Gauss's Principle Of Least Constraint Research Articles

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.

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.

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.

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.

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.

Dynamical model of Cosserat elastic rod based on Gauss principle

Based on the generalized principles of dynamics, the feature of Gauss principle of least constraint is that the motion law can be directly obtained by using the variation method of seeking the minimal value of the constraint function without establishing any dynamic differential equations. According to the Kirchhoff's dynamic analogy, the configuration of an elastic rod can be described by the rotation of rigid cross section of the rod along the centerline. Since the local small change of the attitude of cross section can be accumulated infinitely along the arc-coordinate, the Kirchhoff's model is suited to describe the super-large deformation of elastic rod. Therefore the analytical mechanics of elastic rod with arc-coordinate s and time t as double arguments has been developed. The Cosserat model of elastic rod takes into consideration the factors neglected by the Kirchhoff model, such as the shear deformation of cross section, the tensile deformation of centerline, and distributed load, so it is more suitable to modeling a real elastic rod. In this paper, the model of the Cosserat rod is established based on the Gauss principle, and the constraint function of the rod is derived in the general form. The plane motion of the rod is discussed as a special case. As regards the special problem that different parts of the rod in space are unable to self-invade each other, a constraint condition is derived to restrict the possible configurations in variation calculation so as to avoid the invading possibility.

A geometric approach to the Gibbs–Appell equations in Lagrangian mechanics

We present a geometrical derivation of the Gibbs–Appell equations in constrained Lagrangian mechanics using the affine structures existing in the spaces of velocities and accelerations. The theoretical basis is the Gauss principle of least constraint, and the necessary conditions for the minimum expressed in terms of the Gibbs function give rise to the Gibbs–Appell equations. We give a second-order formulation of the constraint equation, and using the concept of Moore–Penrose inverse, we provide an explicit intrinsic construction of the projector on the space of the effective or non-ideal forces. The Gibbs–Appell equations are expressed in an intrinsic way in terms of the fibre derivative of the Gibbs function and the effective part of the applied force.

Variational principles for systems with unilateral constraints

Derivations and formulations are given of the variational principles of analytical mechanics for systems with unilateral ideal smooth constraints, originally established for systems with bilateral constraints. The virtual work principle, the Fourier inequality, the d’Alembert–Lagrange principle, the Gauss principle of least constraint and its modification – the Chetayev principle of maximum work, the Jourdain principle, the Hamilton–Ostrogradskii principle, the principle of least action in Lagrangian and Jacobian forms, and the Suslov–Voronets principle are described.

Reflections on the Gauss Principle of Least Constraint

The Gauss principle of least constraint is derived from a new point of view. Then, an extended principle of least constraint is derived to cover the case of nonideal constraints. Finally, a version of the principle for general underdetermined systems is adumbrated. Throughout, the notion of generalized inverses of matices plays a prominent role.

Application of Gauss principle of least constraint to the infiltration into unsaturated soil

This paper presents a new method for solving the problem of infiltration into unsaturated soils using the Gauss principle of least constraint. This technique gives a reasonable qualitative as well as quantitative picture of moisture condition during the unsteady infiltration from both semicircular furrows and buried pipes. The results are in good agreement with those obtained by the alternating direction implicit (ADI) difference method. The computation time is reduced to about 1/900 of that of ADI method.