Lower-mobility Parallel Mechanisms Research Articles

Design and Experimental Research of 3-RRS Parallel Ankle Rehabilitation Robot.

The ankle is a crucial joint that supports the human body weight. An ankle sprain will adversely affect the patient’s daily life, so it is of great significance to ensure its strength. To help patients with ankle dysfunction to carry out effective rehabilitation training, the bone structure and motion mechanism of the ankle were analyzed in this paper. Referring to the configuration of the lower-mobility parallel mechanism, a 3-RRS (R and S denote revolute and spherical joint respectively) parallel ankle rehabilitation robot (PARR) was proposed. The robot can realize both single and compound ankle rehabilitation training. The structure of the robot was introduced, and the kinematics model was established. The freedom of movement of the robot was analyzed using the screw theory, and the robot kinematics were analyzed using spherical analytics theory. A circular composite rehabilitation trajectory was planned, and the accuracy of the kinematics model was verified by virtual prototype simulation. The Multibody simulation results show that the trajectory of the target point is basically the same as the expected trajectory. The maximum trajectory error is about 2.5 mm in the simulation process, which is within the controllable range. The experimental results of the virtual prototype simulation show that the maximum angular deflection error of the three motors is 2° when running a circular trajectory, which meets the experimental requirements. Finally, a control strategy for passive rehabilitation training was designed, and the effectiveness of this control strategy was verified by a prototype experiment.

Constraint characteristics and type synthesis of two families of 1T2R parallel mechanism

AbstractPU- and P*U*-equivalent parallel mechanisms (PMs) are critical families of PMs with one translational and two rotational (1T2R) degree of freedoms and have always been a research hotspot among lower mobility PMs. However, researches on these two families of PMs remain to be inadequate, and existing types are few and uncomprehensive. In this study, first, general wrench systems of PU-equivalent are derived based on virtual-chain approach and screw theory, revealing its constraint characteristics under general configuration. Cause of one parasitic motion is put forward and general wrench systems of P*U*-equivalent PMs with one parasitic motion are obtained. In addition, constraint analysis has been carried out to figure out constraint characteristics of P*U*-equivalent PMs with three parasitic motions. Second, branch chains are divided by generating constraints, and their structures are synthesized based on the presented rules. Then, the process for type synthesis of PU-equivalent PMs and P*U*-equivalent PMs and a series of novel 1T2R PMs are attained based on this. Finally, a novel PU-equivalent PM, 2PRU-PRUPc, and a novel P*U*-equivalent PM with one parasitic motion, 2PRU-PUU PM, are analyzed, demonstrating the effectiveness of the proposed type synthesis method.

Analysis of rotation capacity of a novel 2R1T mechanism based on origami of thick panels

A foldable and symmetrical lower-mobility parallel mechanism was proposed based on Waterbomb origami of thick panels. It consists of a moving platform, a base plate and three deployable foldable legs between moving platform and base plate. Firstly, constraint wrenches of each leg were formulated based on screw theory and the results illustrated that the moving platform is in possession of two degrees of orientation freedom and one translational degree of freedom. Secondly, it was approved that base and moving platform are always symmetrical about a middle plane and the moving platform can rotate continuously about any axis chosen freely on this plane. Solving models including forward and inverse position problems were established to determine the maximum rotational angle and workspace. Finally, performance indexe of maximum rotational angle of the PM was analyzed, and effects of two structural variables to the performance were summarized. Conclusions obtained can provide a theoretical basis for the structural design and engineering application of this 2T1R parallel mechanism.

Configuration synthesis and structural design of lower-mobility parallel external fixators based on corrective degree-of-freedom classification

Foot and ankle deformity remains a common presenting complaint in orthopedic surgery. External fixator is a key medical device to assist gradual correction. Its configuration selection has a significant influence on correction results. Among two external fixators currently used in clinic, conventional Ilizarov external fixator (IEF) has high use difficulty and low accuracy, whereas the single configuration of later hexapod external fixator (HEF) leads to inadequate adaptation to diverse deformity types in foot and ankle. To solve the above problems, this paper presents a novel parallel distraction apparatus (PDA). First, corrective degree-of-freedom (c-DOF) is defined to determine expected posture transformation of external fixator required by corrective manipulation. By summarizing and classifying c-DOF, predominant foot and ankle deformities with fewer correction requirements are divided into six types. Subsequently, six lower-mobility parallel mechanisms are introduced as the configurations of external fixators via the c-DOF based configuration synthesis approach. After performing reconfigurable structural design, PDA can respectively form six types of parallel external fixators for six deformity types. Finally, clinical case simulations are provided to verify the application potential and advantages of PDA.

Screw rolling between moving and fixed axodes traced by lower-mobility parallel mechanism

Analogous to the null space map, this work proposes a “unit space map” to identify the generalized transmission wrench screw, which plays an important role in the general method determining the multi-parameter instantaneous screw of lower-mobility parallel mechanism which can be expressed as the linear combination of several proper basis screws of a twist screw system. Based on the fundamental equation, the screw rolling between moving and fixed axodes is decomposed into the linear superposition of several coupling screw rollings associated with several pairs of moving and fixed envelope axodes of proper basis screw, respectively. Moreover, several numerical examples of SP+SPR+SPU and 3-UPU parallel mechanisms are presented to verify the linear superposition, unit space map, and the method determining the multi-parameter instantaneous screw. Finally, the graph of the velocity-ratio parameter is employed to analyze the influence of input velocities on the axode shape of the 3-UPU PM when it undergoes two motion patterns: constant velocity-ratio and variable velocity-ratio.

Actuation Spaces Synthesis of Lower-Mobility Parallel Mechanisms Based on Screw Theory

The lower-mobility parallel mechanism has been widely used in the engineering field due to its numerous excellent characteristics. However, little work has been devoted to the actuator selection and placement that best satisfy the system’s functional requirements during concept design. In this study, a unified approach for synthesizing the actuation spaces of both rigid and flexure parallel mechanisms has been presented, and all possible combinations of inputs could be obtained, laying a theoretical foundation for the subsequent optimization of inputs. According to the linear independence of actuation space and constraint space of the lower-mobility parallel mechanism, a general expression of actuation spaces in the format of screw systems is deduced, a unified synthesis process for the lower-mobility parallel mechanism is derived, and the efficiency of the method is validated with two selective examples based on screw theory. This study presents a theoretical framework for the input selection problems of parallel mechanisms, aiming to help designers select and place actuators in a correct and even optimal way after the configuration design.

A Systematic Approach for Accuracy Design of Lower-Mobility Parallel Mechanism

SUMMARYGeometric accuracy is a critical performance factor for parallel robots, and regardless of error compensation, accuracy design or tolerance allocation is another way to ensure the pose accuracy of a robot at design stage. A general method of both geometric error modeling and accuracy design of lower-mobility parallel mechanisms is presented. First, a general approach for error modeling of lower-mobility parallel mechanism is proposed based on screw theory, and then the geometric errors affecting the compensatable and uncompensatable accuracy of the end-effector are separated using the properties of dual vector space. The pose error aroused by compensatable geometric errors can be compensated via kinematic calibration, while the uncompensatable geometric errors should be minimized during the manufacturing and assembly processes. Based on that, the tolerance allocation method is presented, giving each uncompensatable geometric error a proper tolerance by the use of reliability theory. Compared with the traditional tolerance allocation method, the advantages of the proposed method are as follows: the number of geometric errors to be allocated is greatly reduced; the results of serialized tolerance allocation can be obtained according to different reliability indices of pose accuracy of end-effector for designers to choose; on the premise of guaranteeing the same pose accuracy of end-effector, the allocated tolerances are loose and easy to realize. Finally, the proposed methods are successfully applied to an R(2-RPS&RP)&UPS lower-mobility parallel robot, and the effectiveness and practicability of the proposed method are verified.

Kinematic analysis of a novel 2-PrRS-PR(P)S metamorphic parallel mechanism

Lower mobility parallel mechanisms have been developed in different structures and widely applied in industry, but still have disadvantages in various tasks and functional requirements. A 2-PrRS-PR(P)S metamorphic parallel mechanism with two working configurations and a transiting configuration is presented. First, the architecture and the way of metamorphosis about the mechanism are described in detail. The mobility of the metamorphic parallel mechanism is obtained with screw theory. Furthermore, the parasitic motions in different configurations are derived based on the geometry constraint conditions. Both the inverse kinematic problem and forward kinematic problem of the mechanism are investigated by the closed-loop equation and validated via numerical examples. Then, the velocity and acceleration in two working configurations are obtained by the derivation of the inverse kinematic problem. Finally, the reachable orientation workspaces are discussed using the three-dimensional search method in different configurations. A comparison with the 3-PRS PM without metamorphic mechanism in workspace is presented and an example of the metamorphic parallel mechanism in robotic supporting leg is presented. The above analyses provide theoretical foundations for application of this mechanism.

Operation mode analysis of lower-mobility parallel mechanisms based on dual quaternions

This paper aims to provide an efficient way for analyzing operation modes and revealing corresponding motion characteristics of lower-mobility parallel mechanisms (LMPMs) using unit dual quaternions. Unit dual quaternions are first classified into 132 cases corresponding to 132 elementary operation modes based on the number of constant zero components. Meanwhile, the kinematic interpretation of these cases is presented in detail. Then a general method for analyzing operation modes and revealing motion characteristics of LMPMs is proposed according to unit dual quaternions and geometrical constraints. By this means, operation modes of LMPMs with complicated constraint conditions can also be analyzed, where a prime decomposition of the ideal corresponding to constraint equations in this condition is infeasible. Taken a 3-RSR LMPM and a 3-RPS LMPM as examples, the operation modes and motion characteristics can be obtained by the proposed approach. It is shown that the former LMPM has seven operation modes including two elementary operation modes and five extra operation modes. Under certain operation modes, the zero-torsion motion of the 3-RSR LMPM can not even be achieved. On the other hand, the latter has two operation modes in which the parasitic motion exists. To gain a deeper insight into the physical meaning of the operation modes, axodes are analyzed and drawn by mean of the velocity operator. It is demonstrated that the 3-RSR LMPM can realize an equal-diameter spherical pure rolling movement with variable diameters and the 3-RPS LMPM can achieve a rolling movement accompanied by sliding.

An iterative method for the inverse kinematics of lower-mobility parallel mechanism with three RS or SR chains based on kinematically equivalent mechanism

This paper presents an iterative method for solving the inverse kinematics (IK) of lower-mobility parallel mechanisms (LMPMs) with three RS or SR chains based on the kinematically equivalent mechanism (KEM). KEM is a virtual parallel mechanism with 6 DOF having the same motion as the original LMPM under the actuated joint motion constraints. The method enables us to solve the IK of such parallel mechanisms considering various structural parameters and different sets of output degrees of freedom (DOF) in a unified way, yet without formulating conventional constraint equations. As a demonstration, an iterative procedure for solving the IK of a 3-PRS parallel mechanism based on its KEMs is presented. Numerical examples show the process in solving the IK of the LMPMs considering different sets of output DOFs and various structural parameters using the proposed method.

Task-oriented type synthesis of the lower-mobility parallel mechanisms with a common platform

A systematic procedure for the task-oriented type synthesis of parallel mechanisms (PMs) with a common platform is proposed in this study. The method is composed of two stages. In the first stage, constraint wrenches that are compatible to specific task-oriented output motions of the PM is distributed to several limbs. In the second stage, the geometric conditions and admissible joint structures of PM limbs are identified by employing the concept of reciprocal screw. Then desired symmetric or asymmetric PMs are constructed with the proper combinations of those admissible limbs. Lastly, the suggested synthesis method is verified by identifying various forms of lowermobility PMs, such as planar 1T-type PMs, spatial 2T-type PMs and spatial 1T2R-type PMs.

Kinematic performance evaluation of 2-degree-of-freedom parallel mechanism applied in multilayer garage

In this article, our recent work on a kind of 2-degree-of-freedom lower-mobility parallel mechanism, which has one rotation degree of freedom and one translational degree of freedom, used in multilayer garage is presented. It has the following characteristics: lower-mobility, non-symmetric structure but can realize symmetric movement and a good compatibility for different kinds of lifting work. Kinematic performance should be considered in the first of designing a new kind of mechanism, the optimal kinematic design and analysis of this lower-mobility parallel mechanism are primarily investigated. In process of study, the global conditioning index over workspace is adopted, we establish a new evaluation method for the lower-mobility parallel mechanism, called global symmetry index and simulation results are shown. In addition, the flexible workspace of this lower-mobility parallel mechanism is also proposed. The evaluation index can be also applied on other lower-mobility parallel mechanism, which needs steady and symmetric movement.

A novel numerical approach for workspace determination of parallel mechanisms

In this paper, a novel numerical approach is proposed for workspace determination of parallel mechanisms. Compared with the classical numerical approaches, this presented approach discretizes both location and orientation of the mechanism simultaneously, not only one of the two. This technique makes the presented numerical approach applicable in determining almost all types of workspaces, while traditional numerical approaches are only applicable in determining the constant orientation workspace and orientation workspace. The presented approach and its steps to determine the inclusive orientation workspace and total orientation workspace are described in detail. A lower-mobility parallel mechanism and a six-degrees-of-freedom Stewart platform are set as examples, the workspaces of these mechanisms are estimated and visualized by the proposed numerical approach. Furthermore, the efficiency of the presented approach is discussed. The examples show that the presented approach is applicable in determining the inclusive orientation workspace and total orientation workspace of parallel mechanisms with high efficiency.

Review on 2R1T 3-DOF parallel mechanisms

Mechanisms are the skeletons and end-effectors of manufacturing equipments. The innovation of mechanism is indispensable to the independent innovation of manufacturing equipments. Compared with serial industrial manipulators, parallel mechanisms offer high rigidity, good dynamic response, potential high accuracy and large payload capability with the price of less dexterity, complex distribution of singular configurations and limited workspace. Over the last two decades, lower-mobility parallel mechanisms whose degrees of freedom (DOF) are fewer than six have been successfully implemented in many applications like positioning, orientation, and axis-symmetrical machining, which do not reqiure all six DOFs. With the well optimized architecture of a lower-mobility parallel mechanism, the reduced cost of fabrication, actuation, control and maintenance can be obtained compared with their 6-DOF counterparts. The 2R1T (R: rotation, T: translation) 3-DOF parallel mechanisms are most typical examples of successful applications in advanced manufacturing equipments such as parallel kinematic machine, industrial robot, and articulated tool head. Based on the relative geometrical relations between the two axes of rotation, the 2R1T 3-DOF parallel mechanism is classified into four categories: UP, PU, RPR and P*U*. The motion of the moving platform of a UP (U: universal joint, P: prismatic joint) parallel mechanism can be generated by a UP serial chain, including a rotation about a fixed axis, a rotation about a moving axis perpendicular to the fixed one, and a translation along the normal of the two axes. The motion of the moving platform of a PU parallel mechanism can be generated by a PU serial chain, including a rotation about an orientation-fixed axis, a rotation about a moving axis perpendicular to the orientation-fixed one, and a translation perpendicular to the orientation-fixed one. The motion of the moving platform of a RPR parallel mechanism can be generated by a RPR serial chain, including a rotation about a fixed axis near the base, a translation perpendicular to the fixed revolute axis and a rotation about a moving axis perpendicular to the orientation-fixed one. When the orientation and position of the two rotational axes of a 2R1T parallel mechanism are time varying, the motion of the moving platform cannot be modeled by a 3-DOF serial chain and is denoted by a virtual serial chain P*U*. The current research status of study on type synthesis, performance analysis and dimensional synthesis of the UP, PU, RPR and P*U* parallel mechanisms are reviewed and discussed. It is worth noting that despite the fact that many well-acknowledged progresses have been obtained by Chinese researchers in type synthesis, kinematic analysis and synthesis of lower-mobility parallel mechanisms, there is still a lack of basic principles and methods for selecting the architectures with engineering potentials from the massive outputs of type synthesis. This is an important challenge that the Chinese researchers in the fundamental research area of parallel mechanisms must face to support the self-innovation of manufacturing equipments. In addition, there are many open problems concerning rigid-flexible coupled dynamic design, accuracy calibration, and control of 2R1T 3-DOF parallel mechanisms, which require an immediate deployment of systematic and thorough research.

Singularity analysis of a 3-RPS parallel manipulator using geometric algebra

Singular configurations must be avoided in path planning and control of a parallel manipulator. However, most studies rarely focus on an overall singularity loci distribution of lower-mobility parallel mechanisms. Geometric algebra is employed in analysis of singularity of a 3-RPS parallel manipulator. Twist and wrench in screw theory are represented in geometric algebra. Linear dependency of twists and wrenches are described by outer product in geometric algebra. Reciprocity between twists and constraint wrenches are reflected by duality. To compute the positions of the three spherical joints of the 3-RPS parallel manipulator, Tilt-and-Torsion angles are used to describe the orientation of the moving platform. The outer product of twists and constraint wrenches is used as an index for closeness to singularity(ICS) of the 3-RPS parallel manipulator. An overall and thorough perspective of the singularity loci distribution of the 3-RPS parallel manipulator is disclosed, which is helpful to design, trajectory planning and control of this kind of parallel manipulator.

A method for force analysis of the overconstrained lower mobility parallel mechanism

A novel method for force analysis of the overconstrained lower mobility parallel mechanisms (LMPM) is proposed. Firstly, definitions of the constraint wrenches and overconstraint wrenches are put forward. Then, overconstrained LMPMs are classified into two classes based on characteristic of the elastic deformations generated at the end of the supporting limbs, which are limb stiffness decoupled overconstrained LMPM and limb stiffness coupled overconstrained LMPM, respectively. Corresponding to the limb stiffness decoupled and coupled overconstrained LMPMs, stiffness matrix of the supporting limb's overconstraint wrenches and constraint wrenches are introduced and defined respectively. Next, taking the link's spatial composite elastic deformations into account, the stiffness matrix of the supporting limb's overconstraint wrenches or constraint wrenches is established, and general analytical expression for the solution of each supporting limb's overconstraint wrenches or constraint wrenches is derived. Furthermore, the internal relation between the constraint wrenches and the actual joint reactions is discussed. Finally, based on Adams simulation software, a method to set up the force simulation model of the overconstrained LMPM is put forward, and the simulation results and the theoretical calculation ones are basically consistent, which effectively verifies the correctness of the proposed method for force analysis of the overconstrained LMPM.

Kinematic and Stiffness Analysis of Lower-mobility Parallel Mechanisms Based on QR Decomposition

Kinematic and Stiffness Analysis of Lower-mobility Parallel Mechanisms Based on QR Decomposition

Rigid-Flexibility Modeling and Simulations of Lower-Mobility Parallel Mechanisms

Considering the flexibility performance of lightweight slender axle in Delta parallel mechanisms, the effects of the elastic deformation on the kinematic accuracy of the mechanism are calculated based on the virtual prototype technology. Firstly, the 3D model of the mechanism is built via ADAMS software. Secondly, the geometric model of the driving rod is imported into ANSYS software to get meshed and generate the modal neutral files. Then, getting back to ADAMS, the original rigid component is replaced with the flexible body which constraint and driving is added to lately. At last, the rigid-flexibility hybrid model of the mechanism is built. Each frequency and vibration mode of the flexible body is calculated, using the simulation capabilities of the software. Furthermore, changes in motion accuracy of the mechanism under the effect of the force-elastic coupling are analyzed comparatively.

Coupling 3-PSR/PSU 5-axis compensation mechanism for stabilized platform and its analysis

This article presents a coupling 3-PSR/PSU parallel mechanism for ship-based stabilized platform based on the coupling characteristics of the ship motions. With coupling characteristics, the goal of 3-axis drives and 5-axis compensation of the moving platform is achieved. From the screw theory, degrees-of-freedom and coupling kinematic characteristics of the mechanism are analyzed. A method for equivalent transformation of motion parameters is proposed. This presents a mapping relationship between the independent degrees-of-freedom and the coupling degrees-of-freedom. The investigations are further extended to the coupling dynamics characteristics of the mechanism. In particular, the effects produced by the coupling force/torque to the system are analyzed. Hence, taking minimum system driving force as the optimization goal, the coupling dynamic equation of the parallel mechanism is established. The numerical example reveals that this coupling parallel mechanism can be applied to the multi-axis compensation of ship-based stabilized platform. The study also enriches the application area of lower-mobility parallel mechanism.

Analysis and Design of Lower-Mobility Parallel Mechanism of Non-Symmetrical Based on Variable Topology Theory

First, the branched-chain of parallel mechanism was Classified according to reciprocal screw theory. Then, the introduction of variable topology mechanism theory, with the characteristics of parallel mechanisms themselves, the definition and basic variable topology means of variable topology parallel mechanism were given. With evolutionary theory, the method to design lower-mobility parallel mechanisms of non-asymmetric was proposed based on variable topology mechanism theory .Taking 3-RPS as ideal mechanism and topology synthesis was carried out, besides 2-RPS mechanism were analyzed. The introduction of variable topology mechanism theory provided a theoretical basis and innovative approaches for the synthesis configuration of Lower-mobility parallel mechanisms of non-asymmetric.