A fundamental problem in robotics is to characterize the kinematics of the robotic mechanism, i.e. to infer the relationship between the joint configuration and the position of the end-effector of the robot, typically the gripper. Motions of robotics mechanisms, essentially composed by rigid links connected by joints, are often characterized using the group of rigid body motions SE(3). Exploiting Lie algebra properties, kinematics problems can be formulated as systems of polynomial equations that can be solved using algebraic geometry tools. Algebraic geometry can further be used to study the dynamics properties of robotics mechanisms, i.e. the effect of forces and torques on the robot motions.
The goal of this minisymposium is to show the practical interest of algebraic geometry to analyze and control kinematic and dynamic motions of robotic systems in various applications such as solving inverse kinematic and dynamic problems, tracking manipulability ellipsoids or analyzing robots workspace. Furthermore, this minisymposium aims at bringing together mathematicians and roboticists to discuss further challenges in robotics involving application and development of algebraic geometry tools.

## Program

### 10-10.30am: Prof. Jon Selig - Some Applications of Classical Algebraic Geometry in Robotics.

The purpose of this talk is to give a brief overview of some problems in Robotics and how they can be viewed in terms of classical algebraic geometry. The group of proper rigid-body displacements plays a fundamental role in Robotics and a lot of Mechanical Engineering. Although this is not an algebraic group, using dual quaternions, it can be modelled as an open set in a six-dimensional non-singular quadric. Many of the linear subspaces of this quadric have important interpretations in terms of Robotics. In particular, lines through the identity element correspond to either rotational or translational one-parameter subgroups. In turn these correspond to mechanical joints in a robot or mechanism. Some other linear subspaces will be discussed. Series composition of joints then give rise to Segre varieties in the quadric and intersections of these varieties solve some important enumerative problems in Robotics. For some serial chains the displacements of the final link are the solution to a purely geometrical problem. For example, the displacements that maintain the contact between a point and a fixed plane. In these cases the solution lies in the intersection of the quadric representing the group of all rigid-body displacements and another non-singular quadric. This can be used to study parallel robots. Leading us to consider other realisations of the group. The standard 4-by-4 representation of the group gives a variety of degree 8 in P12. The rotation group SO(3) here is mapped to a Veronese variety. To look at the Gough-Stewart platform, a particular type of parallel robot, it is useful to look at an old idea, pentaspherical coodinates. This is equivalent to a realisation of the group of rigid-body displacements as a subgroup of the conformal group of R3. Finally some other application to robot motion and dynamics will be briefly discussed.

### 10.30-11am: Shivesh Kumar and Prof. Andreas Müller - A modular approach for kinematic and dynamic modeling of complex robotic systems using algebraic geometry.

Parallel mechanisms are increasingly being used as a modular subsystem units in the design of modern robotic systems for their superior stiffness and payload to weight ration. This leads to series-parallel hybrid robots which combine the advantages of both serial and parallel topologies but also inherit their kinematic complexity. One of the main challenges in modeling and simulation of these complex robotic systems is the existence of kinematic loops. Standard approaches in multi-body kinematics and dynamics adopt numerical resolution of loop closure constraints which leads to accuracy and inefficiency problems. These approaches give you a limited understanding of the geometry of the system. Recently, approaches from computational algebraic geometry have enabled a global description of the kinematic behavior of these complex systems. In this talk, we present a modular and analytical approach towards exploiting these algebraic methods for kinematics and dynamics modeling. This approach forms the basis of a software workbench called Hybrid Robot Dynamics (HyRoDyn). Further, we demonstrate its application in multi-body simulation and control of a complex series-parallel humanoid.

### 11-11.30am: Dr. Zijia Li - Kinematics Analysis of Serial Manipulators via Computational Algebraic Geometry.

Kinematic singularities of a redundant serial manipulator with 7 rotational joints are analyzed and their effects on the possible self-motion are studied. We obtain the numerical kinematic singularities through algebraic varieties and demonstrate this on the kinematically redundant serial manipulator KUKA LBR iiwa. The algebraic equations for determining the variety are derived by taking the determinant of the 6-by-6 submatrix of the Jacobian matrix of the forward kinematics. By the primary decomposition, the singularities can be classified. Further analysis of the kinematic singularities including the inverse kinematics of the redundant manipulator provides us with valuable insights. Firstly, there are kinematic singularities where the inverse kinematics has no effect on the self-motion and cannot be used to avoid obstacles. Secondly, there are kinematic singularities, which lead to a single closed-loop connection with the serial redundant manipulator, so that a kinematotropic mechanism is achieved. Then we show the result of kinematic singularities of several industry robots which are obtained similarly. A special inverse kinematics analysis of a (2n+1)R serial manipulator is also presented in the end.

### 11.30am-12pm: Noémie Jaquier and Dr. Sylvain Calinon - Robot manipulability tracking and transfer.

Body posture influences human and robots performance in manipulation tasks, as appropriate poses facilitate motion or force exertion along different axes. In robotics, manipulability ellipsoids are used to analyze, control and design the robot dexterity as a function of the articulatory joints configuration. These ellipsoids can be designed according to different task requirements, such as tracking a desired position or applying a specific force. In the first part of this talk, we present a manipulability tracking formulation inspired by the classical inverse kinematics problem in robotics. Our formulation uses the Jacobian of the map from the joint space to the manipulability space. This relationship demands to consider that manipulability ellipsoids lie on the manifold of symmetric positive definite matrices, which is here tackled by exploiting tensor-based representations and Riemannian geometry. In the second part of the talk, we show how this tracking formulation can be combined with learning from demonstration techniques to transfer manipulability ellipsoids between robots. The presented approaches are illustrated with various robotic systems, including robotic hands, humanoids and dual-arm manipulators.

## Invited Speakers

### Prof. Jon Selig (London South Bank University, UK)

Jon Selig graduated from the University of York, with a BSc in Physics in 1980. He went on to study in the Department of Applied Mathematics and Theoretical Physics at the University of Liverpool and was awarded a PhD in 1984. From 1984 to 1987 he was a postdoctoral research fellow in the design discipline of the Open University, studying robot gripping. He joined the Department of Electrical and Electronic Engineering at South Bank Polytechnic in 1987. In 1992 the Polytechnic became a University, and in 1999 he transferred to the School of Computing, Information Systems and Mathematics. In 2008 this school became part of the Faculty of Business, and in 2015 he transferred to the School of Engineering. Jon's research interests can be summarised as the applications of modern geometry to problems in robotics.

### Shivesh Kumar (DFKI Bremen, Germany)

Shivesh Kumar is a researcher in the Robot Control team at the DFKI Robotics Innovation Center, Bremen. He obtained his Masters degree in Control Engineering, Robotics, and Applied Informatics with specialization in Advanced Robotics from Ecole Centrale de Nantes, France in 2015. He was also an Erasmus Mundus HERITAGE scholar there. Priorly, he holds a Bachelor in Technology degree in Mechanical Engineering from National Institute of Technology Karnataka, India in 2013. His research interests spans kinematics, dynamics and control of serial, parallel and hybrid robots with applications in the fields of exoskeletons, humanoids, rehabilitation and industrial automation.

### Prof. Andreas Müller (Johannes Kepler University, Austria)

Andreas Mueller is full professor and head of the Institute of Robotic at the Johannes Kepler University Linz, Austria. Prior appointments include positions as researcher University Duisburg-Essen, Germany and at the Institute of Mechatronics, Chemnitz, Germany (also deputy CEO) and as associate professor at the Michigan University – Jiao Tong University Joint Institute in Shanghai. His research interests include geometric approaches to the modeling and analysis of robotic systems that shall allow better understand their mobility and singularities. Part of this research aims at tailored modular dynamics modeling combining Lie group theory with algebraic geometry. He served as associate editor for various journals including IEEE Transactions on Robotics, IEEE Robotics and Automation Letters, ASME Journal of Mechanisms and Robotics, Meccanica, Mechanism and Machine Theory, Mechanical Sciences.

### Dr. Zijia Li (Johannes Kepler University, Austria)

Zijia Li is currently a PostDoc at Research Institute for Symbolic Computation (RISC) from Johannes Kepler University. He started his research in computer algebra from 2008 under the supervision of Prof. Lihong Zhi in Chinese Academy of Science. He got a doctorate degree with symbolic computation for kinematics in Jan 2016 from Johannes Kepler University supervised by Prof. Josef Schicho and Prof. Hans-Peter Schröcker. After his PhD, he went Joanneum Research for two-year research stays with focusing on redundant manipulators as a senior researcher. Then he stayed at TUWien as a PostDoc for one year from 03.2018 to 02.2019. Now he is currently supported by an FWF project in RISC for the next two years.

### Noémie Jaquier (Idiap Research Institute, Switzerland)

Noémie Jaquier received the B.Sc on Microengineering and the M.Sc in Robotics with a Minor in Computational Neurosciences from the Ecole Polytechnique Federale de Lausanne (EPFL) in 2014 and 2016, respectively. She is a Ph.D student at the Idiap Research Institute. Her research interest cover human-robot interactions, machine learning and differential geometry in robotics.

### Dr. Sylvain Calinon (Idiap Research Institute, Switzerland)

Sylvain Calinon received the Ph.D. degree in robotics from Learning Algorithms and Systems Laboratory, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland, in 2007. He is a Senior Researcher at the Idiap Research Institute, and a Lecturer at the EPFL. From 2009 to 2014, he was a Team Leader at the Department of Advanced Robotics, Italian Institute of Technology. From 2007 to 2009, he was a Postdoc at EPFL. Dr Calinon is the author of 100+ publications in robot learning and human-robot interaction, with recognition including Best Paper Awards in the journal of Intelligent Service Robotics (2017) and at the IEEE Ro-Man’2007, and Best Paper Award Finalist at ICRA’2016, ICIRA’2015, IROS’2013 and Humanoids’2009. He currently serves as an Associate Editor in IEEE Transactions on Robotics and IEEE Robotics and Automation Letters.