A dynamical approach to the variational inequality on modified elastic graphs

We consider the variational inequality on modified elastic graphs. Since the variational inequality is derived from the minimization problem for the modified elastic energy defined on graphs with the unilateral constraint, a solution to the variational inequality can be constructed by the direct method of calculus of variations. In this paper we prove the existence of solutions to the variational inequality via a dynamical approach. More precisely, we construct an L2-type gradient flow corresponding to the variational inequality and prove the existence of solutions to the variational inequality via the study on the limit of the flow.

Unilaterally Constrained Motion of a Curved Surgical Tool

SUMMARYConstrained motion is essential for varying robotics tasks, especially in surgical robotics, for instance, the case of minimally invasive interventions. This article proposes generic formulations of the classical bilateral constrained motion (i.e., when the incision hole has almost the same diameter as that of the tool) as well as unilaterally constrained motion (i.e., when the hole incision has a larger diameter compared to the tool diameter). One of the latter constraints is combined with another surgical task such as incision/ablation or suturing a wound (modeled here by 3D geometric paths). The developed control methods based on the hierarchical task approach are able to manage simultaneously the constrained motion (depending on the configuration case, i.e., bilateral or unilateral constraint) and a 3D path following. In addition, the proposed methods can operate with both straight or curved surgical tools. The proposed methods were successfully validated in various scenarios. Foremost, a simulation framework was proposed to access the performances of each proposed controller. Thereafter, several experimental validations were carried out. Both the simulation and experimental results have demonstrated the relevance of the proposed approach, as well as promising performances in terms of behavior as well as accuracy.

A minimisation problem in L∞ with PDE and unilateral constraints

We study the minimisation of a cost functional which measures the misfit on the boundary of a domain between a component of the solution to a certain parametric elliptic PDE system and a prediction of the values of this solution. We pose this problem as a PDE-constrained minimisation problem for a supremal cost functional in L∞, where except for the PDE constraint there is also a unilateral constraint on the parameter. We utilise approximation by PDE-constrained minimisation problems in Lp as p →∞ and the generalised Kuhn-Tucker theory to derive the relevant variational inequalities in Lp and L∞. These results are motivated by the mathematical modelling of the novel bio-medical imaging method of Fluorescent Optical Tomography.

A Multi-Scale Model of Soft Imperfect Interface with Nonlocal Damage

The aim of this paper is to propose a model of bonded interface including nonlocal damage and unilateral conditions. The model is derived from the problem of a composite structure made by two adherents and a thin adhesive. The adhesive is damaged at microscopic level and is subjected to two regimes, one in traction and one in compression. The model of interface is derived by matched asymptotic expansions. In this paper, two cases corresponding to the two regimes are discussed. Moreover, this model can be considered as a model of contact with adhesion and unilateral constraint. At the end of the paper, a simple numerical example is presented to show the evolution of the model.

A Boundary Layer Approach to Multibody Systems Involving Single Frictional Impacts

A systematic theoretical approach is presented, revealing dynamics of a class of multibody systems. Specifically, the motion is restricted by a set of bilateral constraints, acting simultaneously with a unilateral constraint, representing a frictional impact. The analysis is carried out within the framework of Analytical Dynamics and uses some concepts of differential geometry, which provides a foundation for applying Newton's second law. This permits a successful and illuminating description of the dynamics. Starting from the unilateral constraint, a boundary is defined, providing a subspace of allowable motions within the original configuration manifold. Then, the emphasis is focused on a thin boundary layer. In addition to the usual restrictions imposed on the tangent space, the bilateral constraints cause a correction of the direction where the main impulse occurs. When friction effects are negligible, the dominant action occurs along this direction and is described by a single nonlinear ordinary differential equation (ODE), independent of the number of the original generalized coordinates. The presence of friction increases this to a system of three ODEs, capturing the essential dynamics in an appropriate subspace, arising by bringing the image of the friction cone from the physical to the configuration space. Moreover, it is shown that the classical Darboux–Keller approach corresponds to a special case of the new method. Finally, the theoretical results are complemented by a selected set of numerical results for three examples.

Kinetic Responses to Unilateral Constraint of the Metatarsophalangeal Joints During Bipedal Walking

The paper was published without confirmation from a ‘contributor’, and after mutual agreement they decided to retract the paper.

Variable Structure Control of a Mass Spring Damper Subjected to a Unilateral Constraint: Application to Radio-Frequency MEMS Switches

A feedback control strategy is presented for improving the transient response of the ubiquitous mass-spring-damper (MSD) system; the closed-loop system has a small settling time with no overshoot for a step input. This type of response is ideal for MSD systems subjected to a unilateral constraint such as radio-frequency micro-electro-mechanical-system (RF MEMS) switches, which are required to close in a short period of time without bouncing. The control strategy switches the stiffness of the MSD between its nominal value and a negative value, resulting in a hybrid dynamical system. A phase portrait analysis of the hybrid system is carried out to establish the asymptotic stability property of the equilibrium and quantify the transient response. Simulation results are presented using parameter values of a real RF MEMS switch from the literature. As compared to open-loop strategies that are currently used, the proposed feedback control strategy promises to provide comparable switch-closing times with robust performance and eliminate bouncing.