变质量非线性非完整系统的Gibbs-Appell方程

本文首先将Gibbs-Appell方程推广到最一般的变质量非完整系统.得到变质量非线性非完整系统在广义坐标、准坐标下的Gibbs-Appell方程和积分变分原理,最后给出一个例子.     

Mathematical modeling and trajectory planning of mobile manipulators with flexible links and joints

This paper is concerned with mathematical modeling and optimal motion designing of flexible mobile manipulators. The system is composed of a multiple flexible links and flexible revolute joints manipulator mounted on a mobile platform. First, analyzing on kinematics and dynamics of the model is carried out then; open-loop optimal control approach is presented for optimal motion designing of the system. The problem is known to be complex since combined motion of the base and manipulator, non-holonomic constraint of the base and highly non-linear and complicated dynamic equations as a result of the flexible nature of both links and joints are taken into account. In the proposed method, the generalized coordinates and additional kinematic constraints are selected in such a way that the base motion coordination along the predefined path is guaranteed while the optimal motion trajectory of the end-effector is generated. This method by using Pontryagin’s minimum principle and deriving the optimality conditions converts the optimal control problem into a two point boundary value problem. A comparative assessment of the dynamic model is validated through computer simulations, and then additional simulations are done for trajectory planning of a two-link flexible mobile manipulator to demonstrate effectiveness and capability of the proposed approach.     

The general equations of analytical dynamics

It is shown that the generalized Poincaré and Chetayev equations, which represent the equations of motion of mechanical systems using a certain closed system of infinitesimal linear operators, are related to the fundamental equations of analytical dynamics. Equations are derived in quasi-coordinates for the case of redundant variables; it is shown that when an energy integral exists the operator X₀ = ∂/∂t satisfies the Chetayev cyclic-displacement conditions. Using the energy integral the order of the system of equations of motion is reduced, and generalized Jacobi-Whittaker equations are derived from the Chetayev equations. It is shown that the Poincaré-Chetayev equations are equivalent to a number of equations of motion of non-holonomic systems, in particular, the Maggi, Volterra, Kane, and so on, equations. On the basis of these, and also of other previously obtained results, the Poincaré and Chetayev equations in redundant variables, applicable both to holonomic and non-holonomic systems, can be regarded as general equations of classical dynamics, equivalent to the well-known fundamental forms of the equations of motion, a number of which follow as special cases from the Poincaré and Chetayev equations.     

Inverse dynamics of underactuated multibody systems

The present work deals with controlled mechanical systems subject to holonomic constraints. In particular, we focus on underactuated systems, defined as systems in which the number of degrees of freedom exceeds the number of inputs. The governing equations of motion can be written in the form of differential-algebraic equations (DAEs) with a mixed set of holonomic and control constraints. The rotationless formulation of multibody dynamics will be considered [1]. To this end, we apply a specific projection method to the DAEs in terms of redundant coordinates. A similar projection approach has been previously developed in the framework of generalized coordinates by Blajer & Kołodziejczyk [2]. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical integration of differential-algebraic equations with mixed holonomic and control constraints

The present work aims at the incorporation of control (or servo) constraints into finite–dimensional mechanical systems subject to holonomic constraints. In particular, we focus on underactuated systems, defined as systems in which the number of degrees of freedom exceeds the number of inputs. The corresponding equations of motion can be written in the form of differential–algebraic equations (DAEs) with a mixed set of holonomic and control constraints. Apart from closed–loop multibody systems, the present formulation accommodates the so–called rotationless formulation of multibody dynamics. To this end, we apply a specific projection method to the DAEs in terms of redundant coordinates. A similar projection approach has been previously developed in the framework of generalized coordinates by Blajer & Kołodziejczyk [1]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computed torque control of an under-actuated service robot platform modeled by natural coordinates

The paper investigates the motion planning of a suspended service robot platform equipped with ducted fan actuators. The platform consists of an RRT robot and a cable suspended swinging actuator that form a subsequent parallel kinematic chain and it is equipped with ducted fan actuators. In spite of the complementary ducted fan actuators, the system is under-actuated. The method of computed torques is applied to control the motion of the robot.The under-actuated systems have less control inputs than degrees of freedom. We assume that the investigated under-actuated system has desired outputs of the same number as inputs. In spite of the fact that the inverse dynamical calculation leads to the solution of a system of differential–algebraic equations (DAE), the desired control inputs can be determined uniquely by the method of computed torques.We use natural (Cartesian) coordinates to describe the configuration of the robot, while a set of algebraic equations represents the geometric constraints. In this modeling approach the mathematical model of the dynamical system itself is also a DAE.The paper discusses the inverse dynamics problem of the complex hybrid robotic system. The results include the desired actuator forces as well as the nominal coordinates corresponding to the desired motion of the carried payload. The method of computed torque control with a PD controller is applied to under-actuated systems described by natural coordinates, while the inverse dynamics is solved via the backward Euler discretization of the DAE system for which a general formalism is proposed. The results are compared with the closed form results obtained by simplified models of the system. Numerical simulation and experiments demonstrate the applicability of the presented concepts.     

Forms of Hamilton's principle in quasi-coordinates

In a previous paper [1], certain conditions, due to Hp̈lder, Voronets and Suslov in the case of linear constraints, for deriving three forms of Hamilton's principle in generalized coordinates and velocities for the general case of non-linear non-holonomic constraints were analysed. It was shown that these three forms are equivalent and transform into one another. As a sequel to that analysis, similar issues are investigated for the case of non-linear quasi-coordinates and quasi-velocities and, in addition, the three forms of Hamilton's principle are exhibited in the case of a Legendre transformation, which transforms the equation of motion to canonical form in quasi-coordinates.     

多变形体体系动力学

无论对于刚体或是变形体,Gibbs-Appell方程是建立动力方程的行之有效的工具.本文将采用Gibbs-Appell方程来建立多变形体体系的动力方程.我们将指出采用该方法的优越性.文中给出了具有显式表达的多变形体体系的动力方程.此外也采用了一些新的及近代发展的概念,诸如欧拉参数,准坐标及相对坐标等.     

Adaptive synchronization for two identical generalized Lorenz chaotic systems via a single controller

This paper presents a systematic design procedure to synchronize two identical generalized Lorenz chaotic systems based on a sliding mode control. In contrast to the previous works, this approach only needs a single controller to realize synchronization, which has considerable significance in reducing the cost and complexity for controller implementation. A switching surface only including partial states is adopted to ensure the stability of the error dynamics in the sliding mode. Then an adaptive sliding mode controller (ASMC) is derived to guarantee the occurrence of the sliding motion even when the parameters of the drive and response generalized Lorenz systems are unknown. Last, an example is included to illustrate the results developed in this paper.     

Optimal feedback control for constrained mechanical systems

An approach to minimize the control costs and ensuring a stable deviation control is the Riccati controller and we want to use it to control constrained dynamical systems (differential algebraic equations of Index 3). To describe their discrete dynamics, a constrained variational integrators [1] is used. Using a discrete version of the Lagrange-d’Alembert principle yields a forced constrained discrete Euler-Lagrange equation in a position-momentum form that depends on the current and future time steps [2]. The desired optimal trajectory (q_opt, p_opt) and according control input u_opt is determined solving the discrete mechanics and optimal control (DMOC) algorithm [3] based on the variational integrator. Then, during time stepping of the perturbed system, the discrete Riccati equation yields the optimal deviation control input u_R. Adding u_opt and u_R to the discrete Euler-Lagrange equation causes a structure preserving trajectory as both DMOC and Riccati equations are based on the same variational integrator. Furthermore, coordinate transformations are implemented (minimal, redundant and nullspace) enabling the choice of different coordinates in the feedback loop and in the optimal control problem. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

受冲击性约束作用的系统的运动方程

当具有n个自由度的系统加有P个冲击性的约束时,要求解系统的运动,一般都需要解含n+P个方程的方程组.本文提出以待定乘子法为基础,分别就取广义坐标和伪坐标的二种情况,从n个碰撞方程中消去未知的待定乘子,将碰撞方程简化为n-P个,它和P个冲击性约束方程一起组成了含n个方程的方程组,就能求解具有冲击性约束的碰撞问题,这比一般方法更为简便.     

Simulation of Multibody Systems with Nonholonomic Constraints

The paper deals with the nonholonomic multibody system dynamics from a point of view resulting from some present applications in high‐tec areas like high‐speed train technology or biomechanics of some disciplines in high‐performance sports. A formulation of nonholonomic constraints which are linear related to generalized velocities is based on a derivative‐free approach for generating Lagrangian motion equations of multibody systems with kinematical tree structure as well as for constrained multibody systems. This has been done by using di.erential‐geometric concepts in a Riemannian space. The ideas are illustrated by the classical edge condition on double‐curved surfaces. The surfaces are described by C²‐vector functions, for example by NURBS‐approximation. As an example a bobsleigh is regarded moving on a double‐curved surface.     

Systematic Modelling Using Lagrangian DAEs

A treatment for formulating equations of motion for discrete engineering systems using a differential-algebraic form of Lagrange's equation is presented. The distinguishing characteristics of this approach are the retention of constraints in the mathematical model and the consequent use of dependent coordinates. A derivation of Lagrange's equation based on the first law of thermodynamics is featured. Nontraditional constraint classifications for Lagrangian differential-algebraic equations (DAEs) are defined. Model formulation is systematic and lays a foundation for developing DAE-based tools and algorithms for applications in dynamic systems and control.     

A unified approach to the modelling of holonomic and nonholonomic mechanical systems

An alternative technique, called projection method, for solving constrained system problems is presented. This approach can be used to derive equations of motion of both holonomic and nonholonomic systems, and the dynamic equations can be expressed in generalized velocities and/or quasi-velocities. Compared against the other methods of classical mechanics (Lagrange's, Gibbs-Appell, Kane's,...), the present method turns out to be extraordinarily short, elementary and general. As such, it deserves to be promoted as a generally accepted method in academic and engineering applications. Three examples are reported to illustrate advantages of the technique     

Modeling a heavy rope using redundant coordinates

A generalized Lagrange formalism using redundant coordinates can be applied to systems with distributed parameters. This is illustrated through an example of a heavy rope. The resulting mathematical model for the rope in three-dimensional space is singularity-free, at the expense of redundant coordinates constrained by nonlinear equations. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

One method for solving systems of nonlinear partial differential equations

A method for reducing systems of partial differential equations to corresponding systems of ordinary differential equations is proposed. A system of equations describing two-dimensional, cylindrical, and spherical flows of a polytropic gas; a system of dimensionless Stokes equations for the dynamics of a viscous incompressible fluid; a system of Maxwell’s equations for vacuum; and a system of gas dynamics equations in cylindrical coordinates are studied. It is shown how this approach can be used for solving certain problems (shockless compression, turbulence, etc.).     

Model-based control strategies for systems with constraints of the program type

The paper presents a model-based tracking control strategy for constrained mechanical systems. Constraints we consider can be material and non-material ones referred to as program constraints. The program constraint equations represent tasks put upon system motions and they can be differential equations of orders higher than one or two, and be non-integrable. The tracking control strategy relies upon two dynamic models: a reference model, which is a dynamic model of a system with arbitrary order differential constraints and a dynamic control model. The reference model serves as a motion planner, which generates inputs to the dynamic control model. It is based upon a generalized program motion equations (GPME) method. The method enables to combine material and program constraints and merge them both into the motion equations. Lagrange’s equations with multipliers are the peculiar case of the GPME, since they can be applied to systems with constraints of first orders. Our tracking strategy referred to as a model reference program motion tracking control strategy enables tracking of any program motion predefined by the program constraints. It extends the “trajectory tracking” to the “program motion tracking”. We also demonstrate that our tracking strategy can be extended to a hybrid program motion/force tracking.