Polynomial Invariants of the Euclidean Group Action on Multiple Screws

<p>In this work, we examine the polynomial invariants of the special Euclidean group in three dimensions, SE(3), in its action on multiple screw systems. We look at the problem of finding generating sets for these invariant subalgebras, and also briefly describe the invariants for the standard actions on R^n of both SE(3) and SO(3). The problem of the screw system action is then approached using SAGBI basis techniques, which are used to find invariants for the translational subaction of SE(3), including a full basis in the one and two-screw cases. These are then compared to the known invariants of the rotational subaction. In the one and two-screw cases, we successfully derive a full basis for the SE(3) invariants, while in the three-screw case, we suggest some possible lines of approach.</p>

Polynomial Invariants of the Euclidean Group Action on Multiple Screws

<p>In this work, we examine the polynomial invariants of the special Euclidean group in three dimensions, SE(3), in its action on multiple screw systems. We look at the problem of finding generating sets for these invariant subalgebras, and also briefly describe the invariants for the standard actions on R^n of both SE(3) and SO(3). The problem of the screw system action is then approached using SAGBI basis techniques, which are used to find invariants for the translational subaction of SE(3), including a full basis in the one and two-screw cases. These are then compared to the known invariants of the rotational subaction. In the one and two-screw cases, we successfully derive a full basis for the SE(3) invariants, while in the three-screw case, we suggest some possible lines of approach.</p>

Numerical Accuracy of Ladder Schemes for Parallel Transport on Manifolds

Parallel transport is a fundamental tool to perform statistics on Riemannian manifolds. Since closed formulae do not exist in general, practitioners often have to resort to numerical schemes. Ladder methods are a popular class of algorithms that rely on iterative constructions of geodesic parallelograms. And yet, the literature lacks a clear analysis of their convergence performance. In this work, we give Taylor approximations of the elementary constructions of Schild’s ladder and the pole ladder with respect to the Riemann curvature of the underlying space. We then prove that these methods can be iterated to converge with quadratic speed, even when geodesics are approximated by numerical schemes. We also contribute a new link between Schild’s ladder and the Fanning scheme which explains why the latter naturally converges only linearly. The extra computational cost of ladder methods is thus easily compensated by a drastic reduction of the number of steps needed to achieve the requested accuracy. Illustrations on the 2-sphere, the space of symmetric positive definite matrices and the special Euclidean group show that the theoretical errors we have established are measured with a high accuracy in practice. The special Euclidean group with an anisotropic left-invariant metric is of particular interest as it is a tractable example of a non-symmetric space in general, which reduces to a Riemannian symmetric space in a particular case. As a secondary contribution, we compute the covariant derivative of the curvature in this space.

A Mobility Determination Method for Parallel Platforms Based on the Lie Algebra of SE(3) and Its Subspaces

This contribution presents a screw theory-based method for determining the mobility of fully parallel platforms. The method is based in the application of three stages. The first stage involves the application of the intersection of the subalgebras of Lie algebra, se(3), of the special Euclidean group, SE(3), associated with the legs of the platform. The second stage analyzes the possibility of the legs of the platform generating a sum or direct sum of two subalgebras of the Lie algebra, se(3). The last stage, if necessary, considers the possibility of the kinematic pairs of the legs satisfying certain velocity conditions; these conditions allow to reduce the platform's mobility analysis to one that can solved using one of the two previous stages.

Design of Robotic Manipulator to Hollow Out Zucchini

Robotics entered the food industry starting from packaging to cooking. Zucchini is an important dish in the Middle Eastern kitchen. The eventual challenge of hollowing out Zucchini is to avoid poking its bottom with the corer. This paper introduces a novel robotic mechanism for hollowing out zucchini tasks precisely and efficiently. The mobility of the robot arm ensured a smooth hollowing while the corer is put in motion. Moreover, frames are assigned to the tip of the corer, holder, zucchini and camera to avoid stabbing the zucchini bottom. Accordingly, the special Euclidean group of the homogenous transformation matrices derived between different links are discussed in the context of their properties. The kinematic analysis is based on Mozzi-Chasles’ theorem rather than using the traditional Denavit Hartenberg convention for better task-oriented planning of hollowing out zucchini mechanism. Results indicate a promising mechanism that is well designed, simple and easy to build, maintained and most importantly to perform the required task. This paper forms an important block in building the whole automated system to hollow out Zucchini.

A Mobility Determination Method for Parallel Platforms Based on the Lie Algebra of SE(3) and its Subspaces

This contribution presents a screw theory-based method for determining the mobility of fully parallel platforms. The method is based in the application of three stages. The first stage involves the application of the intersection of the subalgebras of Lie algebra, se(3), of the special Euclidean group, SE(3), associated with the legs of the platform. The second stage analyzes the possibility of the legs of the platform generating a sum or direct sum of two subalgebras of the Lie algebra, se(3). The last stage, if necessary, considers the possibility of the kinematic pairs of the legs satisfying certain velocity conditions; these conditions allow to reduce the platform’s mobility analysis to one that can solved using one of the two previous stages.