Martin Pfurner

Algebraic methods in mechanism analysis and synthesis

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 Studys 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.

Path Planning in Kinematic Image Space Without the Study Condition

This article proposes a new dual quaternion based approach for motion interpolation . The highlight is that dual quaternions act in the usual way on points, even if the Study condition is not fulfilled. This induces a fibration of kinematic image space into straight lines that describe the same rigid body displacement. This allows to use standard interpolation schemes for (piecewise) rational curves in a linear space rather than on the curved Study quadric.

Closed form inverse kinematics solution for a redundant anthropomorphic robot arm

An inverse kinematics solution of a redundant 7R serial chain that mimics the human arm is presented. Such manipulators are composed of two spherical wrists with one revolute joint in between. In the case of non-redundant manipulators the inverse kinematics yields a discrete set of solutions for the joint axes to reach a given end effector position and orientation. For redundant arms, however, the solution consists of a one parameter set. In contrast to other solutions to this problem herein a closed form solution is given without the need of specifying the design, task nor the redundant parameter in the solution process. That gives the possibility to use the degree of freedom provided by the additional joint for different applications, such as avoiding singularities, joint limits and collisions or to apply an optimization algorithm to achieve, for example, minimal joint velocities or joint movements. Closed form inverse kinematics solution of a redundant manipulator mimicking the human arm.No substitution of the redundancy parameter while solving the problem is needed.Solutions are implicit equations in joint parameters depending on the design and task motion parameters.

A New Family of Overconstrained 6R-Mechanisms

Using algebraic techniques an algorithm to detect new overconstrained 6R-mechanisms is developed. Many of the known overconstrained mechanisms were found rather synthetically or intuitively than by analytical methods. Within the last years analytical methods were used also to find more such mechanisms. With help of this algorithm a whole family of new overconstrained 6R-mechanisms is found. Some members of this family were already known before.

Multiple-Mode Closed 7-Link Chains Based on Overconstrained 4-Link Mechanisms

This article completes the algebraic analysis of multiple-mode 7-link chains based on the concatenation of two overconstrained 4-link mechanisms with only revolute and prismatic joints, i.e. the Bennett or the overconstrained RPRP mechanisms. Both initial mechanisms are locked in one pose of their one parameter motion. Then they are transformed to a position where one joint coincides and all the links of the basic 4-link mechanisms are deleted. New links that can be arbitrarily inserted between the seven joints. The only possible types of mechanisms in this manner are the 7R, the 5R2P and the 4R3P mechanisms. Those chains have the property, that they can fulfill an arbitrary one parameter motion of the 7-link chain and, additionally, both sub-motions of the basic 4-link chains. In special configurations it is possible to switch between the modes without the need of disconnecting and reassembling. Here an algebraic approach to those mechanisms is presented that gives the opportunity to identify the motion and the transitional configurations exactly. In all of the analyzed types of mechanisms such transition configurations exist.

Implementation of a New and Efficient Algorithm for the Inverse Kinematics of Serial 6R Chains

The aim of the reported project was to implement a new and efficient algorithm that yields simultaneously all solutions of the inverse kinematics of general 6R chains in a fast software prototype based on a C# code. The algorithm itself was developed in the working group of the authors and was previously only running in a computer algebra system. It is well known that the inverse kinematics problem of general 6R chains is highly nonlinear and yields in general 16 solutions. Using geometric preprocessing the new algorithm reduces the initial mathematical description to several linear and only two nonlinear equations. This paper recalls the algorithm, discusses the software and shows how this tool can be used in path planning and singularity detection along a path.

Algebraic Analysis of a New Variable-DOF 7R Mechanism

A new variable-DOF (degree-of-freedom) single-loop mechanism with seven revolute joints is proposed. The mechanism is constructed as a combination of two Bennett mechanisms and has two distinct 1-DOF 7R modes, where it is able to perform general 1-DOF 7R motion, and one 2-DOF double-Bennett mode, which is composed of two 1-DOF Bennett 4R motions. This paper shows the construction of this mechanism and analyzes all the motion modes and the transition configurations. Through these transition configurations, the mechanism can switch from one 1-DOF 7R mode to the 2-DOF double-Bennett mode and then to the other 1-DOF 7R mode.

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.

A New and Efficient Algorithm for the Inverse Kinematics of a General Serial 6R

In this paper a new and very efficient algorithm to compute the inverse kinematics of a general 6R serial kinematic chain is presented. The main idea is to make use of classical multidimensional geometry to structure the problem and to use the geometric information before starting the elimination process. For the geometric preprocessing we use the Study model of Euclidean displacements, sometimes called kinematic image, which identifies a displacement with a point on a six dimensional quadric S6 2 in seven dimensional projective space P7 . The 6R chain is broken up in the middle to form two open 3R chains. The kinematic image of a 3R chain turns out to be a Segre-manifold consisting of a one parameter set of 3-spaces. The intersection of two Segre-manifolds and S6 2 yields 16 points which are the kinematic images representing the 16 solutions of the inverse kinematics. Algebraically this procedure means that we have to solve a system of seven linear equations and one resultant to arrive at the univariate 16 degree polynomial. From this step in the algorithm we get two out of the six joint angles and the remaining 4 angles are obtained straight forward by solving the inverse kinematics of two 2R chains.Copyright

Input-Output Equation for Planar Four-Bar Linkages

In this paper the generalised input-output (I-O) equation for planar 4R function generators is derived in a new way, leading to the algebraic form of the well known Freudenstein equation. The long term goal is to develop a generalised method to derive constraint based algebraic I-O equations that can be used for continuous approximate synthesis, where the synthesis equations are integrated between minimum and maximum input angle values resulting in a linkage whose objective function has been optimised over every output angle. In this paper we use a planar projection of Study’s soma and the Cartesian displacement constraints for the dyads. These are mapped to the image space leading to four constraint equations in terms of the image space coordinates and the sines and cosines of the input and output angles. Using the tangent of the half angle substitution the trigonometric equations are converted to algebraic ones. Algebraic methods are used to eliminate the image space coordinates, then the polynomial resultants are found to obtain common roots leading to the desired equations.