Development of Alternative Methods for Robot Kinematics

The problem of finding mathematical tools to represent rigid body motions in space has long been on the agenda of physicists and mathematicians and is considered to be a well-researched and well-understood problem. Robotics, computer vision, graphics, and other engineering disciplines require concise and efficient means of representing and applying generalized coordinate transformations in three dimensions. Robotics requires systematic ways to represent the relative position or orientation of a manipulator rigid links and objects. However, with the advent of high-speed computers and their application to the generation of animated graphical images and control of robot manipulators, new interest arose in identifying compact and computationally efficient representations of spatial transformations. The traditional methods for representing forward kinematics of manipulators have been the homogeneous matrix in line with the D-H algorithm. In robotics, this matrix is used to describe one coordinate system with respect to another one. However for online operation and manipulation of the robotic manipulator in a flexible manner the computational time plays an important role. Although this method is used extensively in kinematic analysis but it is relatively neglected in practical robotic systems due to some complications in dealing with the problem of orientation representation. On the other hand, such matrices are highly redundant to represent six independent degrees of freedom. This redundancy can introduce numerical problems in calculations, wastes storage, and often increases the computational cost of algorithms. Keeping these drawbacks in mind, alternative methods are being sought by various researchers for representing the same and reducing the computational time to make the system fast responsive in a flexible environment. Researchers in robot kinematics tried alternative methods in order to represent rigid body transformations based on concepts introduced by mathematicians and physicists such as Euler angle or Epsilon algebra. In the present work alternative representations, using quaternion algebra and lie algebra are proposed, tried and compared

Inverse Kinematic Analysis of Robot Manipulators

An important part of industrial robot manipulators is to achieve desired position and orientation of end effector or tool so as to complete the pre-specified task. To achieve the above stated goal one should have the sound knowledge of inverse kinematic problem. The problem of getting inverse kinematic solution has been on the outline of various researchers and is deliberated as thorough researched and mature problem. There are many fields of applications of robot manipulators to execute the given tasks such as material handling, pick-n-place, planetary and undersea explorations, space manipulation, and hazardous field etc. Moreover, medical field robotics catches applications in rehabilitation and surgery that involve kinematic, dynamic and control operations. Therefore, industrial robot manipulators are required to have proper knowledge of its joint variables as well as understanding of kinematic parameters. The motion of the end effector or manipulator is controlled by their joint actuator and this produces the required motion in each joints. Therefore, the controller should always supply an accurate value of joint variables analogous to the end effector position. Even though industrial robots are in the advanced stage, some of the basic problems in kinematics are still unsolved and constitute an active focus for research. Among these unsolved problems, the direct kinematics problem for parallel mechanism and inverse kinematics for serial chains constitute a decent share of research domain. The forward kinematics of robot manipulator is simpler problem and it has unique or closed form solution. The forward kinematics can be given by the conversion of joint space to Cartesian space of the manipulator. On the other hand inverse kinematics can be determined by the conversion of Cartesian space to joint space. The inverse kinematic of the robot manipulator does not provide the closed form solution. Hence, industrial manipulator can achieve a desired task or end effector position in more than one configuration. Therefore, to achieve exact solution of the joint variables has been the main concern to the researchers. A brief introduction of industrial robot manipulators, evolution and classification is presented. The basic configurations of robot manipulator are demonstrated and their benefits and drawbacks are deliberated along with the applications. The difficulties to solve forward and inverse kinematics of robot manipulator are discussed and solution of inverse kinematic is introduced through conventional methods. In order to accomplish the desired objective of the work and attain the solution of inverse kinematic problem an efficient study of the existing tools and techniques has been done. A review of literature survey and various tools used to solve inverse kinematic problem on different aspects is discussed. The various approaches of inverse kinematic solution is categorized in four sections namely structural analysis of mechanism, conventional approaches, intelligence or soft computing approaches and optimization based approaches. A portion of important and more significant literatures are thoroughly discussed and brief investigation is made on conclusions and gaps with respect to the inverse kinematic solution of industrial robot manipulators. Based on the survey of tools and techniques used for the kinematic analysis the broad objective of the present research work is presented as; to carry out the kinematic analyses of different configurations of industrial robot manipulators. The mathematical modelling of selected robot manipulator using existing tools and techniques has to be made for the comparative study of proposed method. On the other hand, development of new algorithm and their mathematical modelling for the solution of inverse kinematic problem has to be made for the analysis of quality and efficiency of the obtained solutions. Therefore, the study of appropriate tools and techniques used for the solution of inverse kinematic problems and comparison with proposed method is considered. Moreover, recommendation of the appropriate method for the solution of inverse kinematic problem is presented in the work. Apart from the forward kinematic analysis, the inverse kinematic analysis is quite complex, due to its non-linear formulations and having multiple solutions. There is no unique solution for the inverse kinematics thus necessitating application of appropriate predictive models from the soft computing domain. Artificial neural network (ANN) can be gainfully used to yield the desired results. Therefore, in the present work several models of artificial neural network (ANN) are used for the solution of the inverse kinematic problem. This model of ANN does not rely on higher mathematical formulations and are adept to solve NP-hard, non-linear and higher degree of polynomial equations. Although intelligent approaches are not new in this field but some selected models of ANN and their hybridization has been presented for the comparative evaluation of inverse kinematic. The hybridization scheme of ANN and an investigation has been made on accuracies of adopted algorithms. On the other hand, any Optimization algorithms which are capable of solving various multimodal functions can be implemented to solve the inverse kinematic problem. To overcome the problem of conventional tool and intelligent based method the optimization based approach can be implemented. In general, the optimization based approaches are more stable and often converge to the global solution. The major problem of ANN based approaches are its slow convergence and often stuck in local optimum point. Therefore, in present work different optimization based approaches are considered. The formulation of the objective function and associated constrained are discussed thoroughly. The comparison of all adopted algorithms on the basis of number of solutions, mathematical operations and computational time has been presented. The thesis concludes the summary with contributions and scope of the future research work

A New Approach To Inverse Kinematic Solutions Of Serial Robot Arms Based On Quaterninons In The Screw Theory Framework

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2009Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 2009Screw teori, üç boyutlu uzayda dönme ve öteleme hareketlerinin birleşimi ile oluşan, genel hareket, hız, kuvvet ve torkların ifade edilmesini sağlayan bir yöntemdir. Genel olarak screw hareketi bir doğru etrafında dönme ve yine aynı doğru boyunca öteleme hareketlerinin bir birleşimidir. Katı cisimlerin tüm hareketleri bu yaklaşımla ifade edilebilir. Genel olarak üç boyutlu uzayda screw hareketi bir doğru ve bir oran (pitch) kullanılarak ifade edilir. (Burada kullanılan oran (pitch), dönme başına meydana gelen öteleme miktarıdır). Genel screw hareketi toplamda dört eleman kullanılarak tanımlanabilir. Bunlardan üç tanesi dönme ve ötelemenin meydana geldiği doğruyu, bir tanesi de doğru etrafında meydana gelen dönme miktarını ifade etmek için kullanılır. Katı cisimlerin hareketinde kullanılan en geleneksel yöntem Euler açılarıdır. Euler açıları bir katı cismin hareketini 6 eleman kullanarak ifade eder. Bunlardan üç tanesi kartezyen koordinatlarda öteleme hareketinin ifadesinde kullanılırken, diğer üç tanesi de bu koordinat sistemlerinde meydana gelen dönmelerin ifadesinde kullanılır. Screw teorinin robot kinematiğinde çeşitli uygulamaları vardır. Diğer yöntemlere kıyasla screw teorinin robot kinematiğinde şu üstünlükleri vardır; yalnız iki koordinat sistemiyle kinematik analiz yapılır, geometrik olarak çok anlaşılırdır ve ters kinematik çözümlemede tekil nokta probleminden etkilenmez. Bu nedenlerden dolayı screw teorinin robot kinematiğinde çok önemli bir yeri vardır. En genel anlamda bu tezin amaçlarını iki temel başlık altında toplayabiliriz. Bunlardan birincisi seri robotların ters kinematiğinde tekil nokta problemlerinden etkilenmeden çözümlerin elde edilmesidir. Bunun için önerilen yöntemler screw teori tabanlı olarak seçilmiştir. İkinci temel amaç ise kinematik problemin etkin bir cebir kullanılarak ifade edilmesidir. Bunun içinde önerilen yöntemlerde kuaterniyon cebiri kullanılmıştır. Kuaterniyonlar rankı dört olan hiper-kompleks sayılardır. Kuaterniyon cebirinde bu dört eleman kullanılarak bir doğru tanımlanır ve bu doğru etrafında herhangi bir dönme temsil edilebilir. Fakat genel katı cisim hareketi tek bir kuaterniyon ile ifade edilemez. Bunun için ya iki kuaterniyon (bunlardan bir tanesi “birim kuaternion” dönmeyi ifade etmede, diğeri ötelemeyi ifade etmede kullanılır) ya da dual kuaterniyonlar kullanılmalıdır. Dual operatörler screw hareketi ifade etmede kullanılabilecek en iyi operatörlerdir. Aynı zamanda dual operatörlerin içinde de dual kuaterniyon operatörü screw hareketin temsilinde kullanılabilecek en verimli ve en az parametreli dual operatördür. Bu tezde seri robot kollarının ters kinematik çözümlerine yönelik screw teori tabanlı yöntemler incelenmiştir. Bunlardan ilki ekponensiyel haritalama yöntemidir. Bu yöntemde screw teori ve matris cebiri kullanılır. Bu nedenle tekil nokta problemi olmamasına karşın denklemler çok fazla parametre ile ifade edilmiştir. Bu durumu ortadan kaldırmaya yönelik iki farklı ters kinematik çözümü önerilmiştir. Bunlardan birincisi birim kuaterniyon (dönme operatörü) ve bir kuaterniyon (öteleme oerpatörü) kullanılarak elde edilmiştir. İkinci çözüm ise dual kuaterniyonlar kullanılarak elde edilmiştir. Bu üç yöntem ve robot kinematiğinde en çok kullanılan yöntem olan D-H yöntemi tekil nokta problemleri, hesaplama verimi, dizayn zorluğu ve çözüm doğruluğu açısından karşılaştırılmışlardır. Simulasyon çalışmaları Matlab ortamında geçekleştirilmiştir. Animasyon uygulamaları ise Matlabın sanal gerçeklik araç kutusu kullanılarak gerçekleştirilmiştir (VRML). Simulasyon denemelerinde Staubli TXL60 seri robotunun tek ve kooperatif çalışma örnekleri yapılmıştır.Screw theory is a way to express displacements, velocities, forces and torques in three dimensional space combining both rotational and translational parts. Any motion along a screw can be decomposed into a rotation about an axis followed by a translation along that axis. Any general displacement of a rigid body can therefore be described by a screw. In general, a three dimensional motion can be defined using a screw with a given direction and pitch. Four parameters are required to fully define a screw motion, the 3 components of a direction vector and the angle rotated about that line. In contrast, the traditional method of characterizing 3-D motion using Euler Angles requires 6 parameters, 3 rotation angles and a 3x1 translation vector. Several application of screw theory has been introduced in robot kinematic. Compared with other methods, screws theory method just establish two coordinates, its geometrical meaning is obvious and it avoids singularities due to the use of the local coordinates. Therefore, screw theory has regained importance and has become an important method in robot kinematic. The major intents of this thesis are to formulize inverse kinematic problem in a compact closed form and to avoid singularity problem. Non-singular inverse kinematic solutions are obtained by using screw theory. Quaternion algebra is used to formulize kinematic problem in a compact closed form. Quaternions are hyper-complex numbers of rank 4, constituting a four dimensional vector space over the field of real numbers. Any rotation can be represented by unit-quaternion and also any screw motion can be defined by unit dual-quaternion. Screw motion can also be defined by using two quaternions however dual operators are the best way to describe screw motion and also the dual-quaternion is the most compact and efficient dual operator to express screw displacement. In this thesis, three inverse kinematic solution methods of 6-DOF serial robot manipulator, which are based on screw theory is presented. The first one is exponential mapping method. This method uses matrices as a screw operator. There are 16 parameters to describe screw motion in matrix operators while just 6 parameters are needed. Thus, however this method is singularity avoding, it is not compact closed. And also two new formulations of the inverse kinematic solution of the 6-DOF serial robot manipulator are proposed by using quaternion algebra. In these two new formulation methods, one of them uses quaternions as a screw operator which combines a unit quaternion plus pure quaternion and the other one uses dual-quaternions as a screw operator. These three methods and also the D-H convantion, which is the most common method in robot kinematic are compared with respect to singularity, computation efficiency, design complexity and accuracy. Simulation results are obtained by using Matlab and animation applications are obtained by using the virtual reality toolbox of MATLAB (VRML). Simulation experiments are made for single and cooperative working of Staubli TXL60 serial robot arm.Yüksek LisansM.Sc

Kinematics and Robot Design I, KaRD2018

This volume collects the papers published on the Special Issue “Kinematics and Robot Design I, KaRD2018” (https://www.mdpi.com/journal/robotics/special_issues/KARD), which is the first issue of the KaRD Special Issue series, hosted by the open access journal “MDPI Robotics”. The KaRD series aims at creating an open environment where researchers can present their works and discuss all the topics focused on the many aspects that involve kinematics in the design of robotic/automatic systems. Kinematics is so intimately related to the design of robotic/automatic systems that the admitted topics of the KaRD series practically cover all the subjects normally present in well-established international conferences on “mechanisms and robotics”. KaRD2018 received 22 papers and, after the peer-review process, accepted only 14 papers. The accepted papers cover some theoretical and many design/applicative aspects

A Survey on Dual-Quaternions

Over the past few years, the applications of dual-quaternions have not only developed in many different directions but has also evolved in exciting ways in several areas. As dual-quaternions offer an efficient and compact symbolic form with unique mathematical properties. While dual-quaternions are now common place in many aspects of research and implementation, such as, robotics and engineering through to computer graphics and animation, there are still a large number of avenues for exploration with huge potential benefits. This article is the first to provide a comprehensive review of the dual-quaternion landscape. In this survey, we present a review of dual-quaternion techniques and applications developed over the years while providing insights into current and future directions. The article starts with the definition of dual-quaternions, their mathematical formulation, while explaining key aspects of importance (e.g., compression and ambiguities). The literature review in this article is divided into categories to help manage and visualize the application of dual-quaternions for solving specific problems. A timeline illustrating key methods is presented, explaining how dual-quaternion approaches have progressed over the years. The most popular dual-quaternion methods are discussed with regard to their impact in the literature, performance, computational cost and their real-world results (compared to associated models). Finally, we indicate the limitations of dual-quaternion methodologies and propose future research directions.Comment: arXiv admin note: text overlap with arXiv:2303.1339

Forward dynamics of continuum and soft robots: a strain parametrization based approach

soumis à IEEE TROIn this article we propose a new solution to the forward dynamics of Cosserat beams with in perspective, its application to continuum and soft robotics manipulation and locomotion. In contrast to usual approaches, it is based on the non-linear parametrization of the beam shape by its strain fields and their discretization on a functional basis of strain modes. While remaining geometrically exact, the approach provides a minimal set of ordinary differential equations in the usual Lagrange matrix form that can be solved with standard explicit time-integrators. Inspired from rigid robotics, the calculation of the matrices of the Lagrange model is performed with a continuous inverse Newton-Euler algorithm. The approach is tested on several numerical benches of non-linear structural statics, as well as further examples illustrating its capabilities for dynamics

Line-of-sight-stabilization and tracking control for inertial platforms

Nowadays, line of sight stabilization and tracking using inertially stabilized platforms (ISPs) are still challenging engineering problems. With a growing demand for high-precision applications, more involved control techniques are necessary to achieve better performance. In this work, kinematic and dynamic models for a three degrees-of-freedom ISP are presented. These models are based in the vehicle-manipulator system (VMS) framework for modeling of robot manipulators operating in a mobile base (vehicles). The dynamic model follows the Euler-Lagrange formulation and is implemented by numeric simulations using the iterative Newton-Euler method. Two distinct control strategies for both stabilization and tracking are proposed: (i) computed torque control and (ii) sliding mode control using the recent SuperTwisting Algorithm (STA) combined with a High-Order Sliding Mode Observer (HOSMO). Simulations using data from a simulated vessel allow us to compare the performance of the computed torque controllers with respect to the commonly used P-PI controller. Besides, the results obtained for the sliding mode controllers indicate that the Super-Twisting algorithm offers ideal robustness to the vehicle motion disturbances and also to parametric uncertainties, resulting in a stabilization precision of approximately 0,8 mrad.Hoje em dia, a estabilização e o rastreamento da linha de visada utilizando plataformas inerciais continuam a constituir desafiadores problemas de engenharia. Com a crescente demanda por aplicações de alta precisão, técnicas de controle complexas são necessárias para atingir melhor desempenho. Neste trabalho, modelos cinemáticos e dinâmicos para uma plataforma mecânica de estabilização inercial são apresentados. Tais modelos se baseiam no formalismo para sistemas veículo-manipulator para a modelagem de manipuladores robóticos operando em uma base móvel (veículo). O modelo dinâmico apresentado segue a formulação analítica de Euler-Lagrange e é implementado em simulações numéricas através do método iterativo de Newton-Euler. Duas estratégias de controle distintas para estabilização e rastreamento são propostas: (i) controle por torque-computado e (ii) controle por modos deslizantes utilizando o recente algoritmo Super-Twisting combinado com um observador baseado em modos deslizantes de alta ordem. Simulações utilizando dados de movimentação de um navio simulado permitem comparar o desempenho dos controladores por torque computado em relação a um tipo comum de controlador linear utilizado na literatura: o P-PI. Além disso, os resultados obtidos para o controle por modos deslizantes permitem concluir que o algoritmo Super-Twisting apresenta rejeição ideal a perturbações provenientes do movimento do veículo e também a incertezas paramétricas, resultando em precisão de estabilização de aproximadamente 0,8 mrad

A way of relating instantaneous and finite screws based on the screw triangle product

It has been a desire to unify the models for structural and parametric analyses and design in the field of robotic mechanisms. This requires a mathematical tool that enables analytical description, formulation and operation possible for both finite and instantaneous motions. This paper presents a method to investigate the algebraic structures of finite screws represented in a quasi-vector form and instantaneous screws represented in a vector form. By revisiting algebraic operations of screw compositions, this paper examines associativity and derivative properties of the screw triangle product of finite screws and produces a vigorous proof that a derivative of a screw triangle product can be expressed as a linear combination of instantaneous screws. It is proved that the entire set of finite screws forms an algebraic structure as Lie group under the screw triangle product and its time derivative at the initial pose forms the corresponding Lie algebra under the screw cross product, allowing the algebraic structures of finite screws in quasi-vector form and instantaneous screws in vector form to be revealed.