NURBS Interpolation Research Articles

A robot motion position and posture control method for freeform surface laser treatment based on NURBS interpolation

A robot motion position and posture control method for freeform surface laser treatment based on NURBS interpolation

NURBS Interpolator with Scheduling Scheme Combining Cubic and Quartic S-shaped Feedrate Profiles Under Drive and Chord Error Constraints

Traditional S-shaped feedrate profiles only account for the initial and end feedrates during scheduling; they fail to consider whether the feed rate should continuously satisfy the constraints of the chord error in the processing step. To overcome this problem, this paper proposes a feedrate scheduling scheme that processes S-shaped quartic feedrate profiles at the initial stage followed by S-shaped cubic feedrate profiles to ensure machining quality and efficiency. First, the scheme traverses the feedrate points that satisfy the chord error to determine a slope suitable for the jerk of S-shaped quartic feedrate profile, in order to ensure that the increase in the feedrates does not exceed its constraints. Second, an iterative method is used to optimally allocate processing time between the S-shaped cubic and quartic feedrate profiles to optimize processing efficiency. Finally, a method for determining the new maximum feedrate under feedrate interference is discussed. Simulations and experiments are performed to verify the effectiveness of the proposed method relative to other methods. The results reveal that, when using the proposed approach, the processing and running times of the computer are reduced by 39.48% and 118.34%, respectively, as compared with those of other methods. • Combining S-shaped quartic and cubic feedrate profile, and then calculating the optimal time allocation between them. • Ensure that the speed, acceleration and jerk meet the given constraints. • The method avoids the problem when the profile is complex the maximum feedrate can’t be updated and reduces the caculation. • The problem of low machining efficiency caused by uniform motion near the minimum feedrate point is avoided.

Development of look-ahead NURBS interpolation based on the exponential function

Aiming at the problem of large number of points and complex calculation of NURBS pre-interpolation, this paper puts forward a look-ahead interpolation with offline feedrate optimization. The NURBS interpolator is divided into two stages: pre-interpolation and real-time interpolation. The pre-interpolation preprocesses the curve segment by segment according to the curvature characteristics, at the same time the exponential function method is adopted during the pre-interpolation in the look-ahead module. This method makes the step increment of interpolation parameters change exponentially in the area with gentle curvature, and greatly reduces the number of pre-interpolation points, which reduces the amount of calculation and improves the real-time performance. The real-time interpolation stage adopts the bidirectional adaptive acceleration and deceleration control method to realize speed smoothing without needing to calculate the deceleration point. The simulation results show that the real-time performance of the algorithm is greatly improved.

A NURBS-based degenerated stiffener element for isogeometric static and buckling analysis

Complex shells and curvilinear stiffeners are promising for the lightweight design of aerospace structures. Along with the increasing of structural size and geometric characteristics, traditional finite element analysis (FEA) requires fine grids to content the analysis accuracy, which is extremely time-consuming. In this study, a novel NURBS-based degenerated stiffener element is proposed. Based on the 6 degree of freedom (DOF) degenerated beam element, the proposed stiffener element solves the issue of direct coupling between the beam element and 5 DOF degenerated shell element. Skin and stiffener are characterized by NURBS surface and curve, respectively. Meanwhile, the buckling behavior of stiffened panels is investigated based on isogeometric analysis (IGA), which reduces the geometric discrete error and realizes the integration of modeling and analysis. In addition, since the grids of skin and stiffeners are always unmatched, a strong coupling method based on NURBS interpolation and projection algorithm is constructed to merge non-conforming grids. The degrees-of-freedom (DOFs) of stiffeners is eliminated in the equilibrium equation, which greatly reduces the calculation scale. Furthermore, an improved point-to-surface projection algorithm is proposed, which improves the computational efficiency and solves the problem of unstable convergence of Newton iteration. Finally, the static and buckling analyses are performed for different types of stiffened plates and shells. Displacement solutions, von Mises stress solutions and buckling modes are discussed in detail. The prediction accuracy and meshes are compared with the shell-type model and beam-shell-type model in ABAQUS software, which indicates the high applicability and efficiency of the proposed analysis method for various types of stiffened panels.

Building Information Protection Method of Urban Historical Features Based on BIM Technology

In order to improve the building information protection method of urban historical features, this paper combines BIM technology to study the building information protection method of urban historical features and studies the least-squares fitting technology of NURBS curves. On this basis, this paper proposes a fitting method of the constrained least-squares method. Moreover, this paper designs a constrained least-squares method for the requirement of fitting accuracy and gives a method for setting the initial value. In addition, this paper proposes a NURBS interpolation method based on arc length parameter compensation technology. The experimental study verifies that the building information protection method of urban historical features based on BIM technology has a good effect and can effectively promote the building information protection of urban historical features.

NURBS interpolator with pre-compensation based on discrete inverse transfer function for CNC high-precision machining

In the field of computerized numerical control (CNC) machining, high-speed and high-precision machining has been regarded as the key research by many scholars. In conventional methods, high-speed machining and high-precision machining are contradictory. It is inevitable to reduce the feedrate to improve the processing accuracy. In the paper, a pre-compensation based on discrete inverse transfer function (PDIT) theory is proposed. PDIT is able to improve machining contour accuracy without decreasing feedrate. The proposed PDIT theory is divided into three parts: NURBS interpolator, feedrate scheduling, and interpolator with pre-compensation. The NURBS interpolator has great advantage of directly interpolating the parametric curve. Therefore, the paper adopts the NURBS interpolator to accomplish interpolation. In the feedrate scheduling, S-type flexible acceleration and deceleration are used for path planning, and the maximum starting feedrate is obtained under the feedrate constraint. In the interpolator with pre-compensation, the NURBS interpolator is pre-compensated by PDIT. For the input, the response of the transfer function reaches a steady-state response within periods of time. Before steady-state response, the unsteady-state response exists in the transfer function. The unsteady-state response usually sustains dozens of interpolation periods, which inevitably lead to contour error in machining. Hence, the PDIT theory is employed to compensate the contour error caused by the unsteady-state response of transfer function to NURBS interpolator. The drive system is a transfer function, so the unsteady-state response of drive system can also cause machining errors before the steady-state response. In the paper, the NURBS interpolator is pre-compensated by PDIT theory before drive system to reduce contour errors and improve machining accuracy. Finally, the performance of the proposed PDIT is evaluated by simulations and experiments. The experimental results illustrate that PDIT theory obviously improve the machining accuracy.

Real-Time NURBS Interpolation under Multiple Constraints.

NURBS interpolation is superior to traditional linear or circular interpolation in terms of code size, surface quality, and machining efficiency. However, with the increasing demands for high-accuracy and efficient machining, NURBS interpolation has faced a growing number of challenges. Many researchers are actively involved in this field with great interest. Due to the special form of NURBS curve, there is a nonlinear relationship between its curve and arc length; feed fluctuations and mechanical shocks which are caused during the interpolation process will seriously affect the surface accuracy and quality of machined parts. To solve these problems, a real-time NURBS interpolation is proposed under multiple constraints (RNIC) in this paper. First, the formulas of the constrained feedrate under geometric errors, kinematic constraints, drive constraints, and contour errors are given. Then, the two stages for the proposed interpolation are established. The former stage is offline preprocessing stage, which aims to quickly find feedrate sensitive areas (FSAs), while the latter online stage is the real-time interpolation, which is responsible for smoothing the velocity. In the preprocessing stage, we utilized FSA scan module and feedrate adjustment module to detect the FSAs and adjust the feedrate at the start/end of each subsegment by a bidirectional scanning algorithm. Each segment contains acceleration and deceleration (some contains uniform speed) stages, which can be well matched with the processing process of acceleration and deceleration. Finally, according to the proposed method and the adaptive speed adjustment method, the simulation of a “butterfly-shaped” NURBS curve using the S-shaped ACC/DEC algorithm is carried out, which verifies the reliability and effectiveness of the proposed algorithm.

Planar tool radius compensation for CNC systems based on NURBS interpolation

This paper introduces the realization of a tool radius compensation algorithm for NURBS trajectory. First, a single-segment NURBS trajectory tool radius compensation algorithm is developed. Different from the straight line and arc trajectory, the self-intersection phenomenon is prone to happen when calculating a single NURBS tool center trajectory, and the self-intersection will cause the overcut of workpiece. To avoid this situation, the algorithm introduced in this paper can detect whether the NURBS tool center track has caused overcut, and deal with the self-processing. Second, the tool radius compensation algorithm with multi-segment NURBS trajectory is implemented. The focus of this part is the tool radius compensation of the trajectory transfer, and the trajectory transfer is divided into two types: the extension type and the shortened type. For the shortened type transfer, cross-processing is needed to avoid the overcut of workpiece at the transfer. When calculating the tool radius compensation of the shortened type, we not only need to find the intersection of the tool center trajectory of two adjacent NURBS curves, but also need to select the intersection we need when a number of intersections exist. For the extension type transfer, in order to ensure the continuity of the tool center trajectory, we need to extend the tool center trajectory or add arc-segment at the transfer. The proposed algorithm can automatically decide where to extend the tool center trajectory or add arc-segment to achieve the best efficiency. Finally, the algorithm can output the calculated NURBS tool center trajectory in the form of linear segment interpolation G code or NURBS interpolation G code according to the processing needs. Simulations on VERICUT and experiments on three-axis CNC machine tool shows the effectiveness and validation of the tool path compensation algorithm.

Calculation of Space NURBS Curve Interpolation Error

Aiming at the problem of low accuracy of interpolation error calculation of existing NURBS curves, an approximate method for the distance between a point and a NURBS interpolation curve is proposed while satisfying the accuracy of the solution. Firstly, the minimum parameter interval of the node vector corresponding to the data point under test in the original data point sequence is determined, and the parameter interval is subdivided according to the corresponding step size, and the corresponding parameter value is obtained. Secondly, the distance from the measured point to the NURBS curve is calculated, and the nearest distance is found out. The node interval is subdivided again on one side of the nearest distance. Finally, the distance between the data point to be measured and each subdivision point is calculated again, and the minimum distance is taken as the interpolation error between the point and the NURBS curve. The simulation results of actual tool position data show that this method can more accurately obtain the error of spatial NURBS interpolation curve.

NURBS Interpolator with Adaptive Smooth Feedrate Scheduling and Minimal Feedrate Fluctuation

For an effective NURBS interpolator, generating the smooth feedrate profile and minimizing the feedrate fluctuation are two main tasks, which have a great influence on the machining quality. In this paper, a NURBS interpolator is proposed to solve the problem with offline-online two stages, which contain curving splitting, feedrate scheduling, and real-time interpolation modules. At first, in offline module, the limited feedrate of the curve is calculated with consideration of geometric characteristics and dynamic constraints, and the curve is split into several segments according to the key regions. Thus the necessary information is stored. Then the feedrate scheduling algorithm is applied with constant feedrate scheduling method at each segment, where tangential acceleration and jerk are taken into consideration to generate an adaptive smooth S-type feedrate profile. Moreover, in real-time interpolation module, an average feedrate calculation method is proposed to determine the sampling step size. And to minimize the feedrate fluctuation, a double interpolation algorithm based on the cosine theorem is presented, which use the second-order Taylor’s expansion method two times in one interpolation period. Finally, simulations and experiments are conducted to verify the performance of the proposed interpolator.

Surface reconstruction of laser-cladding remanufacturing blade using in adaptive belt grinding

The remanufacturing technology of aero-engine blade is one of the effective ways to save expensive materials such as titanium alloy, make full use of resources, and solve the problem of resource hiding. Laser-cladding technology has been widely used in the remanufacturing of blades due to its ability to fabricate high-performance alloy surfaces on metal substrates without affecting the properties of the matrix. In order to solve the problem that the grinding precision is difficult to guarantee due to the deformation and residual distribution of the blade in laser cladding remanufacturing, the self-adaptive grinding process of abrasive belt is repaired based on online inspection and reconstruction model. Firstly, the distribution of theoretical measuring points is planned based on the curvature of cross-section curve, and then the theoretical measurement points are adaptively extracted according to the improved firefly algorithm, so that the CMM measurement path planning of laser cladding and remanufacturing blades is realized. Taking the layer as the unit, the measurement data are processed, and the iterative nearest point algorithm is used for data registration to realize the laser-cladding remanufacturing blade measurement data processing. On this basis, the blade surface model is obtained based on NURBS interpolation. Finally, a laser cladding and remanufacturing blade of an aero-engine was tested. The experimental results show that the blade surface roughness is 0.4 μm, the blade edge accuracy is less than 0.05 mm, and the blade edge error is less than 10%. The experimental results show that this method can effectively solve the grinding process of laser-cladding remanufacturing blade.

Open and portable NURBS interpolator based on quantum framework

In order to solve the embedded NURBS interpolator closed structure problem, this paper proposes a NURBS interpolator based on quantum framework software bus architecture. Firstly, the NURBS interpolator trajectory interpolation algorithm and feedrate scheduling algorithm are presented. The embedded NC system systematic architecture is analyzed based on quantum framework. Secondly, the high-speed and high-precision NURBS curve interpolator and platform based on quantum framework are constructed. Finally, software simulation and actual machining experiment verify the correctness and validity of NURBS interpolator structure based on quantum framework by this paper.

An isogeometric formulation of the Koiter’s theory for buckling and initial post-buckling analysis of composite shells

Numerical formulations of the Koiter theory allow the efficient prediction, through a reduced model, of the behavior of shell structures when failure is dominated by buckling. In this work, we propose an isogeometric version of the method based on a solid-shell model. A NURBS interpolation is employed on the middle surface of the shell to accurately describe the geometry and the high continuity typical of the displacement field in buckling problems and to directly link the CAD model to the structural one. A linear interpolation is then adopted through the thickness together with a modified generalized constitutive matrix, which allows us to easily eliminate thickness locking and model multi-layered composites. Reduced integration schemes, which take into account the continuity of the shape functions, are used to avoid interpolation locking and make the integration faster. A Mixed Integration Point strategy makes it possible to transform the displacement model into a mixed (stress–displacement) one, required by the Koiter method to obtain accurate predictions, without introducing stress interpolation functions. The result is an efficient numerical tool for buckling and initial post-buckling analysis of composite shells, characterized by a low number of DOFs and integration points and by a simple and quick construction of the reduced model.

An efficient isogeometric solid-shell formulation for geometrically nonlinear analysis of elastic shells

In this work an isogeometric solid-shell model for geometrically nonlinear analyses is proposed. It is based on a linear interpolation through the thickness and a NURBS interpolation on the middle surface of the shell for both the geometry and the displacement field. The Green–Lagrange strains are linearized along the thickness direction and a modified generalized constitutive matrix is adopted to easily eliminate thickness locking without introducing any additional unknowns and to model multi-layered composite shells. Reduced integration schemes, which take into account the high continuity of the shape functions, are investigated to avoid interpolation locking and to increase the computational efficiency. The relaxation of the constitutive equations at each integration point is adopted in the iterative scheme in order to reconstruct the equilibrium path using large steps and a low number of iterations, even for very slender structures. This strategy makes it possible to minimize the number of stiffness matrix evaluations and decompositions and it turns out to be particularly convenient in isogeometric analyses.

A stabilized finite element formulation for liquid shells and its application to lipid bilayers

This paper presents a new finite element (FE) formulation for liquid shells that is based on an explicit, 3D surface discretization using C1-continuous finite elements constructed from NURBS interpolation. Both displacement-based and mixed displacement/pressure FE formulations are proposed. The latter is needed for area-incompressible material behavior, where penalty-type regularizations can lead to misleading results. In order to obtain quasi-static solutions for liquid shells devoid of shear stiffness, several numerical stabilization schemes are proposed based on adding stiffness, adding viscosity or using projection. Several numerical examples are considered in order to illustrate the accuracy and the capabilities of the proposed formulation, and to compare the different stabilization schemes. The presented formulation is capable of simulating non-trivial surface shapes associated with tube formation and protein-induced budding of lipid bilayers. In the latter case, the presented formulation yields non-axisymmetric solutions, which have not been observed in previous simulations. It is shown that those non-axisymmetric shapes are preferred over axisymmetric ones.

Motion Planning of a Humanoid Robot in a Complex Environment Using RRT and Spatiotemporal Post-Processing Techniques

This paper presents a motion planning method which provides a complete whole body trajectory of a humanoid robot traveling in a complex environment with various life items, steps, slopes, gates, walls, etc. A two-stage planning scheme is employed to simplify the problem of discontinuously changing dynamical constraints, where a sequence of double-support postures is planned in the first stage, and a continuous and smooth trajectory interpolates it in the second stage. RRT is utilized in the both stages in order to exploit the whole body with a large degree-of-freedom and perform in such a three-dimensionally intricate field. An inevitable issue of that random-sampling-based approach is that the searched trajectory is necessarily jaggy and detouring. The proposed method includes effective post-processing techniques for each stage, which are mandatory in practice. In the first stage, a necessary condition is that any pair of adjacent postures has to share one supporting foot as the pivot. A key idea is to insert intermediate postures between a pair of non-adjacent milestones with a cross-combination of the support feet to bypass and average a sequence of double-support transitions. The optimal sequence of the possible transitions is found by applying Dijkstra's method. In the second stage, an idea of the center of mass (COM)-enforcement to artificially constrain the trajectory of COM on one based on the mass-concentrated approximation is proposed, expecting that a solution can be found on this manifold. A continuous and smooth trajectory which satisfies the geometric constraint is planned by RRT and NURBS interpolation, and then, the temporal property of the trajectory is adjusted based on the dynamic programming to satisfy the dynamical constraint without breaking the geometric constraint. The performance was examined in a mock field of a living room. Comparing with a naive method only with RRT-connect, it substantially shortened the total travel distance in many cases. It also succeeded to plan dynamically feasible walking trajectories on that complex terrain.

Design and Implementation of High-Performance Real-Time Free-Form NURBS Interpolator in Micro CNC Machine Tool

This paper provides a suitable methodology to implement a high-performance and precise NURBS interpolator considering machine dynamics. The NURBS tool path curve is realized in real time by mapping the feedrate with a cubic curve with respect to time satisfying the machine dynamics and chord error. This is done by analyzing the feedrate profile offline and scheduling it with respect to time using a time optimal jerk limited algorithm. The proposed and the conventional NURBS interpolators are implemented in a three axis micro machine tool and the machining performance with respect to surface finish at different feedrates is compared. It is found that the feedrate profile obtained using the proposed interpolator scheme results in improved surface finish. Experimental evidences regarding the improvement in surface finish using a NURBS interpolator are shown for the first time according the authors knowledge. The proposed interpolator is designed with flexibility so that it can be implemented in all types of conventional CNC 3 axis machines also.

SOPC-Based Motion Controller with NURBS Interpolator

SOPC becomes popular nowadays, which combines software and hardware in a single chip. In this paper, a multi-axis motion controller is implemented based on a SOPC system with a NIOSII processor in a single FPGA chip. The motion controller includes DDA module, spindle module and NURBS calculation module realized in Verilog HDL. DDA algorithm is used to drive the motors while spindle module is to drive the spindle. Because of NURBS recursive nature, NURBS calculation module is used to accelerate the calculation which takes the advantage of parallel structure. The NIOSII processor realizes the feedrate scheduling, in which constant feedrate is adopted. The second-order interpolation algorithm is used to control the feedrate.

Novel path generation algorithm for high-speed pocket milling

A novel tool path generation algorithm is presented for high-speed pocket milling. A spiral fashion polyline is obtained by linear interpolation which is offset from the pocket boundary. The control points are then inserted into the polyline under exponential regularisation. Finally, the smooth tool path is generated via B-spline curve fitting. The tool path generated through this method can be directly used by the machine controller with a NURBS interpolation function, which avoids the time-consuming computation of dividing them into many short line segments. Compared with conventional methods of direction-parallel and contour-parallel path generation, the tool paths obtained by this algorithm have less cutting force fluctuation because they are Cncontinuous, smooth and have no direction jump. The engineering applications proved that the presented method can not only save machining time over the conventional algorithm, but also can avoid solving time-consuming scalar elliptic partial differential equations which are needed by existing spiral tool path generation method.

Precision NURBS interpolator based on recursive characteristics of NURBS

In NURBS interpolation, real-time parameter update is an indispensable step which affects not only feedrate fluctuation but also contour error. Using Taylor approximation interpolation method to find the next interpolation point causes a large feedrate fluctuation due to the accumulation and truncation errors. This paper presents a new, simple, and precise NURBS interpolator for CNC systems. The proposed interpolation algorithm does not use Taylor’s expansion, but the recursive equation of the NURBS formula. A simulation study is conducted to demonstrate the advantages of this proposed interpolator compared with those using Taylor’s equation. It is readily seen that this interpolator using the new concept of interpolation for modern CNC systems is simple and precise. The proposed method can be used for interpolating a continuous NURBS curve.