Spatial Kinematic Chains Research Articles

Hypercomplex Quaternions and Higher-Order Analysis of Spatial Kinematic Chains

Abstract This paper introduces a novel computational method for analyzing the higher-order acceleration field of spatial kinematics chains. The method is based on vector and quaternionic calculus, as well as dual and multidual algebra. A closed-form coordinate-free solution generated by the morphism between the Lie group of rigid body displacements and the unit multidual quaternions is presented. Presented solution is used for higher-order kinematics investigation of lower-pair serial chains. Additionally, a general method for studying the vector field of arbitrary higher-order accelerations is discribed. The method utilizes the “automatic differentiation” feature of multidual and hyper-multidual functions to obtain the higher-order derivative of a rigid body pose without need in further differentiation of the body pose regarding time. Also is proved that all information regarding the properties of the distribution of higher-order accelerations is contained in the specified unit hyper-multidual quaternion.

Synthesis and Analysis of a Novel Linkage Mechanism with the Helical Motion of the End-Effector

The proposed study is focused on the development of a novel modification of the slider-crank mechanism, the kinematic scheme of which is organized by coupling planar and spatial (screw) kinematic chains. The structure of this mechanism provides the end-effector with a cyclic movement along a helicoidal trajectory, while all other links have planar motions. A kinematic analysis of the synthesized mechanism is discussed in this study. The motion variables of the end-effector at different pitches in the screw joints are determined analytically and numerically. Based on the mechanism’s kinematic scheme, its assembling computer-aided design (CAD) model has been created, in accordance with which a physical prototype has been built. A functional performance of the prototype is studied by comparing the displacements of its end-effector with analogical displacements of the CAD model. The question of the technological application of the developed mechanism for spring winding and various objects’ screwing has been considered in this study.

FEA-based optimization of a complete structure of a monolithic z/tip/tilt micromanipulator

In micro/nano manipulation mechanisms, the compliant z/tip/tilt stages have proved to have enormous advantages for out-of-plane positioning. Small workspace is a challenge in designing these out-of-plane positioning systems. This deficiency can be overcome significantly by a new insight into the optimization approach of a compliant mechanism that can be implemented easily on the entire structure of any spatial mechanism and/or any spatial mechanism with n-symmetrical spatial kinematic chains. Finite Element Method (FEM) adopted by the engineering software, ANSYS, is utilized to perform the structural optimization of an entire three-way symmetric spatial compliant mechanism to achieve the best performances and overcoming deficiencies. Therefore, this paper introduces a compliant monolithic three degrees of freedom (DOFs) z/tip/tilt piezo-driven micromanipulator with a comparable large workspace. Furthermore, as compact as 201 mm × 180 mm × 75 mm and made of Acrylonitrile Butadiene Styrene (ABS) with the first natural frequency of 115.42 Hz make this design superior in comparison to its peers. Regarding the structural aspects, three Scott-Russell and six leaf parallelogram mechanisms are employed to transform horizontal piezoelectric actuators’ inputs (PEAs) to vertical inputs and transfer them to the stage. The best performances of the proposed structure are achieved by optimizing the mechanism using FEM. Due to the optimization, a large amplification ratio of 5.3 is achieved. Moreover, finite element analyses and analytical modeling are performed to verify the performance of the proposed mechanism. Finally, in a comparative study, the captured workspace and resonant frequency of the proposed micromanipulator are compared against those available in theliterature and provide an insight into the performances of the proposed monolithic mechanism.

A Computational Geometric Approach for Motion Generation of Spatial Linkages With Sphere and Plane Constraints

This paper studies the problem of spatial linkage synthesis for motion generation from the perspective of extracting geometric constraints from a set of specified spatial displacements. In previous work, we have developed a computational geometric framework for integrated type and dimensional synthesis of planar and spherical linkages, the main feature of which is to extract the mechanically realizable geometric constraints from task positions, and thus reduce the motion synthesis problem to that of identifying kinematic dyads and triads associated with the resulting geometric constraints. The proposed approach herein extends this data-driven paradigm to spatial cases, with the focus on acquiring the point-on-a-sphere and point-on-a-plane geometric constraints which are associated with those spatial kinematic chains commonly encountered in spatial mechanism design. Using the theory of kinematic mapping and dual quaternions, we develop a unified version of design equations that represents both types of geometric constraints, and present a simple and efficient algorithm for uncovering them from the given motion.

Kinematic Synthesis of Mechanism for System with a Technical Vision

A solution to the problem of synthesizing an initial three-dimensional kinematic chain with spherical and rotary kinematic pairs is presented. It is shown that this chain can be used as a structural module for structural-kinematic synthesis of three-dimensional four-link motion generating lever mechanisms by the preset positions of the in-and output links. This paper affects the actual today’s problem of optimal synthesis of spatial link mechanisms. In this regard, the task of developing methods for the synthesis of complex spatial link mechanisms with the desired laws of motion of the input and output elements allowing automatizing the implementation of all design phases with the use of computer is quite relevant. The authors develop machine-oriented method of structural and kinematic synthesis of spatial link mechanisms based on the use of spatial initial kinematic chains (IKC) realizing prescribed motions. A new approach to the design of spatial mechanisms is suggested, according to which the design process is based on the kinematic synthesis of four-link initial kinematic chain (IKC) and associable kinematic chains (AKC).

A new approach for obtaining the dynamic balancing conditions in serial mechanisms

Adaptive balancing means that the mechanical structure of the manipulator is modified in order to achieve the decoupling of dynamic equations. This work deals with a systematic formulation for the adaptive balancing. Basically, two traditional balancing techniques are employed here: the addition of counterweight and counter-rotating disks coupled to the moving links. In addition, the feasibility of the dynamic decoupling for three distinct types of serial manipulators is discussed regarding the achievement of such balancing and the complexity level of the modified mechanical structure. The balancing conditions are developed here for 3-dof spatial and planar open-loop kinematic chain mechanisms, whose topologies are composed of revolute and prismatic joints.

Kinematics of Self-Reconfigurable Robotic System

Theory of mechanisms with variable structure in robotic applications self-reconfigurable structures is one of the directions of development of robotics. On our faculty is developed self-reconfigurable modular robotic system capable of rebuilding a structure according to the requirements of the current task. To perform kinematic changes of its structure we needs to have information on where to find the modules and their submodules. Therefore it is necessary to know the mathematical relationship between the positions of its individual parts. Spatial kinematic chains is investigated based on the theory of simultaneous movements, where the position of each member is specified in the global coordinate system using local coordinate systems associated with individual members. To identify the position of the individual local coordinate systems were used transformation matrix, the accuracy of which is verified in mathematical programs and results from these programs are then compared with results of movement analysis notified by simulation procedures.

Modelling, simulation, and experimental verification of the kinematics and dynamics of a double wishbone suspension configuration

This paper describes the non-linear modelling of a double wishbone suspension developed to investigate the non-linear kinematics and dynamics in the closed, spatial kinematic chain configuration. Analytical and ADAMS models are generated and kinematic and dynamic characteristics of the models are investigated. The analytical model of the suspension mechanism is an idealized four degree-of-freedom (DOF) model, with suspension members considered as rigid links and bushings taken as linear spring—damper elements. The simulation results of the model subjected to a virtual kinematics and compliance (K&C) test are compared with results generated by an ADAMS model, developed based on parameters obtained from the vehicle manufacturer, subjected to the same virtual test. The experimental K&C testing on the test vehicle presents a method of capturing the kinematic characteristics of the suspension mechanism. The comparison of the simulation and experimental results presented shows that the models are capable of simulating the characteristics of the pre-existing suspension configuration of the test vehicle.

Kinematic Representations of Pop-Up Paper Mechanisms

Pop-up paper mechanisms use techniques similar to the well-studied paper folding techniques of origami. However, pop-ups differ in both the manner of construction and the target uses, warranting further study. This paper outlines the use of planar and spherical kinematics to model commonly used pop-up paper mechanisms. A survey of common joint types is given, including folds, interlocking slots, bends, pivots, sliders, and rotating sliders. Also included is an overview of common one-piece and layered mechanisms, including single-slit, double-slit, V-fold, tent, tube strap, and arch mechanisms. Each mechanism or joint is shown using both a paper representation and either a rigid-body or pseudo-rigid-body representation. In addition, this paper shows that more complex mechanisms may be created by combining simple mechanisms in various ways. The principles presented are applied to the creation of new pop-up joints and mechanisms. The new mechanisms employ both spherical and spatial kinematic chains. Understanding pop-up mechanism kinematics could lead to new applications in deployable structures, packaging, and instruments for minimally invasive surgery.

Inverse kinematics of variable geometry parallel manipulator

The variable geometry parallel manipulator (VGPM) is a kind of manipulator that is suitable to change the spatial distribution of a variable geometry body (VGB). In the VGPM, the Stewart platform is utilized as a driving mechanism, and a number of spatial RSRR kinematic chains, which connect with the base frame and the top platform through revolute pairs, respectively, guide the rigid plates with spatially shaped inner surfaces to form the envelope of the VGB. The inverse kinematics problem of the VGPM is to determine every actuator displacement of the Stewart platform for given the outlet area and deflection angle of the VGB. In this paper, we present a method to solve this kind of inverse kinematics problem. At first, we present an assumption of the double semi-ellipses approximate distribution model of the VGB to determine the approximate position and orientation of the top platform. Through analysis of the deviations between the approximate distribution model and the given distribution parameters of the VGB, after making the compensation of the deviations, we obtain the corrected inverse kinematic solution of the VGPM with acceptable tolerance.

Using the Skeleton Method to Define a Preliminary Geometrical Model for Three-Dimensional Speed Reducers

This paper deals with the first design phase of three-dimensional speed reducers. Concepts to help designers during the entry-level model definition are presented and, more particularly, those related to the task of geometrical synthesis. From these principles, a specific CAD tool has been built and is also described. A rough model, called a skeleton, is introduced to represent each conventional reducing stage category, thus enabling the automatic formation of entire geometrical models of speed reducers. Shaft orientations and positions are calculated from products of transformation matrices, in the same way as for spatial kinematic chain closure conventional problems. Optimisation techniques are used to obtain a preliminary dimensioning of the structure. The numerical processing is achieved progressively in three steps in order to improve the final convergence and, if necessary, to enlighten further the designer on failure origins within specification data. Three examples are given to illustrate both the creation of the geometrical model and the way results are obtained.

Application of the principle of breaking ocean waves to mixing technology

Turbulent mixing in the multiphase system of breaking ocean waves and whitecaps is known to play an important role in the mass exchange between ocean and atmosphere. Thus engineering applications to this naturally occurring dynamic exchange process appear to be of interest in various areas of applied mixing technology. Starting from the fact that ocean waves break after having reached their point of instability, a three-dimensional collapsible swivel mechanism (CSM) was developed for simulating by mechanical means the highly dynamic movement of breaking ocean waves. The CSM, employing reversion kinematics of a six-link spatial kinematic chain, has been realized in the construction of a new mixing technology (called swing mixer) that can either move the fluid by the use of mixing tools in a vessel (stirrer principle) or by moving the entire vessel (shaker principle). A first description of swing mixers has recently been given.1 A special characteristic of swing mixers is their three-dimensional reversing motion, the forward and backward paths being nonsuperimposable mirror images of one another. During the mixing process in swing mixers, the efficient mixing principle of repeated stretching and folding2 also takes place in the third dimension. Details of the mixing technology of swing mixers will be discussed together with some suggestions as to how spatial and temporal changes in the concentration may be determined with the help of CCD cameras in a given multiphase system agitated by swing mixers.

Computer-aided structure synthesis of spatial kinematic chains

This paper presents a new form of structure formula for spatial mechanisms, which is expressed by using N bars with corresponding pair elements rather than by using pairs. The method of computer-aided structure synthesis of multi-loop, multi-degree-of-freedom, spatial kinematic chains is applied, and all possible kinematic chains are calculated under given conditions.

A theorem on axes of relative screw motions of three rigid bodies of a spatial kinematic chain

A theorem on axes of relative screw motions of three rigid bodies of a spatial kinematic chain

On the Flexibility of Spatial Kinematic Chains

A number of distinct or non-isomorphic kinematic chains exist for a specified number of links and joints. For example, sixteen distinct chains can be obtained with eight links and two hundred and thirty chains with ten links having a single degree of freedom. Similarly, many space mechanisms can be formed with four links and joints having different degrees of freedom. So far no measure is available to know which of these possesses greater mobility or flexibility. Flexibility is not to be confused with the degree of freedom. Intuitively one feels that a six-link chain has greater flexibility than a four-bar chain both having the same degrees of freedom. Though the mobility of a chain increases with the number of links one is not sure how the structural arrangement, type of links and joints, their numbers and sequence etc. influence the same. Combining graph theory with the concepts of probability, simple formulae are developed to investigate the relative merits of spatial and planar kinematic chains. The greater the flexibility or mobility of the chain, the higher is the ability to meet the motion requirements, i.e., a chain having greater entropy can be expected, say, to reproduce a given function more accurately.

Discrete positions method in kinematics and control of spatial linkages

Using the screw theory in the field of infinitesimal displacements, the first and second order transfer functions of spatial kinematic chains are algebrised. Based on this, a method of finite increments of the relative displacements in the joints is offered. Finally, an algorithm for the control strategy of open kinematic chains of manipulators and robots was evolved using an imaginary closing of the kinematic chain while bearing in mind the conditions for instantaneous mobility (transitory).

Application of Linear and Nonlinear Graphs in Structural Synthesis of Kinematic Chains

Using graph theory and Polya’s theory of counting, the present paper performs structural synthesis and analysis of planar and three-dimensional kinematic chains. The Section 2 of the paper develops a mathematical model that permits one to perform structural analysis and synthesis of planar kinematic chains with kinematic elements such as revolute pairs, cam pairs, springs, belt-pulley, piston-cylinder, and gears. The theory developed is applied to enumerate eight-link kinematic chains with these kinematic elements. The Section 3 of the paper develops a mathematical model that permits one to perform structural analysis and synthesis of multi-loop spatial kinematic chains with higher and lower kinematic pairs. The theory developed is applied to enumerate all possible two-loop kinematic chains with or without general constraints.

On a General Method of Spatial Kinematic Synthesis by Means of a Stretch-Rotation Tensor

One of the key concepts in a general method of spatial kinematic synthesis is a stretch-rotation operator applied to members of a general spatial kinematic chain. The latter consists of one or more interconnected loops of successively ball-jointed bar-slideball members. Each member is represented by a vector free to stretch-rotate with the motion of the chain. In the mathematical model of the general chain, displacement is simulated by means of stretch-rotation tensors operating on each member vector. Appropriate mathematical constraints render the general chain and its mathematical model equivalent to a particular mechanism. With this approach and by taking derivatives, first, second, and higher-order loop equations can be developed which form the basis for a general method of spatial kinematic synthesis, applicable to path, function and motion generation (body guidance) with first, second, and higher-order as well as for combined “point-order” approximations.