Dynamic Analysis of Multi-Rigid-Body Systems

A method for dynamic analysis of large-scale constrained system of mixed rigid and flexible bodies with intermittent motion is presented. The system equations of motion are written in the Lagrangian formulation using a finite set of coupled reference position and local elastic generalized coordinates. Equations of motion are computer generated and integrated forward in time using an explicit-implicit integration algorithm. Points in time at which sudden events of the intermittent behavior occur are monitored by an event predictor which controls the integration algorithm and forces a solution for the system impulse-momentum relation at those points. Solutions of impulse-momentum relations define the jump discontinuities in the composite velocity vector as well as the generalized impulses of the reaction forces at different joints of the mechanical system.

Abstract. In the last few years, the 3D GIS domain has developed rapidly, and has become increasingly accessible to different disciplines. 3D Spatial analysis of Built-up areas seems to be one of the most challenging topics in the communities currently dealing with spatial data. One of the most basic problems in spatial analysis is related to visibility computation in such an environment. Visibility calculation methods aim to identify the parts visible from a single point, or multiple points, of objects in the environment. In this work, we present a unique method combining visibility analysis in 3D environments with dynamic motion planning algorithm, named Visibility Velocity Obstacles (VVO) with Markov process defined as spatial visibility analysis for routes in 3D dense city environment. Based on our VVO analysis, we use Reinforcement Learning (RL) method in order to find an optimal action policy in dense 3D city environment described as Markov decision process, navigating in the most visible routes. As far as we know, we present for the first time a Reinforcement Learning (RL) solution to the visibility analysis in 3D dense environment problem, generating a sequence of viewpoints that allows an optimal visibility in different routes in urban city. Our analysis is based on fast and unique solution for visibility boundaries, formulating the problem with RL methods.

The scalar problem of plane wave diffraction by a system of bodies and infinitely thin screens is considered in a quasi-classical formulation. The solution is sought in the classical sense but is defined not in the entire space ℝ3 but rather everywhere except for the screen edges. The original boundary value problem for the Helmholtz equation is reduced to a system of weakly singular integral equations in the regions occupied by the bodies and on the screen surfaces. The equivalence of the integral and differential formulations is proven, and the solvability of the system in the Sobolev spaces is established. The integral equations are approximately solved by the Bubnov-Galerkin method. The convergence of the method is proved, its software implementation is described, and numerical results are presented.

Rather than the traditional vector differential equation, this paper introduces rigid body dynamics in a new form, as a matrix differential equation. We focus on axisymmetric rigid bodies which are adequate to describe a large class of problems, including tensegrity systems. For a system of beta rigid bodies, the forces are characterized in terms of network theory, and the kinematics are characterized in terms of the bar vectors (directed connections between two nodes attached to a rigid body). The dynamics are characterized by a second order differential equation in a 3 times 2beta configuration matrix. The first contribution of the paper is the dynamic model of a broad class of systems of rigid bodies, characterized in a compact form, requiring no inversion of a variable mass matrix. The second contribution is the derivation of all equilibria as a linear algebra problem in the control variables. The third contribution is the derivation of a linear model of the system of rigid bodies. One significance of these equations is the exact characterization of the statics and dynamics of all class 1 tensegrity structures, where rigid bar lengths are constant and the string force densities are control variables, which appear linearly. This will offer a significant advantage in control design tasks

In the present study, the equations of motion for generalized planar linkages that consist of a system of rigid bodies with all common types of kinematic joints are derived using a recursive approach. The system of rigid bodies is replaced by a dynamically equivalent constrained system of particles. Then for the resulting equivalent system of particles, the concepts of linear and angular momentums are used to generate the equations of motion without either introducing any rotational coordinates or distributing the external forces and force couples over the particles. For the open loop case, the equations of motion are generated recursively along the open chains. For the closed loop case, the system is transformed to open loops by cutting suitable kinematic joints and introducing cut-joints kinematic constraints. An example of a multi-branch closed-loop system is chosen to demonstrate the generality and simplicity of the proposed method.

The time-optimal control of a system of rigid bodies connected in series by single-degree-of-freedom joints is studied. The dynamical equations of the system are highly nonlinear, and a closed-form representation of the minimum-time feedback control is not possible. However, a suboptimal feedback control, which provides a close approximation to the optimal control, is developed. The suboptimal control is expressed in terms of switching curves for each of the system controls. These curves are obtained from the linearized equations of motion for the system. Approximations are made for the effects of gravity loads and angular velocity terms in the nonlinear equations of motion. Digital simulation is used to obtain a comparison of response times of the optimal and suboptimal controls. The speed of response of the suboptimal control is found to compare quite favorably with the response speed of the optimal control.

In the present study, the equations of motion for planar mechanical systems that consist of interconnected rigid bodies with common types of kinematic joint are derived on the basis of particle dynamics and a recursive approach. The system of rigid bodies is replaced by a dynamically equivalent constrained system of particles. Then, for the resulting equivalent system of particles, the concepts of linear and angular momenta are used to generate the equations of motion without either introducing any rotational coordinates or distributing the external forces and force couples over the particles. For an open-loop system, the equations of motion are generated recursively along the open chains. For a closed-loop system, the system is transformed to open chains by cutting suitable kinematic joints with the addition of cut-joint kinematic constraints. Examples of multibranch closed-loop systems are chosen to demonstrate the generality and simplicity of the proposed method.

Any articulated system of rigid bodies defines a statically equivalent serial chain (SESC). The SESC is a virtual chain that terminates at the center of mass (CoM) of the original system of bodies. An SESC may be generated experimentally without knowing the mass, CoM, or length of each link in the system given that its joint angles and overall CoM may be measured. This paper presents three developments toward recognizing the SESC as a practical modeling technique. Two of the three developments improve utilizing the technique in practical applications where the arrangement of the joints impacts the derivation of the SESC. The final development provides insight into the number of poses needed to create a usable SESC in the presence of data collection errors. First, modifications to a matrix necessary in computing the SESC are proposed, followed by the experimental validation of SESC modeling. Second, the problem of generating an SESC experimentally when the system of bodies includes a mass fixed in the ground frame are presented and a remedy is proposed for humanoid-like systems. Third, an investigation of the error of the experimental SESC versus the number of data readings collected in the presence of errors in joint readings and CoM data is conducted. By conducting the method on three different systems with various levels of data error, a general form of the function for estimating the error of the experimental SESC is proposed.

Outlier detection is an important problem in spatial analysis which involves finding a region of spatial locations with features significantly different from the rest of the population. In this paper, we used fast fourier transform to highlight the areas with high frequency change. The spatial points identified by the fourier transform are then reconfirmed with Z-value test and outlier regions are identified. We performed several experiments to highlight the accuracy and efficiency of the approach and compared it with some other existing approaches.

A central problem in spatial analysis is the mapping of data for complex spatial fields using relatively simple data structures, such as those of a conventional GIS. This complexity can be measured using such indices as multi-scale variance, which reflects spatial autocorrelation, and multi-fractal dimension, which characterizes the values of fields. These indices are computed for three spatial processes: Gaussian noise, a simple mathematical function, and data for a random walk. Fractal analysis is then used to produce a vegetation map of the central region of California based on a satellite image. This analysis suggests that real world data lie on a continuum between the simple and the random, and that a major GIS challenge is the scientific representation and understanding of rapidly changing multi-scale fields.

Analytical mechanics is used to derive original 3D equations of motion that represent impact at a point in a system of rigid bodies. For oblique impact between rough bodies in an eccentric (unbalanced) configuration, these equations are used to compare the calculations of energy dissipation obtained using either the kinematic, the kinetic, or the energetic coefficient of restitution (COR); eN,eP, or e*. Examples demonstrate that for equal energy dissipation by nonfrictional sources, either eN≤e*≤eP or eP≤e*≤eN depending on whether the unbalance of the impact configuration is positive or negative relative to the initial direction of slip. Consequently, when friction brings initial slip to rest during the contact period, calculations that show energy gains from impact can result from either the kinematic or the kinetic COR. On the other hand, the energetic COR always correctly accounts for energy dissipation due to both hysteresis of the normal contact force and friction, i.e., it is energetically consistent.

The new finite element studied here is of triangular type and is used in sheet stamping simulation. The number of equilibrium equations is half that of known methods owing to an original displacement description. The new trianglular element does not use any nodal rotation, and only the translations are useful. This is allowed by the theory of screw elements, which transform a case of rotation into a translational displacement with an offset, and by analogy with a system of rigid bodies. All the steps of the method are presented up to the evaluation of the stiffness matrix. Some examples of simulations are then given.

Asymptotic properties of the motions of a heavy gyroscope with internal dissipation

An investigation on the dynamic modeling and analysis of spatial mechanisms with spherical clearance joints including friction is presented. For this purpose, the ball and the socket, which compose a spherical joint, are modeled as two individual colliding components. The normal contact-impact forces that develop at the spherical clearance joint are determined by using a continuous force model. A continuous analysis approach is used here with a Hertzian-based contact force model, which includes a dissipative term representing the energy dissipation during the contact process. The pseudopenetration that occurs between the potential contact points of the ball and the socket surface, as well as the indentation rate play a crucial role in the evaluation of the normal contact forces. In addition, several different friction force models based on the Coulomb's law are revisited in this work. The friction models utilized here can accommodate the various friction regimens and phenomena that take place at the contact interface between the ball and the socket. Both the normal and tangential contact forces are evaluated and included into the systems' dynamics equation of motion, developed under the framework of multibody systems formulations. A spatial four-bar mechanism, which includes a spherical joint with clearance, is used as an application example to examine and quantify the effects of various friction force models, clearance sizes, and the friction coefficients.

Analytical Mechanics. Foundations of Engineering Mechanics Series