Matrix Analysis of Second-Order Kinematic Constraints of Single-Loop Linkages in Screw Coordinates

Volume 5B: 42nd Mechanisms and Robotics Conference ◽

10.1115/detc2018-85433 ◽

2018 ◽

Author(s):

Liheng Wu ◽

Andreas Müller ◽

Jian S. Dai

Keyword(s):

Quadratic Forms ◽

Hessian Matrix ◽

Coefficient Matrix ◽

Higher Order ◽

Second Order ◽

Matrix Analysis ◽

Kinematic Constraints ◽

Single Loop ◽

Order Analysis ◽

Order Constraints

Higher order loop constraints play a key role in the local mobility, singularity and dynamic analysis of closed loop linkages. Recently, closed forms of higher order kinematic constraints have been achieved with nested Lie product in screw coordinates, and are purely algebraic operations. However, the complexity of expressions makes the higher order analysis complicated and highly reliant on computer implementations. In this paper matrix expressions of first and second-order kinematic constraints, i.e. involving the Jacobian and Hessian matrix, are formulated explicitly for single-loop linkages in terms of screw coordinates. For overconstrained linkages, which possess self-stress, the first- and second-order constraints are reduced to a set of quadratic forms. The test for the order of mobility relies on solutions of higher order constraints. Second-order mobility analysis boils down to testing the property of coefficient matrix of the quadratic forms (i.e. the Hessian) rather than to solving them. Thus, the second-order analysis is simplified.

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Adjoint-based exact Hessian computation

BIT Numerical Mathematics ◽

10.1007/s10543-020-00833-0 ◽

2021 ◽

Cited By ~ 1

Author(s):

Shin-ichi Ito ◽

Takeru Matsuda ◽

Yuto Miyatake

Keyword(s):

Krylov Subspace ◽

Hessian Matrix ◽

Adjoint System ◽

Coefficient Matrix ◽

Scalar Function ◽

Second Order ◽

Subspace Method ◽

Initial Value ◽

Research Fields ◽

Memory Efficiency

AbstractWe consider a scalar function depending on a numerical solution of an initial value problem, and its second-derivative (Hessian) matrix for the initial value. The need to extract the information of the Hessian or to solve a linear system having the Hessian as a coefficient matrix arises in many research fields such as optimization, Bayesian estimation, and uncertainty quantification. From the perspective of memory efficiency, these tasks often employ a Krylov subspace method that does not need to hold the Hessian matrix explicitly and only requires computing the multiplication of the Hessian and a given vector. One of the ways to obtain an approximation of such Hessian-vector multiplication is to integrate the so-called second-order adjoint system numerically. However, the error in the approximation could be significant even if the numerical integration to the second-order adjoint system is sufficiently accurate. This paper presents a novel algorithm that computes the intended Hessian-vector multiplication exactly and efficiently. For this aim, we give a new concise derivation of the second-order adjoint system and show that the intended multiplication can be computed exactly by applying a particular numerical method to the second-order adjoint system. In the discussion, symplectic partitioned Runge–Kutta methods play an essential role.

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Effects of Higher Order Retarded Gravity

Universe ◽

10.3390/universe7070207 ◽

2021 ◽

Vol 7 (7) ◽

pp. 207

Author(s):

Asher Yahalom

Keyword(s):

Taylor Expansion ◽

Temporal Change ◽

Higher Order ◽

Second Order ◽

Galactic Rotation ◽

Retardation Time ◽

Rotation Curves ◽

Order Analysis ◽

Galactic Rotation Curves ◽

Higher Order Terms

In a recent paper, we have a shown that the flattening of galactic rotation curves can be explained by retardation. However, this will rely on a temporal change of galactic mass. In our previous work, we kept only second order terms of the retardation time in our analysis, while higher terms in the Taylor expansion where not considered. Here we consider analysis to all orders and show that a second order analysis will indeed suffice, and higher order terms can be neglected.

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LOOKING FOR CHAOS.

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127496000205 ◽

1996 ◽

Vol 06 (03) ◽

pp. 485-496 ◽

Cited By ~ 2

Author(s):

HARRY DANKOWICZ

Keyword(s):

Melnikov Method ◽

Homoclinic Orbits ◽

Melnikov Function ◽

Higher Order ◽

Second Order ◽

Order Analysis ◽

First Order ◽

Order Calculation ◽

Alternative Approach

This paper derives an alternative approach to the Melnikov method, which greatly reduces the amount of algebra involved in higher-order calculations. To illustrate this, a particular system is studied for which such a higher-order analysis is necessary, due to an identically vanishing first-order Melnikov function. The results of a second-order calculation imply the existence of transverse homoclinic orbits and, consequently, the existence of a horseshoe.

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EA-CG: An Approximate Second-Order Method for Training Fully-Connected Neural Networks

Proceedings of the AAAI Conference on Artificial Intelligence ◽

10.1609/aaai.v33i01.33013337 ◽

2019 ◽

Vol 33 ◽

pp. 3337-3346

Author(s):

Sheng-Wei Chen ◽

Chun-Nan Chou ◽

Edward Y. Chang

Keyword(s):

Neural Networks ◽

Empirical Studies ◽

Hessian Matrix ◽

Coefficient Matrix ◽

Second Order ◽

Order Method ◽

Criterion Functions ◽

Rank Approximation ◽

Fully Connected ◽

Approximate Hessian

For training fully-connected neural networks (FCNNs), we propose a practical approximate second-order method including: 1) an approximation of the Hessian matrix and 2) a conjugate gradient (CG) based method. Our proposed approximate Hessian matrix is memory-efficient and can be applied to any FCNNs where the activation and criterion functions are twice differentiable. We devise a CG-based method incorporating one-rank approximation to derive Newton directions for training FCNNs, which significantly reduces both space and time complexity. This CG-based method can be employed to solve any linear equation where the coefficient matrix is Kroneckerfactored, symmetric and positive definite. Empirical studies show the efficacy and efficiency of our proposed method.

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Effects of Higher Order Retarded Gravity

10.20944/preprints202105.0518.v1 ◽

2021 ◽

Author(s):

Asher Yahalom

Keyword(s):

Taylor Expansion ◽

Temporal Change ◽

Higher Order ◽

Second Order ◽

Galactic Rotation ◽

Retardation Time ◽

Rotation Curves ◽

Order Analysis ◽

Galactic Rotation Curves ◽

Higher Order Terms

In a recent paper we have a shown that the flattening of galactic rotation curves can be explained by retardation. However, this will rely on a temporal change of galactic mass. In our previous work we have kept only second order terms of the retardation time in our analysis, while higher terms in the Taylor expansion where not considered. Here we consider analysis to all orders and show that indeed a second order analysis will suffice, and higher order terms can be neglected.

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Properties of Constrained Generalization Algorithms

10.29007/xtb8 ◽

2018 ◽

Author(s):

Thierry Boy de La Tour

Keyword(s):

Lambda Calculus ◽

Higher Order ◽

Relative Strength ◽

Second Order ◽

The Other ◽

First Order ◽

Deterministic Algorithms ◽

Typed Lambda Calculus ◽

Order Constraints

Two non deterministic algorithms for generalizing a solution of a constraint expressed in second order typed lambda-calculus are presented. One algorithm derives from the proof of completeness of the higher order unification rules by D. C. Jensen and T. Pietrzykowski, the other is abstracted from an algorithm by N. Peltier and the author for generalizing proofs. A framework is developed in which such constrained generalization algorithms can be designed, allowing a uniform presentation for the two algorithms. Their relative strength at generalization is then analyzed through some properties of interest: their behaviour on valid and first order constraints, or whether they may be iterated or composed.

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Optimal control of higher order differential inclusions with functional constraints

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2019018 ◽

2020 ◽

Vol 26 ◽

pp. 37 ◽

Cited By ~ 3

Author(s):

Elimhan N. Mahmudov

Keyword(s):

Optimal Control ◽

Optimality Conditions ◽

Differential Inclusion ◽

Differential Inclusions ◽

Higher Order ◽

Second Order ◽

Sufficient Optimality Conditions ◽

Functional Constraints ◽

Complementary Slackness ◽

Complementary Slackness Conditions

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

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