Principle Of Least Constraint Research Articles

A variational approach to deriving Hamiltonian functions under non-holonomic constraints — the case of the Gaussian thermostats

The Hamiltonian function (H), if it exists, provides a powerful platform for studying the dynamics of constrained many-particle systems. For systems under holonomic constraints, H can be derived from the principle of least action (PLA) formulated in terms of a suitable set of generalized coordinates. In molecular dynamics (MD) simulations, where the system temperature often prescribes a non-holonomic constraint, the challenge is to find the system’s Hamiltonian. The Gaussian Isokinetic (GIK) thermostat, the first deterministic physics-preserving thermostat in MD, was derived from the principle of least constraint (PLC) in the early 1980s: its H, interestingly, was discovered almost 15 years later. A mathematically rigorous connection of the GIK Hamiltonian with PLA, however, has been missing until now. In this article, we adopt a variational approach to establish the equivalence of PLC with PLA for a class of non-holonomically constrained many-particle systems, and present the general formulation of the Euler–Lagrange equations of motion through time rescaling. Making use of the particular forms of the Lagrange multiplier obtained from the principle of least constraint, we derive H for the GIK and the Gaussian Isoenergetic thermostats.

Magnus Force Estimation Using Gauss’s Principle of Least Constraint

The flow around a rotating cylinder is one of the fundamental problems that have piqued the interests of many venerable fluid mechanicians since the time of Rayleigh. The force caused by the rotation of the cylinder has always been considered as an immediate consequence of viscosity, since the potential flow model failed entirely to predict the value of the circulation due to the lack of a Kutta-like condition. On the other hand, Glauert modeled the flow outside the boundary layer of a rotating cylinder as a potential flow with an unknown circulation. He then obtained an approximate solution of Prandtl’s boundary-layer equations and applied the no-slip condition to estimate the circulation in the outer flow. Interestingly, for rapidly rotating cylinders (α=(ωR/U)≫1), up to fourth-order in the small parameter ϵ=(1/α), the obtained circulation is independent of viscosity. In this work, we use Glauert’s model of the outer flow (i.e., a potential flow with an unknown circulation). However, instead of the tedious boundary-layer calculations, we rely on Gauss’s principle of least constraint to obtain the unknown circulation. A perfect match with Glauert’s solution is found. Moreover, our solution, in contrast to Glauert’s, points to the existence of different physics at small rotational speeds. The obtained results, given their perfect matching with Glauert’s solution (relying on the no-slip condition), point to a potential equivalence between the no-slip condition and fluid body forces.

Efficient Constrained Dynamics Algorithms Based on an Equivalent LQR Formulation Using Gauss' Principle of Least Constraint

Efficient Constrained Dynamics Algorithms Based on an Equivalent LQR Formulation Using Gauss' Principle of Least Constraint

A minimization principle for incompressible fluid mechanics

Most variational principles in classical mechanics are based on the principle of least action, which is only a stationary principle. In contrast, Gauss' principle of least constraint is a true minimum principle. In this paper, we apply Gauss' principle to the mechanics of incompressible flows, thereby discovering the fundamental quantity that Nature minimizes in most flows encountered in everyday life. We show that the magnitude of the pressure gradient over the domain is minimum at every instant of time. We call it the principle of minimum pressure gradient (PMPG). It turns a fluid mechanics problem into a minimization one. We demonstrate this intriguing property by solving four classical problems in fluid mechanics using the PMPG without resorting to Navier–Stokes' equation. In some cases, the PMPG minimization approach is not any more efficient than solving Navier–Stokes'. However, in other cases, it is more insightful and efficient. In fact, the inviscid version of the PMPG allowed solving the long-standing problem of the aerohydrodynamic lift over smooth cylindrical shapes where Euler's equation fails to provide a unique answer. The PMPG transcends Navier–Stokes' equations in its applicability to non-Newtonian fluids with arbitrary constitutive relations and fluids subject to arbitrary forcing (e.g., electromagnetic).

The General Gauss Principle of Least Constraint

Abstract This paper develops a general form of Gauss’s Principle of Least Constraint, which deals with the manner in which Nature appears to orchestrate the motion of constrained mechanical systems. The theory of constrained motion has been at the heart of classical mechanics since the days of Lagrange, and it is used in various areas of science and engineering like analytical dynamics, quantum mechanics, statistical physics, and nonequilibrium thermodynamics. The new principle permits the constraints on any mechanical system to be inconsistent and shows that Nature handles these inconsistent constraints in the least squares sense. This broadening of Gauss’s original principle leads to two forms of the General Gauss Principle obtained in this paper. They explain why the motion that Nature generates is robust with respect to inaccuracies with which constraints are often specified in modeling naturally occurring and engineered systems since their specification in dynamical systems are often only approximate, and many physical systems may not exactly satisfy them at every instant of time. An important byproduct of the new principle is a refinement of the notion of what constitutes a virtual displacement, a foundational concept in all classical mechanics.

Statics and Dynamics of Continuum Robots Based on Cosserat Rods and Optimal Control Theories

This article explores the relationship between optimal control and Cosserat beam theory from the perspective of solving the forward and inverse dynamics (and statics as a subcase) of continuous manipulators and snake-like bioinspired locomotors. By invoking the principle of minimum potential energy and the Gauss principle of least constraint, it is shown that the quasi-static and dynamic evolutions of these robots are the solutions of optimal control problems in the space variable, which can be solved at each step (of loading or time) of a simulation with the shooting method. In addition to offering an alternative viewpoint on several simulation approaches proposed in the recent past, the optimal control viewpoint allows us to improve some of them while providing a better understanding of their numerical properties. The approach and its properties are illustrated through a set of numerical examples validated against a reference simulator.

Dynamics of mobile robots based on the principle of least Gauss coordination taking into account random disturbing forces

In this paper, the derivation of the equations of dynamics of a four-wheeled mobile robot is carried out using the variational principle of least constraint, known as the Gauss principle. Equations of nonholonomic constraints are obtained. The function of the measure of coercion of the four-wheeled mobile robot is composed. Dynamic equations based on the Gauss principle are obtained taking into account the dynamic characteristics of two DC motors. Methods for taking into account the friction forces on the wheels and random perturbations due to the unevenness of the canvas are proposed. On the Maple platform, an algorithm and a program for modeling the dynamics of a mobile robot based on the Gauss principle were developed the correctness of the obtained equations of robot motion were proved.

Dynamics of Mobile Manipulators Using Dual Quaternion Algebra

Abstract This article presents two approaches to obtain the dynamical equations of mobile manipulators using dual quaternion algebra. The first one is based on a general recursive Newton–Euler formulation and uses twists and wrenches, which are propagated through high-level algebraic operations and works for any type of joints and arbitrary parameterizations. The second approach is based on Gauss’s Principle of Least Constraint (GPLC) and includes arbitrary equality constraints. In addition to showing the connections of GPLC with Gibbs–Appell and Kane’s equations, we use it to model a nonholonomic mobile manipulator. Our current formulations are more general than their counterparts in the state of the art, although GPLC is more computationally expensive, and simulation results show that they are as accurate as the classic recursive Newton–Euler algorithm.

Decentralized Control of Multiagent Navigation Systems

In this letter, we introduce a decentralized, nonlinear, discontinuous, and computationally simple control law for large scale multiagent navigation systems. The control is based on extending Gauss's principle of least constraint with a dynamic incorporation of inequality constraints, actuator saturation, and actuator dynamics. With no individual path planner, each agent executes its motion and generates its control actions by reacting solely to the evolution of its constrained dynamics, which is equivalent to solving a linear matrix equation with a dimension up to around 20 without iteration at each time instant. Numerical experiments are conducted on hundreds of two-dimensional (2-D) double integrators subjected to path and collision constraints, demonstrating the promise of the proposed method.

Nonequilibrium molecular dynamics method based on coarse-graining formalism: Application to a nonuniform temperature field system.

A nonequilibrium molecular dynamics method is proposed to produce nonequilibrium states flexibly. In this method, virtual points are set in a simulation box, and coarse-grained physical quantities at these points are constrained using Gauss's principle of least constraint. The coarse-grained physical quantities are evaluated by averaging microscopic quantities with an appropriate weight. To obtain the weight to evaluate the coarse-grained physical quantities, a shape function matrix is initially constructed from the particle configuration. This matrix expresses an interpolation of the physical quantities at particle positions from the coarse-grained quantities at the virtual points. Then, a matrix form of the weight is calculated as the Moore-Penrose pseudoinverse matrix for the shape function matrix. This method is applied to constrain the coarse-grained kinetic energy and produce a nonuniform temperature field in the system. The temperature profile at a nonequilibrium steady state depends on the method for constructing the shape function matrix. In particular, a local temperature coincides with the coarse-grained temperature when the shape function matrix is constructed based on a higher-order interpolation.

Gauss's Principle With Inequality Constraints for Multiagent Navigation and Control

Multiagent navigation systems present opportunities for many applications due to their agility and cooperation. In any multiagent navigation system, it is critical that actual interagent collisions are strictly prevented. In this article, we present a solution to the 2-D multiagent navigation problem with collision avoidance. Our solution to this problem is based on a novel extension to Gauss's principle of least constraint (GPLC), in which a fixed set of strict equality constraints is replaced by time-varying sets of active inequality constraints. To the best of our knowledge, this is the first instance that extends GPLC with dynamic incorporation and stabilization of active inequality constraints and with actuator delay and saturation. Herein, the dynamics of a collision-free multiagent system satisfies the Karush-Kuhn-Tucker conditions. Active inequality constraints enforce collision avoidance, leader following, and agglomeration behaviors, and they are stabilized using Baumgarte's error stabilization approach. We show that in dense configurations, the positional arrangement of the agents can lead to linearly dependent constraints, and we propose specialized solutions involving QR decomposition and regularization. The efficacy and efficiency of the proposed method are demonstrated by a dimensional analysis of a worst-case scenario and numerical studies of up to 100 agents tracking a prescribed virtual leader.

Gauss optimization method for the dynamics of unilateral contact of rigid multibody systems

The discontinuous dynamical problem of multi-point contact and collision in multi-body system has always been a hot and difficult issue in this field. Based on the Gauss’ principle of least constraint, a unified optimization model for multibody system dynamics with multi-point contact and collision is established. The paper presents the study of the numerical solution scheme, in which particle swarm optimization method is used to deal with the corresponding optimization model. The article also presents the comparison of the Gauss optimization method (GOM) and the hybrid linear complementarity method (i.e. combining differential algebraic equations (DAEs) and linear complementarity problems (LCP)), commonly used to solve the dynamic contact problem of multibody systems with bilateral constraints. The results illustrate that, the GOM has the same advantage of dynamical modelling with LCP and when the redundant constraint exists, the GOM always has a unique solution and so no additional processing is needed, whereas the corresponding DAE-LCP method may have singular cases with multiple solutions or no solutions. Using numerical examples, the GOM is verified to effectively solve the dynamics of multibody systems with redundant unilateral and bilateral constraints without additional redundancy processing. The GOM can also be applied to the optimal control of systems in the future and combined with the parameter optimization of systems to handle dynamic problems. The work given provides the dynamics and control of the complex system with a new train of thought and method. The paper establishes a unified optimization model and the corresponding numerical solution scheme for multibody system dynamics with multi-point contact and collision based on Gauss’ principle of least constraint. Comparisons between Gauss optimization method (GOM) and the hybrid linear complementarity method (HLCM) are carried out through theoretical analysis and numerical examples, which show that GOM has the same advantage of dynamical modelling as HLCM and when redundant constraints exist, GOM has unique solutions without additional processes, whereas HLCM may have singular cases with multiple or no solutions.

Gauss’s principle and tracking control of underactuated mechanical systems

Abstract Gauss’s principle of least constraint is used to derive equations of motion. The principle is then used to motivate a template for the closed loop of the (tracking) controlled system. The control law arises from minimization of the distance between the controlled model and the template in the sense of least constraint. The control template is derived and analysed for the example of a free rigid body. This leads to quite natural formulations of its controlled inertia, damping and stiffness. This template is then used to compute a controller for a quadcopter, an underactuated system. A simulation result for an aerobatic maneuver demonstrates quite good performance of the proposed controller even though there is no formal proof of stability.

Application of Gauss principle of least constraint in multibody systems with redundant constraints

Redundancy in the constrained mechanical systems often occurs in complex multibody mechanic systems in the existence of excessive constraints and singular positions due to system motion. In this work, Gauss principle of least constraint (GPLC) is applied to solve the dynamic motion of system with redundant constraints without changing the physics of system. Furthermore, the particle swarm optimization method is used to handle the minimization optimization problem. Eventually, the effectiveness of GPLC is validated through the dynamic modelling and simulation of two numerical examples (a planar four-bar mechanism and a spatial parallelogram mechanism). The simulation results are analyzed and compared with those obtained from Udwaia-Phohomsiri formulation and augmented Lagrangian formulation, in terms of constraint violation, computational efficiency and variation of the mechanical energy. From the viewpoint of computational efficiency and accuracy, GPLC can be regarded as a practical real-time simulation method for multibody systems with redundant constraints.

Extended constraint enforcement formulations for finite-DOF systems based on Gauss’s principle of least constraint

This paper aims to contribute to formulations of mathematical models for finite-DOF systems based on the Gauss’s principle of least constraint. A new theorem allows to extend the definition of the Gaussian deviation function, overcoming, without requiring any change of variables, algebraic limitations that arise when this principle is applied to systems for which the generalized mass matrices become singular under relaxation of constraints. As a result of this new theorem, the recursive constraint enforcement algorithm based on Udwadia–Kalaba equation, proposed within the scope of the modular modeling methodology for finite-DOF systems, is generalized. Finally, in a case study based on the classical problem of a knife-edge disc rolling in a plane, three constraint enforcement strategies that combine the application of the theorem herein proposed with conventional numerical integration methods have its results confronted, in two simulation scenarios, with those obtained from a benchmark model of this system.

A regularization method for solving the redundant problems in multibody dynamic system

Redundant constraints may cause great difficulties during the numerical simulation, when solving the differential algebraic equations (DAEs). The article presents a regularization method based on optimization form of the Gauss principle of least constraint and the Udwadia and Kalaba (UK) method, which expresses the solution of the dynamic motion in an explicit expression. Compared with Lagrange method, the proposed method does not need to solve the Lagrange multipliers any more for the complex multibody mechanical systems. In the end, a numerical example is given to illustrate that the proposed method not only has the ability to handle the problems with redundant constraints, but also has higher accuracy and keep a stable constraint violation level than the traditional methods.

Canonical Kane’s Equations of Motion for Discrete Dynamical Systems

The generalized inversion-based canonical equations of motion (CEOMs) for ideally constrained discrete dynamical systems are introduced in the framework of Kane’s method. Upon selection of the inertia-scaled canonical generalized acceleration variables, the proposed formulation employs the acceleration form of constraints and the generalized inversion Greville formula for parameterized solutions of underdetermined algebraic equations together with the nonminimal constrained momentum balance equations. The resulting CEOMs are explicit and full order in the acceleration variables. Moreover, the geometry of constrained motion is revealed by the CEOMs intuitively by partitioning the canonical accelerations column matrix into two portions at every time instant: a portion that drives the dynamical system to abide by the constraints, and a portion that generates the momentum balance dynamics such that the system abides by the Newton–Euler laws of motion. Some insightful geometrical perspectives of the CEOMs are illustrated via vectorial visualizations, which lead to verifying Gauss’s principle of least constraints and its Udwadia–Kalaba interpretation. The procedure is illustrated by formulating a third-order CEOMs dynamic model of two particles experiencing a single degree of freedom, as well as a sixth-order CEOMs dynamic model of a three-degrees-of-freedom disk that is rolling without slipping on an inertial plane under the effect of friction.

Locally Linear Embedding and fMRI Feature Selection in Psychiatric Classification.

Background: Functional magnetic resonance imaging (fMRI) provides non-invasive measures of neuronal activity using an endogenous Blood Oxygenation-Level Dependent (BOLD) contrast. This article introduces a nonlinear dimensionality reduction (Locally Linear Embedding) to extract informative measures of the underlying neuronal activity from BOLD time-series. The method is validated using the Leave-One-Out-Cross-Validation (LOOCV) accuracy of classifying psychiatric diagnoses using resting-state and task-related fMRI. Methods: Locally Linear Embedding of BOLD time-series (into each voxel’s respective tensor) was used to optimise feature selection. This uses Gauß’ Principle of Least Constraint to conserve quantities over both space and time. This conservation was assessed using LOOCV to greedily select time points in an incremental fashion on training data that was categorised in terms of psychiatric diagnoses. Findings: The embedded fMRI gave highly diagnostic performances (> 80%) on eleven publicly-available datasets containing healthy controls and patients with either Schizophrenia, Attention-Deficit Hyperactivity Disorder (ADHD), or Autism Spectrum Disorder (ASD). Furthermore, unlike the original fMRI data before or after using Principal Component Analysis (PCA) for artefact reduction, the embedded fMRI furnished significantly better than chance classification (defined as the majority class proportion) on ten of eleven datasets. Interpretation: Locally Linear Embedding appears to be a useful feature extraction procedure that retains important information about patterns of brain activity distinguishing among psychiatric cohorts.

基于高斯最小拘束原理的广义质量矩阵奇异性问题研究

高斯最小拘束原理为含约束的多体系统动力学提供了一种新的建模思路：将以求解微分代数方程为主的动力学问题引入到求函数最小值的优化问题的框架中。质量矩阵奇异性问题是在常规多体系统动力学求解框架下经常遇到的难点问题。本文从建模方式出发研究了经典动力学框架下的难点问题，通过引入广义逆，建立了广义质量矩阵奇异情形下的高斯最小拘束原理，研究了针对奇异性问题的优化方法的数值求解策略。算例中分别采用了优化方法及第一类拉格朗日方程进行了建模及数值模拟。算例表明了文中方法在解决该类奇异性问题时的有效性。 Gauss principle of least constraints provides a new modeling method for dynamics of multibody systems, which changes dynamical problem of mainly solving differential-algebra equation into the frame of solving minimum. The problem of the mass singular matrix is the difficulty in tradi-tional dynamics of multibody systems. We start with new modeling method for solving the singular problems. By introducing the generalized inverse, the Gauss principle of least constraints for matrix singular matrix and solving strategies for optimization method are established. In the example, the optimization and the Lagrange equation of the first kind are used to model and calculate. The example validates the optimization method for this kind of singular problems.

A natural absolute coordinate formulation for the kinematic and dynamic analysis of rigid multibody systems

The purpose of this paper was to present the key features of a novel coordinate formulation for the analytical description of the motion of rigid multibody systems, namely the natural absolute coordinate formulation (NACF). As it is shown in this work, the kinematic and dynamic analysis of rigid multibody systems can be significantly enhanced employing the NACF. In particular, this formulation combines the main advantages of the natural coordinate formulation (NCF), such as the remarkable property of leading to a constant mass matrix and to zero centrifugal and Coriolis generalized inertia forces, with the generality and the effectiveness of the reference point coordinate formulation (RPCF), which is essentially represented by the possibility to develop and assemble the equations of motion of a multibody system together with the algebraic equations which model the joint constraints in a systematic manner. Moreover, a new computational method hereinafter referred to as the robust generalized coordinate partitioning algorithm is also introduced in this work. The robust generalized coordinate partitioning algorithm can be successfully utilized to numerically solve the index-one form of the multibody system equations of motion formulated by using the proposed NACF as well as the well-known RPCF. In particular, the computational procedure presented in this paper owes its robustness to the combination of the main ideas of the well-established generalized coordinate partitioning method, which is commonly employed to cope with the drift phenomenon of the constraint equations at the position and velocity levels when an index-one formulation of the equations of motion is considered, with the more general and advanced constraint enforcement technique at the acceleration level represented by the fundamental equations of constrained motion. In fact, the fundamental equations of constrained motion represent an effective and efficient method able to calculate analytically the generalized constraint forces relative to a multibody system subjected to a general set of redundant holonomic and/or nonholonomic constraint equations by using the Gauss principle of least constraint, thus avoiding the definition of the Lagrange multipliers. The fundamental equations of constrained motion are remarkably effective when used for modeling the dynamic behavior of rigid multibody systems mathematically represented employing the NACF, as it is shown in this paper. Four simple benchmark multibody systems are also examined in order to exemplify the application of the principal concepts developed in the paper.