Elisha Sacks

Parametric kinematic tolerance analysis of general planar systems

We present an algorithm for functional kinematic tolerance analysis of general planar mechanical systems with parametric tolerances. The algorithm performs worst-case analysis of systems of curved parts with contact changes, including open and closed kinematic chains. It computes quantitative variations and helps designers detect qualitative variations, such as blocking and under-cutting. The algorithm constructs a variation model for each interacting pair of parts: a mapping from the part tolerances and configurations to the kinematic variation of the pair. These models generalize the configuration space representation of nominal kinematics to toleranced parts. They are composed via sensitivity analysis and linear programming to derive the system variation at a given configuration. The variation relative to the nominal system function is computed by sampling the system variation. We demonstrate the algorithm on detailed parametric models of a movie camera film advance and of a micro-mechanical gear discriminator.

Automated modeling and kinematic simulation of mechanisms

Abstract An analysis program is presented for rigid part mechanisms, such as feeders, locks and brakes. The program performs a kinematic simulation of driving motions and part contacts, with a limited dynamic simulation of gravity, springs and friction. It produces a realistic, 3D animation and a concise, symbolic interpretation of the simulation. It derives the kinematic motion equations for a large class of mechanisms, including ones with complex part shape, varying part contacts, and multiple driving motions. It avoids collision detection during simulation by precomputing pairwise part interactions. It uses a simple model of dynamics that captures the steady-state effect of forces without the conceptual and computational cost of full dynamic simulation. It is demonstrated that the simulation algorithm captures the workings of most mechanisms via a survey of 2500 mechanisms from an engineering encyclopedia.

Algorithmic issues in modeling motion

This article is a survey of research areas in which motion plays a pivotal role. The aim of the article is to review current approaches to modeling motion together with related data structures and algorithms, and to summarize the challenges that lie ahead in producing a more unified theory of motion representation that would be useful across several disciplines.

Automatic analysis of one-parameter planar ordinary differential equations by intelligent numeric simulation

Abstract This paper describes research on automating the analysis of physical systems modeled by ordinary differential equations. It describes a program called POINCARE that analyzes one-parameter autonomous planar systems at the level of experts through a combination of theoretical dynamics, numerical simulation, and geometric reasoning. The input is the system, a bounding box for the system state, a bounding interval for the parameter, and error tolerances. POINCARE partitions the parameter interval into open subintervals of equivalent behavior bounded by bifurcation points, classifies the bifurcation points, and constructs representative phase diagrams for the subintervals. It detects local and global generic one-parameter bifurcations. It constructs the phase diagrams by identifying fixed points, saddle manifolds, and limit cycles and partitioning the remaining trajectories into open regions of uniform asymptotic behavior. It obtains the region boundaries by numeric simulation, guided by theoretical knowledge. It determines the behavior near fixed points from the Jacobian of the system, scans outward to find basins of attractors and limit cycles, and stops at the bounding box.

Automatic qualitative analysis of dynamic systems using piecewise linear approximations

Abstract This paper explores automating the qualitative analysis of physical systems. It describes a program, called PLR (for Piecewise Linear Reasoner), that formalizes an analysis strategy employed by experts. PLR takes parameterized ordinary differential equations as input and produces a qualitative description of the solutions for all initial values. It approximates intractable nonlinear systems with piecewise linear ones, analyzes the approximations, and draws conclusions about the original systems. It chooses approximations that are accurate enough to reproduce the essential properties of their nonlinear prototypes, yet simple enough to be analyzed completely and efficiently. It derives additional properties, such as boundedness or periodicity, by theoretical methods. PLRs analysis depends on abstract properties of systems rather than on specific numeric values. This makes its conclusions more robust and enables it to handle parameterized equations transparently. I demonstrate PLR on several common nonlinear systems and on published examples from mechanical engineering.

The Occlusion Camera

We introduce the occlusion camera: a non-pinhole camera with 3D distorted rays. Some of the rays sample surfaces that are occluded in the reference view, while the rest sample visible surfaces. The extra samples alleviate disocclusion errors. The silhouette curves are pushed back, so nearly visible samples become visible. A single occlusion camera covers the entire silhouette of an object, whereas many depth images are required to achieve the same effect. Like regular depth images, occlusion-camera images have a single layer thus the number of samples they contain is bounded by the image resolution, and connectivity is defined implicitly. We construct and use occlusion-camera images in hardware. An occlusion-camera image does not guarantee that all disocclusion errors are avoided. Objects with complex geometry are rendered using the union of the samples stored by a planar pinhole camera and an occlusion camera depth image.

Computational Kinematic Analysis of Higher Pairs with Multiple Contacts

We present a computational kinematic theory of higher pairs with multiple contacts, including simultaneous contacts, intermittent contacts, and changing contacts. The theory systematizes single- and multiple-contact kinematic analysis by mapping it into geometric computation in configuration space. It derives the contact conditions, contact functions, and relations between contacts from the shapes and degrees of freedom of the parts. It helps identify common design flaws, such as undercutting, interference, and jamming, that cannot be systematically identified with current methods. We describe a program for the most common pairs: planar higher pairs with two degrees of freedom.

Sliced Configuration Spaces for Curved Planar Bodies

We present the first practical, implemented configuration-space computation algorithm for a curved, planar object translating and rotating amidst stationary obstacles. The bodies are rigid, compact, regular, and bounded by a finite number of rational parametric curve segments. The algorithm represents the three-dimensional configu ration space as two-dimensional slices in which the moving object has a fixed orientation. It discretizes the configuration space into in tervals of equivalent slices separated by critical slices. The output is topologically correct and accurate to within a specified toler ance. We have implemented the algorithm for objects bounded by line segments and circular arcs, which is an important class for applications. The program is simple, fast, and robust. The slice representation is a natural and efficient abstract data type for geo metric computations in robotics and engineering.

Practical Sliced Configuration Spaces for Curved Planar Pairs

In this article, Ipresenta practical configuration-space computation algorithm for pairs of cwvedplanarparts, based on the general algorithm developed by Bajaj and me. The general algorithm advances the theoretical understanding of configuration-space computation, but is too slow andfragilefor some applications. The new algorithm solves these problems by restricting the analysis to parts bounded by line segments and circular arms, whereas the general algorithm handles rationalparametric curves. The trade-off is worthwhile, because the restricted class handles most robotics and mechanical engineering applications. The algorithm reduces run time by a factor of 60 on nine representative engineering pairs, and by a factor of 9 on two human-knee pairs. It also handles common specialpairs by specialized methods. A survey of 2,500 mechanisms shows that these methods cover 90%/O ofpairs and yield an additionalfactor of 10 reduction in average run time. The theme of this article is that application requirements, as well as intrinsic theoretical interest, should drive configuration-space research.

A dynamic systems perspective on qualitative simulation

Abstract This paper presents a simple geometric interpretation of qualitative simulation (QS), based on the phase space representation of dynamic systems theory. QS consists of two steps: transition analysis determines the sequence of qualitative states that a system traverses and global interpretation derives its long-term behavior. I recast transition analysis as a search problem in phase space and replace the assorted transition rules with two algebraic conditions. The first condition determines transitions between arbitrarily shaped regions in phase space, as opposed to QS, which only handles n-dimensional rectangles. It also provides more accurate results by considering only the boundaries between regions. The second condition determines whether nearby trajectories approach a fixed point asymptotically. It obtains better results than QS by exploiting local stability properties. I recast global interpretation of dissipative systems as a search for attractors in phase space and present a global interpretation strategy for the subset of these systems whose local behavior determines global behavior uniquely. Although limited in scope, this strategy handles many systems that appear in the QS literature.