Pierre M. Larochelle

Spherical Mechanism Synthesis in Virtual Reality

This paper presents a new approach to using virtual reality (VR) to design spherical mechanisms. VR provides a three-dimensional (3-D) design space where a designer can input design positions using a combination of hand gestures and motions and view the resultant mechanism in stereo using natural head movement to change the viewpoint. Because of the three-dimensional nature of the design and verification of spherical mechanisms, VR is examined as a new design interface in this research. In addition to providing a VR environment for design, the research presented in this paper has focused on developing a design in context approach to spherical mechanism design. Previous design methods have involved placing coordinate frames along the surface of a constraint sphere. The new design in context approach allows a designer to freely place geometric models of movable objects inside an environment consisting of fixed objects. The fixed objects could either act as a base for a mechanism or be potential sources of interference with the motion of the mechanism. This approach allows a designer to perform kinematic synthesis of a mechanism while giving consideration to the interaction of that mechanism with its application environment.

Spatial Mechanism Design in Virtual Reality With Networking

Mechanisms are used in many devices to move a rigid body through a finite sequence of prescribed locations in space. The most commonly used mechanisms are four-bar planar mechanisms that move an object in one plane in space. Spatial mechanisms allow motion in three-dimensions (3D). Spatial 4C mechanisms are two degree of freedom kinematic closed-chains consisting of four rigid links simply connected in series by cylindrical (C) joints. A cylindrical joint is a two degree of freedom joint which allows translation along and rotation about a line in space. This paper describes a synthesis process for the design of 4C spatial mechanisms in a virtual environment. Virtual reality allows the user to view and interact with digital models in a more intuitive way than using the traditional humancomputer interface (HCI). The software developed as part of this research also allows multiple users to network and share the designed mechanism. Networking tools have the potential to greatly enhance communication between members of the design team at different industrial sites and therefore reduce design costs.

Approximating Spatial Locations With Spherical Orientations for Spherical Mechanism Design

In this paper we present a novel method for approximating a finite set of n spatial locations 1 with n spherical orientations. This is accomplished by determining a design sphere and the associated orientations on this design sphere which are nearest the n spatial locations. The design sphere and the orientations on it are optimized such that the sum, of the distances between each spatial location and its approximating spherical orientation is minimized. The result is a design sphere and n spherical orientations which best approximate a set of n spatial locations. In addition, we include a modification to the method which enables the designer to require that one of the n desired spatial locations be exactly preserved. This method for approximating spatial locations with spherical orientations is directly applicable to the synthesis of spherical mechanisms for motion generation. Here we demonstrate the utility of the method for motion generation task specification in spherical mechanism design.

Planar Motion Synthesis Using an Approximate Bi-Invariant Metric

In this paper we present a technique for using a bi-invariant metric in the image space of spherical displacements for designing planar mechanisms for n (> 5) position rigid body guidance. The goal is to perform the dimensional synthesis of the mechanism such that the distance between the position and orientation of the guided body to each of the n goal positions is minimized. Rather than measure these distances in the plane, we introduce an approximating sphere and identify rotations which are equivalent to the planar displacements to a specified tolerance. We then measure distances between the rigid body and the goal positions using a bi-invariant metric on the image space of spherical displacements. The optimal linkage is obtained by minimizing this distance for each of the n goal positions.

Collision detection of cylindrical rigid bodies for motion planning

This paper presents a novel methodology for detecting collisions of cylindrically shaped rigid bodies moving in three dimensions. This algorithm uses line geometry and dual number algebra to exploit the geometry of right circular cylindrical objects to facilitate the detection of collisions. First, the rigid bodies are modelled with infinite cylinders and an efficient necessary condition for collision is evaluated. If the necessary condition is not satisfied then the two bodies do not collide. If the necessary condition is satisfied then a collision between the bodies may occur and we proceed to the next stage of the algorithm. In the second stage the bodies are modelled with finite cylinders and a definitive necessary and sufficient collision detection algorithm is employed. The result is a straight-forward and efficient means of detecting collisions of cylindrically shaped bodies moving in three dimensions. This methodology has applications in robot motion planning, spatial mechanism design, and workspace analysis of parallel kinematic machines such as Stewart-Gough platforms. A case study of motion planning for an industrial robot is included

Algebraic Motion Approximation With NURBS Motions and Its Application to Spherical Mechanism Synthesis

In this work we bring together classical mechanism theory with recent works in the area of Computer Aided Geometric Design(CAGD) of rational motions as well as curve approximation techniques in CAGD to study the problem of mechanism motion approximation from a computational geometric viewpoint. We present a framework for approximating algebraic motions of spherical mechanisms with rational B-Spline spherical motions. Algebraic spherical motions and rational B-spline spherical motions are represented as algebraic curves and rational B-Spline curves in the space of quaternions (or the image space). Thus the problem of motion approximation is transformed into a curve approximation problem, where concepts and techniques in the field of Computer Aided Geometric Design and Computational Geometry may be applied. An example is included at the end to show how a NURBS motion can be used for synthesizing spherical four-bar linkages.

Collision Detection of Cylindrical Rigid Bodies Using Line Geometry

This paper presents a novel methodology for detecting collisions of cylindrically shaped rigid bodies moving in three dimensions. This algorithm uses line geometry and dual number algebra to exploit the geometry of right circular cylindrical objects to facilitate the detection of collisions. First, the rigid bodies are modelled with infinite length cylinders and a necessary condition for collision is evaluated. If the necessary condition is not satisfied then the two bodies are not capable of collision. If the necessary condition is satisfied then a collision between the bodies may occur and we proceed to the next stage of the algorithm. In the second stage the bodies are modelled with finite length cylinders and a definitive necessary and sufficient collision detection algorithm is employed. The result is a straight-forward and efficient means of detecting collisions of cylindrically shaped bodies moving in three dimensions. This methodology has applications in spatial mechanism design, robot motion planning, workspace analysis of parallel kinematic machines such as Stewart-Gough platforms, nuclear physics, medical research, computer graphics and well drilling. A case study examining a spatial 4C robotic mechanism for self collisions is included.Copyright

Approximate Motion Synthesis of Open and Closed Chains via Parametric Constraint Manifold Fitting: Preliminary Results

In this paper we present a novel dyad dimensional synthesis technique for approximate motion synthesis. The methodology utilizes an analytic representation of the dyad’s constraint manifold that is parameterized by its dimensional synthesis variables. Nonlinear optimization techniques are then employed to minimize the distance from the dyad’s constraint manifold to a finite number of desired locations of the workpiece. The result is an approximate motion dimensional synthesis technique that is applicable to planar, spherical, and spatial dyads. Here, we specifically address the planar RR, spherical RR and spatial CC dyads since these are often found in the kinematic structure of robotic systems and mechanisms. These dyads may be combined serially to form a complex open chain (e.g. a robot) or when connected back to the fixed link they may be joined so as to form one or more closed chains (e.g. a linkage, a parallel mechanism, or a platform). Finally, we present some initial numerical design case studies that demonstrate the utility of the synthesis technique.Copyright

A distance metric for finite sets of rigid-body displacements via the polar decomposition

An open research question is how to define a useful metric on the special Euclidean group SE(n) with respect to: (1) the choice of coordinate frames and (2) the units used to measure linear and angular distances that is useful for the synthesis and analysis of mechanical systems. We discuss a technique for approximating elements of SE(n) with elements of the special orthogonal group SO(n+1). This technique is based on using the singular value decomposition (SVD) and the polar decompositions (PD) of the homogeneous transform representation of the elements of SE(n). The embedding of the elements of SE(n) into SO(n+1) yields hyperdimensional rotations that approximate the rigid-body displacements. The bi-invariant metric on SO(n+1) is then used to measure the distance between any two displacements. The result is a left invariant PD based metric on SE(n).

SVD and PD Based Projection Metrics on SE(n)

An open research question is how to define a metric on SE(n) that is as invariant as possible with respect to (1) the choice of coordinate frames and (2) the units used to measure linear and angular distances. We present two techniques for approximating elements of the special Euclidean group SE(n) with elements of the special orthogonal group S0(n+1). These techniques are based on the singular value and polar decompositions (denoted as SVD and PD respectively) of the homogeneous transform representation of the elements of SE(n). The projection of the elements of SE(n) onto S0(n+1) yields hyperdimensional rotations that approximate the rigid-body displacements. Any of the infinite bi-invariant metrics on SO(n-fl) may then be used to measure the distance between any two spatial displacements. The results are PD and SVD based projection techniques that yield two approximately bi-invariant metrics on SE(n). These metrics have applications in motion synthesis, robot calibration, motion interpolation, and hybrid robot control.