Composite rigid body formalism for flexible multibody systems

The paper describes the extension of the composite rigid body formalism for the flexible multibody systems. The extension has been done in such a way that all advantages of the formalism with respect to the coordinates of large motion of rigid bodies are extended to the flexible degrees of freedom, e.g. the same recursive treatment of both coordinates and no appearance of O(n ³) computational complexity terms due to the flexibility. This extension has been derived for both open loop and closed loop systems of flexible bodies. The comparison of the computational complexity of this formalism with other known approaches has shown that the described formalism of composite rigid body and the residual algorithm based on it are more efficient formalisms for small number of bodies in the chains and deformation modes than the usual recursive formalism of articulated body inertia.     

DCT- and DST-based splitting methods for Toeplitz systems

New splitting iterative methods for Toeplitz systems are proposed by means of recently developed matrix splittings based on discrete sine and cosine transforms due to Kailath and Olshevsky [Displacement structure approach to discrete-trigonometric transform-based preconditioners of G. Strang type and of T. Chan type, SIAM J. Matrix Anal. Appl. 26 (2005), pp. 706–734]. Theoretical analysis shows that new splitting iterative methods converge to the unique solution of a symmetric Toeplitz linear system. Moreover, an upper bound of the contraction factor of our new splitting iterations is derived. Numerical examples are reported to illustrate the effectiveness of new splitting iterative methods.     

Trigonometric transform splitting methods for real symmetric Toeplitz systems

In this paper we study efficient iterative methods for real symmetric Toeplitz systems based on the trigonometric transformation splitting (TTS) of the real symmetric Toeplitz matrix $A$. Theoretical analyses show that if the generating function $f$ of the $n\times n$ Toeplitz matrix $A$ is a real positive even function, then the TTS iterative methods converge to the unique solution of the linear system of equations for sufficient large $n$. Moreover, we derive an upper bound of the contraction factor of the TTS iteration which is dependent solely on the spectra of the two TTS matrices involved.Different from the CSCS iterative method in Ng (2003) in which all operations counts concern complex operations when the DFTs are employed, even if the Toeplitz matrix $A$ is real and symmetric, our method only involves real arithmetics when the DCTs and DSTs are used. The numerical experiments show that our method works better than CSCS iterative method and much better than the positive definite and skew-symmetric splitting (PSS) iterative method in Bai et al. (2005) and the symmetric Gauss–Seidel (SGS) iterative method.     

On convergence of splitting iteration methods for non-Hermitian positive-definite linear systems

A new splitting iteration method is presented for the system of linear equations when the coefficient matrix is a non-Hermitian positive-definite matrix. The spectral radius, the optimal parameter, and some norm properties of the iteration matrix for the new method are discussed in detail. Based on these results, the new method is convergent under reasonable conditions for any non-Hermitian positive-definite linear system. Finally, the numerical examples show that the new method is more effective than the Hermitian and skew-Hermitian splitting iterative (or positive-definite and skew-Hermitian splitting iterative) method in central processing unit time.     

Convergence of P-regular splitting iterative methods for non-Hermitian positive semidefinite linear systems

In this paper, we study the convergence of P-regular splitting iterative methods for singular and non-singular non-Hermitian positive semidefinite linear systems, which generalize the known results. Some numerical experiments are performed to illustrate the convergence results.     

On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems

For the singular, non-Hermitian, and positive semidefinite systems of linear equations, we derive necessary and sufficient conditions for guaranteeing the semi-convergence of the Hermitian and skew-Hermitian splitting (HSS) iteration methods. We then investigate the semi-convergence factor and estimate its upper bound for the HSS iteration method. If the semi-convergence condition is satisfied, it is shown that the semi-convergence rate is the same as that of the HSS iteration method applied to a linear system with the coefficient matrix equal to the compression of the original matrix on the range space of its Hermitian part, that is, the matrix obtained from the original matrix by restricting the domain and projecting the range space to the range space of the Hermitian part. In particular, an upper bound is obtained in terms of the largest and the smallest nonzero eigenvalues of the Hermitian part of the coefficient matrix. In addition, applications of the HSS iteration method as a preconditioner for Krylov subspace methods such as GMRES are investigated in detail, and several examples are used to illustrate the theoretical results and examine the numerical effectiveness of the HSS iteration method served either as a preconditioner for GMRES or as a solver.     

Dynamic simulation of three-dimensional rigid body systems using Vector-Network techniques

This paper describes a new approach to the problem of motion prediction. An extension of the “Vector-Network” method to rigid body systems in 3-D is introduced. This procedure, with its computational implementation, is shown to eliminate the work of analytically modelling a dynamic system in order to generate its time response. The entire procedure (Vector-Network) is a basic application of concepts of graph theory in which laws of vector dynamics are combined. The principle of orthogonality is reviewed and a specific application to the generation of equations of motion is presented. Moreover, various new terminal equations are shown to be necessary in the derivation of this algorithm. It is observed that this procedure is very attractive and may prove to be helpful due to its simplicity and basic application of Newton's law.     

Vision-based control for rigid body stabilization

This paper addresses the problem of stabilizing to a desired equilibrium point an eye-in-hand system, which consists of a single camera mounted on a rigid body free to move on . It is assumed that there is a collection of landmarks fixed in the environment and that the image coordinates of those landmarks are provided to the system by an on-board CCD camera. The proposed method addresses not only the problem of stabilization but also that of maintaining feature visibility along the system’s trajectory. The resulting solution consists of a feedback control law based on the current and desired image coordinates and reconstructed attitude and depth ratio information, which guarantees that (i) the desired equilibrium point is an almost global attractor; (ii) a set of necessary conditions for feature visibility holds throughout the system’s trajectories; and (iii) the image of a predefined feature point is kept inside the camera’s field of view.     

通信时延和联合连通拓扑下多刚体系统分布式姿态一致性控制

本文研究了通信时延和联合连通切换拓扑条件下的多刚体系统分布式姿态一致性控制问题. 通过构建有 效的辅助向量并选择合适的Lyapunov-Krasovskii函数, 分别对恒定通信时延和时变通信时延两种不同情况下的控 制器进行了设计. 数值仿真结果表明, 本文提出的方法能够有效地解决这类分布式姿态一致性控制问题. 多刚体; 姿态一致性; 联合连通拓扑; 时变时延; Lyapunov函数     

Time-varying formation control with attitude synchronization of multiple rigid body systems

In this article, we study the leader-following formation control problem for a group of rigid body systems whose followers' motions are described by dual quaternion equations. A few features are as follows. First, we introduce an exosystem to generate the leader's trajectory as well as the formation configuration, which can produce a large class of time-varying signals so that we can achieve a variety of time-varying formations. Second, to overcome the communication constraint described by a digraph, we extend the distributed observer to estimate not only the desired attitude and angular velocity but also the leader's position and linear velocity. Third, a novel distributed control law is synthesized to furnish a rigorous performance analysis of the closed-loop system. The effectiveness of our design is illustrated by a numerical example.     

Closed-form time derivatives of the equations of motion of rigid body systems

Multibody System Dynamics - Derivatives of equations of motion (EOM) describing the dynamics of rigid body systems are becoming increasingly relevant for the robotics community and find many...     

Large-scale rigid body simulations

For decades, rigid body dynamics has been used in several active research fields to simulate the behavior of completely undeformable, rigid bodies. Due to the focus of the simulations to either high physical accuracy or real time environments, the state-of-the-art algorithms cannot be used in excess of several thousand to several ten thousand rigid bodies. Either the complexity of the algorithms would result in infeasible runtimes, or the simulation could no longer satisfy the real time aspects.     

One-step splitting methods for semi-discrete parabolic equations

The main purpose of the paper is to discuss splitting methods for parabolic equations via the method of lines. Firstly, we deal with the formulation of these methods for autonomous semi-discrete equations $$\frac{{dy}}{{dt}} = f(y),{\rm E}f{\rm E}non - linear,$$ f satisfying a linear splitting relation \(f(y) = \sum\limits_{i = 1}^k {f_i (y)} \) . A class of one-step integration formulas is defined, which is shown to contain all known splitting methods, provided the functionsf _i are defined appropriately. For a number of methods stability results are given. Secondly, attention is paid to alternating direction methods for problems with an arbitrary non-linear coupling between space derivatives.     

Explicit momentum conserving algorithms for rigid body dynamics

Two new explicit time integration algorithms are presented for solving the equations of motion of rigid body dynamics that identically preserve angular momentum in the absence of applied torques. This is achieved by expressing the equations of motion in conservation form. Both algorithms also eliminate the need for computing the angular acceleration. The first algorithm employs a one-pass predictor-corrector scheme while the second algorithm is based upon the staggered time integration approach of Park. Numerical results are presented comparing the new algorithms to the algorithms of Simo and Wong and Park et al. The predictor-corrector algorithm is shown to suffer weak instabilities while the staggered conserving algorithm exhibits improved performance compared to the staggered algorithm of Park et al.     

Autonomous rigid body attitude synchronization

Control laws to synchronize attitudes in a swarm of fully actuated rigid bodies, in the absence of a common reference attitude or hierarchy in the swarm, are proposed in [Smith, T. R., Hanssmann, H., & Leonard, N.E. (2001). Orientation control of multiple underwater vehicles with symmetry-breaking potentials. In Proc. 40th IEEE conf. decision and control (pp. 4598-4603); Nair, S., Leonard, N. E. (2007). Stable synchronization of rigid body networks. Networks and Heterogeneous Media, 2(4), 595-624]. The present paper studies two separate extensions with the same energy shaping approach: (i) locally synchronizing the rigid bodies’ attitudes, but without restricting their final motion and (ii) relaxing the communication topology from undirected, fixed and connected to directed, varying and uniformly connected. The specific strategies that must be developed for these extensions illustrate the limitations of attitude control with reduced information.     

A fast impulsive contact suite for rigid body simulation

A suite of algorithms is presented for contact resolution in rigid body simulation under the Coulomb friction model: Given a set of rigid bodies with many contacts among them, resolve dynamic contacts (collisions) and static (persistent) contacts. The suite consists of four algorithms: 1) partial sequential collision resolution, 2) final resolution of collisions through the solution of a single convex QP (positive semidefinite quadratic program), 3) resolution of static contacts through the solution of a single convex QP, 4) freezing of "stationary" bodies. This suite can generate realistic-looking results for simple examples yet, for the first time, can also tractably resolve contacts for a simulation as large as 1,000 cubes in an "hourglass". Freezing speeds up this simulation by more than 25 times. Thanks to excellent commercial QP technology, the contact resolution suite is simple to implement and can be "plugged into" any simulation algorithm to provide fast and realistic-looking animations of rigid bodies.     

Brachistochrone for a rigid body sliding down a curve

Extremality conditions in the brachistochrone problem for a perfectly rigid body sliding without friction down an unknown (to be determined) curve in a vertical plane are found. In this case, the body has to track the tangent to the trajectory. According to the principle of constraint release, the reaction of the support creates a torque used as control. For dimensionless Appell’s equations with a single similarity coefficient, the standard problem of the fastest descent from a given initial point to a given terminal point assuming that the initial velocity is zero is formulated. The Okhotsimskii-Pontryagin method is used to analyze the differential of the objective function. Necessary optimality conditions are found, and a formula for the optimal control that does not involve adjoint variables is derived from them. Properties of the optimal trajectories are investigated analytically both in the general case and for the limiting (zero and infinite) values of the similarity coefficient. It is found that cycloid-shaped brachistochrones occur as the similarity coefficient tends to infinity. For some values of the similarity coefficient, numerical results are presented that demonstrate the shape of the corresponding brachistochrones and the optimal time of motion. The results are compared with those obtained by solving the classical brachistochrone problem.     

Operator splitting methods for the computation of reacting flows

If the modeling of the chemistry of a reacting flow system is to consider more than a simple, one-step, global reaction the attainment of a numerical solution of the relevant governing equations is generally jeopardised by the problem of stiffness. In the present work a general approach to overcoming the stiffness hazard is presented. It makes use of operator-splitting techniques, the novel feature being that splitting of the chemical source terms themselves is permitted, with the aid of a new chemical splitting parameter. For the case of elliptic flow with a unimolecular reversible chemical reaction a convergence analysis provides an insight into the conditions under which a converged solution will be obtained. The existence of an optimum value of the chemical splitting parameter indicates the possibility of accelerating convergence. Finally, calculated results for some test problems, involving linear and nonlinear chemical kinetic models, illustrate the viability and usefulness of methods based on the new approach.     

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.     

Adaptive control of rigid body satellite

The minimal controller synthesis （MCS） is an extension of the hyperstable model reference adaptive control algorithm. The aim of minimal controller synthesis is to achieve excellent closed-loop control despite the presence of plant parameter variations, external disturbances, dynamic coupling within the plant and plant nonlinearities. The minimal controller synthesis algorithm was successfully applied to the problem of decentralized adaptive schemes. The decentralized minimal controller synthesis adaptive control strategy for controlling the attitude of a rigid body satellite is adopted in this paper. A model reference adaptive control strategy which uses one single three-axis slew is proposed for the purpose of controlling the attitude of a rigid body satellite. The simulation results are excellent and show that the controlled system is robust against disturbances.