Stewart-Gough Platform Research Articles

Improving the Performance of a New Storage and Retrieval Machine Based on a Parallel Manipulator Using FMEA Analysis

Automated Storage/Retrieval Systems (AS/RS) are computer-controlled material handling systems that can automatically store and retrieve loads with high throughput. In this work, we estimate important design parameters for a newly designed storage/retrieval (SGP-AS/RS) machine based on the specific input details to come up with an optimized design solution. The new system is based on cable- driven parallel manipulator (or wire robot) technology that is similar to a parallel Stewart-Gough platform architecture. For the reliability of such a system, a Failure Mode Effect Analysis (FMEA) is carried out to find out how much reliable is this machine as a whole based on the of DIN's EN 60812 standards. This International Standard describes methods for FMEA and helps not only to optimize the machine, but also to achieve a disturbance free operation.

On the Numerical Modelling and Error Compensation for General Gough-Stewart Platform

Parallel robots are specially designed to perform high-precision tasks. Nevertheless, manufacturing, assembling and control issues can reduce their capacity to perform adequately. Observing the acquired measurement data with high-precision devices - such as laser-based instruments - it is not surprising that the error data follows patterns or have a structure because, in many cases, the greatest error comes from a mechanical bias introduced by manufacturing issues. Even though we cannot determine with certainty where the error comes from, a pattern in the measured data suggests that it is feasible that it can be modelled and corrected - in a significant proportion - by purely software applications, without the need of disassembling or re-manufacturing any component. This work deals with the problem of finding a mathematical model which adequately fits the error data from the legs of a general Gough-Stewart platform. Hence, we obtain an expression which can be subtracted from the control parameters in order to compensate the inherent mechanical error in the legs. The purpose of this article is two-fold: 1) to present numerical results of the beneficial effects of the error compensation in the legs as well as in the end-effector, and 2) to introduce a numerical methodology to find a model for error compensation and to numerically simulate its effects. Numerical, graphical and statistical evidence of the error improvements, according this methodology, is provided.

Parametric damped vibrations of Gough–Stewart platforms for symmetric configurations

Modal behavior of a Gough–Stewart Platform (GSP) is sensitive to several variables related to its inertia, damping and stiffness as well as its complex 3-D geometry. To optimize its dynamical performance, due to the complications of this system, it is crucial to have the equations parametrically at the neutral configuration. However, in the literature, no complete parametric solution to this problem is presented. In this paper, we establish a fully-parametric and closed-form model for the damped vibrations of GSPs. In particular, this analytical model can be used in order to design, optimize and control GSPs in high-precision/bandwidth applications. Parametric expressions of the damped eigenfrequencies and the corresponding eigenvectors as well as the Jacobian, stiffness and damping matrices are developed. Interestingly, despite the complexity of the system, it is shown how well-structured algebraic expressions are obtained using the Cartesian-space approach. Having analytically studied the eigenvectors, the conditions for decoupled vibrations are also analytically formulated. Finally, using a reference GSP, the sensitivity of the damped eigenfrequencies to stiffness and damping variations are investigated accompanied by a cross-check with an ABAQUS® simulation.

Singularity-free workspace analysis of general 6-UPS parallel mechanisms via convex optimization

This paper investigates the singularity-free zones in the workspace of general 6-UPS parallel mechanisms. The emphasis is placed on obtaining the maximum volume ellipsoid or sphere in the singularity-free subregions, while taking into account the actuator strokes. The proposed algorithms are based on the convex optimization and have several advantages over the existing methods such as Lagrange multipliers approach, which makes it appropriate for other applicable optimization problems. For determining the maximal singularity-free zones, in order to find the maximum volume ellipsoid, a judicious iterative procedure, referred to as Improved Lower Bound Semidefinite Programming, is proposed. Additionally, different reformulations of the problem are proposed in order to solve other interesting problems in the robotics community such as obtaining the maximum volume sphere. Furthermore, an approach based on the sum of squares method is proposed to solve this problem for a general Gough–Stewart platform with any arbitrary geometric parameter which is conducive to a polynomial optimization problem. The computational time for the proposed algorithms are considerably low compared to other methods proposed in the literature for obtaining the singularity-free workspace which opens an avenue to use them as systematic algorithms for on-line purposes.

Solution of direct kinematic problem for Stewart-Gough platform with the use of analytical equation of plane

The paper concerns the solution of direct kinematic problem for the Stewart-Gough platform of the type 6-3. The article represents a detailed analysis of methods of direct kinematic problem solution for platform mechanisms based on the parallel structures. The complexity of the problem solution is estimated for the mechanisms of parallel kinematics in comparison with the classic manipulators, characterized by the open kinematic chain. The method for the solution of this problem is suggested. It consists in setting up the correspondence between the functional dependence of Cartesian coordinates and the orientation of the moving platform centre on the values of generalized coordinates of the manipulator, which may be represented, in the case of platform manipulators, by the lengths of extensible arms to connect the foundation and the moving platform of the manipulator. The method is constructed in such a way that the solution of the direct kinematic problem reduces to solution of the analytical equation of plane where the moving platform is situated. The equation of the required plane is built according to three points which in this case are attachment points of moving platform joints. To define joints coordinates values it is necessary to generate a system of nine nonlinear equations. It ought to be noted that in generating a system of equation are used the equations with the same type of nonlinearity. The physical meaning of all nine equations of the system is Euclidean distance between the points of the manipulator. The location and orientation of the moving platform is represented as a homogeneous transformation matrix. The components of translation and rotation of this matrix can be defined through the required plane. The obtained theoretical results are supposed to be used in the decision support system during the complex research of multi-sectional manipulators of parallel kinematics to describe the geometrically similar 3D-prototype of the manipulator. In the process of work it is planned to make the optimal choice of numerical approach to the solution of the suggested equation system, considering the nonlinearity type etc., to develop the methodology for evaluation of received solutions, to adjust the suggested method for the solution of direct kinematic problem for the Stewart-Gough platform of the type 6-6.

Optimization Algorithms for Kinematically Optimal Design of Parallel Manipulators

Optimal design is an inevitable step for parallel manipulators. The formulated optimal design problems are generally constrained, nonlinear, multimodal, and even without closed-form analytical expressions. Numerical optimization algorithms are thus applied to solve the problems. However, the optimization algorithms are usually chosen ad arbitrium. This paper aims to provide a guideline to choose algorithms for optimal design problems. Typical algorithms, the sequential quadratic programming (SQP) with multiple initial points, the controlled random search (CRS), the genetic algorithm (GA), the differential evolution (DE), and the particle swarm optimization (PSO), are investigated in detail for their convergence performances by using two canonical design examples, the Delta robot and the Gough-Stewart platform. It is shown that SQP with multiple initial points can be efficient for simple design problems, while DE and PSO perform effectively and steadily for all design problems. CRS can be used to generate good initial points since it exhibits excellent convergence evolution in the starting period.

New Geometric Approaches to the Analysis and Design of Stewart–Gough Platforms

In general, rearranging the legs of a Stewart-Gough platform, i.e., changing the locations of its leg attachments, modifies the platform singularity locus in a rather unexpected way. Nevertheless, some leg rearrangements have been recently found to leave singularities invariant. Identification of such rearrangements is useful not only for the kinematic analysis of the platforms, but also as a tool to redesign manipulators avoiding the implementation of multiple spherical joints, which are difficult to construct and have a small motion range. In this study, a summary of these singularity-invariant leg rearrangements is presented, and their practical implications are illustrated with several examples including well-known architectures.

Tracking performance improvements of an electrohydraulic Gough-Stewart platform using a fuzzy incremental controller

Purpose– Sinusoidal signals are often used as the inputs of the six degree of freedom (DOF) motion simulator platforms. The purpose of this paper is to propose a fuzzy incremental controller (FIC) to improve sinusoidal signal tracking performances of an electrohydraulic Gough-Stewart platform (GSP).Design/methodology/approach– An FIC is proposed to control an electrohydraulic GSP without any model parameters. The FIC output can be self-organized by only using the hydraulic actuator position information. The control rules are determined by a systematic deterministic method.Findings– Experimental results show that the proposed FIC is valid and can achieve better tracking performances compared with classical PID controller and a decoupling controller (a model-based controller).Originality/value– An FIC using a systematic deterministic rule-base determination method is proposed to improve sinusoidal signal tracking performances of electrohydraulic GSP.

Actuation Torque Reduction in Parallel Robots Using Joint Compliance

This work studies in detail how the judicial application of compliance in parallel manipulators can produce manipulators that require significantly lower actuator effort within a range of desired operating conditions. We propose a framework that uses the Jacobian matrices of redundant parallel manipulators to consider the influence of compliance both in parallel with the actuated joints as well as the passive joints, greatly simplifying previous approaches. We also propose a simple optimization procedure to maximize the motor force reduction for desired regions of the workspace and range of external forces. We then apply the method to a Stewart-Gough platform and to a 3-URS (universal rotational and spherical joint) manipulator. Our results show that parallel manipulators with tasks that involve a preferred external force direction, as for instance, big weights in the platform, can see large reductions in actuator effort through the judicial use of compliant joints without significantly losing rigidity.

Optimum calibration of a parallel kinematic manipulator using digital indicators

This paper presents a calibration method for parallel manipulators using a measurement system specially installed on an external fixed frame. The external fixed frame is important as an error reference for calibration in certain operations, such as in the configuration of a parallel manipulator functioning as a machine tool where the workpiece is fixed to a worktable. The pose of the end-effector is measured using three digital indicators installed on the external fixed frame. To enable measurement, the end-effector is assumed to be a plane large enough that all digital indicators could touch. The error is defined as the difference between the theoretical and actual readings of the digital indicators. The geometric parameters of the parallel manipulator are optimized to minimize this error. This calibration method is low cost and feasible for compensating geometric parameter errors for a parallel manipulator. Optimal pose selection for the calibration is achieved using a swarm intelligence search algorithm. The method is implemented on a prototype of a six degrees-of-freedom (DOFs) Gough-Stewart platform constructed to function as a machine tool.

Dimensional optimization of 6-DOF 3-CCC type asymmetric parallel manipulator

In this paper, dimensional optimization of a six-degrees-of-freedom (DOF) 3-CCC (C: cylindrical joint) type asymmetric parallel manipulator (APM) is performed by using particle swarm optimization (PSO). The 3-CCC APM constructed by defining three angle and three distance constraints between base and moving platforms is a member of 3D3A generalized Stewart–Gough platform (GSP) type parallel manipulators. The dimensional optimization purposes to find the optimum limb lengths, lengths of line segments on the base and moving platforms, attachment points of the line segments on the base platform, the orientation angles of the moving platform, and position of the end-effector in the reachable workspace in order to maximize the translational and orientational dexterous workspaces of the 3-CCC APM, separately. The dexterous workspaces are obtained by applying condition number and minimum singular values of the Jacobian matrix. The optimization results are compared with the traditional GSP manipulator for illustrating the kinematic performance of 3-CCC APM. Optimizations show that 3-CCC APM have superior dexterous workspace characteristics than the traditional GSP manipulator.

Evaluation and optimization of dynamic stiffness values of the PKMs: Collinear stiffness value approach

Dynamic stiffness of the parallel-kinematics machines is considered and a new engineering index – dynamic collinear stiffness value (DynCSV) – is proposed. The DynCSV presents an extension of a concept derived by Portman for evaluation of the static stiffness values. The DynCSV depends on robot-configuration and inertia parameters and presents the function of the vibration frequency. Mathematically, the DynCSV is a quadratic form associated with the real symmetric matrix of the Newton–Euler equation. The minimal and maximal values of the DynCSV give natural criteria for evaluation, limitation, and optimization of dynamic systems of machines and robots. In the frequency range, where the dynamic stiffness matrix has a semipositive definition, the minimum DynCSV is applied to construct a PKM workspace satisfying preliminary given stiffness limits and compute a virtual protective barrier, keeping the PKM mechanism from approaching both singular configurations and resonance frequencies. As an application example, the DynCSV-based local and global stiffness features of the Gough–Stewart platform mechanism are simulated, investigated, and visualized.

Correcting Duporcq's theorem

In 1898, Ernest Duporcq stated a famous theorem about rigid-body motions with spherical trajectories, without giving a rigorous proof. Today, this theorem is again of interest, as it is strongly connected with the topic of self-motions of planar Stewart–Gough platforms. We discuss Duporcq's theorem from this point of view and demonstrate that it is not correct. Moreover, we also present a revised version of this theorem.

Some geometric, kinematic, and dynamic considerations on Stewart-Gough platforms with singularity analysis

SUMMARYIn this study, some geometric, kinematic, and dynamic aspects of the design of a Stewart-Gough platform are examined. The focus of the analyses is on a particular Stewart-Gough platform that we have constructed. The analysis begins with workspace simulations for different moving platform orientations. The computations extend to a parametric study of some geometric and kinematic constraints: Joint angle, rotation angle, and limb length. Actuator force is another parameter considered; and the triple relationship between workspace, joint angle, and actuator force is discussed. Parametric analyses are culminated with a brief discussion of the real design parameters.

Introducing the Theory of Bonds for Stewart Gough Platforms With Self-Motions1

We transfer the basic idea of bonds, introduced by Hegedüs, Schicho, and Schröcker for overconstrained closed chains with rotational joints, to the theory of self-motions of parallel manipulators of Stewart Gough (SG) type. Moreover, we present some basic facts and results on bonds and demonstrate the potential of this theory on the basis of several examples. As a by-product we give a geometric characterization of all SG platforms with a pure translational self-motion and of all spherical three-degrees of freedom (DOF) RPR manipulators with self-motions.

Singularity analysis and detection of 6-UCU parallel manipulator

6-UCU kind Gough-Stewart platform (GSP) has been used extensively in practice. The singularity of GSP has been studied by many scholars, but their works mainly focused on finding the methods to divide the cases of singularity or searching the solutions with Jacobian matrices. On the other hand, this paper studies the singularities of 6-UCU parallel manipulator caused by not only the active joints but also passive universal joints. Two types of singularity are derived based on a degree of freedom method by using screw theory. Singularity detection is essential to certify the absence of singularity within a prescribed workspace or a reachable workspace for a practical use of the 6-UCU parallel manipulator. Algorithms are proposed by using evolutionary strategy to detect the singularity in the desired or reachable workspace of the 6-UCU parallel manipulator. Case studies are presented to demonstrate the effectiveness of the proposed singularity detection methods.

Systematic optimal design procedures for the Gough-Stewart platform used as motion simulators

Purpose – The purpose of this paper is to present systematic optimal design procedures for the Gough-Stewart platforms used as engineering motion simulators. Design/methodology/approach – Three systematic optimal design procedures are proposed to solve the engineering design problems for the Gough-Stewart platform used as motion simulators. In these systematic optimal design procedures, two contradicting design optimality criteria with good representations of performances of the Gough-Stewart platforms are chosen as the objective functions. In addition, the two objective function optimization problems are solved by using the multi-objective evolutionary algorithms. Findings – In the systematic optimal design procedures, multiple compromised design solutions are found by using Elitist Non-Dominated Sorting Genetic Algorithm version II in the primary design stage, and many candidates can be used in the secondary design stage for higher decisions. Two higher decision methods have been presented to choose the final solutions. Originality/value – This paper proposes three systematic optimal design procedures to solve the practical design problems of the Gough-Stewart platforms used as motion simulators, which are very important for the engineering designers.

Trajectory generation and tracking control of a multi-level hybrid support manipulator in FAST

The Feed Support System (FSS) of Five-hundred-meter Aperture Spherical radio Telescope (FAST) is a multi-level redundant support manipulator, which consists of a cable driven parallel manipulator, an A–B rotation mechanism and a Gough–Stewart platform. In this article, we report our work on FAST in the following aspects: first, kinematic model and trajectory generation strategy of the FSS are established. Second, considering preventing the pseudo-drag problem of flexible cable and realizing the accurate pose control for the cable driven parallel manipulator, the hybrid position/force control is implemented and validated. Then, the prediction control is adopted in the Gough–Stewart platform to improve the terminal accuracy. Finally, with the 1:15 similarity model of FSS, experiments are carried out to prove the control accuracy of the cable driven parallel manipulator, showing that the terminal error is within 10mm and the cable tension is kept in the given range. Further experiments on tracking control of the entire FSS illustrate terminal tracking accuracy of the astronomical observation is less than 2mm, meeting the design requirement. Trajectory generation and tracking control given in this paper lay the foundation for the FAST prototype.

Udwadia-Kalaba Approach for Parallel Manipulator Dynamics

A novel Udwadia-Kalaba approach for parallel manipulator dynamics analysis is presented. The approach segments a parallel manipulator system into several leg-subsystems and the platform subsystem, which are connected by kinematic constraints. The Udwadia-Kalaba equation is then used to calculate the constraint forces due to the constraints. Based on this, the equation of motion, which is an explicit (i.e., closed) form, can be formulated. The method allows a systematic procedure to generate the dynamic model for both direct dynamics and inverse dynamics without invoking additional variables (such as multipliers or quasi-variables), nor does it require projection. A classical parallel Stewart-Gough platform is chosen to demonstrate the feasibility and advantages of this approach.

Novel horseback riding simulator based on 6-DOF motion measurement, a motion base, and interactive control of gaits

This article describes the development process of a horseback riding simulator capable of generating a close to natural riding sensation. The development process started with 6-DOF motion measurement of the dynamics of the rider’s saddle during riding. Six-dimensional measurements during riding are required to fully understand the character of the saddle dynamics during different gaits. The measured quantities, i.e. translation accelerations and angular velocities, were transformed and filtered to form position references for a hydraulic Gough–Stewart platform. The presented novel motion measurement method is easy to use and makes it possible to measure gaits of several individual horses in a short time. It also provides a straightforward approach to produce motion data for the motion platform. In the second phase of the study, an artificial horse body made of glass fiber was mounted on the motion platform. The developed simulator realizes all three basic gaits walk, trot, and gallop. The rider is able to control the behavior of the simulator via sensors and an interactive interface. The control aids sensed by the interface are reins tension, calf pressure, and stirrup angle. A novel intelligent algorithm mimicking the real rider–horse interaction to vary the gaits and their frequencies was developed and tested.