Algebraic Elimination Research Articles

Computing Circuit Polynomials in the Algebraic Rigidity Matroid

We present an algorithm for computing circuit polynomials in the algebraic rigidity matroid associated to the Cayley–Menger ideal for points in 2D. It relies on combinatorial resultants, a new operation on graphs that captures properties of the Sylvester resultant of two polynomials in this ideal. We show that every rigidity circuit has a construction tree from graphs based on this operation. Our algorithm performs an algebraic elimination guided by such a construction tree and uses classical resultants, factorization, and ideal membership. To highlight its effectiveness, we implemented the algorithm in Mathematica: it took less than 15 seconds on an example where a Gröbner basis calculation took 5 days and 6 hours. Additional speed-ups are obtained using non- generators of the Cayley–Menger ideal and simple variations on our main algorithm.

Differential Elimination for Dynamical Models via Projections with Applications to Structural Identifiability

Elimination of unknowns in a system of differential equations is often required when analyzing (possibly nonlinear) dynamical systems models, where only a subset of variables are observable. One such analysis, identifiability, often relies on computing input-output relations via differential algebraic elimination. Determining identifiability, a natural prerequisite for meaningful parameter estimation, is often prohibitively expensive for medium to large systems due to the computationally expensive task of elimination. We propose an algorithm that computes a description of the set of differential-algebraic relations between the input and output variables of a dynamical system model. The resulting algorithm outperforms general-purpose software for differential elimination on a set of benchmark models from the literature. We use the designed elimination algorithm to build a new randomized algorithm for assessing structural identifiability of a parameter in a parametric model. A parameter is said to be identifiable if its value can be uniquely determined from input-output data assuming the absence of noise and sufficiently exciting inputs. Our new algorithm allows the identification of models that could not be tackled before. Our implementation is publicly available online as a Julia package.

Planar and spherical four-bar linkage [formula omitted] algebraic input–output equations

The algebraic polynomial input–output (IO) equations relating any two of the relative joint displacement parameters, called vi and vj, between any of the six distinct pairs of rigid links in arbitrary planar and spherical four-bar mechanisms are derived. First, the forward kinematics transformation matrices of the corresponding serial kinematic chains are computed in terms of their Denavit–Hartenberg parameters, but with all angles converted to tangent half-angle parameters. These matrices are mapped to their corresponding Study soma coordinates. The serial kinematic chain is closed by equating the soma coordinates to the identity array. Algebraic polynomial elimination methods are then used to obtain a single polynomial in terms of only the design and the selected IO joint displacement parameters. This yields six independent algebraic IO Equations for each of the planar and spherical 4R linkages; the same techniques are applied to derive six additional algebraic IO equations for each of the RRRP and PRRP planar linkages providing a catalogue of 24. The utility of these IO equation sets is demonstrated via discussion of the associated mobility and design parameter spaces.

An Algebraic Elimination by Substitution Algorithm for Vehicle–Bridge Interaction Problems

This paper proposes an algorithm to solve vehicle–bridge interaction (VBI) problems. It describes the VBI system in a linear second-order differential-algebraic equation (DAE) manner, incorporating the motions of both the vehicle and the bridge subsystems, together with the contact forces. On account of the conjugation between the VBI system’s motions and contact forces, the key feature of the proposed algorithm is to solve the DAE with the the algebraic method of elimination by substitution, avoiding the singularity of the differential operator. It enables a versatile and consensus solution scheme for different types of contact mechanisms, including the rigid contact model with constraints, the elastic contact model with the compliance method, and the creep tangential contact model, as well as different wheel–rail profiles. Combined with an appropriate numerical integration scheme, it allows an explicit expression of the contact forces, subsequently leading to the partitioned non-iterative analysis of the motions of both the vehicle and the bridge subsystems. In addition, by incorporating the dynamic condensation of wheels’ degrees of freedom into those of upper components in a matrix form, i.e. a reduced system approach, the proposed algorithm can avoid the numerical drift associated with the kinematic constraints on the acceleration level. To demonstrate the accuracy of the proposed approach, this study first examines a benchmark example of a single degree of freedom sprung mass model traversing a simply supported bridge. Further, it considers a more realistic problem of one and eight train vehicles crossing three three-span continuous bridges. The comparisons and the agreements with existing algorithms prove the accuracy and computational efficiency of the proposed analysis scheme.

CGA-based novel modeling method for solving the forward displacement analysis of 3-RPR planar parallel mechanism

This paper put forward a new geometric modeling method for forward displacement analysis of 3-RPR planar parallel mechanisms (PPMs) under the conformal geometric algebra (CGA) framework. A geometric constraint relationship of four lines intersecting at a point is first given. Then, a new coordinate-invariant geometric constraint equation of 3-RPR PPMs is elaborated via CGA operation. Next, an univariate polynomial equation without algebraic elimination is published. At last, the coordinates of three points on the mobile platform are figured out. The best part of this research is that a complete geometric modeling method by geometric constraint relationship of four lines intersecting at a point is formulated under the CGA framework, which has good intuition and offers a novel idea for solving the other complex mechanisms. Three numerical examples are utilized to confirm the effectiveness of the proposed method.

Trust-Region Solver of a Nonlinear Magnetometer Disturbance Estimation Problem

Earth geomagnetic field provides very important information for autonomous navigation. However, in practice, magnetometer measurements are easily distorted by outer disturbances, as shown in the right diagram. In a recent study, the real-time magnetometer disturbance estimation problem has been solved via a constrained nonlinear programming for better underwater navigation accuracy. However, the employed interior-point optimizer will require considerable computational resources and cannot always guarantee optimality during optimization updates. This paper further investigates this problem and refines the solution by introducing the trust-region method. The challenge of the designed approach mainly falls into finding out the feasible regions of the possible solutions. Geometric analysis and algebraic elimination methods are taken into account to give globally optimal solutions to the trust-region solver. Experimentations are conducted to verify the correctness of the proposed method. Moreover, it is also confirmed that the trust-region technique can continuously and accurately propagate so that high-frequency dynamic magnetic disturbances can be efficiently estimated.

A Novel Geometric Modeling and Calculation Method for Forward Displacement Analysis of 6-3 Stewart Platforms

A novel geometric modeling and calculation method for forward displacement analysis of the 6-3 Stewart platforms is proposed by using the conformal geometric algebra (CGA) framework. Firstly, two formulas between 2-blade and 1-blade are formulated. Secondly, the expressions for two spherical joints of the moving platform are given via CGA operation. Thirdly, a coordinate-invariant geometric constraint equation is deduced. Fourthly, a 16-degree univariate polynomial equation without algebraic elimination by using the Euler angle substitution is presented. Fifthly, the coordinates of three spherical joints on the moving platform are calculated without judging the radical symbols. Finally, two numerical examples are used to verify the method. The highlight of this paper is that a new geometric modeling and calculation method without algebraic elimination is obtained by using the determinant form of the CGA inner product algorithm, which provides a new idea to solve a more complex spatial parallel mechanism in the future.

Identification method for polynomially parametrized LTI systems based on exhaustive modelling with algebraic elimination

Identification method for polynomially parametrized LTI systems based on exhaustive modelling with algebraic elimination

Spectral aspects of Nitsche’s method on nonconforming meshes

Nitsche’s method is an effective framework for the solution of problems involving embedded domains. Weak enforcement of Dirichlet boundary and interface conditions engenders additional active degrees of freedom compared to the corresponding kinematically admissible formulation, and hence additional solutions in eigenvalue problems. The original and added eigenpairs are designated proper and complementary, respectively. The number of complementary pairs equals the number of degrees of freedom that would be constrained in the kinematically admissible formulation. We investigate the number of complementary pairs that arise in representative nonconforming configurations of bilinear quadrilaterals. Algebraic elimination of the added degrees of freedom from the Nitsche formulation yields a formulation with several advantageous features. Practical procedures for solving eigenvalue problems based on reduced methods are proposed.

Application of a Novel Elimination Algorithm with Developed Continuation Method for Nonlinear Forward Kinematics Solution of Modular Hybrid Manipulators

SUMMARYThis paper addresses the application of a novel elimination algorithm with a newly developed homotopy continuation method (HCM) for forward kinematics of a specific hybrid modular manipulator known as n-(6UPS). First, the kinematic model of n-(6UPS) was extracted using a homogenous transformation matrix method. Then, a novel algebraic elimination algorithm was developed to transform the highly nonlinear proposed kinematic model into a system of polynomial equations for each module. Next, the HCM is considered to solve the system of equations. Comparison of the results from the proposed approach with experimental data and other methods demonstrates the efficiency of the proposed contribution.

Complementary Solutions of Nitsche’s Method

Embedded methods that are based on Nitsche’s approach can facilitate the task of mesh generation in many configurations. The basic workings of the method are well understood, in terms of a bound on the stabilization parameter. However, its spectral behavior has not been explored in depth. In addition to the eigenpairs which approximate the exact ones, as in the standard formulation, Nitsche’s method gives rise to mesh-dependent complementary pairs. The dependence of the eigenvalues on the Nitsche parameter is related to a boundary quotient of the eigenfunctions, explaining the manner in which stabilization engenders coercivity without degrading the accuracy of the discrete eigenpairs. The boundary quotient proves to be useful for separating the two types of solutions. The quotient space is handy for determining the number of eigenpairs and complementary pairs. The complementary solutions approximate functions in the orthogonal complement of the kinematically admissible subspace. A global result for errors in the Galerkin approximation of the eigenvalue problem that pertains to all modes of the discretization, is extended to the Nitsche formulation. Numerical studies on non-conforming aligned meshes confirm the dependence of the eigenvalues on the parameter, in line with the corresponding boundary quotients. The spectrum of a reduced system obtained by algebraic elimination is free of complementary solutions, warranting its use in the solution of boundary-value problems. The reduced system offers an incompatible discretization of eigenvalue problems that is suitable for engineering applications. Using Irons–Guyan reduction yields a spectrum that is virtually insensitive to stabilization, with high accuracy in both eigenvalues and eigenfunctions.

CGA-Based approach to direct kinematics of parallel mechanisms with the 3-RS structure

This paper presents a unified geometric modeling and solution procedure for direct kinematic analysis of a class of parallel mechanisms based on conformal geometric algebra (CGA). After locking the actuated joints, such parallel mechanisms will be turned into a 3-RS structure, which is composed of two triangular platforms connected by three RS serial chains in parallel. Using the proposed approach, the univariate polynomial equation for the direct kinematic analysis of these parallel mechanisms can be derived in three steps. Firstly, the positions of two of the three spherical joints on the moving platform are formulated by the intersection, dissection and dual of the basic geometric entities under the frame of CGA. Secondly, a coordinate-invariant equation expressed in terms of geometric entities is derived via CGA operation. Thirdly, a univariate polynomial equation is obtained directly from the aforementioned coordinate-invariant equation by using tangent-half-angle substitution. Several case studies are then presented to verify the solution procedure. The novelties of this approach lie in that: (1) The formulation is concise and coordinate-invariant and has intrinsic geometric intuition due to the use of CGA; (2) No algebraic elimination procedure is required to derive the univariate polynomial equation; and (3) The proposed approach is applicable to the direct kinematics of this family of parallel mechanisms with any link parameters.

Development and application of a standstill parameter identification technique for the synchronous generator

This work presents the development of an offline standstill estimation technique, where the synchronous machine is locked at an arbitrary (but known) angle and is excited over a short period of time. The proposed time domain method requires few seconds of captured data in contrast to the well-known standard Standstill Frequency Response (SSFR) technique that could take more than 6h to conduct. This is based on nonlinear least squares estimation and algebraic elimination theory. The resulting algorithm is non-iterative where the data is used to construct polynomials that are solved for a finite number of roots which determine the electrical parameter values. Experimental results are presented showing the efficacy of the technique in furnishing the parameters of a salient pole synchronous machine.

Algebraic solution for the forward displacement analysis of the general 6-6 stewart mechanism

The solution for the forward displacement analysis(FDA) of the general 6-6 Stewart mechanism(i.e., the connection points of the moving and fixed platforms are not restricted to lying in a plane) has been extensively studied, but the efficiency of the solution remains to be effectively addressed. To this end, an algebraic elimination method is proposed for the FDA of the general 6-6 Stewart mechanism. The kinematic constraint equations are built using conformal geometric algebra(CGA). The kinematic constraint equations are transformed by a substitution of variables into seven equations with seven unknown variables. According to the characteristic of anti-symmetric matrices, the aforementioned seven equations can be further transformed into seven equations with four unknown variables by a substitution of variables using the Grobner basis. Its elimination weight is increased through changing the degree of one variable, and sixteen equations with four unknown variables can be obtained using the Grobner basis. A 40th-degree univariate polynomial equation is derived by constructing a relatively small-sized 9´9 Sylvester resultant matrix. Finally, two numerical examples are employed to verify the proposed method. The results indicate that the proposed method can effectively improve the efficiency of solution and reduce the computational burden because of the small-sized resultant matrix.

Incremental light bundle adjustment for structure from motion and robotics

Bundle adjustment (BA) is essential in many robotics and structure-from-motion applications. In robotics, often a bundle adjustment solution is desired to be available incrementally as new poses and 3D points are observed. Similarly in batch structure from motion, cameras are typically added incrementally to allow good initializations. Current incremental BA methods quickly become computationally expensive as more camera poses and 3D points are added into the optimization. In this paper we introduce incremental light bundle adjustment (iLBA), an efficient optimization framework that substantially reduces computational complexity compared to incremental bundle adjustment. First, the number of variables in the optimization is reduced by algebraic elimination of observed 3D points, leading to a structureless BA. The resulting cost function is formulated in terms of three-view constraints instead of re-projection errors and only the camera poses are optimized. Second, the optimization problem is represented using graphical models and incremental inference is applied, updating the solution using adaptive partial calculations each time a new camera is incorporated into the optimization. Typically, only a small fraction of the camera poses are recalculated in each optimization step. The 3D points, although not explicitly optimized, can be reconstructed based on the optimized camera poses at any time. We study probabilistic and computational aspects of iLBA and compare its accuracy against incremental BA and another recent structureless method using real–imagery and synthetic datasets. Results indicate iLBA is 2–10 times faster than incremental BA, depending on number of image observations per frame.

A new closed-form solution to inverse static force analysis of a spatial three-spring system

In this paper, a new closed-form solution to the inverse static force analysis of a spatial three-spring system is presented. The system is formed by three springs each of which connects the ground at one end and joins a common point at the other. When a known force is applied to the common point of the system, the goal of inverse static analysis is to determine all the equilibrium configurations. A system of three polynomial equations in three variables is derived based on the geometric constraint and static force balancing. A 20 by 20 Dixon resultant matrices firstly derived from these three polynomials and then reduced to an 18 by 18 matrix. A 46th-degree univariate polynomial equation is yielded from the above 18 by 18 matrix. By further analysis, we found that 24 roots were degenerated and only the remaining 22 roots are the ones for the three-spring system. The result agrees with previous results. At last, two numerical examples are given to verify the elimination procedure. The presented algebraic elimination solution reveals some intrinsic geometry nature of this challenging problem.

Commissioning rules for optimal velocity controller damping of servo axes using elimination methods

We present a new and algebraic approach to the optimal damping of servo axes during commissioning. The approach is based on control of root loci of the denominator of the transfer function using algebraic elimination. The results are either explicit formulas for simple systems or descriptions of the optima via roots of univariate polynomials. The power of the result is demonstrated in examples.

On Time-Optimal Trajectories in Non-Uniform Mediums

This paper addresses the problem of finding time-optimal trajectories in two-dimensional space, where the mover's speed monotonically decreases or increases in one of the space's coordinates. We address such problems in different settings for the velocity function and in the presence of obstacles. First, we consider the problem without any obstacles. We show that the problem with linear speed decrease is reducible to the de l'Hopital's problem, and that, for this case, the time-optimal trajectory is a circular segment. Next, we show that the problem with linear speed decreases and rectilinear obstacles can be solved in polynomial time by a dynamic program. Finally, we consider the case without obstacles, where the medium is non-uniform and the mover's velocity is a piecewise linear function. We reduce this problem to that of solving a system of polynomial equations of fixed degree, for which algebraic elimination theory allows us to solve the problem to optimality.

Equimultiplicity, algebraic elimination, and blowing-up

Given a variety X over a perfect field, we study the partition defined on X by the multiplicity (into equimultiple points), and the effect of blowing up at smooth equimultiple centers. Over fields of characteristic zero we prove resolution of singularities by using the multiplicity as an invariant, instead of the Hilbert Samuel function.

Устранение циклов во вторичной структуре алгебраической байесовской сети на основе анализа ее четвертичной структуры

Algebraic Bayesian Network (ABN) is one of the logical and probabilistic graphical models of bases of knowledge patterns uncertainty. Algorithms for global logical and probabilistic inference in ABN can be applied only under the condition of acyclicity of its secondary structure — join graph. The existing method for join graph transformation into join tree application is restrictedly applicable. The goal of the work is to offer new methods for the transformation cyclic ABN into an acyclic one, based on a structure theorem for minimal join graphs cycles. Two methods for eliminating cycles are proposed and their correctness.is proved. Keywords: algebraic Bayesian networks, quaternary structure, machine learning, probabilistic graphical knowledge models, global structure, primary structure acyclicity.