Specifity of Including of Structural Nonlinearity in Model of Dynamics of Cable-Driven Robot

The paper deals with a problem of modeling of the dynamics of a parallel cable-driven robot with the inclusion of structural nonlinearity of cables in a mathematical model. Mathematical model is implemented in a computer model with the possibility of using of symbolic calculations. Parallel cable robots as a type of robotics have been developing in the last two or three decades. The research in the theoretical field was being carried out and the mathematical model of the cable system was being refined with the spread of the practical use of cable robots. This is a non-trivial task to draw up a dynamic model of a cable-driven robot. Cable-driven robots are highly nonlinear systems, because of the main reason for the nonlinearity is the properties of the cable system. As an element of a mechanical system, the cable or the wire rope is a unilateral constraint, since the cable works only for stretching, but not for compression. Thus, the cables are structurally nonlinear elements of the system. On the other hand, cables have the property of sagging under their own weight. Thus, the cables are geometrically nonlinear elements of the system. Under the condition of a payload mass that is utterly greater than the mass of each cable, the cables can be considered strained without sagging and geometric nonlinearity can be neglected. Since symbolic computations can be used in a computer model which implements a mathematical model of the dynamics of a robot, in such a way it must provide the possibility of symbolic computations with the condition of structural nonlinearity. The main aim of this work is to develop a method that ensures the inclusion of the structural nonlinearity of the cable system in the mathematical model. It is supposed to consider the possibility of implementation of the computer model with symbolic computations. The problem of including a mathematical model of cables as unilateral constraints in the model of highly loaded cable robots is considered. The justification for including the activation functions in a system of differential equations of dynamics of cable-driven robot is formulated. A model of wire ropes as unilateral constraints is represented via including the activation functions in a system of differential equations. With using of the proposed method, numerical solution of a problem of forward dynamics has been obtained for high-loaded parallel cable-driven robot.

Structures With Multiple Rigid Configurations Due to Prestress and Unilateral Contacts

This paper presents structures with multiple equilibrium configurations arising from the combination of a state of prestress and unilateral contacts. A design problem is posed where preloaded elastic springs and unilateral constraints are embedded throughout a mechanism. The spring parameters are designed such that multiple target configurations are immobilized due to contact. In each of these configurations, the spring forces maintain compressive reaction forces, immobilizing the structure. Each immobilized configuration can rigidly resist perturbation forces up to some finite magnitude where contact is lost. Hence, this case of multiple configurations in equilibrium due to the combination of prestress and contact is referred to as multi-configuration rigidity. Two examples of structures exhibiting multi-configuration rigidity are presented. First, a four bar linkage with a single kinematic degree of freedom is used to introduce the concept. In the context of the linkage, multi-configuration rigidity is compared to multi-stability, exhibiting the key differences between the two concepts. Then, a 24-degree-of-freedom kirigami surface is presented that can morph between flat and spherical configurations, motivated by RF antenna applications. By embedding torsional springs and fold angle stops throughout the structure, flat and spherical configurations are made rigid. Actuation between the configurations can easily be achieved by snapping the structure between the rigid configurations.

Steps Towards Non-Smooth Multibody Dynamics

Before the background of many thoughts about contact and impact behavior with and without friction in the past centuries a comprehensive theory appeared not before the second half of the last century, mainly connected with the names of Moreau in Montpellier and Panagiotopoulos in Thessaloniki. My former Institute has been part of this evolution focusing on non-smooth multibody dynamics and on large systems. The local development from simple impact to complex contact systems including all possible contact details will be subject of the paper, considering also the necessary mathematical evolution from classical multibody system theory with bilateral constraints and single-valued forces to non-smooth multibody system theory with unilateral constraints and set-valued forces. Paper will be illustrated by practical examples.

Usability of Haptic Volumetric Assistance for Surgical Navigation Tasks

Manual control of surgical instruments represents a sensorimotor control task with at least 3-6 degrees of freedom (DoF). The impact of haptic guidance on volumetric navigation tasks, such as milling of planned volumes for prosthesis fits or preserving sensitive tissues, is investigated. Interaction centered studies are performed to evaluate the usability of the assistance modes for navigation within a volume, along the surface of a volume and around forbidden regions. Results show that haptic assistance can reduce the number of constraint violations, if the virtual stiffness is high enough. However, haptic assistance also can increase error rates when counterforces are close to the absolute perception threshold, as a false sense of security can arise. For navigation along complex surfaces bilateral haptic constraints should be preferred, while unilateral constraints are sufficient for simple geometries. This study complements previous publications as a basis for a flexible rule-based selection or adaptation of modular haptic assistance systems.

Modelling and Numerical Solution of the Problem with Unilateral Constraints and Friction under Dynamic Action of the Load

Problems with unilateral constraints are not uncommon in the practice of calculating building construction and structures. Certain difficulties in solving them arise during contact friction, as well as the dynamic action of the load. It is known that such problems from a mathematical point of view s are not correct enough, their solution becomes more complicated and depends on the history of loading and deformation of the structure. At the same time, the ability to take into account the complex working conditions of the structure makes its calculation more complete and accurate. The paper considers the solution of a dynamic contact problem on the basis of the finite element method and the step-by-step loading method. Unilateral constraints with Coulomb friction are modeled using contact finite elements of a frame-rod type. The method of compensating loads is applied in order to comply with the limitations under ultimate friction-sliding conditions. Based on the considered discrete contact model and the step-by-step analysis method, a numerical algorithm has been developed, which allows in one step-by-step process to integrate simultaneously the equations of motion and implement contact conditions with Coulomb friction. With the help of the proposed approach, numerical solutions of the problem pertaining to a structure contact with the base have been obtained and analyzed at various parameters of dynamic load. Comparison of the results with the solution obtained by the well-known iteration method on the ultimate friction forces allows to conclude about the efficiency and reliability of the developed algorithm under complex contact conditions and dynamic loading.

NONLINEAR PROBLEM OF INTERFACE CRACK BEHAVIOR UNDER THE ACTION OF SHEARING WAVE

Solution of the problem for an interface crack under the action of a harmonic shear wave is presented. It is shown that the same problems solutions of other authors were performed without taking into account the crack faces contact, and results obtained indicate the interpenetration of the faces, that is not possible. Thus, it is proved that the problem is nonlinear because the positions and sizes of the contact zone are unknown and variable during the loading. The solution is obtained by the boundary integral equations method taking into account the contact interaction of the crack faces: using the Somigliana dynamic identity and the boundary equations arising from them, the transition from the two-dimensional problem to the equivalent problem at the boundaries of the domain is realized; the vector components in the boundary integral equations are presented by Fourier series, to prevent the interpenetration of the crack faces and the emergence of tensile forces in the contact zone the Signorini unilateral constraints are involved. The numerical solution is performed by the method of boundary elements with constant approximation of the problem parameters on an element. Numerical researches of the shear wave frequency influence onto the crack faces and adjoining surface displacements, opening and extent of crack faces contact zone are carried out. The quantitative difference between the maximum tangential and normal components of adhesion line and the crack faces displacements is shown. It is shown that the position and length of the contact area change during the load period, and the magnitudes of the contact forces vary along the crack length.

Solution stability to parametric distributed optimal control problems with finite unilateral constraints

<p style='text-indent:20px;'>This paper deals with stability of solution map to a parametric control problem governed by semilinear elliptic equations with finite unilateral constraints, where the objective functional is not convex. By using the first-order necessary optimality conditions, we derive some sufficient conditions under which the solution map is upper semicontinuous with respect to parameters.</p>

Tykhonov Well-Posedness and Convergence Results for Contact Problems with Unilateral Constraints

This work presents a unified approach to the analysis of contact problems with various interface laws that model the processes involved in contact between a deformable body and a rigid or reactive foundation. These laws are then used in the formulation of a general static frictional contact problem with unilateral constraints for elastic materials, which is governed by three parameters. A weak formulation of the problem is derived, which is in the form of an elliptic variational inequality, and the Tykhonov well-posedness of the problem is established, under appropriate assumptions on the data and parameters, with respect to a special Tykhonov triple. The proof is based on arguments on coercivity, compactness, and lower-semicontinuity. This abstract result leads to different convergence results, which establish the continuous dependence of the weak solution on the data and the parameters. Moreover, these results elucidate the links among the weak solutions of the different models. Finally, the corresponding mechanical interpretations of the conditions and the results are provided. The novelty in this work is the application of the Tykhonov well-posedness concept, which allows a unified and elegant framework for this class of static contact problems.