Dynamic Modeling and Experimental Validation of a 3-PRR Parallel Manipulator with Flexible Intermediate Links

This paper presents the development of structural dynamic equations of motion for a 3-PRR parallel manipulator with three flexible intermediate links, based on the assumed mode method. Lagrange’s equation is used to derive the dynamic model of the manipulator system. Flexible intermediate links are modeled as Euler–Bernoulli beams with pinned–pinned boundary conditions. Dynamic equations of motion of a 3-PRR parallel manipulator with three flexible links are developed by adopting the assumed mode method. The effect of concentrated rotational inertia at both ends of intermediate links is included in this model. Numerical simulations of vibration responses, coupling forces and inertial forces are presented. The corresponding frequency spectra analysis is performed using the Fast Fourier Transform (FFT). Experimental modal tests are performed to validate the theoretical model through comparison and analysis of modal characteristics of the flexible manipulator system.     

Coupling characteristics of rigid body motion and elastic deformation of a 3-PRR parallel manipulator with flexible links

Modeling of multibody dynamics with flexible links is a challenging task, which not only involves the effect of rigid body motion on elastic deformations, but also includes the influence of elastic deformations on rigid body motion. This paper presents coupling characteristics of rigid body motions and elastic motions of a 3-PRR parallel manipulator with three flexible intermediate links. The intermediate links are modeled as Euler–Bernoulli beams with pinned-pinned boundary conditions based on the assumed mode method (AMM). Using Lagrange multipliers, the fully coupled equations of motions of the flexible parallel manipulator are developed by incorporating the rigid body motions with elastic motions. The mutual dependence of elastic deformations and rigid body motions are investigated from the analysis of the derived equations of motion. Open-loop simulation without joint motion controls and closed-loop simulation with joint motion controls are performed to illustrate the effect of elastic motion on rigid body motions and the coupling effect amongst flexible links. These analyses and results provide valuable insight to the design and control of the parallel manipulator with flexible intermediate links.     

A recursive,numerically stable,and efficient simulation algorithm for serial robots

Traditionally, the dynamic model, i.e., the equations of motion, of a robotic system is derived from Euler–Lagrange (EL) or Newton–Euler (NE) equations. The EL equations begin with a set of generally independent generalized coordinates, whereas the NE equations are based on the Cartesian coordinates. The NE equations consider various forces and moments on the free body diagram of each link of the robotic system at hand, and, hence, require the calculation of the constrained forces and moments that eventually do not participate in the motion of the coupled system. Hence, the principle of elimination of constraint forces has been proposed in the literature. One such methodology is based on the Decoupled Natural Orthogonal Complement (DeNOC) matrices, reported elsewhere. It is shown in this paper that one can also begin with the EL equations of motion based on the kinetic and potential energies of the system, and use the DeNOC matrices to obtain the independent equations of motion. The advantage of the proposed approach is that a computationally more efficient forward dynamics algorithm for the serial robots having slender rods is obtained, which is numerically stable. The typical six-degree-of-freedom PUMA robot is considered here to illustrate the advantages of the proposed algorithm.     

Dual dynamics: Designing behavior systems for autonomous robots

This paper describes the “dual dynamics” (DD) design scheme for robotic behavior control systems. Behaviors are formally specified as dynamical systems using differential equations. A key idea for the DD scheme is that a robotic agent can work in different “modes,” which lead to qualitatively different behavioral patterns. Mathematically, transitions between modes are bifurcations in the control system. This work was presented, in part, at the Second International Symposium on Artificial Life and Robotics, Oita, Japan, February 18–20, 1997     

Buckling, flutter and vibration analyses of beams by integral equation formulations

This paper presents a model for the investigation of buckling, flutter and vibration analyses of beams using the integral equation formulation. A mathematical formulation based on Euler–Bernoulli beam theory is presented for beams with variable sections on elastic foundations and subjected to lateral excitation, conservative and non-conservative loads. Using the boundary element method and radial basis functions, the equation of motion is reduced to an algebraic system related to internal and boundary unknowns. Eigenvalue problems related to buckling and vibrations are formulated and numerically solved. Buckling loads, natural frequencies and associated eigenmodes are computed. The corresponding slope, bending and shear forces can be directly obtained. The load-frequency dependence is investigated for various elastic foundations and the divergence critical loads are evidenced. Under non-conservative loads, a dynamic stability analysis is illustrated numerically based on the coalescence of eigenfrequencies. The flutter load and instability regions with respect to the elastic concentrated and distributed foundations are identified. Using the eigenmodes, numerically computed, non-linear vibrations of beams are investigated based on one mode analysis. The presented model is quite general and the obtained numerical results are in agreement with available data.     

Decay Rates for a Beam with Pointwise Force and Moment Feedback

We consider the Rayleigh beam equation and the Euler–Bernoulli beam equation with pointwise feedback shear force and bending moment at the position ξ in a bounded domain (0,π) with certain boundary conditions. The energy decay rate in both cases is investigated. In the case of the Rayleigh beam, we show that the decay rate is exponential if and only if ξ/π is a rational number with coprime factorization ξ/π=p/q, where q is odd. Moreover, for any other location of the actuator we give explicit polynomial decay estimates valid for regular initial data. In the case of the Euler–Bernoulli beam, even for a nonhomogeneous material, exponential decay of the energy is proved, independently of the position of the actuator. Date received: October 30, 2000. Date revised: December 20, 2001.     

Reduced Stochastic Models for Complex Molecular Systems

We present a new numerical method for the identification of the most important metastable states of a system with complicated dynamical behavior from time series information. The approach is based on the representation of the effective dynamics of the full system by a Markov jump process between metastable states, and the dynamics within each of these metastable states by rather simple stochastic differential equations (SDEs). Its algorithmic realization exploits the concept of hidden Markov models (HMMs) with output behavior given by SDEs. A first complete algorithm including an explicit Euler–Maruyama-based likelihood estimator has already been presented in Horenko et al. (MMS, 2006a). Herein, we present a semi-implicit exponential estimator that, in contrast to the Euler–Maruyama-based estimator, also allows for reliable parameter optimization for time series where the time steps between single observations are large. The performance of the resulting method is demonstrated for some generic examples, in detail compared to the Euler–Maruyama-based estimator, and finally applied to time series originating from a 100 ns B-DNA molecular dynamics simulation.Dedicated to Peter Deuflhard on the occassion of his sixtieth birthday.Supported by the SfB 450 and DFG research center “Mathematics for key technologies” (FZT 86) in Berlin.     

Analysis of Large Flexible Body Deformation in Multibody Systems Using Absolute Coordinates

To consider large deformation problems in multibody system simulations afinite element approach, called absolute nodal coordinate.formulation,has been proposed. In this formulation absolute nodal coordinates andtheir material derivatives are applied to represent both deformation andrigid body motion. The choice of nodal variables allows a fullynonlinear representation of rigid body motion and can provide the exactrigid body inertia in the case of large rotations. The methodology isespecially suited for but not limited to modeling of beams, cables andshells in multibody dynamics.This paper summarizes the absolute nodal coordinate formulation for a 3D Euler–Bernoulli beam model, in particular the definition of nodal variables, corresponding generalized elastic and inertia forces and equations of motion. The element stiffness matrix is a nonlinear function of the nodal variables even in the case of linearized strain/displacement relations. Nonlinear strain/displacement relations can be calculated from the global displacements using quadrature formulae.Computational examples are given which demonstrate the capabilities of the applied methodology. Consequences of the choice of shape.functions on the representation of internal forces are discussed. Linearized strain/displacement modeling is compared to the nonlinear approach and significant advantages of the latter, when using the absolute nodal coordinate formulation, are outlined.     

Using singular extremals to obtain new equations of motion and unknown constants

The paper uses the concept of a singular extremal in the classical dimensional analysis, which considerably extends the capabilities of this analysis and allows finding the unknown constants and new motion equations. The study was carried out in line with the program of the Russian Academy of Sciences “The fundamentals of information technologies and systems,” Project No. 1–3. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 115–124, July–August 2009.     

Dynamic features of flexible spacecraft control in process of its transformation into a large space structure

Consideration was given to dynamics of angular motion control of the flexible spacecraft reconstructed into a large space structure. In formal terms, this transformation lies in gradual reduction of the constructive rigidity to small values giving rise to low-frequency ( $ \tilde f Consideration was given to dynamics of angular motion control of the flexible spacecraft reconstructed into a large space structure. In formal terms, this transformation lies in gradual reduction of the constructive rigidity to small values giving rise to low-frequency ( ≤ 0.05 Hz) oscillations which represent one of the attributes of the class of large space structures. The existing quantitative definition of the large space structure was specified. It was demonstrated that as the frequencies of structure’s elastic oscillations approach those of the control of object “rigid” motion, a new kind of interrelations between the motions of both types, the so-called “capture” of the controller frequency by that of the elastic oscillations, arises which impairs control efficiency to the point of losing system stability. Analytical (for the linear control systems) and computer-aided (for the discrete systems) methods for determination of the boundaries separating the two qualitatively different forms of existence of the transformed elastic object were proposed. Some results of computer simulation of the orientation control of variable objects such as flexible spacecraft and large space structure were presented. Original Russian Text ? I.N. Krutova, V.M. Sukhanov, 2008, published in Avtomatika i Telemekhanika, 2008, No. 5, pp. 41–56. This work was supported by the Russian Foundation for Basic Research, project no. 05-08-18175.     

Nonlinear vibration of nanobeam with attached mass at the free end via nonlocal elasticity theory

In the present study, nonlinear free and forced vibration of Euler–Bernoulli nanobeam with attached nanoparticle at the free end is investigated based on nonlocal elasticity theory. The effects of the different nonlocal parameters (γ) and mass ratios (α) as well as effects of fixed-free boundary conditions on the vibrations are determined. To obtain the equation of motion of the system, the Hamilton’s principle is employed. The stretching of neutral axis which introduces cubic nonlinearity is included into the equation for deriving nonlinear equation. And also effects of damping and forcing are included into the equations. The approximate solutions of the equations are derived by using the multiple scale method. Fundamental frequencies, frequency shift and mode shapes for the linear problem are estimated for a nonlocal Euler–Bernoulli nanobeam with an attached nanoparticle and graphically represented the frequency shift and mode shapes. Nonlinear frequencies are derived depending on amplitude and phase modulation. Frequency–response curves are drawn for different nonlocal parameters and different modes.

     

Local exact controllability for Berger plate equation

We study the exact controllability of a nonlinear plate equation by the means of a control which acts on an internal region of the plate. The main result asserts that this system is locally exactly controllable if the associated linear Euler–Bernoulli system is exactly controllable. In particular, for rectangular domains, we obtain that the Berger system is locally exactly controllable in arbitrarily small time and for every open and nonempty control region.     

Humanoid Robotic System with and without Elasticity Elements Walking on an Immobile/Mobile Platform

This work is concerned with the modeling and analysis of a complex humanoid robotic system walking on an immobile/mobile platform. For this purpose, a software package was synthesized which allows one to select configuration of both the humanoid and the platform. Each joint of the biped and platform can be defined by the user via the motor state (active or locked) and gear type (rigid or elastic). The user can also form very diverse configurations of the humanoid and platform. The software package forms a mathematical model. By selecting system’s parameters the simulation allows user to analyze dynamic behavior of the biped of selected configuration, walking on either an immobile or mobile platform of selected configuration. In the moment when the biped steps on the platform, the latter, by its dynamics, acts on the biped dynamics, and the biped on the other hand, by its characteristics, influences dynamics of the platform motion. These two complex contacting systems form a more complex system, whose mathematical model has to encompass all the elements of coupling between the humanoid joints and platform joints. The phenomenon of coupling is analyzed first on a humanoid robotic system with all rigid elements, which is in contact with the platform mechanism having also all rigid elements. It has been shown that coupling is more influenced when elasticity elements are included into the configuration. Insufficient knowledge of coupling characteristics may present a serious disturbance to the system in the robotic task realization. The deviation of the ZMP (Zero-Moment Point) from the reference trajectory is presented, which implies the need for the synthesis of new control structures for stabilizing biped motion on the immobile/mobile platform. The reference trajectory may be defined in very different ways and from several aspects. Reference trajectory of each joint can be defined so to encompass or not encompass elastic deformations. The control structure for the biped walking on the platform should be defined so that it satisfies the requirement for the ZMP to be within the given boundaries in every sampling instant, which guarantees dynamic balance of the locomotion mechanism in the real regime. The control is defined as CR (Centralized Reference control, calculated from the reference state), plus LO (control via local feedbacks of motor motion with respect to position and velocity). In the case of the biped motion on a mobile platform CR control is defined separately under the real conditions of unknown characteristics of coupling between the two complex systems, as well as unknown elasticity properties. The analysis of simulation results of the humanoid robot motion on a mobile platform gives evidence for all the complexity of this system and shows how much system parameters (choice of trajectory, configuration, geometry, elasticity characteristics, motor, etc.) influence stabilization of its humanoid motion.     

Registration of 3D Points Using Geometric Algebra and Tensor Voting

We address the problem of finding the correspondences of two point sets in 3D undergoing a rigid transformation. Using these correspondences the motion between the two sets can be computed to perform registration. Our approach is based on the analysis of the rigid motion equations as expressed in the Geometric Algebra framework. Through this analysis it was apparent that this problem could be cast into a problem of finding a certain 3D plane in a different space that satisfies certain geometric constraints. In order to find this plane in a robust way, the Tensor Voting methodology was used. Unlike other common algorithms for point registration (like the Iterated Closest Points algorithm), ours does not require an initialization, works equally well with small and large transformations, it cannot be trapped in “local minima” and works even in the presence of large amounts of outliers. We also show that this algorithm is easily extended to account for multiple motions and certain non-rigid or elastic transformations.     

Using adaptation methods to stabilize elastic oscillations of large variable-parameter satellites

Consideration was given to providing stability and some performance indices,—in particular, the transient time at reorientation—of the dynamics of large flexible-structure information satellites under a varying spectrum of its natural frequencies. A generalized system structure including the loop of adaptively adjusted additional feedback (AAF) was considered with the aim of providing the desirable system dynamics in the cases where the lower frequency approaches that of control of the “rigid” spacecraft motion. A procedure was presented to design an algorithm for adaptation of the coefficients of the AAF subsystem realizing the desirable dynamics of control of the large elastic satellites. The results of mathematical modeling corroborating efficiency of the proposed system were given.     

Higher order geodesics in Lie groups

For all n > 2, we study nth order generalisations of Riemannian cubics, which are second-order variational curves used for interpolation in semi-Riemannian manifolds M. After finding two scalar constants of motion, one for all M, the other when M is locally symmetric, we take M to be a Lie group G with bi-invariant semi-Riemannian metric. The Euler–Lagrange equation is reduced to a system consisting of a linking equation and an equation in the Lie algebra. A Lax pair form of the second equation is found, as is an additional vector constant of motion, and a duality theory, based on the invariance of the Euler–Lagrange equation under group inversion, is developed. When G is semisimple, these results allow the linking equation to be solved by quadrature using methods of two recent papers; the solution is presented in the case of the rotation group SO(3), which is important in rigid body motion planning.     

Analysis of the properties of a linear system using the method of artificial basis matrices

Formulas relating elements of the method in adjacent basis matrices are used to solve a system of linear algebraic equations and to represent analytically the general solutions to the corresponding system of linear algebraic inequalities for a nondegenerate constraint matrix. Sponsored by the ICS NATO program of April 18 2006, in line with the Project “Optimal replacement of information technologies and stable development (in Kazakhstan, Ukraine, and the USA),” NATO Grant CLG 982209. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 119–127, July–August 2007.     

Finite element modeling of low speed reaming vibrations with reamer geometry modifications

Reaming is a finishing process used to remove a small amount of material from a predrilled hole. In low speed cutting processes, it is the formation of lobed or multi-cornered holes that is of concern, rather than tool chatter, which occurs at high speed near the natural frequency of the tool. Using a quasi-static model in the characteristic form for the reaming process, a finite element modeling for the low speed reaming process, based on the Euler–Bernoulli beam model, was developed. Cutting and rubbing forces were applied as concentrated and distributed forces on a variable engagement length of the reamer. The variable engagement length is considered to simulate the actual applied forces length as the reamer advances to the workpiece. The time dependant changes in the bending stiffness of the reamer were included in the governing equation of the equilibrium of the reamer, and its stability analysis was performed at different time steps. Using this model, the vibration damping effect of uneven spacing of reamer teeth was investigated. The results demonstrate that uneven spacing of reamer teeth reduces the tool vibration, and therefore leads to a more stable condition. Finally, the optimum configuration of uneven tooth pitch angles for a six-flute reamer, in order to have the highest vibration decay rate during the reaming, was presented.     

Study of a stochastic model of the“dangling spider” problem with an infinite previous history and poisson switchings

To study the stability of the stochastic“dangling spider” model, the second Lyapunov method is substantiated for stochastic functional differential equations with the entire previous history. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 79–105, July–August, 2000.