Finite Motion Research Articles

Explicit isogeometric collocation for the dynamics of three-dimensional beams undergoing finite motions

We initiate the study of three-dimensional shear-deformable geometrically exact beam dynamics through explicit isogeometric collocation methods. The formulation we propose is based on a natural combination of the chosen finite rotations representation with an explicit, geometrically consistent Lie group time integrator. We focus on extending the integration scheme, originally proposed for rigid body dynamics, to our nonlinear initial–boundary value problem, where special attention is required by Neumann boundary conditions. The overall formulation is simple and only relies on a geometrically consistent procedure to compute the internal forces once control angular and linear accelerations of the beam cross sections are obtained from the previous time step. The capabilities of the method are shown through numerical applications involving very large displacements and rotations and different boundary conditions.

A formulation for modeling levitation based vibration energy harvesters undergoing finite motion

This paper presents a geometrically exact formulation for modeling electromagnetic vibration energy harvesters undergoing finite displacements and rotations. The formulation is conceived to model uniaxial harvesters that extract energy from the vibration of flexible multibody systems that are moving arbitrarily in space. The finite rotation framework is based on an updated Lagrangian procedure; the geometrically exact dynamic equilibrium equations of the harvester are written in terms of local director vectors using a material description. The resulting coupled electro-mechanical equations of motion preserve its classical one-dimensional form. Numerical tests are conducted to address the response of the harvester under specific finite displacements-finite rotations excitations.

Tetrahedron formation of nanosatellites with single-input control

The present paper studies the formation flight of four nanosatellites forming a tetrahedron. The main goal of this research is to find the relative orbits of these satellites that, at least in the linear Hill–Clohessy–Wiltshire model, ensure finite relative motion and keep the volume and shape of the tetrahedron configuration. Since real motions of these satellites will differ from the linear ones, especially under the influence of the $J_{2}$ perturbation, active control is necessary. In addition, the limited size of the satellites does not allow us to use a complex 3-axis attitude control system. In the present paper we consider the passive magnetic attitude control system and suppose that the thrust direction is always aligned with the local geomagnetic field. In order to increase mission lifetime the control algorithm that minimizes the propellant consumption and keeps the tetrahedron volume and shape is investigated.

Type synthesis of 1T2R parallel mechanisms with parasitic motions

Finite motion analysis and type synthesis of parallel mechanisms with parasitic motions are great challenges in the fields of mechanisms and robotics. This paper proposes a finite screw based approach to handle this problem by taking one translational and two rotational (1T2R) parallel mechanisms with parasitic motions as examples. Firstly, 1T2R finite motion of the moving platform is categorized into four cases whose finite motions are described by finite screw analytically and then the reasons for producing parasitic motions are explored in this process. Based on this, available limbs are obtained algebraically by deriving formulas formed by triangle screw product. Finally, the assembly conditions among limbs, fixed base and moving platform are acquired in an analytical manner, leading to synthesize 1T2R parallel mechanisms with parasitic motions systematically. This paper solves a long-term problem that describes and analyzes finite motions of parallel mechanisms with parasitic motions and synthesizes their topology structures.

Study on Finite Element Analysis of Breeze Vibration of Transmission Lines Based on Three-Node Cable Elements

In order to study the mechanism of the breeze vibration of transmission lines, based on the non-linear model of Scanlan's experience, the finite element motion equation of the transmission line's breeze vibration is derived, and the dynamic solution is carried out. The results show that the parameters of Scanlan vortex force have a significant influence on the stable amplitude of the breeze vibration of transmission lines. Due to the neglect of the bending stiffness of the cable elements, the finite element calculation results of the cable breeze vibration are slightly less than the energy balance method, and as the wind speed increases, the error increases gradually; when the wind speed is 3m/s, the frequency of the wire vortex-induced vibration is 30Hz, which is consistent with the calculation of the Stokehal number.

Elimination of Nonstationary Oscillations of an Elastic System at the Stopping Time after Finite Rotation by the Given Law via the Tuning of Eigenfrequencies

The article deals with an arbitrary elastic 3D-system (body) that performs a controlled finite rotation with respect to some fixed axis and small nonstationary oscillations. The system oscillations occur due to external load (power control) or inertial load of the rotational transportation of the carrying body (kinematic control). The linear equations of oscillations are used in normal coordinates, in which motion is represented by eigenmodes of vibrations for system that is free in the rotation angle (including system rotation as a solid body in the case of power control) and for system fixed in rotation angle in the case of kinematic control. It is assumed that the (power or inertial)load acting on the system is proportional to some controlling finite time function from a certain class. The purpose of this article is to solve the problem of system rotation for a certain time from one rest position to another at a given finite angle using the given control function and to eliminate the elastic oscillations on the several lowest eigenmodes at the stopping time. The relations between the time of the system rotation under the action of a given control function and the eigenmodes frequencies for oscillations being eliminated are obtained on the basis of the exact solutions of the equations in normal coordinates. These relations satisfy the zero initial and final conditions. They are “tuned” by minimizing the positive definite quadratic form written for them by varying the system parameters to fulfill these relations simultaneously for several eigenfrequencies. As an example, the calculations for a model of a symmetrical spacecraft with two identical elastic solar cell panels consisting of four planar non-deformable sections connected by elastic hinges are carried out for comparison and analysis of the results accuracy. The finite rolling motion of the system with the damping at the stopping time of rotation for several (from one to three) lowest eigenmodes of antisymmetric vibrations is considered. The comparisons of the initial equations of motion for the system in generalized coordinates using several simple control functions and the found parameters of the “tuned” system with numerical solutions are accomplished.

Development and application of CATIA-based interference simulation system for tube bending

Aiming at the urgent demands for precise and intelligent forming of complex bent tubular components, this paper reviews the developments of tube bending related simulation technologies. Combined with the advantages of finite element (FE) simulation analysis and geometry motion simulation analysis, the present work is proposed to establish a knowledge base of tube bending simulation based on the knowledge engineering technology, and to develop an interference simulation system of tube bending based on CATIA software. Through the application of interference simulation system in continuous tube multi-bending in civil aircrafts, the extraction of model data, selection of process parameters, FE simulation of single bending segment, geometry simulation of continuous multi-bending, generation of simulation reports and numerical control codes can be automatically realized, and the bending quality and interference situation can be simultaneously obtained. Using the interference simulation system, the product efficiency of aircraft duct components can be greatly improved, and the labour cost and working intensity of technicians can be significantly reduced.

Higher-order Taylor approximation of finite motions in mechanisms

SUMMARYHigher-order derivatives of kinematic mappings give insight into the motion characteristics of complex mechanisms. Screw theory and its associated Lie group theory have been used to find these derivatives of loop closure equations up to an arbitrary order. In this paper, this is extended to the higher-order derivatives of the solution to these loop closure equations to provide an approximation of the finite motion of serial and parallel mechanisms. This recursive algorithm, consisting solely of matrix operations, relies on a simplified representation of the higher-order derivatives of open chains. The method is applied to a serial, a multi-DOF parallel, and an overconstrained mechanism. In all cases, adequate approximation is obtained over a large portion of the workspace.

Towards real-time simulation of deformable structures by means of co-rotational finite element formulation

Whereas typical Finite Element (FE) computations are performed off-line, many virtual-reality (VR) applications put a demand for interactive simulations involving deformable objects. Interactive simulation implies real-time or nearly real-time computation and graphical representation of modeled deformable objects. The growing computational power of modern conventional hardware calls for FEM developments in this direction. Depending on specific VR applications, the developments need to account for different aspects of physical behavior, with geometrically nonlinear deformations emerging as one of most important and frequent. This paper proposes a simplified co-rotational FE formulation that considers the overall finite element motion as a superposition of rigid-body rotation and deformation described by a linear model with respect to the co-rotated reference frame. By neglecting the stress stiffening effect and the dependence of the element stiffness matrix on the deformational displacements, the formulation aims at meeting the objectives of highly efficient simulation, under certain conditions even real-time simulation, and with acceptable deviation in accuracy compared to the rigorous nonlinear FE formulation. Computations with both solid and shell elements are addressed. A set of examples is provided to illustrate and discuss the aspects of accuracy and achievable simulation speed.

Compliant four-bar linkage synthesis with second-order flexure hinge approximation

The paper discusses an original methodology for addressing the synthesis of a compliant four-bar mechanism for prescribed finite coupler motion. The proposed method is based on the modelling of the four flexure hinges by using a planetary arrangement (two external cylinders rolling without slipping one on the other) able to accurately reproduce the relative motion between connected parts for a large range of deformation. The adopted approach allows a trade-off between accuracy and limited complexity in deducing the closure loop equations of the whole mechanism. The methodology is applied on a test case of a four-bar compliant linkage whose coupler occupies a prescribed set of finitely separated positions. The synthesized linkage has been analysed both with a pseudo-rigid model (built with mid-sized hinges) and a nonlinear finite element model.

A Finite and Instantaneous Screw Based Approach for Topology Design and Kinematic Analysis of 5-Axis Parallel Kinematic Machines

Unifying the models for topology design and kinematic analysis has long been a desire for the research of parallel kinematic machines (PKMs). This requires that analytical description, formulation and operation for both finite and instantaneous motions are performed by the same mathematical tool. Based upon finite and instantaneous screw theory, a unified and systematic approach for topology design and kinematic analysis of PKMs is proposed in this paper. Using the derivative mapping between finite and instantaneous screws built in the authors’ previous work, the finite and instantaneous motions of PKMs are analytically described by the simple and non-redundant screws in quasi-vector and vector forms. And topological and parametric models of PKMs are algebraically formulated and related. These related topological and parametric models are ready to do type synthesis and kinematic analysis of PKMs under the unified framework of screw theory. In order to show the validity of the proposed approach, a kind of two-translational and three-rotational (2T3R) 5-axis PKMs is taken as example. Numerous new structures of the 2T3R PKMs are synthesized as the results of topology design, and their Jacobian matrix is obtained easily for parameter optimization and performance evaluation. Some of the synthesized PKMs have outstanding capabilities in terms of large workspaces and flexible orientations, and have great potential for industrial applications of machining and manufacture. Among them, METROM PKM is a typical example which has attracted a lot of attention from global companies and already been developed as commercial products. The approach is a general and unified approach that can be used in the innovative design of different kinds of PKMs.

A Multi-Time-Scale Finite Time Controller for the Quadrotor UAVs with Uncertainties

A control method with a multi-time-scale structure is proposed to perform finite time motion control of the quadrotor unmanned aerial vehicles (UAVs) with uncertainties. In order to facilitate the controller-design, finite time extended state observer (ESO) in the first and second time scales is applied to estimate the system uncertainties; the attitude controllers and the height controller are designed for finite time stabilization of the equilibriums in the third time scale; and finally, the output feedback technique is utilized to design the horizontal auxiliary controllers, meanwhile the reference angles are settled for the attitude dynamics in the fourth and slowest time scale. It allows us to analyze the system dynamics in each time scale independently, and conclusions on finite time stabilization are achieved gradually with a large region of attraction. Simulation and experimental results on the Qball2 platform are given to verify the efficacy of the strategy and establish the feasibility of implementation.

Geometric Error Modeling of Parallel Manipulators Based on Conformal Geometric Algebra

An approach for geometric error modeling of parallel manipulators (PMs) based on the visual representation and direct calculation of conformal geometric algebra is introduced in this paper. In this method, the finite motion of an open-loop chain is firstly formulated. Through linearization of the finite motion, error propagation of the open-loop chain is analyzed. Then the error sources are separated in terms of joint perturbations and geometric errors. Next, motions and constraints of PMs are analyzed visually by their reciprocal property. Finally geometric error model of PMs are formulated considering the actuations and constraints. The merits of this new approach are twofold: (1) complete and continuous geometric error modeling can be achieved since finite motions are considered, (2) visual and analytical computation of motions and constraints are applied for transferring geometric errors from the open-loop chain to the PM. A 2-DoF rotational PM is applied to demonstrate the geometric error modeling process. Comparisons between simulation and analytical models show that this approach is highly effective.

Finite Motion Analysis of Parallel Mechanisms with Parasitic Motions Based on Conformal Geometric Algebra

Finite motion analysis of parallel mechanisms (PMs) denotes formulating the map between finite motion of end-effector and those of its component limbs. By employing conformal geometric algebra (CGA), this paper presents an analytical and accurate method to analyze the finite motions of PMs with parasitic motions. Herein, parasitic motions are defined as the dependent motions in the constraint Degrees-of-Freedom (DoFs) of PMs. Firstly, description of rigid body transformations based on CGA is reviewed. Then, the intersection algorithm of finite motions is introduced by exploiting the algebraic properties of CGA. Based on this, a method to formulate the finite motions of PMs with parasitic motions is proposed. Finally, Z3 mechanism is sketched as example by utilizing the approach. This method facilitates the invention of new mechanisms and can also be applied in the finite motion analysis of other kinds of PMs.

Fault-tolerant feature-based estimation of space debris rotational motion during active removal missions

One of the key functionalities required by an Active Debris Removal mission is the assessment of the target kinematics and inertial properties. Passive sensors, such as stereo cameras, are often included in the onboard instrumentation of a chaser spacecraft for capturing sequential photographs and for tracking features of the target surface. A plenty of methods, based on Kalman filtering, are available for the estimation of the target's state from feature positions; however, to guarantee the filter convergence, they typically require continuity of measurements and the capability of tracking a fixed set of pre-defined features of the object. These requirements clash with the actual tracking conditions: failures in feature detection often occur and the assumption of having some a-priori knowledge about the shape of the target could be restrictive in certain cases. The aim of the presented work is to propose a fault-tolerant alternative method for estimating the angular velocity and the relative magnitudes of the principal moments of inertia of the target. Raw data regarding the positions of the tracked features are processed to evaluate corrupted values of a 3-dimentional parameter which entirely describes the finite screw motion of the debris and which primarily is invariant on the particular set of considered features of the object. Missing values of the parameter are completely restored exploiting the typical periodicity of the rotational motion of an uncontrolled satellite: compressed sensing techniques, typically adopted for recovering images or for prognostic applications, are herein used in a completely original fashion for retrieving a kinematic signal that appears sparse in the frequency domain. Due to its invariance about the features, no assumptions are needed about the target's shape and continuity of the tracking. The obtained signal is useful for the indirect evaluation of an attitude signal that feeds an unscented Kalman filter for the estimation of the global rotational state of the target. The results of the computer simulations showed a good robustness of the method and its potential applicability for general motion conditions of the target.

Tex3: An 2R1T Parallel Manipulator with Minimum DOF of Joints and Fixed Linear Actuators

A new 2R1T (R: rotation, T: translation) parallel manipulator (PM), called Tex3, is proposed. The Tex3 PM is a 2PUR-PRU PM consisting of 9 joints with 12 DOFs (degrees of freedom), and can be actuated by fixed linear actuators. Mobility analysis shows that it is an RPR-equivalent PM whose finite motion is the product of a rotation (R) followed by a translation (P) and another rotation (R). Inverse position and velocity analyses are discussed. Furthermore, the local transmission index and good transmission workspace are used to evaluate the motion/force transmissibility of the proposed PM. The singularity loci of the proposed PM are obtained according to the motion/force transmissibility, corresponding to the inverse, forward and combined kinematic singularity. The constraint singularity is also discussed. Finally, link parameters are optimized to obtain an improved good transmission workspace. Because of its minimum DOF of joints and fixed linear actuators, the Tex3 PM has good kinematic performance and is suited to the high-speed machining of workpieces with curved surfaces.

Large deviations for small noise diffusions in a fast markovian environment

A large deviation principle is established for a two-scale stochastic system in which the slow component is a continuous process given by a small noise finite dimensional Ito stochastic differential equation, and the fast component is a finite state pure jump process. Previous works have considered settings where the coupling between the components is weak in a certain sense. In the current work we study a fully coupled system in which the drift and diffusion coefficient of the slow component and the jump intensity function and jump distribution of the fast process depend on the states of both components. In addition, the diffusion can be degenerate. Our proofs use certain stochastic control representations for expectations of exponential functionals of finite dimensional Brownian motions and Poisson random measures together with weak convergence arguments. A key challenge is in the proof of the large deviation lower bound where, due to the interplay between the degeneracy of the diffusion and the full dependence of the coefficients on the two components, the associated local rate function has poor regularity properties.

Nonlinear resonances in the ABC-flow.

In this paper, we study resonances of the ABC-flow in the near integrable case ( C≪1). This is an interesting example of a Hamiltonian system with 3/2 degrees of freedom in which simultaneous existence of two resonances of the same order is possible. Analytical conditions of the resonance existence are received. It is shown numerically that the largest n:1 (n = 1, 2, 3) resonances exist, and their energies are equal to theoretical energies in the near integrable case. We provide analytical and numerical evidences for existence of two branches of the two largest n:1 (n = 1, 2) resonances in the region of finite motion.

A nonlinear theory for fibre-reinforced magneto-elastic rods

We derive a model for the finite motion of a fibre-reinforced magneto-elastic rod. The reinforcing particles are assumed weakly and uniformly magnetized, rigid and firmly embedded into the elastomeric matrix. We deduce closed-form expressions of the quasi-static motion of the rod in terms of the external magnetic field and of the body forces. The dependences of the motion on the shape of the inclusions, their orientation, their anisotropic magnetic properties and the Young modulus of the matrix are analysed and discussed. Two case studies are presented, in which the rod is used as an actuator suspended in a cantilever configuration. This work can foster new applications in the field of soft-actuators.

Type synthesis of 1T2R and 2R1T parallel mechanisms employing conformal geometric algebra

Abstracts This paper proposes an analytical method based on conformal geometrical algebra to implement and unify type synthesis of one translational and two rotational (1T2R) parallel mechanisms and two rotational and one translational (2R1T) parallel mechanisms. Firstly, conformal geometrical algebra is introduced in this paper to describe finite motions and carry out type synthesis of parallel mechanisms in an algebraic way. Secondly, the finite motions of 1T2R and 2R1T parallel mechanisms are described and their relations are explored, and this results in unifying the type synthesis of these two types of parallel mechanisms in the same procedure. Then, a parametric and algebraic generation method is proposed to obtain available limbs and axes layout of joints with these limbs. Finally, the assembly conditions among available limbs are given to synthesize systematically 2R1T parallel mechanisms with non-overconstrained or overconstrained property, as well as 1T2R parallel mechanisms with the same properties. This method proposed in this paper is valid in carrying out finite motion representation and type synthesis of parallel mechanisms in an algebraic and analytical way, and then is beneficial to an automatic manner using computer programming languages. Furthermore, the method may solve type synthesis of special parallel mechanisms, such as parallel mechanisms with varied output rotational axes.