A deterministic application θ:ℝ2→ℝ2 deforms bijectively and regularly the plane and allows the construction of a deformed random field X∘θ:ℝ2→ℝ from a regular, stationary, and isotropic random field X:ℝ2→ℝ. The deformed field X∘θ is, in general, not isotropic (and not even stationary), however, we provide an explicit characterization of the deformations θ that preserve the isotropy. Further assuming that X is Gaussian, we introduce a weak form of isotropy of the field X∘θ, defined by an invariance property of the mean Euler characteristic of some of its excursion sets. We prove that deformed fields satisfying this property are strictly isotropic. In addition, we are able to identify θ, assuming that the mean Euler characteristic of excursion sets of X∘θ over some basic domain is known.

Performance evaluation of a parallel robot is a multicriteria problem. By taking Delta robot as an object of study, this paper presents the kinematic performance evaluation of a three translational degrees-of-freedom parallel robot from the viewpoint of singularity, isotropy, and velocity transmission. It is shown that the determinant of a Jacobian matrix cannot measure the distance from the singular configuration due to the existing inverse kinematic singularity of a Delta robot. The determinants of inverse and direct kinematic Jacobian matrices are adopted for the measurement of distance from the singular configuration based on the theory of numerical linear dependence. The denominator of the Jacobian matrix will be lost in the computation of the condition number when the end-effector is on the centerline of the workspace, so the Delta robot may also be nearly at a singular configuration when the condition number of the Jacobian matrix is equal to 1. The velocity transmission index whose physical meaning is the maximum input angular velocity when the end-effector translates in the unit velocity is presented. The evaluation of singularity, isotropy, and velocity transmission of a Delta robot is investigated by simulation. The velocity transmission index can also be used for the velocity transmission evaluation of a parallel robot with pure rotational degrees-of-freedom based on the principle of similarity. The physical meaning is modified to be the maximum input velocity when the end-effector rotates in the unit angular velocity.

In this paper, a family of discriminant functions is defined for searching isotropic configurations in robot manipulators. A subfamily of the functions extensively exploits properties of symmetric polynomials derived from a manipulability matrix. The complexity analysis of computing the discriminant functions is provided. Possible applications of the functions are mentioned and illustrated.

Dexterity and workspace are two of the most important design criteria in developing manipulators. This paper presents algorithms for developing contour surfaces with specified dexterities and evaluating the area of the surfaces or the volume of the enclosed regions. The obtained results can be utilized to evaluate the dexterity and the rate of change of dexterity. Any closed curve or surface can be used to determine the singularity-free workspace of a manipulator with better dexterity. The proposed algorithms can be employed to study the dexterity and singularity-free workspace of 3-DOF manipulators and 4-DOF redundant serial manipulators. The contour surfaces in some subspaces of 6-DOF manipulators can also be investigated.

Isotropic manipulators are generally considered as designs with optimum dexterity. Currently, many 6-DOF (degrees-of-freedom) isotropic parallel manipulators can be developed by numerical or analytical methods. At an isotropic configuration, a manipulator is equidistant from its neighboring singular points. The distance, however, can be very small, so an isotropic design might have relatively smaller singularity-free workspace. This paper presents methods to develop traditional 6-DOF parallel manipulators with better dexterity and larger singularity-free workspace. Some fully symmetric nontraditional designs are then proposed. The evaluation of kinematic properties shows that the fully symmetric designs have very good global dexterity, better rotatability, and relatively larger singularity-free workspace. The manipulators are suitable for some special tasks requiring higher precision, better rotatability, or larger workspace.

This research presents a comprehensive and useful survey of the kinematic models of wheeled mobile robots and their optimal configurations. The kinematic modeling of wheeled mobile robots with no-slip is presented, by considering four common types of wheels: fixed, orientable, castor, and Swedish. Next, the accuracy of the kinematic models is discussed considering their sensitivity or relative error amplification, giving rise to the isotropy concept. As practical application of the previous theory, all types of three-wheeled mobile robots are modeled and their optimal (isotropic) configurations for no error amplification are obtained. Finally, three practical examples of error amplification are developed for several types of wheeled mobile robots in order to illustrate the benefits and limitations of the isotropic configurations.

We introduce an invariant, called the contact number, associated with each Euclidean submanifold. We show that this invariant is, surprisingly, closely related to the notions of isotropic submanifolds and holomorphic curves. We are able to establish a simple criterion for a submanifold to have any given contact number. Moreover, we completely classify codimension-$2$ submanifolds with contact number ${\geq}3$. We also study surfaces in $\mathbb{E}^6$ with contact number ${\geq}4$. As an immediate consequence, we obtain the first explicit examples of non-spherical pseudo-umbilical surfaces in Euclidean spaces.

AMS 2000 Mathematics subject classification: Primary 53C40; 53A10. Secondary 53B25; 53C42

The iteration of random tessellations in ℝd is considered, where each cell of an initial tessellation is further subdivided into smaller cells by so-called component tessellations. Sufficient conditions for stationarity and isotropy of iterated tessellations are given. Formulae are derived for the intensities of their facet processes, and for the expected intrinsic volumes of their typical facets. Particular emphasis is put on two special cases: superposition and nesting of tessellations. Bernoulli thinning of iterated tessellations is also considered.

For modelling non-stationary spatial random fields Z = {Z(x) : x∊ℝn, n≥2} a recent method has been proposed to deform bijectively the index space so that the spatial dispersion D(x,y) = var[Z(x)-Z(y)], (x,y)∊ℝnxℝn, depends only on the Euclidean distance in the deformed space through an isotropic variogram γ. We prove uniqueness of this model in two different cases: (i) γ is strictly increasing; (ii) γ(u) is differentiable for u > 0.

The concept of θ-stationarity for a simple second-order point process in R2 is introduced. This concept is closely related to that of isotropy. Some θ-stationary processes are defined. Techniques are given for simulating realisations of these processes. The second-order analysis of these processes which have an obvious point of reference or origin is considered. Methods are suggested for modelling spatial patterns which are realisations of such processes. These methods are illustrated using simulated data. The ideas are extended to multitype point processes.

Formulae relating properties of a two-phase structure to properties of a section through that structure are well known in stereology. An exact description of the error introduced into these formulae when the projection of a thick slice is observed rather than a true section (i.e. a slice of zero thickness) is presented for a model of opaque particles distributed according to a homogeneous Poisson point process in En.

This paper formulates the random process of line-segments in the Euclidean plane. Under conditions more general than Poisson, expressions are obtained, for Borel A ⊂ R2, for the first moments of M(A), the number of segment mid-points in A; N(A), the number of segments which intersect with convex A; S(A), the total length within A of segments crossing A; and C(A) the number of segment-segment crossings within A. In the case of Poisson mid-points, the distribution of the rth nearest line-segment to a given point is found.