The connection between geometry and mechanics goes back a long way perhaps even as far back as Archimedes. In the 19th century, at the height of the industrial revolution, many of the prominent mathematicians of the day, Chebyshev, Darboux and Sylvester to name just a few, were studying the geometry of mechanisms and machines. Some of the foundations of modern geometry date back to this period, although it is difficult to tell this from a reading of modern texts in geometry. For most of the 20th century, however, this connection was largely ignored by mathematicians and engineers. But the past few years have seen an increasing number of mathematicians looking at geometrical problems in robotics. A few of them have published their work in robotics journals but most have been content to publish their work in maths journals or at specialised conferences. This special issue of Robotica seeks to expose some of this work to a more general robotics audience and begin a mutually beneficial discussion between the mathematicians, computer scientists and engineers. The idea is to present some of the more advanced material in this area to demonstrate to the wider robotics community the power, range and sophistication of these modern ideas. It is hoped that this special issue will show that modern mathematical techniques have a great deal to offer for the practising engineers. Also that many problems in engineering can give rise to interesting and sophisticated mathematics.

The significance of singularities in the design and control of robot manipulators is well known, and there is an extensive literature on the determination and analysis of singularities for a wide variety of serial and parallel manipulators—indeed such an analysis is an essential part of manipulator design. Singularity theory provides methodologies for a deeper analysis with the aim of classifying singularities, providing local models and local and global invariants. This paper surveys applications of singularity-theoretic methods in robot kinematics and presents some new results.

Algebraic methods in connection with classical multidimensional geometry have proven to be very efficient in the computation of direct and inverse kinematics of mechanisms as well as the explanation of strange, pathological behavior. In this paper, we give an overview of the results achieved within the last few years using the algebraic geometric method, geometric preprocessing, and numerical analysis. We provide the mathematical and geometrical background, like Study's parametrization of the Euclidean motion group, the ideals belonging to mechanism constraints, and methods to solve polynomial equations. The methods are explained with different examples from mechanism analysis and synthesis.

This article synthezises the most important results on the kinematics of cuspidal manipulators i.e. nonredundant manipulators that can change posture without meeting a singularity. The characteristic surfaces, the uniqueness domains and the regions of feasible paths in the workspace are defined. Then, several sufficient geometric conditions for a manipulator to be noncuspidal are enumerated and a general necessary and sufficient condition for a manipulator to be cuspidal is provided. An explicit DH-parameter-based condition for an orthogonal manipulator to be cuspidal is derived. The full classification of 3R orthogonal manipulators is provided and all types of cuspidal and noncuspidal orthogonal manipulators are enumerated. Finally, some facts about cuspidal and noncuspidal 6R manipulators are reported.

There have been many interesting recent results in the area of geometrical methods for path planning in robotics. So it seems very timely to attempt a description of mathematical developments surrounding very elementary engineering tasks. Even with such limited scope, there is too much to cover in detail. Inevitably, our knowledge and personal preferences have a lot to do with what is emphasised, included, or left out.

Part I is introductory, elementary in tone, and important for understanding the need for geometrical methods in path planning. Part II describes the results on geometrical constructions that imitate well-known constructions from classical approximation theory. Part III reviews a class of methods where classical criteria are extended to curves in Riemannian manifolds, including several recent mathematical results that have not yet found their way into the literature.

We propose a new, control theoretic methodology for defining performance measures of mobile manipulators. As a guiding principle, we assume that the kinematics or the dynamics of a mobile manipulator are represented by the end point map of a control system with outputs, and that a locally controllable system yields nontrivial performance measures. In the paper, we focus on two categories of dynamic performance measures: the compliance measure and the admittance measure. In both these categories, the following local and global performance characteristics are introduced: the agility ellipsoid, the agility and mobility, the condition number and the distortion. The usefulness of new local measures is demonstrated on the example of determining optimal motion patterns of a wheeled mobile robot.

This paper investigates the singular curves in the joint space of a family of planar parallel manipulators. It focuses on special points, referred to as cusp points, which may appear on these curves. Cusp points play an important role in the kinematic behavior of parallel manipulators since they make possible a nonsingular change of assembly mode. The purpose of this study is twofold. First, it exposes a method to compute joint space singular curves of 3-RPR planar parallel manipulators. Second, it presents an algorithm for detecting and computing all cusp points in the joint space of these same manipulators.

We address general filtering problems on the Euclidean group SE(3). We first generalize, to stochastic nonlinear systems evolving on SE(3), the particle filter of Liu and West for simultaneous estimation of the state and covariance. The filter is constructed in a coordinate-invariant way, and explicitly takes into account the geometry of SE(3) and P(n), the space of symmetric positive definite matrices. Some basic results for bilinear systems on SE(3) with linear and quadratic measurements are also derived. Three examples—GPS attitude estimation, needle tip location, and vision-based robot end-effector pose estimation—are presented to illustrate the framework.

Over the past several decades, a number of O(n) methods for forward and inverse dynamics computations have been developed in the multibody dynamics and robotics literature. A method was developed by Fixman in 1974 for O(n) computation of the mass-matrix determinant for a serial polymer chain consisting of point masses. In other of our recent papers, we extended this method in order to compute the inverse of the mass matrix for serial chains consisting of point masses. In the present paper, we extend these ideas further and address the case of serial chains composed of rigid-bodies. This requires the use of relatively deep mathematics associated with the rotation group, SO(3), and the special Euclidean group, SE(3), and specifically, it requires that one differentiates real-valued functions of Lie-group-valued argument.

This paper describes the methods applicable to the modeling and control of mechanical contact, particularly those systems that experience uncertain stick/slip phenomena. Geometric kinematic reductions are used to reduce a system's description from a second-order dynamic model with frictional disturbances coming from a function space to a first-order model with frictional disturbances coming from a space of finite automata over a finite set. As a result, modeling for purposes of control is made more straight-forward by getting rid of some dependencies on low-level mechanics (in particular, the details of friction modeling). Moreover, the online estimation of the uncertain, discrete-valued variables has reduced sensing requirements. The primary contributions of this paper are the introduction of a simplifying representation of friction and formal tests for kinematic reducibility. Results are illustrated using a slip-steered vehicle model and an actuator array model.

Evidence is presented to indicate that the answer is, “Yes, sometimes.”