Tracing sub-surface swept profiles of tapered toroidal end mills between level cuts

Abstract Development of closed-form solutions and algorithms for constructing sub-surface swept profiles (SWP) of toroidal and conical bodies is presented in this paper. While the problem of identifying the entire SWP of such surfaces has been extensively investigated in extant studies, construction of subsurface SWPs has rarely been addressed despite the subject being of great significance to machining process employing nonstandard-shaped NC tools. Torus shapes considered in extant literature are restricted to the fourth quadrant of a tube cross section. In industrial applications, however, profile cutters contain different regions of a toroidal surface. To identify SWP elements in the proposed study, a single analytical expression in one variable has been deduced using two moving frames. The basic idea behind such a formulation is to employ the one-to-many strategy, which greatly reduces the computational cost and effort. Algorithms to identify feasible domains of SWP parameters at each level cut, where toroidal and conical surfaces meet, have also been proposed in this study. This is important, since cutting a tool surfaces along the rotation axis divides SWP-parameter domains into non overlapping sets of intervals that must be addressed for each tool posture. In addition, this study demonstrates that for certain tool postures, while C1 continuity between sub-surfaces is satisfied, the SWP connectivity is lost at some points. To locate these so called singular-characteristic points, some precomputation steps have been performed. Lastly, several factors affecting the smoothness of SWPs have been identified and discussed. Highlights Closed form solutions have been derived for constructing the sub-swept profiles of toroidal tools. Three algorithms have been presented to identify the feasible domains of swept profile parameters. In order to locate the singular-characteristic points some precomputation steps have been carried out. Finally, several factors, affecting the smoothness of the swept profiles, have been identified.

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Five Axis Swept Profiles of Torus Like Cutters via Separation of Inner and Outer Characteristic Curves

Volume 1A: 38th Computers and Information in Engineering Conference ◽

10.1115/detc2018-85619 ◽

2018 ◽

Cited By ~ 1

Author(s):

Eyyup Aras

Keyword(s):

Tool Path ◽

Rotation Axis ◽

Linear Interpolation ◽

Unit Vector ◽

Industrial Applications ◽

Body Motion ◽

Moving Frames ◽

Closed Form Solutions ◽

Toroidal Surface ◽

Five Axis

A broadly applicable formulation for identifying the swept profiles (SWP) generated by subsets of a toroidal surface is presented. While the problem of locating the entire SWP of a torus has been extensively addressed in the literature, this rarely addressed problem is of significance to NC machining with non-standard shape of milling tools. A torus, generated by revolving a circle about an axis coplanar with the circle, is made up of inner and outer parts of a tube. The common use of the torus is in a fillet-end mill which contains only the fourth quadrant of a cross section of the tube. However, in the industrial applications the different regions of the torus geometry appear. Especially we can see this on the profile cutters, such as the corner-rounding and concave-radius end mills. Also to the best of our knowledge, the interior of the torus-tube is either neglected or represented by B-spline curves in literature. In case of common milling tool surfaces such as sphere, cylinder and frustum there exists only one SWP in any instance of a tool movement. But, in case of the toroidal surface there exist two sophisticated SWPs and we need to consider only one of them in tool swept envelope generation. Therefore, considering the complexity of five-axis tool motions there is a need not only to distinguish the front from the rear of the cutter but also the exterior from the interior of a tube. This paper presents a methodology and algorithms for analytically formulating the SWP of any sub-set of the torus in five-axis tool motions. By introducing the rigid body motion theory, two moving frames along with a fixed frame are defined. Arbitrary poses of a tool between tool path locations are interpolated by a spherical linear interpolation (slerp) whose effect is a rotation with uniform angular velocity around a fixed rotation axis. For the problem of NC simulation, by using the envelope theory the closed-form solutions of swept profiles are formulated as two-unit vector functions.

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Generalized Truncated Methods for an Efficient Solution of Retrial Systems

Mathematical Problems in Engineering ◽

10.1155/2008/183089 ◽

2008 ◽

Vol 2008 ◽

pp. 1-15 ◽

Cited By ~ 6

Author(s):

Ma Jose Domenech-Benlloch ◽

Jose Manuel Gimenez-Guzman ◽

Vicent Pla ◽

Jorge Martinez-Bauset ◽

Vicente Casares-Giner

Keyword(s):

Closed Form ◽

State Space ◽

Efficient Solution ◽

Analytic Solution ◽

Computational Cost ◽

Retrial Queues ◽

Performance Parameters ◽

Approximate Methods ◽

Closed Form Solutions ◽

Wide Range

We are concerned with the analytic solution of multiserver retrial queues including the impatience phenomenon. As there are not closed-form solutions to these systems, approximate methods are required. We propose two different generalized truncated methods to effectively solve this type of systems. The methods proposed are based on the homogenization of the state space beyond a given number of users in the retrial orbit. We compare the proposed methods with the most well-known methods appeared in the literature in a wide range of scenarios. We conclude that the proposed methods generally outperform previous proposals in terms of accuracy for the most common performance parameters used in retrial systems with a moderated growth in the computational cost.

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Dynamic Catalysis Fundamentals: I. Fast calculation of limit cycles in dynamic catalysis

10.26434/chemrxiv-2021-10hk4 ◽

2021 ◽

Author(s):

Brandon Foley ◽

Neil Razdan

Keyword(s):

Steady State ◽

Differential Equations ◽

Closed Form ◽

Linear Systems ◽

Computational Cost ◽

Kinetic Equations ◽

Reaction Rates ◽

Closed Form Solutions ◽

Non Linear ◽

Non Linear Systems

Dynamic catalysis—the forced oscillation of catalytic reaction coordinate potential energy surfaces (PES)—has recently emerged as a promising method for the acceleration of heterogeneously-catalyzed reactions. Theoretical study of enhancement of rates and supra-equilibrium product yield via dynamic catalysis has, to-date, been severely limited by onerous computational demands of forward integration of stiff, coupled ordinary differential equations (ODEs) that are necessary to quantitatively describe periodic cycling between PESs. We establish a new approach that reduces, by ≳108×, the computational cost of finding the time-averaged rate at dynamic steady state (i.e. the limit cycle for linear and nonlinear systems of kinetic equations). Our developments are motivated by and conceived from physical and mathematical insight drawn from examination of a simple, didactic case study for which closed-form solutions of rate enhancement are derived in explicit terms of periods of oscillation and elementary step rate constants. Generalization of such closed-form solutions to more complex catalytic systems is achieved by introducing a periodic boundary condition requiring the dynamic steady state solution to have the same periodicity as the kinetic oscillations and solving the corresponding differential equations by linear algebra or Newton-Raphson-based approaches. The methodology is well-suited to extension to non-linear systems for which we detail the potential for multiple solutions or solutions with different periodicities. For linear and non-linear systems alike, the acute decrement in computational expense enables rapid optimization of oscillation waveforms and, consequently, accelerates understanding of the key catalyst properties that enable maximization of reaction rates, conversions, and selectivities during dynamic catalysis.

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A Semidefinite Relaxation Method for Elliptical Location

Electronics ◽

10.3390/electronics9010128 ◽

2020 ◽

Vol 9 (1) ◽

pp. 128

Author(s):

Xin Wang ◽

Ying Ding ◽

Le Yang

Keyword(s):

Maximum Likelihood ◽

Closed Form ◽

Maximum Likelihood Estimator ◽

Communication Systems ◽

Computational Cost ◽

Semidefinite Relaxation ◽

Noise Levels ◽

Likelihood Estimator ◽

Closed Form Solutions ◽

Wide Range

Wireless location is a supporting technology in many application scenarios of wireless communication systems. Recently, an increasing number of studies have been conducted on range-based elliptical location in a variety of backgrounds. Specifically, the design and implementation of position estimators are of great significance. The difficulties arising from implementing a maximum likelihood estimator for elliptical location come from the nonconvexity of the negative log-likelihood functions. The need for computational efficiency further enhances the difficulties. Traditional algorithms suffer from the problems of high computational cost and low initialization justifiability. On the other hand, existing closed-form solutions are sensitive to the measurement noise levels. We recognize that the root of these drawbacks lies in an oversimplified linear approximation of the nonconvex model, and accordingly design a maximum likelihood estimator through semidefinite relaxation for elliptical location. We relax the elliptical location problems to semidefinite programs, which can be solved efficiently with interior-point methods. Additionally, we theoretically analyze the complexity of the proposed algorithm. Finally, we design and carry out a series of simulation experiments, showing that the proposed algorithm outperforms several widely used closed-form solutions at a wide range of noise levels. Extensive results under extreme noise conditions verify the deployability of the algorithm.

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Closed-form preintegration methods for graph-based visual–inertial navigation

The International Journal of Robotics Research ◽

10.1177/0278364919835021 ◽

2019 ◽

Vol 38 (5) ◽

pp. 563-586 ◽

Cited By ~ 12

Author(s):

Kevin Eckenhoff ◽

Patrick Geneva ◽

Guoquan Huang

Keyword(s):

Closed Form ◽

Computational Cost ◽

Inertial Navigation ◽

Measurement Unit ◽

Intensity Difference ◽

Navigation Systems ◽

Camera Motion ◽

Piecewise Constant ◽

Closed Form Solutions ◽

Tightly Coupled

In this paper, we propose a new analytical preintegration theory for graph-based sensor fusion with an inertial measurement unit (IMU) and a camera (or other aiding sensors). Rather than using discrete sampling of the measurement dynamics as in current methods, we derive the closed-form solutions to the preintegration equations, yielding improved accuracy in state estimation. We advocate two new different inertial models for preintegration: (i) the model that assumes piecewise constant measurements; and (ii) the model that assumes piecewise constant local true acceleration. Through extensive Monte Carlo simulations, we show the effect that the choice of preintegration model has on estimation performance. To validate the proposed preintegration theory, we develop both direct and indirect visual–inertial navigation systems (VINSs) that leverage our preintegration. In the first, within a tightly coupled, sliding-window optimization framework, we jointly estimate the features in the window and the IMU states while performing marginalization to bound the computational cost. In the second, we loosely couple the IMU preintegration with a direct image alignment that estimates relative camera motion by minimizing the photometric errors (i.e., image intensity difference), allowing for efficient and informative loop closures. Both systems are extensively validated in real-world experiments and are shown to offer competitive performance to state-of-the-art methods.

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