Stewart platform

This research is geared towards shaping, simulating, designing and developing a movement system, which is based on the principles of the Stewart platform. From the point of view of mobility, it is considered a modified Stewart platform so that as mobility increases, the positions of the static and movable points of the platform change until a form compatible with the human shoulder is reached. Using modelling and simulation, the operation of the modified platform and its compatibility with the human shoulder are justified. To make the real model, there have been used technically and economically accessible materials, to obtain a low-cost product. Unlike the classic Stewart platform, the approach covered by this paper is to simplify, by horizontal optimization, the motion system so that it can synthesize the movement of the shoulder joint, with degrees of freedom similar to the human shoulder, while simultaneously maintaining a reliable force level. The main purpose of this work is to reproduce, first by design, then by calculation, a physical model of the biological mechanism of the shoulder. The model designed in this research replaces the muscles with wires and actuators. In this process, the selection and arrangement of components was simplified in positioning and as number, favoring anatomical mimicry. There was studied and constructed the mechanism according to the anatomical descriptions, which were described from a medical point of view, and there can be noticed that by applying technical mechanics, results the same kind of movements, but not necessarily the same type of joint.

Stewart platform-based Parallel Kinematic Machines (PKM) have been extensively studied by researchers due to their inherent finer control characteristics. This has opened its potential deployment opportunities in versatile critical applications like the medical field, engineering machines, space research, electronic chip manufacturing, automobile manufacturing, etc. All these precise, complicated, and repeatable motion applications require micro and nano-scale movement control in 3D space; a 6-DOF PKM can take this challenge smartly. For this, the PKM must be more accurate than the desired application accuracy level and thus proper calibration for a PKM robot is essential. Forward kinematics-based calibration for such hexapod machines becomes unnecessarily complex and inverse kinematics complete this task with much ease. To analyze different techniques, an external instrument-based, constraint-based, and auto or self-calibration-based approaches have been used for calibration. This ...

Accurate calibration of a Stewart platform is important for their precise and efficient operation. However, the calibration of these platforms using forward kinematics is a challenge for researchers because forward kinematics normally generates multiple feasible and unfeasible solutions for any pose of the moving platform. The complex kinematic relations among the six actuator paths connecting the fixed base to the moving platform further compound the difficulty in establishing a straightforward and efficient calibration method. The authors developed a new forward kinematics-based calibration method using Denavit-Hartenberg (DH) convention and used the Stewart platform “Tiger 66.1” developed in their lab for experimenting with the photogrammetry-based calibration strategies described in this paper. This system became operational upon completion of construction, marking its inaugural use. The authors used their calibration model for estimating the errors in the system and adopted three compensation options or strategies as per Least Square method to improve the accuracy of the system. These strategies leveraged a high-resolution digital camera and off-the-shelf software to capture the poses of the moving platform's center. This process is non-invasive and does not need any additional equipment to be attached to the hexapod or any alteration of the hexapod hardware. This photogrammetry-based calibration process involves multiple high-resolution images from different angles to measure the position and orientation of the platform center in the three-dimensional space. The Target poses and Actual poses are then compared, and the error compensations are estimated using the Least-Squared methods to calculate the Predicted poses. Results from each of the three compensation approaches demonstrated noticeable enhancements in platform pose accuracies, suggesting room for further improvements. Given that "Tiger 66.1" is based on the general Stewart Platform structure, the proposed calibration method holds promise for extension to machines operating on similar principles where non-invasive calibration is desirable. This study contributes to advancing the field of Stewart platform calibration, paving the way for more precise and efficient applications in various domains.

An inverse model for the kinematics and dynamics of a 6-SBU structured Stewart platform is generally derived for prediction of leg lengths and the force actuation to meet the desired pose within prescribed kinematic and dynamic constraints. The feedback control generated from the instantaneous difference between the measured leg lengths and predicted leg lengths at each instant of time is found to be effective along with the feedforward estimation for force actuation to the legs. In this paper an attempt has been made to accommodate intricacies involved in inverse modeling that makes the feedforward estimations precise and reduce the feedback control effort.

Researchers have studied Stewart-Gough platforms, also known as Gough-Stewart platforms or hexapod platforms extensively for their inherent fine control characteristics. Their studies led to the potential deployment opportunities of Stewart-Gough Platforms in many critical applications such as the medical field, engineering machines, space research, electronic chip manufacturing, automobile manufacturing, etc. Some of these applications need micro and nano-level movement control in 3D space for the motions to be precise, complicated, and repeatable; a Stewart-Gough platform fulfills these challenges smartly. For this, the platform must be more accurate than the specified application accuracy level and thus proper calibration for a parallel robot is crucial. Forward kinematicsbased calibration for these hexapod machines becomes unnecessarily complex and inverse kinematics complete this task with much ease. To experiment with different calibration techniques, various calibration approaches were implemented by using external instruments, constraining one or more motions of the system, and using extra sensors for auto or self-calibration. This survey paid attention to those key methodologies, their outcome, and important details related to inverse kinematic-based parallel robot calibrations. It was observed during this study that the researchers focused on improving the accuracy of the platform position and orientation considering the errors contributed by one source or multiple sources. The error sources considered are mainly kinematic and structural, in some cases, environmental factors also are reviewed, however, those calibrations are done under no-load conditions. This study aims to review the present state of the art in this field and highlight the processes and errors considered for the calibration of Stewart-Gough platforms.

This paper presents a method to generate unique, feasible forward kinematic solutions for a general Stewart platform. This is done by using inverse kinematics to obtain valid workspace data and corresponding actuator lengths for the moving platform. For parallel kinematic machines, such as the Stewart Platform, inverse kinematics are straight forward, but the forward kinematics are complex and generates multiple solutions due to the closed loop structure of the kinematic links. In this method, a simple iterative algorithm has been used employing modified Denavit-Hartenberg convention. The outcome is encouraging as this method generates a single feasible forward kinematic solution for each valid pose with the solved DH parameters and unlike earlier forward kinematics solutions, this unique solution does not need to be manually verified. Therefore, the forward kinematic solutions can be used directly for further calculations without the need for manual pose verification. This capability is essential for the six degree of freedom materials testing system developed by the authors in their laboratory. The developed system is aimed at characterizing additively manufactured materials under complex combined multiple loading conditions. The material characterization is done by enabling high precision force control on the moving platform via in situ calibration of the as-built kinematics of the Stewart Gough Platform.

I would like to thank my colleagues: Frédéric Bossens for putting my feet on the right way, Mihaita Horodinca for his big efforts in manufacturing the Stewart platform, Vin-cent Piefort for his advices in the finite element modelling, Ioanica Burda and Salvator Dusabimana for their efforts in ...

As future astronomic missions will require more and more stringent resolution requirements, the high demand for an environment clean of vibrations and disturbance appears. This also leads to the need for high precision steering devices for fine pointing of sensitive optics with the highest possible accuracy. Several methods exist to reduce vibration levels: the first consists in isolating the sensitive system from the perturbation and the second in damping the structure vibration modes. Therefore, two Stewart platforms have been designed, manufactured and tested. The first is a soft hexapod that provides 6 degree-of-freedom (DOF) active isolation and the second is a stiff hexapod that provides active damping to whatever flexible payload attached/mounted to it. In addition, both hexapods have steering capabilities.

This paper reports on a six-axis vibration isolator for space applications. It is divided into three parts. The first part recalls the principles of active isolation and summarizes the main theoretical results for multiple-axis decentralized control based on force feedback. The second part discusses the technology and describes the evolution of the design over the five years of this project. The third part is devoted to the identification of the transmissibility matrix and the performance evaluation. Zero-gravity tests in parabolic flight are reported. The isolator is proved efficient in a frequency band between 5 Hz and 400 Hz, with a maximum attenuation of -40 dB between 50 Hz and 200 Hz.

As future astronomic missions will require more and more stringent resolution requirements, the high demand for an environment clean of vibrations and disturbance appears. This also leads to the need for high precision steering devices for fine pointing of sensitive optics with the highest possible accuracy. Several methods exist to reduce vibration levels : the first consists in isolating the sensitive system from the perturbation and the second in damping the structure vibration modes. Therefore, two Stewart platforms have been designed, manufactured and tested. The first is a soft hexapod that provides 6 degree-of-freedom (DOF) active isolation and the second is a stiff hexapod that provides active damping to whatever flexible payload attached/mounted to it. In addition, both hexapods have steering capabilities.

In this paper we consider two situations. In the first, all kinematic chains are elastic, while the second situation is characterized by one rigid kinematic chain, with the rest of them being elastic. In addition, the kinematic joints are considered to be rigid. The calculations are performed using the screw coordinates. For the free vibrations of the rigid solid we determined the rigidity matrix and the eigenpulsations in both cases. It was proved that the results in the second case cannot be considered as limits for the results of the first situation, putting infinite values for the elements of the rigidity matrix of one kinematic chain. We also developed the theory for the forced vibrations of the system. A numerical application is considered and a great variety of cases are developed and discussed. The results obtained for the forced vibrations are presented and discussed. The paper combines elastic and rigid kinematic chains, as well as general configurations of the kinematic c...

This paper presents a computational geometric solution to the kinematic registration problem using computational line geometry. Kinematic registration involves computation of the screw parameters of a motion from specified positions of geometric features of the moving body. The problem is formulated using a special complex of lines associated with kinematics namely the bisecting linear line complex. In this fashion the problem is reduced to an approximation problem in the line space and a slightly modified version of the line approximation method developed by Pottmann, Peternell, and Ravani (1999) is used to find the solution.

The VES is a six DOF position-controlled platform which mimics the motions of the vehicle being emulated. A manipulator is bolted to the top of the platform through an in-line, six DOF force sensor. The force sensor detects the inertial and gravitational forces reflected to the base of the manipulator caused by motions of the manipulator links, and by the interaction of the manipulator endpoint with the external environment. The platform is then servocontrolled to match the motions of the emulated vehicle if that vehicle were to be subjected to the same forces exerted through the base of the manipulator. The design of the mechanical system was dictated by the platform payload and motion requirements, which include the capability of moving up to 1000 pound loads through a workspace of @Math[@PM]12 inches in each of three linear directions and @Math[@PM]30 degrees in three rotations at frequencies of up to 10 Hz. The VES mechanism is based on a Stewart platform configuration, where th...

We are developing an autonomous mobile robotic system to emulate six degree of freedom relative spacecraft motion during proximity operations. A mobile omni-directional base robot provides x, y, and yaw planar motion with moderate accuracy through six independently driven motors. With a six degree of freedom micro-positioning Stewart platform on top of the moving base, six degree of freedom spacecraft motion can be emulated with high accuracy. This paper presents our approach to dynamic modeling, control, and simulation for the overall system. Compared with other simulations that introduced significant simplifications, we believe that our rigorous modeling approach is crucial for the high fidelity hardware in-the-loop emulation.

This paper presents kinematic and dynamic modeling techniques for flexible robots. . The main emphasis is to discretize whole flexible link in very small parts each considering as a rigid link. This approach is based on a "discretization method". In kinematics position or deflection is solved by deflection of cantilever beam theory while orientation is solved by forward kinematics. Dynamics is solved by applying Lagrange-Euler formulation.

This paper describes the dynamic investigation of a 3-component and a 6-component force- moment sensors using the facilities in the PTB (Physikalisch-Technische Bundesanstalt) in Germany. The 3-component sensor has the transverse forces Fx and Fy of each having 200 N capacity and a twisting moment Mz of 10 Nm capacity and the 6-component sensor has force components Fx, Fy and Fy each having 200 N capacity and moment components Mx, My and Mz each having 20 Nm capacity. The sensitivity of the sensors decreases as the frequency increases and the sensors show almost 90 o symmetry due to their geometry.

Micromachining is meant for small accurate and precise parts production for many industrial applications. These parts vary from very simple to complicated shapes. As applications grow in complexity and shrink in size, machines need to be designed to meet the desired precision and accuracy. In this paper it is proposed to give a six degree of freedom to the bed of a machine instead to tool box which carry a heavy load. A parallel mechanism consists of six legs of variable lengths is proposed for the machine bed. The objective is to enhance accuracy and precision in micromachining in comparison to conventional machines. The paper presents a design of the machine bed, the kinematics, dynamics and a sim-mechanics model of the bed. Simulation results verify the working of a 6-DOF machine bed for micromachining.

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 con...

Yakıt çalkalanma dinamiğinin uçak dinamiğine etkisi olabilmektedir. Uçağın hareketleri neticesinde oluşan ivmeler yakıtın belli bir yöne doğru yığılmasına veya çalkalanma hareketi yapmasına yol açabilmektedir. Bu hareket neticesinde uçağın ağırlık merkezi ve atalet değerleri değişebilmektedir. Bu çalışmada bu etkinin yansıtılabilmesi için öncelikle yakıtın çeşitli koşullardaki denge durumları hesaplanmış, sonrasında da ikinci derece bir yakıt dinamiği modeli eklenerek yakıt yüzeyinin anlık açıları bulunmuştur. Bu anlık açılar neticesinde oluşan ağırlık merkezi ve atalet değişimleri hesaplanarak uçak dinamiğine etkileri incelenmiştir.

The paper deals with machines employing parallel-kinematics structures (PKS). They represent a relatively new generation of machine tools. Depending on the number of struts, the machines are referred to as hexapod or tripod machines. Such machines offer several advantages comparing to the conventional machine tools with serial kinematics, such as high flexibility, high stiffness, and high accuracy. It is very suitable for High- Speed-Machining (HSM), light machining and has received a wide interesting in manufacture industry. To achieve a desired positioning accuracy and stability, the static and dynamic properties of the machine must be searched and mathematically described. The calculation of the estimate of positioning deviation, including respective uncertainty and covariances, is much more complicated task comparing to the serial kinematics.

Bu calismada oncelikle delta robot tabanli ornek bir kaotik karistirici tasarimi yapilmistir. Tasarlanan prototip karistirici uzerinde farkli kaotik sistemlere ait yorungelerin karsilastirmalarinin yapilabilmesi icin Matlab programi kullanilarak, kaotik sistemlerden alinan verileri delta robotun yorunge koordinatlarina donusturen bir kontrol yazilimi gelistirilmistir.

In this paper, a computational method for obtaining the maximum Dynamic Load Carrying Capacity (DLCC) for the 6-UPS Stewart platform manipulator is developed. In this paper, the manipulator is assumed to be non-rigid and the joint actuator torque capacity and accuracy of motion a r e considered major limiting factors in determining the maximum payload. The maximum dynamic payload carrying capacity of the manipulator is established, while the dynamic model of a typical hydraulic actuator system is used in the joint actuator force capacity for a given trajectory. The exibility of the manipulator is assumed to be eventuated from the manipulator's joints exibility. According to the high complexity of the dynamic equations system of the exible joints parallel manipulators, the e ects of the exibility of the prismatic joints are considered in a static situation to show the considerable e ects of the joint's exibility on the motion accuracy of the 6UPS-Stewart platform. This method can be used for determining the maximum dynamic payload, which acts as an end-e ector for the mechanical design of the manipulator and the optimized selection of the actuator, such as machine tools, based on the hexapod mechanism.

This paper deals with the modeling, simulation, and experimental validation of a modified Gough-Stewart platform (MGSP) for vibration isolation, where the first six natural frequencies corresponding to the first six degreeof-freedom are nearly the same, enabling effective attenuation of the first six modes. The configuration is termed as dynamically isotropic and this work presents a geometry-based analytical approach to obtain the design parameters of the MGSP at its neutral position. The approach accommodates various payload configurations, including variable center of mass and mass/inertia properties. The validation of the design is demonstrated using a finite element software ANSYS ® , and the model is further refined to incorporate flexural joints and structural damping. A prototype of the MGSP featuring flexural joints was tested, and it yielded experimental outcomes in close agreement with the finite element analysis results-the first six natural frequencies were close to the expected 29 Hz and vibration isolation of about 22 dB/octave. The close agreement among analytical, finite element, and experimental outcomes underscores the efficacy of our design approach and the suitability of an MGSP for micro-vibration isolation applications in spacecraft.

A new hybrid rotary platform with three degrees of freedom that possesses both a large workspace and the ability to bear heavy loads is presented. The degrees of freedom and rotary characteristics of the hybrid rotary platform are analyzed, and its solution of inverse position is obtained. The dimensional optimization, structural design, and prototype development of the hybrid rotary platform were carried out, an electrical control system based on a multi-axis motion controller and a three-axis gyroscope was developed, and a calibration experiment on the hybrid rotary platform was performed. After the key kinematic parameters were calibrated, the overall accuracy of the hybrid rotary platform greatly improved.

This paper proposes a novel hyperstatic six-component force/torque sensor structure based on the Stewart platform, and presents the parameter optimization of the sensor structure with genetic algorithms. The structure characteristic of the hyperstatic sensor is analyzed compared with the traditional Stewart platform-based sensor. The mathematical expression of the sensor's force mapping matrix is built by using the screw theory. In this paper, the condition number and generalized amplifying index are used to evaluate the isotropy and sensitivity of the sensor, respectively. Based on genetic algorithms, the optimal design of the sensor structure is performed with the objective of achieving high measurement sensitivity and well-conditioned transformation between the input and output forces. The computed results prove the better performance of the proposed sensor structure and the effectiveness of the optimization algorithm.

This paper proposes a new configuration of six-component force/torque sensor based on Stewart platform, and discusses the performance analysis and comprehensive index optimization of the six-component force sensor. Based on the classical Stewart platform-based force sensor, a modified sensor structure is proposed, and their statics mathematical models are built by using the screw theory. The condition number and generalized amplifying coefficient defined by singular values of force Jacobian matrix are used to evaluate the performance of sensor, such as isotropy, sensitivity, stiffness, etc. The effect of spherical joint frictional moment on the sensor is established by using the related force mapping matrix. With the combination of the defined performance indices, the comprehensive index optimization of the structural parameters of the modified Stewart six-component force sensor is achieved by using the indices atlases and genetic algorithm. The research results of the paper are useful for the design and further research of the six-component force sensor.

Yes, George, I'm happy to do that. I arrived in South Australia on 1st April – April ... Fool's Day – 1975, having spent three years previously in Tasmania as a tutor in the ... Politics Department at the University of Tasmania. Before then, of course, I'd been ... Catholic background, ...

It is widely acknowledged that precision design and error compensation are two basic methods to achieve high geometrical accuracy for machine tools. And for these two methods, error modeling and sensitivity analysis are key issues. In this paper, an error modeling method based on multibody system (MBS) is used to construct the error mapping between error sources and the cutting tool pose error for a five-axis machine tool. According to the error mapping, the traditional definitions of local sensitivity indices (LSIs) and global sensitivity indices (GSIs) are introduced. Based on the LSIs and GSIs, the general local sensitivity indices (GLSIs), general global sensitivity indices (GGSIs), and general global sensitivity fluctuation indices (GGSFIs) are proposed. By using these new indices, error sensitivity analysis of the fiveaxis machine tool is conducted. According to the error sensitivity analysis results, the precision design of angular error components is conducted in numerical simulation. The results show that by using the proposed sensitivity analysis method, we can improve less error components but get more improvement for the cutting tool accuracy. It indicates that the proposed sensitivity indices and sensitivity analysis method are very effective and meaningful. The proposed sensitivity indices and sensitivity analysis method can also be used in the precision design and error compensation for other machine tools.

There are so many applications of ellipse equation, such that graphical representation and finding problem of any elliptical shapes object, computer graphics, space science, engineering design and so on. In this paper state about general equation of ellipse. Generally, there are various equations for drawing an ellipse in 2d plane but not exactly the equations are general, must have some condition and rule. So, here observed those things and found a problem to create a general equation of ellipse. And tried to invent a polynomial equation of curve that can represent an ellipse and focus of ellipse act as variable of equation (and a starting point of major axis (vertex) also act as a variable).

The field of deformity correction is advancing at a rapid pace and with the advent of newer techniques, the doctors are able to provide accurate correction with lesser and lesser time entailed in the process. It has become imperative that the surgeons can learn the latest techniques and instruments so that the patient can benefit from it. In this review article, we have tried to go through the latest in the field of deformity correction.

In this paper, we propose a class of Generalized Gough-Stewart Platforms (GGSPs) used, as a novel approach, to eliminate the classical isotropic constraint of GSPs (hexapods). GGSPs are based on the standard GSP architecture with additional rotations of the three strut-pairs. Despite the architectural generalization introduced in GGSPs, they do not require much more effort in order to be fabricated. This is due to the fact that all the struts (actuators) can be chosen identical, similar to standard GSPs. We analytically show how effectively the classical isotropic constraint is removed and that still sufficient simplicity is retained. Furthermore, this paper gives an intuitive understanding of dynamic isotropy in GGSPs as well as GSPs.

The optimum design of spherical parallel manipulators (SPM) is studied for a prescribed workspace. A numerical method is developed to find optimal design parameters including link dimensions and architecture parameters for a maximum dexterity. In the method, the objective function is formulated in such a way that the optimal problem is converted to a nonlinear least squares problem, which can be readily solved. Moreover, the problem of design space is addressed. A system of inequalities in terms of link dimensions is derived to describe the design space for feasible SPMs. Examples are included to illustrate the application of the method.

Bu araştırma çalışmasında, seralar için Bluetooth tabanlı kablosuz ölçüm sistemi tasarlanmıştır. Sistemin prototip uygulaması sera içinde kurulan deneylerle test edilmiştir. Sistem yönetici bilgisayar, CB-RSPA333 seri port adaptörü ve 2 kablosuz iletişim biriminden oluşmuştur. Her kablosuz iletişim biriminde PIC16F876 mikrodenetleyicisi, CB-SPA331 Bluetooth modülü ve birer adet SHT11 ve DS18B20 sensörleri bulunmaktadır. Sistem sera dışındaki yönetici bilgisayardan gönderilen komutlara göre, sera içindeki kablosuz birimlerin kontrolü, bilgisayara verilerin aktarılması ve kaydedilmesi, gerçek zamanlı monitörleme işlemlerini yapacak şekilde programlanmıştır. Test çalışmalarında bitkinin tepe ve alt seviyelerindeki hava, toprak sıcaklıkları ve bağıl nem verileri ölçülmüştür. Tipik bir yaz gününde kablosuz olarak kaydedilen verilere göre, bitkinin tepe ve alt taraflarındaki hava sıcaklıkları sırasıyla 12.1°C-36.2°C ile 13.9°C-31.7°C aralıklarında değişmiştir. Bağıl nem değişimleri sırasıyla % 19.9-67.3 ile %25.9-%72.1 aralığında ölçülmüştür.

Dokuma kumaslarin uretilmesinde agizlik acma islemi oldukca onemlidir. Gunumuzde, agizlik acma islemi icin yuksek hizli dokuma makinelerinde rotatif armurlu mekanizmalar kullanilmaktadir. Rotatif armurlerin temel ozelliklerinden birisi makinenin ana miline tek yonlu beklemeli donme hareketinin iletilmesidir. Boyle bir hareketin elde edilmesi icin ileri teknoloji ve maliyet gerektirmektedir. Bu calismanin amaci, rotatif armurlerde armur miline 1800 aci altinda calisacak esit zamanli, beklemeli-salinim hareketini ileten ve rotatif armurlerin tahrik mekanizmalarinin gorevini saglayan daha basit konstruksiyonlu ve daha dusuk maliyetli bir tahrik mekanizmasinin tasarlanmasidir.

The Gough Stewart Robotic manipulator is a parallel manipulator with six-degree of freedom, which has six equations of Kinematics (Inverse and forward), with six variables (Lengths, Position, and Orientation). In this work derived the inverse equations, which used to compute the lengths of the linkages and its changes depended on the position and orientation of the platform's center, then derived the forward equations to calculate the position and orientation of the moving platform in terms of the lengths. This theoretical model of the kinematics analysis of the Gough Stewart has been built into the Simulink package in Matlab to obtain the lengths, position, and orientation for the manipulator at any time of motion. The input parameters (Position and Orientation) in inverse blocks compared with the output parameters (Position and Orientation) in the forward blocks, which show good results.

This work deals with Gough-Stewart robot manipulator, which has six degrees of freedom, six actuators, fixed base, and moving platforms. Here, the Jacobian matrix derived to detect the singular point in the workspace for manipulator at determinant of Jacobian matrix equal to zero, then derived the equation of motion from the dynamic analysis by Lagrange method to verify the singular points with Jacobian where the forces increase rapidly at this point. Finally, design blocks in Simulink include the Jacobian matrix and the equations of motion to detection the singularities at any time for current input parameters (X, Y, Z, α, β, γ), where the determinant of the Jacobian equal to zero at maximum forces.

The novelty of the paper lies in an extended nonlinear model for the Stewart platform with moving payload. The models found in the existing literature are most of the time only valid for static loads with symmetric configurations, as reflected by the presumed assumption on Moments of Inertia (MOI) being static and symmetric. Such assumption precludes a wide range of systems having asymmetric or moving payloads e.g. stabilized platforms used to stabilize satellite antenna trackers or surgical tables etc. In this paper the actual MOI of the Stabilizing Stewart platform with moving payload are computed and used to parameterize the extended nonlinear model. This model is subsequently used for designing sliding mode controller. The control performance of the proposed algorithm is verified with computer simulations. These simulations demonstrate better stability properties and performance of the proposed dynamic model with much lower uncertainty bounds, as compared to that of the controller designed on the prevalent nonlinear model.

Any set of two legs in a Gough-Stewart platform sharing an attachment is defined as a ∆component. This component links a point in the platform (base) to a line in the base (platform). Thus, if the two legs, which are involved in a ∆ component, are rearranged without altering the location of the line and the point in their base and platform local reference frames, the singularity locus of the Gough-Stewart platform remains the same, provided that no architectural singularities are introduced. Such leg rearrangements are defined as ∆-transforms, and they can be applied sequentially and simultaneously. Although it may seem counterintuitive at first glance, the rearrangement of legs using simultaneous ∆-transforms does not necessarily lead to leg configurations containing a ∆component. As a consequence, the application of ∆-transforms reveals itself as a simple, yet powerful, technique for the kinematic analysis of large families of Gough-Stewart platforms. It is also shown that these transforms shed new light on the characterization of architectural singularities and their associated self-motions.