Routh Reduction Research Articles

Delay Effects in the Dynamics of Human Controlled Towing of Vehicles

Abstract A simplified mechanical model of vehicle towing is constructed. The towing procedure is modeled by means of two blocks connected with a linear spring that represents the elastic drawbar. The driver in the towing car controls the velocity of towing, while the driver of the towed vehicle tries to maintain a constant tension in the drawbar. Both control forces are subjected to a constant time delay that represents the reaction time of the operators. Utilizing the translational symmetry of the system, Routh reduction is carried out to make the governing equations as compact as possible. Stability charts are constructed in the space of the control parameters and the fastest decay of signals is identified. As time delay increases, the stable domain disappears and reappears periodically. Guidelines are provided for the design of the elasticity of the drawbar, which takes into account the masses of the vehicles and the average reaction time of the drivers.

Derivation and Properties of the Angular Momentum Relative Amended Potential

AbstractThe paper concerns the determination of the Angular Momentum Relative Amended Potential (AMR potential) in the framework of the Full n-Body problem and some of its basic properties are discussed. The AMR potential is derived using two different approaches : first using a Routh reduction of the system relative to rotation about the total angular momentum, second as a variation of the Sundman Inequality, using the Cauchy inequality.

Cotangent bundle reduction and Routh reduction for polysymplectic manifolds

We discuss Lagrangian and Hamiltonian field theories that are invariant under a symmetry group. We apply the polysymplectic reduction theorem for both types of field equations and we investigate aspects of the corresponding reconstruction process. We identify the polysymplectic structures that lie at the basis of cotangent bundle reduction and Routh reduction in this setting and we relate them by means of the Routhian function and its associated Legendre transformation. Throughout the paper we provide examples that illustrate various aspects of the results.

Routh reducibility and controllability of unstable mechanical systems

Routh reduction presents the minimum number of differential equations that uniquely describe the state of nonlinear mechanical systems where the state variables can be separated into essential ones and cyclic ones. This work extends Routh reducibility for a relevant set of controlled mechanical systems. A chain of theorems is presented for identifying the conditions when reduced order rank conditions can be applied for determining the Kalman controllability of Routh reducible mechanical systems where actuation takes place along the cyclic coordinates only, while some of the essential coordinates and their derivatives are observed. Four mechanical examples represent the advantages of using reduced rank conditions to check and/or to exclude linear controllability in such systems.

Routh Reduction of Palatini Gravity in Vacuum

An interpretation of Einstein-Hilbert gravity equations as Lagrangian reduction of Palatini gravity is made. The main technique involved in this task consists in representing the equations of motion as a set of differential forms on a suitable bundle. In this setting Einstein-Hilbert gravity can be considered as a kind of Routh reduction of the underlying field theory for Palatini gravity. As a byproduct of this approach, a novel set of conditions for the existence of a vielbein for a given metric is found.

Dynamic Analysis of Tippe Top on Cylinder’s Inner Surface With and Without Friction based on Routh Reduction

Physics computing can be used to help to solve complex dynamic equations, both translation and rotation. The purpose of this study was to obtain differences in the dynamics of the tippe top with and without friction moving on inner surface of a cylindrical with varying initial state based of Routhian Reduction. The equation of tippe top in flat fields with and without friction has been reduced by the Routhian reduction method with the Poincare equation with computational in the previous research, and computation has also been carried out in the search for numerical solutions to the dynamics of tippe top with friction in the Maple program. In this study the reduction used is a Routhian reduction, so that the equation used in determining the equations of tippe top motion with and without friction that moves in a curved plane in the form of a cylindrical surface with varying initial state based on maple is Poincaré’s equation based on Routhian reduction with and without friction. The effect of friction can be seen clearly through the dynamics and graph equations in the return top. This method can reduce the equation of backward motion with and without friction that moves on the surface of the cylinder clearly in the form of a set of differential equations. This research can be continued by solving the dynamic equations of the tippe top in other curved fields such as the torus and ball. The findings of this study are dynamic equations and graphs of friction with and without friction equations that move in curved fields in the inner of surfaces in cylinders with varying initial state based on maple.

A note on Hybrid Routh reduction for time-dependent Lagrangian systems

This note discusses Routh reduction for hybrid time-dependent mechanical systems. We give general conditions on whether it is possible to reduce by symmetries a hybrid time-dependent Lagrangian system extending and unifying previous results for continuous-time systems. We illustrate the applicability of the method using the example of a billiard with moving walls.

Remarks on the Covering of the Possible Motion Area by Solutions in Rigid Body Systems

Abstract The problem of motion of a rigid body with a fixed point is considered. We study qualitatively the solutions of the system after Routh reduction. For the Lagrange integrable case, we show that the trajectories of solutions starting at the boundary of a possible motion area can both cover and not cover the entire possible motion area. It distinguishes these systems from the systems without gyroscopic forces, where the trajectories always cover the possible motion area. We also present some numerical and analytical results on the same matter for the Kovalevskaya case.

Geometry of Routh reduction

The Routh reduction for Lagrangian systems with cyclic variable is presented as an example of a Lagrangian reduction. It appears that the Routhian, which is a generating object of reduced dynamics, is not a function any more but a section of a bundle of affine values.

Research on robust reduction control method of steering-by-wire based on MBSE

Based on the theory and method of MBSE, a model of Steering-By-Wire (SBW) system with mechanical and electrical integration is established. Based on the model, a robust control strategy with H∞ mixed sensitivity is used, and a robust controller is designed. The high order robust controller is poor in real-time and has high cost. Routh, Hankel norm and ISE index reduction method are used to reduce the controller. Through time domain and frequency domain analysis, the effects of various reduction methods on the controller are discussed. Simulation results show that the ISE index reduction method has better reduction effect and has good robust performance and robust stability.

Routh reduction for first-order Lagrangian field theories

We present a reduction theory for first order Lagrangian field theories which takes into account the conservation of momenta. The relation between the solutions of the original problem with a prescribed value of the momentum and the solutions of the reduced problem is established. An illustrative example is discussed in detail.

Lagrangian Mechanics and Reduction on Fibered Manifolds

This paper develops a generalized formulation of Lagrangian mechanics on fibered manifolds, together with a reduction theory for symmetries corresponding to Lie groupoid actions. As special cases, this theory includes not only Lagrangian reduction (including reduction by stages) for Lie group actions, but also classical Routh reduction, which we show is naturally posed in this fibered setting. Along the way, we also develop some new results for Lagrangian mechanics on Lie algebroids, most notably a new, coordinate-free formulation of the equations of motion. Finally, we extend the foregoing to include fibered and Lie algebroid generalizations of the Hamilton-Pontryagin principle of Yoshimura and Marsden, along with the associated reduction theory.

Routh reduction and Cartan mechanics

In the present work a Cartan mechanics version for Routh reduction is considered, as an intermediate step toward Routh reduction in field theory. Motivation for this generalization comes from an scheme for integrable systems [12], used for understanding the occurrence of Toda field theories in so called Hamiltonian reduction of WZNW field theories [11]. As a way to accomplish with this intermediate aim, this article also contains a formulation of the Lagrangian Adler-Kostant-Symes systems discussed in [12] in terms of Routh reduction.

Finsler Geodesics of Lagrangian Systems Through Routh Reduction

We make use of a symmetry reduction technique called Routh reduction to show that the solutions of the Euler–Lagrange equations of a strongly convex autonomous Lagrangian which lie on a specific energy level can be thought of as geodesics of an associated Finsler function.

Aspects of reduction and transformation of Lagrangian systems with symmetry

This paper contains results on geometric Routh reduction and it is a continuation of a previous paper [7] where a new class of transformations is introduced between Lagrangian systems obtained after Routh reduction. In general, these reduced Lagrangian systems have magnetic force terms and are singular in the sense that the Lagrangian does not depend on some velocity components. The main purpose of this paper is to show that the Routh reduction process itself is entirely captured by the application of such a new transformation on the initial Lagrangian system with symmetry.

Characterizing the reachable set for a spacecraft with two rotors

A commonly employed actuation device to change the orientation and angular velocity of a spacecraft is the reaction wheel (or internal rotor). With this mode of actuation, it has been shown that three internal rotors are sufficient to reach any arbitrary orientation and angular velocity. A problem of long standing interest however, is the aspect of controllability when there are only two rotors, resulting either by design or due to the failure of one rotor. An early result on this problem is by Crouch (1984) [1], which affirms that the system is not accessible when there are two rotors. However, there is no attempt to characterize the reachable set in this particular case.In this paper, we analyze the dynamics of the system on a level set of the angular momentum, which is a constant of motion. We derive the dynamic equations on the level set using a geometric framework based on the Lagrange–Routh reduction procedure. We show that there is a submanifold of this level set on which every point is reachable. Using this result we characterize the reachable set of a spacecraft with two rotors.

Routh reduction and the class of magnetic Lagrangian systems

In this paper, some new aspects related to Routh reduction of Lagrangian systems with symmetry are discussed. The main result of this paper is the introduction of a new concept of transformation that is applicable to systems obtained after Routh reduction of Lagrangian systems with symmetry, so-called magnetic Lagrangian systems. We use these transformations in order to show that, under suitable conditions, the reduction with respect to a (full) semi-direct product group is equivalent to the reduction with respect to an Abelian normal subgroup. The results in this paper are closely related to the more general theory of Routh reduction by stages.

On the existence of J 2 invariant relative orbits from the dynamical system point of view

The present paper addresses the existence of J2 invariant relative orbits with arbitrary relative magnitude over the infinite time using the Routh reduction and Poincare techniques in the J2 Hamiltonian problem. The current research also proposes a novel numerical searching approach for J2 invariant relative orbits from the dynamical system point of view. A new type of Poincare mapping is defined from different central manifolds of the pseudo-circular orbits (parameterized by the Jacobi energy E, the polar component of momentum Hz and the measure of distance Δr between the fixed point and its central manifolds) to the nodal periods Td and the drifts of longitude of the ascending node during one period (ΔΩ), which differs from Koon et al.’s (AIAA 2001) definition on central manifolds parameterized by the same fixed point. The Poincare mapping is surjective because it compresses the three-dimensional variables into two-dimensional images, and the mapping degenerates into a bijective mapping in consideration of the fixed points. An iteration algorithm to the degenerated bijective mapping is proposed from the continuation procedure to perform the ergodic representation of E- and Hz-contour maps on the space of Td–ΔΩ. For the surjective mapping with Δr ≠ 0, different pseudo-circular or elliptical orbits may share the same images. Hence, the inverse surjective mapping may achieve non-unique variables from a single image, which makes the generation of J2 invariant relative orbits possible. The pseudo-circular or elliptical orbits generated from the surjective mapping will be defined in different meridian planes. Hence, the critical contribution of the present paper is the assignment of J2 invariant relative orbits to different invariant parameters E and Hz depending on the E- and Hz-contour map, which will hold J2 invariant relative orbits for extended durations. To investigate the high-order nonlinearity neglected by previous studies, a formation configuration with a large magnitude of 500 km is successfully generated from the theory developed in the present work, which is beyond the scope of the linear conditions of J2 invariant relative orbits. Therefore, the existence of J2 invariant relative orbit with an arbitrary relative magnitude over the infinite time is achieved from the dynamical system point of view.

Routh Reduction by Stages

This paper deals with the Lagrangian analogue of symplectic or point reduction by stages. We develop Routh reduction as a reduction technique that preserves the Lagrangian nature of the dynamics. To do so we heavily rely on the relation between Routh reduction and cotangent symplectic reduction. The main results in this paper are: (i) we develop a class of so called magnetic Lagrangian systems and this class has the property that it is closed under Routh reduction; (ii) we construct a transformation relating the magnetic Lagrangian system obtained after two subsequent Routh reductions and the magnetic Lagrangian system obtained after Routh reduction w.r.t. to the full symmetry group.

A Route to Routh—The Parametric Problem

There is a well known principle in classical mechanic stating that a variational problem independent of a space variable w (so called cyclic variable), but dependent on the velocity w? can be expressed without both w and w?. This is the Routh reduction principle. We develop a geometrical approach to the problem and deal with general first order variational integrals admitting a Lie symmetry group of point transformations. In the classical setting the Routh reduction is applied only to the symmetries that preserve the independent variable. In this article we remove such restriction.