A New Theoretical and Numerical Approach to Finite Rigid Body Rotations Based on Lie Group Theory

A systematic theoretical approach is presented first, in an effort to provide a complete and illuminating study on motion of a rigid body rotating about a fixed point. Since the configuration space of this motion is a differentiable manifold possessing group properties, this approach is based on some fundamental concepts of differential geometry. A key idea is the introduction of a canonical connection, matching the manifold and group properties of the configuration space. Next, following the selection of an appropriate metric, the dynamics is also carried over. The present approach is theoretically more demanding than the traditional treatments but brings substantial benefits. In particular, an elegant interpretation can be provided for all the quantities with fundamental importance in rigid body motion. It also leads to a correction of some misconceptions and geometrical inconsistencies in the field. Finally, it provides powerful insight and a strong basis for the development of efficient numerical techniques in problems involving large rotations. This is demonstrated by an example, including the basic characteristics of the class of systems examined.

Three-Dimensional Modeling of Container Cranes

This work deals with three-dimensional (3D) modeling of container cranes including the hoisting cable length commands. The proposed models allow to effectively study the 3D motion caused by the eccentricity of initial conditions or loading conditions such as those induced by wind. The container is modeled as a 3D rigid body while the hoisting cables are treated either as inextensible trusses or as linearly elastic straight, taut cables. The 3D model with inextensible cables is shown to coalesce into existing two-dimensional models under the relevant planarity constraints. Details about the treatment of the internal inextensibility constraints are discussed. Time-marching simulations are carried out to show representative 2D and 3D responses to initial conditions and commanded motion of the trolley. The main differences between the constrained model and that with the elasticity of the cables are highlighted in the framework of a few significant design scenarios.

Aeroelastic Tailoring of Helicopter Blades

The dynamic loads transmitted from the rotor to the airframe are responsible for vibrations, discomfort and alternate stress of components. A new and promising way to minimize vibration is to reduce dynamic loads at their source by performing an aeroelastic optimization of the rotor. This optimization is done thanks to couplings between the flapwise-bending motion and the torsion motion. The impact of elastic couplings (composite anisotropy) on the blade dynamic behaviour and on dynamic loads are evaluated in this paper. Firstly, analytical results, based on a purely linear modal approach, are given to understand the influence of those couplings in terms of frequency placement, aerodynamic lift load and vertical shear modification. Then, those elastic couplings are introduced on a simplified but representative blade (homogeneous beam with constant chord) and results are presented.

Time-Differentiable Null Space Method for Constrained Mechanical Systems (Application to a Rheonomic System)

The governing equations of multibody systems are, in general, formulated in the form of differential algebraic equations (DAEs) involving the Lagrange multipliers. For efficient and accurate analysis, it is desirable to eliminate the Lagrange multipliers and dependent variables. Methods called null space method and Maggi’s method eliminate the Lagrange multipliers by using the null space matrix for the coefficient matrix which appears in the constraint equation in velocity level. In a previous report, the author presented a method to obtain a time differentiable null space matrix for scleronomic systems, whose constraint does not depend on time explicitly. In this report, the method is generalized to rheonomic systems, whose constraint depends on time explicitly. Finally, the presented method is applied to four-bar linkages.

A Smooth Contact Algorithm Using Cubic Spline Surface Interpolation for Rigid and Flexible Bodies

The contact analysis of multi-flexible-body dynamics (MFBD) has been an important issue in the area of computational dynamics because the realistic dynamic analysis of many mechanical systems includes the contacts among rigid and flexible bodies. But, until now, the contact analysis in the multi-flexible-body dynamics has still remained as a big, challenging area. Especially, the most of contact algorithms have been developed based on the facetted triangles. As a result, the contact force based on the facetted surface was not accurate and smooth because the geometrical error is already included in the contact surface representation stage. This kind of error can be very important in the precise mechanism such as gear contact or cam-valve contact problems. In order to resolve this problem, this study suggests a cubic spline surface representation method and related contact algorithms. The proposed contact algorithms are using the compliant contact force model based on the Hertzian contact theory. In order to evaluate the smooth contact force, the penetration depth and contact normal directions are evaluated by using the cubic spline surface interpolation. Also, for the robust and efficient contact algorithm development, the contact algorithms are divided into four main parts which are a surface representation, a pre-search, a detailed search and a contact force generation. In the surface representation part, we propose a smooth surface representation method which can be used for smooth rigid and flexible bodies. In the pre-search, the algorithm performs collision detection and composes the expected contact pairs for the detailed search. In the detailed search, the penetration depth and contact reference frame are calculated with the cubic spline surface interpolation in order to generate the accurate and smooth contact force. Finally in the contact force generation part, we evaluate the contact force and Jacobian matrix for the implicit time integrator.

Open-Source Real-Time Robot Operation and Control System for Highly Dynamic, Modular Machines

The control system of a highly dynamic robot requires the ability to respond quickly to changes in the robot’s state. This type of system is needed in varying fields such as dynamic locomotion, multicopter control, and human-robot interaction. Robots in these fields require software and hardware capable of hard real-time, high frequency control. In addition, the application outlined in this paper requires modular components, remote guidance, and mobile control. The described system integrates a computer on the robot for running a control algorithm, a bus for communicating with microcontrollers connected to sensors and actuators, and a remote user interface for interacting with the robot. Current commercial solutions can be expensive, and open source solutions are often time consuming. The key innovation described in this paper is the building of a control system from existing — mostly open source — components that can provide realtime, high frequency control of the robot. This paper covers the development of such a control system based on ROS, OROCOS, and EtherCAT, its implementation on a dynamic bipedal robot, and system performance test results.

Massively Parallel Discrete Element Modeling of Legged Mobility on Granular Terrain

Legged mobility of robotic systems is an active area of research. Quantitatively understanding mobility of these systems on natural terrain is critical for design and operations of these systems. In this paper, we present results of computational simulations of legged mobility on granular terrain using massively parallel Discrete Element Method. We model the interactions of a leg from a micro ground vehicle with sandy terrain made of polydispersed granular media. In these simulations, we model the interactions between millions of granules and the leg to quantify ground reactions and associated qualitative behaviors. The simulations are run on parallel computers to overcome the severe computational complexity of simulating these large problems in physically feasible time-frames. We are using high fidelity first-principles approaches to model emergent complex behavior that cannot otherwise be modeled. We present results from a parametric sweep where different leg speeds and penetrations are used to understand differences in ground reaction.

Nondegenerate Continuation Problems for the Excitation Response of Nonlinear Beam Structures

This paper investigates the dynamics of a slender beam subjected to transverse periodic excitation. Of particular interest is the formulation of nondegenerate continuation problems that may be analyzed numerically, in order to explore the parameter-dependence of the steady-state excitation response, while accounting for geometric nonlinearities. Several candidate formulations are presented, including finite-difference (FD) and finite-element (FE) discretizations of the governing scalar, integro-partial differential boundary-value problem (BVP), as well as of a corresponding first-order-in-space, mixed formulation. As an example, a periodic BVP — obtained from a Galerkin-type, FE discretization with continuously differentiable, piecewise-polynomial trial and test functions, and an elimination of Lagrange multipliers associated with spatial boundary conditions — is analyzed to determine the beam response via numerical continuation using a MATLAB-based software suite. In the case of an FE discretization of the mixed formulation with continuous, piecewise-polynomial trial and test functions, it is shown that the choice of spatial boundary conditions may render the resultant index-1, differential-algebraic BVP equivariant under a symmetry group of state-space translations. The paper demonstrates several methods for breaking the equivariance in order to obtain a nondegenerate continuation problem, including a projection onto a symmetry-reduced state space or the introduction of an artificial continuation parameter. As is further demonstrated, an orthogonal collocation discretization in time of the BVP gives rise to ghost solutions, corresponding to arbitrary drift in the algebraic variables. This singularity is resolved by using an asymmetric discretization in time.

On a Consistent Application of Newton’s Law to Constrained Mechanical Systems

An investigation is carried out for deriving conditions on the correct application of Newton’s law of motion to mechanical systems subjected to constraints. It utilizes some fundamental concepts of differential geometry and treats both holonomic and anholonomic constraints. The focus is on establishment of conditions, so that the form of Newton’s law remains invariant when imposing an additional set of motion constraints on a system. Based on this requirement, two conditions are derived, specifying the metric and the form of the connection on the new manifold. The latter is weaker than a similar condition employed frequently in the literature, but holding on Riemannian manifolds only. This is shown to have several practical implications. First, it provides a valuable freedom for selecting the connection on the manifold describing large rigid body rotation, so that the group properties of this manifold are preserved. Moreover, it is used to state clearly the conditions for expressing Newton’s law on the tangent space (and not on the dual space) of a manifold. Finally, the Euler-Lagrange operator is examined and issues related to equations of motion for anholonomic and vakonomic systems are investigated.

Basic Optimization Methodology for the Design of Friction Damping in Blade Shrouds

This paper is focused on the optimization of friction element parameters in blade shrouds for various types of excitation. In order to create and validate a proper modelling methodology an experimental stand and a numerical simulation of blades interaction by means of a friction element placed in the shrouds were prepared. Mathematical models are based on the finite element method combined with rigid bodies. The interaction of the friction element and blades is described by normal contact and tangential friction forces derived for particular geometrical parameters of the studied mechanical system. The models can be analyzed both in frequency domain (by the harmonic balance method) or in time domain (by the numerical integration). The results of the optimization of friction element parameters with respect to the bending vibration suppression are documented in the paper. Another contact modelling approach intended for more complex contact surfaces is based on the decomposition of a contact surface into a set of elementary areas and on the expression of contact and friction forces between these areas. All methodologies are implemented in the MATLAB system and the results for the chosen test cases are compared with the results obtained by a measurement or by the ANSYS software.