Jourdain's Principle Research Articles

Design and analysis of a size-tunable tetrahedron rolling mechanism based on deployable RRR chains

This paper proposes the design and analysis of a size-tunable tetrahedron rolling mechanism, which has five-degrees-of-freedoms (DOF). Aiming to pass complex terrains, the mechanism consists of four deployable nodes and six planar deployable RRR chains, which are composed of scissor-like elements. Consequently, the radially reciprocating motion of the scissor-like elements enables the size of the mechanism to be adjustable on a large scale. The constraint equations of the mechanism are carried out based on the geometric parameters. The dynamics of the mechanism are then analyzed using the Natural Coordinates Method and the Jourdain Principle. In addition, the dynamic rolling gait of the mechanism is designed in various sizes based on the dynamic analysis. The validity of the theoretical analysis and the functionality of the mechanism is verified by a manufactured prototype, whose practical experiments are compared with the simulation.

An Unknown-Input HOSM Approach to Estimate Lean and Steering Motorcycle Dynamics

This paper deals with state estimation of powered single-track vehicle and robust reconstruction of related unknown inputs. For this purpose, we consider an unknown-input high-order sliding-mode observer (UIHOSMO). First, a motorcycle dynamic model is derived using Jourdain's principle. The strong observability of the obtained model is illustrated. Then, we consider both the observation of the powered two-wheeled (PTW) dynamic states and the reconstruction of the lean dynamics and the rider's torque applied on the handlebar. Finally, several simulation cases are provided to illustrate the efficiency of the observer.

Dynamic modelling of a two-wheeled vehicle: Jourdain formalism

This paper presents a motorcycle direct dynamic formulation by the Jourdain's principle approach. This vehicle is considered as an assembly of six rigid bodies and the resulting equation of motion allows to simulate 11 degrees of freedom. The vehicle geometry is described and a step-by-step procedure is introduced to evaluate the kinematics and the generalised efforts of the considered vehicle. In addition, to simulate the equation of motion, a Lyapunov-based stabilisation is developed to assess the vehicle behaviour in response to a propulsion/braking torque applied on the vehicle's wheels and a rider torque exerted on the motorcycle's handlebar. Simulation results reveal some dynamic features such as load transfer and counter-steering phenomena.

Jourdain Principle of a Super-Thin Elastic Rod Dynamics

A super thin elastic rod is modeled with a background of DNA super coiling structure, and its dynamics is discussed based on the Jourdain variation. The cross section of the rod is taken as the object of this study and two velocity spaces about arc coordinate and the time are obtained respectively. Virtual displacements of the section on the two velocity spaces are defined and can be expressed in terms of Jourdain variation. Jourdain principles of a super thin elastic rod dynamics on arc coordinate and the time velocity space are established, respectively, which show that there are two ways to realize the constraint conditions. If the constitutive relation of the rod is linear, the Jourdain principle takes the Euler–Lagrange form with generalized coordinates. The Kirchhoff equation, Lagrange equation and Appell equation can be derived from the present Jourdain principle. While the rod subjected to a surface constraint, Lagrange equation with undetermined multipliers may be derived.

Modeling and Analysis of the Dynamics of an Omni-directional Mobile Manipulators System

This work studies the dynamic modeling method for a service robot with Omni-directional Mobile ManipulatorS configuration. Based on screw theory, Lie group notations, reciprocal product of twist and wrench, and Jourdain principle, the robot's motion equations including the whole body manipulation are formulated with left invariant representation. A legible and canonical dynamic model representing the relation between the inputs and the generalized dynamic load wrenches is presented. Considering the tradeoff between the symbolic concision, the modularization in code realization and the computation load, the dynamic model is decomposed into succinct block factorizations, and the basic computation unites are boiled down to the adjoint map corresponding to each joint. The traditional Lie bracket operation is extended to a generalized form. Computation efficiency, for the coefficient matrixes of the system motion equation, is discussed based on its special representation form. The generalization of the modeling method with Lie group and algebra tool is also summarized.

Differential variational principles of mechanical systems in the event space

In this paper, the differential variational principles of mechanical systems in the event space are studied. The D'Alembert-Lagrange principle, the Jourdain principle, the Gauss principle and the universal D'Alembert principle in the event space are established on the basis of the D'Alembert principle of the system. The parametric forms of Euler-Lagrange, Nielsen and Appell for these principles are given, and the parametric form of Mangeron-Deleanu for the universal D'Alembert principle is deduced.

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.

Identification of Friction and Rigid-Body Dynamics of Parallel Kinematic Structures for Model-Based Control

Until now, the experimental identification of the dynamics of parallelrobots is restricted to simple models in combination with adaptivecontrol algorithms. This gap is closed by a new approach presented inthis paper, which is suited for even complex parallel kinematicstructures. The approach consists of two steps and utilizes simplepoint-to-point (PTP) motions that lead to a separation of friction andrigid-body dynamics. In the first step, local models are determined fora lot of different configurations, i.e. end-effector positions. In thesecond step, the overall friction model and the overall rigid-bodymodel are calculated from the local models by linear Least-Squaresestimators. The use of linear estimators is based on a formulation ofthe dynamic equations, which is linear with respect to a dynamicparameter vector of minimal dimension. This formulation is automaticallyobtained by an algorithm, which utilizes Jourdain's principle of virtualpower. The experimental application of the identified model tomodel-based feedforward control of the innovative hexapod PaLiDA, whichhas been developed by the Institute of Production Engineering andMachine Tools of the University of Hannover, proves the capability andefficiency of the presented algorithms.

Modelling and identification of rigid‐body dynamics of parallel kinematic structures

AbstractA new approach for modelling the dynamics of parallel kinematic (PKM) structures is presented in this paper. It leads to a formulation of the dynamic equations which is linear with respect to a dynamic parameter vector of minimal dimension. Thus, the equations can be directly used for parameter identi.cation by linear estimation techniques. The algorithm utilises Jourdain's principle of virtual power which leads to very efficient resulting code. The parameter reduction is based on opening the kinematic loops so that analytic rules known from serial robots can be implemented. Additionally, a new approach for dynamic parameter identification is suggested. The application to modelling the PKM PaLiDa and identifying its gravitational parameters proves the capacity of the presented approaches.

Noether's theorem of a rotational relativistic variable mass system

Noether's theory of a rotational relativistic variable mass system is studied. Firstly, Jourdain's principle of the rotational relativistic variable mass system is given. Secondly, on the basis of the invariance of the Jourdain's principle under the infinitesimal transformations of groups, Noether's theorem and its inverse theorem of the rotational relativistic variable mass system are presented. Finally, an example is given to illustrate the application of the result.

Finite and impulsive motion of constrained mechanical systems via Jourdain's principle: discrete and hybrid parameter models

The main purpose of this paper is to present a unified analytical dynamics framework for the analysis of finite and impulsive motion of mechanical systems using Jourdain's principle. Emphasis is given to the general case when a mechanical system is described by a hybrid (discrete-distributed) parameter model. A large group of finite and impulsive, generally non-holonomic, constraints are analysed in detail and a so-called extended Appellian classification is presented for these constrained motion problems. The fundamental dynamic equation of constrained systems is developed in terms of velocity variations (Jourdain's principle). Based on this equation and the constraints, the methods of quasivelocities and Lagrangian multipliers are adopted and interpreted for the finite motion of hybrid parameter models of mechanical systems; and the methods of independent quasivelocity variations and Lagrangian multipliers are introduced for the analysis of impulsive motion of such models. To illustrate the proposed material, an example of a one-link flexible arm intercepting and capturing a moving target is considered.

ON THE FORESHORTENING EFFECTS OF A ROTATING FLEXIBLE BEAM USING DIFFERENT MODELING METHODS14780001*

ABSTRACT This paper presents six methods of modeling a flexible beam rotating in the vertical plane. The first three methods, Hamilton's principle, Lagrange's equations for quasi-coordinates, and Newton–Euler equations, give a continuous model. The last three methods, Lagrange's equations, Jourdain's principle, and Newton–Euler equations, give a discrete model. The main conclusion of the paper is that, if foreshortening is not properly included in the beam model, then different methods give different equations. The paper also discusses several issues concerning symbolic modeling of mechanical structures within Maple. *Communicated by J. McPhee.

NONHOLONOMIC DYNAMICS AND JOURDAIN'S PRINCIPLE

For linear nonholonomic constraint systems, we prove that Jourdain's principle is not derived from D'Alembert's principle but equivalent to it if the variation is defined in the phase space rather than the configuration space. For nonlinear nonholonomic constraint systems, that the virtual displacements are undefined causes the inability of D'Alembert's principle to treat them; Jourdain's principle can be proved to be applicable through the application of Gauss's principle. The Newton-Euler equations of motion for rigid bodies in the form of Jourdain variational principle is first derived. The merit of Jourdain's principle relative to Gauss's principle is that the former does not need to evaluate the acceleration in the derivation of the equations of motion as does the latter.

Dynamic Modeling and Analysis of a Robot Manipulator Intercepting and Capturing a Moving Object, with the Consideration of Structural Flexibility

In this paper we investigate the dynamics of robotic interception and capture of a moving object. This problem, i.e., the interception and capture of a moving object by a robot, is called dynamic mass capture. The effects of structural flexibility of the robot is taken into consideration. In terms of time, the analysis is divided into three phases: before capture (finite motion), at the vicinity of interception and capture (impulsive motion), and after capture (finite motion). Special attention is paid to the modeling of the second phase when the robot captures the target and it becomes part of the end effector, thus, the system's degrees of freedom suddenly change. To decribe this event, a novel approach is proposed. This is based on the use of a class of impulsive constraints, the so-called inert constraints. Jourdain's principle is employed to derive the dynamic equations for both finite and impulsive motions. Simulation results are presented for two examples: a single flexible link and a two-link manipulator capturing moving objects. In the example of the single link, the results are compared with the observations of an experiment, and good agreement is found between experimental and simulation results.

A distributed parameter model for the dynamics of flexible‐link robots

In this paper a model is developed for kinematic and dynamic analysis of flexible robots undergoing general three-dimensional motion. For modeling robotic links, distributed mass and flexibility are considered without discretization. Some modeling issues are discussed, and parameters characterizing the real design of a robot are introduced into the analysis. The concept of a fictitious rigid link is presented to consider the rigid body motion of a link separately, and to account for possibly complex link shapes. Based on Jourdain's principle, an alternative formulation is proposed to derive the dynamic equations of flexible robots. The equations of motion are developed and analyzed in detail. The vibrations of links are described by linear, inhomogeneous partial differential equations, with homogeneous, nonlinear, time-dependent boundary conditions. © 1998 John Wiley & Sons, Inc.

A distributed parameter model for the dynamics of flexible-link robots

In this paper a model is developed for kinematic and dynamic analysis of flexible robots undergoing general three-dimensional motion. For modeling robotic links, distributed mass and flexibility are considered without discretization. Some modeling issues are discussed, and parameters characterizing the real design of a robot are introduced into the analysis. The concept of a fictitious rigid link is presented to consider the rigid body motion of a link separately, and to account for possibly complex link shapes. Based on Jourdain's principle, an alternative formulation is proposed to derive the dynamic equations of flexible robots. The equations of motion are developed and analyzed in detail. The vibrations of links are described by linear, inhomogeneous partial differential equations, with homogeneous, nonlinear, time-dependent boundary conditions. © 1998 John Wiley & Sons, Inc.

Developing algorithms for efficient simulation of flexible space manipulator operations

A brief description is given of the European Robot Arm (ERA), and its intended application to assembly, maintenance and servicing of the Russian segment of the International Space Station Alpha. The need for manipulator simulation is recalled, and ERA simulator development is outlined. System characteristics that must be modelled for incorporation in the dynamics kernel for the simulator are mentioned. A brief review is presented of efficient Order- N 3 and Order- N algorithms for the simulation of manipulator dynamics. Parenthetically the existence of a link between Kane's method, the natural orthogonal complement method, and Jourdain's principle is indicated. An algorithm of the Order- N type has then been developed and implemented in the ERA simulation facility. Some numerical results are presented, displaying accuracy as well as computational efficiency.

Dynamics of flexible multibody systems with tree topologies

The dynamic equations of flexible multibody systems with tree topological configuration are derived by using the Jourdain's principle. The independent joint coordinates are introduced to describe the large displacements of the bodies, and the modal coordinates are used to describe small deformations of flexible bodies based on the consistent mass finite element method and normal vibration mode analysis. The minimum differential equations are developed, which are compatible with the equations of multi-rigid body systems or structural dynamics. The stiff problem in the numerical integration is thus alleviated effectively. The method used in this paper can be extended to deal with systems with other topological configurations. Finally, the validity and feasibility of the presented mathematical model are demonstrated by a numerical example of a manipulator with two elastic links.

On Jourdain's principle

This paper reexamines the effect of the validity, or the lack thereof, of the commutation rule for kinematically possible, or admissible, and virtual variations i.e. dδ(…) vs δd(…), on the form of Jourdain's principle (JP), in general quasivariables. It is shown that JP is independent of any such rule. A similar independence holds in the derivation of the equations of motion from Lagrange's principle.

COMPARISONS OF A LINEAR WITH A NON-LINEAR MULTIBODY SIMULATION OF AN OFF-ROAD VEHICLE

SUMMARY Over the past two decades there has been a great deal of research in the area of multibody dynamics and its application to the modelling of vehicles. Multibody dynamics formalisms can be categorized based upon several criteria. Of these, whether linear or non-linear equations-of-motion are generated by the technique is one of the most significant. In the Dynamics Laboratory in the Department of Mechanical Engineering at Queen's University both linear (A'GEM) and non-linear (REVS) programs for modelling road vehicles are used. A'GEM is based upon an influence coefficient technique and generates the linear equations-of-motion for systems with general topology. REVS is based upon Kane's Equation (The Principal of Virtual Power or Jourdain's Principle) and generates the non-linear equations-of-motion for systems with general topology. Both programs are implemented on personal computers. In this paper the results from models of a 1/4 ton off-road vehicle (the Iltis) developed with A'GEM and REVS are compared with each other. Comparisons between the models are made on the basis of accuracy and execution time which are the fundamental tradeoff variables between linear and non-linear equations.