Joint Motion Planning of Industrial Robot Based on Modified Cubic Hermite Interpolation with Velocity Constraint

As for industrial robots’ point-to-point joint motion planning with constrained velocity, cubic polynomial planning has the problem of discontinuous acceleration; quintic polynomial planning requires acceleration to be specified in advance, which will likely cause velocity to fluctuate largely because appropriate acceleration assigned in advance is hardly acquired. Aiming at these problems, a modified cubic Hermite interpolation for joint motion planning was proposed. In the proposed methodology, knots of cubic Hermite interpolation need to be reconfigured according to the initial knots. The formulas for how to build new knots were put forward after derivation. Using the newly-built knots instead of initial knots for cubic Hermite interpolation, joint motion planning was carried out. The purpose was that the joint planning not only satisfied the displacement and velocity constraints at the initial knots but also guaranteed C2 continuity and less velocity fluctuation. A study case was given to verify the rationality and effectiveness of the methodology. Compared with the other two planning methods, it proved that the raised problems can be solved effectively via the proposed methodology, which is beneficial to the working performance and service life of industrial robots.

Geometric Piecewise Cubic Bézier Interpolating Polynomial with C2 Continuity

Bézier curve is a parametric polynomial that is applied to produce good piecewise interpolation methods with more advantage over the other piecewise polynomials. It is, therefore, crucial to construct Bézier curves that are smooth and able to increase the accuracy of the solutions. Most of the known strategies for determining internal control points for piecewise Bezier curves achieve only partial smoothness, satisfying the first order of continuity. Some solutions allow you to construct interpolation polynomials with smoothness in width along the approximating curve. However, they are still unable to handle the locations of the inner control points. The partial smoothness and non-controlling locations of inner control points may affect the accuracy of the approximate curve of the dataset. In order to improve the smoothness and accuracy of the previous strategies, а new piecewise cubic Bézier polynomial with second-order of continuity C2 is proposed in this study to estimate missing values. The proposed method employs geometric construction to find the inner control points for each adjacent subinterval of the given dataset. Not only the proposed method preserves stability and smoothness, the error analysis of numerical results also indicates that the resultant interpolating polynomial is more accurate than the ones produced by the existing methods.

Surface Modeling from 2D Contours with an Application to Craniofacial Fracture Construction

Treating trauma to the cranio-maxillofacial region is a great challenge and requires expert clinical skills and sophisticated radiological imaging. The aim of reconstruction of the facial fractures is to rehabilitate the patient both functionally and aesthetically. Bio-modeling is an important tool for constructing surfaces using 2D cross sections. The aim of this manuscript was to show 3D construction using 2D CT scan contours. The fractured part of the cranial vault were constructed using a Ball curve with two shape parameters, later the 2D contours were flipped into 3D with an equidistant z component. The surface created was represented by a bi-cubic rational Ball surface with C2 continuity. At the end of this article, we present two real cases, in which we had constructed the frontal and parietal bone fractures using a bi-cubic rational Ball surface. The proposed method was validated by constructing the non-fractured part.

Efficient C2 Continuous Surface Creation Technique Based on Ordinary Differential Equation

In order to reduce the data size and simplify the process of creating characters’ 3D models, a new and interactive ordinary differential equation (ODE)-based C2 continuous surface creation algorithm is introduced in this paper. With this approach, the creation of a three-dimensional surface is transformed into generating two boundary curves plus four control curves and solving a vector-valued sixth order ordinary differential equation subjected to boundary constraints consisting of boundary curves, and first and second partial derivatives at the boundary curves. Unlike the existing patch modeling approaches which require tedious and time-consuming manual operations to stitch two separate patches together to achieve continuity between two stitched patches, the proposed technique maintains the C2 continuity between adjacent surface patches naturally, which avoids manual stitching operations. Besides, compared with polygon surface modeling, our ODE C2 surface creation method can significantly reduce and compress the data size, deform the surface easily by simply changing the first and second partial derivatives, and shape control parameters instead of manipulating loads of polygon points.

C2 Continuous Blending of Time-Dependent Parametric Surfaces

Surface blending is widely applied in mechanical engineering. Creating a smooth transition surface of C2 continuity between time-dependent parametric surfaces that change their positions and shapes with time is an important and unsolved topic in surface blending. In order to address this issue, this paper develops a new approach to unify both time-dependent and time-independent surface blending with C2 continuity. It proposes a new surface blending mathematical model consisting of a vector-valued sixth-order partial differential equation and blending boundary constraints and investigates a simple and efficient approximate analytical solution of the mathematical model. A number of examples are presented to demonstrate the effectiveness and applications. The proposed approach has the advantages of (1) unifying time-independent and time-dependent surface blending, (2) always maintaining C2 continuity at trimlines when parametric surfaces change their positions and shapes with time, (3) providing effective shape control handles to achieve the expected shapes of blending surfaces but still exactly satisfy the given blending boundary constraints, and (4) quickly generating C2 continuous blending surfaces from the approximate analytical solution with easiness, good accuracy, and high efficiency.

C2-continuous orientation trajectory planning for robot based on spline quaternion curve

Purpose To realize the smooth interpolation of orientation on robot end-effector, this paper aims to propose a novel algorithm based on the unit quaternion spline curve. Design/methodology/approach This algorithm combines the spherical linear quaternion interpolation and the cubic B-spline quaternion curve. With this method, a C2-continuous smooth trajectory of multiple teaching orientations is obtained. To achieve the visualization of quaternion curves on a unit sphere, a mapping algorithm between a unit quaternion and a point on the spherical surface is given based on the physical meaning of the unit quaternion. Findings Finally, the curvature analysis of a practical case shows that the orientation trajectory (OT) constructed by this algorithm satisfied the C2-continuity. Originality/value This OT satisfies the requirement of smooth interpolation among multiple orientations on robots in industrial applications.

A point projection approach for improving the accuracy of the multilevel B-spline approximation

In this study, we present a method for improving the accuracy of the multilevel B-spline approximation (MBA) method. We combine a point projection method with the MBA method for reducing the approximation error by directly adjusting the control points in the local area. An initial surface is generated by the MBA method, and grid points are produced on the surface. These grid points are projected onto the scattered point set, and the distances between the grid points and the projected points are computed. The control points are then modified based on the distances. The proposed method shows better approximations even with the same number of control points and ensures C2-continuity. The experimental results with examples verify the validity of the proposed method. Highlights We propose a method for improving the multilevel B-spline approximation method. We use a point projection method for computing the amount of errors. The computed errors are directly applied to the control points for reducing the approximation error.