Motion equations of cooperative multi flexible mobile manipulator via recursive Gibbs–Appell formulation

In this paper, the developed model of an N-flexible-link mobile manipulator with revolute-prismatic joints is presented for the cooperative flexible multi mobile manipulator. In this model, the deformation of flexible links is calculated by using the assumed modes method. In additions, non-holonomic constraints of the robots’ mobile platforms that bound its locomotion are considered. This limitation is alleviated through the concurrent motion of revolute and prismatic joints, although it results in computational complexity and changes the final motion equations to time-varying form. Not only is the proposed dynamic model implemented for the multi-mobile manipulators with arms having independent motion, but also for multi-mobile manipulators in cooperation after defining gripper's kinematic constraints. These constraints are imported to the dynamic equations by defining Lagrange multipliers. The recursive Gibbs–Appell formulation is preferred over other similar approaches owing to the capability of solving the equations without the need to use Lagrange multipliers for eliminating non-holonomic constraints in addition to the novel optimized process of obtaining system equations. Hence, cumbersome simultaneous computations for eliminating the constraints of platform and arms are circumvented. Therefore, this formulation is improved for the first time by importing Lagrange multipliers for solving kinematic constrained systems. In the simulation section, the results of forward dynamics solution for two flexible single-arm manipulators with revolute-prismatic joints while carrying a rigid object are presented. Inverse dynamics equations of the system are also presented to obtain the maximum dynamic load-carrying capacity of the two-rigid-link mobile manipulators on a predefined path. Two constraints, namely the capacity of joint motors torque and robot motion stability are considered as the limitation criteria. The concluded motion equations are used to accurately control the movement of sensitive bodies, which is not achievable through the use of one platform.     

Two-step generation of the equations of motion of closed-chains

This paper presents a two-step generation of the equations of motion of planar mechanisms using point and joint coordinates. First, the formulation replaces a rigid body by a dynamically equivalent constrained system of particles and uses Newton’s second law to study the motion of the particles without introducing any rotational coordinates. Then, the equations of motion are transformed to a reduced set in terms of selected relative joint variables using a velocity transformation matrix. For an open-chain, this process automatically eliminates all of the non-working constraint forces and leads to an efficient integration of the equations of motion. For a closed-chain, suitable joints should be cut and few cut-joints constraint equations are included. An example of a closed-chain is used to demonstrate the generality and efficiency of the proposed method.     

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.     

Kinematic and dynamic analysis of the Gantry-Tau,a 3-DoF translational parallel manipulator

This paper deals with the mathematical modeling of kinematics and dynamics of the 3-degrees-of-freedom Gantry-Tau manipulator. Compared to many other parallel robots, Gantry-Tau offers a large accessible workspace and high stiffness. The kinematics of Gantry-Tau is presented which includes inverse kinematics formulation for the position, velocity and acceleration of the mechanism. Also, based on the obtained Jacobin matrices, singular configurations of the robot are studied. Afterwards, the equations of the inverse dynamic model of the Gantry-Tau are obtained through two different methods, i.e., virtual work and Newton–Euler. Finally, a case study is performed to verify the correctness of the derived models and investigate their computational efficiency.     

Parametric optimization of a four-link close-chain manipulator with active and passive actuators

We investigate the problem of optimization of motion laws and design parameters of a four-link manipulator with a closed-chain kinematic structure. The manipulator performs cyclic transfer operations in a horizontal plane under the action of active and passive (springs and dampers) actuators. As a minimization criterion, we take a quadratic (with respect to control moments of forces) functional. An algorithm is proposed for constructing a suboptimal solution of the formulated problem based on parametrization of the generalized coordinates of the manipulator with a family of given functions and on the use of numerical procedures of mathematical programming.     

Modelling Flexible Manipulators With Motors at the Joints

A computationally efficient recursive model of a flexible manipulator with motors at the joints is described in this paper. The model adopts a mixed Eulerian and Lagrangian formulation of the equations of a flexible body and exploits the chained structure of the equations for a serial manipulator. The dynamic effects of the motors at the joints, including gyroscopic terms, are fully taken into account. Symbolic manipulation is used in a newly developed package, whose performance in detailed reproduction of the dynamic effects due to the interplay between the motors and the flexible links is assessed through simulation. A comparison between the complete model and a simplified one, where the motors are considered as simple inertias rotating around their own axis, has been carried out, using both a time domain analysis and a frequency domain analysis, in order to show the relevance of gyroscopic effects in modelling flexible robots.     

Computer-assisted generation of robot dynamic models in an analytical form

An algorithm for constructing dynamic models of single-arm robots is presented in this paper. Motion equations of robots in analytical form are derived applying a fully automated procedure. It is shown that the solution of a direct and/or inverse problem based on the analytical model requires considerably fewer floating-point multiplications/additions than is the case with previously-developed numerical methods. The developed method is therefore very suitable for real-time application of robot dynamic models. The developed program package is illustrated using the example of Stanford manipulator.     

Vehicle dynamics simulations with coupled multibody and finite element models

With the advent of multibody system simulations (MSS) programs, it has become common practice to use computer modeling to evaluate vehicle dynamics performance. This approach has proved to be very effective for predicting the handling performance of vehicles; however, it has proved less successful for predicting the vehicle response at frequencies that are of interest in ride harshness and durability applications. The lack of correlation between theory and experiment can be partially traced back to tire models that are inadequate for rough road simulation. This paper presents a comprehensive vehicle dynamics model for simulating the dynamic response of ground vehicles on rough surfaces. This approach uses a MSS program to simulate the vehicle and a nonlinear FE program for the tires. Parallel processing of the tire models improves the efficiency of the overall simulation. Applications for this technology include vehicle ride and harshness analysis and durability loads simulation. This paper describes the MSS vehicle model, the tire FE model, and the interface which transfers data between the two simulations. Simulation and experiment results for a single tire without a vehicle encountering an obstacle and for a vehicle with four tires driving across a pot hole are presented. Conclusions and opportunities for further research end the paper.     

多体机械手的一般动力学模型

本文建立了多体机械手的一般动力学方程.设多体系统是由任意数目的刚体组成的树形拓扑结构,并认为铰是柱铰链,允许具有相对转动和滑动.考虑到实际问题中摩擦力的影响,采用Newton-Euler方法,建立了运动方程.进一步通过构造分配矩阵,将动力学方程分离,得到了一组实用的力方程和运动方程.     

Global mode method for dynamic modeling of a flexible-link flexible-joint manipulator with tip mass

In this paper, the global mode method (GMM) is proposed to obtain a reduced-order analytical dynamic model for a signal flexible-link flexible-joint (SFF) manipulator. Firstly, the nonlinear partial differential equations (PDE) that govern the motion of the flexible link and flexible joint, respectively, are derived by applying the Hamilton principle. By combining the linearized governing equations of motion for a flexible link and the equation of motion for the flexible joint, the characteristic equation is obtained for the whole system. The natural frequencies and global mode shapes of the linearized model of the SFF manipulator are determined, and orthogonality relations of the global mode shapes are established. Then, the global mode shapes and their orthogonality relations are used to truncate the nonlinear PDEs of the SFF manipulator to a nonlinear ordinary differential equation with a few degrees-of-freedom (DOF). For comparison, two other dynamic models of the SFF are derived by employing the assumed mode method (AMM) and finite element method (FEM). To verify the method proposed, the results from the GMM are compared with those obtained from the FEM. The effects of the link length and payload mass on the convergence of AMM model for the first two frequencies are investigated. Based on the dynamic models, obtained by GMM and AMM, dynamical responses for the system with different numbers of modes are worked out numerically, which are compared with those obtained from FEM. These comparisons show a good agreement between the results of the GMM and that of the FEM model, which indeed proved the accuracy and applicability of the GMM model.     

Dynamic analysis of cable-driven parallel manipulators with time-varying cable lengths

In conventional researches, cables of cable-driven parallel manipulators are treated as simple linear elements that can only work in tension. This results in the fact that the effect of cable dynamics on the positioning precision of the end-effector is not adequately taken into account. To overcome this shortcoming, a dynamic model for cable-driven parallel manipulators with cables of slowly time-varying length is presented in this paper. The partial differential equation characterizing the dynamics of a cable with varying-length is deduced, and converted into ordinary differential equations through spatial discretization by finite difference approximation. Then, the dynamic model for cable-driven parallel manipulators is achieved considering the relationship between the motion of the end-effector and the cable end force, in which the degrees of freedom of cables and the end-effector are all involved. Two numerical examples are demonstrated to validate the dynamic model, and also show that it is necessary to take into consideration the cable dynamics for manipulators of long-span cables.     

The Behavioral Approach to Systems and Modeling

Passive and active limited-slip differentials are used in high-performance cars to optimize the torque distribution on the driving wheels for traction maximization, driving comfort, stability and active safety of the vehicle. In this paper, detailed and reduced dynamic models for the simulation of four kinds of differential are presented. The models refer to the limited-slip steering differential with two clutches. The model of the conventional differential, of the mechanical limited-slip differential and of the controlled limited-slip differential can be obtained by simplification. The detailed model allows the simulation of the internal phenomena that influence the differential dynamics. The reduced model focuses only on the main dynamic behaviour of the differential. Some simulations show the use of the reduced model to compare the effects of the four differentials on the vehicle dynamics.     

The numerical simulation of a typical fluid-structure interaction problem

To determine the dynamic response of a structure under the influence of the fluid flow one must solve a coupled computational fluid dynamics (CFD) and computational structural dynamics (CSD) mathematical problem. This paper presents the comparison of two methods for the calculation of the fluid-structure interaction. The first one is of explicit-implicit type and uses a staggered time advancement of the fluid and structure problems. The second uses a fully implicit discretization in the physical time of the fluid-structure equations and an explicit advancement in the dual-time. The physical fluid-structure problem is accompanied by the equations of the mesh motion, which are written as for a pseudo-structural system with its own dynamics. Representative numerical results are presented for the two degrees of freedom tipical section in unsteady transonic flow. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A novel dynamics model for railway ballastless track with medium-thick slabs

Motivated by the requirements for elaborated slab ballastless track dynamics analysis in practical engineering application, a novel dynamic model for the railway ballastless tracks with medium-thick slabs is proposed in this work based on the Reissner–Mindlin plate theory, and it is implemented into the coupled dynamics analysis of a vehicle and the ballastless track. First, an efficient and easily programmable computational algorithm is adopted to solve the transverse deflection of the Reissner–Mindlin plate, in which the displacements and shear strains are chosen as the independent variables and subsequently constructed by spline functions, resulting in no shear-locking effect. The involved partial differential equations are transformed into ordinary ones by using the energy variation principle. Further, a mathematical model for the ballastless track dynamics analysis is established, which can consider the effects of the shear deformation and moment of inertia involved in the medium-thick track slab. Experimental verification and comparative analysis with other models demonstrate the accuracy and efficiency of the proposed model. Finally, a spatially coupled dynamics model of a vehicle and the ballastless track is developed, and it is efficiently solved by using the hybrid explicit-implicit time integration method. Compared with the widely used modelling the track slab by elastic thin plate, the reliability and advantages of the proposed vehicle-slab track coupled dynamics model are demonstrated.     

A complete analysis of axial piston pump leakage and output flow ripples

The paper is focused on understanding the flow losses and the resulting flow/pressure dynamics in a piston pump. Initially, equations to evaluate leakages in all piston pump gaps will be presented and tested against numerical models, later the equations will be linked to determine the general pressure/flow pump dynamic characteristics. The model will also provide the temporal pressure in each piston/cylinder chamber and the temporal leakage in all pump clearances. A test rig able to measure the dynamic pressure inside a piston chamber was build and employed to evaluate pressure ripple dynamics as a function of turning speed, outlet pressure and swash plate angle. The comparison between experimental and simulated results is very good, giving confidence to the model presented. The advantage of using the analytical approach is that explicit equations allow a more direct understanding of the effect of dimension changes and operating conditions on pump dynamics. Fluid used hydraulic oil ISO 32.     

Partial difference equation based model reference control of a multiagent network of underactuated aquatic vehicles with strongly nonlinear dynamics

In a recent work, the authors presented an extension of robust model reference adaptive control (MRAC) laws for spatially varying partial differential equations (PDEs) proposed by them earlier for the decentralized adaptive control of heterogeneous multiagent networks with agent parameter uncertainty using the partial difference equations (PdEs) on graphs framework. The examples provided demonstrated the capabilities of this approach under the assumption that individual vehicles executing coordinated maneuvers were fully actuated and characterized by linear dynamics. However, detailed models for autonomous vehicles–whether terrestrial, aerial, or aquatic–are often underactuated and strongly nonlinear. Using this approach, but assuming the plant parameters to be known, this work presents the model reference (MR) control laws without adaptation for the coordination of underactuated aquatic vehicles modeled individually in terms of strongly nonlinear dynamic equations arising from ideal planar hydrodynamics. The case of unknown plant parameters for this class of underactuated agents with complex dynamics is an open problem. The paper is based on an invited talk on adaptive control presented at the 2008 World Congress of Nonlinear Analysts.     

Transversal vibration analysis of the upper span of nonlinear closed-loop track systems

In order to analyze the effect of track pre-tension on the transversal vibration of the upper track, a multibody dynamics model of a tracked vehicle needs to be established. In view of the complex structure of tracked vehicles, the computational efficiency of conventional methods cannot meet engineering design requirements. In this paper, the Riccati Transfer Matrix Method for nonlinear closed-loop multibody systems is proposed and applied to the modeling and simulation of tracked vehicle systems. It avoids the overall dynamic equations of the system and achieves efficient computation with a smaller matrix scale. By selecting accelerations and forces as elements of the state vectors instead of coordinates and forces, linearization is avoided. The track chain is cut in a revolute joint. Its effect on the closed-loop system is represented by a set of constraint equations and internal constraint forces. An improved Riccati transformation for closed-loop systems is proposed to connect the internal force and acceleration parts in the state vector with the system constraint internal forces. The track pre-tension can be adjusted by moving the position of the idler relative to the chassis. The root mean square of vertical acceleration of mid-span with respect to the chassis is selected to evaluate the transversal vibration of the upper track. Simulation of a tracked vehicle driving on different road conditions is carried out with different idler displacements. Finally, a reasonable adjustment strategy for the track tensioner is proposed corresponding to different running conditions.     

Dynamic modeling of a Stewart platform using the generalized momentum approach

Dynamic modeling of parallel manipulators presents an inherent complexity, mainly due to system closed-loop structure and kinematic constraints.In this paper, an approach based on the manipulator generalized momentum is explored and applied to the dynamic modeling of a Stewart platform. The generalized momentum is used to compute the kinetic component of the generalized force acting on each manipulator rigid body. Analytic expressions for the rigid bodies inertia and Coriolis and centripetal terms matrices are obtained, which can be added, as they are expressed in the same frame. Gravitational part of the generalized force is obtained using the manipulator potential energy. The computational load of the dynamic model is evaluated, measured by the number of arithmetic operations involved in the computation of the inertia and Coriolis and centripetal terms matrices. It is shown the model obtained using the proposed approach presents a low computational load. This could be an important advantage if fast simulation or model-based real-time control are envisaged.     

Dynamics and stability of a hybrid serial-parallel mobile robot

Kinematics, dynamics, and stability analysis of a hybrid serial-parallel wheeled mobile robot is detailed in this paper. Privileging the advantages of both serial and parallel robots, the suggested structure will provide higher stability for heavy object manipulation by a mobile robotic system. The proposed system is made of a differentially-driven wheeled platform, a planar parallel manipulator, which is called here as star-triangle (ST) mechanism, and a serial Puma-type manipulator arm. In order to develop a comprehensive kinematics model of the robot; first it is divided into three modules, i.e. a mobile platform, a parallel ST mechanism, and a serial robot. Next, a closed-form dynamics model is derived for the whole hybrid system based on a combined Newton–Euler and Lagrange formulation. Then, a careful validation procedure is presented to verify the obtained dynamics model. Finally, using the new postural stability metric named as moment-height stability (MHS), the important role of the parallel ST mechanism for stabilizing the mobile robotic system is demonstrated. The obtained results show that the proposed hybrid serial–parallel arrangement effectively enhances the tip-over stability of the overall mobile robotic system. Hence, it can be successfully exploited to prevent tip-over instability particularly during heavy object manipulation tasks.