Forces of reaction and neighbouring Hamilton's principle in the tracking control of manipulators via a sliding scheme

Robotica ◽  
1993 ◽  
Vol 11 (3) ◽  
pp. 227-232 ◽  
Author(s):  
Guy Jumarie
Keyword(s):  
Variational Principle ◽  
Tracking Control ◽  
Taylor Expansion ◽  
Perturbation Approach ◽  
Hamilton’S Principle ◽  
Sliding Surface ◽  
Hamilton's Principle ◽  
Control Effort ◽  
Sliding Surfaces ◽  
Control Schemes

SUMMARYIt is shown that if one comes back to the formulation of the Hamilton's variational principle, it is then possible to obtain new viewpoints on the tracking control of robot manipulators. First, the Lagrange multiplier associated to the sliding surface can be interpretated in terms of control effort and/or forces of reaction of the mechanical system. Secondly, one can use the Taylor expansion of the mechanical Lagrangian, combined with a neighbour- ing Hamilton's principle, to obtain control schemes via sliding surfaces. Thirdly, a perturbation approach combined with the neighbouring Hamilton's principle provides results on the robustness of the control.

Stochastic Hamilton's principle and noisy sliding surfaces in the tracking control of manipulators

Robotica ◽  
1995 ◽  
Vol 13 (2) ◽  
pp. 209-213
Author(s):  
Guy Jumarie
Keyword(s):  
Dynamic Programming ◽  
Tracking Control ◽  
Programming Approach ◽  
Hamilton’S Principle ◽  
Sliding Surface ◽  
Hamilton's Principle ◽  
Dynamic Programming Approach ◽  
Sliding Surfaces ◽  
Definition Of

SummaryIn the tracking control of manipulators via the sliding scheme, it may happen that sometimes, because of various inaccuracies, the definition of the actual sliding surface involves errors terms which may be either deterministic or, on the contrary, stochastic. This paper considers this last case and shows how one can estimate the new performances of the system so disturbed. A stochastic Hamilton's principle is applied, by combining the Lagrange parameter technique with results of the dynamic programming approach.

scholarly journals Hamilton-type principles applied to ice-sheet dynamics: new approximations for large-scale ice-sheet flow

2010 ◽  
Vol 56 (197) ◽  
pp. 497-513 ◽  
Author(s):  
J.N. Bassis
Keyword(s):  
Variational Principle ◽  
Large Scale ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Ice Sheet ◽  
Alternative Formulation ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Ice Sheet Dynamics ◽  
Large Scale Flow

AbstractIce-sheet modelers tend to be more familiar with the Newtonian, vectorial formulation of continuum mechanics, in which the motion of an ice sheet or glacier is determined by the balance of stresses acting on the ice at any instant in time. However, there is also an equivalent and alternative formulation of mechanics where the equations of motion are instead found by invoking a variational principle, often called Hamilton’s principle. In this study, we show that a slightly modified version of Hamilton’s principle can be used to derive the equations of ice-sheet motion. Moreover, Hamilton’s principle provides a pathway in which analytic and numeric approximations can be made directly to the variational principle using the Rayleigh–Ritz method. To this end, we use the Rayleigh–Ritz method to derive a variational principle describing the large-scale flow of ice sheets that stitches the shallow-ice and shallow-shelf approximations together. Numerical examples show that the approximation yields realistic steady-state ice-sheet configurations for a variety of basal tractions and sliding laws. Small parameter expansions show that the approximation reduces to the appropriate asymptotic limits of shallow ice and shallow stream for large and small values of the basal traction number.

On the Reciprocal Form of Hamilton’s Principle

1973 ◽  
Vol 40 (1) ◽  
pp. 93-100 ◽  
Author(s):  
Z. M. Elias
Keyword(s):  
Variational Principle ◽  
Continuous Systems ◽  
Sufficient Condition ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Energy Principle ◽  
Spurious Solutions ◽  
Complementary Energy Principle ◽  
Necessary And Sufficient ◽  
Geometric Compatibility

A complementary energy principle for dynamic analysis due to Toupin is critically examined. It is found that the variational principle is a necessary but not a sufficient condition for geometric compatibility and that consequently it allows the occurrence of spurious solutions. A necessary and sufficient condition of compatibility is obtained through the reciprocal form of Hamilton’s principle which is derived for discrete and continuous systems. Additional terms appearing in the derived principle insure that spurious solutions cannot occur. The derived variational principle can be expressed in terms of stresses and velocities or in terms of impulses.

Hamilton’s Principle for External Viscous Fluid-Structure Interaction

2000 ◽  
Author(s):  
Haym Benaroya ◽  
Timothy Wei
Keyword(s):  
Variational Principle ◽  
Equation Of Motion ◽  
External Flow ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Structure Interaction ◽  
Future Paper ◽  
The Subject ◽  
End Times ◽  
Transport Theorem

Abstract In this paper, Hamilton’s principle is extended so as to be able to model external flow-structure interaction. This is accomplished by using Reynold’s Transport theorem. In this form, Hamilton’s principle is hybrid in the sense that it has an analytical part as well as a part that depends on experimentally derived functions. Examples are presented. A discussion on implications and extensions is extensive. In this work, the general theory is developed for the case where the configuration is not prescribed at the end times of the variational principle. This leads to a single governing equation of motion. This limitation can be removed by prescribing the end times, as is usual. This is outlined in the present paper, and will be the subject of a future paper.

Hamiltonian Fluid Dynamics

1998 ◽  
Author(s):  
Rick Salmon
Keyword(s):  
Variational Principle ◽  
Conservation Laws ◽  
Variational Principles ◽  
Symmetry And Conservation Laws ◽  
The Body ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Symmetry Properties

In this final chapter, we return to the subject of the first: the fundamental principles of fluid mechanics. In chapter 1, we derived the equations of fluid motion from Hamilton’s principle of stationary action, emphasizing its logical simplicity and the resulting close correspondence between mechanics and thermodynamics. Now we explore the Hamiltonian approach more fully, discovering its other advantages. The most important of these advantages arise from the correspondence between the symmetry properties of the Lagrangian and the conservation laws of the resulting dynamical equations. Therefore, we begin with a very brief introduction to symmetry and conservation laws. Noether’s theorem applies to the equations that arise from variational principles like Hamilton’s principle. According to Noether’s theorem : If a variational principle is invariant to a continuous transformation of its dependent and independent variables, then the equations arising from the variational principle possess a divergence-form conservation law. The invariance property is also called a symmetry property. Thus Noether’s theorem connects symmetry properties and conservation laws. We shall neither state nor prove the general form of Noether’s theorem; to do so would require a lengthy digression on continuous groups. Instead we illustrate the connection between symmetry and conservation laws with a series of increasingly complex and important examples. These examples convey the flavor of the general theory. Our first example is very simple. Consider a body of mass m moving in one dimension. The body is attached to the end of a spring with spring-constant K. Let x(t) be the displacement of the body from its location when the spring is unstretched.

Space-Discretized Verlet-Algorithm from a Variational Principle

1997 ◽  
Vol 52 (8-9) ◽  
pp. 585-587
Author(s):  
Walter Nadler ◽  
Hans H. Diebner ◽  
Otto E. Rössler
Keyword(s):  
Variational Principle ◽  
Digital Computer ◽  
Equations Of Motion ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Space And Time ◽  
Space Discretization

Abstract A form of the Verlet-algorithm for the integration of Newton’s equations of motion is derived from Hamilton's principle in discretized space and time. It allows the computation of exactly time-reversible trajectories on a digital computer, offers the possibility of systematically investigating the effects of space discretization, and provides a criterion as to when a trajectory ceases to be physical.

scholarly journals Hamilton’s Principle for Circuits with Dissipative Elements

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-7 ◽  
Author(s):  
Zdeněk Biolek ◽  
Dalibor Biolek ◽  
Viera Biolková
Keyword(s):  
Variational Principle ◽  
Higher Order ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Frequency Dependent ◽  
Nearest Neighbours ◽  
Classic Form

The classic form of Hamilton’s variational principle does not hold for circuits with dissipative elements. It is shown in the paper that this may not be true in the case of systems consisting of the so-called higher-order elements. Hamilton’s principle is then extended to circuits containing the classical resistors and Frequency Dependent Negative Resistors (FDNRs). The extension is also made to any pair of elements which are the nearest neighbours on any Σ-diagonal of Chua’s table.

Variational principles in continuum mechanics

1968 ◽  
Vol 305 (1480) ◽  
pp. 1-25 ◽  
Keyword(s):  
Variational Principle ◽  
Continuum Mechanics ◽  
General Problem ◽  
Water Waves ◽  
Variational Principles ◽  
Differential Forms ◽  
Lagrangian Description ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Vector Potentials

Variational principles for problems in fluid dynamics, plasma dynamics and elasticity are discussed in the context of the general problem of finding a variational principle for a given system of equations. In continuum mechanics, the difficulties arise when the Eulerian description is used; the extension of Hamilton’s principle is straightforward in the Lagrangian description. It is found that the solution to these difficulties is to represent the Eulerian velocity v by expressions of the type v = ∇ X + λ∇ μ introduced by Clebsch (1859) for the case of isentropic fluid flow. The relation with Hamilton’s principle is elucidated following work by Lin (1963). It is also shown that the potential representation of electromagnetic fields and the variational principle for Maxwell’s equations can be fitted into the same overall scheme. The equations for water waves, waves in rotating and stratified fluids, Rossby waves, and plasma waves are given particular attention since the need for variational formulations of these equations has arisen in recent work on wave propagation (Whitham 1967). The idea of solving some of the equations by ‘potential representations’ (such as the Clebsch representation in continuum mechanics and the scalar and vector potentials in electromagnetism), and then finding a variational principle for the remaining equations, seems to be the crucial one for the general problem. An analogy with Pfaff’s problem in differential forms is given to support this idea.

A note on Hamilton's principle for perfect fluids

1970 ◽  
Vol 44 (1) ◽  
pp. 19-31 ◽  
Author(s):  
Francis P. Bretherton
Keyword(s):  
Variational Principle ◽  
Equations Of Motion ◽  
Velocity Fields ◽  
Lagrangian Formulation ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Barotropic Fluid ◽  
Perfect Fluids ◽  
Vortex Lines ◽  
Equal Density

A derivation is given of the Eulerian equations of motion directly from the Lagrangian formulation of Hamilton's principle. The circulation round a circuit of material particles of uniform entropy appears as a constant of the motion associated with the indistinguishability of fluid elements with equal density, entropy and velocity. A discussion is given of the Lin constraint, and it is pointed out that, for a barotropic fluid, the variational principle recently suggested by Seliger & Whitham does not permit velocity fields in which the vortex lines are knotted.

Variational principles for perfect and dissipative fluid flows

1982 ◽  
Vol 381 (1781) ◽  
pp. 457-468 ◽  
Keyword(s):  
Variational Principle ◽  
Perfect Fluid ◽  
Variational Principles ◽  
Fluid Flows ◽  
Viscous Flows ◽  
Conducting Fluid ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Dissipative Fluid ◽  
Thermally Conducting

Many versions of an extension of Hamilton’s principle to perfect fluid flows exist in the literature. In this paper the most general form, due to Serrin, is identified and the limitations of some of the others discussed. An extension of the variational principle to viscous, thermally conducting fluid flows is suggested. This is compared with some other variational principles for viscous flows, and the relative merits of the various approaches are discussed.

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