main-content

## Über dieses Buch

Computational kinematics is an enthralling area of science with a rich spectrum of problems at the junction of mechanics, robotics, computer science, mathematics, and computer graphics. The present book collects up-to-date methods as presented during the Fifth International Workshop on Computational Kinematics (CK2009) held at the University of Duisburg-Essen, Germany. The covered topics include design and optimization of cable-driven robots, analysis of parallel manipulators, motion planning, numerical methods for mechanism calibration and optimization, geometric approaches to mechanism analysis and design, synthesis of mechanisms, kinematical issues in biomechanics, balancing and construction of novel mechanical devices, detection and treatment of singularities, as well as computational methods for gear design. The results should be of interest for practicing and research engineers as well as Ph.D. students from the fields of mechanical and electrical engineering, computer science, and computer graphics.

## Inhaltsverzeichnis

### Kinematic analysis of a spatial four-wire driven parallel crane without constraining mechanism

We are interested in wire-driven parallel robot with four wires and at least two distinct attachment points on the end-effector. Such type of robot is non redundant, it exhibits 4 d.o.f. and can be used as a crane. This paper addresses the inverse and forward kinematics problem, taking into account the mechanical equilibrium equations. We show that surprisingly the forward kinematics can be solved, either for determining all solutions or in a real-time context, but that the inverse kinematics is still an open issue.

J-P. Merlet, D. Daney

### Extension of Antipodal Theorem to Workspace Analysis of Planar Wire-Actuated Manipulators

The wrench-closure workspace of planar wire-actuated parallel manipulators is investigated utilizing the antipodal method from multi-finger grasping and verified by the null space method from static analysis. The antipodal theorem is extended to analyzing planar manipulators with wires at distinct attachment points and external force/gravity as an additional wire. It is discussed that the null space method gives a more realistic workspace formulation as it takes into account wire tension limits. The antipodal method is superior for the workspaces analysis if large wire tensions are possible.

Derek McColl, Leila Notash

### Modelling and Simulation of a Cable-Based Parallel Manipulator as an Assisting Device

This paper presents an investigation on a robotic system as assisting device for elderly or people with motion disabilities. There is a high level of motivation for elderly (or disabled people) to perform basic daily-living activities independently. Therefore, it is of great interest to design safe and reliable assisting devices that are able to help end-users in daily life activities. Cable-based robots can accomplish the requirements of safety and flexibility because of their main characteristics. A 4-cable based parallel manipulator has been proposed in this paper as an assisting device to help people in the sit to stand transfer. A modelling and simulation in ADAMS environment are presented and discussed for the system operation.

Gianni Castelli, Erika Ottaviano

### Closed-form Force Distribution for Parallel Wire Robots

This paper presents an algorithm to determine feasible force distributions for parallel wire robots in closed-form. The force distributions are continuous along trajectories and differentiable at most of the points. The computational efforts are strictly bounded and small even for large numbers of wires. The algorithm is compared to other approaches for calculation of force distribution in terms of the numerical effort and their applicability for control purposes.

Andreas Pott, Tobias Bruckmann, Lars Mikelsons

### Computing the Configuration Space for Motion Planning between Assembly Modes

In this paper, the authors will show a method to visualize the configuration space so that the relation between input and output variables can be easily assessed. Making use of concepts such as the configuration space with constant input, it will be possible to obtain all the Direct Kinematic Problem solutions in parallel manipulators, and analyse how to plan motions between them. Visualizing the entity called reduced configuration space, allows an effective motion planning to reach a wider operational workspace.

Monica Urizar, Victor Petuya, Oscar Altuzarra, Alfonso Hernandez

### Kinematic analysis of a class of analytic planar 3-RPR parallel manipulators

A class of analytic planar 3-R

P

R manipulators is analyzed in this paper. These manipulators have congruent base and moving platforms and the moving platform is rotated of 180 deg about an axis in the plane. The forward kinematics is reduced to the solution of a 3

rd

-degree polynomial and a quadratic equation in sequence. The singularities are calculated and plotted in the joint space. The second-order singularities (cups points), which play an important role in non-singular change of assembly-mode motions, are also analyzed.

Philippe Wenger, Damien Chablat

### Non-singular assembly mode change in 3-RPR-parallel manipulators

Non singular assembly mode change of parallel manipulators has been discussed for a while within the robotics community. This term means that a parallel robot can pass from one solution of the direct kinematics into another without crossing a singularity. In this paper we will show that opposed to the accepted opinion all general planar 3-RPR parallel manipulators have this ability. Using geometric properties of the singularity surface of this manipulator we will give a rigorous mathematical proof for this proposition. This proof will use the fact that the singularity surface is a fourth order surface having only very special singularities. A secondary result of this proof will be the first proof for the widespread used property that the singularity surface divides the workspace of the manipulator into two aspects that are path connected. We derive a simple technique how to construct singularity free trajectories that join all assembly modes of one connected component.

Manfred L. Husty

### Kinetostatic and Singularity Analyses of the 3- UPU Translational Parallel Robot

This paper deals with the kinetostatic and singularity analyses of the 3- UPU parallel robot. The kinetostatic model was derived analytically and two types of singularity were identified, i.e., architecture singularity and configuration singulariy. First, the actuators forces needed to balance an external load on the platform were calculated, then and thanks to its analytical form, the singularity of the kinetostatic model was studied as a function of the different design parameters of the robot.

A-H. Chebbi, Z. Affi, L. Romdhane

### Forward displacement analysis of a 3-RPR planar parallel manipulator revisited

This paper revisits the forward displacement analysis (also forward kinematics) of a class of 3-

R

PR planar parallel manipulator with congruent equilateral base and moving platform using an algebraic approach. Conditions on the inputs under which the manipulator has different number of solutions to the forward displacement analysis, including infinite solutions (self-motion), a double solution, and two distinct solutions are re-derived in an elegant way. In the first two cases, the parallel manipulator is in singular configurations. The geometric interpretation of these conditions on the input is presented.

Xianwen Kong

### Sensitivity and Dexterity Comparison of 3-RRR planar parallel manipulators

This paper deals with the sensitivity and dexterity comparison of 3-RRR planar parallel manipulators. First, the sensitivity coefficients of the pose of the moving platform of the manipulator to variations in its geometric parameters and actuated variables are derived and expressed algebraically. Moreover, two global sensitivity indices are determined, one related to the orientation of the moving platform of the manipulator and another one related to its position. The dexterity of the manipulator is also studied by means of the conditioning number of its normalized kinematic Jacobian matrix. Finally, the sensitivity of a 3-

R

RR PPM is analyzed in detail to compare the sensitivity of its best working mode to its dexterity.

Nicolas Binaud, Stéphane Caro, Philippe Wenger

### Inverse Kinematics and Motion Simulation of a 2-DOF Parallel Manipulator with 3-PUP Legs

This paper presents a position analysis of a novel parallel manipulator with three

${\underline P}UP$

legs. By investigating the loop-closure equation for the manipulator, the paper presents a closed-form solution of the inverse kinematics problem. The platform of the parallel mechanism under investigation has two independent degrees of freedom, namely, a translation and a rotation about a skew axis. The paper also shows that the platform is subjected to one parasitic motion, which is an additional translation. The paper validates the obtained closed-form equations with a numerical example. Finally, the results of this example are compared with the results of a kinematics simulation using a commercially available software package.

Ernesto Rodriguez-Leal, Jian S. Dai, Gordon R. Pennock

### ON NEW CLASS OF PARALLEL-CROSS MECHANISMS

This paper deals with the conceptual design and analysis of a new class of parallel-cross mechanisms. In the conceptual design two cases are considered in function of the position of the actuators: the actuators are located upon the base and the actuators are located under the base. In order to appreciate the performances of the proposed mechanisms, they are compared with the traditional architecture of parallel manipulators. It is shown that the workspaces of the proposed parallel-cross mechanisms are larger than the traditional architectures of parallel manipulators. The simulations for comparative analysis are carried out taking into account the geometrical limitations of structures and pressure angle.

V. A. Glazunov, S. Briot, V. Arakelian, Minh Thanh Nguyen

### Motion Interpolation with Bennett Biarcs

We present an interpolation scheme for first order Hermite motion data (two positions with associated instantaneous screws specifying the tangent vector fields) that is based on a generalization of the classic biarc construction to curves on quadrics. The result is a sequence of Bennett motions. These motions possess several properties that make them particularly useful for motion interpolation, especially for applications requiring collision detection. We suggest methods for choosing the free parameter that determines the interpolating pair of Bennett motions and we demonstrate how to obtain an interpolation algorithm which is invariant with respect to changes in the moving and the fixed coordinate frame.

Hans-Peter Schröcker, Bert Jüttler

### Motion Estimation using a Statistical Solid Dynamic Method

The most frequently used method in three dimensional human gait analysis involves placing markers on the skin of the analyzed segment. The measured motion is composed of the rigid bone motion and the surrounding soft tissues deformations. Soft tissue deformations introduce a significant artifact which strongly influences the bone position and orientation and joint kinematics estimation. In this study, we approached the problem of soft tissue artifacts using a statistical solid dynamics method. The statistical solid dynamics method is a combination of several previously reported tools. The first tool is called Point Cluster Technique (PCT). It is based on a least squares optimization of markers’ position and orientation. The second tool is a Kalman filter, which was added to the PCT in this study. The methods were tested and evaluated on a controlled human movement, when the subject was asked to move his arm in a simple planar motion while his arm was constrained to a designed arm handle. Eighteen subjects participated in the experiment in order to get statistically significant results. The result of these experiments indicated that adding Kalman filter to the PCT method produced a more accurate signal. However, it could not be concluded that the proposed Kalman filter is better than the low pass filter in the estimation of the human arm motion. Addition of a Kalman filter to the PCT method in the estimation procedure of rigid body motion results in a smoother signal that better represents the real motion. However, implementation of the Kalman filter with a better biomechanical motion model will probably improve the results.

Alon Wolf, Merav Senesh

### Spatial Generalization of the Planar Path Generation Problem

This paper deals with the dimensional synthesis of a 4C spatial mechanism. Several positions of a line in space are specified, and the goal is to design a 4C mechanism whose coupler can guide the line to pass through these positions. This problem is a spatial generalization of the planar 4R path generation problem. The maximum number of positions of lines that can be specified is found to be nine, which is identical to the maximum number of design points in the planar path generation problem. In order to avoid the complexity of obtaining numerical solutions, we use the screw triangle formulation for one CC dyad and matrix formulations for the other CC dyad to derive the design equations.We then use the Newton-Raphson method to solve the design equations, and a numerical example is provided. In this paper, the similarities between the planar and spatial path generation problems are established. A point in the planar path generation problem corresponds to a line in space, while revolute joints are replaced by cylindrical joints in the spatial path generation problem. Furthermore, the maximum numbers of allowable design positions of lines and points are both nine.

Chintien Huang, Bingrong Huang

### Motion Planning of Nonholonomic Systems with Dynamics

In the framework of control theory the motion planning problem of a robotic system amounts to determining a control function that steers the system from an initial state to a prescribed desirable state in such a way that the resulting state or output trajectory stays within an admissible region, free from obstacles. Basically, motion planning algorithms are devised to solve the problem without obstacles, and then suitable obstacle avoidance mechanisms are added. In this paper we shall concentrate on motion planning algorithms without obstacles for nonholonomic robotic systems. A comprehensive overview of approaches to the motion planning problem for the holonomic and the nonholonomic kinematics is contained in [7].

Krzysztof Tchoń, Janusz Jakubiak, Łukasz Małek

### Optimal Kinematic Calibration of Robots Based on Maximum Positioning-Error Estimation (Theory and Application to a Parallel-Mechanism Pipe Bender)

To enable optimization of the kinemetic calibration conditions of a robot to its task, we devised a method of estimating the output pose error after calibration and an error evaluation index. The index is based on the linearized relationship between the output pose error and measurement error. It depends on the kinematic parameters of the robot, calibration conditions and error evaluation conditions specified by the robot’s task. An experimental application to a parallelmechanism pipe bender was also undertaken.

Junichi Imoto, Yukio Takeda, Hidenobu Saito, Ken Ichiryu

### Fast Distance Computation Using Quadratically Supported Surfaces

We use the class of surfaces with quadratic polynomial support functions in order to define bounding geometric primitives for shortest distance computation. The common normals of two such surfaces can be computed by solving a single polynomial equation of degree six. Based on this observation, we formulate an algorithm for computing the shortest distance between enclosures of two moving or static objects by surfaces of this type. It is demonstrated that the performance of this algorithm compares favourably with methods for computing the distance between two ellipsoids, which can also be used as bounding primitives for distance computation and collision detection.

Margot Rabl, Bert Jüttler

### On the Computation of the Home Posture of the McGill Schönflies-Motion Generator

The McGill Schönflies-Motion Generator (SMG) is a two-limb parallel robot which is capable of producing three independent translations in the Cartesian space and one rotation about a fixed axis. In this paper, the

home posture

of the McGill SMG, defined as the configuration at which the robot rests, while it is not in operation, is determined. For this matter, the geometry and the velocity analysis of the McGill SMG is recalled from a previous publication. By making intensive use of both linear-algebra identities and results specific to the kinematics of the Schönflies subgroup, the normality conditions associated with the minimization of the condition number of the forward-kinematics Jacobian of the robot are derived in frame-invariant form. This form lends itself to geometric interpretations that would not be possible with lengthy componentwise expressions if commercial algebraic software packages were used.

D. Alizadeh, J. Angeles, S. Nokleby

### Hardware-in-the-Loop Simulation of Constraint Elements in Mechanical Systems

To simulate constraint elements of mechanical systems like joints or bearings in their environment a Hardware-in-the-Loop (HiL) simulator is proposed. It couples a real joint with a numerical simulation of its environment by means of an actuator system and sensors. The actual application standing behind this investigation is HiL testing of hip endoprostheses with respect to their dislocation behavior. HiL simulations allow the analysis of real system components in a virtual environment controlled by a computer and are thereby particularly advantageous if test conditions in the real environment of the system component are either too complex or impossible to apply like in vivo investigations of dislocation scenarios. As a preliminary research two fundamental variants for the coupling between the real joint and the simulation of the environment are discussed and experimentally demonstrated.

Michael Kähler, Christoph Woernle, Rainer Bader

### Explicit Algebraic Solution of Geometrically Simple Serial Manipulators

An algorithm is developed to solve the inverse kinematics of special serial manipulators that contain a spherical or planar sub-chain anywhere within an entire six joint sequence. It is known, for such cases, that the inverse kinematics is solvable in closed form, i.e., with a univariate polynomial of degree four or less; sometimes even with a quadratic equation. This algorithm yields explicit algebraic solutions for these kind of manipulators even when the design or the end-effector pose is not explicitly given.

Martin Pfurner

### 3R Wrist Positioning - a Classical Problem and its Geometric Background

The

wrist centre positioning problem

(WP) peculiar to the motion produced by the first three joints of a general six revolute jointed (6R), wrist partitioned serial robot and the underlying geometry is reexamined. Conventionally a sequence of six rotational operations, alternately in terms of known geometric parameters and unknown joint angles, expresses the desired position. However the solution can be represented by four intersection points between a fourth order cyclid, and a circle. Properties of the curves of intersection of the cyclid with the absolute plane reveal why the univariate polynomial (UVP) is of fourth degree rather than eighth as indicated by the Bezout number. Simple cyclid geometry makes it convenient to investigate specific 3R positioning architectures and expose degenerate cases.

P. Zsombor-Murray, A. Gfrerrer

### A Geometric Newton-Raphson Method for Gough-Stewart Platforms

A geometric version of the well known Newton-Raphson methods is introduced. This root finding method is adapted to find the zero of a function defined on the group of rigid body displacements. At each step of the algorithm a rigid displacement is found that approximates the solution. The method is applied to the forward kinematics problem of the Gough-Stewart platform.

J. M. Selig, Hui Li

### Aspects of Clifford Algebra for Screw Theory

Some aspects of Clifford Algebra are presented in the context of rigid body mechanics. Multivectors are discussed together with their inner, outer and geometric products. Their role in representing concepts from screw theory in naturally geometrical terms is briefly explored.

Joe Rooney

### Uncoupled 6-dof Tripods via Group Theory

Using kinematic equivalencies originating from the closure of the product in displacement Lie subgroups, we have synthesized seven 6-dof tripods that allow the actuation of the positioning of a center of spherical motion and the independent actuation of the spherical motion around the center. These two actuations are uncoupled. One of these tripods, 3-P

RPP

(RR), is truly amazing by fully actuating the inputs closely near the base.

Chung-Ching Lee, Jacques M. Hervé

### A new approach towards the synthesis of six-bar double dwell mechanisms

This paper presents a new formulation for the synthesis of planar six-bar linkages with only rotary joints showing two dwells in each cycle of the input crank. The formulation combines the concepts of instantaneous kinematics and optimisation to produce a simple, one-step synthesis method. The method is illustrated with a numerical example, where it is seen to produce better results in terms of accuracy and computational efficiency with respect to reported works.

### Application of higher order derivatives in the synthesis of crank and cam mechanisms

This paper describes the creation of straight line and circular guidance mechanisms based on curvature theory. The goal of this approach is the development of efficient algorithms for the determination of specific points on a arbitrarily moved coupler link, which satisfy certain curvature characteristics. Collectively these points form specific curves, in particular a Ball, an inflection and a Burmester curve. Because the calculation of these curves requires derivatives up to the 4th order, the approach using complex numbers results in a compact and universal description, which can be cast into efficient algorithms. This allows the swift realization of desired properties in an interactive mechanism synthesis, which is demonstrated with the aid of some examples.

H. Lederer, G. Lonij, B. Corves

### Interactive design of a robotic gripper system with the geometry program “GECKO”

In this paper, the spatial geometry program called “GECKO” currently under development at the IGM RWTH-Aachen University is presented. “GECKO” allows the interactive study and execution of graphical design procedures well known in mechanism theory. Particularly, several new program features for the definition of spatial trajectories and the analysis of drive values of a six bar serial structure used in a robotic gripper system are discussed.

G. Lonij, S. -W. Choi, B. Corves

### The Axis Constraint Equation and a General 6R Double-Spherical Overconstrained Mechanism

This paper investigates a 6

R

double-spherical overconstrained mechanism with a general arrangement and proves the axis constraint equation in this general type mechanism. The equation is derived with a relaxation of four geometric constraints existing in previous literature which requires that the consecutive axes be perpendicular at each spherical corner.

Lei Cui, Jian S. Dai

### Two methods for force balancing of Bennett linkages

In this paper, we present two methods for the force balancing of Bennett linkages. In the first approach, we formulate the closure equation and the force balancing equations algebraically in terms of complex variables. Using this formulation, the problem corresponds to a factorization problem of polynomials. In the second approach, we use redistribution of masses for the coupler onto the input and output crank, to decouple the problem which leads to an elegant problem formulation and solution.

Brian Moore, Josef Schicho

### Regular Polygonal and Regular Spherical Polyhedral Linkages Comprising Bennett Loops

In this study, assemblies of Bennett loops constructing regular polygonal linkages and regular polyhedral linkages are presented. The regular polyhedral linkages, necessarily, depend on spherical polyhedral shapes. Most of the resulting linkages have single degree of freedom, but there are exceptions such as a cubic linkage.

Gökhan Kiper, Eres Söylemez

### Kinematic analysis of an adjustable slider-crank mechanism

In this paper a kinematic analysis of an adjustable slider-crank mechanism is presented. The proposed mechanism is formed by an output member, i.e. the slider, by a connecting rod and by an equivalent crank mechanism, consisting of a pair of identical gears and a connecting link assembled in a typical epicyclical configuration. One point of the planet gear moves along an epicycloidal path, while the other gear is held stationary and the planet arm rotates around a fixed hinge. Such epicycloidal motion is converted into a reciprocating motion of the slider by means of the connecting rod, as in a traditional slider-crank mechanism. By holding the planet arm stationary and modifying the relative angular position of the two gears, an adjustment of the entire mechanism is achieved, in such a way that, if the arm is allowed to rotate again, the slider starts moving according to a different law of motion.

D. Mundo, G. Gatti, G. Danieli, D. B. Dooner

### Improving marker based inverse kinematics solutions for under-determined spinal models

A well known problem in biomechanics is the inverse kinematics based motion reconstruction from segments in a mechanical model with motion capture data. This contribution presents an optimization based inverse kinematics approach to determine solutions for under-determined systems. This paper presents the application of a constrained nonlinear least-squares optimization approach for underdetermined systems. It is suitable for large multibody chains with only few marker information available, e.g. for a detailed model of the human spine. Results of a successful motion reconstruction will be shown for a spinal model with 75dof.

Christian Simonidis, Wolfgang Seemann

### Kinematical Analysis and Design of a New Surgical Parallel Robot

Robots in surgery are used because of their superior advantages: improved accuracy, tremor elimination etc, which exceeds human capabilities. Unlike serial robots, parallel robots offer higher stiffness, reduced masses in motion, higher rigidity, all of these leading more precise manipulations that fit medical applications. In this paper, a new innovative parallel structure for laparoscopy is presented. It is studied the forward and inverse kinematics, the singularities, it is generated the workspace and are presented some simulation results.

D. Pisla, N. Plitea, B. Gherman, A. Pisla, C. Vaida

### Design Improvements on a Carotid Blood Flow Measurement System

This paper investigates mechanical design improvements for a carotid blood flow measurement robotic system. This robotic system is composed of a serial robot with a wrist having a 6-DOF parallel mechanism. Main focus is to enhance the workspace performance of the wrist. Thus, a workspace analysis is carried out by referring to a built prototype that is named as WTA-1R (Waseda-Tokyo women’s medical Aloka blood flow measurement system no.1 Refined). Then, mechanical design enhancements are proposed.

G. Carbone, R. Nakadate, J. Solis, M. Ceccarelli, A. Takanishi, E. Minagawa, M. Sugawara, K. Niki

### Comparison of Pose Selection Criteria for Kinematic Calibration through Simulation

This paper explores the effectiveness of criteria to select a set of poses, which improve the calibration accuracy, out of a larger pool of possible poses. Five different criteria have been suggested in the literature and are investigated in this article. The effectiveness of the pose selection criteria is explored with a simulation of a two degrees of freedom planar manipulator. The results of this simulation show that using a larger number of poses, irregardless of any pose selection criteria, is preferable. When a small number of poses are used the pose selection criteria does have a significant effect.

Andrew Horne, Leila Notash

### A New Procedure for the Optimization of a Dielectric Elastomer Actuator

A novel mathematical procedure is proposed, which makes it possible to optimize lozenge-shaped dielectric-elastomer-based linear actuators for known materials and desired force/stroke requirements. Simulation results are provided which both demonstrate the efficacy of the novel procedure with respect to traditional design approaches and show that simpler, cheaper, lighter and betterbehaved lozenge-shaped actuators can be conceived which do not require any integration of compliant frame elements.

Rocco Vertechy, Giovanni Berselli, Vincenzo Parenti Castelli, Gabriele Vassura

### Light–Weight High Dynamic Camera Orientation System

This paper suggests a system that is able to orient a camera around its pan and tilt axis with the same high dynamic as a human orients his eyes. Since the device is a part of a gaze–driven head–mounted stereo camera system, beside the high dynamic also the compactness of the device is important. Therefore a small and light–weight system is developed which is mounted on a humans head near the ears. Based on the experience of a previous version a small piezo–actuator driven parallel kinematics is developed. The kinematic solution, a dynamic simulation and the step response behavior of the new device are presented.

Thomas Villgrattner, Thomas Thümmel, Heinz Ulbrich

### Dynamic Balancing of Clavel’s Delta Robot

The Delta robot has shown to be a useful device in many applications. Due to large accelerations however, vibrations can decrease the accuracy and performance considerably. Instead of common techniques to reduce vibrations such as damping or including waiting times in the motion cycle, this article shows how the Delta robot can be dynamically balanced in a practical way by which all vibrations are eliminated. Because of its specific architecture, the Delta robot can be force balanced with only three counter-masses and two additional links. Moment balancing can be achieved by active actuation of three additional rotating inertias.

V. van der Wijk, J. L. Herder

### Kinematical Solution by Structural Approximation

The paper deals with the new method for positional kinematical solution of mechanisms with loops. The method is based on the concept of structural approximation, i.e. the structure of the mechanism being solved is simplified in such a way that the mechanism with simplified structure is analytically solvable. The analytical solution is the basis of the iteration. This method has been successfully applied for the inverse kinematical solution of non-simple serial robots. This paper extends this method for mechanisms with loops and for forward kinematical solution of parallel kinematical structures. The method of structural approximation is demonstrated on Hexapod, Sliding Star and planar multiloop non-simple mechanism.

Pavel Kukula, Michael Valasek

### Singularity Analysis of a 3 Degrees-of-Freedom Parallel Manipulator

The paper deals with the kinematic study of a parallel manipulator with 3 degrees-of-freedom named HPR (heave, pith, roll). The inverse kinematics is solved and jacobian matrixes are evaluated. Then singularity loci are analyzed within a dimensionless geometrical description of the parallel structure. Singularity loci determination is revealed scaling factor independent for this kind of mechanism. Besides, workspace boundaries based on the concept of first and second type singularity are studied varying characteristic geometrical ratios of the manipulator. Finally a methodology to design HPR mechanism with desired workspace free of singularities is proposed.

Stefano Pastorelli, Alessandro Battezzato

### Branching singularities in kinematotropic parallel mechanisms

The paper addresses the branching singularities in kinematotropic parallel mechanisms. The new formulae of mobility, connectivity, overconstraint and redundancy of parallel robots, recently proposed by the author, are used to characterize the behavior of the kinematotropic parallel mechanisms in the branching singularity and in each distinct branch. Four types of branching singularities are identified starting from the new formulae. The structural parameters associated with the four types of branching singularities and the distinct branches as well can be easily identified by inspection with no need to calculate the Jacobian matrix.

Grigore Gogu

### A new approach to the classification of architecturally singular parallel manipulators

This is a new approach to the classification of architecturally singular parallel manipulators.We prove that a complete classification is possible in a geometric way if we distinguish the cases whether the linear line complex spanned by the carrier lines of the legs is always singular or not. The proof is based on the result that 5-legged planar parallel manipulators of Stewart Gough type which belong in every possible configuration to a singular linear line complex must possess 4 collinear anchor points. Moreover we list all types of 5-legged planar parallel manipulators with this property.

Georg Nawratil

### Straightening-Free Algorithm for the Singularity Analysis of Stewart-Gough Platforms with Collinear/Coplanar Attachments

An algorithm to derive the pure condition of any double-planar Stewart-Gough platform in a standard form suitable for comparison is presented. By applying the multilinear properties of brackets directly to the superbracket encoding of the pure condition, no straightening is required. It is then shown that any 3-3 platform has a corresponding 6-6 platform having its same superbracket, meaning that they have identical singularity loci. In general, the superbracket of any doubleplanar platform can be written as a linear combination of the superbrackets of 3-3 platforms, leading to a direct singularity assessment by inspecting the resulting decomposition.

Júlia Borràs, Federico Thomas, Carme Torras

### A Computational Approach for the Evaluation of Single d.o.f. Planetary Gear Efficiency

The paper presents a new computational method for the determination of formulas for the efficiency of single d.o.f. planetary gear trains. The method is based on kinematic modelling of the gear train and on explicit and systematic operations, easy to implement, for example, by a computer algebra package. After a presentation of the required background for the kinematic and static analyses, the proposed method is justified and described. The application of the method to a sample gear train is discussed in details.

Pietro Fanghella

### The Computational Fundamentals of Spatial Cycloidal Gearing

The tooth flanks of bevel gears with involute teeth are still cut using approximations such as Tredgold’s and octoid curves, while the geometry of the exact spherical involute is well known. The modeling of the tooth flanks of gears with skew axes, however, still represents a challenge to geometers. Hence, there is a need to develop algorithms for the geometric modeling of these gears. As a matter of fact, the modeling of gears with skew axes and involute teeth is still an open question, as it is not even known whether it makes sense to speak of such tooth geometries. This paper is a contribution along these lines, as pertaining to gears with skew axes and cycloid teeth. To this end, the authors follow and extend results reported by Martin Disteli at the turn of the 20th century concerning the general synthesis of gears with skew axes. The main goal is to shed light on the geometry of the tooth flanks of gears with skew axes. The dualization of the tooth profiles of spherical cycloidal gears leads to ruled surfaces as conjugate tooth flanks such that at any instant the contact points are located on a straight line. A main result reported herein is Theorem 5, which is original. All results are proven by means of a consistent use of dual vectors representing directed lines and rigid-body twists.

Giorgio Figliolini, Hellmuth Stachel, Jorge Angeles

### Geometrical dimensions of helical gears with equalized relative velocities at the beginning and the ending of the meshing

The paper gives a new method for the determination of the geometrical dimensions of the helical gears, with specific addendum modifications. The method is based on the equalization of the relative velocities at the points where the meshing of the teeth begins and ends. The numerical values were obtained for some given gears using a MATLAB application.