Improved DAE formulation for inverse dynamics simulation of cranes

Cranes are underactuated systems with less control inputs than degrees of freedom. Dynamics and control of such systems is a challenging task, and the existence of solution to the inverse dynamics simulation problem in which an r-degree-of-freedom system with m actuators, m<r, is subject to m specified motion task (servo-constraints) is conditioned upon the system is differentially flat (all the system states and control inputs can be algebraically expressed in terms of the outputs and their time derivatives up to a certain order). The outputs are often designed as specified in time load coordinates to model a rest-to-rest maneuver along a trajectory in the working space, from the initial load position to its desired destination. The flatness-based methodology results then in the required control inputs determined in terms of the fourth time derivatives of the imposed outputs, and the derivations are featured by substantial complexity. The DAE formulation motivated in this contribution offers a more convenient approach to the prediction of dynamics and control of cranes executing prescribed load motions, and only the second time derivatives of the specified outputs are involved. While most of the inverse simulation formulations, both flatness-based and DAE ones, are performed using independent state variables, the use of dependent coordinates and velocities may lead to substantial modeling simplifications and gains in computational efficiency. An improved DAE formulation of this type is presented in this paper.     

Index reduction by minimal extension for the inverse dynamics simulation of cranes

The present work deals with the inverse dynamics simulation of underactuated mechanical systems relying on servo constraints. The servo-constraint problem of discrete mechanical systems is governed by differential–algebraic equations (DAEs) with high index. We propose a new index reduction approach, which makes possible the stable numerical integration of the DAEs. The new method is developed in the framework of a specific crane formulation and facilitates a reduction from index five to index three and even to index one. Particular attention is placed on the special case in which the reduced index-1 formulation is purely algebraic. In this case the system at hand can be classified as differentially flat system. Both redundant coordinates and minimal coordinates can be employed within the newly developed approach. The success of the proposed method is demonstrated with two representative numerical examples.     

Reduced-order observer-based output feedback regulation for a class of nonlinear systems with iISS inverse dynamics

Global asymptotic regulation via output feedback based on a reduced-order observer is considered for a class of nonlinear systems with integral input-to-state stable (iISS) inverse dynamics. The contributions are that: (1) we further consider the conditions on changing supply rates in iISS systems, and discuss the necessity of some conditions by constructing a counter-example; (2) For a more general class of nonlinear systems with iISS inverse dynamics, a reduced-order observer is introduced, and by the combined approach of backstepping and changing iISS supply rates, an output feedback controller is given to drive the output of system to the origin asymptotically and maintain other closed-loop signals bounded.     

Adaptive finite-time stabilisation of a class of high-order nonlinear systems with inverse dynamics

This paper investigates the adaptive finite-time stabilisation of a class of high-order nonlinear systems with inverse dynamics and time-varying control coefficients. Under the assumptions that the inverse dynamics of the subsystem are input-to-state stable , and nonlinear terms depend upon the inverse dynamics, the states and unknown parameters, we present a systematic design procedure for an adaptive finite-time state-feedback control law by using a recursive design approach. It is shown that under the proposed adaptive control law, the state of the closed-loop system is finite-time convergent. Two examples are provided to show the effectiveness of the proposed method.     

Global set‐point tracking control for a class of nonlinear systems with iISS inverse dynamics

In this paper, we consider the problem of global set‐point tracking control for a class of nonlinear systems with dynamic uncertainty. Unlike the existing works, the investigated system is with the integral input‐to‐state stable (iISS) inverse dynamics and more general uncertain nonlinearities. By using a recursive design method, a partial‐state feedback controller is designed. The tuning function technique is applied in this procedure to avoid the overparametrization. It is shown that the developed control procedure could guarantee that the tracking error is driven to the origin and the other signals are bounded. In addition, it can also reduce to a linear or even a classical PI control law under some sufficient conditions. Simulation results are illustrated to show the effectiveness of the proposed algorithm. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Numerical integration for the Choquet integral

Choquet integrals with respect to non-additive (or fuzzy measures) have been used in a large number of applications because they permit us to integrate information from different sources when there are interactions. Successful applications use a discrete reference set.In the case of measures on a continuous reference set, as e.g. the real line, few results have been obtained that permit us to have an analytical expression of the integral. However, in most of the cases there is no such analytical expression.In this paper we describe how to perform the numerical integration of a Choquet integral with respect to a non-additive measure.     

Investigation of a method for the three-dimensional rigid body dynamics inverse problem

The three-dimensional rigid body dynamics inverse problem is cast as a discrete optimal control problem and solved using dynamic programming. The optimal control problem uses a forward dynamic model to provide estimates of unknown forces while matching noisy measurement histories. An L curve analysis is used to objectively select the amount of smoothing by trading off the magnitude of the estimated unknown forces with the fit to the noisy measurements. The forward dynamic model is derived using finite-element methodology and accounts for the inertia and mass properties of rigid bodies with the use of natural coordinates. The advantages of this forward model are that the large displacements’ nonlinearities can be routinely represented in the inverse problem and that it also allows the use of an exact energy conserving method to numerically integrate the equations of motion. A numerical example of a large displacement three body model is included to demonstrate the performance of the methodology.     

Numerical integration on a tetrahedron

Generalizing toR ³ a previous work inR ² [1], we give approximations with any arbitrary degree, of integrals on a tetrahedron, as a product (different from Stroud's conical product formulas) of one dimensional Gauss-Jacobi formulas.     

四元数与欧拉角刚体动力学数值积分算法及其比较

为推广四元数保辛积分在工程中的应用,对欧拉角表示的状态方程数值积分与四元数的保辛积分进行比较.重陀螺的数值仿真结果表明四元数保辛积分的数值结果明显优于欧拉角状态方程积分.与欧拉角状态方程积分相比,四元数保辛积分在刚体动力学的数值仿真中更具优势.     

四元数与欧拉角刚体动力学数值积分算法及其比较

为推广四元数保辛积分在工程中的应用,对欧拉角表示的状态方程数值积分与四元数的保辛积分进行比较.重陀螺的数值仿真结果表明四元数保辛积分的数值结果明显优于欧拉角状态方程积分.与欧拉角状态方程积分相比,四元数保辛积分在刚体动力学的数值仿真中更具优势.     

A unified solution scheme for inverse dynamics

In this paper, a completely new solution scheme for inverse dynamics, which can be commonly applied in different types of link systems such as open- or closed-loop mechanisms, or ones constituting rigid or flexible link members, is presented. The scheme is developed using the finite element method (FEM), which evaluates the entire system as a continuum with the equation of motion in Cartesian coordinates and in dimension of force. The inverse dynamics is calculated by using a matrix-form relation to the nodal forces obtained by the FEM. The matrix-form equations are divided individually into terms of force, transformation between coordinates and length, which makes the scheme potentially better in terms of applicability and expansibility. The scheme cannot only deal with open- and closed-loop link systems independently, but it can also deal seamlessly with those that gradually change their forms and dynamics. There is also no need to revise the basic numerical algorithm of the scheme, regardless of the stiffness of the constituting link member, i. e. rigid or flexible. The main objective of this paper is to present the extensive ability of the scheme as a unified scheme, by carrying out calculations on several types of rigid and flexible manipulators, along with an application to feed-forward control of a link mechanism which continuously changes its form from an open- to a closed-loop.     

A weighted LS-SVM approach for the identification of a class of nonlinear inverse systems

In this paper, a weighted least square support vector machine algorithm for identification is proposed based on the T-S model. The method adopts fuzzy c-means clustering to identify the structure. Based on clustering, the original input/output space is divided into several subspaces and submodels are identified by least square support vector machine (LS-SVM). Then, a regression model is constructed by combining these submodels with a weighted mechanism. Furthermore we adopt the method to identify a class of inverse systems with immeasurable state variables. In the process of identification, an allied inverse system is constructed to obtain enough information for modeling. Simulation experiments show that the proposed method can identify the nonlinear allied inverse system effectively and provides satisfactory accuracy and good generalization. Supported by the National Natural Science Foundation of China (Grant No. 60874013) and the Doctoral Project of the Ministry of Education of China (Grant No. 20070286001)     

A class of inverse problems for discontinuous systems

We analyze a class of inverse parametric problems for dynamic processes described by systems of ordinary differential equations whose form and piecewise-constant parameters depend on what subdomain in the state space the state of the process belongs to. The study was supported by the INTAS (Project No. 06-1000017-8909). Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 142–152, November–December 2008.     

Numerical inverse kinematics for modular reconfigurable robots

The inverse kinematics solutions of a reconfigurable robot system built upon a collection of standardized components is difficult to obtain because of its varying configurations. This article addresses the formulation of a generic numerical inverse kinematics model and automatic generation of the model for arbitrary robot geometry including serial and tree‐typed geometries. Both revolute and prismatic types of joints are considered. The inverse kinematics is obtained through the differential kinematics equations based on the product‐of‐exponential (POE) formulas. The Newton–Raphson iteration method is employed for solution. The automated model generation is accomplished by using the kinematic graph representation of a modular robot assembly configuration and the related accessibility matrix and path matrix. Examples of the inverse kinematics solutions for different types of modular robots are given to demonstrate the applicability and effectiveness of the proposed algorithm. ©1999 John Wiley & Sons, Inc.     

Inverse kinematics and inverse dynamics for control of a biped walking machine

Analytical techniques are presented for the motion planning and control of a 12 degree-of-freedom biped walking machine. From the Newton-Euler equations, joint torques are obtained in terms of joint trajectories, and the inverse dynamics are developed for both the single-support and double-support cases. Physical admissibility of the biped trajectory is characterized in terms of the equivalent force-moment and zero-moment point. This methodology has been used to obtain reference inputs and implement the feedforward control of walking robots. A simulation example illustrates the application of the techniques to plan the forward-walking trajectory of the biped robot. The implementation of a prototype mechanism and controller is also described.     

Numerical contour integration for loop integrals

A fully numerical method to calculate loop integrals, a numerical contour-integration method, is proposed. Loop integrals can be interpreted as a contour integral in a complex plane for an integrand with multi-poles in the plane. Stable and efficient numerical integrations an along appropriate contour can be performed for scalar and tensor integrals appearing in loop calculations of the standard model. Examples of 3- and 4-point diagrams in 1-loop integrals and 2- and 3-point diagrams in 2-loop integrals with arbitrary masses are shown.Moreover it is shown that numerical evaluations of the Hypergeometric function, which often appears in the loop integrals, can be performed using the numerical contour-integration method.     

Numerical methods for a class of nonlinear integro-differential equations

In a previous article (Glowinski, J. Math. Anal. Appl. 41, 67–96, 1973) the first author discussed several methods for the numerical solution of nonlinear equations of the integro-differential type with periodic boundary conditions. In this article we discuss an alternative methodology largely based on the Strang’s symmetrized operator-splitting scheme. Several numerical experiments suggest that the new method is robust and accurate. It is also easier to implement than the various methods discussed by Glowinski in J. Math. Anal. Appl. 41, 67–96 (1973).     

Approximation algorithms for the bandwidth minimization problem for a large class of trees

We present approximation algorithms for the bandwidth minimization problem (BMP) for a large class of trees. The BMP is NP-hard, even for trees of maximum node degree 3. The problem finds applications in many areas, including VLSI layout, multiprocessor scheduling, and matrix processing, and has been studied for both graphs and matrices. We study the problem on trees having the following property: given any tree nodev, the depth difference of any two nonempty subtrees rooted atv is bounded by a constantk. We call such treesh(k)trees orgeneralized height-balanced (GHB)trees. The above definition extends the class of balanced trees to trees with depthd=Θ(\N\). For any tree in the above defined class, anO (logd) times optimal algorithm is presented. Furthermore, we extend the application of the algorithm to trees that simulate theh(k) property, which we callh(k)-like trees, and also provide intuitive ideas for an approximation algorithm for general trees. This work has been supported in part by the Computer Learning Research (CLEAR) Center at the University of Texas at Dallas.     

Approximation algorithms for the bandwidth minimization problem for a large class of trees

We present approximation algorithms for the bandwidth minimization problem (BMP) for a large class of trees. The BMP is NP-hard, even for trees of maximum node degree 3. The problem finds applications in many areas, including VLSI layout, multiprocessor scheduling, and matrix processing, and has been studied for both graphs and matrices. We study the problem on trees having the following property: given any tree nodev, the depth difference of any two nonempty subtrees rooted atv is bounded by a constantk. We call such treesh(k)trees orgeneralized height-balanced (GHB)trees. The above definition extends the class of balanced trees to trees with depthd=Θ(\N\). For any tree in the above defined class, anO (logd) times optimal algorithm is presented. Furthermore, we extend the application of the algorithm to trees that simulate theh(k) property, which we callh(k)-like trees, and also provide intuitive ideas for an approximation algorithm for general trees.