Quaternionic Curves Research Articles

High-precision trajectory tracking design and simulation for six degree of freedom robot based on improved active disturbance rejection control

High-precision trajectory tracking control is an important factor in the performance of industrial robots. In this study, a high-precision trajectory tracking strategy was proposed for controlling a degree of freedom serial robot on the basis of improved active disturbance rejection control. An independent control strategy of a single joint was adopted, and the corresponding decoupling control law was designed. An attitude trajectory-planning algorithm based on the circular-blending quaternion curve was improved. The position and attitude trajectories were transformed into the joint trajectory by using a kinematics equation and inverse velocity Jacobian matrix. The above-mentioned transformation link was used as a preprocessing link of the active disturbance rejection control, which is used for replacing the tracking differentiator of a typical active disturbance rejection control to eliminate the effect of the tracking delay. An experimental simulation was conducted by combining MATLAB and ADAMS. Simulation results show that the proposed control strategy can perform the tracking control of a task-space trajectory. The tracking precision of position and attitude trajectories were 0.01 mm and 0.01 s, respectively.

On quaternionic tori and their moduli space

Quaternionic tori are defined as quotients of the skew field \mathbb H of quaternions by rank-4 lattices. Using slice regular functions, these tori are endowed with natural structures of quaternionic manifolds (in fact quaternionic curves), and a fundamental region in a 12-dimensional real subspace is then constructed to classify them up to biregular diffeomorphisms. The points of the moduli space correspond to suitable special bases of rank-4 lattices, which are studied with respect to the action of the group GL (4, \mathbb Z) , and up to biregular diffeomeorphisms. All tori with a non trivial group of biregular automorphisms – and all possible groups of their biregular automorphisms – are then identified, and recognized to correspond to five different subsets of boundary points of the moduli space.

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

PurposeTo 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/approachThis 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.FindingsFinally, the curvature analysis of a practical case shows that the orientation trajectory (OT) constructed by this algorithm satisfied the C2-continuity.Originality/valueThis OT satisfies the requirement of smooth interpolation among multiple orientations on robots in industrial applications.

The quaternionic expression of ruled surfaces

In this paper, firstly, the ruled surface is expressed as a spatial quaternionic. Also, the spatial quaternionic definitions of the Striction curve, the distribution parameter, angle of pitch and the pitch are given. Finally, integral invariants of the closed spatial quaternionic ruled surfaces drawn by the motion of the Frenet vectors {t,n1,n2} belonging to the spatial quaternionic curve ? are calculated.

Inextensible flow of a semi-real quaternionic curve in semi-euclidean space R42

In this paper, we investigate a general formulation for inextensible*ows of semi-real quaternionic curve in R2. We obtain necessary and su¢ cientconditions for inextensible *ow of semi-real quaternionic curves. Moreover, wegive the evolution equation of curvatures as a partial diğerential equation

Characterizations for the Dual Split Quaternionic Curves

Characterizations for the Dual Split Quaternionic Curves

Characterizations for the Dual Split Quaternionic Curves

Characterizations for the Dual Split Quaternionic Curves

On the Quaternionic Focal Curves

In this study, a brief summary about quaternions and quaternionic curves are firstly presented. Also, the definition of focal curve is given. The focal curve of a smooth curve consists of the centers of its osculating hypersphere. By using this definition and the quaternionic osculating hyperspheres of these curves, the quaternionic focal curves in the spaces $\Q$ and $\Q_\nu$ with index $\nu=\{1,2\}$ are discussed. Some relations about spatial semi-real quaternionic curves and semi-real quaternionic curves are examined by using focal curvatures and scalar Frenet equations between the focal curvatures. Then, the notions: such as vertex, flattenings, a symmetry point are defined for these curves. Moreover, the relation between the Frenet apparatus of a quaternionic curve and the Frenet apparatus of its quaternionic focal curve are presented.

Null Quaternionic Bertrand Partner Curves

In this paper, we study null quaternionic Bertrand curves in the semi-Euclidean spaces \(E_{1}^{3}\) and \(E_{1}^{4}\), respectively. We obtain some characterizations for null quaternionic Bertrand partner curves. We show that the constant distance between the quaternionic partner curves is independent from the first curvatures of the curves in both spaces \(E_{1}^{3}\) and \(E_{1}^{4}\).

A Study On Quaternionic \(B_2\)-Slant Helix In Semi-Euclidean 4-Space

In this work, we defined a quaternionic \(B_2\)-slant helix in semi-Euclidean space \(\mathbb{E}_2^4\). Then we gave Frenet formulae for the quaternionic curve in semi-Euclidean space \(\mathbb{E}_2^4\). Also, we investigated some necessary and sufficient conditions for a space curve to be a quaternionic \(B_2\)-slant helix according to quaternionic curves in semi-Euclidean space \(\mathbb{E}_2^4\).

Constant Ratio Quaternionic Curves in Euclidean Spaces

In this paper, we give some characterizations of quaternionic curves whose position vectors can be written as linear combinations of their Serret-Frenet vectors in Euclidean 3-space \({\mathbb{E}^{3}}\) and Euclidean 4-space \({\mathbb{E}^{4}}\). We characterize such curves in terms of their curvature functions. Finally, we give the necessary and sufficient conditions for these types of curves to become constant ratio, T-constant, and N-constant.

Null Quaternionic Bertrand Curves in Minkowski Space R14

The geometry of null curves in Minkowski spacetime has played an important role in the development of general relativity, as well as in mathematics and physics of gravitation. Many scientists have used Minkowski space to apply general relativity. There has been an increase in research on null curves in geometry and physics [1]. Bonnor [2] has introduced a Cartan frame for null curves in R 1 and proved the fundamental existence and congruence theorems. Bejancu [3] has given a method for the general study of the geometry of null curves in lightlike manifolds and in semi-Riemannian manifolds. A. Ferrandez, A. Gimenez and P. Lucas [4] have shown that a null Frenet curve, parametrized by the pseudo-arc parameter, is a null helix, if its lightlike curvature is constant. Coken and Ciftci [5] have reconstructed the Cartan frame of a null curve in Minkowski spacetime for an arbitrary parameter, and have characterized the Bertrand null curves. And then, Duggal and Jin [6] have studied major developments of null curves, hypersurfaces and their physical use, in their recent book with voluminous bibliography. Quaternions were discovered by Hamilton as an extension to the complex numbers in 1843. Quaternions have found a broad application in many scientific areas: in mechanics of a solid body, for the description of rotation in space, in computer animation, etc. One of the most important tools used to analyze a quaternionic curve is the Frenet frame. Therefore, in [7], Bharathi and Nagaraj have defined Serret-Frenet formulas for a quaternionic curve in E and E, and then Coken and Tuna have studied Serret-Frenet formulas for quaternionic curves and quaternionic inclined curves in semi-euclidean spaces [8]. Moreover, we have studied the differential geometry of null quaternionic curves in semi-euclidean 3-spaces R v and gave the Frenet formula for null quaternionic curves by using spatial quaternions. We have constructed recently the Cartan frame for a null quaternionic curve in the 4-dimensional Minkowski space R 1 [9]. Then, we have established a relation of Bertrand pairs with null quaternionic Cartan helices in R v [10].

Pseudo-Spherical Null Quaternionic Curves in Minkowski Space R14

Pseudo-Spherical Null Quaternionic Curves in Minkowski Space R₁⁴

On Spatial Quaternionic Involute Curve A New View

In this study, the normal vector and the unit Darboux vector of spatial involute curve of the spatial quaternionic curve are taken as the position vector, the curvature and torsion of obtained smarandahce curve were calculeted.

Quaternionic osculating curves in Euclidean and semi-Euclidean space

In this study, the osculating curves in Euclidean space E3 and E4, well known in differential geometry, are studied through the instrumentality of quaternions. We inoculate sundry delineations for quaternionic osculating curves in the Euclidean space E3, then we portray the quaternionic osculating curve in E4 as a quaternionic curve whose position vector every time reclines in the orthogonal complement N½ (or N⅓) of its first binormal vector field N2 (or N3), where {T,N1,N2,N3} be the Frenet instrumentations of the quaternionic curve in the Euclidean space E4. We feature quaternionic oscu­lating curves from the point of view their curvature functions K, k and (r — K) and serve the necessary and the sufficient conditions for arbitrary quaternionic curve in E4 to be a quaternionic osculating. Moreover, we gain an explicit equation of a quaternionic osculating curve in E4. In the last two section, we described quaternionic osculating curves in the semi-Euclidean space and some theorems are testified.

A class of generalized B-spline quaternion curves

Unit quaternion curves have gained considerable attention in the fields of robot control and computer animation. Kim et al. proposed a general construction method of unit quaternion curves which can transform the closed form equation for kth order B-spline basis functions in R3 into its unit quaternion analogue in SO(3) while preserving the Ck−2-continuity. Juhász and Róth generalized the classical B-spline functions by means of monotone increasing continuously differentiable core functions based on the recurrence formula of B-spline functions. In order to extend the applications of the generalized B-spline functions in computer animation, the definition and construction scheme of generalized B-spline quaternion curves in S3 are put forward in this paper. The introduced nonlinear core functions are not only theoretically interesting, but also offer a large variety of shapes. Some properties of this class of unit quaternion curves, such as continuity and local controllability are also discussed. Experimental results show the effectiveness and usefulness of our construction methods of generalized B-spline quaternion curves.

An Application According to Spatial Quaternionic Smarandache Curve

In this paper, we found the Darboux vector of the spatial quaternionic curve according to the Frenet frame. Then, the curvature and torsion of the spatial quaternionic smarandache curve formed by the unit Darboux vector with the normal vector was calculated. Finally; these values are expressed depending upon the spatial quaternionic curve.

Null Quaternionic Curves in Semi-Euclidean 3-Space of Index ν

The quaternions were first described by Irish mathematician Sir William Rowan Hamilton in 1843. The space of quaternions Q are isomorphic to E, 4dimensional vector space over the real numbers. The set of the quaternions, which was introduced by Hamilton, can be represented as Q = {ae1 + be2 + ce3 + d; a, b, c, d ∈ R}. Here, ei = −1, ei × ej = ek, 1 ≤ i ≤ 3 and (ijk) is an even permutation of (123). There are a lot of papers associated with quaternions. Also they are used in both theoretical and applied mathematics. Hence, the importance of the study of quaternions and its presence in the physical theories are clear. In 1987, the Serret-Frenet formulae for quaternionic curves in E and E were given by Bharathi and Nagaraj [1]. After that, split quaternions are identified with semi-Euclidean space E 2 , while the vector part of split quaternions are identified with Minkowski 3-space by Inoguchi [2]. Many studies have been published on the quaternionic curves using this results. Among them Tuna has defined the Serret–Frenet formulae for a quaternionic curve in the semi-Euclidean space E 2 [3]. We have studied quaternion valued functions and quaternionic inclined curves in the semi-Euclidean space E 2 [4]. But, to our knowledge, there has been no study on the Serret– Frenet formulae for null quaternionic curves in the semiEuclidean spaces. In this paper, we will study Serret– Frenet formulae for null quaternionic curve in R1 and null quaternionic curve in R2. It is well known that there exist spacelike quaternionic curve and timelike quaternionic curve in the semiEuclidean spaces. However, null quaternionic curves have many properties which are very different from spacelike quaternionic and timelike quaternionic curves. In geometry of null quaternionic curves difficulties arise because the arc length vanishes, so that it is impossible to normalize the tangent vector in the usual way. Since the length

An application according to spatial quaternionic Smarandache curve

In this paper, we found the Darboux vector of the spatial quaternionic curve according to the Frenet frame. Then, the curvature and torsion of the spatial quaternionic smarandache curve formed by the unit Darboux vector with the normal vector was calculated. Finally; these values are expressed depending upon the spatial quaternionic curve. Mathematics Subject Classification: 11R52, 53A04

A note on the quaternionic curves

A note on the quaternionic curves