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A geometric algebra approach to determine motion/constraint, mobility and singularity of parallel mechanism

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Abstract

The crucial procedure of mobility and singularity identification of parallel mechanisms is widely recognized as how to determine their motions (constraints) concisely and visually. In this paper, we propose a geometric algebra (GA) based approach to determine the motions/constraints, mobility and singularity of parallel mechanisms mainly utilizing the geometric and algebraic relations. Firstly, the motions, constraints and their relations are represented by conformal geometric algebra (CGA) formulas in a concise form by employing the characterized geometric elements with . Secondly, the mobility of parallel mechanism, including its number and property and the axes of motions, not only at origin configuration but also in the prescribed workspace, is obtained by the procedure proposed in this paper. Thirdly, the singularity of parallel mechanism is identified by the two indices proposed in this paper with shuffle and outer products. Finally, a typical example is given to illustrate the motions/constraints, mobility and singularity analysis. This approach is beneficial to kinematic analysis and optimal design of parallel mechanisms, especially for which would be carried out in automatic and visual manner using computer programming languages.

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... Compared with multi-input and multi-output (MIMO) of parallel prototype manipulator, the singularity of driver configuration should be considered at the beginning of topology optimization process of spatial compliant mechanism. X Huo et al. 19 proposed a geometric algebra (GA)-based approach to determine the motions/constraints, mobility, and singularity of parallel mechanisms mainly utilizing the geometric and algebraic relations. H Yao et al. 20 presented a method based on GA for the singularity of 3-DOF redundant constraints of 3-Revolute-Prismatic-Revolute (3-RPR) planar parallel manipulators. ...
... Derivative of the volume constraint v(x) in equation (19) with respect to the design variable x yields ...
... When gradients are required by the optimization algorithm employed to solve equation (19), these are easily derived for the objectives and constraints involving only x. These expressions will then contain derivatives of the displacement, which in turn can be obtained by taking the derivative of the equilibrium equation f = Ku. ...
Article
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Considering that the Jacobian matrix maps the velocity of the joint space of the robot to the end velocity of the Cartesian space, a novel topology optimization approach is proposed in this article for the design of three-translational degree-of-freedom spatial compliant mechanisms by combining the optimized Jacobian mapping matrix with the solid isotropic material with penalization topological method. First, by using the condition number method, the structural parameters of Universal-Prismatic-Universal (UPU)-type parallel prototype manipulator are optimized, and then, the differential Jacobian mapping matrix is calculated by using equivalent infinitesimal method. Second, comparing with the driver configuration and twist/wrench constraint conditions of the UPU-type parallel prototype manipulator, the topological algorithm combining solid isotropic material with penalization with optimized Jacobian mapping matrix is proposed. Finally, a novel spatial compliant mechanism with three-translational degree-of-freedom is derived, and numerical simulation results are reported to demonstrate the effectiveness of the proposed method.
... To describe the parasitic motion of a 3-PRS PM, a point-plane geometric method has been introduced [1, [8][9][10] . If a point on a rigid body preserves incidence with a known plane, the point is said to be constrained by the plane. ...
... However, a geometric method cannot be generalized because the geometric constraints obtained differ from mechanism to mechanism except for manipulator having the same constraint. Although the parasitic motion is widely studied [1,6,8,11,12] , it is derived from geometric position constraints, and thus, it is critical to accurately identify geometric constraints. Unfortunately, a systematic method for this purpose does not exist. ...
... The coupling relation represents a parasitic motion as a function of independent motion. Usually, this problem is addressed at the position level [1,8,9] even when dealing with the velocity [4] and dynamics [25] . This section will derive the coupling relation directly from the analytic constraint matrix at the velocity level. ...
Article
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This study derives the equation of parasitic motion of 3-DoFs parallel manipulator from the velocity-level analytic-constraint equation and compares it with a well-known position-level geometric method. The velocity-level constraint is formulated based on the extended Jacobian, derived from the instantaneous motion space (IMS) and the instantaneous restriction space (IRS) for free motion and constraint. In contrast, the position-level constraint, adopted in previous studies, is geometrically obtained by analyzing the moving platform and limb motions. The velocity-level analytic-constraint matrix is used to further analyze the task-space motion. In this paper, the procedure of detecting and identifying the parasitic terms from the independent terms is introduced utilizing the property that comes from the virtue of analytic constraint and inverse rate kinematics algorithm. Then, an equation for the constraint-compatible task motion and a coupling relation between the parasitic and independent motions are derived during further analysis. The paper derived the parasitic motion from position constraint, and an algebraic equivalency with the velocity constraint is shown by taking the time derivative of a point-plane position level constraint for comparison. Finally, numerical simulations are provided to validate the proposed approach and demonstrate the effect of the constraints on the given input velocity within the entire rotational workspace.
... In recent decades, CGA has been mostly applied to solve the inverse kinematics problem of the serial mechanisms [11][12][13][14] via CGA operation of the geometric entities. In addition, Tanev [15,16], Kim et al. [17], and Huo et al. [18] employed CGA to study the singularity analysis of PMs. Huo et al. [18] and Li et al. [19] proposed a mobility analysis approach for PMs based on geometric algebra. ...
... In addition, Tanev [15,16], Kim et al. [17], and Huo et al. [18] employed CGA to study the singularity analysis of PMs. Huo et al. [18] and Li et al. [19] proposed a mobility analysis approach for PMs based on geometric algebra. Zhang et al. [20,21] and Wei et al. [22] applied CGA to solve the direct kinematics of parallel mechanisms. ...
Article
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A geometric modeling and solution procedure for direct kinematic analysis of 6-4 Stewart platforms with any link parameters is proposed based on conformal geometric algebra (CGA). Firstly, the positions of the two single spherical joints on the moving platform are formulated by the intersection, dissection, and dual of the basic entities under the frame of CGA. Secondly, a coordinate-invariant equation is derived via CGA operation in the positions of the other two pairwise spherical joints. Thirdly, the other five equations are formulated in terms of geometric constraints. Fourthly, a 32-degree univariate polynomial equation is reduced from a constructed 7 by 7 matrix which is relatively small in size by using a Gröbner-Sylvester hybrid method. Finally, a numerical example is employed to verify the solution procedure. The novelty of the paper lies in that (1) the formulation is concise and coordinate-invariant and has intrinsic geometric intuition due to the use of CGA and (2) the size of the resultant matrix is smaller than those existed.
... Parallel manipulators (PMs) are recognized as promising solutions in various industrial domains due to their inherent merits in terms of high stiffness, good accuracy/dynamic performance, large load-weight ratio and easy integration with long guideways [2,6,19,30,31]. Intensive researches have been carried out for the analysis and design of PMs, for instance, topology synthesis [14,24], performance evaluation [18,33], and accuracy improvement [12,21]. ...
... By employing CGA, complete and continuous description of geometric error model can be achieved. Moreover, unknown constraints of PMs can be obtained only by drawing some auxiliary geometric entities such as points, lines and planes [14]. Hence, elimination of perturbations of passive joints can be easily performed. ...
Article
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An approach for geometric error modeling of parallel manipulators (PMs) based on the visual representation and direct calculation of conformal geometric algebra is introduced in this paper. In this method, the finite motion of an open-loop chain is firstly formulated. Through linearization of the finite motion, error propagation of the open-loop chain is analyzed. Then the error sources are separated in terms of joint perturbations and geometric errors. Next, motions and constraints of PMs are analyzed visually by their reciprocal property. Finally geometric error model of PMs are formulated considering the actuations and constraints. The merits of this new approach are twofold: (1) complete and continuous geometric error modeling can be achieved since finite motions are considered, (2) visual and analytical computation of motions and constraints are applied for transferring geometric errors from the open-loop chain to the PM. A 2-DoF rotational PM is applied to demonstrate the geometric error modeling process. Comparisons between simulation and analytical models show that this approach is highly effective.
... Huo et al. [19] present a mobility analysis applying conformal geometric algebra, and a singularity analysis using an idea similar to the ones presented in the above-mentioned contributions. A mobility analysis of overconstrained parallel mechanisms is performed using Grassmann-Cayley algebra by Chai et al. [20], while Yang and Li [21] propose a novel identification method for the constraint singularities of parallel robots based on differential manifolds. ...
... Proof. According to equation (19), it follows that: ...
Preprint
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The singularities of serial robotic manipulators are those configurations in which the robot loses the ability to move in at least one direction. Hence, their identification is fundamental to enhance the performance of current control and motion planning strategies. While classical approaches entail the computation of the determinant of either a 6x n or nxn matrix for an n degrees of freedom serial robot, this work addresses a novel singularity identification method based on modelling the twists defined by the joint axes of the robot as vectors of the six-dimensional and three-dimensional geometric algebras. In particular, it consists of identifying which configurations cause the exterior product of these twists to vanish. In addition, since rotors represent rotations in geometric algebra, once these singularities have been identified, a distance function is defined in the configuration space C such that its restriction to the set of singular configurations S allows us to compute the distance of any configuration to a given singularity. This distance function is used to enhance how the singularities are handled in three different scenarios, namely motion planning, motion control and bilateral teleoperation.
... The dark-gray joints are actuated Fig. 6. Two connected edges in bearing rigidity graphs with two bearing measurements only mechanisms, formulas can be found in [37,38]): ...
Article
Research on formation control and cooperative localization for multirobot systems has been an active field over the last years. A powerful theoretical framework for addressing formation control and localization, especially when exploiting onboard sensing, is that of formation rigidity (mainly studied for the cases of distance and bearing measurements). Rigidity of a formation depends not only on the topology of the sensing/communication graph but also on the spatial arrangement of the robots, since special configurations ("singularities" of the rigidity matrix), which are hard to detect in general, can cause a rigidity loss and prevent convergence of formation control/localization algorithms based on formation rigidity. The aim of this paper is to gain additional insights into the internal structure of bearing rigid formations by considering an alternative characterization of formation rigidity using tools borrowed from the mechanical engineering community. In particular, we show that bearing rigid graphs can be given a physical interpretation related to virtual mechanisms, whose mobility and singularities can be analyzed and detected in an analytical way by using tools from the mechanical engineering community (screw theory, Grassmann geometry, and Grassmann-Cayley algebra). These tools offer a powerful alternative to the evaluation of the mobility and singularities typically obtained by numerically determining the spectral properties of the bearing rigidity matrix (which typically prevents drawing general conclusions). We apply the proposed machinery to several case formations with different degrees of actuation and discuss known (and unknown) singularity cases for representative formations. The impact on the localization problem is also discussed.
... Additionally, Song et al. [35] and Qi et al. [36] used CGA theory in topology synthesis of mechanisms. Furthermore, Tanev [37,38], Jin et al. [30] and Huo et al. [39] utilized CGA to probe into the singularity analysis of PMs. Huo and Song [40] and Huo et al. [41] also used CGA to analyze the finite motion of PMs with parasitic motions and solve the inverse kinematics problem of the serial mechanisms. ...
Article
This paper demonstrates a novel geometric modeling and computational method of the family of spatial parallel mechanisms with 3-R(P)S structure for direct kinematic analysis based on the point pair relationship. The point pair relationship, which is derived from the framework of conformal geometric algebra (CGA), consists of the relationship between the point and the point pair and two point pairs. The first research is on the distance relationship between the point and the point pair. Secondly, the derivation of the distance relationship between two point pairs is based on the aforementioned result, which shows the mathematical homogeneity. Thirdly, two formulations for a point of the point pairs that satisfy the distance relationship between two point pairs are reduced. Fourthly, the point pair relationship is applied to solve the direct kinematic analysis of the spatial parallel mechanism with 3-R(P)S structure. Finally, four numerical examples are provided to verify the validity of the proposed algorithm. Overall, the proposed method can be generalized for the direct kinematics of a series of spatial parallel mechanisms with 3-R(P)S structure.
... The surface on which all the lower nodes are located is the reflection surface, whereas the surface on which all the upper nodes are located is the supporting surface. According to the DOF analysis of parallel mechanisms [22][23][24], each URU chain provides a constraint force that restricts the independence of the two translational DOFs of the two adjacent nodes, and the six URU chains restrict six DOFs of the basic combination unit. As a result, the DOFs of the basic combination unit are 3 9 6 21    . ...
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Space deployable structures with large calibers, high accuracy, and large folding ratios are indispensable equipment in the aerospace field. Given that the single-DOF 3RR-3RRR deployable unit cannot be fully folded, this study proposes a 3UU-3URU deployable unit with two kinds of DOF: folding movement and orientation adjustment. First, based on the G-K formula, the DOF of the 3UU-3URU unit is analyzed. Then, the 3UU-3URU unit is used to construct a deployable truss antenna with a curved surface, and the DOF of the whole deployable antenna containing multiple 3UU-3URU units is calculated. The structural design of a deployable antenna with two loops is carried out with specific parameters and geometric relations. Next, a DOF simulation of a basic combination unit composed of three 3UU-3URU units is performed. Finally, a prototype of the basic combination unit is manufactured, and the DOF of the mechanism is experimentally verified.
... Husty and Schröcker [25] used methods from algebraic geometry to define the DOF of a PM as Hilbert Dimension of a set of nonlinear polynomial kinematic constraint equations. By introducing Clifford algebra, Huo et al., [26,27] proposed an analytical approach for the determination of the motions/constraints, mobility and singularity of PMs. Based on the screw theory and the Chasles' theorem, Yang [28] presented a method for the type synthesis of 3T1R PMs with variable rotational axis. ...
Article
Full-text available
To determine mobility properties is one of the most challenging issue in the analysis and synthesis of parallel manipulators (PMs). However, currently, the most used methods mainly rely on experiences and manual analysis, which led to inefficient implementation. The motivation of this paper is to present an automatic mobility analysis algorithm and software package for the researchers and designers with an effective and practical means. According to the topological design theory of PMs based on position and orientation characteristic (POC) equations, this paper proposes a set of computer algorithmic rules and procedures for automated mobility analysis of PMs in the most user-friendly and efficient way. Firstly, a complete digital information model for topological structures which has a mapping relationship with the POC of a PM is proposed. This model not only describes the dimension of the motion outputs, but also includes the mapping relationship between the output orientation and the axes of the kinematic joints. Secondly, algorithmic rules are established that convert the union and intersection operations of POC into binary logical operations and then the detailed algorithmic procedures for an automatic mobility analysis are presented. In what follows, a corresponding software for automatic mobility analysis is described. The software package is equipped with a GUI that facilitates the input and allows the visualization of the results. Finally, four typical examples are provided to show the effectiveness of the software package for most of parallel mechanisms (not including some paradoxical mechanisms).
... In this regard, Sun [3] discussed the kinematic constraints within the ATP and proposed a group of PM-ATPs that can be applied as tracking mechanism, docking equipment, or machine tools. The parameterized topological models were further analyzed [21][22][23]. By filling in the gap between finite and instantaneous screw theory, Sun [24,25] succeeded in connecting topology analysis to the following performance analysis and even optimal design, which is a milestone in the topology synthesis of parallel manipulators. ...
Article
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This paper deals with the elastodynamic modeling and parameter sensitivity analysis of a parallel manipulator with articulated traveling plate (PM-ATP) for assembling large components in aviation and aerospace. In the elastodynamic modeling, the PM-ATP is divided into four levels, i.e., element, part, substructure, and the whole mechanism. Herein, three substructures, including translation, bar, and ATP, are categorized according to the composition of the PM-ATP. Based on the kineto-elastodynamic (KED) method, differential motion equations of lower levels are formulated and assembled to build the elastodynamic model of the upper level. Degrees of freedom (DoFs) at connecting nodes of parts and deformation compatibility conditions of substructures are considered in the assembling. The proposed layer-by-layer method makes the modeling process more explicit, especially for the ATP having complex structures and multiple joints. Simulations by finite element software and experiments by dynamic testing system are carried out to verify the natural frequencies of the PM-ATP, which show consistency with the results from the analytical model. In the parameter sensitivity analysis, response surface method (RSM) is applied to formulate the surrogate model between the elastic dynamic performances and parameters. On this basis, differentiation of performance reliability to the parameter mean value and standard variance are adopted as the sensitivity indices, from which the main parameters that greatly affect the elastic dynamic performances can be selected as the design variables. The present works are necessary preparations for future optimal design. They can also provide reference for the analysis and evaluation of other PM-ATPs.
... The research methods of singularity are divided into geometric algorithm and analytical algorithm. Geometric algorithm [12][13][14][15] is mainly used to study the singularity of the manipulator, explore the mechanism of singularity, and so on. However, the results with the geometric method are incomplete in some cases. ...
Article
Full-text available
Singularity analysis is one of the basic problems for parallel manipulators. When a manipulator moves in a singular configuration, the motion and transmission performance are poor. In certain serious cases, the normal operation could be damaged. Based on the topology structure and kinematics analysis of a 2(3HUS+S) parallel manipulator, the Jacobian matrices were established. Then, the singular locus surface was obtained by numerical simulation. In addition, the relationship between the motion path curve and the singular locus surface was analyzed. In this study, α, β, and γ are the attitude angles that describe the motion of moving platforms. There is a nonsingular attitude space in singular locus surfaces, and the singular locus surface is a single surface in a small attitude angle range. The nonsingular attitude space increases as the absolute value of γ increases, and singularity could be avoided when γ is large. Furthermore, the motion path curve passes through the singular locus surface two times, and the two intersection points are consistent with the positions where the motion dexterity is equal to zero. This study provides new insights on the singularity analysis of parallel manipulators, particularly for the structure parameter optimization of the nonsingular attitude space.
... In terms of motion and constraint determination for analyzing the typical parallel mechanism [34][35][36], methods based on the screw theory have been proposed [37][38][39][40][41][42][43][44][45]. The main procedure of these methods is: (1) to obtain the motion-screw system of each limb, (2) to determine the constraint-screw systems of each limb, (3) to calculate the union of the constraint-screw systems of limbs, which is the constraint-screw system of the end effector or the platform, and (4) to attain the motion-screw system of the platform by using the reciprocal product [46]. ...
Article
The parallel mechanism with a reconfigurable platform retains all advantages of parallel mechanisms and provides additional functions due to the reconfigurable platform, leading to kinematic coupling between limbs that restricts the development of the mechanism. This paper aims at dealing with kinematic coupling between limbs by investigating the transferability of limb constraints and their degree of relevance to the platform-constraints based on the geometric model of the mechanism. The paper applies the screw system theory to verifying degree of relevance between limb constraint wrenches and platform constraint wrenches, and reveals the transferability of limb constraints, to obtain the final resultant wrenches and twists of the end-effector. The proposed method is extended to parallel mechanisms with planar n-bar reconfigurable platforms, spherical n-bar reconfigurable platforms and other spatial reconfigurable platforms and lends itself to way of studying a parallel mechanism with a reconfigurable platform.
... Huo et al. [23] present a mobility analysis applying conformal geometric algebra and a singularity analysis using an idea similar to the ones presented in the above-mentioned contributions. A mobility analysis of overconstrained parallel mechanisms is performed using Grassmann-Cayley algebra by Chai et al. [24], while Yang and Li [25] propose a novel identification method for the constraint singularities of parallel robots based on differential manifolds. ...
Article
Full-text available
The singularities of serial robotic manipulators are those configurations in which the robot loses the ability to move in at least one direction. Hence, their identification is fundamental to enhance the performance of current control and motion planning strategies. While classical approaches entail the computation of the determinant of either a 6×n or n×n matrix for an n-degrees-of-freedom serial robot, this work addresses a novel singularity identification method based on modelling the twists defined by the joint axes of the robot as vectors of the six-dimensional and three-dimensional geometric algebras. In particular, it consists of identifying which configurations cause the exterior product of these twists to vanish. In addition, since rotors represent rotations in geometric algebra, once these singularities have been identified, a distance function is defined in the configuration space C, such that its restriction to the set of singular configurations S allows us to compute the distance of any configuration to a given singularity. This distance function is used to enhance how the singularities are handled in three different scenarios, namely, motion planning, motion control and bilateral teleoperation.
... Since small rigidity is not beneficial for carrying large and heavy aircraft parts, large numbers of serial structures are applied to build the pose adjusting platform in the current docking equipment. Besides compensating the shortages of serial mechanisms by applying multiple units, another alternative is to build the positioning platform by parallel mechanisms (PMs) [7][8][9][10]. ...
Article
Having potentially high stiffness and good dynamic response, a parallel pose adjusting mechanism was proposed for being an attachment to a big serial robot of a macro-micro robotic system. This paper addresses its design optimization problem mainly concerning arrangements of design variables and objectives. Parameter changes during construction are added to the design variables in order to prevent the negative effects to the physical prototype. These parameter changes are interpreted as parameter uncertainty and modeled by probabilistic theory. For the objectives, both static and dynamic performances are simultaneously optimized by Pareto-based method. The involved performance indices are instantaneous energy based stiffness index, first natural frequency and execution mass. The optimization procedure is implemented as: (1) carrying out performance modeling and defining performance indices, (2) reformulating statistical objectives and probabilistic constraints considering parameter uncertainty, (3) conducting Pareto-based optimization with the aid of response surface method (RSM) and particle swarm optimization (PSO), (4) selecting optimal solution by searching for cooperative equilibrium point (CEP). By addressing parameter uncertainty and the best compromise among multiple objectives, the presented optimization procedure provides more reliable optimal parameters that would not be affected by minor parameter changes during construction, and less biased optimum between static and dynamic performances comparing with the conventional optimization methods. The proposed optimization method can also be applied to the other similar mechanisms.
... parallel mechanisms [15] , especially based on screw theory. In the works that use Geometric Algebra, first by Tanev [16] , the singularity is determined as vanishing of the six-blade, which is yet another form of the Jacobian [17,18] . ...
Article
A novel criterion for singularity analysis of parallel robots is presented. It relies on screw theory, the 3-dimensional Kennedy theorem, and the singular properties of minimal parallel robots. A parallel robot is minimal if in any generic configuration, activating any leg/limb causes a motion in all its joints and links. For any link of the robot, a pair of legs is removed. In the resulting 2 degrees-of-freedom mechanism, all possible instantaneous screw axes belong to a cylindroid. A center axis of this cylindroid is determined. This algorithm is performed for three different pairs of legs. The position is singular, if the instantaneous screw axis of the chosen link crosses and is perpendicular to three center axes of the cylindroids. This criterion is applied to a 6/6 Stewart Platform and validated on a 3/6 Stewart Platform using results known in the literature. It is also applied to two-platform minimal parallel robots and verified through the Jacobian; hence demonstrating its general applicability to minimal robots. Since any parallel robot is decomposable into minimal robots, the criterion applies to all constrained parallel mechanisms.
Article
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Although the serial-parallel manipulators(S-PMs) formed by the symmetric 3-RPS and 3-SPR PMs have been widely studied, the terminal constraint and mobility criterion for this class of manipulators are still open questions. This paper solves the terminal constraint and mobility problems of the (3-SPR) + (3-RPS) and the (3-RPS) + (3-SPR) S-PMs. In this paper, the terminal mobility composition principle for the two S-PMs is presented. By analyzing the rank of two types screw systems composed of six constraint forces concerned with the two S-PMs under different geometry conditions, the degree of freedom (DOF) of the two S-PMs are determined. From algebraic and geometric concepts, the approaches for solving the intersection of two sub three order screw systems in determining the terminal constraint are proposed. In addition, the terminal constraint form and mobility property for the two S-PMs under different conditions are discussed. The results show that the terminal constraint of the (3-SPR) + (3-RPS) S-PM is one finite-non-zero pitch screw and the terminal constraint of the (3-RPS) + (3-SPR) S-PM is empty in the general case. This research provides a foundation for the terminal constraint and mobility analysis for S-PMs.
Article
Object manipulation without change of orientation is an important task in various applications. The use of parallelogram-based mechanisms to attain this has been limited by the fact that most of those are planar mechanisms. The work presented here describes a two degree of freedom, spatial parallelogram mechanism with only revolute joints which can be employed for object manipulation. The proposed mechanism has several advantages over other mechanisms for similar applications—large ratio of the workspace area to the fixed link’s dimensions; all the singularity positions form a well-defined curve, etc. A method to avoid uncontrolled motion at singular positions has also been proposed. Two methods for converting this concept to a three degree-of-freedom mechanism with a solid spheroid workspace have also proposed. A condition under which compact folding of the mechanism can be achieved has also been discussed which opens the possibility of applying it as a deployable mechanism.
Chapter
Robot kinematics, the study of the motions, is the prerequisite for statics, dynamics, accuracy analysis, and design of robotic mechanism [1, 2, 3]. Kinematics of robotic mechanism concerns the motion transmissions from the input to the output. It is divided into two main topics, forward and inverse kinematics [4, 5].
Article
This paper predicts the dynamic characteristics of a 5-DOF hybrid machine tool by using scale model, and the geometric distortion of bearings is considered in the study. As the distortion causes scaling laws related to rotational joint properties to vary between the full-size system and scale model, the effects of revolute mass and stiffness on the dynamic characteristics of the 5-DOF hybrid machine tool are investigated. On this basis, the scaling laws related to rotational joint mass and axial stiffness are relaxed as they have little effect on the dynamic characteristics of hybrid machine tool. Moreover, bearing radial stiffness has significant influence on the dynamic characteristics, and it is selected as the fundamental scaling parameter, because the standard bearing cannot be scaled down in arbitrary ratio. Thus, the scaling factors of parameters shown in the scaling laws are determined on the basis of bearing radial stiffness. Considering the effect of the complex contact between roller and raceway on the radial stiffness, the scaling factor of bearing radial stiffness is obtained through experiments on bearing axial stiffness. Subsequently, dimension parameters of the scale model are obtained, and the scaling factor of the dynamic characteristics of the hybrid machine tool are derived. Finally, dynamic characteristics of the full-size hybrid machine tool and its scale model are simulated through structural dynamics to validate the dynamic characteristics prediction.
Article
This paper focuses on the establishment of the kinematic diagram and the mobility of Rubik's Cube. Owing to the complex and variable characteristics of Rubik's Cube mechanism, a method of separating the inner and outer loops of Rubik's Cube mechanism and analyzing the Rubik's Cube with 1/8 module as a unit is proposed. Based on the topological graph and adjacency matrix, the kinematic diagram of Rubik's Cube unit which is in the first octant in both aligned and non-aligned states is constructed. On this basis, an analytical method for mobility due to the characteristics of strong coupling between the loops is proposed which is to separate the constraint of the internal and external of the mechanism, to successively decompose the mechanism by layer, then to process the front and back basic loops in proper order. Besides, a method is proposed to classify the sub-pieces and to decompose the coupling loops according to the direction of motion and the directed graph. Based on the screw theory and the “modified K-G formula”, the mobility of Rubik's Cube is obtained when it is in aligned and non-aligned states. Further, the theoretical analysis results are verified by the ADAMS.
Article
Hybrid manipulators formed by one 2R1T parallel manipulator (PM) and one RR serial manipulator (SM) have been widely studied. However, the previous researches only focused on the PM part. The basic terminal constraint/mobility and kinematics issues of the integrated hybrid manipulators have been ignored and thus should be reconsidered. The paper studies the basic motion and constraint issues of the integrated hybrid manipulators by using the Exechon hybrid manipulator as a case. First, using Grassmann–Cayley algebra(GCA), a method for deriving the terminal constraint of the 5-DOF hybrid manipulator is proposed and the terminal DOF property is solved. Second, the terminal position and orientation coupling relation in the hybrid manipulator is revealed based on the elimination method and the inverse displacement is solved. Third, the 6 × 6 form full Jacobian containing all actuation and constraint information of the manipulator is derived. The established models in this paper are applicable for the 5-DOF hybrid manipulators.
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This paper presents an efficient graphical representation geometric algebra approach for the forward displacement analysis (FDA) of 3-PPS parallel mechanisms (PMs). Our approach based on conformal geometric algebra (CGA) reveals interesting intuitions and insight of the modelling and solution for FDA of this type PMs and reduces considerably computational burden and time. In this work, we first deduce the position of an arbitrary joint among three spherical joints on the moving platform through motors and represent it in terms of only one variable. Then, we formulate the position of either one joint of two remained spherical joints using the intersection of a sphere and two planes under the CGA framework. To obtain the geometric formulation of the third spherical joint position, we construct the outer product of two spatial balls with centers at two formulated spherical joints centers and two known planes. The positions of three spherical joints are all explicit geometric formulations of geometric entities of the parallel mechanisms. We achieve a coordinate-invariant equation expressed in terms of geometric entities via inner product of the last joint by itself. The symbolic 8th-degree univariate polynomial input-output equation is first proposed and all 8 sets of closed-form solutions can be obtained. The algorithm is the geometric algebra computation in a clear and coordinates-free way that avoids the use of rotation matrices, and complex algebraic modelling and nonlinear and multivariable elimination computations as most current approaches do. Finally, two numerical examples have been applied to demonstrate the efficiency of the algorithm comparing with the Dixon resultant method. The results show that this algorithm can simplify complexity of the problem and reduce computation time dramatically with strong geometrical intuition. The achieved symbolic univariate polynomial input-output equation can be applied directly to solve the DPA for all types of 3-PPS PMs instead of using any other methods. This work provides a novel geometric modeling and solution approach for the forward displacement analysis of spatial parallel mechanism and be of greatly useful for roboticists or engineers to design and develop 3-PPS PMs for kinds of applications.
Article
In this paper, the dynamic modeling and generalized force analysis of three-(rotation pair)-(prismatic pair)-spherical pair (3-RPS) parallel mechanism were carried out for the first time based on the five-dimensional geometric algebra space—(4,0,1) conformal geometric. Compared with the traditional homogeneous matrix method, the maximum error values of generalized force of the three branch chains are 1.90×10-4N,1.39×10-4N and 6.0×10-5N,respectively. The results are basically consistent with the homogeneous matrix method. For composite rigid body transformation of two rotational motions, the rotation matrix method needs 27 times of multiplication and 18 times of addition, while the conformal geometric method only needs 16 times of multiplication and 15 times of addition. The computational efficiency of this method can be improved. In five-dimensional space, derivative operation can be linearly mapped to multiplication operation in three-dimensional space, so that dynamic equation has no derivative term. The dynamic model can separate variables of known and unknown, and realize parallel computation. This method provides a new idea for dynamic modeling and solving of parallel mechanism.
Chapter
This paper purposes a new kinematic analysis approach of closed-loop mechanisms based on the linear dependence of twists. First, the generalized velocity equation of closed-loop mechanisms is established by means of the exponential product form of rigid body motion. Second, the condition of the decoupling of closed-loop mechanism is obtained. Third, a simple analysis approach for decoupling analysis of the mechanism is proposed. Next, the approach is proven by the analysis of the decoupling mechanism and partial decoupling mechanisms and the synthesis of a decoupling spherical mechanism. The examples indicate that the actuation wrench is related with the linear dependence of the kinematic joints of the close-loop mechanism. This broadens the traditional knowledge about the actuation wrenches based on the reciprocal screw theory.
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The constraint performance analysis in the limited kinetostatic subspace of parallel manipulators is a significant but ignored issue. The motion/force constrainability analysis, with focus on lower-mobility parallel manipulators, is the subject of this study. Via the theory of screws, three generalized frame-invariant constraint indices are proposed based on the concept of the power coefficient. The introduced indices can not only identify the singularity and the fully constrained property but also measure the closeness between a particular pose and an unconstrained configuration (or fully constrained configuration). In order to demonstrate the feasibility and the validity of the analysis methods and indices, the detailed evaluation of two typical industrial parallel manipulators are presented, the Sprint Z3 head and the Tricept mechanism.
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This paper introduces a novel family of singularity-free kinematically redundant planar parallel mechanisms that have unlimited rotational capabilities. The proposed mechanisms are akin to conventional three-degree-of-freedom planar parallel mechanisms. By introducing a novel kinematically redundant arrangement, four-degree-of-freedom parallel mechanisms are obtained that can completely alleviate singularities and provide unlimited rotational capabilities. The kinematics of the mechanisms are derived, and the Jacobian matrices are obtained. It is shown that the singularities of this type of mechanism are governed by the orientation of a passive link connecting the redundant leg to the platform and that the latter orientation is easily controlled using the kinematic redundancy, thereby alleviating all direct kinematic singularities. An example mechanism is proposed, and a prototype is demonstrated. Example trajectories that include full cycle rotations are shown. The prototype also illustrates the use of the kinematic redundancy for an auxiliary task, namely grasping.
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Singularity is an inherent characteristic of parallel robots and is also a typical mathematical problem in engineering application. In general, to identify singularity configuration, the singular solution in mathematics should be derived. This work introduces an alternative approach to the singularity identification of lower-mobility parallel robots considering the motion/force transmissibility and constrainability. The theory of screws is used as the mathematic tool to define the transmission and constraint indices of parallel robots. The singularity is hereby classified into four types concerning both input and output members of a parallel robot, that is, input transmission singularity, output transmission singularity, input constraint singularity, and output constraint singularity. Furthermore, we take several typical parallel robots as examples to illustrate the process of singularity analysis. Particularly, the input and output constraint singularities which are firstly proposed in this work are depicted in detail. The results demonstrate that the method can not only identify all possible singular configurations, but also explain their physical meanings. Therefore, the proposed approach is proved to be comprehensible and effective in solving singularity problems in parallel mechanisms.
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This paper presents the idea of constructing reconfigurable limbs by integrating metamorphic linkages as subchains. The planar five-bar metamorphic linkages that have three phases resulting from locking of motors are considered. Under the assumption that the constraint exerted by the reconfigurable limb can switch between no constraint, a constraint force, and a constraint couple, the output motions of the metamorphic linkage in its two planar four-bar linkage phases are identified. By adding an appropriate joint to planar four-bar linkages with translational output, four planar five-bar linkages that can be employed in the construction of reconfigurable limbs are enumerated. Serial chains that can provide a constraint couple and a constraint force are synthesized based on screw theory. Reconfigurable limbs that have three configurations associated with the three distinct phases of the metamorphic linkages are assembled with planar five-bar metamorphic linkages and serial chains with four degrees of freedom. A class of reconfigurable parallel mechanisms are constructed by connecting a moving platform and a base with three identical reconfigurable limbs. The degrees of freedom of the reconfigurable parallel mechanism in different configurations with the metamorphic linkages in different phases are given. Finally, the actuation scheme for this kind of mechanisms is addressed.
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This paper presents a generalized approach to the singularity analysis of mechanisms with arbitrary kinematic chains and an equal number of inputs and outputs. The instantaneous kinematics of a mechanism is described by means of a velocity equation, explicitly including not only the input and output velocities but also the passive-joint velocities. A precise definition of singularity of a general mechanism is provided. On the basis of the six types of singular configurations and the motion space interpretation of kinematic singularity introduced in the paper, a comprehensive singularity classification is proposed.
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The application of geometric algebra to the engineering sciences is a young, active subject of research. The promise of this field is that the mathematical structure of geometric algebra together with its descriptive power will result in intuitive and more robust algorithms. This book examines all aspects essential for a successful application of geometric algebra: the theoretical foundations, the representation of geometric constraints, and the numerical estimation from uncertain data. Formally, the book consists of two parts: theoretical foundations and applications. The first part includes chapters on random variables in geometric algebra, linear estimation methods that incorporate the uncertainty of algebraic elements, and the representation of geometry in Euclidean, projective, conformal and conic space. The second part is dedicated to applications of geometric algebra, which include uncertain geometry and transformations, a generalized camera model, and pose estimation. Graduate students, scientists, researchers and practitioners will benefit from this book. The examples given in the text are mostly recent research results, so practitioners can see how to apply geometric algebra to real tasks, while researchers note starting points for future investigations. Students will profit from the detailed introduction to geometric algebra, while the text is supported by the author's visualization software, CLUCalc, freely available online, and a website that includes downloadable exercises, slides and tutorials.
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This paper introduces a methodology to analyze geometrically the singularities of manipulators, of which legs apply both actuation forces and constraint moments to their moving platform. Lower-mobility parallel manipulators and parallel manipulators, of which some legs do not have any spherical joint, are such manipulators. The geometric conditions associated with the dependency of six Pl\"ucker vectors of finite lines or lines at infinity constituting the rows of the inverse Jacobian matrix are formulated using Grassmann-Cayley Algebra. Accordingly, the singularity conditions are obtained in vector form. This study is illustrated with the singularity analysis of four manipulators.
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This paper presents a generalization of Kutzbach-Grübler criterion for mobility analysis of kinematic chains based on group theory. This new mobility criterion applies to exceptional linkages and reduces to a group theoretic representation of Kutzbach-Grübler criterion for trivial linkages. Furthermore, it is shown that using sets and groups of Euclidean displacements, one can distinguish between trivial, exceptional, and paradoxical linkages. Using these concepts, formal definitions of trivial, exceptional, and paradoxical linkages are presented and it is shown that there are two classes of paradoxical linkages. In addition, a new class of linkages is identified that have partitioned mobility. This work builds upon and extends the work of Hervé and Fanghella and Galletti in application of group theory to analysis of kinematic chains.
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Chapter
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We propose a 3-SPS/S redundant motion mechanism with kinematic redundancy in this paper. The proposed mechanism with redundancy enables redundant motion as it works like pan/tilt systems to reduce the unnecessary degree of freedom for the yaw and pitch motions in the configuration space. By reducing one degree of freedom, this robot has self-motion that causes the inverse kinematics to have infinite solutions, so that the proposed research performs an optimization process for obtaining the optimal solution using geometric approaches. This optimal solution provides the fastest response among the infinite solutions by minimizing each link movement to the target orientation. In addition, we perform a geometric singularity analysis due to limited link length and avoidance using conformal geometric algebra for the feasibility and correctness of the inverse kinematics.
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Conference Paper
The goal of this paper is to demonstrate that the Modified Grübler-Kutzbach Criterion when combined with a simple procedure for determining the reciprocal screws offers a direct and simple method for analysing highly complex mechanisms including the over-constrained parallel manipulators. Since the scalar product of screws is not dependent on the choice of the origin, one can quickly obtain a simple expression of screws by selecting an appropriate coordinate system. In such simple expression, the coordinates of a screw would include 0 or 1, and thus greatly simplifies the procedure for determine the number of constraints in a mechanism. Seven rules have been presented to help simplify the analysis process. The advocated approach makes it possible to determine, within minutes, the mobility of a highly complex mechanism by using a pencil and a paper. Many over-constrained mechanisms, including three parallel mechanisms, are presented as examples.
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This paper uses geometric algebra to formulate, in a single framework, the kinematics of a three finger robotic hand, a binocular robotic head, and the interactions between 3D objects, all of which are seen in stereo images. The main objective is the formulation of a kinematic control law to close the loop between perception and actions, which allows to perform a smooth visually guided object manipulation.
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This paper describes the topology, geometry and kinematics of a translational parallel mechanism with three degrees of freedom. Basic topological and geometric conditions for ensuring exclusively translational motion of the driven link relative to the base are formulated. Fundamental relations for the kinematic analysis of the system are introduced. The equations are needed to describe and solve the direct and inverse problem of the kinematics and to provide a basis for the analytical definition of the mechanism's singular configurations and their determination.
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The analysis of singularities is central to the development and control of a manipulator. However, existing methods for singularity set computation still concentrate on specific classes of manipulators. The absence of general methods able to perform such computation on a large class of manipulators is problematic because it hinders the analysis of unconventional manipulators and the development of new robot topologies. The purpose of this paper is to provide such a method for nonredundant mechanisms with algebraic lower pairs and designated input and output speeds. We formulate systems of equations that describe the whole singularity set and each one of the singularity types independently, and show how to compute the configurations in each type using a numerical technique based on linear relaxations. The method can be used to analyze manipulators with arbitrary geometry, and it isolates the singularities with the desired accuracy. We illustrate the formulation of the conditions and their numerical solution with examples, and use 3-D projections to visualize the complex partitions of the configuration space induced by the singularities.
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The paper presents a geometric algebra (Clifford algebra) approach to singularity analysis of a spatial parallel manipulator with four degrees of freedom. The geometric algebra provides a good geometrical insight in identifying the singularities of parallel manipulators with fewer than six degrees of freedom.
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1. Linear spaces 2. Real and complex algebras 3. Exact sequences 4. Real quadratic spaces 5. The classification of quadratic spaces 6. Anti-involutions of R(n) 7. Anti-involutions of C(n) 8. Quarternions 9. Quarternionic linear spaces 10. Anti-involutions of H(n) 11. Tensor products of algebras 12. Anti-involutions of 2K(n) 13. The classical groups 14. Quadric Grassmannians 15. Clifford algebras 16. Spin groups 17. Conjugation 18. 2x2 Clifford matrices 19. The Cayley algebra 20. Topological spaces 21. Manifolds 22. Lie groups 23. Conformal groups 24. Triality.
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This paper investigates the behavior of a type of parallel mechanisms with a central strut. The mechanism is of lower mobility, redundantly actuated, and used for sprained ankle rehabilitation. Singularity and dexterity are investigated for this type of parallel mechanisms based on the Jacobian matrix in terms of rank deficiency and condition number, throughout the workspace. The nonredundant cases with three and two limbs are compared with the redundantly actuated case with three limbs. The analysis demonstrates the advantage of introducing the actuation redundancy to eliminate singularities and to improve dexterity and justifies the choice of the presented mechanism for ankle rehabilitation.
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This paper develops an approach for constructing the null space N(A) of a linear system of homogeneous equations using the cofactors of an augmented coefficient matrix A. The relationship between the row space R(AT) and null space is exploited by introducing an augmenting vector which is linearly independent of the row space and dependent on the null space. The resultant null space is shown to be a vector of cofactors of the augmenting row of the coefficient matrix and is invariant. This provides a straightforward solution to a linear system of homogeneous equations without going through Gauss-Seidel elimination. The approach is derived from a onedimensional null space and is extended to a multidimensional one by partitioning the coefficient matrix and consequently constructing a set of (n-m) null-space vectors based on cofactors. Examples are given and accuracy is compared with Gauss-Seidel elimination. The approach is further used in a screw-algebra context with a simple procedure to obtain a system of reciprocal screws representing a set of constraint wrenches from a set of twists of freedom, in the form of a linear system of homogeneous equations in R6. The paper provides rigorous proofs and applications in both linear algebra and advanced kinematics.
Article
This paper proposes a generalized transmission index for spatial mechanisms, based on the virtual coefficient between the transmission wrench screw and the output twist screw. This index is applicable to single-loop spatial linkages, with fixed output and single or multiple degrees of freedom. We show that the transmission index introduced by Sutherland and Roth, the pressure angle of higher-pair mechanisms and the transmission angle of linkages are special cases of our index. A method is developed to compute the transmission wrench screw in spatial single-loop linkages. Furthermore, the transmission quality is defined based on the generalized transmission index to evaluate the global performance of a mechanism.
Chapter
Geometric algebra is employed for the analysis of the singularity of parallel manipulators with limited mobility. The rotations and translations of vectors and screws are performed in the degenerate geometric algebra G 3,0,1. The condition for singularity is obtained using the language of geometric algebra. The approach is applied to two parallel manipulators with limited mobility. Key wordsparallel manipulator-geometric algebra-kinematics-singularity
Article
Type synthesis of both rigid and compliant parallel mechanisms has become a hot issue in the field of mechanisms and robotics research in recent years. A unified approach to type synthesis of the two classes of mechanisms, however, has not been referred and investigated up to date. Based on the state-of-art analysis for several major type synthesis approaches related to rigid and compliant mechanisms, respectively, it proves feasible to establish a unified methodology for type synthesis of these two classes of mechanisms. That is a synthesis philosophy in terms of the hierarchy mapping between mathematic, physical, and mechanical building blocks in the framework of screw theory, as addressed in this paper. The key point of the proposed method lies in establishing the mapping among three different building blocks (i.e. geometric building block, kinematic or constraint building block, and mechanical building block). As a result, it makes the whole type synthesis process simple and visible. By using the proposed method, two examples are taken to verify the effectiveness for the type synthesis of both rigid and flexure mechanisms. The content of this paper may provide a theoretical frame for constructing a visualized algorithm or software about the unified type synthesis (or conceptual design) of both rigid and flexure parallel mechanisms. Keywordstype synthesis–flexure mechanism–parallel mechanism–screw system–building block
Article
Singularity analysis is a basic problem of parallel mechanism, and this problem cannot be avoided in both workspace and motion planning. How to express the singularity locus in an analytical form is the research emphasis for many researchers for a long time. This paper presents a new method for the singularity analysis of the 6-SPS parallel mechanism. The rotation matrix is described by quaternion, and both the rotation matrix and the coordinate vectors have been expanded to four-dimensional forms. Through analyzing the coupling relationship between the position variables and the orientation variables, utilizing properties of the quaternion, eight equivalent equations can be obtained. A new kind of Jacobian matrix is derived from those equations, and the analytical expression of the singularity locus is obtained by calculating the determinant of the new Jacobian matrix. The singularity analysis of parallel mechanisms, whose moving platform actuated by 6 links and the vertices of both the base and the moving platforms has been placed on a circle respectively, can be solved by this analytical expression. Keywordsparallel mechanism–singularity analysis–quaternion–Jacobian matrix
Article
This paper presents a critical review on the calculation of the mobility, the main structural and kinematic parameter of a mechanism. We focus on a brief presentation and a critical analysis of various methods presented in the literature in the last 150 years, to clearly situate the different contributions to this very important subject of the theory of mechanisms. Thirty five approaches/formulas for mobility calculation are presented and their genesis, similarities and limitations are investigated. The various methods proposed in the literature for mobility calculation are grouped in two categories: approaches based on setting up the kinematic constraint equations and their rank calculation and formulas for a quick calculation of mobility without need to develop the constraint equations. We emphasize on the limits of formulas for quick calculation of mobility by applying them to a parallel robot with elementary legs and ascertaining that the results are erroneous. In fact, the formulas for quick calculation of mobility known in the literature do not fit for many classical or modern mechanisms. We explain why these formulas do not work for certain mechanisms and we propose a new formula for quick mobility calculation of parallel mechanisms with elementary legs.
Article
Computational kinematic analysis of mechanisms is a promising tool for the development of new classes of manipulators. In this paper, the authors present a velocity equation to be compiled by general-purpose software and applicable to any mechanism topology. First, the approach to model the mechanism is introduced. The method uses a set of kinematic restrictions applied to characteristic points of the mechanism. The resultant velocity equation is not an input–output equation, but a comprehensive one. In addition, the Jacobian characterizing the equation is dimensionless hence extremely useful for singularity and performance indicators. The motion space of the manipulator is obtained from the velocity equation. Angular velocities are compiled out of three non-collinear point velocities. The procedure is applied to a 3-DoF parallel manipulator to illustrate.
Article
This paper presents a linear algebraic procedure in obtaining (6−n) reciprocal screws from an n-screw system of a robotic manipulator. The procedure starts by formulating the reciprocal relationship between a screw system and its reciprocal screw system in matrix form, and augmenting the screw matrix which is assembled with n screws in the screw system. Vectors of cofactors of the augmented row are then produced and the partitioning of the screw matrix is implemented to generate each of the (6−n) reciprocal screws. The theory developed in this paper provides a novel and simple procedure to obtain reciprocal screws from any screw system. The paper illustrates applications to cases with different screw systems. © 2003 Wiley Periodicals, Inc.
Article
A double parallel manipulator (DPM) is composed of two parallel mechanisms with a central aids. The device is expected to have large workspace by reducing interference between links and to avoid singularity by constraining its motion. To prove this, workspace and singularity are analyzed. The workspace of the device is decoupled into a positional and an orientational workspace, each of which is computed and compared with that of a Stewart platform. The singularities occurring outside the workspace are analytically found at the configurations where a Jacobian matrix becomes rank deficient. To ascertain the analytical results, they are geometrically examined at the configurations where the duty for resisting forces and moments is not properly shared by a central axis and linear actuators due to losing static equilibrium. The geometrical results are coincident to the analytical results, which proves the DPM is free from the singularity problem
Theory of Parallel Mechanisms
  • Z Huang
  • Q C Li
  • H F Ding
Z. Huang, Q.C. Li, H.F. Ding, Theory of Parallel Mechanisms, Springer, Netherlands, 2013.
Matrix Analysis and Applied Linear Algebra
  • C D Meyer
C.D. Meyer, Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics, Philadelphia, PA, 20 0 0.
  • G Sommers
  • G Computergrafik
  • Datenverarbeitung
G. Sommers, Computergrafik, G. Datenverarbeitung, et al., Geometric Computing with Clifford Algebras, Springer, Berlin Heidelberg, 2001.