Minisymposium on Algebraic Geometry for Kinematics and Dynamics in Robotics

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.

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.