Error propagation on the Euclidean motion group arises in a number of areas such as in dead reckoning errors in mobile robot navigation and joint errors that accumulate from the base to the distal end of kinematic chains such as manipulators and biological macromolecules. We address error propagation in rigid-body poses in a coordinate-free way. In this paper we show how errors propagated by convolution on the Euclidean motion group, SE133, can be approximated to second order using the theory of Lie algebras and Lie groups. We then show how errors that are small (but not so small that linearization is valid) can be propagated by a recursive formula derived here. This formula takes into account errors to second order, whereas prior efforts only considered the first-order case. Our formulation is non-parametric in the sense that it will work for probability density functions of any form (not only Gaussians). Numerical tests demonstrate the accuracy of this second-order theory in the context of a manipulator arm and a flexible needle with bevel tip.

A quantity of importance in coherent optical communications is the probability density of a filtered signal in the presence of phase noise (PN). The Fokker-Planck(FP) approach has been recognized as a rigorous way to describe these statistical properties. However, computational difficulties in solving these FP equation shave prevented their widespread application. In this paper, we present a new and simple computational solution method based on techniques from noncommutative harmonic analysis on motion groups. This proposed method can easily solve all the PN FP equations with any kind of intermediate frequency filter. We also present a new derivation of PN FP equations from the viewpoint of stochastic processes.

This paper presents a simple numerical method for forward kinematics of a general 6-DOF parallel manipulators, which can generate a unique solution directly. This method utilizes the trivial nature of the inverse kinematics of parallel manipulators, and derives a straightforward linear relationship between the small change of joint variables (leg lengths) and the resulting small motion of the platform. The solution to the forward kinematics is then achieved through a series of small changes in joint variables. Numerical examples validate and confirm the efficiency of the method

Hyper-redundant manipulators have a very large number of redundant degrees of freedom. They have been recognized as a means to improve manipulator performance in complex and unstructured environments. However, the high degree of redundancy also poses new challenges when performing inverse kinematics calculations. Prior works have shown that the workspace density (generated by sampling the joint space and evaluating the frequency of occurrence of the resulting end-effector reference frames) is a valuable quantity for use in ${\cal O}(P)$ inverse kinematics algorithms. Here $P$ is the number of modules in a manipulator constructed of a serial cascade of modules. This paper develops a new “divide-and-conquer” method for inverse kinematics using the workspace density. This method does not involve a high-dimensional Jacobian matrix and offers high accuracy. And its computational complexity is only ${\cal O}({\rm log}_2\,P)$, which makes it ideal for very high degree-of-freedom systems. Numerical simulations are performed to demonstrate this new method on a planar example, and a detailed comparison with a breadth-first search method is conducted.

Biomimetics and biorobotics form a thriving field of modern scientific research endeavors. The development of wall-climbing robots boasts a wide array of potential real-world applications. With this technology still in progression, a simple and fundamentally sound robot has been created to serve as a test bed and proof-of-concept robot. The developed robot is inspired by the concept of Mini-WhegsTM robot. It features more robust compliant adhesive feet, a preloading tail and modular design with inexpensive educational parts. Experiments have demonstrated the capabilities of this robot such as traversing horizontal, oblique, and vertical smooth surfaces, "walking" upside down and making the transition between orthogonal surfaces.