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Singularity Analysis and Representation of 6DOF Parallel Robot Using Natural Coordinates
Singularity Analysis and Representation of 6DOF Parallel Robot Using Natural Coordinates
Zou, Shangyuan;Liu, Hairui;Liu, Yanli;Yao, Jiafeng;Wu, Hongtao
Hindawi Journal of Robotics Volume 2021, Article ID 9935794, 11 pages https://doi.org/10.1155/2021/9935794 Research Article Singularity Analysis and Representation of 6DOF Parallel Robot Using Natural Coordinates 1 2 2,3 3 3 Shangyuan Zou , Hairui Liu , Yanli Liu , Jiafeng Yao , and Hongtao Wu College of Transportation and Safty, Jiangsu College of Safety Technology, Xuzhou 221011, China College of Electrical Engineering, Jiangsu College of Safety Technology, Xuzhou 221011, China College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics & Astronautics, Nanjing 210016, China Correspondence should be addressed to Shangyuan Zou; firstname.lastname@example.org Received 4 March 2021; Accepted 24 May 2021; Published 10 June 2021 Academic Editor: Ruthber Rodriguez Serrezuela Copyright © 2021 Shangyuan Zou et al. &is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Singularity research is carried out. &e problem, which is about six-dimensional parameters of position and orientation can not realizethree-dimensionalvisualizationfor6DOFparallelrobot,hasbeensolved.Firstly,accordingtothestructuralcharacteristics of the 6DOF parallel robot with the planar platform, the position and orientation of the mobile platform are described, re- spectively,andthesixequationsofforwardkinematicsareestablishedbychoosingthenaturalcoordinatesofthreerepresentative points as parameters. &en, the singularities of the 6DOF parallel robot with a planar platform are divided into input singularity andoutputsingularity.Aimingattheoutputsingularity,incombinationwithsixconstraintequationsamongthepositionvectors of three representative points, an analytical algorithm is proposed to express the coupling singularity of position and orientation andtheanalyticalexpressionisderived.Infurtherresearch,threekindsofoutputsingularitiesarefound,thespatialdistributionof the output singular trajectory is determined, and a uniﬁed three-dimensional fully visualized description of six-dimensional coupling variables isrealizedfortheﬁrst time.&e problemsof ﬁnding thesingularorientationata givenposition orthesingular position at a given orientation are solved. &e analysis of the singularity lays a solid foundation for the description of the three- dimensionalcompletevisualizationofasix-dimensionalsingularity-freeworkspacebasedonforwardkinematics.Whatismore,it has great signiﬁcance for both trajectory planning and control design of the parallel robot. &eﬁrstmethodtostudythesingularityofthe6DOFparallel 1. Introduction robot is based on the screw theory. In the 1970s, Hunt  In recent years, 6DOF parallel robots are more and more ﬁrst used the screw theory to analyze the singularity of the widely used in VR, entertainment, medical and aerospace parallel mechanism. Yan Wen et al.  proposed a singular simulators, wave compensation simulators, radio telescopes kinematic theory by using the screw theory. Cao et al.  (FAST), and so on. &e requirements of high speed, high also used the screw theory to study the singularity. Lar- precision, high rigidity, high dynamic performance, low yushkin et al.  used the screw theory to study the sin- gularity of the 3DOF translational parallel mechanism and a inertia, small structure size, and other performance are also higher and higher. If the parallel robot meets these per- planar parallel mechanism. &e problem of singularity is an formance requirements, it is necessary to avoid singular important mechanical characteristic of the parallel robot, conﬁguration. which is one of the research hotspots for the parallel robot. When the mechanism is in a special conﬁguration, the What is more, the study of singularity is very crucial for the normal degree of freedom changes instantaneously, which design and control of the parallel robot. Gosselin and means the mechanism exits singularity. &ere are many Angeles  ﬁrst proposed an analysis method based on the methods to study the singularity of the 6DOF parallel robot. input-output speed and divided the singularities into 2 Journal of Robotics boundary singularity, conﬁguration singularity, and struc- to divide the traditional research of singularity into input ture singularity, in which the conﬁguration singularity is the singularity and output singularity, the kinematic equations are divided into two parts and combined with two diﬀerent main problem that we studied. When the mechanism is singular, the corresponding sets of constraint equations. Aiming at the output singu- conﬁgurationissingular.Singularconﬁgurationisanimportant larity, the analytical expressions of output pose singular basis for the determination of performance indexes such as trajectory are obtained, and the types of output singularity nonsingular workspace, ﬂexibility, and isotropy. Singular are analyzed. What is more, the complete visualization of conﬁguration includes input and output singularities, which six-dimensional position and orientation parameter singu- need to be considered and avoided in solving the maximum larityinthree-dimensionalspaceisrealizedfortheﬁrsttime. nonsingular workspace [6, 7], trajectory planning, control, and &e study of output singularities lays a solid foundation for other stages. In order to better understand the nature of solving the maximum nonsingular workspace, trajectory mechanismsingularconﬁgurationandbetteravoidthesingular planning , and control . conﬁgurationanditssurroundingareasinpracticalapplication, many scholars at home and abroad have launched the research 2. Kinematics Foundation of 6DOF on singular conﬁguration. Parallel Robot &e singular conﬁguration was ﬁrst discovered by Hunt .&eclassicalmethodstostudysingularconﬁgurationare 2.1. Structure. &e 6DOF parallel robot with the planar the Grassmann line geometry method and the Jacobian platformanditscoordinatesystemisshowninFigure1.&e matrix method. Merlet  introduced the Grassmann line mechanism consists of two platforms, the base and the geometry method to analyze the singularity of 6–3 platform mobile platform, and six legs with identical structures. A and established the geometric conditions of singular con- and B are the centers of spherical joint S and universal joint ﬁguration.Wenetal.studiedthesingularitiesof3-DOF U, respectively. All A and B are restricted to a plane, re- i i planar parallel manipulators using Grassmann–Cayley al- spectively; that is, this 6DOF parallel robot belongs to the gebra.Ma etal. studied 3/6-SPSStewartmanipulator by planar type. In order to facilitate the analysis, O xyz, the using the method of Grassmann–Cayley algebra. &e position- absolutestaticframe,isselectedtobeﬁxedlyconnectedwith singularitylociandorientation-singularitylociaredrawnbased the base, and O αβc, the relative moving frame, is ﬁxedly on the polynomial obtained from the coeﬃcient of the outer connectedwiththemobileplatform,respectively. O and O a b product of all 6 line vectors. Although it is convenient and are the centers of the mobile platform and the base, re- intuitivetoverifythesingularitiesbythemethodofGrassmann spectively. &e z and c axes are perpendicular to their re- linear geometry, it is diﬃcult to ﬁnd the singularities of the spective planes, respectively. whole distribution. Scholars such as Choi et al.  and Choi &ree points, P , P , P , are selected as natural coordi- 1 2 3 and Ryu  used the Jacobian matrix method to study the nates, which are located at the coordinate origin, α axis singular conﬁguration of the 4-DOF parallel robot and 4-DOF endpoint and β axis endpoint of the moving frame O αβc, parallel manipulator, respectively. respectively, as shown in Figure 1. &e vectors, P � x , 1 1 A proper model is established by selecting the natural T T y , z } , P � x , y , z , and P � x , y , z , are used 1 1 2 2 2 2 3 3 3 3 coordinates formed by the appropriate points and orien- to represent three position coordinates of three natural tationvectorstodescribethemultibodysystem.Andthe coordinates, P , P , and P , in the static frame. As a result, th 1 2 3 order analytical polynomial of singular trajectory for a nine variables to be solved are included in the forward ki- kind of parallel mechanism is derived using the half-angle nematics model. conversion method . At present, Li et al.  were all given orientation pa- rameters to ﬁnd position singularity. Cao et al.  had also 2.2. Represent Position P and Orientation R with Natural studied how to ﬁnd the orientation singularity at the given Coordinates. &ere exists the following relationship among position parameters. In addition, singular trajectories P , P , and P , as follows: P � P + R · x, P � P + R · y. 2 3 1 2 1 3 1 [16, 17] are represented by three-dimensional parameters in &e vector coordinates of α axis, β axis, and c axis in the three-dimensional space. Although position and orientation moving frame O αβc are remembered as α � α ,α ,α , a x y z T T parameters are coupled, the complete visualization of po- β � β ,β ,β , and c � c , c , c . &en, α, β, c are x y z x y z sition and orientation singular trajectories is independent of represented with natural coordinates of the three repre- eachother,whichdoesnotrealizethecompletevisualization sentative points P , P , P , as follows: 1 2 3 of the six parameters in three-dimensional space. &erefore, T T the study of position and orientation singularities greatly α � P − P � x , y , z − x , y , z 2 1 2 2 2 1 1 1 (1) aﬀectsthecompletevisualizationofpositionandorientation �