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The theory of screws clarifies many analogies between apparently unrelated notions in mechanics, including the duality between forces and angular velocities. It is known that the real 6-dimensional space of screws can be endowed with an operator E , E 2 = 0 , that converts it into a rank 3 free module over the dual numbers. In this paper we prove the converse, namely, given a rank 3 free module over the dual numbers, endowed with orientation and a suitable scalar product (D -module geometry), we show that it is possible to define, in a canonical way, a Euclidean space so that each element of the module is represented by a screw vector field over it. The new approach has the effectiveness of motor calculus while being independent of any reduction point. It gives insights into the transference principle by showing that affine space geometry is basically vector space geometry over the dual numbers. The main results of screw theory are then recovered by using this point of view. [ABSTRACT FROM AUTHOR]

We consider a family of Riemannian manifolds M such that for each unit speed geodesic γ of M there exists a distinguished bijective correspondence L between infinitesimal translations along γ and infinitesimal rotations around it. The simplest examples are R 3 , S 3 and hyperbolic 3-space, with L defined in terms of the cross product. More generally, M is a connected compact semisimple Lie group, or its non-compact dual, or Euclidean space acted on transitively by some group which is contained properly in the full group of rigid motions. Let G be the identity component of the isometry group of M. A curve in G may be thought of as a motion of a body in M. Given λ ∈ R , we define a left invariant distribution on G accounting for infinitesimal roto-translations of M of pitch λ . We give conditions for the controllability of the associated control system on G and find explicitly all the geodesics of the natural sub-Riemannian structure. We also study a similar system on R 7 ⋊ S O 7 involving the octonionic cross product. In an appendix we give a friendly presentation of the non-compact dual of a compact classical group, as a set of “small rotations”. [ABSTRACT FROM AUTHOR]

In this paper, we determine elliptical motions that occur on any given ellipsoid in 3D space without using affine transformations. To this end, first, we define the generalized Euclidean inner product whose sphere is the given ellipsoid, and determine skew symmetric matrices and the generalized vector product related to the 3D generalized Euclidean inner product space. Finally, we generate elliptical rotation matrices in 3D generalized Euclidean space using the famous Rodrigues, Cayley, and Householder methods. The formulas and results obtained are supported with numerical examples. We also give an algorithm for generalized elliptical rotation. [ABSTRACT FROM AUTHOR]

We consider bivariate polynomials over the skew field of quaternions, where the indeterminates commute with all coefficients and with each other. We analyze existence of univariate factorizations, that is, factorizations with univariate linear factors. A necessary condition for existence of univariate factorizations is factorization of the norm polynomial into a product of univariate polynomials. This condition is, however, not sufficient. Our central result states that univariate factorizations exist after multiplication with a suitable univariate real polynomial as long as the necessary factorization condition is fulfilled. We present an algorithm for computing this real polynomial and a corresponding univariate factorization. If a univariate factorization of the original polynomial exists, a suitable input of the algorithm produces a constant multiplication factor, thus giving an a posteriori condition for existence of univariate factorizations. Some factorizations obtained in this way are of interest in mechanism science. We present an example of a curious closed-loop mechanism with eight revolute joints. [ABSTRACT FROM AUTHOR]

In this paper, we define and classify split quaternion operators. Then, we show that the split quaternion product of a split quaternion operator and a curve, which lies on Lorentzian unit sphere or on hyperbolic unit sphere, parametrizes a ruled surface in the 3-dimensional Minkowski space E 1 3 if the vector part of the operator is perpendicular to the position vector of the spherical curve. Moreover, the ruled surfaces are represented as 2-parameter homothetic motions in E 1 3 by using semi-orthogonal matrices corresponding to the split quaternion operators. Finally, some examples are given to illustrate some applications of our main results. [ABSTRACT FROM AUTHOR]

We present an algorithm to compute all factorizations into linear factors of univariate polynomials over the split quaternions, provided such a factorization exists. Failure of the algorithm is equivalent to non-factorizability for which we present also geometric interpretations in terms of rulings on the quadric of non-invertible split quaternions. However, suitable real polynomial multiples of split quaternion polynomials can still be factorized and we describe how to find these real polynomials. Split quaternion polynomials describe rational motions in the hyperbolic plane. Factorization with linear factors corresponds to the decomposition of the rational motion into hyperbolic rotations. Since multiplication with a real polynomial does not change the motion, this decomposition is always possible. Some of our ideas can be transferred to the factorization theory of motion polynomials. These are polynomials over the dual quaternions with real norm polynomial and they describe rational motions in Euclidean kinematics. We transfer techniques developed for split quaternions to compute new factorizations of certain dual quaternion polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we study kinematics of the motion of a two-wheeled caster board. Using the theory of finite rotations, we propose an elementary derivation of the formula connecting the angles of slope of the decks of the caster board with the angles of rotation of the wheels. [ABSTRACT FROM AUTHOR]

It is known that velocity averaged over time is less than or equal to velocity averaged over distance. This inequality is illustrated with some routine applied mathematics problems involving simple harmonic motion and free fall motion. Generalizations to motion along space curves are explored, and inequalities involving spatial and temporal averages of other kinematic/dynamic quantities are derived from Chebyshev's sum inequality. These quantities include kinetic energy, tangential and normal components of acceleration, work, and impulse. [ABSTRACT FROM AUTHOR]

We investigate factorizability of a quadratic split quaternion polynomial. In addition to inequality conditions for existence of such factorization, we provide lucid geometric interpretations in the projective space over the split quaternions. [ABSTRACT FROM AUTHOR]

Among many physical and mechanical engineering problems, parallel car parking has drawn the attention of many researchers as it shows how methods of Lie algebra and differential geometry can be applied elegantly to explain the behavior of such a system. In this paper we have reviewed some aspects of this problem in this context and analyse the stability and controllability of such a system by some unconventional integrators like Kahan and Lie trotter integrator. At the same time we have also given the results obtained through conventional Runge Kutta integrator to see a comparison. [ABSTRACT FROM AUTHOR]

In this study, we define a new non-commutative number system called hybrid numbers. This number system can be accepted as a generalization of the complex $$\left( {\mathbf {i}}^{2}=-1\right) $$ , hyperbolic $$\left( {\mathbf {h}} ^{2}=1\right) $$ and dual number $$\left( \varvec{\varepsilon }^{2}=0\right) $$ systems. A hybrid number is a number created with any combination of the complex, hyperbolic and dual numbers satisfying the relation $$\mathbf { ih=-hi=i}+\varvec{\varepsilon }.$$ Because these numbers are a composition of dual, complex and hyperbolic numbers, we think that it would be better to call them hybrid numbers instead of the generalized complex numbers. In this paper, we give some algebraic and geometric properties of this number set with some classifications. In addition, we examined the roots of a hybrid number according to its type and character. [ABSTRACT FROM AUTHOR]

We derive an optimal control formulation for a nonholonomic mechanical system using the nonholonomic constraint itself as the control. We focus on Suslov's problem, which is defined as the motion of a rigid body with a vanishing projection of the body frame angular velocity on a given direction $$\varvec{\xi }$$ . We derive the optimal control formulation, first for an arbitrary group, and then in the classical realization of Suslov's problem for the rotation group $$\textit{SO}(3)$$ . We show that it is possible to control the system using the constraint $$\varvec{\xi }(t)$$ and demonstrate numerical examples in which the system tracks quite complex trajectories such as a spiral. [ABSTRACT FROM AUTHOR]

Dimensional synthesisis a type of inverse problem in linkage kinematics where the objective is to calculate the linkage dimensions required to achieve prescribed linkage output motion.Motion generationis a particular category of dimensional synthesis where the objective is to calculate the linkage dimensions required to achieve a group of prescribed link positions. In motion generation for afour-bar linkage, positions are prescribed for the coupler link. While early motion generation methods were primarilyqualitative, ongoing advancements in computing hardware and software continue to makequantitativemotion generation more practical. By providing overviews of works representative of developments in quantitative four-bar motion generation since 1970, this work is essentially an overview that spans over 40 years of developments in quantitative four-bar motion generation. [ABSTRACT FROM PUBLISHER]

In this paper the generalization of the rotations on any lightcone in Minkowski 3-space $${\mathbb {R}}_{g}^{1,2}$$ is given. The rotation motion on the lightcone is examined by means of a bilinear form and Lorentzian notions. We use the corresponding Rodrigues and Cayley formulas and benefit from the hyperbolic split quaternion product to obtain the corresponding rotation matrix. [ABSTRACT FROM AUTHOR]

We present a new different polar representation of split and dual split quaternions inspired by the Cayley-Dickson representation. In this new polar form representation, a split quaternion is represented by a pair of complex numbers, and a dual split quaternion is represented by a pair of dual complex numbers as in the Cayley-Dickson form. Here, in a split quaternion these two complex numbers are a complex modulus and a complex argument while in a dual split quaternion two dual complex numbers are a dual complex modulus and a dual complex argument. The modulus and argument are calculated from an arbitrary split quaternion in Cayley-Dickson form. Also, the dual modulus and dual argument are calculated from an arbitrary dual split quaternion in Cayley-Dickson form. By the help of polar representation for a dual split quaternion, we show that a Lorentzian screw operator can be written as product of two Lorentzian screw operators. One of these operators is in the two-dimensional space produced by 1 and i vectors. The other is in the three-dimensional space generated by 1, j and k vectors. Thus, an operator in a four-dimensional space is expressed by means of two operators in two and three-dimensional spaces. Here, vector 1 is in the intersection of these spaces. [ABSTRACT FROM AUTHOR]

The objective of this study is to take advantage of using the concept of complex numbers for instantaneous geometric properties of planar motion of rigid bodies. We consider that both the fixed and the moving planes are Gaussian. Then we give Bottema’s instantaneous invariants in complex forms and we study the kinematic geometry of infinitesimally separated positions of a moving Gaussian plane regarding this formulation. The followed analytical method provides a straightforward way to describe order properties of motion. We obtain the complex forms of the inflection circle, cubic of stationary curvature and cubic of twice stationary curvature. Moreover, we give the existence conditions of Ball, Burmester and Ball–Burmester points. [ABSTRACT FROM AUTHOR]

Involutions and anti-involutions are self-inverse linear mappings. In three-dimensional Euclidean space $${\mathbb{R}^{3}}$$ , a reflection of a vector in a plane can be represented by an involution or anti-involution mapping obtained by real-quaternions. A reflection of a line about a line in $${\mathbb{R}^{3}}$$ can also be represented by an involution or anti-involution mapping obtained by dual real-quaternions. In this paper, we will represent involution and anti-involution mappings obtaind by dual split-quaternions and a geometric interpretation of each as rigid-body (screw) motion in three-dimensional Lorentzian space $${\mathbb{R}_1^{3} }$$ . [ABSTRACT FROM AUTHOR]

Hyperbolic rotation is hyperbolically the motion of a smooth object on general hyperboloids given by − a 1 x 2 + a 2 y 2 + a 3 z 2 = ± λ , λ ∈ R + . In this paper, we investigate the hyperbolical rotation matrices in order to get the motion of a point about a fixed point or axis on the general hyperboloids by defining the Lorentzian Scalar Product Space R a 1 a 2 a 3 2 , 1 such that the general hyperboloids are the pseudo-spheres of R a 1 a 2 a 3 2 , 1 . We adapt the Rodrigues, Cayley, and Householder methods to R a 1 a 2 a 3 2 , 1 and define hyperbolic split quaternions to obtain an hyperbolical rotation matrix. [ABSTRACT FROM AUTHOR]

Elliptical rotation is the motion of a point on an ellipse through some angle about a vector. The purpose of this paper is to examine the generation of elliptical rotations and to interpret the motion of a point on an ellipsoid using elliptic inner product and elliptic vector product. To generate an elliptical rotation matrix, first we define an elliptical ortogonal matrix and an elliptical skew symmetric matrix using the associated inner product. Then we use elliptic versions of the famous Rodrigues, Cayley, and Householder methods to construct an elliptical rotation matrix. Finally, we define elliptic quaternions and generate an elliptical rotation matrix using those quaternions. Each method is proven and is provided with several numerical examples. [ABSTRACT FROM AUTHOR]

The aim of the paper is to employ the dual numbers in the multi axes machine error modelling in order to apply the algebraic methods in computations. The calculus of higher order dual numbers allows us to calculate with the appropriate geometric parametrization effectively. We test the model on the phantom data based on the real machine tool. The results of our analysis are used for the geometric manufacturing accuracy description of the work space, together with the reduction of the measuring time. [ABSTRACT FROM AUTHOR]

Evaluation of the performance of three controllers: adaptive PID, model reference adaptive controller, and fuzzy controller, is presented, applied to a mobile robot for autonomous path tracking in a crop. Different tests are run in a simulation environment in order to compare the same trajectory performed under each of those controllers. The tests are designed and implemented using Simulink® programming tools, making the robot follow a desired path across the crop. The results are compared considering the different error indices, curves and actuator requirements of each controller, in order to arrange them in different categories. [ABSTRACT FROM AUTHOR]

A finite sequence of rotations is closed if a sequential application of all the rotations from the sequence results in no net orientation change. A complete characterization of closed rotation sequences involving a given set of rotation axes is presented, and the set of such sequences is shown to be a smooth manifold under a nondegeneracy condition on the rotation axes. The characterization is used to derive several examples of closed rotation sequences, some of which are then shown to specialize to classical examples of such sequences provided by the Rodrigues-Hamilton theorem and the Donkin's theorem. Discrete versions of the Goodman-Robinson and Ishlinskii theorems are also derived and illustrated using the so-called Codman's paradox. [ABSTRACT FROM AUTHOR]

This paper deals with the image-based control of a satellite for remote sensing. Approach is demonstrated by simulation where the position of the satellite is obtained with the Simplified General Perturbations Version 3 model and its orientation by simulating its dynamic and kinematic models. For a known position and orientation of the satellite the images are obtained using the satellite's onboard camera, simulated by the Google Earth application. The orientation of the satellite is governed by reaction wheels, which produce the required moments to the satellite. The image-based control law using SIFT image features is applied to achieve an automatic reference-point observation on the Earth's surface. Main contributions of the paper are the following: use of the same sensor for Earth observation and attitude control, simplicity of the approach, no need for explicit calibration of camera parameters and good tracking accuracy. Demonstrated simulation results and performance analysis confirm the approach applicability. [ABSTRACT FROM AUTHOR]

In this paper we discuss Kendall's problem on a sphere. In rather physical than mathematical terms, a sphere Σ of radius 1, which rolls without slipping and without twisting over a sphere M of radius R > 1, is considered, and a fixed orthonormal frame $${\mathcal {R}}$$ attached to Σ. Given two points p and q on M, not necessarily different, and two states of $${\mathcal {R}}$$ , $${\mathcal {R}_{p}}$$ and $${\mathcal {R}_{q}}$$ , also not necessarily different, it is supposed that Σ is on p with the frame $${\mathcal {R}}$$ in the position $${\mathcal {R}_{p}}$$ , and one wishes to move Σ by a sequence of displacements without slipping and without twisting over geodesics of M to the point q, where the frame $${\mathcal {R}}$$ must be in the state $${\mathcal {R}_{q}}$$ . We prove that it is possible to do this in a sequence of no more than four movements. [ABSTRACT FROM AUTHOR]

The dynamic response of a flexible spinning vehicle with the varying thrust was investigated. The varying thrust was modeled as a series of simple harmonic components superimposed on a constant thrust. The analytical solutions for longitudinal motion were obtained by combining the motion equations and boundary conditions, and then, the transverse bending of rocket was examined solved by considering a coupled forcing vibration with varying parameters. Numerical solutions for a rocket with end-varying thrust were studied. It is shown that the lateral vibrations and motion attitudes of rocket depend on the frequencies of varying thrust to great extent, and an important phenomenon, beating, maybe take place. The proposed approach is more accurate and efficient in comparison to those standard numerical techniques. [ABSTRACT FROM AUTHOR]

In this paper, a new method is introduced for finding a near-optimal path of a nonholonomic robot moving in a 2D environment cluttered with static obstacles. The method is based on the Bump-Surfaces concept and is able to deal with robots represented by a translating and rotating rigid body. The proposed approach is applied to car-like robots. [ABSTRACT FROM AUTHOR]