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The Screw Calculus and its Applications to Mechanics
Abstract
The author sets forth the basic propositions of screw calculus on the basis of the elementary apparatus of modern vector algebra and indicates certain of its applications. The book sets forth material from the theory of sliding vectors, the algebra of complex numbers of the form alpha + omega (alpha superscript 0) with a special multiplier omega that possesses the property omega squares = 0, the algebra of screws, fundamentals of the differential geometry of the rules surface, which are necessary for the kinematics of solids, the foundations of screw analysis, and, finally, certain data from the classical theory of screws in its geometrical aspect, with indication of a number of applications in mechanics.
... It originated from the work of Sir R. Ball [1] as a method for unifying several geometrical notions from classical mechanics and received, in its early development at the end of the nineteenth century and beginning of twentieth century, contributions from eminent mathematicians including Clifford [7], Study [27], Kotel'nikov [14] and von Mises [30,31,33,34]. It was revived in the last decades for its powerfulness in dealing with applications such as robotics, particularly via von Mises motor (screw) calculus [3,9]. ...
... It goes back, at least, to Lovell III's PhD thesis [16] and has long been used in French engineering universities where screws are called torseurs [4,6,19,23,24]. We stress that in this work the screws will always be elements of a 6-dimensional real vector space, hence our notion of screw is equivalent to that of torseur, and conforms with the terminology used by Dimentberg [9]. This terminology differs from the original terminology used by Sir R. Ball. ...
... Clearly, via Eq. (2) a screw can be recovered from the knowledge of the pair (s, s(P)) where P is any point of E. For any P ∈ E the pair (s, s(P)) is called motor and the map S → V × V , s → (s, s(P)), is called motor reduction at P (and P is also referred as the reduction point) [9]. A choice of basis for V provides us with six real numbers called Plücker coordinates of the screw. ...
The theory of screws clarifies many analogies between apparently unrelated notions in mechanics, including the duality between forces and angular velocities. It is known that the real 6-dimensional space of screws can be endowed with an operator E , E 2 = 0 , that converts it into a rank 3 free module over the dual numbers. In this paper we prove the converse, namely, given a rank 3 free module over the dual numbers, endowed with orientation and a suitable scalar product ( D -module geometry), we show that it is possible to define, in a canonical way, a Euclidean space so that each element of the module is represented by a screw vector field over it. The new approach has the effectiveness of motor calculus while being independent of any reduction point. It gives insights into the transference principle by showing that affine space geometry is basically vector space geometry over the dual numbers. The main results of screw theory are then recovered by using this point of view.
... Robert S. Ball introduced screws to unify the descriptions of motions and forces [1] by geometrical means. In the wake of his work, mathematicians and physicists investigated the use of Plücker coordinates [2,3] and of special hypercomplex numbers known as dual vectors [4,5,6] to describe screws. However, in either case a coordinate frame must be specified first. ...
... The introduction of Dimentberg's famous book [6] provides much information on the early history of Screw Theory. A concise chronology of the development of rigid body dynamics in general can also be found in [14]. ...
... William K. Clifford, Alexandr P. Kotelnikov and Eduard Study developed a powerful formalism based on dual numbers [4,5,6]. A dual number is a number of the form a + εb where a, b ∈ R and ε is an imaginary unit such that ε 2 = 0 (dual quantities are denoted by a hat). ...
This paper is intended for students and researchers looking for more insight into Screw Theory. It shows how algebraic considerations lead to both physical and geometrical understanding of screws, and how they can connect affine geometry (what the world is) to linear algebra (what we can easily compute). Various formulations of the theory are first reviewed, as each of them highlights a particular aspect of screws. Their respective qualities and defects are also discussed. Subsequently, the powerful framework of Geometric Algebra (GA) is introduced to elucidate the nature of various physical objects commonly associated with screws, and eventually a new formalism based on GA is proposed, in which traditional screws clearly appear as a special case of more general affine objects. This approach generalizes the concept of a screw in a coordinate-free and origin-independent form. A simultaneous proof of Euler's First and Second Laws is provided to illustrate the use of this formalism.
... Robert S. Ball introduced screws to unify the descriptions of motions and forces [1] by geometrical means. In the wake of his work, mathematicians and physicists investigated the use of Plücker coordinates [2,3] and of special hypercomplex numbers known as dual vectors [4][5][6] to describe screws. However, in either case a coordinate frame must be specified first. ...
... The introduction of Dimentberg's famous book [6] provides much information on the early history of Screw Theory. A concise chronology of the development of rigid body dynamics in general can also be found in [14]. ...
... William K. Clifford, Alexandr P. Kotelnikov and Eduard Study developed a powerful formalism based on dual numbers [4][5][6]. A dual number is a number of the form a + εb where Î a b , and ε is an imaginary unit such that ε 2 = 0 (dual quantities are denoted by a hat). ...
This paper is intended for students and researchers looking for more insight into Screw Theory. It shows how algebraic considerations lead to both physical and geometrical understanding of screws, and how they can connect affine geometry (what the world is) to linear algebra (what we can easily compute). Various formulations of the theory are first reviewed, as each of them highlights a particular aspect of screws. Their respective qualities and defects are also discussed. Subsequently, the powerful framework of Geometric Algebra (GA) is introduced to elucidate the nature of various physical objects commonly associated with screws, and eventually a new formalism based on GA is proposed, in which traditional screws clearly appear as a special case of more general affine objects. This approach generalizes the concept of a screw in a coordinate-free and origin-independent form. A simultaneous proof of Euler's First and Second Laws is provided to illustrate the use of this formalism.
... In the wake of Ball's work (detailed in section 1.1), mathematicians and physicists investigated the use of Plücker coordinates [2] (see section 1.2) and of hypercomplex numbers such as dual vectors [3,4] (see section 1.3). From the second half of the XXth century and with the development of automation and robotism, researchers as F. M. Dimentberg [5] and K. H. Hunt [6] reinvigorated Screw Theory by promoting screws as the main mathematical description of mechanism kinematics. By the same time, Screw Theory underwent a completely different evolution in France [7,8], where it was standardized (and is still widely used) for pedagogical purposes [9]. ...
... The introduction of Dimentberg's famous book [5] provides much information on the early history of Screw Theory. A concise chronology of the development of rigid body dynamics in general can also be found in [12]. ...
... Addition and multiplication of dual numbers are defined from real addition and multiplication. A complete presentation of this formalism is given by Dimentberg [5]. ...
Since it was first developed by Sir Robert S. Ball at the end of the XIXth century, the Theory of Screws has known a considerable variety of reformulations, each of them underlining a different interpretation of screws: as geometrical, mechanical or algebraic objects. Beginning with an overview of the main existing formalisms, this article determines what prerequisites must be satisfied for a mathematical theory to represent screws and conciliate their geometric and algebraic aspects in a clear, elegant and pedagogical way. The mathematical framework of Geometric Algebra was precisely designed to reveal the geometrical significance of the algebraic objects of physics. A new formalism for Screw Theory is hence introduced, based on the geometric algebra G(3,0) and intended to generalize the concept of a screw (and therefore the extent of Screw Theory) and conciliate geometrical insight with algebraic efficiency. Moreover, this approach is coordinate-free and origin-independent, which makes it a powerful tool to treat affine geometry. A simple and straightforward description of finite motions appears as a natural feature of this new formulation. Deposited on HAL open archive as hal-04177875v3. To cite this document:
Loris Delafosse. A New Approach to Screw Theory using Geometric Algebra. 2023. e-print: https://hal.science/hal-04177875v3
... where ξ and ξ 0 are real functions with two real variables x and x * . Dimentberg [3] In this case, the general notation of dual analytic functions is as follows: ...
... This definition allows to write some well-known dual functions as follows ( [3,14]): ...
... Theorem 2.1 ( [3]) Let x = x + εx * be a dual variable. For n ∈ N, ...
... It originated from the work of Sir R. Ball [2] as a method for unifying several geometrical notions from classical mechanics and received, in its early development at the end of the nineteenth century and beginning of twentieth century, contributions from eminent mathematicians including Clifford [8], Study [29], Kotel'nikov [16] and von Mises [32,33,36,37]. It was revived in the last decades for its powerfulness in dealing with applications such as robotics, particularly via von Mises motor (screw) calculus [4,10]. ...
... It goes back, at least, to Lovell III's PhD thesis [15] and has long been used in French engineering universities where screws are called torseurs [5,7,20,24,25]. We stress that in this work the screws will always be elements of a 6-dimensional real vector space, hence our notion of screw is equivalent to that of torseur, and conforms with the terminology in [10]. This terminology differs from the original terminology by Sir R. Ball. ...
... For modern studies pointing out limitations, applications and more precise formalizations of this principle the reader is referred to [6,14,18,26,35]. The traditional approach to this principle is based on the rather surprising and puzzling result that formulas (4)-(5) hold true in screw theory, formulas that are then used to generalize several results in trigonometry [10]. In this approach it is difficult to grasp why (4)-(5) had to hold in the first place, that seems to happen just by chance, and the very fact that defining the dual angle in that peculiar form brings such simplifications appears as quite surprising. ...
It is known that the real 6-dimensional space of screws can be endowed with an operator , , that converts it into a rank 3 module over the dual numbers. In this paper we prove the converse, namely, given a rank 3 module over the dual numbers endowed with orientation and a suitable scalar product (-module geometry), we show that it is possible to define, in a natural way, a Euclidean space so that each element of the module is represented by a screw vector field over it. The new approach has the same effectiveness of motor calculus while being independent of any reduction point. It gives insights into the transference principle by showing that affine space geometry is basically vector space geometry over the dual numbers. The main results of screw theory are then recovered by using this point of view. As a case study, it is shown that -module geometry is effective in deriving results of Euclidean geometry. For instance, in a pretty much algorithmic way we prove classical results in triangle geometry, among which, Ceva's theorem, existence of the Euler line, and Napoleon's theorem.
... Many researchers investigated general compliant behaviors. In the analysis of spatial compliance, screw theory and Jacobian analysis [2][3][4][5][6][7][8][9], and Lie groups [10,11] have been widely used. In the recent work on the synthesis of compliance, mechanisms are designed to realize any specified compliance. ...
... where C 11 ∈ R 2×2 , v ∈ R 2 and c 22 > 0. The location r c of the compliance center C c is determined by Transactions of the ASME r c = Ωv c 22 (9) where the 2 × 2 matrix Ω is defined in Eq. (3). The relationship between the location of the compliance center and the configuration of a mechanism capable of realizing the behavior is presented in Ref. [33] for the general spatial case. ...
... (2) Select the location of J 2 . Based on the selected location of J 1 and Proposition 2, position (3,4) is selected to be the location of J 2 , which is separated from J 1 by l + y . The joint twist of J 2 is expressed as follows: ...
In this article, the synthesis of any specified planar compliance with a serial elastic mechanism having previously determined link lengths is addressed. For a general n-joint serial mechanism, easily assessed necessary conditions on joint locations for the realization of a given compliance are identified. Geometric construction-based synthesis procedures for five-joint and six-joint serial mechanisms having kinematically redundant fixed link lengths are developed. By using these procedures, a given serial manipulator can achieve a large set of different compliant behaviors by using variable stiffness actuation and by adjusting the mechanism configuration.
... The dual notation streamlines the kinematic analysis of rigidbody motion [26,27]. Unfortunately, this notation is not well suited to the representation of virtual work or kinetic and potential energies [28] and hence, the extended notation is used Fig. 2 Illustration of discontinuous Galerkin method; R denotes a generic variable in the following sections. In definition (19), tensor R denotes the rotation tensor, vector r is the position vector of a reference point, notation( ·) indicates the skew-symmetric matrix associated with a vector of size 3 or a matrix of size 6 constructed by two skew-symmetric of size 3, as will be shown in Eqs. ...
... These jump terms do not appear in the classical statement of Hamilton's variational principle [30,34,35]. Introducing the expression of action (28) to Eq. (29) leads to the weak form of Hamilton equations ...
... For the rigid-blade case, an analytical solution of the problem can be obtained: the three frequencies of the time invariant system are shown in Fig. 18. For angular speeds in the range of Ω ∈ [16,28] rad/s, two frequencies coalesce and the system becomes unstable. Figure 19 shows the dominant Floquet multiplier for this problem. ...
The dynamic response of many flexible multibody systems of practical interest is periodic. The investigation of such problems involves two intertwined tasks: first, the determination of the periodic response of the system and second, the analysis of the stability of this periodic solution. Starting from Hamilton’s principle, a unified solution procedure for continuous and discontinuous Galerkin methods is developed for these two tasks. In the proposed finite element formulation, the unknowns consist of the displacement and rotation components at the nodes, which are interpolated via the dual spherical linear interpolation technique. Periodic solutions are obtained by solving the discrete nonlinear equations resulting from continuous and discontinuous Galerkin methods. The monodromy matrix required for stability analysis is constructed directly from the Jacobian matrix of the solution process. Numerical examples are presented to validate the proposed approaches.
... Based on this fundamental advancement, a closed-form solution to a problem of computational kinematics is derived: Applying the principle of transference [23][24][25][26] to the solution approach for the inverse kinematics problem (IKP) of general, spherical three-revolute (3R) chains with three degrees-of-freedom (DOF) and employing the ATRD, a novel analytic method for solving the IKP of 6-DOF general, affine three-cylindric (3C) chains is achieved. The method complements former approaches in which spatial 3C chains are analyzed as subchains in spatial closed RCCC loops [7,[27][28][29][30] and closed 3CCC parallel platforms [31]. In the field of kinematic analysis, spatial kinematic chains equipped with cylindric joints are of significant importance since they appear as subchains in other mechanisms [10] but also since can be regarded as 'relaxed' -in the sense of 'sub-constrained' -versions of all chains featuring 'simple' (revolute, prismatic, and/or helical) joints. ...
... In 1965, Dimentberg [29] reported an extension of Rodrigues' rotation formula, in terms of Rodrigues' rotation vector [43] and its dualized analogue, following the principle of transference. The consistency of the REGG formula (Equation 15) and the ATRD (Equation 33) with Dimentberg's equations follows by means of the (adjoint matrix representation of) Rodrigues' vector ⋅̂⊗ ≔ tan ...
... in coherence with the formulation by Dimentberg [29] and, via the tangent half-angle identities cos( ) = 1 − 2 1 + 2 sin( ) = 2 1 + 2 , in coherence with the REGG formula in Equation (15). Analogously, the adjoint displacement̀is obtained, in terms of the dual tangent-half-angle parameter̃=tan(̃∕2), bỳ = cay ...
Based on the representation of rigid body displacements as adjoint matrices, the article introduces the adjoint trigonometric representation of displacements (ATRD) as a further generalization of the trigonometric representation of rotations. In comparison to the dual Rodrigues–Euler–Gauß–Gelman equation, recently reported for affine screw displacements with arbitrary, fixed pitches, the ATRD is built upon a product of a unit line and a dual angle, instead of upon a product of a unit screw and a real angle. Due to this conceptual difference, the ATRD requires four independent parameters of a unit line instead of five when parametrizing a displacement along a unit screw. As a consequence for computational kinematics, the ATRD permits transferring the analytic solution to the inverse kinematics problem (IKP) of 3‐DOF, general, spherical 3R‐chains into a closed‐form solution to the IKP of 6‐DOF, general, affine 3C‐chains.
... The Transference Principle was first stated by Kotelnikov [5] and soon after by Dimentberg [6]. Still, there have been equivalent statements from others like Rooney [7] and ...
... Dual quaternions are closely related to the concept of screw displacement ( Fig. 1 ). The dual quaternion of a displacement can be written as a function of the dual angle ˆ θ and the dual vector ˆ s [6] : ...
This work explores dual quaternions and their applications. First, a theoretical construction begins at dual numbers, extends to dual vectors, and culminates in dual quaternions. The physical foundations behind the developed theory lie in two important fundamentals: Chasles’ Theorem and the Transference Principle. The former addresses how to represent rigid-body motion whereas the latter provides a method for operating on it. This combination presents dual quaternions as a framework for modeling rigid mechanical systems, both kinematically and kinetically, in a compact, elegant and performant way. Next, a review on the applications of dual quaternions is carried out, providing a general overview of all applications. Important subjects are further detailed, these being the kinematics and dynamics of rigid bodies and mechanisms (both serial and parallel), control and motion interpolation. Discussions regarding dual quaternions and their applications are undertaken, highlighting open questions and research gaps. The advantages and disadvantages of using dual quaternions are summarized. Lastly, conclusions and future directions of research are presented.
... The theory of finite displacement screws began to thrive in the 1960s and 1970s. In 1965, Dimentberg [35] presented the finite screw displacement using a screw combined with the tangent of the dual angle of half the rotation. Woo and Freudenstein [36,37] algebraically developed the laws of the transformation and composition of screw coordinates, and applied them to the kinematics and statics of spatial mechanisms. ...
... According to Eq. (35) or (36), the pitch of the Mozzi-Chasles axis screw is obtained by ...
This paper examines the variations and derivations of the dual Euler-Rodrigues formula from various mathematical forms, including the matrix in 6 × 6, the dual matrix, Lie group SE(3) of the exponential map of the Lie algebra se(3), and the dual quaternion conjugation, and investigates their intrinsic connections. Based on the dual Euler-Rodrigues formula, the axis, the dual rotation angle, and the new traces are obtained by using the properties of the skew-symmetric matrices. In decomposing the Chasles’ motion, this paper examines two ways of realization of the motion based on the Mozzi-Chasles’ axis. With the equivalent motion, the paper relates the finite displacement screw matrix, the exponential map, and the dual quaternion conjugation to the dual Euler-Rodrigues formula and reveals their connection with the Mozzi-Chasles axis screw, whose parameters are used to construct the Lie algebra, the dual Euler-Rodrigues formula, and the dual quaternion. Further, using the Mozzi-Chasles axis screw, the paper presents a complete geometrical interpretation, including both the translation and rotation, and associates it with the algebraic presentation. By decomposing the equivalent translation induced by the rotation, the paper presents the mapping between the compound translation and the secondary part of the Mozzi-Chasles axis screw. With this map and the compound translation, the paper hence reveals the intrinsic connection between various presentations of rigid body transformations by formulating them into the dual Euler-Rodrigues formula and presents the relations of the exponential map of the Mozzi-Chasles axis screw to the finite displacement screw matrix and the dual Euler-Rodrigues formula, leading to the understanding of the various forms of a rigid body displacement in correspondence to the dual Euler-Rodrigues formula.
... Their first applications were given by Kotelnikov [9] and Study [13]. Dual variable functions were introduced by Dimentberg [4]. He investigated the analytic conditions of these functions, and by means of conditions, he described the derivative concept of these functions. ...
... It is seen that the derivative of the function f with respect to dual variable x is equal to the derivative with respect to real variable x [4]. Now, we shall study dual analytic functions f : D n → D, i.e., ...
In this paper, we give how to define the basic concepts of differential geometry on Dual space. For this, dual tangent vectors that have p as dual point of application are defined. Then, the dual analytic functions defined by Dimentberg are examined in detail, and by using the derivative of the these functions, dual directional derivatives and dual tangent maps are introduced.
... The pioneering work of engineering applications of dual algebra is due to Denavit [17] , Dimentberg [18] , Keler [19] , Beyer [20] , Yang [21] , Yang and Freudenstein [22] , Soni and Harrisberger [23] . The concept of the widely adopted Denavit-Hartenberg parameters has likely been inspired by the geometric interpretation of a dual angle. ...
... To do this we assemble the first three terms comprising the matrix S on the left hand side of Eq. (18) and show that their difference from B can never be zero. In order to illustrate the structure of the matrices S 1 , S 2 , S 3 , and S , more concretely, we continue the example at each step below. ...
This paper investigates the question of whether all dual matrices have dual Moore–Penrose generalized inverses. It shows that there are uncountably many dual matrices that do not have them. The proof is constructive and results in the construction of large sets of matrices whose members do not have such an inverse. Various types of generalized inverses of dual matrices are discussed, and the necessary and sufficient conditions for a dual matrix to be a Moore–Penrose generalized inverse of another dual matrix are provided. Necessary and sufficient conditions for other types of generalized inverses of dual matrices are also provided. A necessary condition, which can be easily computed, for a matrix to be a {1,2}-generalized inverse or a Moore–Penrose Inverse of a dual matrix is given. Dual matrices that have no generalized inverses arise in practical situations. This is shown by considering a simple example in kinematics. The paper points out that in other areas of science and engineering where dual matrices are also commonly used, formulations and computations that involve their generalized inverses need to be handled with considerable care. This is because unlike generalized inverses of ordinary matrices, generalized inverses of dual matrices do not always exist.
... See also, [14]. In many areas such as robot mechanics, mechanical design and computational geometry, screw theory has important applications, for instance, [4,9,24]. Moreover, there are applications to the physics of helicoidal (screw) surfaces as an optical vortices, see for example [1,20]. ...
We investigate helicoidal (screw) surfaces generated not only by regular curves but also by curves with singular points. For curves with singular points, it is useful to use frontals in the Euclidean plane. The helicoidal surface of a frontal can naturally be considered as a generalised framed base surface. Moreover, we show that it is also a framed base surface under a mild condition. We give basic invariants and curvatures for helicoidal surfaces of frontals by using the curvatures of Legendre curves. Moreover, we also give criteria for singularities of helicoidal surfaces.
... Kong further develops the variable constraint screw system and virtual-chain method for the topological synthesis of multi-mode mechanism [73,37]; Yu et al. [40,74] develop screw theory into a unified constraint-and kinematics-based method to intuitively design flexure mechanism, etc. Back to talk about finite displacement screw, since Dimentberg puts forward the concept and reveals its correlations with differential geometry and algebraic group [75], some foundation works on continuous displacement representation of screw axis have been made by Roth, Bottema, and Tsai [76][77][78]. Following the discussions on how to define the pitch of a finite twisting motion [79][80][81][82], Huang proposes the most well-recognized quasi-vector form of finite displacement screw [83], and firstly unifies the finite and instantaneous kinematics by linearly representing the screw triangle [84]. ...
Mechanism is a device to transmit and convert motion, force, and energy. However, for a long time, the structural synthesis methodology using the instantaneity and continuity based methods with a total of 11 variants actually just discusses the kinematic geometry significance and losses the mechanic concern. It neglects to extract the force and energy characteristics from task’s requirements, and may produce low-payload, weak-stiffness, or low-efficiency mechanisms. How to
take full consideration of motion, force, and energy in structural synthesis has become a tough challenge for strongly-interactive and physically-intelligent applications in the new-generation robot community. Therefore, this study proposes a new structural synthesis methodology unifying kinematic geometry and statics, integrating continuity and instantaneity by developing the finite and instantaneous screw theory, and taking motion, force, and energy characteristics simultaneously to design moving platform, limb, and joint respectively. As an example, the 2R1T mechanism exerted by 1F1M loads is designed with good payload, interaction and transmission work efficiencies. By utilizing this methodology, the feasible topologies designed are actually very fewer than those obtained by kinematic geometry based methodology, which may behave weak payload and work efficiency during strong interaction.
... In addition to complex numbers, dual and double numbers have found applications in physics, e.g., screw theory [6] or relativistic cosmology [7]. Dual numbers also make it possible to automatically compute derivatives of functions [8][9][10]. ...
The vast majority of existing neural networks operate by rules set within the algebra of real numbers. However, as theoretical understanding of the fundamentals of neural networks and their practical applications grow stronger, new problems arise, which require going beyond such algebra. Various tasks come to light when the original data naturally have complex-valued formats. This situation is encouraging researchers to explore whether neural networks based on complex numbers can provide benefits over the ones limited to real numbers. Multiple recent works have been dedicated to developing the architecture and building blocks of complex-valued neural networks. In this paper, we generalize models by considering other types of hypercomplex numbers of the second order: dual and double numbers. We developed basic operators for these algebras, such as convolution, activation functions, and batch normalization, and rebuilt several real-valued networks to use them with these new algebras. We developed a general methodology for dual and double-valued gradient calculations based on Wirtinger derivatives for complex-valued functions. For classical computer vision (CIFAR-10, CIFAR-100, SVHN) and signal processing (G2Net, MusicNet) classification problems, our benchmarks show that the transition to the hypercomplex domain can be helpful in reaching higher values of metrics, compared to the original real-valued models.
... This limit is independent of the ratio ∆x * ∆x [30]. ...
In this paper, the analyticity conditions of dual functions are clearly examined and the properties of the concept derivative are given in detail. Then, using the dual order relation, the dual analytic regions of dual analytic functions are constructed such that a collection of these regions forms a basis on . Finally, the equivalent of the inverse function theorem in dual space is given by a theorem and proved.
... The demand for a compliant system leads to the study of elastic behaviors. Using screw theory [12], Dimentberg [13] studied the static and small vibrations of rigid platforms suspended by line springs. Lončarić [14] analysed the synthesis problem of stiffness matrix using Lie groups. ...
This paper provides a geometrical insight into the dualities of compliant mechanisms via repelling screws. A method for the construction of the repelling screw system is proposed. By means of
screw theory and linear algebra, the closed-loop relationships among the twist/wrench spaces of both actuation- and constraint-screw systems are identified, upon which the kinematics, statics and stiffness/compliance of both full- and limited-mobility compliant mechanisms are analysed. The internal correlations between repelling screws and dualities of mechanisms are investigated, which reveals both orthogonal and dual properties of mechanisms with either parallel or serial configuration. The repelling-screw-based representation is applied to describing the permitted motions and restricted constraints of the mechanism. In addition, a novel and systematic approach for parallel-to-serial/serial-to-parallel transformation is proposed, which retains the capability of changing the constraints and relative dimensions of the target configuration to better suit a specific task. A few examples conducted demonstrate the feasibility of the proposed approach and the effectiveness of the repelling-screw based interpretations of mechanism geometries.
... Study [26], Mises [18], and Brand [6] advanced dual number algebra and studied line geometry and displacements by means of the 'calculus of motors'. Yang and Freudenstein [29], Dimentberg [11], and Fischer [15] further extended the theory and applied it to analyze problems in rigid body mechanics. ...
The characterization of the workspace for general spatial 3R chains with skew joint axes is refined by describing the variety of singular displacements as the union of the singular configuration manifolds of orientation type, position type, and attitude type. The surface of attitude singularities is revealed by transferring the singularity sets of spherical 3R chains with intersecting joint axes to the geometry of spatial kinematic chains with skew, non-intersecting joint axes. For this purpose, the degeneracy of a screw system is analyzed by means of a particular angle concept for a set of three oriented lines in space. The obtained argumentation is expressed in terms of geometric manipulator Jacobians completing previous results.
... (for detailed information see [1,3,4]). We can write ...
In this study, we obtain triplets from quaternions. First, we obtain triplets from real quaternions. Then, as an application of this, we obtain dual triplets from the dual quaternions. Quaternions, in many areas, it allows ease in calculations and geometric representation. Quaternions are four dimensions. The triplets are in three dimensions. When we express quaternions with triplets, our work is made even easier. Quaternions are very important in the display of rotational movements. Dual quaternions are important in the expression of screw movements. Reducing movements from four dimensions to three dimensions makes our work easier. This simplicity is achieved by obtaining triplets from quaternions.
... The inner product operation in the dual number domain does not yield the expected virtual work. Following the advice of Dimentberg [65], the "extended notation" is introduced, which recast dual vectors and matrices in D 3 to entities in R 6 , i.e., ...
Through a critical review of the various component mode synthesis techniques developed in the past, it is shown that both Craig–Bampton’s and Herting’s methods are particular cases of the mode-acceleration method and furthermore, Rubin’s method is equivalent to Herting’s method. Consequently, the mode-acceleration method is the approach of choice due to its simplicity and because unlike the other methods, it imposes no restriction on the selection of the modes. Next, a general approach to the modal reduction of geometrically nonlinear structures is developed within the framework of the motion formalism, based on the small deformation assumption. The floating frame of reference is defined unequivocally by imposing six linear constraints on the deformation measures, which are defined as the vectorial parameterization of the relative motion tensor that brings the fictitious rigid-body configuration to its deformed counterpart. This approach yields deformation measures that are both objective and tensorial, unlike their classical counterparts that share the first property only. Derivatives are expressed in the material frame, leading to computationally advantageous properties: tangent matrices are functions of the deformation measures only and become nearly constant during the simulation. Numerical examples demonstrate the accuracy, robustness, and numerical efficiency of the proposed approach. With a small number of modal elements, the formulation is able to capture geometrically nonlinear effects accurately, even in the presence of inherently nonlinear phenomena such as buckling.
... The inner product operation in the dual number domain does not yield the expected virtual work. Following the advice of Dimentberg [36], the "extended notation" is introduced, which recast dual vectors and matrices in D 3 to entities in R 6 , i.e., ...
Viscoelasticity plays an important role in the dynamic response of flexible multibody systems. First, single degree-of-freedom joints, such as revolute and prismatic joints, are often equipped with elastomeric components that require complex models to capture their nonlinear behavior under the expected large relative motions found at these joints. Second, flexible joints, often called force or bushing elements, present similar challenges and involve up to six degrees of freedom. Finally, flexible components such as beams, plates, and shells also exhibit viscoelastic behavior. This paper presents a number of viscoelastic models that are suitable for these three types of applications. For single degree-of-freedom joints, models that capture their nonlinear, frequency-dependent, and frequency-independent behavior are necessary. The generalized Maxwell model is a classical model of linear viscoelasticity that can be extended easily to flexible joints. This paper also shows how existing viscoelastic models can be applied to geometrically exact beams, based on a three-dimensional representation of the quasi-static strain field in those structures. The paper presents a number of numerical examples for three types of applications. The shortcomings of the Kelvin–Voigt model, which is often used for flexible multibody systems, are underlined.
... In the 19 th century, Cli¤ord described the dual numbers with in the form A = a+"a , where a; a 2 R, " 2 = 0 and " 6 = 0 [4]. Up to this time, there are number of studies in the literature that concern about the dual numbers and dual complex numbers [1,3,5,[8][9][10][17][18][19][20]. For instance, Fjelstad and Gal examined the extensions of the hyperbolic complex numbers to n dimensions and they presented n dimensional dual complex numbers [9]. ...
... The inner product operation in the dual number domain does not yield the expected virtual work. Following the advice of Dimentberg [6], the "extended notation" is introduced, which recast dual vectors and matrices in D 3 to entities in R 6 , i.e., ...
The application of the finite element method to the mod-eling of Cosserat solids is investigated in detail. In two-and three-dimensional elasticity problems, the nodal unknowns are the components of the displacement vector, which form a linear field. In contrast, when dealing with Cosserat solids, the nodal unknowns form the special Euclidean group SE(3), a nonlinear manifold. This observation has numerous implications on the implementation of the finite element method and raises numerous questions: (1) What is the most suitable representation of this nonlinear manifold? (2) How is it interpolated over one el-ement? (3) How is the associated strain field interpolated? (4) What is the most efficient way to obtain the discrete equations of motion? All these questions are, of course intertwined. This paper shows that reliable schemes are available for the interpolation of the motion and curvature fields. The interpolated fields depend on relative nodal motions only, and hence, are both objective and tensorial. Because these schemes depend on relative nodal motions only, only local parameterization is required, thereby avoiding the occurrence of singularities. For Cosserat solids, it is preferable to perform the discretization operation first, followed by the variation operation. This approach leads to considerable computation efficiency and simplicity.
... Considering the contact moment field, there exists one unique axis called the central axis of contact wrench and denoted by D [2,11], where the moment of contact forces is collinear to f. The projection c Δ of the CoM c onto D is given by: ...
Estimating the center of mass position and the angular momentum derivative of the human body is an important topic in biomechanics, since both quantities are essential to the dynamic description of the motion. In this work, we introduce a novel recursive algorithm to accurately estimate them, by fusing kinetic and kinematic measurements, based on a spectral description of the noise carried by each signal. This method exploits the mathematical relationships that links the center of mass position and the angular momentum derivative to recursively improve their estimation. The effectiveness of the approach is demonstrated on a simulated humanoid avatar, where access to ground truth data is granted. The results show that our method reduces the estimation error on the center of mass position with regard to kinematic estimation alone, in addition to providing a good estimate of the angular momentum variation. The proposed framework is finally applied to a recorded human walking motion in order to illustrate its applicability to real motion analysis data.
... applying the dual inverse tangent function ' Ć atan2' from (4). The geometry of the constraint can be considered as a spatial and affine generalization of the constraint TE1. ...
This article contributes to the conception of oriented dual angles by introducing two geometric representations of the dual-complex unit circle in context of Cayley-Klein geometries. By means of these representations and the principle of transference, line-geometric trigonometric constraint equations are stated and solved analytically. The trigonometric constraints and their solutions build the foundation to obtain closed-form solutions to generalized, line-geometric variants of Paden-Kahan problems.
... The dual numbers occurred in abstract mathematical researches but after a short time they proved their usefulness in applied sciences, and the work of Ball [2] should be mentioned. In the field of theory of mechanisms Dimentberg [3] and Kotelnikov [4] are essential references. In the domain of theory of mechanisms, Yang [5] obtained important results applying the dual quaternions for the kinematical study of spatial mechanisms; thus, he obtained the analytical relations for the displacements from the pairs of the RCCC mechanism. ...
The general methodology of solving a problem of kinematical analysis of a spatial mechanism is presented. The method is based on the system proposed by Hartenberg and Denavit, concerning the notations of the coordinate frames attached to the elements of the mechanism and on the closure matrix equation of a kinematical chain. Carrying out the closure equation for the case of spherical mechanisms and then applying the transfer principle of Kotelnikov, the equations which allow for finding the unknown parameters of the kinematical chain are obtained. Starting from the typical form of closure equation based on homogenous operators Hartenberg-Denavit, the closure matrix equation is obtained in dual format. By identifying two special matrices it is shown that the coefficient of the dual part from the closure equation is in fact a sum in which each term is attained from the real part of the closure equation by performing simple operations involving the special matrices, that reduce the calculus volume.
... Considering the contact moment field, there exists one unique axis called the central axis of contact wrench and denoted by ∆ [1,11], where the moment of contact forces is collinear to f . The projection c ∆ of the CoM c onto ∆ is given by: ...
Estimating the center of mass position and the angular momentum derivative of a the human body is an important topic in biomechanics, since both quantities are essential to the dynamic description of the motion. In this work, we introduce a novel recursive algorithm to accurately estimate them, by fusing kinetic and kinematic measurements, based on a spectral description of the noise carried by each signal. This method exploits the mathematical relationships that links the center of mass position and the angular momentum derivative to recursively improve their estimation. The effectiveness of the approach is demonstrated on a simulated humanoid avatar, where access to ground truth data is granted. The results show that our method reduces the estimation error on the center of mass position with regard to kinematic estimation alone, in addition to providing a good estimate of the angular momentum variation. The proposed framework is finally applied to a recorded human walking motion in order to illustrate its applicability to real motion analysis data.
... 3) From the central axis of the contact wrench: Considering the contact moment field, there exists one unique axis called the central axis of the contact wrench and denoted by ∆ [22], [23], where the moment of contact forces is collinear to f . The projection c ∆ of the CoM c onto ∆ is given by: ...
Estimating the center of mass position and the angular momentum derivative of legged systems is essential for both controlling legged robots and analyzing human motion. In this paper, a novel recursive approach to concurrently and accurately estimate these two quantities together is introduced. The proposed method employs kinetic and kinematic measurements from classic sensors available in robotics and biomechanics, to effectively exploits the accuracy of each measurement in the spectral domain. The soundness of the proposed approach is first validated on a simulated humanoid robot, where ground truth data is available, against an Extend Kalman Filter. The results demonstrate that the proposed method reduces the estimation error on the center of mass position with regard to kinematic estimation alone, whereas at the same time, it provides an accurate estimation of the derivative of angular momentum. Finally, the effectiveness of the proposed method is illustrated on real measurements, obtained from walking experiments with the HRP-2 humanoid robot
... 3) From the central axis of the contact wrench: Considering the contact moment field, there exists one unique axis called the central axis of the contact wrench and denoted by ∆ [22], [23], where the moment of contact forces is collinear to f . The projection c ∆ of the CoM c onto ∆ is given by: ...
Estimating the center of mass position and the angular momentum derivative of legged systems is essential for both controlling legged robots and analyzing human motion. In this paper, a novel recursive approach to concurrently and accurately estimate these two quantities together is introduced. The proposed method employs kinetic and kinematic measurements from classic sensors available in robotics and biomechanics, to effectively exploits the accuracy of each measurement in the spectral domain. The soundness of the proposed approach is first validated on a simulated humanoid robot, where ground truth data is available, against an Extend Kalman Filter. The results demonstrate that the proposed method reduces the estimation error on the center of mass position with regard to kinematic estimation alone, whereas at the same time, it provides an accurate estimation of the derivative of angular momentum. Finally, the effectiveness of the proposed method is illustrated on real measurements, obtained from walking experiments with the HRP-2 humanoid robot.
In this study, we investigate the propagation of polarized light along an optical fiber with the help of screw motion and Clifford's algebra of hyperbolic split dual quaternion in Rσ1,σ2,σ31,2 . The importance of this study is that hyperbolic screw motion allows to us characterize four types of polarization states of polarized light waves in an optical fiber. These are elliptical polarization ((E)-polarization), circular polarization ((C)-polarization), Lorentzian circular polarization ((LC)-polarization), and linear polarization ((L)-polarization). In addition, we determine the parametric equations of the four types of trajectories drawn by the end-points of the polarization vector while propagating in space that are called elliptical-Rytov curves (ER), circular-Rytov curves (CR), hyperbolic-Rytov curves (HR) and linear-Rytov curves (LR). Moreover, we visualize the polarization states and the related Rytov curves via mathematical programs. Furthermore, we use the four Stokes parameters and their matrix formulas to explain polarization states in Rσ1,σ2,σ31,2 .
Workspace and singularity analysis of serial manipulators are the focus of intense research in past decades. The computation of the workspace and its boundary is of significant interest because of their impact on manipulator design, placement in a working environment and trajectory planning. The methodology of geometric and computer modeling of the working space of a two-link planar manipulator is provided. Robot workspace is the set of positions which robot can reach. Workspace is one of most useful measures for the evaluation of robot. It’s usually defined as the reachable space of the end effector in Cartesian coordinate system. It is found, that the working space of the manipulator consists of the two one-parameter families of circles. One of these families consists of concentric circles, while the other is eccentric. In both cases the boundaries of the families are two concentric circles, and the radius of one of them is equal to the difference of the lengths of the elements of the manipulator kinematic mechanism, and the radius of the second circle is equal to the sum of these lengths. Then the workspace is a set of points of two disks (two “clouds”). The geometric image of these sets is a two-dimensional torus. The conducted studies of two families of circles on the plane made it possible to put them in correspondence with 3D models of two surfaces. On the graphs of these surfaces one can get not only the coordinates of the manipulator end-effector or the values of the generalized parameters corresponding to them. The resulting surfaces simulate the working space of the manipulator, and also solve the inverse problem of kinematics. Such surfaces make it possible to study both the parameters of the mechanism and the trajectories of the end-effector movement. The results are important for planning motions in the workspace and configuration space, as well as for the design and kinematic analysis of robots.
In this paper, the stiffness mapping of compliant robotic systems is generalized to the special Euclidean group SE(3). A geometric framework is proposed to unify the existing stiffness models. We analyze the symmetry and exactness relationship between joint and Cartesian stiffness matrices in this framework. To verify the theoretical results, motions of different types of manipulators, including serial and parallel ones, are tested in simulations. Based on the conservative property of the stiffness matrix, an impedance control strategy to achieve variable stiffness is proposed. In addition, a feasible stiffness identification method is developed using the skew-symmetric structure of the stiffness matrix.
In the realm of linkage synthesis the simplest problem is the coordination of the single-degree-of-freedom (single-dof) motion of one link, termed the output link, with the, likewise, single-dof motion of another link, termed the input link. Moreover, the two foregoing links are not coupled directly, but via an intermediate link, termed the coupler link. The input link, in the most common application, is driven by one single motor, either rotational or translational, the motor being supported by one fourth link, which is the fixed link, resting on the linkage frame. In this chapter we study the synthesis of a four-bar linkage for the production of a single-dof motion of the output link, as remotely driven, i.e., by means of the input link, ususally moving at a constant angular or translational velocity, as the case may be.
In this paper, necessary conditions on the locations and orientations of elastic components in a compliant mechanism used to realize any given spatial compliance are identified. The topologies considered are either fully parallel or fully serial mechanisms having an arbitrary number of lumped elastic components. It is shown that the requirements on elastic components are characterized by a sphere for the location distribution and by three cones for the orientation distribution. The easy to assess conditions on the set of components can be used to achieve a more desirable mechanism geometry when used in conjunction with existing spatial compliance synthesis procedures.
The characterization of the workspace for general spatial 3R chains with skew joint axes is refined by describing the variety of singular displacements as the union of the singular configuration manifolds of orientation type, position type, and attitude type. The surface of attitude singularities is revealed by transferring the singularity sets of spherical 3R chains with intersecting joint axes to the geometry of spatial kinematic chains with skew, non-intersecting joint axes. For this purpose, the degeneracy of a screw system is analyzed by means of a particular angle concept for a set of three oriented lines in space. The obtained argumentation is expressed in terms of geometric manipulator Jacobians completing previous results.
This paper presents extended dual quaternions with dual-generalized complex numbers coefficients. The characteristic properties are studied by considering them both as a dual-generalized complex number and as a dual quaternion. Furthermore, special matrix representations of extended dual quaternions are introduced which are used in formulas where these quaternions are involved.
This paper presents extended dual quaternions with dual-generalized complex numbers coefficients. The characteristic properties are studied by considering them both as a dual-generalized complex number and as a dual quaternion. Furthermore, special matrix representations of extended dual quaternions are introduced which are used in formulas where these quaternions are involved.
In an attempt to create a theory for describing such a human phenomenon as a possibility to self-learning, we need to create an instrument for dynamic modeling of sense-to-sense (S2S) [3] associations between heterogeneous objects. This instrument would help understand the nature of the cause-to-effect relationship [4] and the creation of new knowledge. In this article, we describe one of the instruments, sense derivative, that sheds light on the nature of forming new knowledge in the field of Artificial Intelligence.
Like each neuron of the human brain may be connected to up to 10,000 other neurons, passing signals to each other via as many as 1,000 trillion synaptic connections, in Sense Theory there is a possibility for connecting over 1,000 trillion heterogeneous objects. An object in Sense Theory is like a neuron in the human brain. Properties of the object are like dendrites of the neuron. Changing an object in the process of addition or deletion of its properties is like forming new knowledge in the process of synaptic connections of two or more neurons. In Sense Theory, we introduced a mechanism for determining possible semantic relationships between objects by connecting-disconnecting different properties. This mechanism is Sense Integral. In this article, we describe one of the instruments, sense antiderivative, that sheds light on the nature of forming new knowledge in the field of Artificial Intelligence.
Following the conceptual design of compliant mechanisms in Chap. 5, this chapter further looks into the stiffness properties of the developed compliant parallel mechanisms. Two types of stiffness problems are investigated, including the stiffness analysis and synthesis problem. In the stiffness analysis, the reciprocal relationship between motions and wrenches is used to design the layout of constraint limbs and construct the corresponding stiffness matrix. In the stiffness synthesis, the developed stiffness matrix is decomposed to obtain the configuration of constraint limbs based on the stiffness properties of each constraint limb. Existing synthesis algorithms are compared and categorized, including the direct-recursion and matrix-partition algorithm, and it is revealed for the first time the line-vector based matrix-partition algorithm can establish a one-to-one correspondence between the synthesized result and the initial configuration used to construct the compliant parallel mechanism. This is further verified by implementing the algorithms to decompose the constraint stiffness matrix of developed compliant parallel mechanisms.
After developing the compliance/stiffness matrix of a compliant mechanism, the next step is to evaluate the performance of the compliant mechanism based on the developed compliance/stiffness matrix. Sometimes this requires either an explicit or implicit expressions of compliance/stiffness matrix with respect to design parameters of the compliant mechanism, which will lead to further compliance/stiffness parameterization and optimization. As a result, this chapter looks into the Compliance/Stiffness Parameterization and Optimization problems using ortho-planar springs, a typical type of compliant mechanisms. In this chapter, for the first time, the six-dimensional compliance characteristics of ortho-planar springs are investigated using a compliance-matrix based approach, and they are further validated with both finite element (FEM) simulation and experiments. The compliance matrix is developed by treating an ortho-planar spring as a parallel mechanism and is revealed to be diagonal. As a consequence, corresponding diagonal compliance elements are evaluated and analysed in forms of their ratios, revealing that an ortho-planar spring not only has a large linear out-of-plane compliance but also has a large rotational bending compliance. Both FEM simulation and experiments were then conducted to validate the developed compliance matrix. In the FEM simulation, a total number of 30 types of planar-spring models were examined, followed by experiments that examined the typical side-type and radial-type planar springs, presenting a good agreement between the experiment results and analytical models. Further, a planar-spring based continuum manipulator was developed to demonstrate the large-bending capability of its planar-spring modules.
In this chapter, screw theory is introduced as a preliminary study to give readers an overview of screw theory and its application in compliant mechanism designs. The definition of a screw, as well as the key screw operations, are given in Sect. 2.2, such as the interchange operation of a screw, the screw coordinate transformation and the reciprocal product of two screws. Then the concept of the screw system is introduced in Sect. 2.3, which is an extension of the screw in a group form. Several typical screw systems are defined according to the reciprocal relationship between screws, notably the reciprocal screw system and repelling screw system. Further, the concepts of the screw and screw system are extended to describe the twist and wrench in Sect. 2.4, which are the two fundamental concepts in the further study of compliant mechanisms.
This chapter presents the conceptual design of compliant mechanisms. The fundamental idea behind it is to establish the relationship between freedom and constraint space of a compliant mechanism according to the reciprocal relationship between twists and wrenches. By using the constraint-based design approach, a flexible element can be represented by the constraint wrench exerted from it. Thus by knowing the preferred mobility of a compliant mechanism, the configuration of constraints can be determined according to the reciprocal relationship, which further leads to the layout design of the compliant mechanism. Without the loss of generality, compliant parallel mechanisms with single degree-of-freedom flexible elements are selected to verify the proposed design approach. Particularly a physical prototype implemented with shape-memory-alloy (SMA) actuators is built and tested. By employing SMA springs, the single DOF flexible element that resists the translation along its axis can be transformed into a linear actuator that generates a stroke along its axis. Both finite-element-simulations and experimental tests were carried out to verify the mobility of the compliant parallel mechanism, thus validating the initial conceptual design approach.
In this paper, the synthesis of any planar compliance with a 6-component compliant mechanism is addressed. The mechanisms studied are either serial mechanisms with six elastic joints or parallel mechanisms with six springs. For each type of mechanism, conditions on the mechanism configurations that must be satisfied to realize a given compliance are developed. The geometric significance of each condition is identified and graphically represented. Geometric construction based synthesis procedures for both types of mechanism are developed. These procedures allow one to select each elastic component from a restricted space based on its geometry.
Thesis--Columbia University. Bibliography: leaves 232-241. Photocopy. Ann Arbor, Mich., University Microfilms, 1971. 23 cm.
Preliminary Sketch of Biquaternions, Pro-ceed. of the London Mathemat
- W Clifford
Clifford, W., Preliminary Sketch of Biquaternions, Pro-ceed. of the London Mathemat. Soc., Vol. IV, 1873