Rigid-body Motion Constraint Research Articles

Geometric errors identification considering rigid-body motion constraint for rotary axis of multi-axis machine tool using a tracking interferometer

The measurement and identification of the four position-independent geometric errors (PIGEs) and six position-dependent geometric errors (PDGEs) in the rotary axis are necessary to reduce their contributions to the overall machining errors of a multi-axis machine tool. In this paper, a new geometric error identification method using a tracking interferometer is presented by considering the rigid-body motion constraint in multilateration. The rigid-body motion constraint is introduced to establish a new coordinate calculation model for the measurement points, and an identification process is presented to separately identify the PIGEs and PDGEs by deriving identification models based on established geometric error models. The main novelty of the proposed method lies in the consideration of the rigid-body motion constraint, which makes the identification more robust against random factors. Monte Carlo simulations and verifying experiments were performed for validation. The results show that the PIGEs and PDGEs in the rotary axis can be successfully separated and identified by the new method. Improved identification accuracy compared with the traditional method is achieved. The maximum angular positioning error was reduced by 84% after compensation. A lower uncertainty in the identified errors compared with those obtained by the traditional method and instrument software is achieved. The proposed method is validated by comparing the identification results with those obtained by the instrument software and laser interferometers.

Mobile Formation Coordination and Tracking Control for Multiple Nonholonomic Vehicles

This article addresses forward motion control for trajectory tracking and mobile formation coordination for a group of nonholonomic vehicles on $SE(2)$ . First, by constructing an intermediate attitude variable which involves vehicles’ position information and desired attitude, the translational and rotational control inputs are designed in two stages to solve the trajectory tracking problem. Second, the coordination relationships of relative positions and headings are explored thoroughly for a group of nonholonomic vehicles to maintain a mobile formation with rigid-body motion constraints. We prove that, except for the cases of parallel formation and translational straight line formation, a mobile formation with strict rigid-body motion can be achieved if and only if the ratios of linear speed to angular speed for each individual vehicle are constants. Motion properties for mobile formation with weak rigid-body motion are also demonstrated. Thereafter, based on the proposed trajectory tracking approach, a distributed mobile formation control law is designed under a directed tree graph. The performance of the proposed controllers is validated by both numerical simulations and experiments.

Direct Numerical Simulation of Particles in Spatially Varying Electric Fields †

A numerical scheme is developed to simulate the motion of dielectric particles in the uniform and nonuniform electric fields of microfluidic devices. The motion of particles is simulated using a distributed Lagrange multiplier method (DLM) and the electric force acting on the particles is calculated by integrating the Maxwell stress tensor (MST) over the particle surfaces. One of the key features of the DLM method used is that the fluid-particle system is treated implicitly by using a combined weak formulation, where the forces and moments between the particles and fluid cancel, as they are internal to the combined system. The MST is obtained from the electric potential, which, in turn, is obtained by solving the electrostatic problem. In our numerical scheme, the domain is discretized using a finite element scheme and the Marchuk-Yanenko operator-splitting technique is used to decouple the difficulties associated with the incompressibility constraint, the nonlinear convection term, the rigid-body motion constraint and the electric force term. The numerical code is used to study the motion of particles in a dielectrophoretic cage which can be used to trap and hold particles at its center. If the particles moves away from the center of the cage, a resorting force acts on them towards the center. The MST results show that the ratio of the particle-particle interaction and dielectrophoretic forces decreases with increasing particle size. Therefore, larger particles move primarily under the action of the dielectrophoretic (DEP) force, especially in the high electric field gradient regions. Consequently, when the spacing between the electrodes is comparable to the particle size, instead of collecting on the same electrode by forming chains, they collect at different electrodes.

Accuracy of Finite Volume/Staggered Grid Distributed Lagrange Multiplier/Fictitious Domain simulations of particulate flows

Our Distributed Lagrange Multiplier/Fictitious Domain method [1–3] has been recently implemented with a Finite Volume/Staggered Grid discretization scheme [4] and successfully applied to the settling of a single particle in an unbounded domain. Here we scrutinize the space and time accuracy of the computed solution in a broad range of flow configurations of growing complexity. We suggest a 2nd order interpolation operator to impose the rigid body motion constraint at the particle boundary based on Finite Element cubic quadratic basis functions. This enables us to benefit from the best of both discretization schemes: a fast Finite Volume/Staggered Grid Navier & Stokes solver and a high order Finite Element reconstruction at the particle boundary. The resulting space accuracy of the computed solution is improved without the need to employ any hydrodynamic radius calibration procedure for spherical particles. As a result, the method can also accurately be extended to non-spherical and angular particles. The order of space convergence is between linear and quadratic, depending on the flow configuration, and the magnitude of the error has a tendency to increase with the solid volume fraction, the Reynolds number and the angularity of the particles.

Dynamic Analysis of Planar 3-RRR Flexible Parallel Robots with Dynamic Stiffening

In consideration of the second-order coupling quantity of the axial displacement caused by the transverse displacement of flexible beam, the first-order approximation coupling model of planar 3-RRR flexible parallel robots is presented, in which the rigid body motion constraints, elastic deformation motion constraints, and dynamic constraints of the moving platform are considered. Based on the different speed of the moving platform, numerical simulation results using the conventional zero-order approximation coupling model and the proposed firstorder approximation coupling model show that the effect of “dynamic stiffening” term on dynamic characteristics of the system is insignificant and can be neglected, and the zero-order approximation coupling model is enough precisely for catching essentially dynamic characteristics of the system. Then, the commercial software ANSYS 13.0 is used to confirm the validity of the zero-order approximation coupling model.

Surface Representations for 3D Mapping

Point clouds, i.e., sets of 3D coordinates of surface point samples from obstacles, are the predominant form of representation for 3D mapping. They are the raw data format of most 3D sensors and the basis for state of the art algorithms for 3D scan registration. It is argued here that point clouds have severe limitations and a case is made for a necessary paradigm shift to surface based representations. In addition to several conceptual arguments, it is shown how a surface based approach can be used for fast and robust registration of 3D data without the need for robot motion estimates from other sensors. Concretely, a short overview on own work dubbed 3D Plane SLAM is presented. It features an extraction of planes with uncertainties from 3D range scans. Two scans can then be registered by determining the correspondence set that maximizes the global rigid body motion constraint while finding the related optimal decoupled rotations and translations with their underlying uncertainties. The registered scans are embedded in pose-graph SLAM for loop closing and relaxation.

A comparison between a collocation and weak implementation of the rigid‐body motion constraint on a particle surface

AbstractIn this work, we implemented and compared two different methods to impose the rigid‐body motion constraint on a solid particle moving inside a fluid. We consider a fictitious domain method to easily manage the particle motion. As the solid as well as the fluid inertia are neglected, the particle can be discretized through its boundary only. The rigid‐body motion is imposed via Lagrange multipliers on the boundary. In the first method, such constraints are imposed in discrete points on the boundary (collocation), whereas in the second the constraint is imposed in a weak way on elements dividing the particle surface.Two test problems, that is, a spherical and an ellipsoidal particle in a sheared Newtonian fluid, are chosen to compare the methods. In both cases, the analysis is carried out in 2D as well as in 3D.The results show that for the collocation method an optimal number of collocation points exist leading to the smallest error. However, small variations in the optimal value can generate large deviations. In the weak implementation, the error is only mildly affected by the number of elements used to discretize the particle boundary and by the Lagrange multiplier's interpolation space. A further analysis is carried out to study the effect of an approximated integration of weak constraints.A comparison between the two methods showed that the same accuracy can be achieved by using less constraints if the weak discretization is used. Finally, the rigid‐body motion imposed via weak constraints leads to better conditioned linear systems. Copyright © 2009 John Wiley & Sons, Ltd.

A numerical method for fully resolved simulation (FRS) of rigid particle–flow interactions in complex flows

A fictitious-domain based formulation for fully resolved simulations of arbitrary shaped, freely moving rigid particles in unsteady flows is presented. The entire fluid–particle domain is assumed to be an incompressible, but variable density, fluid. The numerical method is based on a finite-volume approach on a co-located, Cartesian grid together with a fractional step method for variable density, low-Mach number flows. The flow inside the fluid region is constrained to be divergence-free for an incompressible fluid, whereas the flow inside the particle domain is constrained to undergo rigid body motion. In this approach, the rigid body motion constraint is imposed by avoiding the explicit calculation of distributed Lagrange multipliers and is based upon the formulation developed by Patankar [N. Patankar, A formulation for fast computations of rigid particulate flows, Center for Turbulence Research Annual Research Briefs 2001 (2001) 185–196]. The rigidity constraint is imposed and the rigid body motion (translation and rotational velocity fields) is obtained directly in the context of a two-stage fractional step scheme. The numerical approach is applied to both imposed particle motion and fluid–particle interaction problems involving freely moving particles. Grid and time-step convergence studies are performed to evaluate the accuracy of the approach. Finally, simulation of rigid particles in a decaying isotropic turbulent flow is performed to study the feasibility of simulations of particle-laden turbulent flows.

Critical evaluation of simple network models of protein dynamics and their comparison with crystallographic B-factors

In recent years, elastic network models (ENM) have been widely used to describe low-frequency collective motions in proteins. These models are often validated and calibrated by fitting mean-square atomic displacements estimated from x-ray crystallography (B-factors). We show that a proper calibration procedure must account for the rigid-body motion and constraints imposed by the crystalline environment on the protein. These fundamental aspects of protein dynamics in crystals are often ignored in currently used ENMs, leading to potentially erroneous network parameters. Here we develop an ENM that properly takes the rigid-body motion and crystalline constraints into account. Its application to the crystallographic B-factors reveals that they are dominated by rigid-body motion and thus are poorly suited for the calibration of models for internal protein dynamics. Furthermore, the translation libration screw (TLS) model that treats proteins as rigid bodies is considerably more successful in interpreting the experimental B-factors than ENMs. This conclusion is reached on the basis of a comparative study of various models of protein dynamics. To evaluate their performance, we used a data set of 330 protein structures that combined the sets previously used in the literature to test and validate different models. We further propose an extended TLS model that treats the bulk of the protein as a rigid body while allowing for flexibility of chain ends. This model outperforms other simple models of protein dynamics in interpreting the crystallographic B-factors.

A fictitious domain method for dynamic simulation of particle sedimentation in Bingham fluids

In this study, we present a fictitious domain method for the dynamic simulation of particle motion in a Bingham viscoplastic fluid at moderate Bingham numbers. Our method is built on the framework established by Glowinski and his coworkers, in the sense that we use their formulation and their operator-splitting idea to simplify the computation, but differs from their method in both spatial and temporal discretizations of the governing equations. Concerning the space scheme, we use a finite-difference method to discretize equations and a collocation-element method to enforce the rigid-body motion constraint inside the particle boundaries. Concerning the time scheme, the combined system is decoupled into three sub-systems: a Navier–Stokes problem, a plasticity problem and a rigid-body-motion problem, and we solve the Navier–Stokes problem with the classic projection method. The plasticity and rigid-body-motion multipliers at the previous time-level are retained in the Navier–Stokes problem to reduce the splitting error. In addition, the present study shows that retaining a viscous diffusion term in the plasticity problem is not necessary for the convergence of the iteration. The method is verified by comparing our results on the lid-driven cavity flow, the drag coefficient for a sphere settling in a tube and the hydrodynamic interactions between two spheres translating along the tube axis to the data available in the literature. Our results confirm that the drag coefficient for a sphere settling in a Bingham fluid at non-zero Reynolds numbers can be well correlated with an effective Reynolds number. For two approaching spheres, there exists a critical separation distance above which a drag-reduction is observed and below which a drag-enhancement takes place, compared to the drag for a single sphere. The drag-reduction however does not happen to a sphere falling towards a solid wall at the moderate Bingham numbers we studied. These observations are explained by a consideration of the competition between a shear-thinning plastic force and a repulsive lubrication force on the sphere occurring in the squeezing flow.

A fictitious domain method for particulate flows with heat transfer

The distributed-Lagrange-multiplier/fictitious-domain (DLM/FD) method of Glowinski et al. [R. Glowinski, T.-W. Pan, T.I. Hesla, D.D. Joseph, A distributed Lagrange multiplier/fictitious domain method for particulate flows, Int. J. Multiphase Flow 25 (1999) 755–794] is extended to deal with heat transfer in particulate flows in two dimensions. The Boussinesq approximation is employed for the coupling between the flow and temperature fields. The fluid-flow equations are solved with the finite-difference projection method on a half-staggered grid. In our operator splitting scheme, the Lagrange multipliers at the previous time level are kept in the fluid equations, and the new Lagrange multipliers for the rigid-body motion constraint and the Dirichlet temperature boundary condition are determined from the reduced saddle-point problem, whereas a very simple scheme based on the fully explicit computation of the Lagrange multiplier is proposed for the problem in which the solid heat conduction inside the particle boundary is also considered. Our code for the case of fixed temperature on the immersed boundary is verified by comparing favorably our results on the natural convection driven by a hot cylinder eccentrically placed in a square box and on the sedimentation of a cold circular particle in a vertical channel to the data in the literature. The code for the case of freely varying temperature on the boundaries of freely moving particles is applied to analyze the motion of a catalyst particle in a box and in particular the heat conductivities of nanofluids and sheared non-colloidal suspensions, respectively. Our preliminary computational results support the argument that the micro-heat-convection in the fluids is primarily responsible for the unusually high heat conductivity of nanofluids. It is shown that the Peclet number plays a negative role in the diffusion-related heat conductivity of a sheared non-colloidal suspension, whereas the Reynolds number does the opposite.

Global alignment of sensor positions with noisy motion measurements

We investigate the global alignment of some 3-D spatial points, knowing the motion between pairwise nearby positions. A common application is to determine the trajectory of a mobile vision-based system from the scene images acquired along its track. Exploiting redundant measurements and the rigid body motion constraint as the observation model, we apply the mixed adjustment model paradigm to develop recursive estimation algorithms under various scenarios. Results of experiments are given to demonstrate the performance for different noise levels in the observations and the improvements in the position estimation. We also present the results of an experiment with underwater images in the construction of the camera platform trajectory.

Spectral distributed Lagrange multiplier method: algorithm and benchmark tests

We extend the formulation of the distributed Lagrange multiplier (DLM) approach for particulate flows to high-order methods within the spectral/ hp element framework. We implement the rigid-body motion constraint inside the particle via a penalty method. The high-order DLM method demonstrates spectral convergence rate, i.e. discretization errors decrease exponentially as the order of spectral polynomials increases. We provide detailed comparisons between the spectral DLM method, direct numerical simulations, and the force coupling method for a number of 2D and 3D benchmark flow problems. We also validate the spectral DLM method with available experimental data for a transient problem. The new DLM method can potentially be very effective in many-moving body problems, where a smaller number of grid points is required in comparison with low-order methods.

Distributed Lagrange multiplier method for particulate flows with collisions

A modified distributed Lagrange multiplier/fictitious domain method (DLM) that allows particles to undergo collisions is developed for particulate flows. In the earlier versions of the DLM method for Newtonian and viscoelastic liquids the particle surfaces were restricted to be more than one velocity element away from each other. A repulsive body force was applied to the particles when the distance between them was smaller than this critical value. This was necessary for ensuring that conflicting rigid body motion constraints from two different particles are not imposed at the same velocity nodes. In the modified DLM method the particles are allowed to come arbitrarily close to each other and even slightly overlap each other. When conflicting rigid body motion constraints from two different particles are applicable on a velocity node, the constraint from the particle that is closer to that node is used and the other constraint is dropped. An elastic repulsive force is applied when the particles overlap each other. In our simulations, the particles are allowed to overlap as much as one hundredth of the velocity element size. The modified DLM method is implemented for both Newtonian and viscoelastic liquids. Our simulations show that when particles are dropped in a channel, and the viscoelastic Mach number ( M) is less than one and the elasticity number ( E) is greater than one, the particles form a chain parallel to the flow direction. As in experiments, the new method allows particles in the chain to approximately touch each other. The particles dropped in a Newtonian liquid, on the other hand, undergo characteristic drafting, kissing and tumbling. During the touching phase, as in experiments, the two particles touch each other. The modified method thus allows hydrodynamic forces to be fully resolved to within the tolerance of the mesh and thus the extra artificial force in a security zone outside the particle which are used in all other methods are not needed.

Variational solid modeling for tolerance analysis

A variational model is a computer representation of a variational class and stands for a collection of different instances of the part or assembly modeled in CAD. The different basic approaches to variational modeling are reviewed. A surface-based approach to variational modeling is discussed. The approach is applied to solving the problems of eliminating rigid-body motion, handling incidence and tangency constraints, and modeling form variations. The application of variational modeling to automated tolerance analysis is also discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>