Wohlhart’s Three-Loop Mechanism: An Overconstrained and Shaky Linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Andreas Müller Invariants for Multi-twists, Screw Systems and Serial Manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Peter Donelan Singularity-Free Extraction of a Dual Quaternion from Orthogonal Dual Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Daniel Condurache Singularities in the Image-Based Visual Servoing of Five Points . . . . . . 150 Abhilash Nayak and Sébastien Briot Real-Time Motion-Planning in Dynamic Environments via Enhanced Velocity Obstacle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Amir Shahidi, Katrin Peitsch, Stefan Klimmek, Mathias Hüsing, and Burkhard Corves Algebraic Analysis of 3-RRC Parallel Manipulators . . . . . . . . . . . . . . . 166 Abhilash Nayak, Martin Pfurner, Huiping Shen, and Manfred Husty xii Contents
- 17. Magneto-Inertial Data Sensory Fusion Based on Jacobian Weighted-Left-Pseudoinverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Janez Podobnik, Marko Munih, and Matjaž Mihelj Evaluating the Snappability of Bar-Joint Frameworks. . . . . . . . . . . . . . 182 G. Nawratil Method for Selecting Self-aligning Mechanisms Enumerated by Matroid Contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Rodrigo Luis Pereira Barreto, Andrea Piga Carboni, Roberto Simoni, and Daniel Martins A Real-Time Capable Forward Kinematics Algorithm for Cable-Driven Parallel Robots Considering Pulley Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 João Cavalcanti Santos and Marc Gouttefarde Stiffness Oriented Tension Distribution Algorithm for Cable-Driven Parallel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Etienne Picard, Stéphane Caro, Franck Plestan, and Fabien Claveau A Forward Kinematic Code for Cable-Driven Parallel Robots Considering Cable Sagging and Pulleys . . . . . . . . . . . . . . . . . . . . . . . . . 218 Marc Fabritius and Andreas Pott Singularity and Workspace Analysis of 3-SPS-U and 4-SPS-U Tensegrity Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 Swaminath Venkateswaran and Damien Chablat Degeneration to Infinity May Provide Information About Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Michel Coste and Nestor Djintelbe Kinematic Synthesis of a Modified Jansen Leg Mechanism . . . . . . . . . . 242 Kevin Chen and J. Michael McCarthy Exponential Displacement Coordinates by Means of the Adjoint Representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Bertold Bongardt and John J. Uicker A Comparative Study on 2-DOF Variable Stiffness Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Christoph Stoeffler, Shivesh Kumar, and Andreas Müller Kinematics and Orientation Workspace of a 3-DOF Parallel Robotic Wrist Actuated by Spherical Four-Bar Linkages . . . . . . . . . . . . . . . . . . 268 Guanglei Wu, Ning Zhang, Chuangchuang Cui, Huiping Shen, and Xuping Zhang Contents xiii
- 18. Analytical Determination of the Longest Cylinder Free of Gain-type Singularities Inside the Workspace of a 3-RPS Spatial Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Argaja Deepak Shende, Bibekananda Patra, Prem Kumar Prasad, and Sandipan Bandyopadhyay Clifford’s Identity and Generalized Cayley-Menger Determinants . . . . . 285 Federico Thomas and Josep M. Porta A New Approach for Continuous Wrapping of a Thick Strand on a Surface — The Planar Case with Constant Length and Free Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Katharina Müller and Andres Kecskemethy Higher Order Path Synthesis of Four-Bar Mechanisms Using Polynomial Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Aravind Baskar and Mark Plecnik Development of a Reconfigurable Four-Bar Mechanism for a Human Robot Collaborative Gripper . . . . . . . . . . . . . . . . . . . . . . 311 Keerthi Sagar, Vishal Ramadoss, Michal Jilich, Matteo Zoppi, Dimiter Zlatanov, and Alessandro Zanella Kinematic Analysis of a Planar Manipulator with Anti-parallelogram Joints and Offsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Philippe Wenger and Matthieu Furet On Singularity and Instability in a Planar Parallel Continuum Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Oscar Altuzarra and Francisco J. Campa Modeling and Simulation of Hybrid Soft Robots Using Finite Element Methods: Brief Overview and Benefits . . . . . . . . . . . . . . . . . . . 335 Stanislao Grazioso, Giuseppe Di Gironimo, Luciano Rosati, and Bruno Siciliano Exoskeleton Control Based on Network of Stable Heteroclinic Channels (SHC) Combined with Gaussian Mixture Models (GMM). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Tadej Petrič, Marko Jamšek, and Jan Babič Model Predictive Controller for a Planar Tensegrity Mechanism with Decoupled Position and Stiffness Control . . . . . . . . . . . . . . . . . . . . 349 JR Jurado Realpe, Salih Abdelaziz, and Philippe Poignet Author Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 xiv Contents
- 19. Advances in Robot Kinematics Facts and Thoughts Jadran Lenarčič(B) Jožef Stefan Institute, Ljubljana, Slovenia jadran.lenarcic@ijs.si Abstract. ARK, Advances in Robot Kinematics, has certainly left an important mark. Many top-notch robotics specialists regularly attended the symposia and contributed with their scientific papers and thoughtful discussions. The accompanying books in the ARK series, published by Kluwer Academic Publishers and later by Springer, gave the series an additional international impact. This is a brief look at how it all started and why it didn’t end until today. Keywords: Symposium on Advances in Robot Kinematics · ARK 1 Introduction The idea of organizing an international symposium specializing in robot kine- matics dates to early 1986. As a young researcher, I had just completed my Ph.D., and I sensed the need for such a conference, since there were quite a few robotic conferences and they were all general in scope. Robot kinematics was seen as a tool in other research topics, and above all, as a first stage in the mathematical modelling of robot dynamics. However, the more I delved into this area, which was not my specialty, as I am an electrical engineer by education, the more I realized that kinematics is an area that addresses specific problems that represent their own scientific field and, therefore, are not necessarily part of something else. As a beginner in the field, I pursued two things in particular: specialized literature on robot kinematics (the idea of launching a specialized journal was also an option) and a community of researchers with whom I could collabo- rate, exchange ideas and results. In 1986 I started to come up with the idea of organizing an international symposium in Ljubljana (back then in socialist Yugoslavia). The main concern for me was how to attract top experts; most of them were located in the United States. I didn’t worry about the funding for the symposium. This was of secondary importance to me. I based my hope on bringing this symposium to life on already-established friendships with important personalities in robotics, among which Bernard Roth was definitely the focal point. We had met two years earlier at the Romansy symposium in Udine, where I was immediately attracted by his kindness. At the c The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. Lenarčič and B. Siciliano (Eds.): ARK 2020, SPAR 15, pp. 1–6, 2021. https://doi.org/10.1007/978-3-030-50975-0_1
- 20. 2 J. Lenarčič same symposium, I also met some other pillars, like Oussama Khatib, Kenneth Waldron, Bruno Siciliano and David Orin. I announced the idea of organizing a specialized symposium on robot kine- matics in 1986 at the Romansy Symposium in Krakow. The interest seemed sufficiently strong, so I decided to press ahead. The title Advances in Robot Kinematics turned out very well. Some things in life do not take much time to think over. One year later, I started organizing the symposium by simply sending out invitation letters to the people I believed to be the most important in the field. It was the birth of Advances in Robot Kinematics, whose short title ARK was surprisingly well accepted by the community. At the symposium held in Bologna in 2018, I announced that the conference in Ljubljana in 2020 would be my last in the capacity of chairman. After twenty- two years, the time has come to step down. This is not a painful or sad moment, but one full of joy, as this scientific event has certainly left an unforgettable footprint on robotics and beyond. I would like to thank everyone who contributed his or her work and enthusiasm. 2 Symposia and Related Books Ljubljana (Slovenia, Yugoslavia) 1988 – Jadran Lenarčič Book: Advances in Robot Kinematics, Cankarjev dom J. Stefan Institute, 22 Contributions Among those invited to the first ARK symposium were researchers who per- sonally fascinated me with their scientific publications. I did not look at where they came from and whether I had met them before. Many of them I had not, but later it turned out that I was extremely lucky in my choices. I saw that a com- munity had begun to build, which was decisive for the future work and success of this symposium. Their positive responses to my invitations were unexpected for me, beyond what I had hoped. Out of the twenty invited, only Ken Waldron apologized, and he later joined and became one of the pillars of the conference. I had to send the invitations by regular mail and waited for at least a month or more for people to respond. There were no emails at that time. The most prominent names among the invitees were Jorge Angeles, Vladimir Lumel- sky, Alberto Rovetta, Roy Featherstone, Michael McCarthy, Vincenzo Parenti- Castelli, Michael Stanisic, and substitutes were sent by Richard Paul, Kazuo Tanie, Philippe Coiffet and Andrew Goldenberg. Adding them to the aforemen- tioned Bernard Roth, Oussama Khatib, Bruno Siciliano, Kenneth Waldron and David Orin, whose support had been the backbone of the whole project, I was fortunate enough to set up a ground-shaking group of international scientists whose research has mostly been related to robot kinematics. The first symposium took place in Ljubljana in September, immediately after the Romansy symposium, held in Udine, Italy. Thus, some participants made use of their presence just over the border. Most importantly, I felt that I had to take care of everyone personally, which was seen as an act of hospitality and respect. With the help of sponsors, I was able to organize lunches and dinners
- 21. Advances in Robot Kinematics Facts and Thoughts 3 in the cost of the conference. It was then that I came to the realization that the social events were at least as important as the technical sessions. It is crucial for the success of a conference that people come together and feel like they are important team members. I organized the oral presentations at the symposium so that everyone had 40 min. The contributions were published in the Proceedings, which was printed in advance and available at the conference. I edited the proceedings myself and had it printed by Cankarjev Dom, the agency which was also the technical orga- nizer of the symposium and all the other events that took place in Slovenia in the following decades. Most important, however, was the unanimous conclusion that these symposia should be continued every other year and that the sym- posium should be organized once in Slovenia and once somewhere else in the Alps-Adria region. The aim was to emphasize the identity of the symposium by its geographical characteristic. There was no going back after that decision. Linz (Austria) 1990 – Sabine Stifter and Jadran Lenarčič Book: Advances in Robot Kinematics With Enmphasis on Symbolic Computation, Springer (1991), 53 Contributions The second symposium moved to Linz by accident. It started with a visit to Ljubljana from a political delegation of Upper Austria, who were looking for options to encourage technical and scientific cooperation between the regions. Shortly thereafter, I received a mail from Linz from Sabine Stifter proposing the co-organization of the symposium with the University of Johannes Keppler in Linz. I met her a few months later at the RISC Institute. We invited two commu- nities to the ARK symposium: a community of kinematicians and a community of mathematicians specialising in symbolic computation. I was convinced that significant synergies could be achieved. The symposium was held in Linz in September. Conference contributions were collected based on extended abstracts and far more submissions came than we expected. We accepted about 130 contributions. Although the conference set important new standards for the future, its focus was, in my opinion, too diffuse. The Proceedings of the abstracts were published at the conference. Based on peer reviews of these, we selected full articles that were published one year later (1991) in an edited book format by Springer. Although this symposium was quite different from the first and subsequent ones, the group of kinematicians who attended the conference, became the core of ARK, thus securing its future and long-term mission. Ferrara (Italy) 1992 – Vincenzo Parenti-Castelli and Jadran Lenarčič Book: 3rd International Workshop on Advances in Robot Kinematics, Antenna Verde University of Ferrara, 44 Contributions In 1992, ARK should have been organised in Slovenia. However, the political situation after Slovenia’s independence process was not the most favourable for the organization of international events. Although there were no problems and matters were cleared up in Slovenia much earlier, I decided to make an exception and organize ARK somewhere else. I was worried about how many participants would attend an event in Slovenia. Vincenzo Parenti-Castelli solved the problem
- 22. 4 J. Lenarčič by proposing to organise ARK in Ferrara, Italy. A more ideal solution could not have been found. Ferrara in many ways laid the foundations for today’s ARK. Among other things, we first agreed with IFToMM to take over the sponsorship that ARK was awarded for all subsequent years. The link with the Linz mathematicians was abandoned, and some important new contributors from the field of robot kine- matics appeared at the symposium, such as Manfred Hiller, Andres Kecskemethy, Josepf Duffy, Carlo Galletti and Jean-Pierre Merlet. The Proceedings were pub- lished and printed by the organizer in the form of full papers. New names were added to the Scientific Committee. The number of oral presentations was signif- icantly fewer than in Linz. This proved to be appropriate, and in later years this number never exceeded 58. By doing so, the conference retained its “boutique” form. Also important was the decision to reposition future symposia to the end of June or the beginning of July. Ljubljana (Slovenia) 1994 – Jadran Lenarčič and Bahram Ravani Book: Advances in Robot Kinematics and Computational Geometry, Kluwer Academic Publishers, 51 Contributions In 1994 ARK took place again in Ljubljana. In preparation for the event, my goal was how to ensure high quality and better public visibility of the pro- ceedings. I decided to try publishing an edited book with the Dutch publishing house Kluwer Academic Publisher, which impressed me with its professionalism and expeditiousness. That’s when I met (via mail) Nathalie Jacobs (Kluwer), first assistant director, later director at Kluwer Academic Publishers, Dordrecht. Working with her contributed to the creation of the impressive series of books that we know today. The work had to be conducted as efficiently as possible, as it was crucial for me to have the book printed and handed over to the conference participants at the event. The conference was held in Ljubljana at the Club of Cankarjev dom. The organizers of the conference strived for an excellent social program in addition to an excellent scientific program. The motivation was to establish a standard for ARKs in the coming years. I trust that returning to Ljubljana was an important contribution to the identity of these symposia. Piran/Portorož (Slovenia) 1996 – Jadran Lenarčič and Vincenzo Parenti-Castelli Book: Recent Advances in Robot Kinematics, Kluwer Academic Pub- lishers, 45 Contributions We agreed to organize the ARK again in Slovenia 1996. The reason was to correct the originally planned order of organizers, which was interrupted due to the political situation in Slovenia a few years before. Piran was, after Ljubljana, the second Slovenian location and turned out to be ideal for the style of the ARK symposia. Certainly, the conference was at an important international level, but the social events were especially well organized by Cankarjev dom. I remember this event as perhaps the liveliest and with a full and friendly atmosphere, sug- gesting that a good conference must first create a community of people who want to participate and want to share their ideas. I do not know if the environment
- 23. Advances in Robot Kinematics Facts and Thoughts 5 contributed to this, but it seems important to me that such an atmosphere was created spontaneously. During this symposium, I decided that all future books will have a different title, in combination with Advances in Robot Kinematics. Strobl/Salzburg (Austria) 1998 – Jadran Lenarčič and Manfred L. Husty Book: Advances in Robot Kinematics: Analysis and Control, Kluwer Academic Publishers, 58 Contributions Piran/Portorož (Slovenia) 2000 – Jadran Lenarčič and Michael M. Stanišić Book: Advances in Robot Kinematics, Kluwer Academic Publishers, 45 Contributions Caldes de Malavella (Spain) 2002 – Jadran Lenarčič and Federico Thomas Book: Advances in Robot kinematics: Theory and Applications, Kluwer Academic publishers, 51 Contributions Sestri Levante (Italy) 2004 – Jadran Lenarčič and Carlo Galletti Book: On Advances in Robot Kinematics, Kluwer Academic publish- ers, 51 Contributions Ljubljana (Slovenia) 2006 – Jadran Lenarčič and Bernard Roth Book: Advances in Robot Kinematics: Mechanisms in Motion, Springer, 53 Contributions Batz-sur-Mer (France) 2008 – Jadran Lenarčič and Philippe Wenger Book: Advances in Robot Kinematics: Analysis and Design, Springer, 48 Contributions This time the conference did not take place in a Mediterranean or Alpine region, nevertheless it impressed us like all its predecessors. Piran/Portorož 2010 (Slovenia) – Jadran Lenarčič and Michael M. Stanišić Book: Advances in Robot Kinematics: Motion in Man and Machine, Springer, 58 Contributions For the first time, articles were not included in sessions based on their topics. I distributed them using a random generator and divided the chapters in the book into neutral Part 1, Part 2, etc. The sessions at the conference were thus of greater interest. Innsbruck 2012 (Austria) – Jadran Lenarčič and Manfred Husty Book: Latest Advances in Robot Kinematics, Springer, 56 Contribu- tions Ljubljana 2014 (Slovenia) – Jadran Lenarčič and Oussama Khatib Book: Advances in Robot Kinematics, Springer, 56 Contributions Grasse 2016 (France) – Jadran Lenarčič and Jean-Pierre Merlet Book: Advances in Robot Kinematics 2016, Springer – SPAAR (2018), 46 Contributions
- 24. 6 J. Lenarčič The proceedings were first published in an electronic version by Inria Sophia Antipolis by the time of the conference in 2016. Later, the ARK Scientific Com- mittee decided to print the book as part of the Springer SPAAR series. Bologna 2018 (Italy) – Jadran Lenarčič and Vincenzo Parenti-Castelli Advances in Robot Kinematics 2018, Springer – SPAAR (2018), 53 Contributions In Bologna, we celebrated the first thirty years of the conference that brought us together and has forged many new friendships. 3 Conclusion I decided that my last conference would take place in Ljubljana in 2020, the city where it all began. I can certainly say that ARK has had a profound impact on my scientific and personal life. Future conferences are sure to follow, and a new generation of researchers is here to lead this initiative. The ARK Scientific Committee has already decided that the conference in 2022 will be held in Spain and will be organized by Oscar Altuzarra. Robot kinematics remains an unlimited source of new scientific and practi- cal challenges. There is scientific material for many more years of research. New technologies, such as high-performance computing and artificial intelligence, will further strengthen this research field and allow solving so far unsolvable prob- lems. In the end, I want to thank all the ARK friends who have contributed with their professionalism and personal engagement.
- 25. Inverse Kinematics Using a Converging Paths Algorithm Omid Heidari1(B) and Alba Perez Gracia1,2(B) 1 Department of Mechanical Engineering, Idaho State University, Pocatello, USA {heidomid,perealba}@isu.edu 2 Remy Robotics, Barcelona, Spain Abstract. This article presents a numerical method for the inverse kine- matics of serial chains by utilizing dual quaternion formulation of the robot kinematics within a converging paths algorithm. The method is inspired by iterative techniques such as FABRIK, however adding infor- mation on the kinematics of the chain to be solved. The method has been tested with 2R, planar 4R and spatial 4R robots. Future work includes optimizing the method to compare with other fast numeric algorithms for inverse kinematics. Keywords: Inverse kinematics · Serial manipulators · FABRIK · Iterative methods 1 Introduction Inverse kinematics (IK) methods are widely used in robotics and computer graphics. Analytic and numerical techniques have been develop for solving the IK problem. Closed-form or analytic methods were among the first to be devel- oped and are successful for simple chains; however, they are developed in a case-by-case strategy and they are hard to generalize. See [7] for a review. Some of the analytic methods use the description of the geometric constraints of the chain. Others use the matrix formulation to the forward kinematics equations of the chain to solve the inverse problem when this is equated to the desired position [4]. Analytic methods have the advantage of avoiding iterative solutions that in principle are more time consuming; however many times they reduce to finding the roots of a univariate polynomial [5], which carries its own numerical problems. New algorithms such as IKFast [3] allow a more systematic approach combining analytic and numerical solutions. A good review for numerical methods, especially those applied to computer graphics, can be found in [2]. Numerical methods are grouped into those based on the Jacobian, Newton-based methods and heuristic methods. Jacobian-based methods [8] are the most popular and a variety of implementations can be found, see for instance [6]. Heuristic methods are developed as a computationally lean alternative to other numerical methods. Most of them consist of a simple app- roach that yields many fast iterations. Among them we highlight Forward And c The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 J. Lenarčič and B. Siciliano (Eds.): ARK 2020, SPAR 15, pp. 7–14, 2021. https://doi.org/10.1007/978-3-030-50975-0_2
- 26. 8 O. Heidari and A. P. Gracia Backward Reaching Inverse Kinematics (FABRIK) [1]. This method is an itera- tive approach where it tries to find the position of each joint by locating a point on line. It is not costly in terms of computation neither it produces complex solutions. However, the algorithm works best with spherical joints, mostly used for IK in computer animations and gaming. Desired properties of IK methods are speed, numerical stability and conver- gence. In addition, being applicable to a broad number of kinematic chains is desirable too. In this work we present a IK algorithm that combines some of the advantages of the numerical methods with the insight of the analytic ones. The method is tested for some simple open-chain robots consisting of revolute joints. The core concept is to break the serial chain and create two legs for a given pose. Then the end-effectors (ee) of these legs move towards each other smoothly till they match and complete the chain. Moving towards a better solution at each step is insured by incorporating pseudo-inverse in Cartesian space. At each step, the solution in the trigonometric joint space is projected to the surface condition for the joint variables to make sure that the constraints are satisfied. 2 Dual Quaternion Inverse Kinematics In this section we describe a method to state the inverse kinematics for serial chains with revolute joints. Let us consider a kinematic serial chain with n degrees of freedom; let the joint space of the chain be defined by the vector θ = {θ1, . . . , θn} and the joint axes be defined by the Plücker coordinates of the lines Si, i = 1, . . . n. 2.1 Product of Exponentials in Dual Quaternion Form In general, for an arbitrary k-degree of freedom robot, kR, given by k joint axes S1 . . . Sk with rotation angles θ1 . . . θk, the forward kinematics of relative displacements is given by the product of exponentials, D = eS1 θ1 2 eS2 θ2 2 . . . eSk θk 2 (1) This can be done using any suitable algebra. In this case, we use Clifford algebra of dual quaternions which can be seen as 2k products of cosines and sines of the angles. Select an order, for instance (c1c2 . . . ck, s1c2 . . . ck, . . . , s1s2 . . . sk). The forward kinematics can be written as a linear combination of the joint axes and Clifford products of joint axes by these coefficients, I S1 S2 . . . S1S2 . . . Sk ⎧ ⎪ ⎨ ⎪ ⎩ cos θ1 2 cos θ2 2 . . . cosθk 2 . . . sin θ1 2 sin θ2 2 . . . sin θk 2 ⎫ ⎪ ⎬ ⎪ ⎭ = P, (2)
- 27. Inverse Kinematics Using a Converging Paths Algorithm 9 When using this expression to solve the inverse kinematics problem, consider a desired pose P. In general, for a robot with k revolute joints and a desired position P we will always have 2k unknowns. The forward kinematics yields 8 linear equations (6 independent). In addition to these, the trigonometric rela- tions among the unknowns are to be considered which are bilinear and spherical relations. 2.2 Inverse Kinematics for the 2R Robot The relative forward kinematics D2R = eS1 θ1 2 eS2 θ2 2 (3) is equated to the target position P. It can be collected in the products of sines and cosines. Denote ci = cos θi 2 and si = sin θi 2 , I S1 S2 S1S2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ c1c2 s1c2 c1s2 s1s2 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ = P, (4) Denote the 22 = 4 unknowns x1 = c1c2, x2 = s1c2, x3 = c1s2, and x4 = s1s2. To the 6 linear independent equations we need to add the bilinear condition and the spherical condition that result from the trigonometric relations, so the final system of equations is Ix1 + S1x2 + S2x3 + S1S2x4 = P, x1x4 = x2x3, x2 1 + x2 2 + x2 3 + x2 4 = 1 (5) This is a total of 8 equations, and the system is over-determined by 4 equa- tions as expected. Considering the rotations only, there are 4 unknowns and 5 equations, overde- termined by 1 as expected. The rotation linear system can be solved for the unknowns, which will comply with the quadratic conditions only if the rotation belongs to the workspace. The system consists of 6 linear equations and two quadratic equations. These quadratic equations can be combined to create the surface where the angles must lie, x2 1 + x2 2 + x2 3 + (x2x3/x1)2 = 1, (6) which is shown in Fig. 1 below. The implicit surface admits a local parameteri- zation at the regular points.
- 28. 10 O. Heidari and A. P. Gracia Fig. 1. Surface condition for the joint variables of the RR chain. 3 Converging Paths Algorithm The proposed algorithm relies on breaking any serial chain into two lower-degree ones which is a well-known approach utilized in different kinematics methods. The end-effectors of these two smaller chains need to get close to each other at each step along an iterative process and reach perfectly at the final step of iteration. In this work, apart chains are called left and right leg. Left is the one having the base of the original robot and right is the one having the ee of the original manipulator as its stationary base at the given pose for IK input. Figure 2 shows a 4R serial chain which is divided into two RR chains. To explain the procedure of the algorithm and different parts of it, this example is illustrated and followed in the rest of this paper. Fig. 2. 4R manipulator as two RR chains
- 29. Inverse Kinematics Using a Converging Paths Algorithm 11 The procedure starts with setting the left leg to a random unit dual quater- nion that satisfies Study’s quadric. Then, this pose is used to calculate the right leg’s configuration that can be the closest. To this end, a pseudo-inverse matrix is computed which may result in a solution that does not satisfy the constraints in Eq. 5. To solve this problem, the algorithm incorporate a method of projection where solutions out of workspace are projected based on the shortest distance they can have to the constraint surface. The projected vector in the joint space is, then, used to compute the Cartesian pose of the right leg and its transforma- tion is set equal to the left leg equations to see if it is in the workspace of the left leg. Equation 1, for a 4R serial chain, can be converted to the following: eS1 θ1 2 eS2 θ2 2 = De−S4 θ1 2 e−S3 θ3 2 (7) Separating the left and right transformations of Eq. 7 by setting them to two different dual quaternion, L = R, results in: eS1 θ1 2 eS2 θ2 2 = L, eS3 θ3 2 eS4 θ4 2 = T (8) where hat symbol is the dual quaternion conjugate operator and T = R̂D. As it was stated in the previous sections, Eq. 8 can be written as a linear combination of Clifford products of joint axes: [L]v = l, [R]u = t (9) where l and t are 8-element vectors representing dual quaternions on each side, L and T respectively. [L] and [R] are corresponding matrices containing joint axes and their Clifford products. It is noteworthy that elements of these matrices are constant depending on a reference pose and do not change by changing the configuration of the robot. Moreover, v and u are 4-element vectors including the products of sines and cosines of angels expressed in Eq. 4. 3.1 Algorithm Description The back and forth procedure explained in the previous section between left and right leg is shown more in details by Algorithm 1. This iterative process repeats till the ee of one leg matches the ee of the other leg in its workspace. To make sure that there is a solution to the inverse kinematics problem, the forward kine- matics is solved so that we have a D that is in the workspace.
- 30. 12 O. Heidari and A. P. Gracia Algorithm 1: Converging Paths Inverse Kinematics Result: v and u that put the left and right leg in a configuration at which their end-effectors reach each other. initialization: L = random unit dual quaternion; δ = 0.0001; while constraints are not satisfied do l = Vector(L); v = [L]−1 l; vprojected = Project(v); l = [L]vprojected; L = DualQuaternion(l); R = L; r = Vector(R̂D); u = [R]−1 r; uprojected = Project(u); t = [R]uprojected; T = DualQuaternion(t); R = DT̂; L = R; if CheckConstraints(v) δ CheckConstraints(u) δ then Exit while loop; end end There are some functions used in this algorithm that affect its performance and speed. Vector() converts a dual quaternion to an 8-element vector and DualQuaternion() does the opposite. CheckConstraint() verifies if the constraints in Eq. 5 are satisfied, returning a value for each constraint that can be used to evaluate the convergence. At last, Project() is the function that finds a solution with the least distance between the given point and the constraint surface. There are different approaches to accomplish this. One method uses vector calculus on the parameterization to compute the normal vector to the surface and then find the line normal to the surface and passing through the given point. This is done by finding a line whose moment is equal to the moment of the point in the direction of the line. Another method that is used in the algorithm is through Lagrange Mul- tipliers. The objective function to minimize is the squared Euclidean distance between the given point and points lying on the constraint surface. The proposed algorithm is in the early stage and the authors are exploring dif- ferent approaches to do the projection. However, the Lagrangian method seems to have a faster response at the current stage. Numerical results suggest con- vergence by iteratively approaching each leg to the other using Moore–Penrose pseudo inverse method, however convergence needs to be formally proved. At each step the pose of one leg’s end effector is set equal to the other one to see if it is in its workspace or not. If not, the corresponding vector in joint space gets projected to the constraint surface to make sure that the ee is in the workspace.
- 31. Inverse Kinematics Using a Converging Paths Algorithm 13 3.2 Numerical Example This section is dedicated to an example of solving the IK of a 4R serial manipu- lator by the proposed method. The first step is to compute matrices [L] and [R] for an arbitrary located joint axes. L = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 −0.639867 0.0305296 0.732783 0 0.512895 −0.510064 0.567498 0 0.572284 0.859595 0.310715 1 0 0 −0.210788 0 −0.13248 1.10913 −0.322849 0 −0.912657 −0.523315 0.545166 0 0.66982 −0.349916 −0.138578 0 0 0 0.14111 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ R = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 −0.0689129 0.84182 0.263595 0 −0.994039 0.491349 −0.0865131 0 −0.0844797 −0.223418 0.802942 1 0 0. 0.527558 0 −1.12461 0.562694 −0.792586 0 0.040865 −0.732085 0.10385 0 0.43654 0.510156 0.0228117 0 0 0 0.378327 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (10) After 9 iterations, the algorithm finds a solution with an error of 0.001. Figure 3 shows the solution and the steps of convergence as well as the constraint surface. The convergence of the right and left leg to the solution in joint space in terms of x1,x2 and x3. Green: uprojected and vprojected, Blue: u and v, Red: expected solution. The convergence of the left (black) and right (green) leg to the solution in Cartesian space. Fig. 3. Convergence in joint and Cartesian spaces. The computation time is relatively high in its current version, as the steps have not been optimized for spped yet. The interest of the method is going to depend on finding efficient processes for each of the steps of the algorithm.
- 32. 14 O. Heidari and A. P. Gracia 4 Conclusion This work presents an inverse kinematics method based on dividing the chain and iteratively projecting the closest point on the workspace. The equations are derived from the forward kinematics and hence it can be applied systematically to serial chains. The method is tested on serial chains with 2, 3 and 4 degrees of freedom and revolute joints. In these examples, the algorithm exhibits fast convergence. Future work will include optimizing the calculations, applying the method to increasingly complex chains and comparing it to the current state of the art as well as examining its behavior at non-regular points. References 1. Aristidou, A., Lasenby, J.: Fabrik: a fast, iterative solver for the inverse kinemat- ics problem. Graph. Models 73(5), 243–260 (2011). http://dblp.uni-trier.de/db/ journals/cvgip/cvgip73.html 2. Aristidou, A., Lasenby, J., Chrysanthou, J., Shamir, A.: Inverse kinematics tech- niques in computer graphics: a survey. Comput. Graph. Forum 37(6), 35–58 (2017) 3. Diankov, R.: Automated construction of robotic manipulation programs. Ph.D. the- sis, Carnegie Mellon University (2010) 4. Manocha, D., Canny, J.F.: Efficient inverse kinematics for general 6R manipulators. IEEE Trans. Robot. Autom. 10(5), 648–657 (1994). https://doi.org/10.1109/70. 326569 5. Raghavan, M., Roth, B.: Inverse kinematics of the general 6R manipulator and related linkages. J. Mech. Des. 115(3), 502–508 (1993). https://doi.org/10.1115/1. 2919218 6. Siciliano, B., Khatib, O.: Springer Handbook of Robotics. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-30301-5 7. Tsai, L.W.: Robot Analysis and Design: The Mechanics of Serial and Parallel Manip- ulators, 1st edn. Wiley, Hoboken (1999) 8. Wampler, C.: Manipulator inverse kinematic solutions based on vector formulations and damped least-squares methods. Proc. IEEE Trans. Syst. Man Cybern. 16(1), 93–101 (1986)
- 33. Another Random Document on Scribd Without Any Related Topics
- 34. impossibility. But Sweden was not prepared to make war on two of the great powers of Europe, especially as no other power was willing to join in an alliance in behalf of Denmark. The change must be made; and was effected, principally because of the persuasive arguments and resolute demeanor of Gripenstedt. King Charles resolved to take the painful measures of a retreat. The standpoint of his government he gave to the Riksdag in the following words: “It cannot be expected from us that we should place our sword on the scale of justice without considering if the object can be attained with the resources at our command.” It was a supreme sacrifice that Charles XV. made when, for the safety of his countries, he was forced to draw back the hand of support and comradeship which he had offered a brother in distress. The noble-hearted king, in one of his poems, has given a touching expression of the sorrow he felt in being unable to assist Denmark in her hour of peril. King Charles might, with proper resources at his command, have proved a formidable enemy. He had given evidence of possessing all the qualities requisite for the make-up of a great general, without doubt an inheritance from his two grandfathers, Prince Bernadotte and Eugene Beauharnais. A few hundred Swedish and Norwegian volunteers took an honorable part in the Danish war, which was the only practical result of the Scandinavian policy. The Swedish press was violent in its attacks upon the government for its change of policy. In March, 1864, the mob of Stockholm assailed the residences of Manderstrœm, Gripenstedt and other cabinet members, breaking the windows with stones. Poor Denmark was left alone. Napoleon III. made the mistake of not attempting to defeat Prussia before she had reached her climax of strength. He was tied up with his Mexican adventure and unwilling to help Denmark. Charles XV. could not endure to see Denmark thus deserted. Privately he offered Christian IX. an alliance which stipulated that the three Scandinavian kingdoms should be joined into a union with one common foreign policy and common defence. Charles was also willing to make the succession one, if necessary. This alliance was to embrace only such parts of Denmark which were
- 35. not to enter the German union. Sweden-Norway would do their utmost to prohibit a separation between Denmark and Schleswig. Denmark refused to accept this offer. Her leading statesman, Monrad, held stubbornly to the idea of an undivided Danish monarchy. For this reason, Denmark was for a second time abandoned to fight out alone her uneven battle. It ended in the loss of Holstein, Lauenburg and the greater part of Schleswig, through the treaty of Vienna, October 30, 1864. In Denmark a hard feeling against the Swedes and Norwegians sprang up as a consequence of the disastrous war fought without allies; and the Scandinavian policy and enthusiasm had received a blow from which they have never fully recovered. Charles XV. did all in his power to revive them. He had the pleasure of uniting the efforts of Sweden, Norway and Denmark in a peaceful work of great significance, the first Scandinavian Exposition of Industry and Art, which was opened at Stockholm in June, 1866. The consequence was a perfect Norwegian conquest of Sweden, in a cultured sense. The painters Tidemand and Gude captured the prizes. The composers Kierulf and Nordraak took the lead in song and music. Ibsen and Bjornson became the craze in literature. The literary contact with Norway was begun in 1861, when Lorenz Dietriechson was appointed a docent at the University of Upsala, and for the first time made the contemporary Norwegian and Danish poets acquainted in Sweden. What Sweden received from Norway was a quaint, late-born Romanticism of a strong national flavor. When this Romanticism was changed into stern Realism its influence upon Swedish culture, especially her literature, was only increased, Swedish literature receiving strong realistic impulses from the neighboring Scandinavian countries. The Norwegian influence ceased, when the Swedes at last became aware that there was in it a deeply pessimistic trait, akin to the stern Norwegian and Scotch Christianity, which is incompatible with the Swedish national temperament, slightly inclined to melancholy, but of a robust and irrepressible desire to live and enjoy. Charles XV. followed up his practical Scandinavian policy by marrying his only daughter Louise to Crown Prince Frederic of Denmark. King
- 36. Charles was as unsuccessful in his noble efforts to unite more closely his two kingdoms as in his foreign policy. The king allowed some time to pass in order to let the ill-feeling, caused by the conflict of 1859 and 1860, die out. In February, 1865, he considered that the moment had arrived to institute the review of the Act of Union. He appointed a committee of Swedes and Norwegians to prepare the proposition of a new Act of Union, on the basis of perfect equality and right to decide separately all matters, except such pertaining to the Union. The committee performed the work, but their proposition was defeated at the Norwegian Storthing of 1871, at the instigation of John Sverdrup and K. Motzfeldt. The Swedish Riksdag for this reason also failed to accept it. At the close of the Riksdag, King Charles made the following utterance in regard to the defeated proposition: “What has now failed to attain success shall perhaps win out without difficulty when the two nations once have learned to place confidence in each other, as the result of a more intimate intercourse.” He saw with great satisfaction the completion of a railway which forever unites the Swedish and the Norwegian capitals with ties of steel. The administration of Charles XV. persevered in its liberal policy concerning questions of economy and jurisprudence. This was particularly noticeable in commercial matters. The idea of free trade had won ascendency in Europe. Napoleon III. had entered a treaty of commerce with England, in strict opposition to the protective system. Other nations were one by one admitted into the free-trade system by means of new treaties. Sweden made a treaty of commerce and navigation in 1865. This step was severely criticised by the Riksdag of 1865-1866, both from a constitutional and financial point of view. Gripenstedt was accused of leading the way over demolished industries, but he defended his position with great eloquence. The treaty was ratified in spite of the powerful opposition in the Riksdag. The press condemned both the treaty and the government in the most violent language.
- 37. The first Riksdag of the new parliamentary system met January 19, 1867. The “Landstings” had sent to the First Chamber the most prominent men of the country. It was a truly representative gathering, a house of peers elected by the people. Lagerbielke, the landtmarshal of the preceding Riksdag, was appointed speaker. The Second Chamber counted a larger number of peasants as representatives than of any other class. Anton Nicolaus Sundberg, then bishop of Carlstad, now archbishop of Sweden, was made speaker of the Second Chamber. The power of the peasants made itself felt at once. There was formed a strong and influential party, the landtmanna, or countrymen’s party, consisting of small landowners. The peasants constituted the majority, but the party also counted many titled and untitled country gentlemen in interests united with them. The founder of the party was Count Arvid Rutger Posse, later minister of state. Emil Key and the peasants Charles Ifvarsson and Liss Olof Larsson were among the leaders of the party. The policy of the Landtmanna party demanded simplification of the administration, economy in the matter of appropriations and a solution of the questions of the defence and taxation in harmony with the interests of the owners of the soil. The party followed up its policy with stern consistency from Riksdag to Riksdag, until in perfect control of the whole government. The opposition consisted of “the Intelligence” or intellectual party, which, without a solid constitution or a fixed policy, has in vain fought the spreading influence and power of the Landtmanna party. The latter has gone almost too far in its endeavors for economical reform, but has also given evidence of appreciation of the material needs of a cultural development, appropriating large sums for the benefit of science and education. The army question was the most important issue of Swedish politics. The events of 1866 had made it evident that a strengthening of the defences was necessary. King Charles was anxious to have the question solved in a satisfactory manner, finding therein the only reliable safeguard for the future independence of Sweden. It was apparent that any attempts to settle the question in accordance with
- 38. the system adopted by Charles XI. would be devoid of result. It was based upon direct taxation of the soil and must be opposed by the strong majority of small landowners of the Landtmanna party. A compromise policy was for this reason begun in 1867, the question of an abolition of the land tax being connected with the army question, although the two ought to have had no connection. The question was started with promises of a reduction or exemption of the duties of the old army system as compensation for the acceptance of a new arrangement for the country’s defence. The government made an army proposition to the Riksdag of 1869, promising several reductions to the landowners who furnished soldiers according to the old system (indelningsverket). The proposition was prepared by a committee, of which the new minister of war, Gustavus Rudolph Abelin, was the chairman. It was based upon the preservation of the old system for the furnishing of the body force of officers and men. The larger force was to be provided for through militia. The militia was to be drilled in the neighborhood of their various homes during sixty days of the year. The proposition was not accepted. The militia compulsory service, as the duty of every citizen for the defence of his country, had nothing to do with the regular army as provided by the stipulations of the old system. But the majority of the Second Chamber confused the two and refused to allow the establishment of the former on a wider basis, because the offers made to reduce the burdens of the old system did not appear to them liberal enough. In 1871 another proposition was made by Abelin to the Riksdag. It was similar to the first one, and its cause was eloquently pleaded by Abelin, Axel Gustavus Adlercreutz, minister of justice, Peter Axel Bergstrœm, minister of civil service, and Gunnar Vennerberg, minister of ecclesiastics. They warned against the mistake of attaching impossible conditions to the acceptance of the proposition. The proposition for an extended militia service was accepted by both Chambers. But when the Second Chamber raised, as a condition for its acceptance, the suspension, for fifteen years, of the old system which provided for the regular army, the government found it impossible to grant this, and the proposition was dropped.
- 39. King Charles was grieved and vexed with the fate of the army bills. The Franco Prussian war made it, in his opinion, of added importance to Sweden to have her defences remodelled. He called an extraordinary session of the Riksdag, in the autumn of 1871, when Abelin brought out a third proposition. It was chiefly of the same contents as the preceding ones. But a remarkable change in the public opinion had now taken place, as to the advisability of retaining the old system. Men who looked upon the question more from a military than an economic point of view entertained doubts as to the practical value of the old regular army as the body force of a compulsory militia. Military officers commenced to attack the old system as the basis of a new army. The Landtmanna party persevered in the request for an abolition of the old system, and this killed the army bill at the extraordinary Riksdag. Together with the request for an abolition of the old army system, demands for redemption from other burdens placed upon the owners of the soil made themselves heard. The land-tax was the principal one of these burdens and caused as much difference of opinion as the army system. The Landtmanna party considered the land-tax to be of the same nature originally as other taxes, which ought to be more evenly distributed and shared by all classes in the same proportion. The Intelligence party was of the opinion that the land-tax in the course of time had come to be rents or mortgages which always were taken into consideration at the exchange of property, as reducing the stock value of the property in question. To free a present generation from the payment of land-tax, was in the eyes of the opposition, an injustice to the other classes whose taxes thereby were to be increased. The Landtmanna party had, in 1869, commenced an agitation for the reduction of the land-tax for shorter periods and on a small scale at first, but with increasing demands at every new Riksdag. The government, whose members had been the champions of parliamentary reform, was soon disregarded by the triumphant party, while its old opponents never forgot it. The earlier advisers of
- 40. the king retired one by one when they saw their influence in the Riksdag vanish. King Charles himself took the defeat of the army bills deep at heart. His health commenced to fail in 1871, and when his faithful consort died, in the same year, having exposed her own health in her attempts to improve the condition of the king, the latter grew worse. After a trip abroad for his health, King Charles XV. died at Malmœ, September 18, 1872, deeply mourned by the two nations. In the following year his youngest brother Nicolaus August, duke of Dalecarlia, died, leaving only two of the children of Oscar I., Oscar Frederic, duke of East Gothland, and Princess Eugenie. The history of Charles XV. carries the principal traits of his character. His sweeping reforms in social, political and economical matters, and his great plans for the future, even if sometimes immature, or high- strung, were always characterized by loftiness of purpose. A typical Swede both in his merits and his faults, this was the secret of the immense popularity of King Charles, which always followed him, although he never sought it. The philosopher Christian Jacob Bostrœm is the most popular of Swedish thinkers and the first who founded a national system and school of philosophy, idealistic and rational, and in strict opposition to the system of Hegel. Bostrœm was born in Pitea, in 1797, was the teacher of the sons of Oscar I., and succeeded the able philosopher Samuel Grubbe, a talented follower of Hœijer, as professor of philosophy at the University of Upsala. Bostrœm was a highly fascinating and suggestive teacher, while he neglected his literary production, which is neither exhaustive nor quite representative of his philosophy. He exerted a considerable influence by his outline of a philosophical state, which pleased the conservatives, while a much more widespread and lasting impression was produced by his criticism of the doctrines of a hell and a devil. A whole literature sprang into life, discussing vehemently the existence or non- existence of the fiend. To this literature and the works and writings of Bostrœm is to be credited the spirit of religious tolerance which characterized life and literature during the reign of Charles XV. It fostered in the cultured few a leaning toward Unitarianism or
- 41. Theosophy, while it gave rise to a shallow materialism and religious indifference in the less cultured classes and individuals. The artistic, literary and musical life bore a decided resemblance to the intellectually interested but dilettantic king. Charles XV. was surrounded by a great number of painters who, although possessing a good deal of talent, succeeded only in the smaller field of genre painting. Remarkable exceptions are J. F. Hœckert, Marcus Larsson and C. H. L. D’Uncker, who possessed sterling genius and acquired great fame. Several promising painters, like George von Rosen, developed later the full scope of their power. The sculptor J. P. Molin was highly talented, a worthy follower of B. E. Fogelberg, who had enriched Swedish art with a number of highly important sculptures. In the world of letters, the spirit of dilettantism was more strongly felt than in art, Swedish literature, after its several glorious epochs, experiencing one of its most stagnant periods. A veritable giant among pygmies was Victor Rydberg, whose remarkable novel, “The Last Athenian,” appeared in 1859, but whose principal productivity as a poet and scientist belongs to a later period. So do, to a great extent, the best works of the poets Eduard Beckstrœm, also an able dramatist, and Count Carl Snoilsky. Zacharias Topelius, the Walter Scott and Hans Christian Andersen of Finland, must be mentioned here. Writing in the Swedish language, and for his principal work using subjects of Swedish history, he was as highly beloved in Sweden as in Finland. His excellent series of historical novels, called “The Surgeon’s Stories,” have been translated into several languages. His juvenile stories are not characterized by the same degree of inventive power as are the tales by Andersen, but Topelius had the latter’s ability of placing himself in intimate contact with the pure minds of all ages. In the most national of Swedish cultural elements, the song, the epoch of dilettantism found its most beautiful and lasting expressions. The quartet and chorus singing at the universities of Upsala and Lund was cultivated to the highest standards of excellence and had a splendid repertory in the songs of Otto
- 42. Lindblad, Vennerberg, Prince Gustavus, Josephsson, Crusell, Cronhamn, etc. The Upsala students caused a great sensation by their singing at the Paris Exposition of 1867, and have repeated their successes at the Paris Exposition of 1878, and in Berlin in 1898. Swedish quartets of men’s and women’s voices have travelled all over the world and made a lasting fame for this minor but bewitching branch of musical art. As dramatic singers of the first rank, Louise Michaëli and Christine Nilsson have been the worthy successors of Jenny Lind. To this period, as well as to the next, belongs Elisa Hvasser, the greatest and most versatile actress Sweden has ever had. This artist was equally at home in the farce and melodrama, but excelled in the tragic parts of the Shakespeare, Schiller, and Ibsen repertory. Indispensable in their positions at the Royal Theatre of Stockholm, Michaëli, the songstress, and Hvasser, the tragedienne, did not travel, thereby losing the fame a world would have been only too glad to give them.
- 43. CHAPTER XVIII Progress and Prosperity—Oscar II Oscar II. ascended the throne at a moment when universal peace was restored after the great conflict between France and Germany, and when an age of commercial prosperity for Sweden seemed to have begun. King Oscar had received the same superior education as his older brothers, is as brilliantly gifted as they were and of a more scholarly mind. As a writer on scientific subjects, a poet and an orator, Oscar II. had distinguished himself before his succession to the throne. The new king offered the best of securities for a sound administration in his thorough and versatile knowledge, wide experience in public affairs, and rich and harmonious endowment. Oscar II. still did not find it easy to gain the love and admiration of the Swedish people, of which he is so eminently worthy. He was the successor of one of the most popular of rulers that the country ever saw, but King Oscar has lived to see his own popularity almost outrival that of his predecessor. King Oscar is, at seventy, a handsome, spirited gentleman, with that dignity which age, rare attainments, high intelligence and a noble soul grant their common possessor. This the most learned and popular monarch of Europe is of a tall, commanding figure, six feet three inches in height, of a handsome, expressive face, with cheeks of a ruddy color and mild blue eyes. Oscar II. has shown great discernment in his arrangement of dynastic matters. Himself married to the fervently religious Princess Sophie of Nassau, the king has married his oldest son, Crown Prince Gustavus Adolphus, to Princess Victoria of Bade, a granddaughter of Emperor William I. of Germany, and a great-granddaughter of Gustavus IV. of Sweden. His third son, Prince Charles, duke of West Gothland, is married to Princess Ingeborg of Denmark, a
- 44. granddaughter of Charles XV. of Sweden. These unions are well calculated to accentuate the increasing political, commercial and cultural intimacy with Germany, the Scandinavian policy of his predecessor and the desire of King Oscar to see the descendants of the old royal line of Sweden as heirs to the crown. In giving his consent to the marriage of his second son, Prince Oscar (Bernadotte), to Lady Ebba Munck, of the Swedish nobility, King Oscar has given evidence of the fact that he is not a match-maker regardless of the feelings of the parties involved. Prince Oscar, formerly Duke of Gothland, upon renouncing his share of inheritance to the two thrones, was allowed to marry the choice of his heart. King Oscar has tried to heal the wounds of the past by opening the vaults of the church of Riddarholm to the sarcophagi of Gustavus IV. and his son, and by giving Queen Carola of Saxony, the only living granddaughter of the former, repeated proofs of esteem and considerate distinction. King Oscar with his crowns had received as an inheritance two important problems to be solved—the reorganization of the Swedish army and the settlement of the difficulties in the relations between the two states of the Union. The latter has not yet found a satisfactory solution, although the king has devoted to it his most strenuous attention and the best of his efforts, in honest application to his royal motto: “The Weal of the Brother Nations.” The reorganization of the Swedish army was not effected until after twenty years of parliamentary struggle. The road of a compromise policy which was opened in 1867 was followed up at the Riksdag of 1873, in all the long chain of years royal army bills being repeatedly rejected. In 1885 the government and Riksdag agreed on a remission of thirty per cent of the military taxes of landowners in exchange for new regulations for the militia compulsory service. In 1887 the Riksdag sanctioned the total abolition of the “indelta,” or cantoned troops, as far as the navy was concerned, which was the first step toward the reorganization of the navy, and the same year the militia law of 1885 went into effect.
- 45. The old Landtmanna, or agrarian party, in 1888 gave place to a new protectionistic party. A contested election of twenty-two members from Stockholm gave a sudden majority to the protectionists, O. R. Themptander, the able minister of state, resigning. The army bill did not fare well at first. In spite of the fact that the Landtmanna party was brushed aside, the old enemies of an army reform, the landowners, nobles and peasants alike, still being strong enough to successfully oppose it. The Riksdag of 1888 passed a grain tariff, which went into effect February 14th of the same year, enforcing several other points of a protective tariff system. King Oscar called an extraordinary, or special, session of the Riksdag, October 18, 1892, when royal propositions were offered and accepted. The land-tax was abolished and a new army bill passed. According to the stipulations of the latter, the beværingstid, or period of liability for every citizen to bear arms, was extended to embrace twenty years instead of twelve, viz., eight years in the first ban of the landtværn, or militia, four years in the second ban, and eight years in the landstorm, or final levy. The first ban of militia is in time of war to form an integral part of the first fighting line, the second ban forming a reserve for the first fighting line. The final levy is to be called out for garrison duty exclusively, and for the defence of the country against foreign invasion. Six military districts have been established, five distributed along the entire coast of Sweden, the sixth inland in the western provinces to be a reserve ready to be used at the point and moment most needed. The reorganized army in active service is composed of værfvade, or enlisted troops, and indelta, or cantoned troops, the expenses also of the latter being paid by the government. The royal guards, chasseurs, hussars, artillery, and engineers are enlisted for two years up to eight. The militia troops are distributed among both the enlisted and the cantoned troops, the length of service with the colors being ninety days in time of peace. The infantry in which all the cantoned troops serve consist of twenty-six regiments and two battalions. The line is armed with Remingtons of 8.8 millimetres calibre. There are eight regiments of cavalry and six regiments and six batteries of field
- 46. artillery, forty batteries in all, with 240 cannon. The effective of the active army, in 1896, was 1,953 officers, 571 employees, 1,779 non- commissioned officers, 1,641 musicians and 38,802 men, with 6,852 horses. The war effective is 272,994 men, besides 180,000 in the landstorm. The chief fortifications of Sweden are Carlscrona, on the south coast; two fortresses outside of Stockholm, viz., Vaxholm and Oscar Fredericsborg; and, in the interior, Carlsborg, near Lake Vetter. The navy comprises 4 turret ships, with 10-inch armor, armed each with 2 10-inch and 4 5.9-inch guns, and having a total displacement of 12,450 tons; 4 armor clad monitors, 9 armored gunboats, 3 corvettes, 9 first-class and 5 second-class gunboats, 2 torpedo cruisers, 7 first-class and 9 second-class torpedo boats, 5 torpedo launches, and 12 school ships. The navy is manned by 267 officers and about 4,500 sailors, not including conscripts to the number of 8,500 men. The entire cost of the defence of Sweden exceeds ten million dollars a year. The movement for a reorganization of the defences has not been caused by any change in the policy of peace, which has faithfully been carried out by all the rulers of the Bernadotte dynasty. The ruler of Sweden and her people desire peace, but not as a gift of mercy from the great powers, but as a self-chosen right which can be effectively defended if necessary. The ever-increasing armament of the European powers has made a strengthening of the Swedish arms unavoidable, but the Swedish government was the first to announce its readiness to accept the invitation of Czar Nicholas II. of Russia to a conference for the discussion of a general reduction of the regular armies. Germany was made the pattern for the reorganization of the army and navy, the Swedish government having followed the German also in the treatment of the labor question, with schemes of accident and old-age insurance, accepted by the Riksdag. King Oscar, at his succession to the throne, gave evidence of his desire to meet the reasonable demands of his Norwegian subjects. He sanctioned, in 1873, the abolition of the office of a governor-
- 47. general of Norway, the government at Christiania to be presided over by a Norwegian minister of state. To the later Norwegian demands for a separate flag, consular service and ministry of foreign affairs, King Oscar has been unyielding. The flag question is of subordinate importance. King Oscar, in 1899, has refused to sanction the resolution of the Storthing, three times passed, for a flag without the mark of Union, for the reason that the flag with that mark was offered to Norway by his father, Oscar I., and gratefully accepted when the country had no colors at all, except the Swedish. The Swedish people will carry their old flag with the mark of Union, irrespective of any changes made in the Norwegian colors. More serious are the questions of consular and diplomatic service. In 1893, the Swedish government offered to compromise by establishing a common ministry of foreign affairs whose head might be indifferently a Swede or a Norwegian. This was rejected by the Norwegian Storthing. The same offer was made in 1837, when the dispute first arose, provided that the Norwegian troops should share the duty of the common defence of both kingdoms. The Swedish Riksdag of 1893 passed a resolution, in compliance with which King Oscar for a second time refused to sanction the bill of Norwegian consulates. The diametrically opposite views which are held in regard to the relations of Sweden and Norway are, to a great extent, caused by a misconception of the nature of the Union. In lack of a Union parliament, it has by many been considered to be only a personal union of two countries under the same king. Such is not the case. It is true that the two countries are both free and independent states and that the king is the only visible bond between them, according to the Act of Union, but the Union is nevertheless an actual and not a personal one. If it was only personal, the king could at will, or when forced to do so, resign his power in one of the countries and continue his reign in the other. The Act of Union cannot be changed except upon a resolution, enacted in both of the respective diets, and with the sanction of the king in behalf of the Union. A change can be made at the same Swedish Riksdag at which it is proposed,
- 48. at the Norwegian Storthing not until the next regular session. As a consequence the Union cannot be dissolved by the representatives of either country alone, and the king cannot dissolve it by exercising any power of his own. The king cannot abdicate one throne without abdicating the other, for the first paragraph of the Act of Union stipulates that the two countries shall be indissolubly and irrevocably united under the rule of the same king. No abdication can be granted, except by common consent of the two diets in joint session. When the two thrones are empty, without an heir-apparent, a new king shall be elected by the two diets in common. What underlies the Norwegian claims of a separate foreign ministry is, besides to own an outward sign of the country’s independence, a desire for a closer constitutional control of diplomatic affairs. From the Swedish side the desirability of a Union parliament and a greater authority for the Union government has been expressed. The Swedes have been found unwilling to grant any change of the constitution of the Union, except the right be added for the Union government to dispose of the military forces of both countries, in equal proportion, for the common defence. King Oscar’s standpoint in the Unionist conflict has contributed much to increase his popularity in Sweden, where his firm refusal to sanction any measure which would cause a weakening to the Union has been received with the highest approval. A committee to review the relations of the Union and propose a revision of its charter was appointed in 1897, but failed to accomplish anything, the views of the Swedish and Norwegian members differing too radically in their opinions. It is to be hoped that the ultimate solution of the unionist conflict, whensoever it come or whatsoever it be, will bring the two countries of the Scandinavian peninsula closer together, without any great sacrifice on either side, least of all of their independence. During the more than eighty years of peace which Sweden has enjoyed under the rule of the Bernadotte dynasty, she has developed her constitutional liberty and her material prosperity in a
- 49. high degree. The dreams of glory by conquest belong to days gone by, but in the fields of peaceable industries she has attained a greatness which the world begins to realize. At the expositions of Paris in 1867, 1878 and 1889, of Vienna in 1873, of Philadelphia in 1876 and of Chicago in 1893, Swedish industry and art have taken part with honor in the international competition. The railways of Sweden have incessantly spun a more and more extended network of steel over the country, opening connections for enterprises in new districts and furthering commerce and industrial art in a wide measure. Oscar II. is an enthusiastic friend of railway improvements, the state having built and acquired a quite considerable length of road at his initiative. The length of Swedish railways, in 1896, was 6,145 miles, of which 2,283 miles belonged to the state, compared to a total of 1,089 miles of Norwegian railways. The post-office, which was made a government department by Axel Oxenstierna, in 1636, annually transmits 130 million letters and parcels. The telegraph lines have not reached a very high state of development; still there are 14,600 miles of telegraph. The telephone has made much more progress, far surpassing that of any other country in Europe. The total length of the connections exceeds 40,000 miles, and the number of apparatus is more than 25,000. Stockholm makes the widest use of the telephone of any city in the world, with her 300,000 inhabitants having a telephone for every thirty. Sweden has developed into a commercial country of no inconsiderable rank, notwithstanding her isolated position. Exports and imports each exceed yearly in value $100,000,000, the imports being 344,290,000 kronor and the exports 311,434,000 kronor in value, in 1895, a Swedish krona being about twenty-eight cents. The commercial value of the foreign trade amounts to thirty-nine dollars in yearly average for each inhabitant of Sweden, which is about as much as in France. The imports chiefly consist of coal, coffee, salt, cotton and wool, while the exports are timber products, about forty per cent of the whole, iron and steel, the best in the world, machinery, butter, cattle, matches, etc. The inland navigation and commerce are very lively. The state finances are in a prosperous
- 50. condition. The budget of 1898 showed total receipts of 120,086,000 kronor, of which 14,229,000 was surplus from proceeding budgets. Thanks to the well equipped and regulated system of instruction, the general education has been so highly advanced that Sweden, in this respect, holds the very front rank among the nations. Besides the national universities of Upsala and Lund and the state medical college of Stockholm, city universities at Stockholm and Gothenburg have been recently founded which are quickly developing. All study at the universities consists of post-graduate work, there being about thirty colleges in various parts of the country which lead their pupils as far as the demands requisite for entering the universities. The Swedish university courses are of unexcelled thoroughness and completeness. The so-called Peasant High Schools are peculiar to Scandinavia, having originated in Denmark. There are twenty-five such high schools in Sweden, which give to young men and women of the peasant class a higher education than is available in the common schools, of which latter there are 10,702, with 692,360 pupils and 13,797 teachers. Scientific research progresses with energy and success, and Sweden possesses to-day a great number of eminent scholars, even if the epoch of men of universal genius appears to be a thing of the past there as elsewhere. Swedish scientists have opened closer relations with their co-workers in all parts of the world. The energy of King Oscar has brought about several congresses of science at Stockholm. In the natural sciences, Sweden still holds an honored place, in physics offering two great names, Eric Edlund and A. J. Angstrœm, the latter celebrated for his work on the solar spectrum, which forms the basis for the spectral analysis. Death has claimed these men and also J. A. H. Gyldén, an eminent astronomer; J. G. Agardh, C. W. Blomstrand, H. O. Nathorst, J. E. Rydquist, able botanist, chemist, agriculturist, and philologist, respectively; Pontus Wikner, the most remarkable of the disciples of the philosopher Bostrœm, and Victor Rydberg, the philosophical poet, novelist and polyhistor.
- 51. Among the most noteworthy of living Swedish scholars are Adolph Norén, Axel Koch and Esaias Tegnér, Junior, philologists; Hans Hildebrand and Oscar Montelius, archæologists; P. Fahlbeck, Nils and Magnus Hœjer, Martin Weibull, Ernest Carlson, historians; A. M. Mittag-Leffler, mathematician; Hugo Hildebrandsson, meteorologist; E. A. H. Key, E. O. T. Westerlund, Anton Wetterstrand, F. J. Biornstrœm, T. F. Hartelius, Curt Wallis, prominent in various branches of medical science. King Oscar with fervent interest and unfailing liberality has encouraged various scientific explorations, and has had the satisfaction to see the greatest geographical discoveries of the century successfully made by Swedes, the circumnavigation of Asia and Europe, and the discovery of the Northeast Passage by Baron N. A. E. Nordenskiold, and the exploration of Central Asia by Sven Hedin, which has forever settled the learned disputes of ages. A third expedition, the most daring of scientific exploits ever attempted, still keeps the world in suspense as to its final outcome. July 11, 1897, S. A. Andrée, a scientifically experienced aëronaut, with two companions, Nils Strindberg and Knut Frænkel, started in a balloon constructed for the purpose, and with provisions for three years, from an island of Spitzbergen, with the purpose of reaching the North Pole. The daring aëronauts have not been heard from since their departure, but authorities like Baron Nordenskiold have expressed the best of hopes that they may have reached Franz Joseph’s Land in safety, whence they might regain settled regions.[6] S. A. Andrée belongs to a class of men, the Swedish engineers, who have won distinction for their ability, and on whom the examples set by Christopher Polhem and John Ericsson have had a stimulating influence. There are among them two inventors of the very first rank, who belong to the reign of Oscar II., Alfred Nobel (d. 1896), the inventor of dynamite, and Gustavus de Laval, the Swedish Edison. The latter is world-famous for his separator and other inventions, which have revolutionized the dairy industry. Alfred Nobel, the disciple of John Ericsson, has not only the glory of having invented one of the most useful helpers of mechanic and industrial
- 52. progress, but also that of having set aside his vast fortune, amounting to something like $12,500,000, for public purposes. The money is so invested as to constitute a fund the interest of which shall be applied to five equal annual prizes, to be awarded for the most important discovery or improvement in chemistry, physics or medicine, for the work in literature highest in the ideal sense, and to the one who shall have acted most and best for the fraternity of nations, the suppression or reduction of standing armies, and the constitution and propagation of peace congresses. The first prize, physics and chemistry, shall be awarded by the Academy of Science of Sweden; that for physiology and medicine by the Carolin Institute of Stockholm; the literary prize by the Swedish Academy; and that for the propagation of peace by a commission of five members elected by the Norwegian Storthing. He especially directed that in distributing these prizes no consideration of nationality shall prevail, so that he who is most worthy of it shall receive the reward, whether he be Scandinavian or not. It seems that the sum of each of the five annual prizes thus instituted will amount to $75,000. The inventor of dynamite was deeply interested in all that was done to promote peace by congresses and societies. He always considered that by improving war material, and thus increasing the dangers of war, he was contributing his share toward the pacification of the world. Alfred Nobel has, by the manner in which the Norwegian Storthing is made an active party in the disposition of his will, indicated his view upon the Union of Sweden and Norway and his hopes for a peaceful solution of their conflicts. Swedish literature, after the period of dilettantism and epigones, has, during the reign of Oscar II., twice been rejuvenated and continues its development on broadened paths and with a wider scope. The eighties were characterized by a strong realistic movement, which went far in daring truth of description and brought problems of a social, religious and political nature under discussion in works of a novelistic or dramatic form. In naturalism, it never went to the extremes of the other Scandinavian literature. The movement was to a great extent brought on by Norwegian and
- 53. Danish influence, and soon subsided for want of solid and fascinating art to maintain it. The Swedish champion of this movement, although without the restrictions of any school, was August Strindberg, a genius of extraordinary endowment. Through the versatility and power of his talent, he created new forms for the Swedish drama, novel, short story and essay. In his battle against reactionary conservatism he went too far; an excitable nature, led into extremes, but he has had the manly courage to confess and regret his mistakes. Strindberg, who is an able historian, ethnographer, naturalist and sinologue, is the most versatile and prolific of contemporary writers. In the wide scope of his genius and originality of his methods, Strindberg is one of the most remarkable dramatists that ever lived. His autobiographical works are of supreme importance, both to the students of literature and psychology. Among his masterpieces are “Master Olof,” the great historic drama of his youth, “Swedish Fates and Adventures,” and “Utopia Realized,” two series of short stories, and “The Father,” a modern drama of unsurpassed tragic grandeur. Several women took an active part in the literary discussion of social problems, with more or less justice considered as the champions of women’s rights. Among these Anne Charlotte Leffler, duchessa di Cajanello, in spite of her premature death, developed into a novelist of merit who will be placed side by side with Bremer, Knorring and Carlén. The golden lyres of Romanticism were silenced and the epigones were hushed by the sarcasms of Realism. Count Snoilsky and Victor Rydberg were the only poets of the earlier period who sang with inspiration and were listened to. After the realistic movement of the eighties came a romantic reaction with new lyrics and new novelists, who avoided the ruthlessness of the realists, but had profited by their merits. This new movement cannot be called a school, for it is marked by its great versatility of subjects and great elasticity of treatment. If the definition of realistic art be “a piece of nature seen through a temperament,” that of the new movement may be “an
- 54. artistic temperament attuned to pieces of nature,” a sensitive and supple talent which has an almost unlimited capacity to tell every story just in the vein its particular subject demands. Pre-eminent in this movement stand Ola Hansson, Selma Lagerlœf, Verner von Heidenstam, Gustaf af Geijerstam, Peter Hallstrœm, Thor Hedberg, Oscar Levertin, all fine novelists, almost all good poets, and Geijerstam, an able dramatist. One of the most interesting and supremely gifted poets Sweden has ever had is Gustaf Frœding, who generally excels, sometimes abuses, his remarkable versatility in finding a true lyric expression for the very widest range of subjects. Sigurd Hedenstierna is the most popular humorist, witty in his sketches, but impossible as a novelist. The greatest humorists are August Strindberg and Gustaf Frœding. Contemporary Sweden has very few and no great literary critics, but some good literary historians in Henric Schueck, Karl Warburg and Oscar Levertin. She has a number of able journalists, most distinguished among whom is their Nestor, S. A. Hedlund, of Gothenburg, a fiery but dignified champion of a liberal government, religious tolerance, social evolution and cultural progress. Swedish literature has a long pedigree compared to Swedish art, which is hardly more than two centuries old. All the more remarkable, then, is its rapid growth and high degree of excellence. The first school of Swedish painters was founded by the German Ehrenstrahl, giving to Swedish art the cosmopolitan character it has preserved to this day, influenced by continental but chiefly French art. Swedish painters early attracted attention abroad. Gustavus Lundberg, with a picture of Boucher and his wife, won the greatest success of the Salon of Paris, in 1743. Peter Adolphus Hall, “painter to the king and the children of France,” has been called the Van Dyck of the miniature painters. He resided in Paris up to the time of the revolution and took part in the storming of the Bastile. Alexander Roslin was, from the year 1760, installed in the Louvre as painter to the king and councillor of the French Academy. In 1771 he carried home a prize which the immortal Greuze could not capture, much to the dismay of Diderot, and died as the most famous and wealthy
- 55. artist of the period. In a later period, Italy attracted many Swedish artists, and later still, in the sixties of the present century, the influence of Germany, especially of the Dusseldorf school, was strongly felt. John Frederic Hœckert won the first prize of the Paris Exposition of 1855 with his large picture “Divine Service in the Lapmark.” When the glories of Hœckert were almost forgotten at home, Edward Wahlberg, in the seventies, was ushered into celebrity as one of the greatest landscape painters of modern times, equally appreciated in Germany, as later in France, and new French laurels were won by Hugo Salmson, William von Gegerfelt and August Hagborg. Since then French influences have become solidly established, with a few important artists of the Munich school, like C. G. Hellquist and Julius Kronberg. The climax of artistic honors was reached by Nils Forsberg, whose picture, “The Death of a Hero,” carried home the first prize of the French Salon in 1888 (not an exposition medal), a distinction which no Swede and exceedingly few non-French artists ever won. The repeated successes which Swedish painters have won at expositions of Europe were more than duplicated by the enthusiastic approval granted it at the World’s Fair in Chicago in 1893. The truth is that Sweden possesses a number of eminent painters in every branch of painting, except the marine, which has been but sparingly represented since the days of Marcus Larsson. The most famous among them are, besides those already mentioned, Richard Bergh, Oscar Biorck, Eva Bonnier, Gustavus Cederstrœm, Prince Eugene, Eugene Jansson, Ernst Josephson, Nils Kreuger, Carl Larsson, Bruno Liljefors, Charles Nordstrœm, Allan Œsterlind, Georg and Hanna Pauli, George von Rosen, Robert Thegerstrom, and A. L. Zorn. It has been said of the Swedish painters, by way of complaint, that they are not, as their brethren in Denmark and Norway, in any marked degree national. Swedish art has, for its characteristic boldness and superiority in modern technique, loftiness of purpose, great individuality of expression and depth of feeling. Be these characteristics national or cosmopolitan, the Swedish painters are certainly a great credit to their country. To King Oscar it must be in a high degree satisfactory to see the artistic tendencies of his family culminate in the works of his youngest son,
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