On the dynamics and Lyapunov stability of constrained and embedded rigid bodies

Lyapunov stability of constrained and embedded rigid bodies is considered. The constraints are of the equality type where the desired motion is to take place on an a priori defined submanifold of movement. Special and augmented state spaces for the representation of systems of rigid bodies are presented. A systematic method of stabilizing these augmented systems and a procedure for constructing Lyapunov functions are presented. The representation is applicable to augmented as well as reduced state spaces of the system defined by the constraints. The augmented state space results from the embedding of the free rigid body system in the larger state space of free rigid body and position control states, and in which the Lyapunov function is constructed. The reduced state space results when the system is restricted and is reduced to lie on the submanifold of movement. It is shown that, for the class of rigid bodies and the physical constraints considered, the projected feedback structures, and the reduced Lyapunov function constitute appropriate stabilizing structures for the constrained system. It is shown that the method applies equally to holonomically constrained and visco-elastically coupled rigid bodies. Digital computer simulations of one single rigid body system are presented to demonstrate the feasibility and effectiveness of the method. Applications to natural systems and the role of cartilage, ligaments and muscles in maintaining the integrity and stability of the joints are noted.     

Position‐based rigid‐body dynamics

We propose a position‐based approach for large‐scale simulations of rigid bodies at interactive frame rates. Our method solves positional constraints between rigid bodies and can therefore be seamlessly integrated into other position‐based methods. Interaction of particles and rigid bodies through common constraints enables two‐way coupling with deformables. The method exhibits exceptional performance and stability while being user controllable and easy to implement. Various results demonstrate the practicability of our method for the resolution of collisions, contacts, stacking and joint constraints. Copyright © 2014 John Wiley & Sons, Ltd.     

Efficient enforcement of hard articulation constraints in the presence of closed loops and contacts

In rigid body simulation, one must distinguish between contacts (so‐called unilateral constraints) and articulations (bilateral constraints). For contacts and friction, iterative solution methods have proven most useful for interactive applications, often in combination with Shock‐Propagation in cases with strong interactions between contacts (such as stacks), prioritizing performance and plausibility over accuracy. For articulation constraints, direct solution methods are preferred, because one can rely on a factorization with linear time complexity for tree‐like systems, even in ill‐conditioned cases caused by large mass‐ratios or high complexity. Despite recent advances, combining the advantages of direct and iterative solution methods wrt. performance has proven difficult and the intricacy of articulations in interactive applications is often limited by the convergence speed of the iterative solution method in the presence of closed kinematic loops (i.e. auxiliary constraints) and contacts. We identify common performance bottlenecks in the dynamic simulation of unilateral and bilateral constraints and are able to present a simulation method, that scales well in the number of constraints even in ill‐conditioned cases with frictional contacts, collisions and closed loops in the kinematic graph. For cases where many joints are connected to a single body, we propose a technique to increase the sparsity of the positive definite linear system. A solution to these bottlenecks is presented in this paper to make the simulation of a wider range of mechanisms possible in real‐time without extensive parameter tuning.     

Network model approach to the analysis of multirigid-body systems

The mathematical model of a rigid body in three dimensional motion developed in [1] is used to formulate the equations of motion for the systems of rigid bodies connected to form a special type of open kinematic chain. In this interconnection pattern of rigid bodies, each rigid body is considered as a 3-port component, and for the sake of generality, initially no constraints are imposed on the joints used to interconnect the rigid bodies. The system is considered as an (m+1)-port component and the corresponding terminal equations are obtained in closed form. As an appliction of these equations, a three-link plane manipulator is considered.     

Constrained rigid body stability and control

Stability and control of a single or three‐body constrained system are considered. Several different types of constrained motion are among them: the impact phase of a free body colliding with the ground, contact with a stationary or moving platform, movement on a frictionless surface or multiple rigid bodies connected by holonomic constraints, and moving as in the human arm. The single body constrained system is controlled by sliding mode control. The stability of the three‐link arm at arbitrary equilibrium points and Lyapunov stability in the vicinity of the equilibrium point are formulated. The formulation and derivations are by computational tools, that is, state space analysis and matrices. The approach can easily be extended to larger systems with many rigid bodies such as skeletal systems. The formulation minimizes human labor in formulations and simulations. The sliding mode behavior of the model on a frictionless surface and the three link arm stability are demonstrated via simulation. Challenges for application to natural systems are outlined. Copyright © 2014 John Wiley & Sons, Ltd.     

Analysis of Rigid Body Interactions for Compliant Motion Tasks Using the Grassmann–Cayley Algebra

This paper studies the statics and the instantaneous kinematics of a rigid body constrained by one to six contacts with a rigid static environment. These properties are analyzed under the frictionless assumption by modeling each contact with a kinematic chain that, instantaneously, is statically and kinematically equivalent to the contact and studying the resulting parallel chain using the Grassmann-Cayley algebra. This algebra provides a complete interpretation of screw theory, in which twist and wrench spaces are expressed by means of the concept of extensor and its inherent duality reflects the reciprocity condition between possible twists and admissible wrenches of partially constrained rigid bodies. Moreover, its join and meet operators are used to compute sum and intersections of the twist and wrench spaces resulting from serial and parallel composition of motion constraints. In particular, it has an explicit formula for the meet operator that gives closed-form expressions of twist and wrench spaces of rigid bodies in contact. The Grassmann-Cayley algebra permits us to work at the symbolic level, that is, in a coordinate-free manner and therefore provides a deeper insight into the kinestatics of rigid body interactions.     

Joint Reaction Forces in Multibody Systems with Redundant Constraints

Redundant constraints are defined as those constraints which can be removed without changing the kinematics of the mechanism. They are usually eliminated from the mathematical model of a multibody system. For a given mechanism the set of redundant constraints can be chosen in many ways. Rigid body systems with redundant constraints do not have a unique solution to the problem of joint reaction forces calculation. If redundant constraints are present in the mechanical system, then the system is statically undetermined. If in the case of dynamics problem the constraints are consistent, all of them are frictionless and we are interested only in positions, velocities and accelerations of the bodies, then the calculation of joint reaction forces is not necessary. In many cases, however, e.g. when we want to take into account friction in joints, the calculation of joint reaction forces cannot be avoided. In some rigid body systems, despite the redundant constraints existence, reaction forces in selected joints can be uniquely determined. The paper presents three methods of finding the constraints for which reaction forces can be uniquely determined using rigid body model. Three different techniques of Jacobian matrix analysis are used.     

Improving performance of rigid body dynamics simulation by removing inaccessible regions from geometric models

Rigid body simulations require collision detection for determining contact points between simulated bodies. Collision detection performance can become dramatically slow, if geometric models of rigid bodies have intricate inaccessible regions close to their boundaries, particularly when bodies are in close proximity. As a result, frame rates of rigid body simulations reduce significantly in the states in which bodies come into close proximity. Thus, removing inaccessible regions from models can significantly improve rigid body simulation performance without influencing the simulation accuracy because inaccessible regions do not come in contact during collisions. This paper presents an automated pair-wise contact preserving model simplification approach based upon detection and removing of inaccessible regions of a given model with respect to another colliding model. We introduce a pose independent data-structure called part section signature to perform accessibility queries on 3D models based on a conservative approximation scheme. The developed approximation scheme is conservative and does not oversimplify but may undersimplify models, which ensures that the contact points determined using simplified and unsimplified models are exactly identical. Also, we present a greedy algorithm to reduce the number of simplified models that are needed to be stored for satisfying memory constraints in case of a simulation scene with more than two models. This paper presents test results of the developed simplification algorithm on a variety of part models. We also report results of collision detection performance tests in rigid body simulations using simplified models, which are generated using developed algorithms, and their comparison with the identical performance tests on respective unsimplified models.     

Performance analysis of direct N-body algorithms for astrophysical simulations on distributed systems

We discuss the performance of direct summation codes used in the simulation of astrophysical stellar systems on highly distributed architectures. These codes compute the gravitational interaction among stars in an exact way and have an O(N²) scaling with the number of particles. They can be applied to a variety of astrophysical problems, like the evolution of star clusters, the dynamics of black holes, the formation of planetary systems, and cosmological simulations. The simulation of realistic star clusters with sufficiently high accuracy cannot be performed on a single workstation but may be possible on parallel computers or grids. We have implemented two parallel schemes for a direct N-body code and we study their performance on general purpose parallel computers and large computational grids. We present the results of timing analyzes conducted on the different architectures and compare them with the predictions from theoretical models. We conclude that the simulation of star clusters with up to a million particles will be possible on large distributed computers in the next decade. Simulating entire galaxies however will in addition require new hybrid methods to speedup the calculation.     

A DAE Approach to Flexible Multibody Dynamics

The present work deals with the dynamics of multibody systems consisting ofrigid bodies and beams. Nonlinear finite element methods are used to devise a frame-indifferent spacediscretization of the underlying geometrically exact beam theory. Both rigid bodies and semi-discrete beams are viewed as finite-dimensional dynamical systems with holonomic constraints. The equations of motion pertaining to the constrained mechanical systems under considerationtake the form of Differential Algebraic Equations (DAEs).The DAEs are discretized directly by applying a Galerkin-based method.It is shown that the proposed DAE approach provides a unified framework for the integration of flexible multibody dynamics.     

Tactics‐Based Behavioural Planning for Goal‐Driven Rigid Body Control

Controlling rigid body dynamic simulations can pose a difficult challenge when constraints exist on the bodies' goal states and the sequence of intermediate states in the resulting animation. Manually adjusting individual rigid body control actions (forces and torques) can become a very labour‐intensive and non‐trivial task, especially if the domain includes a large number of bodies or if it requires complicated chains of inter‐body collisions to achieve the desired goal state. Furthermore, there are some interactive applications that rely on rigid body models where no control guidance by a human animator can be offered at runtime, such as video games. In this work, we present techniques to automatically generate intelligent control actions for rigid body simulations. We introduce sampling‐based motion planning methods that allow us to model goal‐driven behaviour through the use of non‐deterministic Tactics that consist of intelligent, sampling‐based control‐blocks, called Skills. We introduce and compare two variations of a Tactics‐driven planning algorithm, namely behavioural Kinodynamic Rapidly Exploring Random Trees (BK‐RRT) and Behavioural Kinodynamic Balanced Growth Trees (BK‐BGT). We show how our planner can be applied to automatically compute the control sequences for challenging physics‐based domains and that is scalable to solve control problems involving several hundred interacting bodies, each carrying unique goal constraints.     

Contact Conditions for Cylindrical,Prismatic, and Screw Joints in Flexible Multibody Systems

This paper focuses on the modeling of the contact conditionsassociated with cylindrical, prismatic, and screw joints in flexiblemultibody systems. In the classical formulation these joints aredeveloped for rigid bodies, and kinematic constraints are enforcedbetween the kinematic variables of the two bodies. These constraintsexpress the conditions for relative translation and rotation of the twobodies along and about a body-fixed axis, and imply the relative slidingand rotation of the two bodies which remain in constant contact witheach other. However, these kinematic constraints no longer implyrelative sliding with contact when one of the bodies is flexible. Toremedy this situation, a sliding joint and a sliding screwjoint are proposed that involves kinematic constraints at theinstantaneous point of contact between the sliding bodies. For slidingscrew joints, additional constraints are added on the relative rotationof the contacting bodies. Various numerical examples are presented thatdemonstrate the dramatically different behavior of cylindrical,prismatic, or screw joints and of the proposed sliding and sliding screwjoints in the presence of elastic bodies, and the usefulness of theseconstraint elements in the modeling of complex mechanical systems.     

A generalized framework for interactive dynamic simulation for multirigid bodies

This paper presents a generalized framework for dynamic simulation realized in a prototype simulator called the Interactive Generalized Motion Simulator (I-GMS), which can simulate motions of multirigid-body systems with contact interaction in virtual environments. I-GMS is designed to meet two important goals: generality and interactivity. By generality, we mean a dynamic simulator which can easily support various systems of rigid bodies, ranging from a single free-flying rigid object to complex linkages such as those needed for robotic systems or human body simulation. To provide this generality, we have developed I-GMS in an object-oriented framework. The user interactivity is supported through a haptic interface for articulated bodies, introducing interactive dynamic simulation schemes. This user-interaction is achieved by performing push and pull operations via the PHANToM haptic device, which runs as an integrated part of I-GMS. Also, a hybrid scheme was used for simulating internal contacts (between bodies in the multirigid-body system) in the presence of friction, which could avoid the nonexistent solution problem often faced when solving contact problems with Coulomb friction. In our hybrid scheme, two impulse-based methods are exploited so that different methods are applied adaptively, depending on whether the current contact situation is characterized as "bouncing" or "steady." We demonstrate the user-interaction capability of I-GMS through on-line editing of trajectories of a 6-degree of freedom (dof) articulated structure.     

基于面向对象的粒子类模拟并行计算研究

针对经典分子动力学和PIC方法等粒子类模拟方法具有粒子动态移动、粒子计算局部性好等共性,首先,提出了粒子量数据片对象.该对象是单网格片上的一团粒子,其中网格片是包含多个网格单元的矩形区域.然后,设计了并行算法,包括对象之间的粒子迁移和数据交换以及动态负载平衡.最后,在JASMIN框架上具体实现,进而开发了并行经典分子动力学程序和并行PIC程序.在64个处理器上实测表明,并行PIC程序模拟包含3百万个网格、2千万个粒子的复杂物理模型时,获得了80%的并行效率.     

A specified‐time control framework for control‐affine systems and rigid bodies: A time‐rescaling approach

This paper addresses the specified‐time control problem for control‐affine systems and rigid bodies, wherein the specified‐time duration can be designed in advance according to the task requirements. By using the time‐rescaling approach, a novel framework to solve the specified‐time control problem is proposed, and the original systems are converted to the transformation systems based on which the specified‐time control laws for both control‐affine systems and rigid bodies are studied. Compared with the existing approaches, our proposed specified‐time control laws can be derived from the known stabilization control laws. To our best knowledge, it is the first time that transformation system–based specified‐time control framework for control‐affine system and rigid body dynamics is proposed. To further improve the convergence performance of specified‐time control, a finite‐time attitude synchronization control law for rigid bodies on rotation matrices is proposed, and thereby, the finite‐time–based specified‐time control law is designed eventually. In the end, numerical simulations and SimMechanics experiments are provided to illustrate effectiveness of the theoretical results.     

Modeling Spatial Motion of 3D Deformable Multibody Systems with Nonlinearities

A computational strategy for modeling spatial motion of systems of flexible spatial bodies is presented. A new integral formulation of constraints is used in the context of the floating frame of reference approach. We discuss techniques to linearize the equations of motion both with respect to the kinematical coupling between the deformation and rigid body degrees of freedom and with respect to the geometrical nonlinearities (inclusion of stiffening terms). The plastic behavior of bodies is treated by means of plastic multipliers found as the result of fixed-point type iterations within a time step. The time integration is based on implicit Runge Kutta schemes with arbitrary order and of the RadauIIA type. The numerical results show efficiency of the developed techniques.     

GROMACS软件并行计算性能分析

分子动力学模拟是对微观分子原子体系在时间与空间上的运动模拟,是从微观本质上认识体系宏观性质的有力方法.针对如何提升分子动力学并行模拟性能的问题,本文以著名软件GROMACS为例,分析其在分子动力学模拟并行计算方面的实现策略,结合分子动力学模拟关键原理与测试实例,提出MPI+OpenMP并行环境下计算性能的优化策略,为并行计算环境下实现分子动力学模拟的最优化计算性能提供理论和实践参考.对GPU异构并行环境下如何进行MPI、OpenMP、GPU搭配选择以达到性能最优,本文亦给出了一定的理论和实例参考.     

基于约束的刚体碰撞响应仿真研究与应用

为了解决虚拟场景中刚体碰撞与穿透问题,采用了一种基于约束的碰撞响应方法。根据刚体之间的穿透深度和在碰撞点处的相对速度,判定不同的碰撞状态。通过将多刚体碰撞和多点碰撞分割为单个碰撞,以刚体在碰撞点相对速度法向分量构建法向约束,切向分量构建切向摩擦约束。根据不同的碰撞状态以及牛顿恢复系数碰撞模型求取法向误差,并以碰撞点相对速度切向分量作为切向摩擦误差。结合约束与误差,得到碰撞的约束方程,联合刚体的动力学方程,迭代求解,计算碰撞之后刚体的速度,更新刚体的坐标与姿态,并计算刚体顶点在世界坐标系中的坐标。在虚拟矿山场景中进行了应用,仿真效果较好。     

Substructure synthesis methods for dynamic analysis of multi-body systems

The formulation for the dynamic analysis of flexible multi-body systems that undergo large rigid body motion, leads to geometrically non-linear inertia properties due to large rotations. These inertia non-linearities that represent the coupling between gross rigid body motion and small elastic deformation, are dependent on the assumed displacement field. As alternatives to the finite element methods, deformable body shape functions and shape vectors are commonly employed to describe elastic deformation of linear structures. In this paper, substructure shape functions and shape vectors are used to describe elastic deformation of non-linear inertia-variant multi-body systems. This leads to two different representations of inertia nonlinearities; one is based on a consistent mass formulation, while the other is a lumped mass technique. The multi-body systems considered are collections of interconnected rigid and flexible bodies. Open and closed loop systems are permitted.