Spherical Four-bar Mechanism Research Articles

A Linkage-Driven Underactuated Robotic Hand for Adaptive Grasping and In-Hand Manipulation

The development of robotic hand that can imitate human movements has always been an important research topic. In this paper, a linkage-driven underactuated three-finger hand is proposed to imitate the flexion/extension (f/e) and abduction/adduction (a/a) motions of human hand. The robotic hand has three identical underactuated fingers, each of which contains an underactuated planar linkage, a spherical four-bar mechanism, and a set of bevel gears. The spherical four-bar mechanism is designed to provide 2-degree-of-freedom actuation, driving the f/e and a/a motions of the proximal joint simultaneously. Based on screw theory, the kinematic model of the spherical mechanism is established, and the maximum available workspace index (MAW) of the spherical mechanism is proposed to evaluate the workspace with the same adduction and abduction angle ranges. The effects of the parameters of the spherical mechanism on the MAW and the transmission efficiency are obtained, and the parameters of the spherical mechanism are optimized. The optimization results show that the MAW of the spherical mechanism can be increased by up to 3.5 times. Finally, experiments are carried out to show the proposed robotic hand can perform simultaneous adaptive grasping and in-hand manipulation. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The robotic hand is of great significance for replacing workers in heavy, repetitive, and unsafe working environments. Uncertain environments in non-manufacturing fields, such as industries for pick-and-place, sorting, palletizing, assembly and other operations, marine development, and medical service, require that robotic hands can operate tasks like human hands. Inspired by the human hand with the f/e and a/a motions, a linkage-driven underactuated three-finger hand is proposed in this paper to imitate the functions of the human hand. Due to its underactuated characteristics and a/a function, the designed robotic hand is more dexterous and economical. The effects of the design parameters of the spherical mechanism on the MAW and the transmission efficiency are analyzed, and the parameters of the spherical mechanism are optimized to improve the performance of the robotic hand. This approach can be used in other industrial applications related to the robotic hand or the spherical mechanism.

Dimensional synthesis of motion generation in a spherical four-bar mechanism

A new objective function for optimizing spherical, four-bar mechanisms for motion generation is presented. Rigid-body poses of a spherical four-bar mechanism in a standard installation position are investigated, and a normalization processing method is proposed. After processing the coupler points of the mechanism, the feature points generated are located on the circles. The formation principle of the feature coupler circles is analysed, and the internal relationship between the centre angle of the adjacent feature points on the feature coupler circles and the coupler angle of the spherical four-bar mechanism is determined. An input angle determination method is proposed for the spherical four-bar mechanism in a general installation position. An objective function is then established to optimize the basic dimensional types, relative input angles, and installation position parameters of the desired spherical four-bar mechanism. The proposed method is applicable to both prescribed and unprescribed timing problems. Notably, the dimension of the optimization variables is only eight, which is exceptionally low for multiple position motion generation without prescribed timing; therefore, the global optimal solution is more easily obtained. The optimization process is carried out by the genetic algorithm. The feasibility and effectiveness of our proposed method are demonstrated by examples.

Dimensional synthesis of a spherical linkage crank slider mechanism for motion generation using an optimization algorithm

Abstract. In the present study, Fourier theory is applied to establish the expression of rigid-body poses of a spherical four-bar crank slider rigid-body guidance mechanism. According to an analysis of the harmonic components of the trajectory curve and rigid-body rotation angle, it has a certain relationship with the geometric parameters of the mechanism. On this basis, the rigid-body poses are normalized by preprocessing. Then, the rotation angle of the curve around the y axis and z axis is determined, respectively. The theoretical formulas used for calculating the real sizes and the installation position parameters of the desired spherical four-bar crank slider rigid-body guidance mechanism are established. Besides this, a genetic optimization algorithm and theoretical formulas are applied to solve the dimensional synthesis of motion generation for the spherical four-bar crank slider mechanism. The effectiveness of the proposed method is illustrated by an example. The maximum Euclidean distance error of the rigid-body position of the results with the highest similarity is 0.0086, and the average Euclidean distance error is 0.0044. The maximum error of the rigid-body orientation is 0.0179, and the average error is 0.0065.

Planar and spherical four-bar linkage [formula omitted] algebraic input–output equations

The algebraic polynomial input–output (IO) equations relating any two of the relative joint displacement parameters, called vi and vj, between any of the six distinct pairs of rigid links in arbitrary planar and spherical four-bar mechanisms are derived. First, the forward kinematics transformation matrices of the corresponding serial kinematic chains are computed in terms of their Denavit–Hartenberg parameters, but with all angles converted to tangent half-angle parameters. These matrices are mapped to their corresponding Study soma coordinates. The serial kinematic chain is closed by equating the soma coordinates to the identity array. Algebraic polynomial elimination methods are then used to obtain a single polynomial in terms of only the design and the selected IO joint displacement parameters. This yields six independent algebraic IO Equations for each of the planar and spherical 4R linkages; the same techniques are applied to derive six additional algebraic IO equations for each of the RRRP and PRRP planar linkages providing a catalogue of 24. The utility of these IO equation sets is demonstrated via discussion of the associated mobility and design parameter spaces.

SphereWalker: A Hexapod Walking Machine

Abstract This article describes the design and the development of a novel six-legged robotic walking machine named SphereWalker. The six legs are arranged into pairs, and each pair of legs is supported and actuated by a single spherical four-bar mechanism. Two of the four-bar mechanisms are operated in a synchronous fashion, while the middle one is operated at 180 deg out of phase with respect to the other two. A physical prototype has been built, a digital twin has been generated, an actuation and control system has been designed, and the technology has been patented.

On the design of a novel fully compliant spherical four-bar mechanism

In this article, a novel fully compliant spherical four-bar mechanism is introduced and its generalized design methodology is proposed. The original fully compliant mechanism lies on a plane at the free position (undeflected position); therefore, it has the advantages of ease of manufacturing, minimized parts, and no backlash. First, the mobility conditions of the mechanism are obtained. The dimensions of the mechanism are optimally calculated for maximum output rotation, while keeping the deflection of flexural hinges at an acceptable range. Using an optimization method, design tables are prepared to display the relationship between arc lengths and corresponding deflections of flexural hinges. Input–output torque relationship and stresses at compliant segments are obtained analytically. A mechanism dimensioned by this novel design method is analyzed by a finite element analysis method, and the analytical results are verified. Finally, the mechanism is manufactured and it is ensured that the deflections of the compliant segments are consistent with the theoretical results.

Special Unitary Matrices in the Analysis and Synthesis of Spherical Linkages

This work seeks to systematically model and solve the equations associated with the kinematics of spherical mechanisms. The group of special unitary matrices, SU(2), is utilized throughout. Elements of SU(2) are employed here to analyze the three-roll wrist and the spherical Watt I linkage. Additionally, the five orientation synthesis of a spherical four-bar mechanism is solved, and solutions are found for the eight orientation synthesis of the Watt I linkage. Using SU(2) readily allows for the use of a homotopy-continuation-based solver, in this case Bertini. The use of Bertini is motivated by its capacity to calculate every solution to a design problem.

Synthesis of spherical four-bar linkage for open path generation using wavelet feature parameters

A wavelet feature parameters method is proposed for the non-periodic design requirements in the path synthesis of a spherical four-bar linkage mechanism. A mathematical model for the coupler curve of the spherical four-bar linkage is established using a spherical coordinate system. The geometrical characteristics of the open path are described using wavelet feature parameters. Then, a database of wavelet feature parameters is established. Based on the positional relationship of the prescribed curve with the coupler curve generated by the basic dimensional types, the installation positions of the desired linkage can be calculated. Moreover, the open path generation of spherical four-bar linkages can also be performed. Four examples are given. The design results show that the proposed method is effective for the open path generation of spherical four-bar linkages.

Motion generation of spherical four-bar mechanism using harmonic characteristic parameters

In an earlier work, we presented a method to address the path generation problems associated with spherical four-bar mechanism. In this paper, we seek to extend this approach to the synthesis of spherical four-bar linkages for motion generation. Through harmonic analysis, the output characteristics of a spherical four-bar mechanism is observed to be dependent on the rigid body rotation angle operator μp(t). Subsequently, a database that contains 221,968 sets of basic rotation angle dimensional types for a spherical four-bar mechanism was constructed. The theoretical formulas to calculate the actual dimension and installation parameters of spherical four-bar mechanism are deduced from the harmonic parameters for position output and the harmonic parameters for the rigid body rotation angle operator. On the basis of the theoretical formulas and the numerical atlas database, the motion generation problem for a spherical four-bar mechanism was solved. Finally, two examples are provided to demonstrate the validity of this method.

Synthesis of a non-integer periodic function generator of a four-bar mechanism using a Haar wavelet

In this study, a synthesis method for a non-integer periodic function generator of a four-bar mechanism using a Haar wavelet is presented. Using wavelet theory, the mathematical model for the output of a four-bar mechanism is established. Next, the characteristics of the four-bar mechanism output variable are described by the wavelet feature parameters; the relationship of the wavelet feature parameters between the output and its translation or dilation is also investigated. Based on this finding, three numerical atlas databases comprising of 203,500 groups of basic dimensional types of planar four-bar mechanisms, 148,995 groups of basic dimensional types of spherical four-bar mechanism and 11,175 groups of basic dimensional types of planar slider–crank mechanism are compiled. According to the internal relationship between the wavelet feature parameters, the sizes of the objective mechanism can be obtained using fuzzy recognition. Based on a genetic algorithm, optimization of the link parameters can be realized. Three examples are presented to demonstrate the efficacy and practicality of the proposed theory.

A rolling mechanism with two modes of planar and spherical linkages

A novel deformable rolling mechanism is proposed in this paper. The mechanism can switch between two modes: a planar linkage mode and a spherical linkage mode. In the planar linkage mode, the mechanism rolls like a planar four-bar mechanism and moves along a straight line. In the spherical linkage mode, the mechanism rolls as a spherical four-bar mechanism and moves along a polygon. A standing posture, which likes an upright pyramid, of the rolling mechanism is also identified. After the description of the novel mechanism, the kinematic analysis of the mechanism is presented, and appropriate control strategies are proposed. Then, the rolling conditions of the mechanism are investigated based on the zero-moment-point theory. Finally, the various modes of the mechanism are verified by both simulations and prototype experiments.

Variable step-size numerical atlas method for path generation of spherical four-bar crank-slider mechanism

This paper presents a new method for the path generation of spherical four-bar crank-slider mechanism. Based on the established mathematical model of the spherical four-bar crank-slider mechanism, the expression of the coupler curve is determined. Then, depending on Fourier series theory, the immanent connection between coupler curve and its corresponding harmonic characteristic parameters is discovered. Meanwhile, by the analysis of the basic dimensional type and the coupler curve, different step size for each parameter of the basic dimensional type is chosen. Thus, by putting the harmonic characteristic components and associated basic dimensional type of spherical four-bar crank-slider mechanism together, a variable step size numerical atlas database of the spherical four-bar crank-slider mechanism is created. Furthermore, by using fuzzy identification method, the best suitable basic dimensional type of spherical four-bar crank-slider mechanism for prescribed coupler curve is obtained. Finally, an example is given to illustrate the synthesis process and the application of this method.

Least square approximate motion generation synthesis of spherical linkages by using Chebyshev and equal spacing

In this paper, approximate motion synthesis of spherical linkages is presented. Rigid body guidance of a spherical four-bar mechanism is performed by a spherical RR open chain. In the first step, position of a point on rigid body and orientation of a rigid body on a unit sphere are described. Synthesizing function of spherical dyad is derived by means of using unit vectors that describe location of two revolute joints and tip point. Being based on the theory of function approximation and besides the linearization of nonlinear synthesis equations by using superposition method, the design procedure for real solutions of fourth order polynomial equation is developed. In the second step, approximate motion generation synthesis of spherical dyad is presented by using least-square approximation. Chebyshev spacing and equal spacing are used in the determination of poses. In the final step, two numerical examples are given to show how error graph is varied in terms of selected poses. The spherical motion generation synthesis of spherical four-bar mechanism is obtained by the combination of the two real solutions of the synthesis of two spherical dyads.

Design and manufacturing of a controllable miniature flapping wing robotic platform

In this work we present the design of a scalable, controllable, miniature flapping wing robot. The design effort comprises considerations of the number of driving actuators for controllability, body structure for both weight minimization and rigidity, and overall compactness of the robot design. This results in the development of the spherical four-bar transmission mechanism, with a single-wing prototype experimentally characterized. A dual-wing robot design manufactured via the smart composite microstructures technique is presented, featuring independent wing actuation. Finite-element analysis of the final airframe design is presented, ensuring vibration modes out of the operation range of the platform and high rigidity. A working prototype is manufactured and experiments are conducted characterizing the robot’s lift production capabilities and ensuring minimal wing coupling. A scaling law of the proposed design is presented based on momentum theory, predicting an increase of the lift/weight ratio with decreasing size. An optimization methodology for the parameters of a scaled down prototype is presented, based on a developed theoretical simulation and design tool. Finally, a 1/2 scaled down prototype using the optimized parameters is built and tested, featuring a lift/weight ratio of 3/8.

Orientation Order Analysis of Spherical Four-Bar Mechanisms

In this paper, we present a novel methodology for orientation order analysis of spherical RR dyads. The methodology is a spherical generalization of the recent works of Myszka, Murray, and Schmiedeler for assessing position order of planar RR dyads. The objective of the methodology is to determine if a prescribed fixed axis location for a spherical RR dyad will result in the dyad guiding a moving rigid-body through a set of finitely separated spherical orientations in the desired order (e.g., 1, 2, 3, 4, etc.). The planar propeller methodology of Myszka, Murray, and Schmiedeler is extended to yield a spherical hoop methodology. The result is a useful tool to determine if a given spherical RR dyad will guide a moving body through a set of prescribed orientations in the desired order. Finally, we demonstrate the utility of the hoop method for order analysis of spherical RR dyads in two case studies.

Atlas of spherical four-bar mechanisms

An atlas or catalogue of mechanisms provides a useful aid in the synthesis of mechanisms for new applications. The atlas stores mechanisms together with their coupler paths. Fourier techniques can be used as a convenient means for representing, in normalized form, curves for planar mechanisms. This paper looks at the extension to spherical four-bar mechanisms. In particular, a means for projecting a spherical curve onto a plane is discussed which depends only on the geometry of the curve and not on the choice of world coordinate system.

Design of a Bio-Inspired Spherical Four-Bar Mechanism for Flapping-Wing Micro Air-Vehicle Applications

The design of flapping-wing micro air-vehicles presents many engineering challenges. As observed by biologists, insects and birds exhibit complex three-dimensional wing motions. It is believed that these unique patterns of wing motion create favorable aerodynamic forces that enable these species to fly forward, hover, and execute complex motions. From the perspective of micro air-vehicle applications, extremely light-weight designs that accomplish these motions of the wing, using just a single or a few actuators, are preferable. This paper presents a method to design a spherical four-bar flapping mechanism that approximates a given spatial flapping motion of a wing, considered to have favorable aerodynamics. A spherical flapping mechanism was then constructed and its aerodynamic performance was compared to the original spatially moving wing using an instrumented robotic flapper with force sensors.

Numerical atlas method for path generation of spherical four-bar mechanism

In this paper, the relationship between the harmonic component of the coupler curves and the harmonic component of the rotation-angle function is discovered. Based on this finding, a numerical atlas database comprising of more than 6 million of basic dimensional types is built up. The formulas which can compute the position for coupler point, real size and installing dimensions of the spherical four-bar mechanism are deduced. Based on those theoretical formulas and the numerical atlas database of the harmonic characteristic parameters, the problem of path synthesis of spherical four-bar mechanism is solved. Finally, an example is given to show the feasibility of this approach.

SPHERICAL FOUR-BAR MOTION GENERATION WITH A PRESCRIBED RIGID-BODY LOAD

In motion generation, the objective is to calculate the mechanism parameters required to achieve or approximate a set of prescribed rigid-body positions. This work introduces a new design constraint that considers driving link static torque for a given rigid-body load. By incorporating this new constraint into a conventional spherical four-bar motion generation model [1], spherical four-bar mechanisms are synthesized to achieve-not only prescribed rigid-body positions-but also satisfy a maximum driver static torque for a given rigid-body load.

On adjustable spherical four-bar motion generation for expanded prescribed positions

This work formulates and demonstrates a method to synthesize two-phase adjustable spherical four-bar mechanisms to approximate substantially more prescribed rigid-body positions than with conventional motion generation methods [C.H. Suh, C.W. Radcliffe, Kinematics and Mechanism Design, John Wiley and Sons, New York, 1978]. An alternate set of mechanism constraint equations are formulated using the Method of Least Squares and solved for the adjustable spherical mechanism moving pivot variables. The example included demonstrates the synthesis of an adjustable spherical four-bar mechanism to approximate twice the number of prescribed rigid-body positions per phase than with a conventional method [C.H. Suh, C.W. Radcliffe, Kinematics and Mechanism Design, John Wiley and Sons, New York, 1978].