Underconstrained Systems Research Articles

Nonlinear hierarchical control for four-wheel-independent-drive electric vehicle

As under-constrained systems, four-wheel-independent-drive (4WID) electric vehicles have more driving degrees of freedom. In this context, reasonable control and distribution of driving or braking torque to each wheel is extremely important from the vehicle safety perspective. However, it is difficult to provide the optimal wheel torque because of the time-varying characteristics and typical over-actuated nature of the system. In light of these challenges, a novel hierarchical control scheme comprising a top- and bottom-level controller is proposed herein. First, for the top-level controller, a time-varying model-predictive-control (TV-MPC) controller is designed based on an extended 3-degree-of-freedom (3-DOF) reference vehicle model. The total driving force and additional yaw moment can be obtained using the TV-MPC. Second, for the bottom-level controller, the torque expression of each wheel is determined using the equal-adhesion-rate-rule -based algorithm. The co-simulation results obtained herein indicate that the proposed control scheme can effectively improve vehicle safety.

Rigidity transitions in zero-temperature polygons.

We study geometrical clues of a rigidity transition due to the emergence of a system-spanning state of self-stress in underconstrained systems of individual polygons and spring networks constructed from such polygons. When a polygon with harmonic bond edges and an area spring constraint is subject to an expansive strain, we observe that convexity of the polygon is a necessary condition for such a self-stress. We prove that the cyclic configuration of the polygon is a sufficient condition for the self-stress. This correspondence of geometry and rigidity is akin to the straightening of a one dimensional chain of springs to rigidify it. We predict the onset of the rigidity transition and estimate the transition strain using purely geometrical methods. These findings help determine the rigidity of an area-preserving polygon just by knowing its geometry. Since two-dimensional spring networks can be considered as a network of polygons, we look for similar geometric features in underconstrained spring networks under isotropic expansive strain. We observe that all polygons attain convexity at the rigidity transition such that the fraction of convex, but not cyclic, polygons predicts the onset of the rigidity transition. Acyclic polygons in the network correlate with larger tensions, forming effective force chains.

Spatial structure of states of self stress in jammed systems

States of self stress, organizations of internal forces in many-body systems that are in equilibrium with an absence of external forces, can be thought of as the constitutive building blocks of the elastic response of a material. In overconstrained disordered packings they have a natural mathematical correspondence with the zero-energy vibrational modes in underconstrained systems. While substantial attention in the literature has been paid to diverging length scales associated with zero- and finite-energy vibrational modes in jammed systems, less is known about the spatial structure of the states of self stress. In this work we define a natural way in which a unique state of self stress can be associated with each bond in a disordered spring network derived from a jammed packing, and then investigate the spatial structure of these bond-localized states of self stress. This allows for an understanding of how the elastic properties of a system would change upon changing the strength or even existence of any bond in the system.

Determination of nonlinear genetic architecture using compressed sensing

BackgroundOne of the fundamental problems of modern genomics is to extract the genetic architecture of a complex trait from a data set of individual genotypes and trait values. Establishing this important connection between genotype and phenotype is complicated by the large number of candidate genes, the potentially large number of causal loci, and the likely presence of some nonlinear interactions between different genes. Compressed Sensing methods obtain solutions to under-constrained systems of linear equations. These methods can be applied to the problem of determining the best model relating genotype to phenotype, and generally deliver better performance than simply regressing the phenotype against each genetic variant, one at a time. We introduce a Compressed Sensing method that can reconstruct nonlinear genetic models (i.e., including epistasis, or gene-gene interactions) from phenotype-genotype (GWAS) data. Our method uses L1-penalized regression applied to nonlinear functions of the sensing matrix.ResultsThe computational and data resource requirements for our method are similar to those necessary for reconstruction of linear genetic models (or identification of gene-trait associations), assuming a condition of generalized sparsity, which limits the total number of gene-gene interactions. An example of a sparse nonlinear model is one in which a typical locus interacts with several or even many others, but only a small subset of all possible interactions exist. It seems plausible that most genetic architectures fall in this category. We give theoretical arguments suggesting that the method is nearly optimal in performance, and demonstrate its effectiveness on broad classes of nonlinear genetic models using simulated human genomes and the small amount of currently available real data. A phase transition (i.e., dramatic and qualitative change) in the behavior of the algorithm indicates when sufficient data is available for its successful application.ConclusionOur results indicate that predictive models for many complex traits, including a variety of human disease susceptibilities (e.g., with additive heritability h2∼0.5), can be extracted from data sets comprised of n⋆∼100s individuals, where s is the number of distinct causal variants influencing the trait. For example, given a trait controlled by ∼10 k loci, roughly a million individuals would be sufficient for application of the method.

Optimal decomposition and recombination of isostatic geometric constraint systems for designing layered materials

Optimal recursive decomposition (or DR-planning) is crucial for analyzing, designing, solving or finding realizations of geometric constraint systems. While the optimal DR-planning problem is NP-hard even for (general) 2D bar–joint constraint systems, we describe an O(n3) algorithm for a broad class of constraint systems that are isostatic or underconstrained. The algorithm achieves optimality by using the new notion of a canonical DR-plan that also meets various desirable, previously studied criteria. In addition, we leverage recent results on Cayley configuration spaces to show that the indecomposable systems – that are solved at the nodes of the optimal DR-plan by recombining solutions to child systems – can be minimally modified to become decomposable and have a small DR-plan, leading to efficient realization algorithms. We show formal connections to well-known problems such as completion of underconstrained systems. Well suited to these methods are classes of constraint systems that can be used to efficiently model, design and analyze quasi-uniform (aperiodic) and self-similar, layered material structures. We formally illustrate by modeling silica bilayers as body–hyperpin systems and cross-linking microfibrils as pinned line-incidence systems. A software implementation of our algorithms and videos demonstrating the software are publicly available online (visit http://cise.ufl.edu/~tbaker/drp/index.html).

Modeling of Underconstrained Systems with the Force Method

Determination of underconstrained systems using traditional theory may cause a nonconservative solution or even lead to error. The fundamentals of the concept and general formulations for kinematic path are derived such that they can be used for determination of both elastic deformation and change in geometry experienced in underconstrained systems. The condition of orthogonality is used such that the existence of such a space having the property of the general loads orthogonal to geometric displacement is thus proved. Numerical examples in the literature are used to verify the proposed methods. The efficacy of the proposed procedures implemented in engineering application is finally demonstrated under erection simulation.

A new framework for sharp and efficient resolution of NCSP with manifolds of solutions

When numerical CSPs are used to solve systems of n equations with n variables, the preconditioned interval Newton operator plays two key roles: First it allows handling the n equations as a global constraint, hence achieving a powerful contraction. Second it can prove rigorously the existence of solutions. However, none of these advantages can be used for under-constrained systems of equations, which have manifolds of solutions. A new framework is proposed in this paper to extend the advantages of the preconditioned interval Newton to under-constrained systems of equations. This is achieved simply by allowing domains of the NCSP to be parallelepipeds, which generalize the boxes usually used as domains.

Soft Manifold Dynamics behind Negative Thermal Expansion

Minimal models are developed to examine the origin of large negative thermal expansion in underconstrained systems. The dynamics of these models reveals how underconstraint can organize a thermodynamically extensive manifold of low-energy modes which not only drives negative thermal expansion but extends across the Brillioun zone. Mixing of twist and translation in the eigenvectors of these modes, for which in ZrW2O8 there is evidence from infrared and neutron scattering measurements, emerges naturally in our model as a signature of the dynamics of underconstraint.

GPDOF — A FAST ALGORITHM TO DECOMPOSE UNDER-CONSTRAINED GEOMETRIC CONSTRAINT SYSTEMS: APPLICATION TO 3D MODELING

Our approach exploits a general-purpose decomposition algorithm, called GPDOF, and a dictionary of very efficient solving procedures, called r-methods, based on theorems of geometry. GPDOF decomposes an equation system into a sequence of small subsystems solved by r-methods, and produces a set of input parameters.1. Recursive assembly methods (decomposition-recombination), maximum matching based algorithms, and other famous propagation schema are not well-suited or cannot be easily extended to tackle geometric constraint systems that are under-constrained. In this paper, we show experimentally that, provided that redundant constraints have been removed from the system, GPDOF can quickly decompose large under-constrained systems of geometrical constraints. We have validated our approach by reconstructing, from images, 3D models of buildings using interactively introduced geometrical constraints. Models satisfying the set of linear, bilinear and quadratic geometric constraints are optimized to fit the image information. Our models contain several hundreds of equations. The constraint system is decomposed in a few seconds, and can then be solved in hundredths of seconds.

A geometric constraint solver for 3-D assembly modeling

In this paper, we propose a geometric constraint solver for 3-D assembly applications. First, we give a new geometry and constraint expression based on Euler parameters, which can avoid singular points during the solving process and simplify constraint types. Then we present a directed graph based constructive method to geometric constraint system solving that can handle well-, over- and under-constrained systems efficiently. The basic idea of this method is that it first simplifies the constraint graph by pruning those vertices which have only in-arcs from the graph and then reduces the size of strongly connected components (SCCs) left in the graph by DOF-based analysis. The method can solve all kinds of configurations including closed-loops. After that, we apply a hybrid numerical method of Newton–Raphson and Homotopy to solve under-constrained systems. The hybrid method makes use of the high efficiency of the Newton–Raphson method as well as the outstanding convergence of the Homotopy method. Finally, we give a practical example and conclusion.

Singularity analysis of geometric constraint systems

Singularity analysis is an important subject of the geometric constraint satisfaction problem. In this paper, three kinds of singularities are described and corresponding identification methods are presented for both under-constrained systems and over-constrained systems. Another special but common singularity for under-constrained geometric systems, pseudo-singularity, is analyzed. Pseudo-singularity is caused by a variety of constraint matching of under-constrained systems and can be removed by improving constraint distribution. To avoid pseudo-singularity and decide redundant constraints adaptively, a differentiation algorithm is proposed in the paper. Its correctness and efficiency have been validated through its practical applications in a 2D/3D geometric constraint solver CBA.

A constructive approach to solving 3-D geometric constraint systems using dependence analysis

Solving geometric constraint systems in 3-D is much more complicated than that in 2-D because the number of variables is larger and some of the results valid in 2-D cannot be extended for 3-D. In this paper, we propose a new DOF-based graph constructive method to geometric constraint systems solving that can efficiently handle well-, over- and under-constrained systems based on the dependence analysis. The basic idea is that the solutions of some geometric elements depend on some others because of the constraints between them. If some geometric elements depend on each other, they must be solved together. In our approach, we first identify all structurally redundant constraints, then we add some constraints to well constrain the system. And we prove that the order of a constraint system after processing under-constrained cases is not more than that of the original system multiplied by 5. After that, we apply a recursive searching process to identify all the clusters, which is shown to be capable of getting the minimum order-reduction result of a well-constrained system. We also briefly describe the constraint evaluation phase and show the implementation results of our method.

Decomposition Plans for Geometric Constraint Systems, Part I: Performance Measures for CAD

A central issue in dealing with geometric constraint systems for CAD/CAM/CAE is the generation of an optimal decomposition plan that not only aids efficient solution, but also captures design intent and supports conceptual design. Though complex, this issue has evolved and crystallized over the past few years, permitting us to take the next important step: in this paper, we formalize, motivate and explain the decomposition–recombination (DR)-planning problem as well as several performance measures by which DR-planning algorithms can be analyzed and compared. These measures include: generality, validity, completeness, Church–Rosser property, complexity, best- and worst-choice approximation factors, (strict) solvability preservation, ability to deal with underconstrained systems, and ability to incorporate conceptual design decompositions specified by the designer. The problem and several of the performance measures are formally defined here for the first time—they closely reflect specific requirements of CAD/CAM applications. The clear formulation of the problem and performance measures allow us to precisely analyze and compare existing DR-planners that use two well-known types of decomposition methods: SR (constraint shape recognition) and MM (generalized maximum matching) on constraint graphs. This analysis additionally serves to illustrate and provide intuitive substance to the newly formalized measures. In Part II of this article, we use the new performance measures to guide the development of a new DR-planning algorithm which excels with respect to these performance measures.

Plane frames as semi-underconstrained systems

A different approach to the analysis of slender framed structures is presented. It is based on the theory of underconstrained systems. It is shown that frames can be interpreted as semi-underconstrained structures. The approach allows understanding of various features of the mechanical behavior of frames as well as novel computing techniques.

On the evaluation of statical-kinematic stiffness matrix for underconstrained structural systems

Because of singularity of the elastic stiffness matrix, analysis of underconstrained systems requires the comprehensive stiffness matrix, comprising both kinds of first-order structural stiffness—elastic and statical-kinematic. The latter accounts for the role of the member forces, induced in the system by the equilibrium part of the applied load, as the source of the system resistance to the perturbation part of the same load. Evaluation of the sought member forces using orthogonal load resolution into equilibrium and perturbation components is awkward and computationally expensive. A special modification of the singular elastic stiffness matrix simplifies the procedure and makes it amenable to the conventional tools of structural analysis. The efficiency and accuracy of the procedure is illustrated by a detailed numerical example.

Orthogonal load resolution and statical-kinematic stiffness matrix

The structural behavior of underconstrained systems is explored with an emphasis on statically indeterminate ones. The roles of the elastic stiffness and the statical-kinematic stiffness (induced by prestress or an equilibrium load) are investigated and compared. The linearized response of a prestressed underconstrained system is shown to go beyond the simple notion of the first-order stiffness induced by prestress and independent of the member stiffnesses. Some interesting features of structural behavior under a general load and stability under an equilibrium load are revealed and illustrated with an example of a quasi-variant pin-bar system. Finally, a comprehensive stiffness matrix with coefficients representing various sources of first-order stiffness is introduced, to replace, when necessary, the conventional elastic stiffness matrix. The comprehensive matrix is valid for structural systems of any type and is always invertible.

Underconstrained structural systems

Underconstrained structural systems have fewer independent constraints than necessary to be geometrically invariant. Yet these systems can be statically indeterminate, with far-reaching implications for their kinematic mobility. In particular, such a system may possess a unique geometric configuration, in which it lacks finite mobility and allows a stable self-stress. The possibility of stable self-stress is shown to be a statical (as opposed to geometric) criterion for underconstrained systems with only infinitesimal kinematic mobility. Both local and global properties of underconstrained systems are investigated within the context of statical-kinematic interrelations. Among these, the relation between the equilibrium loads and configurations is of special interest as it underlies the concept of statically controlled geometry.