Article

Type Synthesis of Parallel Tracking Mechanism With Varied Axes by Modeling Its Finite Motions Algebraically

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Abstract

Parallel tracking mechanism with varied axes has great potential in actuating antenna to track moving targets. Due to varied rotational axes, its finite motions have not been modeled algebraically. This makes its type synthesis remain a great challenge. Considering these issues, this paper proposes a conformal geometric algebra (CGA) based approach to model its finite motions in an algebraic manner and parametrically generate topological structures of available open-loop limbs. Finite motions of rigid body, articulated joints, and open-loop limbs are formulated by outer product of CGA. Then, finite motions of parallel tracking mechanism with varied axes are modeled algebraically by two independent rotations and four dependent motions with the assistance of kinematic analysis. Afterward, available four degrees-offreedom (4-DoF) open-loop limbs are generated by using revolute joints to realize dependent motions, and available five degrees-offreedom (5-DoF) open-loop limbs are obtained by adding one finite rotation to the generated open-loop limbs. Finally, assembly principles in terms of minimal number and combinations of available open-loop limbs are defined. Typical topological structures are synthesized and illustrated.

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... As differential manifold theory together with screw theory has been successfully used in type synthesis [21,22] , kinematic and dynamic analysis [23,24] of parallel mechanisms, we will use these mathematical tools for finite and instantaneous motion modelling and singularity analysis of parallel mechanisms in this paper. ...
... It should be pointed out that even though the two parallel mechanisms in Figs. 2 and 3 have the same number and types of joints, they are different mechanisms. Through the comparisons between Eqs. (19) and ( 29 ), between Eqs. (22) and ( 33 ), it can be concluded that their finite motions and instantaneous motions are different with each other. After comparing the structures of the mechanisms in the two figures, we can see that the effect after turning the fourth joint in each limb to have the same direction changes the mechanism's motion from bifurcated Schoenflies motion to Schoenflies motion. ...
Article
This paper presents investigations into classification and analysis of constraint singularities for parallel mechanisms. Parallel mechanisms (also called parallel manipulators or parallel robots) have wide applications in industry. The singularities tremendously affect their applications. Existing research works show that constraint singularity causes a mechanism to have instantaneous degree-of-freedoms (DoFs) or bifurcated finite motions. However, the intrinsic differences among the conditions under which the specific constraint singularities happen have not been discussed. This paper is focused on these topics by using differential manifolds as mathematical tools. Firstly, the general mathematical models of parallel mechanisms are formulated by respectively describing their finite motions and instantaneous motions in forms of differential manifolds and their tangent spaces. Then, parallel mechanisms having bifurcated finite motions and instantaneous DoFs are modelled accordingly, and the constraint singularities are thus classified into two kinds by considering their influences on motions of mechanisms in both finite and instantaneous motion levels. Finally, two examples are given to further illustrate the theoretical analysis. This paper lays foundations for mathematical modelling and applications of parallel mechanisms with constraint singularities.
... The latter issue asks for explicitly and concisely analysis of motions and constraints. For this purpose, Conformal Geometric Algebra (CGA) [10,25] is introduced in this paper. It has the merits of visual representation and direct calculation of geometric entities. ...
... .P i Ad T 0,i δε i 1P 2 . . .P i−1 Ad L M,i δq i(25) herein Ad L M,i ∈ R 6×1 is the motion of joint i described in the frame{O 0 }.Finally, geometric error model of an open-loop chain is obtained by rewriting Eq. (25) into matrix form as ...
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.
... In this way, the composition of two dual quaternions could be computed by quaternion multiplication [85], i.e., Euler-Rodrigues' formula with dual angles. As for the intersection algorithms, Sun [86] employed analytical derivations to deal with the intersection of the sets of dual quaternions. Mechanism analysis by dual quaternion was implemented by McAulay [87] for the first time who utilized dual quaternion to describe rigid body displacement. ...
... By taking the advantages of intersection and composition operations, the available limbs and mechanisms would be generated. More details are referred to [35,51,86]. Because the type synthesis is implemented in an algebraic manner, the parameterized topology models are obtained. ...
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Topology and performance are the two main topics dealt in the development of robotic mechanisms. However, it is still a challenge to connect them by integrating the modeling and design process of both parts in a unified frame. As the properties associated with topology and performance, finite motion and instantaneous motion of the robot play key roles in the procedure. On the purpose of providing a fundamental preparation for integrated modeling and design, this paper carries out a review on the existing unified mathematic frameworks for motion description and computation, involving matrix Lie group and Lie algebra, dual quaternion and pure dual quaternion, finite screw and instantaneous screw. Besides the application in robotics, the review of the work from these mathematicians concentrates on the description, composition and intersection operations of the finite and instantaneous motions, especially on the exponential-differential maps which connect the two sides. Furthermore, an in-depth discussion is worked out by investigating the algebraical relationship among these methods and their further progress in integrated robotic development. The presented review offers insightful investigation to the motion description and computation, and therefore would help designers to choose appropriate mathematical tool in the integrated design and modeling and design of mechanisms and robots.
... We assume it as a promising solution for a target tracking system with high precision. 10 Before applying the PTM to build a tracking system, it is found that position and orientation inaccuracies of the PTM have great effects on the precision of tracking trajectory. Therefore, kinematic calibration, the technique to identify and compensate geometric errors, is inevitable to improve the accuracy of the PTM. ...
... For instance, additional constrained forces are applied by two passive limbs in the PTM, and these over-constrained forces contribute to avoidance of kinematic singularity and enhancement of rigidity. 10 An introduction of over constraints is beneficial to the performance of parallel mechanisms, but it also brings difficulties in geometric error modeling by the screw theory. Commonly, a 6 Â 6 non-singular Jacobian matrix consisting actuated and constrained wrenches is applied to eliminate joint displacements in each limb. ...
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This paper deals with geometric error modeling and sensitivity analysis of an over-constrained parallel tracking mechanism. The main contribution is the consideration of over-constrained features that are usually ignored in previous research. The reciprocal property between a motion and a force is applied to tackle this problem in the framework of the screw theory. First of all, a nominal kinematic model of the parallel tracking mechanism is formulated. On this basis, the actual twist of the moving platform is computed through the superposition of the joint twist and geometric errors. The actuation and constrained wrenches of each limb are applied to exclude the joint displacement. After eliminating repeated errors brought by the multiplication of wrenches, a geometric error model of the parallel tracking mechanism is built. Furthermore, two sensitivity indices are defined to select essential geometric errors for future kinematic calibration. Finally, the geometric error model with minimum geometric errors is verified by simulation with SolidWorks software. Two typical poses of the parallel tracking mechanism are selected, and the differences between simulation and calculation results are very small. The results confirm the correctness and accuracy of the geometric error modeling method for over-constrained parallel mechanisms.
... Inventing new mechanisms with new motion patterns is an important and challenging work in mechanism synthesis. Here, motion pattern indicates finite motion pattern (full-cycle motion pattern) [1][2][3] . It means that for the motions generated by a mechanism, the corresponding motion pattern contains not only the number and types (rotations and/or translations) of degree-of-freedoms (DoFs), but also detailed characteristics of each DoF, regardless of the mechanism's dimensions. ...
... The expression of each factor in Eq. (5) can be given using Eqs. (2) and (3) , as ...
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This paper presents research work on synthesis of the mechanisms that generate translations on circular, cylindrical, and conical surfaces. As these three kinds of surfaces are all basic quadrics, the synthesized mechanisms are called motion generators of quadric surfaces. Firstly, the characteristics of these quadrics are analyzed, which result in an easy way to express them. Secondly, the motion sets of one-degree-of-freedom (one-DoF) joints are described by finite screws, leading to a simple and non-redundant manner for mechanisms’ motion description. Based upon this, the motion generators of circular, cylindrical, and conical surfaces are respectively synthesized, and all the serial kinematic chains that generate these quadrics are obtained. The results are verified through simulations in MATLAB software. Finally, as an application of the motion generators of quadrics, closed-loop linkages constituted by the generators of cylindrical and circular surfaces with specific geometric conditions are synthesized, which purely generate one-DOF translations along ellipse curves. Some new serial kinematic chains and closed-loop linkages are invented in this paper. These new mechanisms have simple mechanical structures, and they have potential applications in design of robots used in machining and manufacturing of complex surfaces and curves.
... Besides, Lian [34] put forward a generalized approach for geometric error modeling of PMs based on CGA. 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. ...
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... According to the improved CGK formula, the motion analysis is verified as [35] introduced the concept of parasitic motion in 3PRS parallel mechanisms and concluded that the parasitic motion is a small-value motion on the constrained DOF of 3PRS parallel mechanisms. Although the value is small, parasitic motion significantly influences the practical application of [PP]S parallel mechanisms [36,37]. ...
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Capturing noncooperative targets in space has great prospects for aerospace application. In this work, the knuckle unit of a large-scale reconfigurable space multifingered hand (LSRSMFH) for multitask requirements is studied. A plurality of knuckle units is connected in series to form a finger of the LSRSMFH. First, the lockable spherical (lS) joint, a new metamorphic joint that can function as a Hooke (lS1) or spherical (lS2) joint and is driven by shape memory alloy (SMA) material, is proposed. Based on the lS joint, this paper presents a new metamorphic parallel mechanism (MPM) (i.e., 3RRlS MPM), which has four configurations, namely, 3RRlS1, 3RRlS2, 2RRlS1-RRlS2, and 2RRlS2- RRlS1 configuration. The degree-of-freedom (DOF), overconstraint, and parasitic motion of the 3RRlS MPM are analyzed using screw theory, of which the DOF can be changed from 1 to 3. The 3RRlS1 configuration has a virtual constraint, and the 3RRlS2 configuration has parasitic motions. The results indicate that the mechanism motion screws can qualitatively represent the mechanism parasitic motions, and it is verified by deriving the kinematic equation of the 3RRlS MPM based on its spatial geometric conditions, the workspace of the 3RRlS MPM is further solved. The kinematic analysis indicates that the 3RRlS MPM can realize the folding, capturing, and reconfiguring conditions of the LSRSMFH.
... 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. ...
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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.
... Arbitrary lines in space could be represented by 6 parameters in a visual way. Moreover, the instantaneous and finite rigid body motions could be described in a convenient mathematical notation including all the information as proved in the previous work of Sun and Dai [5,23,31]. Based on these two superior characteristics, motions with coupling relations could be described analytically. Furthermore, the intersection operation of finite motions of limbs with general assembly conditions is prospective to be defined by employing the algorithms in CGA. ...
Article
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Finite motion analysis of parallel mechanisms (PMs) denotes formulating the map between finite motion of end-effector and those of its component limbs. By employing conformal geometric algebra (CGA), this paper presents an analytical and accurate method to analyze the finite motions of PMs with parasitic motions. Herein, parasitic motions are defined as the dependent motions in the constraint Degrees-of-Freedom (DoFs) of PMs. Firstly, description of rigid body transformations based on CGA is reviewed. Then, the intersection algorithm of finite motions is introduced by exploiting the algebraic properties of CGA. Based on this, a method to formulate the finite motions of PMs with parasitic motions is proposed. Finally, Z3 mechanism is sketched as example by utilizing the approach. This method facilitates the invention of new mechanisms and can also be applied in the finite motion analysis of other kinds of PMs.
... Therefore, a prepared tracking mechanism should have excellent performance as large workspace, high stiffness and precision. To this end, a 2-DoF parallel tracking mechanism, named as Helix robot, was proposed in our previous work [4] , which has isotropic kinematic and stiffness performance in a hemisphere workspace. In order to obtain the accuracy tracking trajectory of this robot at high speed under complex force environment, dynamic model-based control strategy is considered as an effective solution. ...
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To improve tracking precision against complex external forces in the working environment, dynamic model-based control algorithms are widely used in tracking mechanisms. As a foundation of the real-time dynamic control system, this paper focuses on the complete predicted dynamic modelling and dynamic parameter identification of a 2-DoF over-constrained tracking parallel mechanism. Firstly, the kinematic model is obtained by differential mapping of the topology models based on finite and instantaneous screws (FIS). Referred to virtual work principle, dynamic equation is constructed. To reduce the computation cost in practice, a linear model involving inertial and friction parameters is formulated for dynamic identification. A direct identification method is applied and all the parameters are obtained at the same time. Experiments are performed to show the accuracy and efficiency of the identification results. The contributions of the paper lie in 1) a concise method for constructing velocity and acceleration models of components is provided without tedious analysis 2) complete dynamic model of an over-constrained tracking parallel mechanism is developed 3) practical dynamic identification process is carried out compromising between precision and efficiency.
Article
This paper presents a novel approach for synthesizing metamorphic parallel mechanisms with an ability of reconfiguring motion between 1R2T and 2R1T based on quotient operation of manifolds. With the theoretical fundamentals of Lie subgroups and submanifolds, the principle of configuration change is proposed in the synthesis with a submanifold switch, that reconfigures the mechanism to provide a various motion, leading to reconfiguration of the metamorphic parallel mechanism platform. The synthesis starts from a quotient operation of manifolds to generate a manifold subchain, the synthesis of these subchains is aiming at the reconfiguration of the platform between 1R2T and 2R1T based on the same composition form of submanifold subchains with various motions. Three new subchains are synthesized with metamorphic joints, revolute joints and prismatic joints based on manifold-operation, which leads to a configuration switch of each subchain in terms of the phase change of the metamorphic joint with respect to relative positions of rotational axis and links, resulting in different constraints and motions of the subchains, leading to a platform with an ability to realize multiple motions. The design of new metamorphic joints is a key factor to realize the different motion of platform in terms of the principle of configuration change and manifold-operation. Further, the special geometrical properties and several specific phases of the metamorphic joint are analyzed. The approach of synthesis based on manifold-operation provides a distinct rationale for this paper. Under this guidance, analysis of motion and constraint conditions transmitted to the platform through subchains is essential, to enable the reconfiguration aimed type synthesis of metamorphic parallel mechanisms.
Article
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Chapter
Motion capability and performance behavior are the essential requirements from the application scenario to the development of robotic mechanism, which lead to three necessary steps including type synthesis, performance modeling, and optimal design. Type synthesis invents all possible topological structures that can realize the desired motions [1, 2]. A promising topological structure is then selected to be the candidate of the mechanism. Corresponding to the required behaviors, performances, such as kinematic, stiffness and/or dynamics of the selected topological structure are modeled and analyzed [3, 4, 5, 6, 7]. These parametric performance models are finally applied to the optimal design [8, 9, 10, 11, 12] which searches for the optimal structural parameters resulting in the robotic mechanism with desired performances.
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Chapter
Mechanisms with invariable and variable rotation axes [1, 2, 3, 4] are of interest in this book because the type synthesis of these mechanisms has not been thoroughly investigated due to the limitation of the adopted mathematical tools.
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Practical laser communication is impeded by the inherent strict requirements on agile and accurate steering of the laser beam over a wide angular range that are addressed by the pointing, acquisition, and tracking (PAT) system. While the vehicle carrying the optical instrumentation is in motion, a PAT system is required to compensate for the vibrations applied to the optical platform through the proper application of advanced control laws. This paper presents a feedforward vibration rejection system interacting with an optical tracking system to successfully perform the PAT task. It features a set of inertial navigation sensors to monitor the optical platform orientation, and the optical tracker monitoring the optical alignment errors. The control effort is defined on the basis of both signals and upon amplification drives the actuators of a novel singularity-free full-hemisphere-range robotic manipulator supporting the optical platform. The resultant technology enables two ground vehicles navigating through a difficult terrain to maintain optical connectivity sufficient for reliable laser communication. The paper presents the development of an extended Kalman filter "fusing" the inertial navigation sensor data, the design and implementation of the disturbance rejection/optical tracking control system, and the results of the experimental evaluation of the overall system performance.
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The forward and inverse kinematics of a two degrees of freedom (DOF) parallel mechanism for beam aiming applications is derived in closed form. The mechanism has three arms attached to a base via revolute joints. Another three arms are also attached to an upper platform via passive revolute joints. Each of the three base arms is connected to its respective opposing passive arm via a spherical joint. A central strut between the platform and the base is connected with spherical joints so that the platform retains the two rotational degrees of freedom with respect to the base. Any two of the three arms on the base are driven to set the required azimuthal and elevation pointing directions.
Introduction to the Highly Reliable Antenna Pedestal of an Unmanned Weather Radar
  • M X Shi
Shi, M. X., 2013, "Introduction to the Highly Reliable Antenna Pedestal of an Unmanned Weather Radar," Electro-Mech. Eng., 29(1), pp. 22-26.