Probability Tolerance Maps: A New Statistical Model for Non-Linear Tolerance Analysis Applied to Rectangular Faces

2010 ◽  
Author(s):  
Nadeem S. Khan ◽  
Jami J. Shah ◽  
Joseph K. Davidson
Keyword(s):  
Probability Distribution ◽  
Statistical Model ◽  
Ray Tracing ◽  
Delaunay Triangulation ◽  
Point Cloud ◽  
Tolerance Analysis ◽  
Minkowski Sum ◽  
Joint Probability Distribution ◽  
Statistical Parameters ◽  
Perfect Form

A new statistical model for the tolerance analysis based upon joint probability distribution of the trivariate normal distributed variables involved in the construction of Tolerance-maps (T-Maps) for rectangular face is presented. Central to the new model is a Tolerance-Map (T-Map) (Patent No. 69638242). It is the range of points resulting from a one-to-one mapping from all the variational possibilities of a perfect-form feature, within its tolerance-zone, to a specially designed Euclidean point-space. The model is fully compatible with the ASME/ANSI/ISO Standard for geometric tolerances. In this research, 4D probability T-Maps (prob T-Maps) have been developed in which the probability value of a point in space is represented by the size of the marker and the associated color. Additionally, 3D prob T-Maps (3D cross sections of the 4D prob T-Maps at pre specified values) are used to represent the probability values of two variables at a time for a constant value of the third variable on a plane. Superposition of the probability point cloud with the T-Map clearly identifies which points are inside and which are outside the T-Map. This represents the pass percentage for parts manufactured with the statistical parameters such as mean and standard deviation as of the assumed trivariate probability distribution. The effect of refinement with form and orientation tolerance is highlighted by calculating the change in pass percentage with the pass percentage for size only. Delaunay triangulation and ray tracing algorithms have been used to automate the process of identifying the points inside and outside the T-Map. Proof of concept software has been implemented to demonstrate this model and to determine pass percentages for various cases. The model is further extended to assemblies by employing convolution algorithms on two trivariate statistical distributions to arrive at the statistical distribution of the assembly. Accumulation T-Maps generated by using Minkowski Sum techniques on the T-Maps of the individual parts is superimposed on the probability point cloud resulting from convolution. Delaunay triangulation and ray tracing algorithms are employed to determine the assemleability percentages for the assembly.

Form Defects Consideration in Polytope-Based Tolerance Analysis

2019 ◽  
Vol 141 (6) ◽  
Author(s):  
Ting Liu ◽  
Laurent Pierre ◽  
Nabil Anwer ◽  
Yanlong Cao ◽  
Jiangxin Yang
Keyword(s):  
Tolerance Analysis ◽  
Significant Loss ◽  
Minkowski Sum ◽  
Skin Model ◽  
Model Combining ◽  
Perfect Form ◽  
Finite Set ◽  
Physical Shape ◽  
Polytope Model

The polytope-based tolerance analysis in design process uses a finite set of constraints to represent specifications and propagates these constraints to any objective point in the Euclidean space. The operations of Minkowski sum and intersection on polytopes are well suited to serial and parallel assemblies. The polytope model has been applied to complex assemblies which contain a large number of joints and geometrical tolerances. However, the previous studies on this model consider toleranced features as surfaces of perfect form. The ignorance of form defects in tolerance analysis would result in a significant loss in accuracy and reliability. In this paper, an extension of the polytope model for tolerance analysis considering form defects is described in which the skin model shape representing the physical shape of the product is adopted to simulate the actual toleranced feature in place of the substitute one used conventionally. The combination of polytope model and skin model shape is expected to inherit many of the advantages of each model, combining easy-to-use tolerance propagation and form defects representation with accuracy guarantees. To demonstrate the method and its respective application, a case study of an assembly is illustrated in detail. The proposed method further enhances the capability of the polytope model in handling form defects and provides more realistic assembly results that approximate the actual assembly conditions for design evaluation.

Statistical Tolerance Analysis With Sensitivities Established With Tolerance-Maps

2018 ◽  
Author(s):  
Aniket N. Chitale ◽  
Joseph K. Davidson ◽  
Jami J. Shah
Keyword(s):  
Tolerance Analysis ◽  
Minkowski Sum ◽  
Coordinate Frame ◽  
Target Feature ◽  
Functional Feature ◽  
Design Function ◽  
Statistical Tolerance ◽  
The One ◽  
The Relationship ◽  
Statistical Tolerance Analysis

The purpose of math models for tolerances is to aid a designer in assessing relationships between tolerances that contribute to variations of a dependent dimension that must be controlled to achieve some design function and which identifies a target (functional) feature. The T-Maps model for representing limits to allowable manufacturing variations is applied to identify the sensitivity of a dependent dimension to each of the contributing tolerances to the relationship. The method is to choose from a library of T-Maps the one that represents, in its own local (canonical) reference frame, each contributing feature and the tolerances specified on it; transform this T-Map to a coordinate frame centered at the target feature; obtain the accumulation T-Map for the assembly with the Minkowski sum; and fit a circumscribing functional T-Map to it. The fitting is accomplished numerically to determine the associated functional tolerance value. The sensitivity for each contributing tolerance-and-feature combination is determined by perturbing the tolerance, refitting the functional map to the accumulation map, and forming a ratio of incremental tolerance values from the two functional T-Maps. Perturbing the tolerance-feature combinations one at a time, the sensitivities for an entire stack of contributing tolerances can be built. For certain classes of loop equations, the same sensitivities result by fitting the functional T-Map to the T-Map for each feature, one-by-one, and forming the overall result as a scalar sum. Sensitivities help a designer to optimize tolerance assignments by identifying those tolerances that most strongly influence the dependent dimension at the target feature. Since the fitting of the functional T-Map is accomplished by intersection of geometric shapes, all the T-Maps are constructed with linear half-spaces.

scholarly journals ANALYSIS OF 3D BUILDING MODELS ACCURACY BASED ON THE AIRBORNE LASER SCANNING POINT CLOUDS

2018 ◽  
Vol XLII-2 ◽  
pp. 797-804 ◽  
Author(s):  
W. Ostrowski ◽  
M. Pilarska ◽  
J. Charyton ◽  
K. Bakuła
Keyword(s):  
Point Cloud ◽  
Laser Scanning ◽  
Large Scale ◽  
Point Clouds ◽  
3D Models ◽  
Airborne Laser Scanning ◽  
Aerial Images ◽  
Statistical Parameters ◽  
Airborne Laser ◽  
Building Models

Creating 3D building models in large scale is becoming more popular and finds many applications. Nowadays, a wide term “3D building models” can be applied to several types of products: well-known CityGML solid models (available on few Levels of Detail), which are mainly generated from Airborne Laser Scanning (ALS) data, as well as 3D mesh models that can be created from both nadir and oblique aerial images. City authorities and national mapping agencies are interested in obtaining the 3D building models. Apart from the completeness of the models, the accuracy aspect is also important. Final accuracy of a building model depends on various factors (accuracy of the source data, complexity of the roof shapes, etc.). In this paper the methodology of inspection of dataset containing 3D models is presented. The proposed approach check all building in dataset with comparison to ALS point clouds testing both: accuracy and level of details. Using analysis of statistical parameters for normal heights for reference point cloud and tested planes and segmentation of point cloud provides the tool that can indicate which building and which roof plane in do not fulfill requirement of model accuracy and detail correctness. Proposed method was tested on two datasets: solid and mesh model.

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