Symmetry Adapted Assur Decompositions

Assur graphs are a tool originally developed by mechanical engineers to decompose mechanisms for simpler analysis and synthesis. Recent work has connected these graphs to strongly directed graphs and decompositions of the pinned rigidity matrix. Many mechanisms have initial configurations, which are symmetric, and other recent work has exploited the orbit matrix as a symmetry adapted form of the rigidity matrix. This paper explores how the decomposition and analysis of symmetric frameworks and their symmetric motions can be supported by the new symmetry adapted tools.

Finite element simulation of a motorway bridge collapse using the concrete damage plasticity model

The aim of this work is to show how the concrete damage plasticity model developed by Lubliner et al. can be applied for calculation of a motorway bridge collapse occurred in the Amur region, Russia. The concrete structural behaviour is highly complex. Being a quasi-brittle material, concrete demonstrates softening behaviour that is numerically complex due to the loss of positive definiteness of the tangent rigidity matrix of the material, and hence the loss of the ellipticity of the equilibrium rate equation. This eventually leads to the loss of well-posedness of the rate boundary value problem. Besides that, concrete behaviour in compression differs from that in tension. There are a few different failure modes of concrete material: tension cracking, compression crushing, spalling of concrete, etc.

Physical Interpretation of Rigidity for Bearing Formations: Application to Mobility and Singularity Analyses

Research on formation control and cooperative localization for multirobot systems has been an active field over the last years. A powerful theoretical framework for addressing formation control and localization, especially when exploiting onboard sensing, is that of formation rigidity (mainly studied for the cases of distance and bearing measurements). Rigidity of a formation depends not only on the topology of the sensing/communication graph but also on the spatial arrangement of the robots, since special configurations (“singularities” of the rigidity matrix), which are hard to detect in general, can cause a rigidity loss and prevent convergence of formation control/localization algorithms based on formation rigidity. The aim of this paper is to gain additional insights into the internal structure of bearing rigid formations by considering an alternative characterization of formation rigidity using tools borrowed from the mechanical engineering community. In particular, we show that bearing rigid graphs can be given a physical interpretation related to virtual mechanisms, whose mobility and singularities can be analyzed and detected in an analytical way by using tools from the mechanical engineering community (screw theory, Grassmann geometry, and Grassmann-Cayley algebra). These tools offer a powerful alternative to the evaluation of the mobility and singularities typically obtained by numerically determining the spectral properties of the bearing rigidity matrix (which typically prevents drawing general conclusions). We apply the proposed machinery to several case formations with different degrees of actuation and discuss known (and unknown) singularity cases for representative formations. The impact on the localization problem is also discussed.

Spectral methods for Langevin dynamics and associated error estimates

We prove the consistency of Galerkin methods to solve Poisson equations where the differential operator under consideration is hypocoercive. We show in particular how the hypocoercive nature of the generator associated with Langevin dynamics can be used at the discrete level to first prove the invertibility of the rigidity matrix, and next provide error bounds on the approximation of the solution of the Poisson equation. We present general convergence results in an abstract setting, as well as explicit convergence rates for a simple example discretized using a tensor basis. Our theoretical findings are illustrated by numerical simulations.

Control of 2D Minimally Persistent Formations with the Fault Tolerance of Three Co-Leaders in Cycle

This paper mainly addresses a novel control law with rigidity matrix based on three co-leaders minimally persistent formations in the plane. This control law particularly considers the fault tolerance of the leaders, and in this way, the three co-leaders model is better than leader-first follower model, leader-remote follower model, etc. in persistent formation. Firstly, the first order kinematic model is adopted for every agent. Then the fundamental moving principal of the leaders and the followers are described in detail. On the basis of these principals, the control law with the rigidity matrix for the whole formation is proposed. Moreover, the stability analysis is also supplied. Finally, simulations show that the proposed controllers ensure the group formation stabilized to maintain the rigid shape, while the distances between the agents remain unchanged.

Rigidity-Based Stabilization of Multi-Agent Formations

This paper is concerned with the decentralized formation control of multi-agent systems moving in the plane using rigid graph theory. Using a double-integrator agent model (as opposed to the simpler, single-integrator model), we propose a new control law to asymptotically stabilize the interagent distance error dynamics. Our approach uses simple backstepping and Lyapunov arguments. The control, which is explicitly dependent on the rigidity matrix of the undirected graph that models the formation, is derived for a class of potential functions. Specific potential functions are then used as a demonstration inclusive of simulation results.

Accuracy of Stress Analysis Using Numerical Integration of Elastic Half-Space

In case of constructions placed on subsoil, it is necessary to create a rigidity matrix for the element subsoil. That rigidity matrix should be then added up in respective positions with the rigidity matrix of an element. To clarify the proposed model of the subsoil, a method is derived for determination of vertical subsoil stress analysis under any shape of a slab construction by means of numerical integration and theory of isoparametric elements using the Jacobian transformation. This approach is rather original and represents the key contribution of this work in interaction solutions. Using the proposed approach, the method can be employed for any shape of a finite element.

Mobility Determination of Mechanisms Based on Rigidity Theory

Rigidity theory deals mostly with the topological computation in mechanical systems, i.e. it aims at making generic statements. Mechanism theory is mainly concerned with the geometrical analysis but again also with generic statements. Even more so for mobility analysis where one is interested in both the generic mobility and that of a particular mechanism. In rigidity theory the mathematical foundation is the topology representation using bar-joint and body-bar graphs, and the corresponding rigidity matrix. In this paper novel geometric rules for constructing the body-bar rigidity matrix are derived for general planar mechanisms comprising revolute and prismatic joints. This allows, for the first time, the treatment of general planar mechanisms with the body-bar approach. The rigidity matrix is also derived for spatial mechanisms with spherical joints. The bar-joint rigidity matrix is shown to be a special case of body-bar representation. It is shown that the rigidity matrices allow for mobility calculation as shown in the paper. This paper is aimed at supplying a unified view and as a result to enable the mechanisms community to employ the theorems and methods used in rigidity theory. An algorithm for mobility determination — the pebble game — is discussed. This algorithm always finds the correct generic mobility if the mechanism can be represented by a body-bar graph.

The Second Order Elastic-Plastic Analysis of Plane Steel Frame Structures with Semi-Rigid Connections

Considering the nonlinear behaviors of semi-rigid connections, a new calculation model for the second order elastic-plastic analysis of plane steel structures is presented by replacing the general element with a new one, which has two rotational springs at its ends. Based on the theory of beam-to-column, the element’s rotation-displacement equations are revised, and the second order elastic-plastic rigidity matrix of this kind of structure is derived in this paper. And the second order elastic-plastic program of this kind of structure is worked out. Two steel frames are calculated with this program. The results of calculations show that this model is reliable and fit to the analysis of all plane steel structures.