Thread parallelism in Trilinos' sparse linear algebra interfaces & linear solvers.
Abstract not provided.
Hoemmen, Mark Frederick↗
SEARCH · Engineering Papers
Results for “linearization”
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Estimating Edge-Local Triangle Count Heavy Hitters in Edge-Linear Time and Almost-Vertex-Linear Space
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Computer science↗
Linear Viscoelastic Properties of Putative Cyclic Polymers Synthesized by Reversible Radical Recombination Polymerization (R3P)
Linear viscoelastic properties in both melt and solution states are reported for a series of poly(3,6-dioxa-1,8-octanedithiol) (polyDODT) made by reversible radical recombination polymerization (R 3 P) under conditions designed to produce linear (LDODT), cyclic (RDODT) and linear-cyclic mixtures (LRDODT). PolyDODT is amorphous (T g < -50 °C) and highly flexible (entanglement molecular weight M e,lin ≈ 1850 g/mol for LDODT). PolyDODT’s low T g and low M e,lin enable characterization over a wide dynamic range and a wide range of dimensionless weight-average molecular weight Z w = M w /M e,lin . Measurements at temperatures from -57 °C to 100 °C provide up to 18 deades of reduced frequency, which is necessary to characterize RDODT melts with Z w from 23 to 300. The two highest molecular weight polymers in the present RDODT series have such high M w (406k and 556k g/mol) that mass spectrometry, NMR spectroscopy, and even chemical assays for chain ends are unable to rule out up to 2 mol% linear contaminant. By studying the samples in solution (using dilution to reduce Z w ) we could compare their dynamics with those of previously established high-purity polystyrene (PS) rings (limited to Z w ≤ 13.6). RDODT solutions with Z w < 15 (concentrations <5wt% for RDODT-406k and 556k) have dynamic moduli G^* that accord with LCCC-purified PS rings in terms of the frequency dependence (including the absence of a plateau), the progression of shapes of G* as a function of Z w , and the linear scaling of their zero-shear viscosity η0 with M w . The shape of G* as a function of Z w for solutions of RDODT-406k and -556k also accord with lower M w RDODT melts (which have ≤ 1.3mol% linear contaminant). Thus, measurement of the linear viscoelastic properties of appropriate concentrations of high M w (>200k g/mol) putative cyclic polymers, in which linear chains evade spectroscopic detection, may provide an alternative means (though not fully proven) of validation of sample purity. When Z w > 15 (including all seven RDODT melts and eight of their solutions), G* has a rubbery plateau. This suggests that the onset of entanglement-like behavior in rings requires 4-5-fold greater Z w than is required for linear chains. Further, the plateau moduli of RDODT samples are indistinguishable from G N o of the corresponding LDODT (melt or matched-concentration solutions). In entangled linear polymers, the observation that G N o is independent of Z w follows from limitations on lateral fluctuations due to neighboring chains becoming independent of position along a given chain. The present results for RDODT suggest that this holds for sufficiently long endless chains, too. While the RDODT have the same G N o as entangled LDODT, when Z w > 60 the terminal relaxation, if reached at all, of RDODT extends to orders of magnitude lower frequency than an entangled linear polymer of the same Z w . Consequently, the viscosity of RDODT with Z w > 60 increases with Z w much more strongly than the 3.4-power observed for entangled linear polymers. Lastly, these novel polymers, with disulfide-linked backbone and broad relaxation time distribution may prove important in relation to biodegradable elastomers and materials with exceptional low-frequency dissipation, extending at least 12 decades below the onset of the rubbery plateau.
32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
Climatology of Linear Mesoscale Convective System Morphology in the United States based on Random Forests Method
This study uses machine learning methods, specifically the random forest (RF), on a radar-based mesoscale convective system (MCS) tracking dataset to classify the five types of linear MCS morphology in the contiguous United States during the period 2004-2016. The algorithm is trained using radar- and satellite-derived spatial and morphological parameters, and reanalysis environmental information from 5-26yr manually identified nonlinear and five linear MCS modes. The algorithm is then used to automate the classification of linear MCSs over 8 years with high accuracy, providing a systematic, long-term climatology of linear MCSs. Results reveal that nearly 40% of MCSs are classified as linear MCSs, in which half of the linear events belong to the type of system having a leading convective line. The occurrence of linear MCSs shows large annual and seasonal variations. On average, 113 linear MCSs occur annually during the warm season (through March to October), with most of these events clustered from May through August in the central eastern Great Plains. MCS characteristics, including duration, propagation speed, orientation, and system cloud size, have large variability among the different linear modes. The systems having a trailing convective line and the systems having a back-building area of convection typically move more slowly and have higher precipitation rate, and thus have higher potential in producing extreme rainfall and flash flooding. Analysis of the environmental conditions associated with linear MCSs show that the storm-relative flow is of most importance in determining the organization mode of linear MCSs.
Cui, Wenjun↗
A comparative numerical analysis of linear and nonlinear aerodynamic sound generation by vortex disturbances in homentropic constant shear flows
Aerodynamic sound generation in shear flows is investigated in the light of the breakthrough in hydrodynamics stability theory in the 1990s, where generic phenomena of non-normal shear flow systems were understood. By applying the thereby emerged short-time/non-modal approach, the sole linear mechanism of wave generation by vortices in shear flows was captured [G. D. Chagelishvili, A. Tevzadze, G. Bodo, and S. S. Moiseev, “Linear mechanism of wave emergence from vortices in smooth shear flows,” Phys. Rev. Lett. 79, 3178-3181 (1997); B. F. Farrell and P. J. Ioannou, “Transient and asymptotic growth of two-dimensional perturbations in viscous compressible shear flow,” Phys. Fluids 12, 3021-3028 (2000); N. A. Bakas, “Mechanism underlying transient growth of planar perturbations in unbounded compressible shear flow,” J. Fluid Mech. 639, 479-507 (2009); and G. Favraud and V. Pagneux, “Superadiabatic evolution of acoustic and vorticity perturbations in Couette flow,” Phys. Rev. E 89, 033012 (2014)]. Its source is the non-normality induced linear mode-coupling, which becomes efficient at moderate Mach numbers that is defined for each perturbation harmonic as the ratio of the shear rate to its characteristic frequency. Based on the results by the non-modal approach, we investigate a two-dimensional homentropic constant shear flow and focus on the dynamical characteristics in the wavenumber plane. This allows to separate from each other the participants of the dynamical processes — vortex and wave modes — and to estimate the efficacy of the process of linear wave-generation. This process is analyzed and visualized on the example of a packet of vortex modes, localized in both, spectral and physical, planes. Further, by employing direct numerical simulations, the wave generation by chaotically distributed vortex modes is analyzed and the involved linear and nonlinear processes are identified. The generated acoustic field is anisotropic in the wavenumber plane, which results in highly directional linear sound radiation, whereas the nonlinearly generated waves are almost omni-directional. As part of this analysis, we compare the effectiveness of the linear and nonlinear mechanisms of wave generation within the range of validity of the rapid distortion theory and show the dominance of the linear aerodynamic sound generation. Finally, topological differences between the linear source term of the acoustic analogy equation and of the anisotropic non-normality induced linear mechanism of wave generation are found.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Frequency-domain full-waveform inversion with non-linear descent directions
Full-waveform inversion (FWI) is a highly non-linear inverse problem, normally solved iteratively, with each iteration involving an update constructed through linear operations on the residuals. Incorporating a flexible degree of non-linearity within each update may have important consequences for convergence rates, determination of low model wavenumbers and discrimination of parameters. In this study, we examine one approach for doing so, wherein higher order scattering terms are included within the sensitivity kernel during the construction of the descent direction, adjusting it away from that of the standard Gauss–Newton approach. These scattering terms are naturally admitted when we construct the sensitivity kernel by varying not the current but the to-be-updated model at each iteration. Linear and/or non-linear inverse scattering methodologies allow these additional sensitivity contributions to be computed from the current data residuals within any given update. We show that in the presence of pre-critical reflection data, the error in a second-order non-linear update to a background of s 0 is, in our scheme, proportional to at most (Δs/s 0 ) 3 in the actual parameter jump Δs causing the reflection. In contrast, the error in a standard Gauss–Newton FWI update is proportional to (Δs/s 0 ) 2 . For numerical implementation of more complex cases, we introduce a non-linear frequency-domain scheme, with an inner and an outer loop. A perturbation is determined from the data residuals within the inner loop, and a descent direction based on the resulting non-linear sensitivity kernel is computed in the outer loop. We examine the response of this non-linear FWI using acoustic single-parameter synthetics derived from the Marmousi model. Lastly, the inverted results vary depending on data frequency ranges and initial models, but we conclude that the non-linear FWI has the capability to generate high-resolution model estimates in both shallow and deep regions, and to converge rapidly, relative to a benchmark FWI approach involving the standard gradient.
58 GEOSCIENCES↗
Predicting oropharyngeal tumor volume throughout the course of radiation therapy from pretreatment computed tomography data using general linear models
Purpose: The purpose of this work was to develop and evaluate the accuracy of several predictive models of variation in tumor volume throughout the course of radiation therapy. Methods: Nineteen patients with oropharyngeal cancers were imaged daily with CT-on-rails for image-guided alignment per an institutional protocol. The daily volumes of 35 tumors in these 19 patients were determined and used to generate (1) a linear model in which tumor volume changed at a constant rate, (2) a general linear model that utilized the power fit relationship between the daily and initial tumor volumes, and (3) a functional general linear model that identified and exploited the primary modes of variation between time series describing the changing tumor volumes. Primary and nodal tumor volumes were examined separately. The accuracy of these models in predicting daily tumor volumes were compared with those of static and linear reference models using leave-one-out cross-validation. Results: In predicting the daily volume of primary tumors, the general linear model and the functional general linear model were more accurate than the static reference model by 9.9% (range: −11.6%–23.8%) and 14.6% (range: −7.3%–27.5%), respectively, and were more accurate than the linear reference model by 14.2% (range: −6.8%–40.3%) and 13.1% (range: −1.5%–52.5%), respectively. In predicting the daily volume of nodal tumors, only the 14.4% (range: −11.1%–20.5%) improvement in accuracy of the functional general linear model compared to the static reference model was statistically significant. Conclusions: A general linear model and a functional general linear model trained on data from a small population of patients can predict the primary tumor volume throughout the course of radiation therapy with greater accuracy than standard reference models. These more accurate models may increase the prognostic value of information about the tumor garnered from pretreatment computed tomography images and facilitate improved treatment management.
62 RADIOLOGY AND NUCLEAR MEDICINE↗
The non-linear power spectrum of the Lyman alpha forest
The Lyman alpha forest power spectrum has been measured on large scales by the BOSS survey in SDSS-III at z∼ 2.3, has been shown to agree well with linear theory predictions, and has provided the first measurement of Baryon Acoustic Oscillations at this redshift. However, the power at small scales, affected by non-linearities, has not been well examined so far. We present results from a variety of hydrodynamic simulations to predict the redshift space non-linear power spectrum of the Lyα transmission for several models, testing the dependence on resolution and box size. A new fitting formula is introduced to facilitate the comparison of our simulation results with observations and other simulations. The non-linear power spectrum has a generic shape determined by a transition scale from linear to non-linear anisotropy, and a Jeans scale below which the power drops rapidly. In addition, we predict the two linear bias factors of the Lyα forest and provide a better physical interpretation of their values and redshift evolution. The dependence of these bias factors and the non-linear power on the amplitude and slope of the primordial fluctuations power spectrum, the temperature-density relation of the intergalactic medium, and the mean Lyα transmission, as well as the redshift evolution, is investigated and discussed in detail. A preliminary comparison to the observations shows that the predicted redshift distortion parameter is in good agreement with the recent determination of Blomqvist et al., but the density bias factor is lower than observed. We make all our results publicly available in the form of tables of the non-linear power spectrum that is directly obtained from all our simulations, and parameters of our fitting formula.
79 ASTRONOMY AND ASTROPHYSICS↗
Non-linear stochastic growth rates and redshift space distortions
The linear growth rate is commonly defined through a simple deterministic relation between the velocity divergence and the matter overdensity in the linear regime. We introduce a formalism that extends this to a non-linear, stochastic relation between θ = ∇ ∙ v(x,t)/aH and δ. This provides a new phenomenological approach that examines the conditional mean <θ|δ>, together with the fluctuations of θ around this mean. We also measure these stochastic components using N-body simulations and find they are non-negative and increase with decreasing scale from ~10 per cent at k < 0.2 h Mpc -1 to 25 per cent at k ~ 0.45 h Mpc -1 at z = 0. Both the stochastic relation and non-linearity are more pronounced for haloes, M ≤ 5 × 10 12 M ⊙ h -1 , compared to the dark matter at z = 0 and 1. Non-linear growth effects manifest themselves as a rotation of the mean <θ|δ> away from the linear theory prediction -f LT δ, where f LT is the linear growth rate. This rotation increases with wavenumber, k, and we show that it can be well-described by second-order Lagrangian perturbation theory (2LPT) fork < 0.1 h Mpc -1 . Furthermore, the stochasticity in the θ – δ relation is not so simply described by 2LPT, and we discuss its impact on measurements of f LT from two-point statistics in redshift space. Furthermore, given that the relationship between δ and θ is stochastic and non-linear, this will have implications for the interpretation and precision of f LT extracted using models which assume a linear, deterministic expression.
79 ASTRONOMY AND ASTROPHYSICS↗
Composite entanglement topology and extensional rheology of symmetric ring-linear polymer blends
Extensive molecular simulations are applied to characterize the equilibrium dynamics, entanglement topology, and nonlinear extensional rheology of symmetric ring-linear polymer blends with systematically varied ring fraction ϕR. Chains with degree of entanglement Z≈14 are mixed to produce 10 well-entangled systems with ϕR varying from neat linear to neat ring melts. Z is large enough that except for very large ϕR, the rings are threaded by multiple linear chains in equilibrium. Primitive path analysis is used to visualize and quantify the structure of the composite ring-linear entanglement network. We measure the quantity of ring-linear threading and linear-linear entanglement as a function of ϕR and identify with simple arguments a ring fraction ϕR≈0.4 where the topological constraints of the composite entanglement network are maximized. These topological analyses are used to rationalize the ϕR-dependence of ring and linear chain dynamics, conformations, and rheology. Simulations of startup uniaxial elongation flows demonstrate the extensional stress overshoot observed in recent filament stretching experiments and characterize how it depends on the blend composition and entanglement topology. The overshoot is driven by an overstretching and recoil of ring polymers due to the convective unthreading of rings from linear chains.
O’Connor, Thomas C.↗
On the Convergence of Inexact Predictor-Corrector Methods for Linear Programming
Interior point methods (IPMs) are a common approach for solving linear programs (LPs) with strong theoretical guarantees and solid empirical performance. The time complexity of these methods is dominated by the cost of solving a linear system of equations at each iteration. In common applications of linear programming, particularly in machine learning and scientific computing, the size of this linear system can become prohibitively large, requiring the use of iterative solvers, which provide an approximate solution to the linear system. However, approximately solving the linear system at each iteration of an IPM invalidates the theoretical guarantees of common IPM analyses. To remedy this, we theoretically and empirically analyze (slightly modified) predictor-corrector IPMs when using approximate linear solvers: our approach guarantees that, when certain conditions are satisfied, the number of IPM iterations does not increase and that the final solution remains feasible. We also provide practical instantiations of approximate linear solvers that satisfy these conditions for special classes of constraint matrices using randomized linear algebra.
Dexter, Gregory↗
The Linear Point Standard Ruler with DESI DR1 and DR2 Data
The linear point, a purely geometric feature in the monopole of the two-point correlation function, has been proposed as an alternative standard ruler. Compared to the peak in the correlation function, it is more robust to late-time nonlinear effects at the percent level. In light of improved simulations and high quality data, we revisit the robustness of the linear point and use it as an alternative to template-based fitting approaches typically used in BAO analyses. We present the linear point measurements on galaxy samples from the first and second data releases (DR1 and DR2) of the DESI survey. We convert the linear point into a dimensionless parameter $α_{iso,LP}$, defined as the ratio of the linear point in the fiducial cosmology and the observed value, analogous to the isotropic BAO scaling parameter $α_{iso}$ used in previous BAO measurements. Using the 2nd generation of AbacusSummit mock catalogs, we find that linear point measurements are more precise when calculated in the post-reconstruction regime with 15-60% smaller uncertainties than those pre-reconstruction. We find a systematic shift in the linear point measurements compared against the isotropic BAO measurements in mocks; we attribute this to the isotropic damping parameter responsible for smearing the linear point in the nonlinear regime. We propose a sample-dependent correction that mitigates the impact of late-time nonlinear effects. While this introduces a cosmology dependence in an otherwise model-independent measurement, this is necessary given the sub-percent precision dictated by current cosmological surveys. Comparing $α_{iso,LP}$ with isotropic BAO measurements made on the DESI DR1 and DR2 galaxy samples, we find excellent agreement after applying this correction, particularly post-reconstruction. We discuss future scope regarding cosmological inference with linear point measurements.
Uberoi, N. [Yale U.] (ORCID:0000000275179629)↗
MO-F-16A-02: Simulation of a Medical Linear Accelerator for Teaching Purposes
Purpose: Detailed functioning of linear accelerator physics is well known. Less well developed is the basic understanding of how the adjustment of the linear accelerator's electrical components affects the resulting radiation beam. Other than the text by Karzmark, there is very little literature devoted to the practical understanding of linear accelerator functionality targeted at the radiotherapy clinic level. The purpose of this work is to describe a simulation environment for medical linear accelerators with the purpose of teaching linear accelerator physics. Methods: Varian type lineacs were simulated. Klystron saturation and peak output were modelled analytically. The energy gain of an electron beam was modelled using load line expressions. The bending magnet was assumed to be a perfect solenoid whose pass through energy varied linearly with solenoid current. The dose rate calculated at depth in water was assumed to be a simple function of the target's beam current. The flattening filter was modelled as an attenuator with conical shape, and the time-averaged dose rate at a depth in water was determined by calculating kerma. Results: Fifteen analytical models were combined into a single model called SIMAC. Performance was verified systematically by adjusting typical linac control parameters. Increasing klystron pulse voltage increased dose rate to a peak, which then decreased as the beam energy was further increased due to the fixed pass through energy of the bending magnet. Increasing accelerator beam current leads to a higher dose per pulse. However, the energy of the electron beam decreases due to beam loading and so the dose rate eventually maximizes and the decreases as beam current was further increased. Conclusion: SIMAC can realistically simulate the functionality of a linear accelerator. It is expected to have value as a teaching tool for both medical physicists and linear accelerator service personnel.
43 PARTICLE ACCELERATORS↗
Jet wake from linearized hydrodynamics
We explore how to improve the hybrid model description of the particles originating from the wake that a jet produced in a heavy ion collision leaves in the droplet of quark-gluon plasma (QGP) through which it propagates, using linearized hydrodynamics on a background Bjorken flow. Jet energy and momentum loss described by the hybrid model become currents sourcing linearized hydrodynamics. By solving the linearized hydrodynamic equations numerically, we investigate the development of the wake in the dynamically evolving droplet of QGP, study the effect of viscosity, scrutinize energy-momentum conservation, and check the validity of the linear approximation. We find that linearized hydrodynamics works better in the viscous case because diffusive modes damp the energy-momentum perturbation produced by the jet. We calculate the distribution of particles produced from the jet wake by using the Cooper-Frye prescription and find that both the transverse momentum spectrum and the distribution of particles in azimuthal angle are similar in shape in linearized hydrodynamics and in the hybrid model. Their normalizations are different because the momentum-rapidity distribution in the linearized hydrodynamics analysis is more spread out, due to sound modes. Since the Bjorken flow has no transverse expansion, we explore the effect of transverse flow by using local boosts to add it into the Cooper-Frye formula. After including the effects of transverse flow in this way, the transverse momentum spectrum becomes harder: more particles with transverse momenta bigger than 2 GeV are produced than in the hybrid model. Although we defer implementing this analysis in a jet Monte Carlo, as would be needed to make quantitative comparisons to data, we gain a qualitative sense of how the jet wake may modify jet observables by computing proxies for two example observables: the lost energy recovered in a cone of varying open angle, and the fragmentation function. We find that linearized hydrodynamics with transverse flow effects added improves the description of the jet wake in the hybrid model in just the way that comparison to data indicates is needed. Our study illuminates a path to improving the description of the wake in the hybrid model, highlighting the need to take into account the effects of both transverse flow and the broadening of the energy-momentum perturbation in spacetime rapidity on particle production.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Threading–Unthreading Transition of Linear-Ring Polymer Blends in Extensional Flow
Adding small amounts of ring polymers to a matrix of their linear counterparts is known to increase the zero-shear-rate viscosity because of linear-ring threading. Uniaxial extensional rheology measurements show that, unlike its pure linear and ring constituents, the blend exhibits an overshoot in the stress growth coefficient. By combining these measurements with ex-situ small-angle neutron scattering and nonequilibrium molecular dynamics simulations, this overshoot is shown here to be driven by a transient threading–unthreading transition of rings embedded within the linear entanglement network. Prior to unthreading, embedded rings deform affinely with the linear entanglement network and produce a measurably stronger elongation of the linear chains in the blend compared to the pure linear melt. Thus, rings uniquely alter the mechanisms of transient elongation in linear polymers.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
A Nth-order linear algorithm for extracting diffuse correlation spectroscopy blood flow indices in heterogeneous tissues
Conventional semi-infinite analytical solutions of correlation diffusion equation may lead to errors when calculating blood flow index (BFI) from diffuse correlation spectroscopy (DCS) measurements in tissues with irregular geometries. Very recently, we created an algorithm integrating a Nth-order linear model of autocorrelation function with the Monte Carlo simulation of photon migrations in homogenous tissues with arbitrary geometries for extraction of BFI (i.e., αD{sub B}). The purpose of this study is to extend the capability of the Nth-order linear algorithm for extracting BFI in heterogeneous tissues with arbitrary geometries. The previous linear algorithm was modified to extract BFIs in different types of tissues simultaneously through utilizing DCS data at multiple source-detector separations. We compared the proposed linear algorithm with the semi-infinite homogenous solution in a computer model of adult head with heterogeneous tissue layers of scalp, skull, cerebrospinal fluid, and brain. To test the capability of the linear algorithm for extracting relative changes of cerebral blood flow (rCBF) in deep brain, we assigned ten levels of αD{sub B} in the brain layer with a step decrement of 10% while maintaining αD{sub B} values constant in other layers. Simulation results demonstrate the accuracy (errors < 3%) of high-order (N ≥ 5) linear algorithm in extracting BFIs in different tissue layers and rCBF in deep brain. By contrast, the semi-infinite homogenous solution resulted in substantial errors in rCBF (34.5% ≤ errors ≤ 60.2%) and BFIs in different layers. The Nth-order linear model simplifies data analysis, thus allowing for online data processing and displaying. Future study will test this linear algorithm in heterogeneous tissues with different levels of blood flow variations and noises.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
How does non-linear dynamics affect the baryon acoustic oscillation?
We study the non-linear behavior of the baryon acoustic oscillation in the power spectrum and the correlation function by decomposing the dark matter perturbations into the short- and long-wavelength modes. The evolution of the dark matter fluctuations can be described as a global coordinate transformation caused by the long-wavelength displacement vector acting on short-wavelength matter perturbation undergoing non-linear growth. Using this feature, we investigate the well known cancellation of the high-k solutions in the standard perturbation theory. While the standard perturbation theory naturally satisfies the cancellation of the high-k solutions, some of the recently proposed improved perturbation theories do not guarantee the cancellation. We show that this cancellation clarifies the success of the standard perturbation theory at the 2-loop order in describing the amplitude of the non-linear power spectrum even at high-k regions. We propose an extension of the standard 2-loop level perturbation theory model of the non-linear power spectrum that more accurately models the non-linear evolution of the baryon acoustic oscillation than the standard perturbation theory. The model consists of simple and intuitive parts: the non-linear evolution of the smoothed power spectrum without the baryon acoustic oscillations and the non-linear evolution of the baryon acoustic oscillations due to the large-scale velocity of dark matter and due to the gravitational attraction between dark matter particles. Our extended model predicts the smoothing parameter of the baryon acoustic oscillation peak at z = 0.35 as ∼ 7.7Mpc/h and describes the small non-linear shift in the peak position due to the galaxy random motions.
79 ASTRONOMY AND ASTROPHYSICS↗
Linear delta systems, hexapod systems, and related methods
A linear delta system includes a support frame, rails mounted to the support frame, linear actuators, each linear actuator configured to translate along a longitudinal length of a respective rail, pairs of parallel rods each coupled to the linear actuators, a platform coupled to a longitudinal end of each of the pairs of parallel rods opposite the respective linear actuator, and an object coupled to the platform. Longitudinal axes of the rails are oriented parallel to each other and lie within a common plane or an uncommon plane. A method of forming a linear delta system includes mounting rails to a support frame, the rails having longitudinal axes that are parallel to each other and lying within a common plane, coupling a linear actuator to each of the rails, coupling a pair of parallel rods to each linear actuator, and coupling a platform to the pairs of parallel rods.
Crawford, Anthony L.↗