6-DOF Parallel Manipulator Research Articles

Kinematics, singularity, and workspace analysis of a spatial parallel robot with pure translational motion in a plane for pick-and-place operations

This research presents a new spatial Parallel Manipulator (PM) with pure translational motion in a plane or 2-DOF PM for pickand-place operation. This spatial 2-DOF PM is constructed by the 2(RRR)-2(PRRR) kinematic chain and is named Ψ2. This Ψ2 PM offers simplicity in its architecture because each limb applies a 1- DOF joint. The applicability of Ψ2 for pick-and-place operations was investigated by performing kinematic and singularity analyses at the preliminary stage. The singularity of Ψ2 was determined by relying on its Jacobian matrices, where three kinds of singularity can be evaluated. Next, the singularity and Jacobian matrix hold a key to identifying the workspace in a regular geometrical shape required for the operation of such an application. The identified operational workspace must be free of singularity and good conditioning workspace. In this study, two predetermined values of Ψ2 kinematic parameters were used to evaluate the kinematic characteristics and singularity of the manipulator as well as to identify its operational workspace. According to the results, Ψ2 has a good prospect of being applied to high-speed pick-and-place operations.

Design and Control of a Pneumatic Muscle Servo Drive Applied to a 6-DoF Parallel Manipulator

Design and Control of a Pneumatic Muscle Servo Drive Applied to a 6-DoF Parallel Manipulator

Unified dynamics analysis of parallel manipulators: A joint-based approach and generalized inertia constraint matrix for parallel manipulators (GICM-P) framework

Parallel manipulators, a distinctive subset of closed-loop multi-body systems, are in high demand due to their precision-centric applications. This research introduces a unified approach to tackle both inverse and forward dynamic analyses of parallel manipulators, rooted in joint-based principles. The methodology dissects a given parallel manipulator into symmetric open-loop subsystems and a mobile body within either a planar or spatial context, depending on the manipulator’s nature. Conventional practices, involving the introduction of joint cuts at relevant locations, are employed to partition the system into multiple open-loop subsystems. Subsequently, the joint coordinate-based approach, typically applied to open-loop systems such as industrial manipulators, is utilized to derive solutions. In particular, this approach focuses on forward dynamics by introducing the generalized inertia constraint matrix for parallel manipulators (GICM-P), a concept built upon the authors’ prior work, originally addressing the GICM for general closed-loop systems. Notably, GICM-P aligns conceptually with the operational space inertia matrix (OSIM) designed for closed-loop systems elsewhere. However, unlike OSIM, which requires mapping joint-space inertia to operational-space inertia, GICM-P leverages acceleration-level constraints between subsystems and the moving platform through straightforward matrix operations. GICM-P offers a deeper understanding of the physics of the problem compared to OSIM, primarily due to its ability to explicitly express subsystem-level interactions via various block matrices – an aspect not previously documented. The paper provides explicit numerical values for GICM-P in the context of a spatial six degrees of freedom (6-DOF) Stewart platform and a planar 3-DOF parallel manipulator along with interpretations.

Structure design and kinematic performance of the deployable translational parallel tape-spring manipulator

AbstractA deployable manipulator has the characteristics of a small installation space and a large workspace, which has great application prospects in small unmanned platforms. Most existing deployable manipulators are designed based on rigid links, whose complexity and mass inevitably increase sharply with increasing numbers of rigid links and joints. Inspired by the remarkable properties of tape springs, this paper proposes novel deployable parallel tape-spring manipulators with low mass, simple mechanics, and a high deployed-to-folded ratio. First, a double C-shaped tape spring is presented to improve the stability of the structure. The combined fixed drive component (CFDC) and combined mobile drive component (CMDC) are designed. Then, novel 2-DOF and 3-DOF deployable translational parallel manipulators are proposed based on the CFDC and CMDC, and their degrees-of-freedom (DOFs), kinematics, and stability are analyzed. The coiled tape spring is regarded as an Archimedean spiral, which can significantly improve the accuracy of kinematic analysis. The correction coefficient of the Euler formula is obtained by comparison with simulation results and experimental results. Furthermore, the stability spaces of the 2-DOF and 3-DOF deployable parallel manipulators are given. Finally, a prototype is fabricated, and experiments are conducted to validate the proposed design and analysis.

Design and testing of a high precision parallel manipulator with hyperbolic–elliptic flexure hinges

A novel six-degree-of-freedom (6-DOF) parallel manipulator (PM) with a 6-PSS configuration is designed as a piston error compensation mechanism in an optical system. To meet the high-precision adjustment requirement, the prismatic hinges are driven by nanometer resolution piezoelectric motors, and hyperbolic–elliptic (H-E) flexure hinges are used as spherical hinges because of their zero clearance and greater precision transmission capability than traditional rigid hinges. The analysis of the closed-loop vector relationship of the kinematic chain of the leg results in the construction of the kinematic equation, which can satisfy the inverse kinematic solution within a certain precision. The Newton–Raphson method is used for the iteration to solve the forward kinematics. A kinematic simulation was performed to verify the accuracy and effectiveness of the kinematic model using the rigid–flexible coupling finite element method. The workspace of the PM was analyzed using inverse kinematics and kinematics simulation. Two test systems were constructed to test the working performance of the PM. The experimental tests demonstrated that the resolution reached the nanometer level and the travel reached the millimeter level.

A Comparative Study on Kinematic Calibration for a 3-DOF Parallel Manipulator Using the Complete-Minimal, Inverse-Kinematic and Geometric-Constraint Error Models

Kinematic calibration is a reliable way to improve the accuracy of parallel manipulators, while the error model dramatically affects the accuracy, reliability, and stability of identification results. In this paper, a comparison study on kinematic calibration for a 3-DOF parallel manipulator with three error models is presented to investigate the relative merits of different error modeling methods. The study takes into consideration the inverse-kinematic error model, which ignores all passive joint errors, the geometric-constraint error model, which is derived by special geometric constraints of the studied RPR-equivalent parallel manipulator, and the complete-minimal error model, which meets the complete, minimal, and continuous criteria. This comparison focuses on aspects such as modeling complexity, identification accuracy, the impact of noise uncertainty, and parameter identifiability. To facilitate a more intuitive comparison, simulations are conducted to draw conclusions in certain aspects, including accuracy, the influence of the S joint, identification with noises, and sensitivity indices. The simulations indicate that the complete-minimal error model exhibits the lowest residual values, and all error models demonstrate stability considering noises. Hereafter, an experiment is conducted on a prototype using a laser tracker, providing further insights into the differences among the three error models. The results show that the residual errors of this machine tool are significantly improved according to the identified parameters, and the complete-minimal error model can approach the measurements by nearly 90% compared to the inverse-kinematic error model. The findings pertaining to the model process, complexity, and limitations are also instructive for other parallel manipulators.

Exploiting the Kinematic Redundancy of a (6 + 3)-Degree-of-Freedom Parallel Manipulator to Produce Unlimited Rotation of the Platform

Abstract Mechanical interference and singularities within the reachable workspace often restrict the orientational workspace of parallel robots. Introducing kinematic redundancy can alleviate this limitation. This paper discusses the possibility to produce unlimited rotation of the platform of a tripedal (6 + 3)-degree-of-freedom kinematically redundant parallel robot. The articulated platform of such a robot has three degrees of mobility. The platforms considered here are planar linkages that contain either revolute or prismatic joints. It is shown that at least two revolute joints are required to produce unlimited rotation with appropriate design and initial configuration, while the platforms with two prismatic joints cannot produce such rotations without crossing a singularity.

Synchronous Sliding Mode Control for a 4-DOF Parallel Manipulator in Practice

This paper proposes a synchronous sliding mode control (SSMC) method for a 4-DOF parallel robot model in practice with the influence of uncertain dynamics such as friction, external noise, and model error. A closed kinematic chain structure characterizes the parallel robot. The moving platform is controlled by four kinematic chains working together to follow a predetermined trajectory. However, traditional control methods only control each joint individually; if one of the actuators fails or has an external force, it can lead to system breakdown. Therefore, synchronous control to coordinate the operation of the arms is necessary. The synchronization algorithm is characterized by the cross-coupling error, which is the combination of tracking and synchronization errors for the system to achieve the synchronization goal. Cross-coupling error is used to design the sliding surface of the SSMC. The advantages of sliding mode control (SMC) are that it resists the influence of uncertain dynamics and it is combined with the synchronization algorithm. A control law is developed based on the sliding surface to ensure simultaneous convergence of both tracking and synchronization errors towards zero. The stability of the system is proven using the Lyapunov theory. In order to verify the effectiveness of the proposed approach, a 4-DOF parallel robot testbench is constructed. The SSMC controller results are compared with the testbench’s proportional-integral-derivative (PID), synchronous proportional-integral-derivative (SPID), and SMC controllers.

Virtual Passive-Joint Space Based Time-Optimal Trajectory Planning for a 4-DOF Parallel Manipulator

Virtual Passive-Joint Space Based Time-Optimal Trajectory Planning for a 4-DOF Parallel Manipulator

Five-axis parallel mechanism system (PMS) CNC partial link control system based on modified inverse kinematic of 6-DOF UPS parallel manipulator

This paper presents a control system algorithm for a five-axis parallel mechanism system (PMS) CNC milling machine based on a 6-DOF Stewart platform parallel manipulator with a universal-prismatic-spherical (UPS) configuration. The control system reads the G-Code commands as standard CNC machine language, then extract data points and interpolates them to generate the robot trajectory patterns as motion references. Then, the control system uses the modified inverse kinematic equation to determine the length of each link to move the end effector to track the trajectory patterns from the previous G-code extraction process. The inverse kinematic equation is modified especially for the five-axis PMS CNC milling machine by including machine-offset and tools-offset parameters so it will be easier for the control system to implement the kinematic equation. As expected, the system simulation results successfully followed the G-Code program moving commands. The average error of the length control system is 0,1 mm, while the average error of the length change rate control system is 1,8 mm/s. The maximum error is 26.9 mm was caused by the system's inability to follow the motion profile in transient. It can be concluded that 6-DOF Stewart platform parallel structures,which provide better performance than serial structures, can be implemented as a new concept for the motion mechanism of five-axis CNC milling machines. The five-axis PMS CNC milling machine also promises better performance than conventional five-axis gantry structures CNC.

A 3-DoF robotic platform for the rehabilitation and assessment of reaction time and balance skills of MS patients.

The central nervous system (CNS) exploits anticipatory (APAs) and compensatory (CPAs) postural adjustments to maintain the balance. The postural adjustments comprising stability of the center of mass (CoM) and the pressure distribution of the body influence each other if there is a lack of performance in either of them. Any predictable or sudden perturbation may pave the way for the divergence of CoM from equilibrium and inhomogeneous pressure distribution of the body. Such a situation is often observed in the daily lives of Multiple Sclerosis (MS) patients due to their poor APAs and CPAs and induces their falls. The way of minimizing the risk of falls in neurological patients is by utilizing perturbation-based rehabilitation, as it is efficient in the recovery of the balance disorder. In light of the findings, we present the design, implementation, and experimental evaluation of a novel 3 DoF parallel manipulator to treat the balance disorder of MS. The robotic platform allows angular motion of the ankle based on its anthropomorphic freedom. Moreover, the end-effector endowed with upper and lower platforms is designed to evaluate both the pressure distribution of each foot and the CoM of the body, respectively. Data gathered from the platforms are utilized to both evaluate the performance of the patients and used in high-level control of the robotic platform to regulate the difficulty level of tasks. In this study, kinematic and dynamic analyses of the robot are derived and validated in the simulation environment. Low-level control of the first prototype is also successfully implemented through the PID controller. The capacity of each platform is evaluated with a set of experiments considering the assessment of pressure distribution and CoM of the foot-like objects on the end-effector. The experimental results indicate that such a system well-address the need for balance skill training and assessment through the APAs and CPAs.

Kinematic Calibration of a 6-DoF Parallel Manipulator With Random and Less Measurements

Parallel manipulator (PM) embraces more kinematic errors than serial mechanism. The substantial errors in PM usually demand more measurements, which leads to low efficiency and affects the accuracy of identification due to the ill-conditioning problem. Having realized this common problem in the kinematic calibration of PMs, we carried out a finite and instantaneous screw (FIS)-based kinematic calibration on a 6- <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</u> UHU PM before application to fracture reduction. <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</u> , U, and H denote actuated revolute joint, universal joint, and helical joint, respectively. The error model is built by the error mapping of serial chains in the FIS framework. Instead of devoting to the measurement planning for more precise identification, the identifiability of kinematic errors is thoroughly analyzed. Redundant errors are categorized as the ones within limb and among limbs. A minimal identification model is thus defined and an error conversion method is proposed for compensation. Calibration experiments show that the FIS method has an improved accuracy of the 6- <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</u> UHU PM at least one magnitude higher than the precision demand, i.e., mean position and orientation errors 0.0574 mm and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$5.1197\,\,{\times }\,\, 10^{-4}$ </tex-math></inline-formula> rad after calibration. Compared with the conventional closed-loop vector (CLV) equation method, the accuracy improvement of FIS method is higher. In addition, comparisons with the commonly used configuration selection methods indicate that our method allows random and a smaller number of measurements and can reach satisfactory accuracy.

Kinematic Modeling and Stiffness Analysis of a 3-DOF 3SPS + 3PRS Parallel Manipulator

The driving mechanism of an axisymmetric vectoring exhaust nozzle (AVEN) is a 3-degrees-of-freedom (3-DOF) parallel manipulator (PM) with a centering device. According to the characteristics of the vectoring nozzle’s movement mechanism, the 3-DOF kinematics model of the 3SPS + 3PRS PM is established. The forward and inverse positional posture models are built based on the mechanism architecture of PM and the closed-loop vector method. The Jacobian matrix and Hessian matrix are derived using screw theory, and velocity and acceleration models are established. Based on the principle of virtual work and combined with the kinematics model of PM, a stiffness model was established to analyze stiffness performance. Finally, the effectiveness of the kinematics model of the PM was verified by a simulation, and the variation of stiffness performances with the positional posture of the PM was quantitatively analyzed. The results show that the translational stiffness decreases with forward axial translation of the moving platform and the rotational stiffness is mainly determined by the posture parameters. This study is expected to offer ideas for the kinematic modeling of 3-DOF PM and provide references for application and optimizing of the AVEN’s driving mechanism.

Bacterial foraging optimization algorithm based on normal cloud model for forward kinematics of a 4-DOF parallel manipulator

For the 4-SPU parallel mechanism, the complex multivariate nonlinear equation of the forward kinematics solution is transformed into a minimization problem. This article proposes a new bacterial foraging optimization algorithm based on normal cloud model to obtain the solution of the forward kinematics. Adaptive dynamic adjustment of walking step size of normal forward cloud generator is adopted to improve the fixed step size. Y conditional cloud generator is used to dynamically adjust the migration probability. The simulation results show that new algorithm is an effective method to solve the forward solution optimization with high precision and high speed.

Nonlinear Computed Torque Control of 6-Dof Parallel Manipulators

Nonlinear Computed Torque Control of 6-Dof Parallel Manipulators

Analysis of coupling effects on hydraulic controlled 6 degrees of freedom parallel manipulator

This paper presents an analysis of the coupling effects between degrees of freedom of a hydraulic controlled six DOF parallel manipulator. Based on the singular value decomposition to the properties of its joint space inverse mass matrix, the method is put forward to analyze coupling effects between degrees of freedom using a transformation matrix, the product of transposed Jabobian matrix and an orthogonal unitary matrix, of which each element represents decoupled modal space coordinates with respect to physical task space frame. The simulation results in frequency domain prove the theoretical analysis right. The analysis will provide useful information to mechanism and controller designer, to asses from concept the coupling effects between DOF in respect of the requirement for a particular application and also for the study on decoupling control strategies.

Comparison of the Dynamic Performance of Planar 3-DOF Parallel Manipulators

This paper presents a comprehensive comparison study on the dynamic performances of three planar 3-DOF parallel manipulators (PPMs): 3-RRR, 3-PRR, and 3-RPR. In this research work, the discrete time transfer matrix method (DT-TMM) is employed for developing dynamic models of the planar parallel manipulators. Numerical simulations using the virtual work principle and ADAMS 2016 software are performed to verify the DT-TMM dynamic model of PPMs. Numerous dynamic performance indices, including dynamic dexterity, the power requirement, energy transmission efficiency, and the joint force/torque margin, are proposed to compare the dynamic performance of three PPMs under the general circular and linear trajectories. The comprehensive analyses and comparisons show that: (1) the 3-RRR PPM has advantages in terms of a circular trajectory, offering the best dynamic dexterity performance, the smallest power requirement, and the second-highest energy transfer efficiency; (2) the 3-PRR PPM performs best in terms of a linear trajectory, offering the best dynamic dexterity, the smallest power requirement range, and the best drive performance; and (3) the 3-RPR PPM has the highest energy transfer efficiency and demonstrates better dynamic performance in a circular trajectory compared to a linear trajectory.

Study on modeling and dynamic performance of a planar flexible parallel manipulator based on finite element method.

The application of a high-speed parallel manipulator necessitates the adoption of a lightweight design to reduce dead weight. However, this increases the elastic deformation of certain components, affecting the dynamic performance of the system. This study examined a 2-DOF planar flexible parallel manipulator. A dynamic model of the parallel manipulator composed of fully flexible links was established using a floating reference coordinate system and a combination of the finite element and augmented Lagrange multiplier methods. A dynamic analysis of the simplified model under three driving torque modes showed that the axial deformation was less than the transverse deformation by three orders of magnitude. Further, the kinematic and dynamic performance of the redundant drive was significantly better than that of the non-redundant drive, and the vibration was well suppressed in the redundant drive mode. In addition, the comprehensive performance of driving Mode 2 was better than that of the other two modes. Finally, the validity of the dynamic model was verified by modeling via Adams. The modular modeling method is conducive to the extension to other models and programming. Furthermore, the dynamic model of the established fully flexible link system can aid in optimizing the lightweight design and dynamic performance of the parallel manipulator.

Kinematics of a 6-DOF parallel manipulator with two limbs actuated by spherical motion generators

Inspired by dual-arm-like manipulation, a novel 6-DOF parallel manipulator with two spherical-universal-revolute limbs is proposed. Compared with general 6-DOF parallel manipulators with six limbs, this new manipulator actuated by spherical motion generators has only two limbs, which brings advantages such as fewer active limbs for avoiding interference, larger reachable and orientational workspace for complex operating, more actuators integrated in active modules for decreasing installation errors and increasing compactness. In this paper, the kinematics of this novel parallel manipulator is solved and illustrated, covering its inverse and forward position analysis, workspace and singularities. The kinematic study reveals interesting features of this manipulator such as multiple working and assembly modes, small footprint and large workspace volume with high dexterity. Numerical examples of kinematic analysis are included. Practical application of the new manipulator is illustrated.

A novel dynamic evaluation method and its application to a 4-DOF parallel manipulator

Taking into account that the velocity and acceleration limits of the end-effector are specified rather than the physical limits of the actuators in the design purpose of a robot, this paper proposes a novel method to evaluate the dynamic performance of the robot along joint-space directions. Based on the given ranges of end-effector velocity and acceleration in task space, the required torque along all joint-space directions are obtained based on the velocity term and the acceleration term in dynamic model, respectively. Furthermore, the actual torque range of each actuator is derived to produce the required torque along all joint-space directions. Considering all the effects of the acceleration term, the velocity term and the gravitational term, two global dynamic performance indices that reflect the minimum range of actuator torque and the ratio of the required torque range to the actual torque range in the whole workspace are presented. This method is validated by utilizing it to evaluate the dynamic performance of a novel 4-DOF parallel manipulator with translational and rotational motion for FSW. This method conforms with the design process of robotic manipulators and would be useful for optimum design, dynamic performance evaluation and actuator selection of a robot.