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Solving the Input/Output Problem for Planar Mechanisms
Abstract
This paper presents a method for solving the input/output problem for all planar mechanisms composed of revolute and slider joints. The solution procedure is a modification of the Dixon resultant method, which was developed to solve sets of polynomial equations; in this paper the method is applied to sets of equations which are linens in the sines and cosines of unknown angles. A particular planar multi-circuit mechanism is analyzed to illustrate the solution procedure, and implementation details are discussed.
... Porta et al. [3] utilized relaxation techniques for position analysis of multiloop linkages. Alternatively, Wampler [4][5][6] presented the use of isotropic coordinates to formulate polynomial equations that describe the position of a general planar linkage, and for single-DOF linkages provided an elimination method, based on the Dixon determinant procedure of Ref. [7], for producing the polynomial equation relating the input to the output. For complex mechanisms, it can be advantageous to solve the polynomial systems using numerical polynomial continuation [8] instead of elimination. ...
... The input variable is designated as x 2 C. Also, designate a design variable, p 2 C, and let y be all the remaining joint variables y 2 C N . (The reason for using complex variables instead of real ones will become clear below [7].) It is assumed that the loop closure equations are formulated as ...
... Similar to this derivation for the four-bar, the procedure in Ref. [5] eliminates half of the variables using the loop equations and then solves the system of bilinear polynomials using the Dixon determinant. See Ref. [7] for a related method based on the Dixon determinant but using tangent-half-angle substitutions. ...
This paper considers single degree-of-freedom (DOF), closed-loop linkages with a designated input angle and one design parameter. For a fixed value of the design parameter, a linkage has input singularities, that is, turning points with respect to the input angle, which break the motion curve into branches. Motion of the linkage along each branch can be driven monotonically from the input. As the design parameter changes, the number of branches and their connections, in short the topology of the motion curve, may change at certain critical points. Allowing the design parameter to vary, the singularities form a curve called the critical curve, whose projection is the singularity trace. Many critical points are the singularities of the critical curve with respect to the design parameter. The critical points have succinct geometric interpretations as transition linkages. This paper presents a general method to compute the singularity trace and its critical points. As an example, the method is used on a Stephenson III linkage, and a range of the design parameter is found where the input angle is able to rotate more than one revolution between singularities. This characteristic is associated with critical points that appear as cusps on the singularity trace.
... If we set θ 3 = 75.75 • , the number of obtained solutions is six [15,16,2]. They are given in Table 1. ...
... We note that, while continuation and elimination methods must filter the solutions among the eighteen possible complex roots, the method given here directly provides the six real solutions shown in the table. The obtained solutions are in accordance with those in [15,16,2]. ...
... From them, only the six shown in Table 1 were labeled as solutions (thus returning the minimum possible number of boxes) and 5 boxes were found to be empty. It is difficult to tell at this point whether the presented algorithm outperforms the previous methods based on Dixon's resultant [15,16], mainly because no statistics are given in this respect in those works, and we have found no publicly available package implementing them. We have checked, however, that our method converges in substantially shorter times than those used by the continuation method in [17,18], using the implementation available at Jan Verschelde's home page, which spent about 3 seconds of CPU time on the same example. ...
This paper presents an interval propagation algorithm for variables in planar single-loop linkages. Given intervals of allowed values for all variables, the algorithm provides, for every variable, the whole set of values, without overestimation, for which the linkage can actually be assembled. We show further how this algorithm can be integrated in a branch-and-prune search scheme, in order to solve the position analysis of general planar multi-loop linkages. Experimental results are included, comparing the method's performance with that of previous techniques given for the same task.
... This reasoning permits to conclude that the curve generated by any point on a plane * Some of the ideas contained in this paper were already presented at the ASME pin-jointed mechanism possessing a finite number of links of finite size is necessarily algebraic [1]. The same result can be attained, in a more compact way, by computing the eliminant of the set of independent loop equations [2], [3], [4]. All coupler curves can be seen as a group of manifold curves joined through singular points usually classified in kinematics as crunodes and cusps [5, 6]. ...
... Unfortunately, this idea cannot be applied , in general, to multi-loop linkages. Actually, the valid configurations of a multi-loop linkage is usually represented by the solution set of an independent set of its vector loop equations [2], [3], [4]. This requires introducing a variable for each link representing its orientation with respect to the fixed link. ...
... The double butterfly linkage has one of the sixteen topologies available for 8-bar single-degree-of-freedom linkages [19]. In the context of classical kinematics of mechanisms, the input-output problem for this linkage leads to either sixteenth order or eighteenth order polynomials depending on the selected fix and input links [2, 4, 22]. This input-output problem, that was solved using continuation in [23] , is equivalent to the position analysis problem of the seven-link Baranov trusses of type II and III [20, 24]. ...
In general, high-order coupler curves of single-degree-of-freedom plane linkages cannot be properly traced by standard predictor-corrector algorithms due to drifting problems and the presence of singularities. Instead of focusing on finding better algorithms for tracing curves, a simple method that first traces the configuration space of planar linkages in a distance space and then maps it onto the mechanism workspace, to obtained the desired coupler curves, is proposed. Tracing the configuration space of a linkage in the proposed distance space is simple because the equation that implicitly defines this space can be straightforwardly obtained from a sequence of bilaterations, and the configuration space embedded in this distance space naturally decomposes into components corresponding to different combinations of signs for the oriented areas of the triangles involved in the bilaterations. The advantages of this two-step method are exemplified by tracing the coupler curves of a double butterfly linkage.
... To obtain the kinematic equations of a planar linkage, we follow the same formulation used in [9], which references the rotation angles of all bars to a fixed, ground coordinate system. With this, every angle θ i assigned to a bar b i defines a unit vector u i = (cos(θ i ), sin(θ i )) that gives the orientation of the bar. ...
... To illustrate the process, and to facilitate the comparison with previous work, we consider the same example as in [9] and [10], a double butterfly linkage, which is the only one ...
... Whereas specific methods for many linkages abound, a few recent methods are already universal, being able to manage arbitrary planar mechanisms. For example, Dhingra used reduced Gröbner-Bases and Sylvester's elimination to obtain a simple polynomial condition describing the solution set [8]. Nielsen and Roth also gave an elimination-based method that uses Dixon's resultant to derive the lowest degree polynomial of the algebraic system under study [9]. This technique was later improved by Wampler [10], who used a complex-plane formulation to reduce the size of the final eigenvalue problem by half. ...
This paper presents a numerical method able to compute all possible configurations of a planar linkage. The procedure is applicable to rigid linkages (i.e., those that can only adopt a finite number of isolated configurations) and to mobile ones (i.e., those that have internal degrees of freedom). The method is based on the fact that this analysis always reduces to finding the roots of a polynomial system of linear, quadratic, and hyperbolic equations, which is here tackled with a new strategy exploiting its structure. The method is conceptually simple, geometric in nature, and easy to implement, yet it provides solutions of the desired accuracy in short computation times. Experiments are included which show its performance on the double butterfly linkage, for which an accurate an complete discretization of its configuration space is obtained
... Also of numerical nature are the polynomial continuation methods [13,14], which, it might be remarked, are based on the Newton-Raphson method. Closed-form solutions are possible if use is made of elimination theories based on resultants [15,16] or Grö bner bases [17]. These, along with polynomial continuation methods, provide all the possible configurations of the mechanism for a given position. ...
... In this case [g] e is the geometric matrix of the element, which in consequence of Eqs. (16) and (17) is only dependent on the rod orientation (one angle, h e , in the plane and two, h 1e and h 2e , in space), but independent of its length, area and material data. The kinematic properties of a linkage depend only on its geometric and kinematic configuration. ...
... From Eq. (16) it follows that the derivative of the geometric matrix of a bar element in the plane is d½g e dh e ¼ À sin 2h e cos 2h e sin 2h e À cos 2h e sin 2h e À cos 2h e À sin 2h e À sin 2h e cos 2h e sin 2h e The input accelerations in Eq. (48) must be compatible, i.e. they must define the solution from a kinematic point of view, since Eq. (48) will yield a mathematically determined solution only if the physical problem is also determined. ...
This paper presents a numerical approach to rigid body linkage kinematics, based on a reduced form of the stiffness matrix and in structural analysis concepts. This matrix may be referred to as geometric stiffness matrix, or simply as geometric matrix. It is derived from basic nodes and length constraints, and provides full information on the kinematic properties of any linkage, including positions, velocities, accelerations, jerks and singular positions. This approach offers a number of major advantages, especially where simplicity and generality are concerned. The computational cost is also very low, because of the simplicity of the numerical calculations and the reduced dimensions of the matrices involved.
... The second one solves the same linkage but assuming that ae is a free variable, yielding a 1-dimensional continuum of solutions. The same benchmarks have been used previously to show the performance of elimination [12], [13], continuation [14], [15], and relaxation techniques [7]. We compare our results with those derived by such techniques, and employ the same linkage dimensions used in these papers. ...
... If we set © ae A ñ è ! ì 3 à è H ì Ò , the number of obtained solutions is six [7], [12], [13] . They are given in Table I. ...
... We note that, while continuation and elimination methods must filter the solutions among the eighteen possible complex roots, the one given here directly provides the six real solutions shown in the table. The obtained solutions are in accordance with those in [7], [12], [13]. Due to the nature of the algorithm all solutions are obtained as intervals that bound them, which allows ...
This paper presents an interval propagation algorithm for variables in single-loop linkages. Given allowed intervals of values for all variables, the algorithm provides, for every variable, the exact interval of values for which the linkage can actually be assembled. We show further how this algorithm can be integrated in a branch-and bound search scheme, in order to solve the position analysis of general multi-loop linkages. Experimental results are included, comparing the method’s performance with that of previous techniques given for the same task. Peer Reviewed
... To obtain the kinematic equations of a planar linkage, we follow the same formulation used in[17], which references the rotation angles of all bars to a fixed, ground coordinate system. With this, every angle θ i assigned to a bar b i defines a unit vector u i = (cos(θ i ), sin(θ i )) that gives the orientation of the bar. ...
... To illustrate the process, and to facilitate the comparison with previous work, we consider the same example as in[17]and[25], a double butterfly linkage, which is the only one of the eight-bar linkages that does not contain a four-bar loop (Fig. 1). Using Laman's theorem[9], it can be shown that this ...
... The second one solves the same problem but assuming that θ 6 is a free variable, yielding a 1-dimensional continuum of solutions. While the former case allows comparing the results with those published in[17]and[25], the latter shows the algorithm's performance for problems rarely addressed in the literature. In both cases, we adopt the geometric parameters used in[17]and[25]: a 0 = 7, a 1 = 7, a 2 = 5, b 0 = 13, b 1 = 6, ...
This paper presents a numerical method able to compute all possible configurations of planar linkages. The procedure is applicable to rigid linkages (i.e., those that can only adopt a finite number of configurations) and to mobile ones (i.e., those that exhibit a continuum of possible configurations). The method is based on the fact that this problem can be reduced to finding the roots of a polynomial system of linear, quadratic, and hyperbolic equations, which is here tackled with a new strategy exploiting its structure. The method is conceptually simple and easy to implement, yet it provides solutions of the desired accuracy in short computation times. Experiments are included that show its performance on the double butterfly linkage and on larger linkages formed by the concatenation of basic patterns. Peer Reviewed
... Analytical methods involve solving a loop closure constraint-based system of nonlinear equations [6]. Most analytical methods use the Polynomial continuation method [7,8], elimination method or Grobner bases [9] to solve the simulation problem. Although, these methods are able to find all the possible assembly configurations of a given mechanism, they are not general in nature. ...
... Here, the homogenous point coordinate a 0 has been assumed as unity without loss in generality. For planar RR link, = 1 The first order partial derivatives for spherical PP constraint given in Eq. (8) can be given as follows ...
This paper presents a geometric constraints driven approach to unified kinematic simulation of n-bar planar and spherical linkage mechanisms consisting of both revolute and prismatic joints. Generalized constraint equations using point, line and plane coordinates have been proposed which unify simulation of planar and spherical linkages and are demonstrably scalable to spatial mechanisms. As opposed to some of the existing approaches, which seek to derive loop-closure equations for each type of mechanism separately, we have shown that the simulation can be made simpler and more efficient by using unified version of the geometric constraints on joints and links. This is facilitated using homogeneous coordinates and constraints on geometric primitives, such as point, line, and plane. Furthermore, the approach enables simpler programming, real-time computation, and ability to handle any type of planar and spherical mechanism. This work facilitates creation of practical and intuitive design tools for mechanism designers.
... En este sentidos son métodos particulares que se concretan en programas de propósito particular. Una vez planteadas las ecuaciones del problema de posición del mecanismo, hay tres maneras de resolver estos sistemas de ecuaciones no lineales [NiR99]: por métodos de continuación polinomial, por métodos de eliminación y las Bases de Gröbner [Buc85]. ...
... Dentro de los métodos de eliminación se pueden distinguir tres tipos: métodos de eliminación simultánea [Wam00], de eliminación sucesiva [NiR99] [DAK00] y de eliminación repetida [DAK01]. Los métodos de eliminación poseen una eficiencia computacional mayor que los de continuación polinomial y las Bases de Gröbner. ...
En esta tesis se diseña un nuevo modelado computacional aplicable a sistemas multicuerpo de seis elementos y un grado de libertad, que son la base de muchas de las máquinas y mecanismos que hoy se emplean con profusión en la industria. Se ha realizado la formulación, desarrollo, implementación y validación de un algoritmo, para el análisis dinámico de cualquier sistema multicuerpo plano de seis elementos y un grado de libertad, basándose en la utilización de un sistema de formulación numérica del movimiento, accesible desde un lenguaje de programación convencional. Se aborda el análisis dinámico considerando las restricciones de posición, cinemáticas y dinámicas a las que se encuentre sometido el sistema que se requiere analizar y que puede ser aplicado sobre cualquier mecanismo constituido por elementos rígidos, independientemente de la dimensión de su movimiento, de su configuración topológica y de los pares cinemáticos que otorguen movimiento relativo a sus eslabones.
... However, their kinematic analysis -one of the most important aspects in linkage design -is still to be solved for more convenience and for greater effectiveness. And yet a lot of work has been done [2][3][4][5][6][7][8][9][10]. Kinematic analysis includes position analysis, velocity analysis and acceleration analysis; of these, position analysis is the most difficult. ...
... Kinematic analysis includes position analysis, velocity analysis and acceleration analysis; of these, position analysis is the most difficult. To analyze the position of a complex planar linkage, there are two categories of methods: the global method [9,10] and the group method [2,[4][5][6][7][8]. The global method takes the linkage as a whole by listing the position equations based on the vector loop, and solving the equations using different techniques. ...
This paper presents a simple and effective method to solve the position of higher-class Assur groups by means of virtual variable
searching. It transforms the higher-class Assur groups into a constraint link, Class II Assur group(s), and virtual driving
link(s), defined by the virtual variable. The constraint link is reassembled by one-dimensional searching of the virtual variable,
and the potential solutions of the position of the higher-class Assur group are achieved with rapid mathematic convergence.
Detailed criteria are set up for complicated higher-class Assur groups, including how to select the virtual driving link and
constraint Link, and how to decide the solving sequence of converted Class II Assur groups. A versatile visual program has
been developed to simulate higher-class planar mechanisms. Finally, an example of the feeding mechanism of a multifunction
domestic sewing machine demonstrates the new method.
... In general, the closed-loop equation of closed-loop mechanisms becomes a system of transcendental equations that cannot be solved easily. Wamper [1] and Nielsen and Roth [2] introduced the calculation method based on the Dixon determinant [3] in order to make the problem into an eigenvalue problem and derived the numerical solution. The numerical solutions were obtained by the iterative calculation. ...
The systematic kinematic analysis method for planar link mechanisms based on their unique procedures can clearly show the analysis process. The analysis procedure is expressed by a combination of many kinds of conversion functions proposed as the minimum calculation units for analyzing a part of the mechanism. When it is desired to perform this systematic kinematics analysis for a specific linkage mechanism, expert researchers can accomplish the analysis by searching for the procedure by themselves, however, it is difficult for non-expert users to find the procedure. This paper proposes the automatic procedure extraction algorithm for the systematic kinematic analysis of closed-loop planar link mechanisms. By limiting the types of conversion functions to only geometric calculations that are related to the two-link chain, the analysis procedure can be represented by only one type transformation function, and the procedure extraction algorithm can be described as a algorithm searching computable 2-link chain. The configuration of mechanism is described as the “LJ-matrix”, which shows the relationship of connections between links with pairs. The algorithm consists of four sub-processes, namely, “LJ-matrix generator”, “Solver process”, “Add-link process”, and “Over-constraint resolver”. Inputting the sketch of the mechanism into the proposed algorithm, it automatically extracts unique analysis procedure and generate a kinematic analysis program as a MATLAB code based on it. Several mechanisms are analyzed as examples to show the usefulness of the proposed method.
... As técnicas de identificação das características dinâmicas de um determinado elemento eram já utilizadas em outras áreas de engenharia, como a Mecânica ou a Elétrica [45,48]. A técnica usual nestes casos é a de vibração forçada (Input-Output), contudo no âmbito de Engenharia Civil, para estruturas de elevadas dimensões torna-se difícil aplicar forças externas de forma a obter a sua resposta. ...
Sendo os terramotos considerados um dos fenómenos da natureza mais violentos
e destrutivos, podendo causar grandes impactos sociais e económicos, em caso
de sismo é importante garantir o acesso dos meios de socorro e de evacuação às
regiões afetadas.
No presente trabalho, foi realizada a modelação de uma passagem pedonal
superior de três vãos. A calibração do modelo numérico, em regime elástico linear
foi feita recorrendo a dados obtidos através de uma campanha experimental de
identificação modal com base em vibrações ambientais. Foram também realizados
ensaios a ligações pré-fabricadas viga-pilar, materializadas por dois varões de aço
(ferrolhos), que permitiram a sua caracterização. Estes resultados serviram de
base a uma calibração em regime inelástico não linear do modelo numérico desta
ligação implementada no programa de cálculo de estruturas SeismoStruct.
O presente trabalho visa a avaliação da vulnerabilidade sísmica do passadiço
pedonal PP 2787, propenso ao descalçamento devido ao insuficiente comprimento
de entrega do tabuleiro na zona dos aparelhos de apoio nos vãos centrais. A
estrutura estudada representa uma série de passadiços pedonais localizados
na zona de Faro. Uma abordagem probabilística permitiu considerar a
variabilidade da ação sísmica, nomeadamente cem cenários sísmicos diferentes,
e incertezas na definição das propriedades dos materiais e/ou comportamento
estrutural. Subsequentemente foram realizadas uma série de análises dinâmicas
incrementais que permitiram a definição de vários estados de dano da estrutura e
a construção das curvas de fragilidade necessárias.
Com base nos resultados obtidos nesta dissertação, foi possível concluir
que, para uma aceleração de projeto de acordo com a regulamentação,
existe uma probabilidade significativa de que a estrutura sofra apenas danos
ligeiros a moderados com probabilidades de ocorrência de 45,41% e 32,79%
respetivamente. Quanto á gestão da fiabilidade concluí-se que só se satisfazem os
requisitos regulamentares se se considerar que a estrutura pertence a uma classe
de importância baixa.
... In order to ensure the design of an eight-bar linkage is usable, we analyze its movement through the five task positions. Our approach uses the Dixon determinant elimination procedure described in Wampler (2001) [12] and Neilson and Roth (1999) [13], to find all the solutions to the forward kinematics problem. They refer to this procedure as solving an input/output problem for planar linkages. ...
This paper formulates a methodology for designing planar eight-bar linkages for five task positions, by adding two RR constraints to a user specified 6R loop. It is known that there are 32 ways in which these constraints can be added, to yield as many as 340 different linkages. The methodology uses a random search within the tolerance zones around the task specifications to increase the number of candidate linkages. These linkages are analyzed using the Dixon determinant approach, to find all possible linkage configurations over the range of motion. These configuration trajectories are sorted into branches. Linkages that have all the five task configurations on one branch, ensure their smooth movement through the five task positions. The result is an array of branch-free useful eight-bar linkage designs. An example is provided to illustrate the results.
... There are three ways to solve to solve the resultant systems of non-linear equations [7] . They may be solved either directly using a numerical procedure (continuation methods [8] ), or using analytical techniques to obtain solutions in close form with elimination theories [9] . Another possibility is the use of Gröbner bases [10] . ...
In this paper a new and general method for the kinematic analysis of planar linkages is presented. This procedure takes advantage of a purely geometric approach different from those general methods based on the use of the Jacobian matrix. The proposed method, named Geometrical-Iterative Method (GIM), solves the initial position and the finite displacement problems of multi-circuit planar linkages. The aim is the construction of the loop equations entirely in a geometric way by transforming the linkage into a series of nodes and geometrical constraints. The geometrical constraints are obtained as a consequence of the application of the rigid body condition to the links of the mechanism. The method is applied to complex linkages with lower pairs, and rotary or linear actuators. This novel method has a high computational efficiency compared to those methods based upon Newton-Raphson.
... General algorithms indeed exists for the closed-form position analysis of multi-loop planar linkages but they invariably rely on resultant elimination techniques applied to sets of kinematic loop equations. For example, Nielsen and Roth [19], and Wampler [20] presented general methods for the analysis of planar linkages using the Dixon's resultant. Although the uniform treatment of these elimination-based methods of all planar linkages is remarkable, the position analysis of Assur kinematic chains based on them has to be carried out on a case-by-case basis because the required variable eliminations change. ...
The real roots of the characteristic polynomial of a planar linkage determine its assembly modes. In this work it is shown how the characteristic polynomial of a Baranov truss derived using a distance-base formulation contains all the necessary and sufficient information for solving the position analysis of the Assur kinematic chains resulting from replacing some of its revolute joints by slider joints. This is a relevant result because it avoids the case-by-case treatment that requires new sets of variable eliminations to obtain the characteristic polynomial of each Assur kinematic chain.
... Although derivations had been found for each of the six-bar linkages [9], until recently, there did not exist a method for deriving these polynomials for general linkages. This limitation is overcome by the general formulations in [6,10,11] for solving input/output problems: each of these can be adapted to produce tracing curve polynomials. The last of these, [11], gives the tracing curve equation in a particularly convenient form for studying the singular foci. ...
The focal points of a curve traced by a planar linkage capture essential information about the curve. Knowledge of the singular foci can be helpful in the design of path-generating linkages and is essential to the determination of path cognates. This paper shows how to determine the singular foci of planar linkages built with rotational links. The method makes use of a general formulation of the tracing curve based on the Dixon determinant of loop equations written in isotropic coordinates. In simple cases, the singular foci can be read off directly from the diagonal of the Dixon matrix, while the worst case requires only the solution of an eigenvalue problem. The method is demonstrated for one inversion each of the Stephenson-3 six-bar and the Watt-1 six-bar.
... A. Dhingra and colleagues used reduced Gröbner bases and Sylvester's elimination to obtain these polynomials [10]. J. Nielsen and B. Roth also gave an elimination-based method that uses Dixon's resultant to derive the lowest degree characteristic polynomials [11]. This technique was later improved by C. Wampler [12], who used a complex-plane formulation to reduce the size of the final eigenvalue problem by half. ...
The position analysis of planar linkages has been dominated by resultant elimination and tangent-half-angle substitution techniques applied to sets of kinematic loop equations. This analysis is thus reduced to finding the roots of a polynomial in one variable, the characteristic polynomial of the linkage. In this paper, by using a new distance-based technique, it is shown that this standard approach becomes unnecessarily involved when applied to the position analysis of the three seven-link Assur kinematic chains. Indeed, it is shown that the characteristic polynomials of these linkages can be straightforwardly derived without relying on variable eliminations nor trigonometric substitutions, and using no others tools than elementary algebra.
... It is less applicable in general studies on a symbolic level. Closed-form solutions can be obtained using elimination methods (resultant methods) [18,19]. These methods are based on an algebraic formulation that allows the elimination of a large number of variables in one single step. ...
In this paper, a new method to solve the forward position problem in planar mechanisms with revolute pairs is presented. This method is based on purely geometrical concepts applying sequentially geometrical restrictions. A searching algorithm defines the order of application of these restrictions to the elements of the mechanism using some developed rules and criteria. The development of the method and its implementation in a simulation program developed by the authors is also explained. The geometric algorithm is compared with the algorithm most commonly used in kinematic analysis, i.e. the Newton–Raphson method in order to evaluate its efficiency. Several illustrative examples are presented using representative mechanisms.RésuméUne nouvelle méthode pour résoudre le problème de position pour mécanismes plans est décrite dans cet article. Cette méthode s’appuie sur des concepts purement géométriques et sur l’application séquentielle de restrictions géométriques. Un algorithme de recherche définit l’ordre d’application de ces restrictions sur les éléments du mécanisme à travers des règles et critères développés. Le développement et l’application de la méthode sont illustrés à l’aide d’un programme de simulation développé par les auteurs. Afin de définir son efficacité l’algorithme géométrique est comparé avec les algorithmes les plus communs utilisés en analyse cinématique, c’est á dire la méthode de Newton–Raphson. Plusieurs exemples sont présentés en utilisant des mécanismes représentatifs.
... The elimination method [14,26,62,64] is analytical, but usually the polynomial must be solved by a numerical method. In Gröbner bases [14,65,66], an invariant polynomial is obtained. Each subsequent equation adds at most one variable. ...
This paper presents an overview of the literature on kinematic and calibration models of parallel mechanisms, the influence of sensors in the mechanism accuracy and parallel mechanisms used as sensors. The most relevant classifications to obtain and solve kinematic models and to identify geometric and non-geometric parameters in the calibration of parallel robots are discussed, examining the advantages and disadvantages of each method, presenting new trends and identifying unsolved problems. This overview tries to answer and show the solutions developed by the most up-to-date research to some of the most frequent questions that appear in the modelling of a parallel mechanism, such as how to measure, the number of sensors and necessary configurations, the type and influence of errors or the number of necessary parameters.
... A few works in the literature already provide general complete linkage solvers, but their applicability is limited to planar or spherical linkages. These include the work by Nielsen and Roth, who gave an algorithm to derive the Dixon resultant of any planar linkage [14], the work by Wampler, which improves on Nielsen and Roth's by applying a complex plane formulation [15], the work by Celaya et al. [16], which provides an interval propagation algorithm, and, finally, the work by Porta et al. [17], which attacks the problem via linear relaxations An examination of these methods shows that [17] is specially amenable for generalization to the spatial case, a task we preliminary addressed in [18] and which we complete in the present paper. As a result, we contribute with a method able to solve the position analysis of any spatial linkage, irrespective of the joint types it involves, of the interconnection pattern of the links, and of the dimension of the solution space. ...
This report presents a new method able to isolate all configurations that a multi-loop linkage can adopt. We tackle the problem by providing formulation and resolution techniques that fit particularly well together. The adopted formulation yields a system of simple equations (only containing linear and bilinear terms, and trivial trigonometric functions for the helical pair exclusively) whose special structure is later exploited by a branch-and-prune method based on linear relaxations. The method is general, as it can be applied to linkages with single or multiple loops with arbitrary topology, involving lower pairs of any kind, and complete, as all possible solutions get accurately bounded, irrespectively of whether the linkage is rigid or mobile.
This paper presents a geometric constraints driven approach to unified kinematic simulation of n-bar planar and spherical linkage mechanisms consisting of both revolute and prismatic joints. Generalized constraint equations using point, line and plane coordinates have been proposed which unify simulation of planar and spherical linkages and are demonstrably scalable to spatial mechanisms. As opposed to some of the existing approaches, which seek to derive loop-closure equations for each type of mechanism separately, we have shown that the simulation can be made simpler and more efficient by using unified version of the geometric constraints on joints and links. This is facilitated using homogeneous coordinates and constraints on geometric primitives, such as point, line, and plane. Furthermore, the approach enables simpler programming, real-time computation, and ability to handle any type of planar and spherical mechanism. This work facilitates creation of practical and intuitive design tools for mechanism designers.
The presented study shows results of structural and kinematic analysis of a planar six-bar quick-return mechanism that is used in shaping and planing machines for transformation of rotational motion of a driving link into prismatic motion of an end-effector. Assur groups of the III and II classes have been separated out from a quick-return mechanism when different driving links have been chosen during a structural analysis. Kinematic analysis has been carried out by grapho-analytical method for the case when a four-bar Assur group is included. Finally, 3D model has been simulated and coordinates of distinguished points of movable links have been found in six positions of the mechanism depending on the rotation of a driving link. The obtained results can be used in kinetostatic and dynamic analysis of the quick-return mechanism. The findings of the study can also be used in a design of planning and shaping machines, in synthesis and analysis of novel planar mechanisms.
Dr. Bernard Roth (see Fig. 1) is the Rodney H. Adams Professor of Engineering at Stanford University and the co-founder and current Academic Director of the Hasso Plattner Institute of Design (also known as the d.school). His long and distinguished academic career at Stanford consists of significant and pioneering accomplishments in teaching, research, and consulting on various aspects of mechanical engineering, with a special emphasis on mechanism design. Bernie has established a worldwide reputation in the kinematic synthesis and analysis of mechanisms and is a pioneer in the field of computer controlled robot manipulators. Bernie has investigated the mathematical theory of rigid body motions and the application of these motions to the kinematic synthesis of mechanisms. He has placed a special emphasis on geometric kinematics over the more traditional time-based formulations which have allowed him to make important contributions to Burmester theory, curvature theory, and screw theory. His text book, Theoretical Kinematics, co-authored with Oene Bottema, is regarded by many kinematicians as the most elegant and rigorous treatment of this applied science. Also, his popular book The Achievement Habit: Stop Wishing, Start Doing, and Take Command of Your Life describes some of the innovative techniques that he employs in the classes, workshops, and short courses that he has offered on creativity and design thinking.
In this paper we present an algorithm that automatically creates the linkage loop equations for planar 1-DoF linkages of any topology with rotating joints, demonstrated up to 8-bars. The algorithm derives the linkage loop equations from the linkage graph by establishing a cycle basis through a single common edge. Divergent and convergent loops are identified and used to establish the fixed angles of the ternary and higher links.
Results demonstrate the automated generation of the linkage loop equations for the five distinct 6-bar mechanisms, Watt I-II and Stephenson I-III, as well as the seventy one distinct 8-bar mechanisms.
The resulting loop equations enable the automatic derivation of the Dixon determinant for linkage kinematic analysis of the position of every possible assembly configuration. The loop equations also enable the automatic derivation of the Jacobian for singularity evaluation and tracking of a particular assembly configuration over the desired range of input angles.
The methodology provides the foundation for the automated configuration analysis of every topology and every assembly configuration of 1-DoF linkages with rotating joints up to 8-bar. The methodology also provides a foundation for automated configuration analysis of 10-bar and higher linkages.
In this paper, we present an algorithm that automatically creates the linkage loop equations for planar one degree of freedom, 1DOF, linkages of any topology with revolute joints, demonstrated up to 8 bar. The algorithm derives the linkage loop equations from the linkage adjacency graph by establishing a rooted cycle basis through a single common edge. Divergent and convergent loops are identified and used to establish the fixed angles of the ternary and higher links. Results demonstrate the automated generation of the linkage loop equations for the nine unique 6-bar linkages with ground-connected inputs that can be constructed from the five distinct 6-bar mechanisms, Watt I-II and Stephenson I-III. Results also automatically produced the loop equations for all 153 unique linkages with a ground-connected input that can be constructed from the 71 distinct 8-bar mechanisms. The resulting loop equations enable the automatic derivation of the Dixon determinant for linkage kinematic analysis of the position of every possible assembly configuration. The loop equations also enable the automatic derivation of the Jacobian for singularity evaluation and tracking of a particular assembly configuration over the desired range of input angles. The methodology provides the foundation for the automated configuration analysis of every topology and every assembly configuration of 1DOF linkages with revolute joints up to 8 bar. The methodology also provides a foundation for automated configuration analysis of 10-bar and higher linkages.
This paper considers single-degree-of-freedom, closed-loop linkages with a designated input angle and one design parameter. For a fixed value of the design parameter, a linkage has turning points (dead-input singularities), which break the motion curve into branches such that the motion along each branch can be driven monotonically from the input. As the design parameter changes, the number of branches and their connections, in short the topology of the motion curve, may change at certain critical points. As the design parameter changes, the turning points sweep out a curve we call the “turning curve,” and the critical points are the singularities in this curve with respect to the design parameter. The critical points have succinct geometric interpretations as transition linkages. We present a general method to compute the turning curve and its critical points. As an example, the method is used on a Stephenson II linkage. Additionally, the Stephenson III linkage is revisited where the input angle is able to rotate more than one revolution between singularities. This characteristic is associated with cusps on the turning point curve.
A new viewpoint on structural composition of mechanism is proposed in this paper: any mechanism can be decomposed into a group of ordered single-open-chains (SOC). Based on the ordered SOC and its constraint property, the following concepts or principles are proposed or established: (1) the criteria for determination of basic kinematic chain (BKC). (2) the coupling degree of BKC, (3) the criteria for determination of types of degree-of-freedom, (4) the formula for calculation of number of mechanism configurations, and (5) a new unified modular method for structural, kinematic and dynamic analysis of mechanism.
The displacement analysis of a kind of nonplanar nine-link Barranov truss is completed by using Dixon resultants together with Sylvester resultant. Firstly, four geometric loop equations are set up by using vector method, and then they are changed into complex number form. Secondly, three constraint equations are used to construct the Dixon resultants, which is a 6times6 matrix and contain two variables to be eliminated. Extract the GCD of two columns and two rows of Dixon matrix and compute its determinant to obtain a new equation. This equation together with the forth constraint equation can be used to construct a Sylvester resultant. A 46 degree univariate polynomial equation is obtained from determinant of Sylvester resultant. Other variables can be computed by Euclidean algorithm and Gaussian elimination. At last a numerical example confirms that assembly configurations number of the Barranov truss is 46.
In this paper, a novel method that solves the Forward displacement problem (FDP) of several common spherical parallel manipulators (SPMs) is presented. The method uses the quaternion algebra to express the FDP as a system of equations and the Dixon determinant procedure to construct univariate polynomials whose roots can be found either numerically or analytically. A case study is solved for a specific SPM, which satisfies certain geometric conditions, having 3 - R(E)R architecture, with R denoting a revolute joint, E a planar motion generator and underlines indicating the actuated joint. In this case, the solutions of the system are obtained analytically by a symbolic method exploiting symmetries.
This paper presents a simple and systematic modular approach for kinematic analysis of complicated Parallel Kinematic Manipulators (in short, PKMs) which coupled degrees are more than 2. (1) Single open chains (in short, SOCs) may be regarded as the basic modules of a PKM. Any PKM can always be decomposed automatically into a set of ordered SOCs, and these SOCs can also be used to recognize the basic kinematic chains contained in it. (2) The kinematic analysis algorithms and the compatibility conditions of the SOC modules are offered. (3) Directly applying the above SOC kinematic modules, the kinematic equations of a PKM can be automatically established. (4) In order to solve kinematic equations of complicated parallel manipulators which coupled degrees are more than 2, a new searching algorithm which requires no initial guess has been presented. The procedural approach is demonstrated in parallel manipulators.
Development of innovative suspension mechanisms and optimization of their elastokinematic performances are usually studied by applying static constraints on the numerical model of the mechanism. During the development of the new Michelin optimized contact patch multilink suspension concept, a two-dimensional semi-analytical model is studied in order to validate it. After writing the loop closure equations and obtaining systems of multivariate polynomials, the method consists in using the determinant of Sylvester's matrix to solve them. The paper presents the design studies to determine the influence of the geometric parameters of the mechanism on the variation of the camber and the half-track under lateral loads.
The position analysis of a nine-link Barranov truss is finished by using Dixon resultants together with Sylvester resultants. Above all, using vector method in complex plane to construct four constraint equations and transform them into complex exponential form, then three constraint equations are used to construct a 6 X 6 Dixon matrix, which contains two variables to be eliminated. We extract the greatest common divisor (GCD) of two columns of Dixon matrix and compute its determinant to obtain a new equation. This equation together with the fourth constraint equation can be used to construct a Sylvester resultant. A 50 deg univariate polynomial equation is obtained from the determinant of Sylvester resultant. Other variables can be computed by Euclidean algorithm and Gaussian elimination. Lastly, a numerical example confirms that the analytical solution number of the Barranov truss is 50. It is the first time to complete analytical solutions of this kind of Barranov truss.
This paper presents a general method for the analysis of planar mechanisms consisting of rigid links connected by rotational and/or translational joints. After describing the links as vectors in the complex plane, a simple recipe is outlined for formulating a set of polynomial equations which determine the locations of the links when the mechanism is assembled. It is then shown how to reduce this system of equations to a generalized eigenvalue problem, or in some cases, a single resultant polynomial. Both input/output problems and tracing-curve equations are treated.
This paper presents the position analysis in analytical form of the Assur group of class 3 and order 3 with four links and six revolute/prismatic joints (triad). The aim of this position analysis is to determine all possible configurations of the Assur group, for a given position of its external joints. Four kinds of the Assur group of class 3, with one, two and three prismatic joints are investigated. The analysis leads to a nonlinear system of three equations with three unknown parameters. Using a successive elimination procedure, a final polynomial equation in one unknown is obtained. The real solutions of the polynomial equation correspond to assembly modes of the Assur group. Three numerical applications of the proposed methods are presented. Finally, a numerical application for position analysis of a planar mechanism with eight links including a triad with three external prismatic joints is also given.
This paper presents position analyses of open normal Assur groups A (3.6). Planar mechanisms can be seen as composed of link groups (Assur groups) with zero mobility relative to the links to which they are successively added. These Assur groups, serving as modules in the synthesis and analysis of complex planar mechanisms, might adopt a certain number of positions which allow choosing different solutions to the engineering task. An open normal Assur group, for which we write A (3,6) in short, is an open (non-closed) linear kinematic chain of ternary (3) links to which 6 binary links are attached. It is found that for a given set of system parameters an open normal Assur group A (3.6) held together exclusively by rotor joints might theoretically adopt 162 different positions (real and complex). If the outer joints at the binary links are prismatic joints, the number of possible positions is reduced to 16.
In this paper, a method to solve the forward position problems of planar linkages with prismatic and revolute joints is presented. These linkages can have any number of degrees of freedom. This method has been named the geometrical iterative method and is based on geometrical concepts. An iteration sequence that corresponds to the system of non-linear equations describing closure of the mechanism loops is defined. This sequence is applied in successive iterations to obtain the position of the mechanism. In order to achieve convergence, the iteration sequence must fulfil two fundamental conditions. A searching algorithm has been developed to obtain a useful iteration sequence. It is based on the use of hierarchical rules and criteria. The method has been implemented in a simulation program developed by the authors. Several illustrative examples are presented using representative linkages.
This paper studies the position analysis of a kind of nine-link Barranov truss. Firstly, vector method is used to derive four geometric constraint equations, which are subsequently transformed into complex exponent form by Euler’s complex number formula. Three of these constraint equations are picked to construct the Dixon resultant, a 6×6 matrix of two variables. Expanding the determinant of this matrix yields an additional equation which is then combined with the forth constraint equation to construct a Sylvester resultant matrix. The determinant of this matrix yields a univariate polynomial of order 56. Solving this polynomial, we obtain all solutions to the suppressed variable. The solutions of other variables can be computed by Euclidean algorithm and Gaussian elimination. A numerical example is provided to verify the result. It is the first time that this kind of Barranov truss is solved analytically.
Spherical linkages, having rotational joints whose axes coincide in a common center point, are sometimes used in multi-degree-of-freedom robot manipulators and in one-degree-of-freedom mechanisms. The forward kinematics of parallel-link robots, the in-verse kinematics of serial-link robots and the input/output motion of single-degree-of-freedom mechanisms are all problems in displacement analysis. In this article, loop equations are formulated and solved for the displacement analysis of all spherical mecha-nisms up to three loops. We show how to solve each mechanism type using either a formulation in terms of rotation matrices or quaternions. In either formulation, the solu-tion method is a modification of Sylvester's elimination method, leading directly to nu-merical calculation via standard eigenvalue routines.
The direct position analysis (DPA) of a manipulator is the computation of the end-effector poses (positions and orientations) compatible with assigned values of the actuated-joint variables. Assigning the actuated-joint variables corresponds to considering the actuated joints locked, which makes the manipulator a structure. The solutions of the DPA of a manipulator one to one correspond to the assembly modes of the structure that is gener-ated by locking the actuated-joint variables of that manipulator. Determining the assem-bly modes of a structure means solving the DPA of a large family of manipulators since the same structure can be generated from different manipulators. This paper provides an algorithm that determines all the assembly modes of two structures with the same topol-ogy that are generated from two families of mechanisms: one planar and the other spherical. The topology of these structures is constituted of nine links (one quaternary link, four ternary links, and four binary links) connected through 12 revolute pairs to form four closed loops.
This paper presents a general method for the analysis of any planar mechanism consisting of rigid links connected by revolute and slider joints. The method combines the complex plane for-mulation of Wampler (1999) with the Dixon determinant proce-dure of Nielsen and Roth (1999). The result is simple to derive and implement, so in addition to providing numerical solutions, the approach facilitates analytical explorations. The procedure leads to a generalized eigenvalue problem of minimal size. Both input/output problems and the derivation of tracing curve equa-tions are addessed. NOMENCLATURE Number of kinematic loops. θ j e iΘj , where Θ j is an angle, in radians. z * Complex conjugate of z.
The displacement analysis problem for planar and spatial mechanisms can be written as a system of algebraic equations in particular as a system of multi-variate polynomial equations. Elimination theory based on resultants and polynomial continuation are some of the methods which have been used to solve this problem. This paper explores an alternate approach, based on Gröbner bases, to solve the displacement analysis problem for planar mechanisms. It is shown that the reduced set of generators obtained using Buchberger's algorithm for Gröbner bases not only yields the input–output polynomial for the mechanism, but also provides comprehensive information on the number of closures and the relationships between various links of the mechanism. Numerical examples illustrating the applicability of Gröbner bases to displacement analysis of 10- and 12-link mechanisms, and determination of coupler curve equation for an 8-link mechanism are presented. It is seen that even though the Gröbner bases method is versatile enough to handle finitely solvable as well as over-constrained systems of equations, it can run into computational problems due to rapidly growing numerical coefficients and/or the set of generators. The examples presented show how these difficulties can be overcome by artificially decoupling complex mechanisms to help facilitate their closed-form analysis.
It is commonly recognized that a convenient formulation for problems in planar kinematics is obtained by considering links to be vectors in the complex plane. However, scant attention has been paid to the natural interpretation of complex vectors as isotropic coordinates. These coordinates, often considered a special trick for analyzing four-bar motion, are in fact uniquely suited to two new techniques for analyzing polynomial systems: the BKK bound and the product-decomposition bound. From this synergistic viewpoint, a fundamental formulation of planar kinematics is developed and used to prove several new results, mostly concerning the degree and circularity of the motion of planar linkages. Useful for both analysis and synthesis of mechanisms, the approach both simplifies theoretical proofs and facilitates the numerical solution of mechanism problems.
It is frequently forgotten that the common methods of position analysis of planar linkages, which are based on dyadic decomposition, are not general. This is true regardless of whether a graphical or analytical approach is used. It is demonstrated in this paper that relatively simple linkages can give rise to problems that cannot be addressed by dyadic decomposition methods, and that solution of these problems can be far more complex.
This paper presents a general method for the analysis of planar mechanisms consisting of rigid links connected by rotational and/or translational joints. After describing the links as vectors in the complex plane, a simple recipe is outlined for formulating a set of polynomial equations which determine the locations of the links when the mechanism is assembled. It is then shown how to reduce this system of equations to a generalized eigenvalue problem, or in some cases, a single resultant polynomial. Both input/output problems and tracing-curve equations are treated.
Solving Polynomial Systems for the Kinematic Analysis and Synthesis of Mechanisms and Robot Manipulators,” Special 50th Anniversary Design Issue
- M Raghavan