Theoretical Kinematics Research Articles

Design and dynamics of a 2PSS+PSR+SP+RUPUR-type parallel rotor of helicopter

A novel parallel rotor of the helicopter is designed by combining a 2PSS+PSR+SP+RUPUR-type parallel manipulator with a rotor of the helicopter. Its kinematics and dynamics models are established. Firstly, the structure principle and the merit performances of the parallel rotor are explained by comparing with conventional and existing rotors of the helicopter, and the degree of freedom of the parallel manipulator is calculated. Secondly, the kinematics formulas are derived for solving the displacement, Jacobian matrix, general velocity, Hessian matrix, and acceleration of the parallel rotor and the kinematic limbs. Thirdly, a dynamics model of the parallel rotor is established based on the derived kinematics formulas. Finally, the theoretical kinematics and dynamics solutions are verified by a simulation mechanism of the parallel rotor of the helicopter.

Kinematics Modeling and Singularity Analysis of a 6-DOF All-Metal Vibration Isolator Based on Dual Quaternions

Driven by the need for impact resistance and vibration reduction for mechanical devices in extreme environments, an all-metal vibration isolator with 6-degree-of-freedom (6-DOF) motion that is horizontally symmetrical was developed. Its stiffness and damping ability are provided by oblique springs in symmetrical arrangement and a metal–rubber elasto-porous damper. The spring is symmetrically distributed in the center axis of the support load surface. It is necessary to investigate the kinematics and the singularity before conducting multi-body dynamics analysis of the vibration isolator. Based on the theory of dual quaternions, the forward kinematics equations of the isolator were successively derived for theoretical kinematics modeling. In addition, an enhanced Broyden numerical iterative algorithm was developed and applied to the numerical solution of the forward kinematics equations of the vibration isolator. Compared with the traditional rotation-matrix method and Newton–Raphson method, the computational efficiency of the enhanced Broyden numerical iterative algorithm was increased by 680% and 290%, respectively. This was due to the enhanced algorithm without the calculations of any inverse matrix and forward kinematics equations. Finally, according to the forward kinematics Jacobian matrix, the position-singularity trajectory at a given orientation and the orientation-singularity space at a given position are calculated, which provides a basis for the algorithm of the 6-DOF vibration isolator to avoid singular positions and orientations.

The Role of Instant Centers in Kinematics and Dynamics of Planar Mechanisms: Review of LaMaViP’s Contributions

Theoretical kinematics and dynamics is one of the research fields where LaMaViP (Laboratory of Mechatronics and Virtual Prototyping) operates, which is the lab leaded by the author at the University of Ferrara. In the last two decades, this research activity at LaMaViP has produced, among others, many novel results that highlight how instant centers’ (ICs) locations condition the kinetostatic and dynamic behaviors of planar mechanisms, and that provide tools suitable for design purposes. This paper reviews/summarizes the tools devised at LaMaViP for PM analysis and synthesis through ICs, and shows that they are a complete set of tools, which make the full description of PMs’ kinematics and dynamics possible, and that the new IC features, identified while setting up these tools, are relevant in machine design.

A toy-inspired kirigami pattern and its kinematic performance by applying mechanisms and machine theory

Abstract. Origami that can form various shapes by setting simple creases on the paper and folding along these creases has a lot of applications from the fields of art to engineering. The inverse problem of origami that determines the distribution of creases based on the desired shape is very complicated. In this paper, we use theoretical kinematics to systematically analyse an inverse folding problem of a toy about how to fold a piece of paper into a disc through a smaller hole without breaking it. The results show that some four-crease and six-crease patterns can achieve the expected function, and they can be easily folded with 1 degree of freedom (DOF). It not only opens up a new way to solve the inverse folding problem but also helps students to understand mechanisms and machine theory.

Development and dynamics of a 2SPU+UPU+SP parallel rotor of helicopter

A prototype of a 2SPS+UPU+SP type parallel rotor of helicopter is developed. Its kinematics model and dynamics model are established. Firstly, the structural principle and the advanced performances of the parallel rotor of the helicopter are explained, and the degree of freedom of the parallel mechanism is calculated. Secondly, the kinematics formulas are derived for solving the displacement, Jacobian matrix, general velocity, Hessian matrix and acceleration of the moving links in the parallel mechanism. Thirdly, a dynamics model of the parallel mechanism for the helicopter rotor is established based on the derived kinematic formulas. Finally, the theoretical kinematics and dynamics solutions are verified by the simulation mechanism of the parallel rotor of the helicopter.

New Joint-Drift-Free Scheme Aided with Projected ZNN for Motion Generation of Redundant Robot Manipulators Perturbed by Disturbances

Joint-drift problems could result in failures in executing task or even damage robots in actual applications and different schemes have been presented to deal with such a knotty problem. However, in these existing schemes, there exists the coupling in coefficients for eliminating the drift in the joint space and the equality constraint for completing the given task in the Cartesian space, thereby, theoretically, leading to a paradox in achieving zero joint drift in the joint space and zero position error in the Cartesian space simultaneously. A novel joint-drift-free (JDF) scheme synthesized by a projected zeroing neural network (PZNN) model for the motion generation and control of redundant robot manipulators perturbed by disturbances is proposed and analyzed in this article. Besides, the PZNN model could adopt saturated or even nonconvex projection functions. The proposed scheme completely decouples the interferences of joint errors in the joint space and position errors in the Cartesian space for the first time. Beyond that, theoretical analysis is conducted in order to validate that the PZNN model is of global convergence to the theoretical kinematics solution to the motion generation of robots, and that the joint-drift problems are thus remedied. Moreover, several simulations and physical experiments on the strength of different robot manipulators are carried out to confirm the superiority, efficiency, and accuracy of the proposed JDF scheme synthesized by the PZNN model for remedying joint-drift problems of redundant robot manipulators in noisy environments.

Type synthesis and kinematics analysis of parallel manipulators with equivalent composite universal joints

Several novel parallel manipulators (PMs) with equivalent composite universal joints Ue are synthesized and their kinematics is studied systematically in this paper. First, the 12 novel PMs with Ue are synthesized, and their degrees of freedom are calculated. Second, the structure and merits of the composite limb with Ue for constructing the PMs with Ue are analyzed; the unified formulas are derived for solving the sub-Jacobian matrix, the sub-Hessian matrix, the angular velocity and the angular acceleration of the composite limb with Ue. Third, the formulas are derived for solving the Jacobian matrix and output velocity, the Hessian matrix and output acceleration of the moving platforms of the 10 novel PMs with Ue. Finally, a 5-DoF PM with 2Ue in the 12 synthesized PMs with Ue is illustrated for solving the kinematics of the PMs with Ue; its reachable workspace is constructed for representing merits of the PMs with Ue. The theoretical kinematics solutions are verified by the simulation solution of the simulation mechanism of the 5-DoF PM with 2Ue.

Kinematic and Static Analysis and Experimental Study for Dual-arm Continuum Robot

The continuum robot, which is different from the traditional rigid robot, consists of a flexible structure, and becomes a research hotspot in the field of robots, because of its good flexibility and adaptability in the complex environment. According to the advantages of the continuum robot, it could be applied in medical, exploration and agricultural operations. This paper proposed a cable-driven dual-arm continuum robot with three-degree-of-freedom. The kinematics of posture parameters and driving parameters, as well as the statics, was established. And the dual-arm coordinated reachable work space was described according to the principle of constant curvature deformation, on the basis of which, the theoretical motion model of the continuum robot was simulated using MATLAB, and experiments of the posture of the dual-arm continuum robot was conducted on the prototype to verify the accuracy of the theoretical kinematics and statics model.

Relationship between fractures patterns and fold kinematics; the case study of Jebel Sehib, a typical fault-propagation fold of southern Tunisia

Fracture patterns in folds are resulting from complex tectonic history. Indeed, in thrust and fold belts fractures occurred because of three different stress tensors: (1) Regional field stress, (2) Tensile stress in hinge area or (3) simple shear in limbs. Therefore, the same regional stress field could create too many different fracture and joint types. Detailed works have characterized fractures observed in nature, but only few studies have linked the fractures to the fold kinematics. On the other hand, we believe that fold kinematics have a key role in fracture amplitude and distribution. For this reason, we studied the fracture network affecting the Jebel Sehib, a typical fault-propagation fold of the Gafsa basin, southern Tunisia. On the same bed, and all over fold parts, we measured the fracture net density. The results showed clear difference in fracture density spread with highest density in the forelimb, medium in the backlimb and lowest in the hinge area. The, we compared the results with the theoretical kinematics of related fold/fracture models. We concluded that the fracture net is mainly anterior to the folding process. In fact, at the beginning of compressive stress, layer parallel shortening (LPS) occurred and generated homogenous fracture net in a pure shear regime attested by conjugated shear joints. During the growth of the fault-propagation fold, the existing fracture net is locally reactivated and amplified by limb simple shear as prospected in the mechanical behaviour of the fault-propagation fold. This leads to higher fracture density in the steeper flank.This hypothesis can be generalised to at least all fault-related folds in which limb simple shear is present. In the Jebel Sehib case study, we observed the flexural flow (internal deformation of the rock) because of the chosen massive limestone reference bed. In other natural examples, most of the beds accommodate simple shear by a flexural slip and rarely by flexural flow. This work have direct consequences on fractured reservoirs exploration in fault-related folds.

A pair of kinematically related space curves

We investigate the relation between two types of space curves, the Mannheim curves and constant-pitch curves and primarily explicate a method of deriving Mannheim curves and constant-pitch curves from each other by means of a suitable deformation of a space curve. We define a “radius” function and a “pitch” function for any arbitrary regular space curve and use these to characterize the two classes of curves. A few non-trivial examples of both Mannheim and constant pitch curves are discussed. The geometric nature of Mannheim curves is established by using the notion of osculating helices. The Frenet–Serret motion of a rigid body in theoretical kinematics is studied for the special case of a Mannheim curve and the axodes in this case are deduced. In particular, we show that the fixed axode is developable if and only if the motion trajectory is a Mannheim curve.

Design and experiment of a millimeter-range and high-frequency compliant mechanism with two output ports

This paper addresses a piezoelectric-actuated compliant mechanism with two output ports to provide tensile and compressive forces along with high dynamic motions for static tension and high-cycle dynamic fatigue testing of small-scale specimens or other precision flexible manipulation applications. A rhombus-type planar multistage displacement amplifier with a pair of guiding flexible beams is designed to realize a millimeter stroke while retaining high fundamental frequency. Improved effect of the guiding flexible beams on the vibration modes and attenuating impact on the output displacement are analyzed respectively by finite element analysis and theoretical kinematics modeling. A prototype is manufactured and the experimental performance evaluation shows a maximum stroke range of 1.44 mm (±720 µm) with the fundamental frequency of 628 Hz.

Three-dimensional structural dynamics of DNA origami Bennett linkages using individual-particle electron tomography

Scaffolded DNA origami has proven to be a powerful and efficient technique to fabricate functional nanomachines by programming the folding of a single-stranded DNA template strand into three-dimensional (3D) nanostructures, designed to be precisely motion-controlled. Although two-dimensional (2D) imaging of DNA nanomachines using transmission electron microscopy and atomic force microscopy suggested these nanomachines are dynamic in 3D, geometric analysis based on 2D imaging was insufficient to uncover the exact motion in 3D. Here we use the individual-particle electron tomography method and reconstruct 129 density maps from 129 individual DNA origami Bennett linkage mechanisms at ~ 6–14 nm resolution. The statistical analyses of these conformations lead to understanding the 3D structural dynamics of Bennett linkage mechanisms. Moreover, our effort provides experimental verification of a theoretical kinematics model of DNA origami, which can be used as feedback to improve the design and control of motion via optimized DNA sequences and routing.

Professor Bernard (“Bernie”) Roth – His journey from kinematics to design thinking

Dr. Bernard Roth (see Fig. 1) is the Rodney H. Adams Professor of Engineering at Stanford University and the co-founder and current Academic Director of the Hasso Plattner Institute of Design (also known as the d.school). His long and distinguished academic career at Stanford consists of significant and pioneering accomplishments in teaching, research, and consulting on various aspects of mechanical engineering, with a special emphasis on mechanism design. Bernie has established a worldwide reputation in the kinematic synthesis and analysis of mechanisms and is a pioneer in the field of computer controlled robot manipulators. Bernie has investigated the mathematical theory of rigid body motions and the application of these motions to the kinematic synthesis of mechanisms. He has placed a special emphasis on geometric kinematics over the more traditional time-based formulations which have allowed him to make important contributions to Burmester theory, curvature theory, and screw theory. His text book, Theoretical Kinematics, co-authored with Oene Bottema, is regarded by many kinematicians as the most elegant and rigorous treatment of this applied science. Also, his popular book The Achievement Habit: Stop Wishing, Start Doing, and Take Command of Your Life describes some of the innovative techniques that he employs in the classes, workshops, and short courses that he has offered on creativity and design thinking.

Upper bound limit analysis of masonry retaining walls using PIV velocity fields

Masonry is the building of structures from individual units, with or without the use of mortar as a bonding component. Stone masonry structures are frequent in the regions where stones are ubiquitous. In Europe, it was massively used until the nineteenth century to build earth retaining walls. For instance, they represent 85% of the retaining walls in Great Britain and 60% in France. Most of these masonry walls are currently perfectly safe, showing an average durability of more than a century. The best economic interest is to maintain the stock in good order by identifying and repairing the structures at risk of collapse and to preserve the currently satisfactory structures at the lowest investment level. Unfortunately, there is not sufficient scientific knowledge to do so reliably. This work aims to develop a design method suitable for existing structures. The analytical model is based upon the yield design theory which provides a rigorous framework and has proven to be effective for this kind of structures. The strength domain of stone masonry is determined using the homogenization theory. The modelisation is carried out in 2D. An experimental campaign was carried out in order to verify the proposed model. The experimental setup is in 2D thanks to the use of Schneebeli rods and is considered to be a physical model and is not scaled. The theoretical failing load and the theoretical kinematics of the failure are compared with the experimental failing load and the actual kinematics of the failure. The discrepancies and differences between the two sets are then discussed.

A Novel Universal Reducer Integrating a Planetary Gear Mechanism with an RCRCR Spatial Mechanism

A novel speed reducer with self-adaptability to variable transmission angles is proposed in this paper, in which a planetary gear mechanism with a difference of a few teeth and an RCRCR (R-revolute pair, C-cylindrical pair) spatial mechanism are integrated. The difference of a few teeth presents the advantages of high transmission ratio and compact structure, while the RCRCR spatial mechanism achieves the translational motion of the planetary gear in space. Theoretical kinematics analyses are carried out to prove the motion of each component in space. Models are established in the Automatic Dynamic Analysis of Mechanical Systems (ADAMS) to verify the performances of universal transmission and torque fluctuation tolerance. The simulation results demonstrate that the proposed speed reducer maintains a stable transmission ratio with variable transmission angles, presenting a high level of tolerance with the position perturbations. Strength analysis and manufacturing and lubrication guidelines of the prototype design are presented.

Modeling of kinematics errors and alignment method of a swing arm ultra-precision diamond turning machine

Ultra-precision machining has been widely used in the manufacturing of optical components. Volumetric error analysis and error compensation are essential issues that must be considered to improve the accuracy of the machining process. However, analysis of the effects of kinematics errors on surface generation by ultra-precision machining using a swing arm ultra-precision diamond turning machine (SA-UDTM) and the alignment methods required have received little research attention to date. This paper presents the development of a kinematics error model of the SA-UDTM that accounts for geometric errors by simplifying the machining system and considering it as a rigid multi-body system. Then, the main error contributions to the processing precision can be derived from sensitivity analysis of error components. Finally, an alignment method is developed to reduce or eliminate the corresponding positioning errors of the mechanical system and thus improves the form accuracy of processed surface. To verify the theoretical kinematics model and the proposed alignment method, a series of contrast machining experiments have been performed, and the experimental results indicate that the proposed method is effective.

Elastokinematics and compliance of a rectilinear rear independent suspension

A rectilinear rear independent suspension has a distinct difference from the traditional ones in that its wheel alignment parameters remain invariable in theoretical kinematics. However, they change within a narrow interval during jounce and rebound when the elasticity of parts, especially the rubber bushings, is taken into account. With the aim of investigating the compliance of the rectilinear rear independent suspension, an elastokinematic model is established in accordance with static equilibrium equations and compatibility conditions. The elastokinematic model has 28 unknowns corresponding to the 5 static equilibrium equations and 23 compatibility equations. Different configurations of rubber bushing and the sensitivity of the suspension stiffness to that of the rubber bushing are analysed. The analytical results indicate that the rubber bushings are best mounted close to the knuckle, and the suspension stiffness is sensitive to the compressive stiffness and the torsional stiffness about the z-axis of the rubber bushing. In addition, the results from kinematic and compliance tests not only verified the elastokinematic model but also revealed the excellent wheel alignment capacity of the rectilinear rear independent suspension compared with that of the MacPherson suspension. This work provided the foundations for the engineering design of a rectilinear rear independent suspension.

Characteristic points and cycles in planar kinematics with application to the human gait.

We present a novel method to process kinematical data typically coming from measurements of joints. This method will be illustrated through two examples. We adopt theoretical kinematics together with the principle of least action. We use motion and inverse motion for describing the whole experimental situation theoretically. By using the principle of least action, the data contain information about inherent reference points, which we call characteristic points. These points are unique for direct and inverse motion. They may be viewed as centers of the fixed and moving reference systems. The respective actions of these characteristic points are analytically calculated. The sum of these actions defines the kinematical action. This sum is by design independent of the choice of reference system. The minimality of the kinematical action can be used again to select numerically one representative cycle in empirically given, approximately periodic motions. Finally, we illustrate the theoretical approach making use of two examples worked out, hinge movement and the sagittal component of the movement of a human leg during gait. This approach enables automatic cycle choices for evaluating large databases in order to compare and to distinguish empirically given movements. The procedure can be extended to three dimensional movements.

Method and Simulation for Kinematics of 6-SPS Parallel Mechanism

Solving forward displacement is the difficulty of the 6-SPS parallel kinematics calculations, to make sure solving forward displacement easily, in this paper, added a initial multi-DOF drive to the 6-SPS parallel mechanism's moving platform in the ADAMS environment to make kinematical simulation, to get simulation of inverse kinematics solution and strike a theoretical inverse kinematics solution in the MATLAB environment, and prove the correctness of simulation of inverse solution. The inverse simulation position be used as input parameters to drive this parallel mechanism, by the solving forward displacement, results showed that the forward displacement is consist with the initial-DOF-driven values, indicating it is correct. It proves the use of these methods can be simply and easily calculate the required 6-SPS parallel mechanism positive solution, which can provide the reference of such institutions' solving forward position and control.

New Low-Speed Motor with Interior Pushrod Harmonic Transmission Device - Theoretical Profile Deduction and Kinematics Analysis

A pushrod harmonic institution which is consisted of harmonic planetary and cam mechanism is proposed. As a result, speed-down device of low-speed motor is changed from external installed to interior installed. Design goals of the transmission with small size, light weight and low cost is achieved. Based on the mechanism principle analysis, the theoretical profile equation of the pushrod roller is deduced, then the displacement, velocity and acceleration equation of a random points are got by the analysis method. It is helpful to analyze the change of each motion parameters and the amplitude for the influence of the movement characteristics, at the same time provides evidence for mechanism simulation, kinematics parameters optimization and high-quality manufacturing of the equipment.