Configuration Analysis of Metamorphic Mechanisms Based on Extended Adjacency Matrix Operations

Multiple bifurcated reconfiguration of double-loop metamorphic mechanisms with prismatic joints

Let GX be a graph obtained from a simple graph G by attaching a self-loop at each vertex of X ⊆ V ( G ) . The general extended adjacency matrix for the graph GX is defined and the bounds for the degree based energy of the graph GX are obtained. The study extends the notion of degree based energy of simple graphs to graphs with self-loops. For the graph GX of order n and size m with σ self-loops, the adjacency energy, E ( G X ) ≥ 4 mn + 2 σ ( n − σ ) n . The spectral radius ρ ( G X ) of its adjacency matrix is always less than or equal to 1 2 [ − 1 + 1 + 4 ( 2 m + σ + Δ − 1 ) ] , where Δ is the maximum degree in the graph GX and the equality conditions are given for 0 < σ < n . Few more bounds for ρ ( G X ) are also obtained. The study shows that, the spectral radius f max ( G X ) of its extended adjacency matrix satisfies: min i ∼ j F ij ( 2 m + σ ) n ≤ f max ( G X ) ≤ max i ∼ j F ij 2 ( 2 m + σ ) . We conclude the article by computing the extended adjacency spectrum of complete graph and complete bipartite graphs with self-loops.

The metamorphic mechanism has the characteristics of multi-configuration, variable constraints, and multi-function. For achieving the variation characteristics of the topological structures in different configurations and the coupling relations intuitively during the process of configuration transformation, a novel comprehensive symbolic matrix for representing the topological structures of the metamorphic mechanism is proposed. The information including variation of components, the relative joint orientations is involved in this new adjacency matrix. The variation characteristics and coupling features of the metamorphic mechanism can be obtained by the generalized operations including intersection and difference on the corresponding symbolic matrices.

The lower-order body matrix method is proposed as to topological configuration issue of planar metamorphic mechanism. Through combining lower-order operator and type code of kinematic pair, the lower-order body matrix is built to describe the information of adjacent body and kinematic pair. In the research on configuration transformation of planar metamorphic mechanism, step-description method is put forward which adapts to the analysis of various planar metamorphic ways. Step mathematical equation is established based on the generalized operational rule of lower-order body matrix. Application example shows that the lower-order matrix built with lower-order body matrix method is simpler and more informative, which has many advantages over the adjacency matrix and the correlation matrix that cannot be combined with the kinematics and dynamics equations, and solve the issue that the low-order sequences can only describe the open chain metamorphic mechanism. Step-description method can describe various planar metamorphic configuration transformation process comprehensively, which the EU matrix method cannot complete, providing a new method on the research of planar metamorphic mechanism.

The structural design of constrained joints is an effective way to realize the type synthesis of constrained metamorphic mechanism. In previous studies, constrained joints are often represented in the form of kinematic diagrams of mechanism, which doesn’t provide enough reference to the practical application of metamorphic mechanisms. On the basis of summarizing the constraint forms and resistance characteristics of the commonly used variable-constraint constrained joints, and according to the force analysis of the augmented Assur group with constrained joints, a new constrained revolute joint with combined variable-constraint of spring force and geometric constraint is designed, which has advantages such as simple structure, reliable working ability and passive adaptive capability to operational conditions. Based on this constrained joint, a two-configuration constrained metamorphic mechanism with under-actuated is designed, the force analysis of the mechanism is completed, and the effectiveness and feasibility of design approach proposed in this paper is verified for the constraint parameters of the invented metamorphic joint. The novel combinatorial variable-stiffness constrained joint can obtain stable and reliable configuration switching ability, which provides a feasible way for promoting the engineering application of metamorphic mechanism.

A note on eigenvalue, spectral radius and energy of extended adjacency matrix

Due to the explosive growth of social network services, friendship inference has been widely adopted by Online Social Service Providers (OSSPs) for friend recommendation. The conventional techniques, however, have limitations in accuracy or scalability to handle such a large yet sparse multi-source data. For example, the OSSPs will be required to manually give the order in which the various information is applied. This unavoidably reduces the applicability of existing friend recommendation systems. To address this issue, we propose a Two-stage Deep learning framework for Friendship Inference (TDFI). This approach can utilize multi-source information simultaneously with low complexity. In particular, we apply an Extended Adjacency Matrix (EAM) to represent the multi-source information. We then adopt an improved Deep AutoEncoder Network (iDAEN) to extract the fused feature vector for each user. The TDFI framework also provides an improved Deep Siamese Network (iDSN) to measure user similarity from iDAEN. Finally, we evaluate the effectiveness and robustness of TDFI on three large-scale real-world datasets. It shows that TDFI can effectively handle the sparse multi-source data while providing better accuracy for friend recommendation.

This paper presents a unique feature of geometric and physical constraint of axes of the reconfigurable-axis (rA) joint and analyses the effectiveness in the constructed limb, resulting in variation of mobility configuration of metamorphic parallel mechanism. This change in mobility is completed by two cases illustrated by a 3(rA)PS metamorphic parallel mechanism having variable mobility from 3 to 6. The underlying principle of the metamorphosis of this rA joint is demonstrated by investigating the dependence of the corresponding screw system comprising of line vectors, leading to evolution of the rA joint from the two types of spherical joints to the three types of variable Hooke joints and the one revolute joint. This new type of metamorphic parallel mechanism has broad application prospects, especially in the field of 3d printing. According to the complexity of the shape of the workpiece, the degree of freedom corresponding to different topologies can be selected, which can reduce the costs and increase the adaptability to different working conditions.

This paper investigates the metamorphic mechanism derived from the hindwing in ladybird beetles, which creates its mathematical models and analyzes kinematics based on screw theory using mathematical software. The hindwing in ladybird beetles is an origami-like structure. Considering the concept of equivalent mechanisms, any origami pattern can be seen as being composed of several spherical multi-bar linkages connecting to each other. Hence, this paper proposes an algebra method to reveal the relations between different joint angles in the spherical multi-bar linkage, using the loop equations and Sylvester’s dialytic elimination method. The results can be used to find the location and the motion trajectory of any desired point in this ladybird wing metamorphic mechanism. The way of analysis of the spherical multi-bar mechanism used in this paper can be widely applied in design and practical applications.

Traditional bionic joints including wrist, waist, ankle and shoulder have three revolute degrees of freedom, which can be treated as spherical joints, and its structure is prone to be damaged with impact force through the center of spherical joint. In this paper, a novel metamorphic parallel mechanism with two configurations used for bionic joint design is presented. The origin kinematic chain for this metamorphic mechanism, which is also the first configuration mechanism, consists of a moving platform, a base plate and four connecting legs between moving platform and base plate. In contrast with traditional ankle joint, a constrained translational degree of freedom can be activated in origin kinematic chain to avoid impact damage. The origin kinematic chain can transform to the second configuration mechanism which is in possession of three revolute degrees of freedom and equivalent to normal spherical joint. The transforming process is represented by new incidence matrix and the motion characteristics of this metamorphic mechanism in different configuration corresponding to each work-stage are analysed using screw theory.

In order to store a graph or digraph in a computer, we need something other than the diagram or the formal definition. This something is the adjacency matrix, a matrix of O's and l's. The l's correspond to the arcs of the digraph. Certain matrix operations will be seen to correspond to digraph concepts.

This paper introduced a new metamorphic parallel mechanism consisting of four reconfigurable rTPS limbs. Based on the reconfigurability of the reconfigurable Hooke (rT) joint, the rTPS limb has two phases while in one phase the limb has no constraint to the platform, in the other it constrains the spherical joint center to lie on a plane. This results in the mechanism to have ability of reconfiguration between different topologies with variable mobility. Geometric constraint equations of the platform rotation matrix and translation vector are set up based on the point-plane constraint, which reveals the bifurcated motion property in the topology with mobility 2 and the geometric condition with mobility change in altering to other mechanism topologies. Following this, a unified kinematics limb modeling is proposed considering the difference between the two phases of the reconfigurable rTPS limb. This is further applied for the mechanism modeling and both the inverse and forward kinematics is analytically solved by combining phases of the four limbs covering all the mechanism topologies.

Mobility analysis of multi-configuration Rubik's Cube mechanism based on the reciprocal screws

We propose a fundamentally new approach to Datalog evaluation. Given a linear Datalog program DB written usingNconstants and binary predicates, we first translate if-and-only-if completions of clauses in DB into a setEq(DB) of matrix equations with a non-linear operation, where relations inMDB, the least Herbrand model of DB, are encoded as adjacency matrices. We then translateEq(DB) into another, but purely linear matrix equationsẼq(DB). It is proved that the least solution ofẼq(DB) in the sense of matrix ordering is converted to the least solution ofEq(DB) and the latter givesMDBas a set of adjacency matrices. Hence, computing the least solution ofẼq(DB) is equivalent to computingMDBspecified by DB. For a class of tail recursive programs and for some other types of programs, our approach achievesO(N3) time complexity irrespective of the number of variables in a clause since only matrix operations costingO(N3) or less are used. We conducted two experiments that compute the least Herbrand models of linear Datalog programs. The first experiment computes transitive closure of artificial data and real network data taken from the Koblenz Network Collection. The second one compared the proposed approach with the state-of-the-art symbolic systems including two Prolog systems and two ASP systems, in terms of computation time for a transitive closure program and the same generation program. In the experiment, it is observed that our linear algebraic approach runs 101~ 104times faster than the symbolic systems when data is not sparse. Our approach is inspired by the emergence of big knowledge graphs and expected to contribute to the realization of rich and scalable logical inference for knowledge graphs.

Bond‐extended stochastic and nonstochastic bilinear indices are introduced in this article as novel bond‐level molecular descriptors (MDs). These novel totals (whole‐molecule) MDs are based on bilinear maps (forms) similar to use defined in linear algebra. The proposed nonstochastic indices try to match molecular structure provided by the molecular topology by using the k th Edge(Bond)‐Adjacency Matrix ( E k , designed here as a nonstochastic E matrix). The stochastic parameters are computed by using the k th stochastic edge‐adjacency matrix, ES k , as matrix operators of bilinear transformations. This new edge (bond)‐adjacency relationship can be obtained directly from E k and can be considered like a new matrix‐transformation strategic to obtain new relations for a molecular graph. In both set of MDs, chemical information is codified by using different pair combinations of atomic weightings (in this case four atomic‐labels: atomic mass, polarizability, van der Waals volume, and electronegativity). In addition, a local‐fragment (bond‐type) formalism was also developed. The k th bond‐type bilinear indices are calculated by summing the k th bond bilinear indices of all bonds of the same bond type in the molecules. The new set of MDs can be easily and quickly calculated in our in house software TOMOCOMD‐CARDD (topological molecular computational design computer‐aided‐rational‐drug design). The reported application and utilization of these MDs for predictive capability correlations of structure with physicochemical and pharmacological properties are reviewed. Three benchmark datasets have been used to evaluate the QSPR/QSAR behavior of the new bond‐level TOMOCOMD‐CARDD MDs. We developed the QSPR models to describe several physicochemical properties of octane isomers (First Case) and, to analyze of the boiling point of 28 alkyl‐alcohols (Second Case) and to examine of the specific rate constant (log k ), the partition coefficient (log P ), as well as the antibacterial activity of 34 derivatives of 2‐furylethylenes (Third Case). For these three rounds, the quantitative models found are significant from a statistical point of view and permit a clear interpretation of the studied properties in terms of the structural features of molecules. A leave‐one‐out cross‐validation procedure revealed that the regression models had a good predictability. The comparison with other approaches reveals good performance of the method proposed. Therefore, it is clearly demonstrated that this suitability is higher than that shown by other 2D/3D well‐known sets of MDs. The approach described here appears to be a very promising structural invariant, useful for QSPR/QSAR studies and shown to provide an excellent alternative or guides for discovery and optimization of new lead compounds, reducing the time and cost of the traditional procedure. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011