Six-bar Linkage Research Articles

A 2 degrees of freedom united propulsive mechanism for amphibious function inspired by frog's hindlimb

Abstract. Amphibious species of frogs are notable candidates to mimic for amphibious robotic design, as their swimming and sprawling locomotion is generated by the united propulsive mechanisms in which the hindlimbs play the dominant role. Although the propulsive system of frogs is not as complex as other amphibians, it is still difficult to employ the propulsive mechanism in robotic design due to the numerous degrees of freedom (DoF). This paper proposes a novel united propulsive mechanism to acquire the amphibious function inspired by the frog's hindlimb. The mechanism is a hybrid design combining a planar six-bar linkage, which is designed based on homotopy continuation and a spatial four-bar linkage. The DoF of the hindlimb-like mechanism are dramatically decreased to 2, with 1 each in the two sub-chains. Forward analysis is conducted to find the pose of the foot when two actuations are input. The improved analysis based on the geometrical features overcomes the multiplicity from the numerical computation. The inverse kinematic analysis is conducted to find the rotation of the input with a given pose of the foot. The aquatic function of the united propulsive mechanism is demonstrated based on the blade element theory, where the rotational speed and the projected area of the foot are fully active. The terrestrial function of the propulsive mechanism is validated with a specific gait.

Legs as linkages: an alternative paradigm for the role of tendons and isometric muscles in facilitating economical gait.

ABSTRACTConsiderable attention has been given to the spring-like behaviour of stretching and recoiling tendons, and how this can reduce the work demanded from muscle for a given loss–return cycling of mechanical energy during high-speed locomotion. However, even completely isometric muscle–tendon units have the potential to act as tension struts, forming links in linkages that avoid the demand for mechanical work-cycling in the first place. Here, forelimb and hindlimb structures and geometries of quadrupeds are considered in terms of linkages that avoid mechanical work at the level of both the whole limb and the individual muscles. The scapula, isometric serratus muscles and forelimb can be viewed as a modified Roberts' straight-line mechanism that supports an approximately horizontal path of the body with vertically orientated forces, resulting in low work demand at the level of both limb and muscle. Modelled isometric triceps brachii inserting to the olecranon form part of a series of four-bar linkages (forelimb) and isometric biceps femoris cranial, rectus femoris and tensor fascia latae form part of a series of six-bar linkages (hindlimb), in both cases potentially resulting in straight-line horizontal motion, generating appropriate moments about shoulder and hip to maintain vertical ground reaction forces and again low mechanical work demand from the limb. Analysing part of the complexity of animal limb structure as linkages that avoid work at the level of both the whole limb and the supporting muscles suggests a new paradigm with which to appreciate the role of isometric muscle–tendon units and multiple muscle origins.

Six-Bar Linkage Models of a Recumbent Tricycle Mechanism to Increase Power Throughput in FES Cycling

This paper presents the kinematic and static analysis of two mechanisms to improve power throughput for persons with tetra- or paraplegia pedaling a performance tricycle via FES. FES, or functional electrical stimulation, activates muscles by passing small electrical currents through the muscle creating a contraction. The use of FES can build muscle in patients, relieve soreness, and promote cardiovascular health. Compared to an able-bodied rider, a cyclist stimulated via FES produces an order of magnitude less power creating some notable pedaling difficulties especially pertaining to inactive zones. An inactive zone occurs when the leg position is unable to produce enough power to propel the tricycle via muscle stimulation. An inactive zone is typically present when one leg is fully bent and the other leg is fully extended. Altering the motion of a cyclist’s legs relative to the crank position can potentially reduce inactive zones and increase power throughput. Some recently marketed bicycles showcase pedal mechanisms utilizing alternate leg motions. This work considers performance tricycle designs based on the Stephenson III and Watt II six-bar mechanisms where the legs define two of the system’s links. The architecture based on the Stephenson III is referred to throughout as the CDT due to the legs’ push acting to coupler-drive the four-bar component of the system. The architecture based on the Watt II is referred to throughout as the CRT due to the legs’ push acting to drive the rocker link of the four-bar component of the system. The unmodified or traditional recumbent tricycle (TRT) provides a benchmarks by which the designs proposed herein may be evaluated. Using knee and hip torques and angular velocities consistent with a previous study, this numerical study using a quasi-static power model of the CRT suggests a roughly 50% increase and the CDT suggests roughly a doubling in average crank power, respectively, for a typical FES cyclist.

Nonlinear dynamic behavior analysis of multi-linkage mechanism with multiple lubrication clearances

Clearance exists in mechanical equipment, which aggravates the friction and wear in moving pairs and reduces the use accuracy of mechanism. So as to reduce the impact of clearance on industrial equipment, lubricating oil is usually added to the clearance of moving pair. The lubricating oil can effectively reduce the collision and vibration between components and reduce noise. Previous studies mainly concentrated on the research of dry clearance joints, but less on clearance lubrication. At the same time, there is little research on nonlinear characteristic of lubrication clearance. For the sake of studying the impact of lubrication clearance with multiple revolute joints on mechanical system, the dynamic response and nonlinear characteristic of planar six-bar linkage are analyzed. The movement model of lubrication clearance is developed, and lubrication force of lubrication clearance is modeled by modified Pinkus–Sternlicht model. Dynamical model of lubrication clearance of planar six-bar linkage is established by Lagrange multiplier method. The impacts of lubrication, viscosity coefficient, clearance value and driving speed on dynamic of mechanism with lubrication clearance are explored. By drawing bifurcation diagram, phase trajectory diagram and Poincare map, the nonlinear characteristic of mechanism with lubrication clearance is studied.

A General Method for Constructing Planar Cognate Mechanisms

Abstract Cognate linkages are mechanisms that share the same motion, a property that can be useful in mechanical design. This article treats planar curve cognates, that is, planar mechanisms with rotational joints whose coupler points draw the same curve, as well as coupler cognates and timed curve cognates. The purpose of this article is to develop a straightforward method based solely on kinematic equations to construct cognates. The approach computes cognates that arise from permuting link rotations and is shown to reproduce all of the known results for cognates of four-bar and six-bar linkages. This approach is then used to construct a cognate of an eight-bar and a ten-bar linkage.

A Miniature Flapping Mechanism Using an Origami-Based Spherical Six-Bar Pattern

In this paper, we suggest a novel transmission for the DC motor-based flapping-wing micro aerial vehicles (FWMAVs). Most DC motor-based FWMAVs employ linkage structures, such as a crank-rocker or a crank-slider, which are designed to transmit the motor’s rotating motion to the wing’s flapping motion. These transmitting linkages have shown successful performance; however, they entail the possibility of mechanical wear originating from the friction between relative moving components and require an onerous assembly process owing to several tiny components. To reduce the assembly process and wear problems, we present a geometrically constrained and origami-based spherical six-bar linkage. The origami-based fabrication method reduces the number of the relative moving components by replacing rigid links and pin joints with facets and folding joints, which shortens the assembly process and reduces friction between components. The constrained spherical six-bar linkage enables us to change the motor’s rotating motion to the linear reciprocating motion. Due to the property that every axis passes through a single central point, the motor’s rotating motion is filtered at the spherical linkage and does not transfer to the flapping wing. Only linear motion, therefore, is passed to the flapping wing. To show the feasibility of the idea, a prototype is fabricated and analyzed by measuring the flapping angle, the wing rotation angle and the thrust.

Synthesis and Application of a Single Degree-of-Freedom Six-Bar Linkage With Mixed Exact and Approximate Pose Constraints

Abstract Existing research on synthesis methods for single degree-of-freedom (DOF) six-bar linkages mainly include four or five exact poses. However, an ideal trajectory cannot be synthesized using only five exact poses, thus, it is necessary to introduce additional poses to constrain the trajectory. If more exact poses are introduced, then the linkage may have no solution. Therefore, the constraints of the approximate pose are considered to make the trajectory conform to the desired trajectory. This paper successfully introduces mixed poses into a six-bar linkage, based on Z (Z < 5) exact poses and K approximate poses of a given error range, and a new synthesis method for single DOF six-bar linkages is proposed. The solution domain of the linkages synthesized by this method is wide and can be adjusted by controlling the error of the approximate poses, which reduces the difficulty of selecting the solution, ensures theoretical feasibility, and enables the trajectory of the final linkage to more closely match the ideal trajectory. Finally, for the coordinated training of multiple joints in human limbs, a rehabilitation device is designed based on the above six-bar linkage, and a prototype is developed and tested. The test results reveal the accuracy of the proposed method and the effectiveness of rehabilitation training.

Finger Clamping Unit: A Clamping Device with a Large Clamping Range

Finger Clamping Unit (FCU) is a small-sized single-point clamp developed to clamp thin object of various thicknesses. It has a six-bar linkage with one compliant link, uses an electric motor as an actuator and induces toggle-lock motion by singular configuration. The main purpose of the FCU is to clamp car bodies in car assembly lines, while adapting to various thicknesses of car body. Unlike other clamping devices in the industry, it has a large clamping range (defined as the thickness range that the FCU can clamp). In this paper, the overall process of FCU development is presented. The six-bar linkage was chosen in type synthesis, and compliance was added for toggle-lock motion. Static analysis of the compliant FCU was completed, and results were plotted in a three-dimensional (3D) plot which expresses the behavior of the FCU. The lengths of each link were optimized in dimension synthesis. An FCU was constructed to reinforce the simulated results. The unique design of both the six-bar linkage FCU and the compliant link were presented. Simulated and experimental 3D plots were compared and analyzed. An appropriate friction model that best fits the experiment data was found. The advantage of the FCU—its clamping range—was proved by experiment.

Mechanically Assisted Neurorehabilitation: A Novel Six-Bar Linkage Mechanism for Gait Rehabilitation.

Repeated and intensive gait training can improve muscle strength and movement coordination of patients with neurological or orthopedic impairments. However, conventional physical therapy by a physiotherapist is laborious and expensive. Therefore, this therapy is not accessible for the majority of patients. This paper presents a six-bar linkage mechanism for human gait rehabilitation with a natural ankle trajectory. Firstly, a six-bar linkage mechanism is selected as the original mechanism to construct a gait rehabilitation device. Then the ankle trajectory is formulated as a function of the crank angle. And the rotation angle of the crank is set as a linear function of time. Therefore, constant speed motor is sufficient to control the mechanism. For the dimensional synthesis, the precise point distances of the gait trajectory and the coupler curve are set as objective functions, with the kinematic constraints including in the optimization procedure. To obtain the optimal structure design parameters, a cooperative dual particle swarm optimization algorithm is developed. The results show that the coupler curve matches well with the gait trajectory. The average distance between the 60 precision points is 3.5 mm.

Mechanism design and analysis of six-bar linkage structure for vegetable transplanting

Mechanism design and analysis of six-bar linkage structure for vegetable transplanting

Synthesis of Six-Bar Timed Curve Generators of Stephenson-Type Using Random Monodromy Loops

AbstractSynthesis of rigid-body mechanisms has traditionally been motivated by the design for kinematic requirements such as rigid-body motions, paths, or functions. A blend of the latter two leads to timed curve synthesis, the goal of which is to produce a path coordinated to the input of a joint variable. This approach has utility for altering the transmission of forces and velocities from an input joint onto an output point path. The design of timed curve generators can be accomplished by setting up a square system of algebraic equations and obtaining all isolated solutions. For a four-bar linkage, obtaining these solutions is routine. The situation becomes much more complicated for the six-bar linkages, but the range of possible output motions is more diverse. The computation of nearly complete solution sets for these six-bar design equations has been facilitated by recent root finding techniques belonging to the field of numerical algebraic geometry. In particular, we implement a method that uses random monodromy loops. In this work, we report these solution sets to all relevant six-bars of the Stephenson topology. The computed solution sets to these generic problems represent a design library, which can be used in a parameter continuation step to design linkages for different subsequent requirements.

A Note on Equivalent Linkages of Direct-Contact Mechanisms

In this paper, the inequivalence of the direct-contact mechanisms and their equivalent four-bar linkages in jerk analysis is discussed. Kinematic analyses for three classical types of direct-contact mechanisms consisting of: (a) higher pairs with permanently invariant curvature centers, (b) higher pairs with suddenly changed curvature, and (c) higher pairs with continuously varying curvature are performed, respectively, through their representative case studies. The analyzed results show that the equivalent four-bar linkage cannot give a correct value of jerk for most situations in the three case studies. Subsequently, the concept of “equivalent six-bar linkage” for direct-contact mechanisms is proposed in order to discuss the infeasibility of the equivalent four-bar linkage for jerk analysis. It is found that the suddenly changed or continuously varying curvature of the higher pairs is not considered in sudden or continuous link-length variations of the equivalent four-bar linkage, which further leads to inconsistency between the angular accelerations of the coupler and the contact normal, and finally results in the infeasibility of the equivalent four-bar linkage for jerk analysis of most direct-contact mechanisms. It is also found that the concept of equivalent six-bar linkage could be applied to evaluate more higher-order time derivatives for most direct-contact mechanisms. The presented case studies and discussion can give demonstrations for understanding the inequivalence of the direct-contact mechanisms and their equivalent four-bar linkages in the aspect of jerk analysis.

Kinematic Design and Evaluation of a Six-Bar Knee-Ankle-Foot Orthosis

Abstract This paper presents a new two-step design procedure and preliminary kinematic evaluation of a novel, passive, six-bar knee-ankle-foot orthosis (KAFO). The kinematic design and preliminary kinematic gait analysis of the KAFO are based on motion capture data from a single healthy male subject. Preliminary kinematic evaluation shows that the designed passive KAFO is capable of supporting flexion and extension of the knee joint during stance and swing phases of walking. The two-step design procedure for the KAFO consists of (1) computational synthesis based on user's motion data and (2) performance optimization. In the computational synthesis step, first the lower leg (knee-ankle-foot) of the subject is approximated as a 2R kinematic chain and its target trajectories are specified from motion capture data. Six-bar linkages are synthesized to coordinate the angular movements of knee and ankle joints of the 2R chain at 11 accuracy points. The first step of the design procedure yields 332 six-bar KAFO design candidates. This is followed by a performance optimization step in which the KAFO design candidates are optimally modified to satisfy specified constraints on end-effector trajectory and shape. This two-step process yields an optimally designed passive six-bar KAFO that shows promising kinematic results at the knee joint of the user during walking. The preliminary prototype manufactured is cost effective, easy to operate, and suitably demonstrates the feasibility of the proposed concept.

An Articulated Closed Kinematic Chain Planar Robotic Leg for High-Speed Locomotion

Abstract This paper presents the design, dynamic modeling, and integration of a single degree of freedom (DOF) robotic leg mechanism intended for tailed quadruped locomotion. The design employs a lightweight six-bar linkage that couples the hip and knee flexion/extension joints mechanically, requiring only a single degree of actuation. By utilizing a parametric optimization, a unique topological arrangement is achieved that results in a foot trajectory that is well suited for dynamic gaits including trot-running, bounding, and galloping. Furthermore, a singular perturbation is introduced to the hybrid dynamic framework to address the lack of robust methods that provide a solution for the differential algebraic equations (DAEs) that characterize closed kinematic chain (CKC) structures as well as the hybrid nature of legged locomotion. By approximating the system dynamics as ordinary differential equations (ODEs) and asymptotically driving the constraint error to zero, CKCs can adopt existing real-time model-based/model-predictive/hybrid-control frameworks. The dynamic model is verified through simulations and the foot trajectory was experimentally validated. Preliminary open-loop planar running demonstrated speeds up to 3.2 m/s. These advantages, accompanied by low-integration costs, warrant this leg as a robust, effective platform for future tailed quadruped research.

A novel methodology for determining the singularities of planar linkages based on Assur groups

In this paper, a simple and effective methodology for judging the singularities of planar linkages is presented, which is based on Assur groups. The methodology determines the geometrical configurations of the Assur groups in singularities. Based on the singularities of a simple two-bar group, the possible singularities of all Assur four-bar and six-bar groups are given by the concept of instant centers. Then the general process and rules of determining singularities are obtained and shown through several examples of the 1-DOF planar linkages. These linkages include the two six-bar linkages and the several eight-bar linkages composed of Assur six-bar groups. The singularities of them verify the proposed methodology, which reveals that the singularities of a linkage only depend on the Assur groups. Therefore, as long as the singularities of Assur groups are determined, all the possible singularities of the linkage composed of these groups can be determined. This methodology is independent of linkages, so it is universal for all planar multi-bar linkages which can be divided into these Assur groups.

Path generation with dwells in the optimum dimensional synthesis of Stephenson III six-bar mechanisms

The optimal dimensional synthesis for Stephenson III six-bar linkages, planar and spherical, is presented. The requirement for dwells of the output link and the path generation task are also addressed. We propose a formulation for the synthesis in which the velocity of the output link is calculated by using dual functions. The condition for dwell is obtained through the minimization of the angular velocity, using the evolutionary algorithm known as Differential Evolution. We perform a numerical comparison between the use of the dual numbers and finite differences for computing the velocity and acceleration coefficients of the output link for the spherical case. The results exhibit the importance of having a precise method for computing derivatives.

Identification design of one-DOF eight-bar based on limb movements

This paper presents an identification design of one-DOF eight-bar based on limb movements by splitting it into a Stephenson II six-bar linkage and a 3R serial chain. They are kinematically identified, followed motion coupling, to form a desired eight-bar model. Specifically, enlightened by the system identification theory which achieves the behavioral simulation instead of focusing on their principles, we extend this idea to the mechanism design field to realize a bionic design. The steps are as follows: (1) Extracting input/output data from a movement, (2) forming a model set with appropriate capacities, (3) defining a criterion as the identification goal, (4) implementing an identification. Two instances involved in the identification of the movements of human fingers and marmoset legs are implemented to show the identification procedures in detail. Then we apply the genetic algorithms (GA) to obtain the expectation of eight-bar models. Ascribe to the identification method, the two models will be developed as a finger grasping device and a jumping robot respectively, achieving some principles that are ignored at the beginning.

Study regarding the implementation of an Ackerman steering geometry in MATLAB

In dynamic vehicle behavior the steering system has an important role. The vehicle stability, maneuverability depends on steering system characteristics. Majority of passenger vehicles uses the rack pinion steering linkage system. In this article, a simplified planar six-bar linkage model of the rack pinion steering system is presented. Kinematic relations are used to determine every linkage position in every possible situation. This simplified system contains a rack, tie rods, steering arms, and wheels. The six-bar linkage model is introduced in MATLAB SIMULINK. In the function of the rack displacement, the wheel angels are determined. The dimensions of this planar model can be estimated due to the minimum error between the calculated wheel angels for the rack and pinion system and an Ackerman principle based system with similar dimensions. For optimizing a system and for verifying the method numerical calculation are used.

A Comparison of Robot Wrist Implementations for the iCub Humanoid †

This article provides a detailed comparative analysis of five orientational, two degrees of freedom (DOF) mechanisms whose envisioned application is the wrist of the iCub humanoid robot. Firstly, the current iCub mk.2 wrist implementation is presented, and the desired design objectives are proposed. Prominent architectures from literature such as the spherical five-bar linkage and spherical six-bar linkage, the OmniWrist-III and the Quaternion joint mechanisms are modeled and analyzed for the said application. Finally, a detailed comparison of their workspace features is presented. The Quaternion joint mechanism emerges as a promising candidate from this study.

Experimental and Numerical Investigation of Six-Bar Linkage Application to Bellow Globe Valve for Compact Design

Bellow sealing in valves provides the characteristic of no emission sealing by extending or compressing within the elastic limit to operate the valve. To keep the motion of the bellow within the elastic limit, a long length of bellow is required, which increases the length of the shaft and height of the valve. Increased height restricts valve usage, especially for large sizes, due to the limited platform space. This study presents the application of a six-bar linkage to operating mechanism by replacing the reciprocating shaft in an attempt to reduce the height of the valve. The mechanism produced aims to reduce the input displacement while maintaining the output displacement in order to reduce the required bellow length. Graphical synthesis of the mechanism was carried out using Autodesk Inventor. The designed mechanism was then subjected to analysis using a simulation tool for the position, force, and flow analysis. The prototype valve with the application of the new mechanism was manufactured using additive manufacturing, which was later used for experimental testing. Application of the mechanism to the bellow valve reduced the required input by 75%, which as a result, reduced the height of the valve up to 50%. The force analysis depicted that the force required to operate the mechanism was approximately eight times higher than the conventional design. Flow analysis of the valve showed that the introduction of the new mechanism had no effect on the flow efficiency of the valve and the flow coefficient of valve remained the same. Application of the six-bar linkage to the valve control mechanism made the design of the valve compact when compared to conventional design in terms of height, which makes it more suitable for use in industry.