Trajectory optimization of the 6-degrees-of-freedom high-speed parallel robot based on B-spline curve

According to the problem that the existing high-speed parallel robot cannot satisfy the operation requirements of non-planar industrial production line, a 6-degrees-of-freedom high-speed parallel robot is proposed to carry out the kinematic and dynamic analyses. Combining with the door-type trajectory commonly used by the parallel robot, it adopts 3-, 5-, and 7-time B-spline curve motion law to conduct the trajectory planning in operation space. Taking the average cumulative effect of joint jerky as the optimization target, a trajectory optimization method is proposed to improve the smoothness of robot end-effector motion with the selected motion law. Furthermore, to solve the deformation problem of the horizontal motion stage of the trajectory, a mapping model between the control point subset of B-spline and the motion point subset of trajectory is established. Based on the main diagonally dominant characteristic of the coefficient matrix, the trajectory deformation evaluation index is constructed to optimize the smoothness and minimum deformation of the robot motion trajectory. Finally, compared to without the optimization, the maximum robot joint jerk decreases by 69.4% and 72.3%, respectively, and the maximum torque decreases by 51.4% and 38.9%, respectively, under a suitable trajectory deformation.

On the Number of Similar Instances of a Pattern in a Finite Set

New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is shown to be no more than $\lfloor{(4 n-1)(n-1)/18}\rfloor$. The number of $k$-term arithmetic progressions that lie within an $n$-point subset of the line is shown to be at most $(n-r)(n+r-k+1)/(2 k-2)$, where $r$ is the remainder when $n$ is divided by $k-1$. This upper bound is achieved when the $n$ points themselves form an arithmetic progression, but for some values of $k$ and $n$, it can also be achieved for other configurations of the $n$ points, and a full classification of such optimal configurations is given. These results are achieved using a new general method based on ordering relations.

Sparse Distance Sets in the Triangular Lattice

A planar point-set $X$ in Euclidean plane is called a $k$-distance set if there are exactly $k$ different distances among the points in $X$. The function $g(k)$ denotes the maximum number of points in the Euclidean plane that is a $k$-distance set. In 1996, Erdős and Fishburn conjectured that for $k\geq 7$, every $g(k)$-point subset of the plane that determines $k$ different distances is similar to a subset of the triangular lattice. We believe that if $g(k)$ is an increasing function of $k$, then the conjecture is false. We present data that supports our claim and a method of construction that unifies known optimal point configurations for $k\geq 3$.

A New Method Applied to the Multi-Parametric Nonlinear Compensations of Computer Numeric Control Bender

Complex Moving Least Squares and its key attributes as basis functions and support region radius were discussed and studied in this paper. Owing to the weak performance of traditional Moving Least Squares when applied to asymmetrical sampling points set, an improved Moving Least Squares named dynamic radius Moving Least Squares was designed, whose support regions radius at fitting point dynamically adjust as the density of local point subset. At last, it is applied to multi-parametric nonlinear compensation of computer numeric control bender through comparative fitting experiments and simulations on couples of data compared to traditional Moving Least Squares. The results of experiments and simulations indicate that the new algorithm makes a great progress on computational complexity, fitting smoothness and fitting accuracy, especially the accuracy in dense points region compared to traditional Moving Least Squares.

Applying Element-Free Galerkin Method to Simulate Die Forging Problems

The analysis for die forging forming problems with finite element method can lose considerable accuracy due to severely distortional meshes. The element-free Galerkin method is suitable for large deformation analysis and provides a higher rate of convergence than that of the conventional finite element methods. A rigid-plastic meshless method based on the element-free Galerkin method has been applied to die forging problems. The arc-tangent friction model is used to handle frictional contact and the penalty method is applied to impose the volumetric incompressibility conditions. By dividing all integration points set into the point subset of the rigid zones and the point subset of the plastic zones, nonsmoothness of the rigid-plastic constitutive relation can be eliminated. A die forging example has been analyzed to demonstrate the performance of the method.

Characterizations of inner product spaces by means of norm one points

Let $X$ be a a real normed linear space of dimension at least three, with unit sphere $S_X$. In this paper we prove that $X$ is an inner product space if and only if every three point subset of $S_X$ has a Chebyshev center in its convex hull. We also give other characterizations expressed in terms of centers of three point subsets of $S_X$ only. We use in these characterizations Chebyshev centers as well as Fermat centers and $p$-centers.