Screw Theory Research Articles

A family of folding single-loop metamorphic mechanisms for aerospace manipulators: Synthesis, network, and analysis

Guided by predetermined degrees of freedom for different desired tasks, the type synthesis of single-loop metamorphic mechanisms (SLMMs) has long posed a challenge. Owing to the complexity and coupling of their overconstraints and the variability of constraints arising from bifurcation motions, it is difficult to ensure geometric coordination and synchronous motion for a network of multiple SLMMs. Here, we propose an approach to type synthesize SLMMs using the atlas method. A novel family of SLMMs is further synthesized. The mobility analysis of a selected 7R SLMM is carried out via screw theory. The kinematics and bifurcation phenomena are analyzed and verified through numerical simulation using computer-aided design models. To evaluate the performance of the SLMMs, the mapping relationship between key design parameters and performance indicators, such as the grasping angle, folding angle, and mechanical advantage, is studied. A networking method is then proposed by considering the geometric coordination conditions and variation of the overconstraints of SLMMs. Prototypes of an SLMM and manipulator with optimal geometrical parameters are developed to verify the feasibility and correctness of the proposed method and kinematic analysis.

Design and performance analysis of the 4UPS-RRR parallel ankle rehabilitation mechanism

Abstract. Ankle sprains are among the most common musculoskeletal injuries. Patients often require systematic rehabilitation training to expedite tissue healing and facilitate joint recovery. However, such training places high demands on medical staff, involves a lengthy process, requires considerable labor, and suffers from a shortage of skilled rehabilitation personnel. To address the need for effective ankle joint dysfunction rehabilitation training, this study analyzes the bone structure and movement mechanism of the ankle. Drawing on the parallel mechanism configuration, we propose a 4UPS-RRR parallel ankle rehabilitation mechanism. The rotation center of the rehabilitation mechanism can be highly coincident with the rotation center of the human ankle joint. The structure and degrees of freedom of the mechanism were designed and analyzed using screw theory. Additionally, a kinematic model of the mechanism was established. The mechanism's workspace was mapped by constraining the linear actuator length and spherical joint rotation angle. Furthermore, the mechanism's dexterity was assessed through the establishment of its Jacobian matrix, with the correctness of the kinematic model verified through simulation. Finally, an experimental platform was utilized to test the maximum range of robot motion, confirming the practicality of the ankle rehabilitation mechanism.

Configuration design of movable heavy-duty reconfigurable posture adjustment platform with dual motion modes

AbstractThe existing single-mode posture adjustment equipment for solar wing docking is only suitable for a limited number of satellite dimensions; it could not meet the diverse development trends of satellite models. The working range requirements are different when different-sized satellites dock with the solar wing, and the docking process is divided into two stages in this paper. While the DOFs required for the two stages are different, a movable heavy-load reconfigurable redundant posture adjustment platform (RrPAP) with dual motion modes is proposed in this paper. The RrPAP consists of a wheeled mobile platform and a reconfigurable parallel posture adjustment mechanism (PAM). The micro-motion PAM limb types are synthesized, and the comprehensive load-bearing index is proposed to select the mechanism types for heavy-load conditions. A decentralized four-limb six-degree-of-freedom (6-DOF) parallel micro-motion PAM is designed. In the macro-motion stage, for the PAM to still have a defined motion after being released from ground constraints, a serial coupling sub-chain is designed between adjacent limbs to restrict relative movement between them. A type synthesis method for symmetrically coupled mechanisms based on mechanism decoupling and motion distribution is proposed. Four types of symmetrically coupled mechanisms with multi-loop consisting of serial coupling sub-chains are synthesized by using this method. The feasibility of the proposed method is demonstrated through an example using the constraint synthesis method based on screw theory. This work provides a foundation for subsequent refinement and expansion of type synthesis theories and the selection of new types of mechanisms.

Novel constant-height deployable mechanisms with only revolute joints and constructed deployable ring trusses for satellite mesh reflector antennas

Deployable mechanisms play an increasingly crucial role in aerospace, yet their design still faces challenges: 1) The folded height increases while its deployed height decreases, which is not conducive to improving the folding performance; 2) Compared with revolute joints, prismatic joints have relatively weak reliability in complex space environments. To address the above challenges, novel constant-height deployable mechanisms with only revolute joints are proposed in this paper. The deployable mechanisms exhibit one degree of freedom, enabling them to attain modular expansion. Subsequently, the mobility properties of these deployable mechanisms are substantiated through screw theory. Assembly strategies for constant-height deployable mechanisms are proposed, and deployable ring trusses are further constructed to provide support structures for satellite mesh reflector antennas. The kinematics are analyzed to investigate the deployable characteristics of the proposed deployable mechanisms and ring trusses. The prototypes of a deployable mechanism and a ring truss are established to validate the designed deployable mechanisms. This work contributes novel types of deployable mechanisms with potential advantages, aiming to function as valuable references for the design and analysis of deployable trusses.

Analysis and synthesis of interconnected hybrid mechanisms using Freedom and Constraint Topologies (FACT)

We introduce visualization-based methods to analyze and synthesize interconnected hybrid (ICH) systems. These systems consist of joints that are not strictly parallel nor serial to one another. We include two types of analysis methods – one that uses joint freedom spaces and another that uses joint constraint spaces. These methods leverage intuition to determine the mobility, under-constraint, over-constraint, transmission, and load paths of any mechanism. The analysis methods are an interpretation of previously established numerical approaches for ICH systems. To adapt the complexities of screw theory and matrix algebra, this work builds off the geometries and systematic rules of the Freedom and Constraint Topologies (FACT) design approach. The synthesis approach is a reversal of the freedom-based analysis method. This synthesis approach takes a designer from any set of desired transmissions to an ICH system capable of achieving them. Both the analysis and synthesis approaches are accompanied by an example compliant mechanism to illustrate their utility and limitations. These methods seek to facilitate the design of emerging mechanism classes, such as metamaterials, and extend the FACT design approach to include ICH systems.

A novel method to enhance the accuracy of parameter identification in elasto-geometrical calibration for industrial robots

Elasto-geometrical calibration is crucial for enhancing the absolute accuracy of robots in machining operations through the identification and compensation of parameter errors. However, the presence of inconsistent measurement units and improper selection of measuring poses can result in the ill-conditioned identification matrix (ICIM) issue, consequently impacting the accuracy of parameter identification. This paper introduces a novel method to tackle this challenge. Initially, an elasto-geometrical error model is developed based on the orientation-independent measurements (OIM), efficiently reducing the impact of mismatched positions and orientations on the ICIM problem. Subsequently, a PSO-SFFS algorithm is proposed to optimize the measurement configurations and minimize the influence of measurement noise. Furthermore, the incorporation of screw theory and the consideration of parallelogram mechanisms enhance the precision and comprehensiveness of the error model. Subsequent to the development of the error model, calibration comparison experiments are conducted on an industrial robot. Both simulation and experimental results validate the effectiveness of the proposed method in improving parameter identification accuracy.

Kinematotropic linkage-based cellular metamaterials with bifurcation motion: Construction and analysis

This paper proposes the design and networking method of a metamorphic Ori-block unit and develops a novel family of cellular mechanical metamaterials. First, a novel 6R (Revolute pair) diamond kinematotropic linkage (DKL) with two bifurcation motions is presented and its kinematic model is analyzed using the improved Denavit–Hartenberg parameter method. Additionally, the screw theory is used to analyze its mobility and verify it has not instantaneity in every configuration. This linkage exhibits two bifurcation motions with one rotational degree-of-freedom (DOF). Moreover, a metamorphic Ori-block unit is designed based on the 6R DKL and optimized to assemble a novel metamaterial cell. According to performance analysis, the first motion mode of this metamaterial cell exhibits a negative Poisson's ratio performance, while the second motion mode involves a large deployed/folded ratio. Based on this metamaterial cell, two methods are proposed for networking cellular metamaterials. On the basis of preserving the kinematic characteristics and performance of the metamaterial cell, an optimized design is conducted, and subsequently the optimized metamaterial cells are assembled into novel cellular metamaterials. This study provides a novel concept for the design and movable network of deployable and programmable cellular metamaterials.

A novel cable configuration method for fully-actuated parallel cable-driven systems: Application in a shoulder rehabilitation exoskeleton

Cable-driven exoskeletons play a significant role in poststroke rehabilitation. Since cables can only pull but not push, conventional parallel cable-driven exoskeletons are usually over-actuated, leading to uncertain dynamics. This paper proposes a cable configuration method utilizing the antagonistic cable pair to achieve full-actuation. The antagonistic cable pair is formed by assigning a passive impedance cable corresponding to an active driving cable. This method is illustrated as feasible through force closure analysis based on screw theory, and is implemented in a shoulder rehabilitation exoskeleton. Subsequently, the dynamics model of the exoskeleton is formulated, and the isomorphic mapping between cable tension space and joint space is established. Additionally, the elastic fluctuation absorption mechanism is presented to absorb the length fluctuation produced by the antagonistic cable pair, through the deformation of elastic elements. The continuity of cable tension is evidenced through its finite and smooth partial derivatives. Finally, the experimental results of prototype validate that the fully-actuated parallel cable-driven exoskeleton can be achieved with the proposed method.

Robotica: decoupled elastostatic stiffness modeling of hybrid robots

AbstractA decoupling method is proposed for the elastic stiffness modeling of hybrid robots based on the rigidity principle, screw theory, strain energy, and Castigliano’s second theorem. It enables the decoupling of parallel and serial modules, as well as the individual contributions of each elastic component to the mechanism’s stiffness performance. The method is implemented as follows: (1) formulate limb constraint wrenches and corresponding limb stiffness matrix based on the screw theory and strain energy, (2) formulate the overall stiffness matrix of parallel and serial modules corresponding to end of the hybrid robots based on the rigidity principle, principle of virtual work, the wrench transfer formula, and strain energy methods, and (3) obtain and decouple the overall stiffness matrix and deflection of the robot based on the Castigliano’s second theorem. Finally, A planar hybrid structure and the 4SRRR + 6R hybrid robot are used as illustrative examples to implement the proposed method. The results indicate that selectively enhancing the stiffness performance of the mechanism is the most effective approach.

Design and kinematic analysis of a deployable mechanism for soft sheets grasping

In the automation process of the garment industry, a notable challenge arises from the presence of flexible materials characterized by irregular shapes, posing difficulties for robotic grasping. This article addresses this challenge by proposing a gripper manipulator centered around the 4 P-RRP quadrilateral cone folding mechanism as the core driving mode, with the angled four-bar linkage mechanism serving as the extension arm. Utilizing screw theory, the developability of the 4 P-RRP four-pyramid mechanism with one degree of freedom is verified, forming the basis for its adoption as the central driving mechanism. Furthermore, an in-depth analysis of the geometric properties of the robot’s extension arm leads to the derivation of an optimal folding ratio structure, which is seamlessly integrated into the robot design. Kinematic equations for the mechanism are formulated and solved using the D-H parameter method to obtain the kinematic solution. The investigation demonstrates that the manipulator utilizing the quadrilateral cone folding mechanism significantly enhances adaptability to irregular shapes. By incorporating this proposed mechanism, robots can efficiently manipulate flexible materials across various garment manufacturing applications.

Design and Experiment of an Ankle Rehabilitation Robot After Fracture Surgery

Abstract In order to address the problem of functional rehabilitation after ankle fracture surgery, this paper presented a novel ankle fracture rehabilitation robot. The robot adopted R-3RRS-P hybrid structure, which was simple in structure and had two working modes: rehabilitation training and motion axis switching. Compared with the existing ankle rehabilitation robot, the proposed robot could simulate more realistic kinematics of the ankle joint complex. Additionally, different body types of patients could be adapted. The kinematic and static models were established in detail using geometric method and screw theory. The coverage of the healthy ankle motion ability was formulated as an optimization problem to improve the robot's performance. Multi-objective optimal design was carried out to determine the dimensional parameters. The interference-free working space was calculated by numerical method. A prototype of the proposed robot was developed, and a series of experiments were performed to evaluate the function and feasibility of the proposed robot.

Geometric Error Identification of Gantry-Type CNC Machine Tool Based on Multi-Station Synchronization Laser Tracers

Laser tracers are a three-dimensional coordinate measurement system that are widely used in industrial measurement. We propose a geometric error identification method based on multi-station synchronization laser tracers to enable the rapid and high-precision measurement of geometric errors for gantry-type computer numerical control (CNC) machine tools. This method also improves on the existing measurement efficiency issues in the single-base station measurement method and multi-base station time-sharing measurement method. We consider a three-axis gantry-type CNC machine tool, and the geometric error mathematical model is derived and established based on the combination of screw theory and a topological analysis of the machine kinematic chain. The four-station laser tracers position and measurement points are realized based on the multi-point positioning principle. A self-calibration algorithm is proposed for the coordinate calibration process of a laser tracer using the Levenberg–Marquardt nonlinear least squares method, and the geometric error is solved using Taylor’s first-order linearization iteration. The experimental results show that the geometric error calculated based on this modeling method is comparable to the results from the Etalon laser tracer. For a volume of 800 mm × 1000 mm × 350 mm, the maximum differences of the linear, angular, and spatial position errors were 2.0 μm, 2.7 μrad, and 12.0 μm, respectively, which verifies the accuracy of the proposed algorithm. This research proposes a modeling method for the precise measurement of errors in machine tools, and the applied nature of this study also makes it relevant both to researchers and those in the industrial sector.

The theory of screws derived from a module over the dual numbers

AbstractThe theory of screws clarifies many analogies between apparently unrelated notions in mechanics, including the duality between forces and angular velocities. It is known that the real 6-dimensional space of screws can be endowed with an operator $$\mathcal {E}$$ E , $$\mathcal {E}^2=0$$ E 2 = 0 , that converts it into a rank 3 free module over the dual numbers. In this paper we prove the converse, namely, given a rank 3 free module over the dual numbers, endowed with orientation and a suitable scalar product ($$\mathbb {D}$$ D -module geometry), we show that it is possible to define, in a canonical way, a Euclidean space so that each element of the module is represented by a screw vector field over it. The new approach has the effectiveness of motor calculus while being independent of any reduction point. It gives insights into the transference principle by showing that affine space geometry is basically vector space geometry over the dual numbers. The main results of screw theory are then recovered by using this point of view.

Screw motion surfaces of constant mean curvature in homogeneous 3-manifolds

We study the geometry of non-minimal surfaces of supercritical constant mean curvature invariant under screw motions in the homogeneous 3-manifolds E(κ,τ) including the space-forms of non-negative curvature. We give a complete classification, thereby unifying and extending various previous results. We give the first classification for the Berger sphere case, and we exhibit a new family of screw motion CMC surfaces, called tubes.

Adaptive backstepping controller based on a novel framework for dynamic solution of an ankle rehabilitation spherical parallel robot

AbstractThis research offers an adaptive model-based methodology for autonomous control of 3-RRR spherical parallel manipulator (RSPM) based on a novel modeling framework. RSPM is an overconstrained parallel mechanism that has a variety of applications in medical procedures such as ankle rehabilitation because of its precision and accuracy. However, obtaining a complete explicit dynamic model of these mechanisms for tracking purposes has been a problematic challenge due to their inherent singularities, coupling effects of the limbs, and redundant constraints imposed by the intermediate joints. This paper presents a novel algorithm to obtain the analytical kinematic solutions of RSPMs based on the closed-loop vector method, which includes constraint analysis. By incorporating constrained kinematics into the dynamic model, a comprehensive explicit dynamic solution of the non-overconstrained version 3-RCC of RSPM is developed in task space, based on screw theory and the linear homogeneous property of algebraic equations on the manipulator twist. Based on the proposed computational framework, a robust self-tuning backstepping control (STBC) strategy is applied to the robot to overcome the effect of external disturbances and time-varying uncertainties. Furthermore, an observer-based compensation (OBC) method is presented for dealing with the nonlinear hysteresis loops of the ankle during trajectory tracking purposes. The closed-loop stability of the whole system including STBC and OBC is theoretically performed by Lyapunov methods. The proposed methodologies are validated by realistic co-simulations in different scenarios. For instant, in the presence of external disturbances, the maximum tracking error norm of STBC is 37.5% less than the sliding mode approach.

Design, analysis, and ground deployment test of space deployable mast based on double-layer deployable unit

The space deployable mast has been successfully applied to space equipment such as solar sails and space telescopes. At present, the deployable units of the space deployable mast mostly adopt the configurations of deployable units of FAST (Folding Articulated Square Truss), HIMAT, and ADAM (Able Deployable Articulated Mast). The configuration of the deployable unit is relatively few. Besides, there is a lack of ground micro-gravity environment simulation test devices for space deployable masts. This paper proposes the space deployable mast based on the double-layer deployable unit. Firstly, the configuration design method of the double-layer deployable unit based on the screw theory has been carried out. Three double-layer hybrid mechanisms serving as the basic units of the deployable mast have been obtained. Then, the kinematic and dynamic models of deployment of the space deployable mast have been established. And the deployment simulation based on MATLAB and Simscape has been performed. Furthermore, modal and static simulations have been performed to obtain the fundamental frequencies and maximum deformation displacements of the deployable masts consisting of 1, 2, 4, 8, and 12 units with and without cables. Finally, the ground test device has been developed to carry out the ground deployment test of the space deployable mast in the ground micro-gravity environment. It has been found that the ground test device could achieve the micro-gravity simulation level of 90.5%.

Innovative design methods for the geometric accuracy of machine tool guide rail oriented to spatial accuracy

The design of geometric accuracy in machine tools is crucial for enhancing machining precision. This study introduces geometric accuracy design methods that are oriented to spatial accuracy to guide the assembly of machine tool guide rails. Firstly, based on screw theory, a spatial error model for the machine tool is constructed, elucidating the relationship between spatial error and geometric error. Secondly, an error transmission model linking the guide rail installation error with the geometric error of moving parts is established, predicated on the rigid body hypothesis. This model reveals the mechanism of the error homogenisation in the machine tool feed system. Then, novel geometric accuracy design methods oriented to spatial accuracy are proposed catering to both non-morphological and morphological design requirements. Finally, the effectiveness and universality of the design methods are verified by simulations and experiments. The results show that more than 95.62 % of the machine tool workspace simulation points are less than 15 μm in spatial accuracy, and the design results outperform the experimental results by 2.05 μm in terms of the body diagonal spatial error.

Formalizing screw theory with 3D geometric algebra

This paper is intended for students and researchers looking for more insight into Screw Theory. It shows how algebraic considerations lead to both physical and geometrical understanding of screws, and how they can connect affine geometry (what the world is) to linear algebra (what we can easily compute). Various formulations of the theory are first reviewed, as each of them highlights a particular aspect of screws. Their respective qualities and defects are also discussed. Subsequently, the powerful framework of Geometric Algebra (GA) is introduced to elucidate the nature of various physical objects commonly associated with screws, and eventually a new formalism based on GA is proposed, in which traditional screws clearly appear as a special case of more general affine objects. This approach generalizes the concept of a screw in a coordinate-free and origin-independent form. A simultaneous proof of Euler’s First and Second Laws is provided to illustrate the use of this formalism.

Research on synthesis method of weakly coupled three-translation parallel mechanism with large bearing capacity

Aiming at the problem of poor bearing capacity in the direction of decoupled DOF of weakly coupled three-translation parallel mechanism, a class of new configurations of large load-bearing capacity weakly coupled three-translation parallel mechanisms with closed-loop unit has been proposed in this paper. Based on the screw theory, configuration evolution method, and graphical approach method, the synthesis method of these kinds of mechanisms has been proposed, and the decoupled criterion of this mechanism and the addition conditions of the kinematic pair diagram are given. The synthesis process has been developed to synthesize the single degree of freedom translational decoupled branch chain and 2T coupled branch chain. To better apply the weakly coupled mechanism to large load occasions, a closed-loop unit construction method has been proposed. Based on the theory of singular value decomposition, a theoretical formula for increasing the load-bearing capacity of a mechanism by constructing a closed-loop unit with redundant drives has been derived for the first time. The closed-loop unit is added to the 2T coupled branch chain and the single DOF decoupled branch chain to construct the large load-bearing capacity single DOF decoupled three-translation parallel mechanism. Kinematic analysis and static analysis have been carried out on the synthesized mechanism. The Jacobian matrix of the synthesized mechanism is a block diagonal matrix, and the bearing capacity is significantly stronger than that of the decoupled mechanism, even higher than that of the coupled parallel mechanism. The synthesized weak coupling three translation parallel mechanism with large load bearing characteristics has a good application prospect in the field of high precision and large load bearing.

Geometric error self-calibration method of five-axis dispensing machine based on the product of exponentials formula

Geometric error identification and kinematic calibration are common and effective methods for improving the machining accuracy of five-axis machines. Compared with the traditional measurement methods characterized by complicated inspection processes and high cost, this paper proposes a self-calibration method for a five-axis dispensing machine based on monocular vision and the product of the exponentials formula. The proposed method can automatically identify geometric errors and complete kinematic calibration, even for simultaneous five-axis motion. First, a ChArUco marker-based pose detection system is designed to achieve a wide range of calibration space for a five-axis dispensing machine. Based on this, the proposed sub-pixel checkpoint detection framework has higher pose detection performance accuracy at different depths of the field. Then, a general kinematic error model of five-axis machines is established using screw theory to avoid singularities. Moreover, the identification method of its geometric error is proposed for an RRTTT-type five-axis dispensing machine. Finally, an arc-shaped dispensing trajectory for the 3D glasses' inner wall is used for experimental validation. The experimental results show that the average orientation quantization error decreases from 0.0112 rad to 0.0039 rad. In contrast, the average position error decreases from 0.1627 mm to 0.0473 mm before and after the self-calibration, respectively. Thus, the convenience, effectiveness, and accuracy of the self-calibration method are validated on this five-axis dispensing machine.