Constructing minimum deflection fixture arrangements using frame invariant norms

This paper describes a fixture planning method that minimizes object deflection under external loads. The method takes into account the natural compliance of the contacting bodies and applies to two-dimensional and three-dimensional quasirigid bodies. The fixturing method is based on a quality measure that characterizes the deflection of a fixtured object in response to unit magnitude wrenches. The object deflection measure is defined in terms of frame-invariant rigid body velocity and wrench norms and is therefore frame invariant. The object deflection measure is applied to the planning of optimal fixture arrangements of polygonal objects. We describe minimum-deflection fixturing algorithms for these objects, and make qualitative observations on the optimal arrangements generated by the algorithms. Concrete examples illustrate the minimum deflection fixturing method. Note to Practitioners-During fixturing, a workpiece needs to not only be stable against external perturbations, but must also stay within a specified tolerance in response to machining or assembly forces. This paper describes a fixture planning approach that minimizes object deflection under applied work loads. The paper describes how to take local material deformation effects into account, using a generic quasirigid contact model. Practical algorithms that compute the optimal fixturing arrangements of polygonal workpieces are described and examples are then presented

Iterative Solutions to the Inverse Geometric Problem for Manipulators with no Closed Form Solution

A set of new iterative solutions to the inverse geometric problem is presented. The approach is general and does not depend on intersecting axes or calculation of the Jacobian. The solution can be applied to any manipulator and is well suited for manipulators for which convergence is poor for conventional Jacobian-based iterative algorithms. For kinematically redundant manipulators, weights can be applied to each joint to introduce stiffness and for collision avoidance. The algorithm uses the unit quaternion to represent the position of each joint and calculates analytically the optimal position of the joint when only the respective joint is considered. This sub-problem is computationally very efficient due to the analytical solution. Several algorithms based on the solution of this sub-problem are presented. For difficult problems, for which the initial condition is far from a solution or the geometry of the manipulator makes the solution hard to reach, it is shown that the algorithm finds a solution fairly close to the solution in only a few iterations

Defining the Pose of any 3D Rigid Object and an Associated Distance

The pose of a rigid object is usually regarded as a rigid transformation, described by a translation and a rotation. However, equating the pose space with the space of rigid transformations is in general abusive, as it does not account for objects with proper symmetries -- which are common among man-made objects.In this article, we define pose as a distinguishable static state of an object, and equate a pose with a set of rigid transformations. Based solely on geometric considerations, we propose a frame-invariant metric on the space of possible poses, valid for any physical rigid object, and requiring no arbitrary tuning. This distance can be evaluated efficiently using a representation of poses within an Euclidean space of at most 12 dimensions depending on the object's symmetries. This makes it possible to efficiently perform neighborhood queries such as radius searches or k-nearest neighbor searches within a large set of poses using off-the-shelf methods. Pose averaging considering this metric can similarly be performed easily, using a projection function from the Euclidean space onto the pose space. The practical value of those theoretical developments is illustrated with an application of pose estimation of instances of a 3D rigid object given an input depth map, via a Mean Shift procedure

Partial Bi-Invariance of SE(3) Metrics 1

In a flurry of articles in the mid to late 1990s, various metrics for the group of rigid-body motions, SE(3), were introduced for measuring distance between any two reference frames or rigid-body motions. During this time, it was shown that one can choose a smooth distance function that is invariant under either all left shifts or all right shifts, but not both. For example, if one defines the distance between two reference frames to be an appropriately weighted Frobenius norm of the difference of the corresponding homogeneous transformation matrices, this will be invariant under left shifts by arbitrary rigidbody motions. However, this is not the full picture-other invariance properties exist. Though the Frobenius norm is not invariant under right shifts by arbitrary rigid-body motions, for an appropriate weighting it is invariant under right shifts by pure rotations. This is also true for metrics based on the Lie-theoretic logarithm. This paper goes further to investigate the full invariance properties of distance functions on SE(3), clarifying the full subsets of motions under which both left and right invariance is possible

Efficient motion planning using generalized penetration depth computation

Motion planning is a fundamental problem in robotics and also arises in other applications including virtual prototyping, navigation, animation and computational structural biology. It has been extensively studied for more than three decades, though most practical algorithms are based on randomized sampling. In this dissertation, we address two main issues that arise with respect to these algorithms: (1) there are no good practical approaches to check for path non-existence even for low degree-of-freedom (DOF) robots; (2) the performance of sampling-based planners can degrade if the free space of a robot has narrow passages. In order to develop effective algorithms to deal with these problems, we use the concept of penetration depth (PD) computation. By quantifying the extent of the intersection between overlapping models (e.g. a robot and an obstacle), PD can provide a distance measure for the configuration space obstacle (C-obstacle). We extend the prior notion of translational PD to generalized PD, which takes into account translational as well as rotational motion to separate two overlapping models. Moreover, we formulate generalized PD computation based on appropriate model-dependent metrics and present two algorithms based on convex decomposition and local optimization. We highlight the efficiency and robustness of our PD algorithms on many complex 3D models. Based on generalized PD computation, we present the first set of practical algorithms for low DOF complete motion planning. Moreover, we use generalized PD computation to develop a retraction-based planner to effectively generate samples in narrow passages for rigid robots. The effectiveness of the resulting planner is shown by alpha puzzle benchmark and part disassembly benchmarks in virtual prototyping

Seconda Giornata di Studio Ettore Funaioli: 18 luglio 2008

In questo volume sono raccolte le memorie presentate in occasione della "Seconda giornata di studio Ettore Funaioli", che si è tenuta il 18 luglio 2008 presso la Facoltà di Ingegneria dell'Alma Mater Studiorum - Università di Bologna. La giornata è stata organizzata dagli ex allievi del Prof. Funaioli con la collaborazione del DIEM, Dipartimento di Ingegneria delle Costruzioni Meccaniche, Nucleari, Aeronautiche e di Metallurgia dell’Alma Mater Studiorum – Università di Bologna e con il patrocinio del GMA – Gruppo di Meccanica Applicata

Seconda Giornata di Studio Ettore Funaioli: 18 luglio 2008

In questo volume sono raccolte le memorie presentate in occasione della "Seconda giornata di studio Ettore Funaioli", che si è tenuta il 18 luglio 2008 presso la Facoltà di Ingegneria dell'Alma Mater Studiorum - Università di Bologna. La giornata è stata organizzata dagli ex allievi del Prof. Funaioli con la collaborazione del DIEM, Dipartimento di Ingegneria delle Costruzioni Meccaniche, Nucleari, Aeronautiche e di Metallurgia dell’Alma Mater Studiorum – Università di Bologna e con il patrocinio del GMA – Gruppo di Meccanica Applicata