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- Hindawi Publishing Corporation
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- Copyright © 2021 Shangyuan Zou et al. This 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.
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Abstract
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; mezousy@126.com 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 [1] 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. [2] proposed a singular simulators, wave compensation simulators, radio telescopes kinematic theory by using the screw theory. Cao et al. [3] (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. [4] 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 [5] 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 [18], and control [19]. 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 [8].&eclassicalmethodstostudysingularconfigurationare 2.1. Structure. &e 6DOF parallel robot with the planar the Grassmann line geometry method and the Jacobian platformanditscoordinatesystemisshowninFigure1.&e matrix method. Merlet [9] 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.[10]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. [11]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. [12] and Choi &ree points, P , P , P , are selected as natural coordi- 1 2 3 and Ryu [13] 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[14].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 [3]. At present, Li et al. [15] were all given orientation pa- rameters to find position singularity. Cao et al. [3] 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 � x − x , y − y , z − z . coupling parameters. What is more, it leads to the solution 2 1 2 1 2 1 and representation of the largest nonsingular workspace. Similarly, we obtain In conclusion, in order to make the six-dimensional parameters of position and orientation be fully visualized in β � x − x , y − y , z − z . (2) 3 1 3 1 3 1 the three-dimensional space, the natural coordinates methodisusedtorepresentthepositionandorientationand Because c is perpendicular to the plane determined by α six kinematic equations are deduced in this paper. In order and β, there is Journal of Robotics 3 γ 3 (P ) 2θ (P ) 5 (P ) α r O 2 a a A P 6 3 P 2 2θ B 6 Figure1:Sketchofthe6DOFparallelrobotwiththeplanarplatform.&erearetwoplatformsandsixlegs.A aretheverticesofthemoving platform.Biaretheverticesofthebase. O αβcistherelativemovingframe. O xyzistheabsolutestaticframe.P ,P ,andP areselectedas 1 2 3 a b natural coordinates. O and O are the circle center of the two platforms. r , r are circumradii of the base and the moving platform, a b a b respectively. θ ,θ are center semiangles corresponding to the short side of the platform, respectively. a b c � α ×β � α β −α β , α β − α β , α β − α β y z z y z x x z x y y x � y z − y z − y z + y z + y z − y z , −x z + x z + x z − x z − x z + x z , 1 2 1 3 2 1 2 3 3 1 3 2 1 2 1 3 2 1 2 3 3 1 3 2 x y − x y − x y + x y + x y − x y } . (3) 1 2 1 3 2 1 2 3 3 1 3 2 &e base vector of the static coordinate system is 3. Kinematics Model of 6DOF Parallel Robot T T T x � 1,0,0 , y � 0,1,0 , z � 0,0,1 . &e position { } { } { } 3.1. Construction Vector Equation and Separation Primary vector, represented by natural coordinates, of the moving and Secondary Variables. As shown in Figure 1, there are 6 platform in the static coordinate system is groupsofclosedvectorsinthemechanism,namely, O O A P � P � x , y , z ; the orientation matrix R is expressed b a 1 1 1 1 1 ⇀ ⇀ ⇀ ⇀ ⇀ as O B A , O O A O B A , . . ., O O A O B A . b 1 1 b a 2 b 2 2 b a 6 b 6 6 According to the vector closed relationship, the six ex- α β c x x x tendable link vectors of six groups of closed vectors are ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ R � [α,β, c] � ⎢α β c ⎥, (4) ⎢ ⎥ expressed as ⎢ ⎥ ⎢ y y y ⎥ ⎣ ⎦ α β c L e � P + Ra − b , (k � 1, . . .), (6) z z z k k k k whereeachcomponentinequation(4)isobtainedaccording where L isthelengthofthekthextendablelink; e istheunit k k to equations (1)–(3), as follows: vector of the kth extendable link; P is the position vector of the moving platform in static frame, namely, α � x − x , x 2 1 P � P � x , y , z ; R is the orientation matrix, shown in 1 1 1 1 equation (4); a is the coordinates of vertices A of the α � y − y , y 2 1 mobile platform in the moving frame O αβc, namely, α � z − z , a � a , a , a ; and b is the coordinates of vertices B z 2 1 k kα kβ kc k of the base platform in the basic frame O xyz, namely, β � x − x , x 3 1 b � b , b , b . As shown in Figure 1, because the ver- k kx ky kz β � y − y , (5) y 3 1 tices of the mobile platform and base platform are all arranged in a plane, the c axis and z axis components of a , b are all β � z − z , k k z 3 1 zero, namely, a � b � 0.&us, a , b are expressed as kc kz k k c � y z − y z − y z + y z + y z − y z , T T x 1 2 1 3 2 1 2 3 3 1 3 2 a � a , a ,0 , b � b , b ,0 . It can be seen that the k kα kβ k kx ky c � −x z + x z + x z − x z − x z + x z , vertex coordinates of mobile and static platforms are also y 1 2 1 3 2 1 2 3 3 1 3 2 expressedbyfourvariables r , r ,θ ,θ ,andthecoordinatesof a b a b c � x y − x y − x y + x y + x y − x y . z 1 2 1 3 2 1 2 3 3 1 3 2 each point are further expressed as follows: 4 Journal of Robotics T Similarity, P � P.y, W � P.α, and W � P.β. α is the y x y x a � a , a , a � r cos φ , r sin φ ,0 , k kα kβ kc a k a k projection of α in direction x, namely, α � α.x. Similarity, (7) x α � α.y, β � β.x, and β � β.y. y x y b � b , b , b � r cos β , r sin β ,0 , k kx ky kz b k b k where 3.2. Kinematic Model Represented by Natural Coordinates. (k − 1)π (k − 1)π Basedonequation(9),thenineunknownsof P , P , P , W , φ � −θ ,β � − θ , k � 1,3,5, p x y x k a k b 3 3 W , α , α , β , and β are transferred and sorted out; then, y x y x y (8) we can obtain the following results: (k −2)π (k − 2)π φ � + θ ,β � +θ , k � 2,4,6. k a k b 3 3 ⎧ ⎪ k α + β ⎪ 0 x y ⎪ P − � C , ⎪ P 1 Let W be the position vector of the mobile platform in ⎪ T ⎪ the moving frame, W � W , W , W ; then, there exists ⎪ x y z ⎪ k α − β 1 x y P � RW. In combination with the orthogonality of R, there P − � C , ⎪ x 2 exists W � R P. By substituting a , b , P, R, and W into 2 k k equation (6) and dot multiplication of the two sides of the equation with their own vector, the six links length square k α +β ⎪ 1 y x scalar equations are obtained as follows (the subscript k is P + � C , y 3 ⎪ 2 omitted here): (10) 2 2 2 ⎪ L − r − r � −2b P − 2b P + 2a W + 2a W −2α a b x x y y x x y y x x x ⎪ a b ⎪ k α −β ⎪ 2 x y ⎪ W − � C , x 4 −2α a b −2β a b − 2β a b + P . y x y x y x y y y p 2 (9) k α +β 2 y x In the previous equation, there are 9 unknowns, in- W + � C , ⎪ y 5 cluding P , P , P , W , W , α , α , β , and β . &e 9 ⎪ p x y x y x y x y ⎪ unknowns are related by the position and orientation pa- rameters of the mobile platform. Furthermore, the coeffi- ⎪ α − β y x � C , cients of the 9 unknowns are determined by the structural parametersandthelengthparametersoftheplatform,where where P is the square of module length of position vector P. P is p x the projection of P in direction x, namely, P � P.x. k � 2r r cosθ − θ , k � r sinθ +2θ Cscθ − θ , k � r sin2θ + θ Cscθ − θ , 0 a b b a 1 a b a b a 2 b b a b a 2 2 2 2 2 2 2 2 C � L + L + L + L + L + L − 6r −6r , 1 2 3 4 5 6 b a √� −1 2 2 2 2 2 2 2 2 2 2 C � Cscθ −θ 3 L − L − L + L cosθ + −2L −2L + L + L + L + L sinθ , 2 b a 3 4 5 6 a 1 2 3 4 5 6 a 12r √� 2 2 2 2 2 2 2 2 2 2 C � Csc θ −θ 2L −2L − L + L − L + L cos θ + 3 L + L − L − L sin θ , 3 b a 1 2 3 4 5 6 a 3 4 5 6 a 12r (11) √� −1 2 2 2 2 2 2 2 2 2 2 C � Cscθ −θ 3 L − L − L + L cosθ + −2L −2L + L + L + L + L sinθ , 4 b a 3 4 5 6 b 1 2 3 4 5 6 b 12r √� 2 2 2 2 2 2 2 2 2 2 C � Cscθ −θ 2L − 2L − L + L − L + L6 cosθ + 3L + L − L − L Sinθ , 5 b a b b 1 2 3 4 5 3 4 5 6 12r 2 2 2 2 2 2 C � Cscθ − θ L − L + L − L + L − L . 6 b a 1 2 3 4 5 6 12r r a b By substituting the representative point expressions of with degree one or quadratic form can be obtained as 9 unknowns into equation (10), six polynomial equations follows: Journal of Robotics 5 2 2 2 x + y + z − k −x + x − y + y � C , 0 1 2 1 3 1 1 1 1 x − k −x + x + y − y � C , 1 1 1 2 1 3 2 y + k −x + x − y + y � C , 1 1 1 3 1 2 3 (12) x −x + x + y −y + y + z −z + z − k −x + x + y − y � C , 1 1 2 1 1 2 1 1 2 2 1 2 1 3 4 x −x + x + y −y + y + z −z + z + k −x + x − y + y � C , 1 1 3 1 1 3 1 1 3 2 1 3 1 2 5 x − x − y + y � C . 1 3 1 2 6 and 4. Output Singularities and Analysis of 6DOF T T Parallel Robot P · P � x , y , z · x , y , z � x x + y y + z z . 1 2 1 1 1 2 2 2 1 2 1 2 1 2 (18) In this paper, the output singularities of planar 6 DOF parallel robot are studied. What is more, the types of output &erefore, there is singularities are discussed. (19) P · P � τ + τ − 1 � x x + y y + z z . 1 2 1 2 1 2 1 2 1 2 4.1. Output Jacobian Matrix Represented by Natural Similarly, we obtain Coordinates. By the relationships among the position vec- tors, P , P , P , represented by natural coordinates, re- 1 2 3 (20) P · P � τ + τ − 1 � x x + y y + z z , 1 3 1 3 1 3 1 3 1 3 member τ � τ ,τ ,τ ; that is, 1 2 3 2 2 2 τ � P · P � x + y + z , 1 1 1 1 1 1 1 (21) P · P � τ + τ − 1 � x x + y y + z z . 2 3 2 3 2 3 2 3 2 3 2 2 2 τ � P · P � x + y + z , (13) 2 2 2 2 2 2 2 2 2 &erefore, we can obtain twelve equations which are τ � P · P � x + y + z . 3 3 3 3 3 3 made up of equations (12), (13), (19), (20), and (21). First, equation (13) is substituted into equation (12). From vector closed relationships in Figure 1, we can &en,thenaturalcoordinates of representativepoints P , P obtain 1 2, and P are also substituted into a new equation (12): P � P, −C − k −x + x − y + y + τ � 0, P � P + α � P + R · x, (14) 1 0 1 2 1 3 1 2 1 P � P + β � P + R · y. 3 1 −C + x − k −x + x + y − y � 0, 2 1 1 1 2 1 3 T T After substituting x � {1,0,0} and y � {0,1,0} into equation (14), P , P , and P become compound variables 1 2 3 including position and orientation parameters. &erefore, −C + y + k −x + x − y + y � 0, 3 1 1 1 3 1 2 τ , τ , and τ are also compound variableswithposition and 1 2 3 orientation parameters. 1 1 −C − k −x + x + y − y + −1 − τ + τ � 0, From equation (14), we obtain 4 2 1 2 1 3 1 2 2 2 P − P � α. (15) 2 1 1 1 −C + k −x + x − y + y + −1 − τ + τ � 0, 5 2 1 3 1 2 1 3 2 2 Dotproductwiththeirownoftwosidesinequation(15). Combination with the unit vector α � {1,0,0}, we obain, −C + x − x − y + y � 0. 6 1 3 1 2 P · P + P · P − 2P · P � 1. (16) 2 2 1 1 1 2 (22) &erefore, there is Equations (13), (19), (20), and (21) are resorted out as 1 1 P · P � P · P + P · P − 1 � τ + τ − 1 , (17) follows: 1 2 1 1 2 2 1 2 2 2 6 Journal of Robotics 2 2 2 By calculating the first partial derivative of time t for 12 x + y + z −τ � 0, 1 1 1 variables x , y , z , x , y , z , x , y , z ,τ ,τ ,τ from the 1 1 1 2 2 2 3 3 3 1 2 3 2 2 2 kinematic equations of equations (22) and (23), equations x + y + z −τ � 0, 2 2 2 (22) and (23) are uniformly expressed as the vector forms as follows: 2 2 2 x + y + z −τ � 0, 3 3 3 3 J · V � V , (24) 2 x c (23) x x + y y + z z + 1 − τ − τ � 0, 1 2 1 2 1 2 1 2 where V � x _ , y _ , z _ , x _ , y _ , z _ , x _ , y _ , z _ ,τ _ ,τ _ ,τ _ and 2 x 1 1 1 2 2 2 3 3 3 1 2 3 V isthevelocitiesaboutsixextendablelinks.&ecoefficient matrix is alinear transformation matrixbetween V and V . x c x x + y y + z z + 1 − τ − τ � 0, 1 3 1 3 1 3 1 3 &erefore, J is the 12 ×12 dimensional Jacobian matrix of the mobile platform. It contains the natural coordinates of representative points in J , with dimensional consistency. x x + y y + z z + 1 − τ − τ � 0. 2 3 2 3 2 3 2 3 So, it does not need to be dimensionless processed. &e specific form of J is as follows: &erefore,12newkinematicequations,representedwith 2 natural coordinates, are obtained by combination with equations (22) and (23). k k k k 0 0 0 0 0 − 0 0 0 − 0 1 0 0 ⎛ ⎜ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 2 2 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ k k k k ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜1 + − 0 − 0 0 0 0 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 2 2 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ k k k k ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ − 1 − 0 0 0 0 0 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 2 2 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ k k k k 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 2 2 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ − 0 − 0 0 0 0 − 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 2 2 2 2 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ k k k k 1 1 ⎟ ⎜ ⎟ ⎜ 2 2 2 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ − − 0 0 0 0 0 − 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 2 2 2 2 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ − 0 0 0 − 0 0 0 0 0 ⎟. (25) J � ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 2 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 2 2 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2x 2y 2z 0 0 0 0 0 0 −1 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 0 2x 2y 2z 0 0 0 0 −1 0 ⎟ ⎜ ⎟ ⎜ 2 2 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 0 0 0 0 2x 2y 2z 0 0 −1 ⎟ ⎜ ⎟ ⎜ 3 3 3 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ x y z x y z 0 0 0 − − 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 2 2 1 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ x y z 0 0 0 x y z − 0 − ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 3 3 3 1 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 2 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 1 1 0 0 0 x y z x y z 0 − − 3 3 3 2 2 2 2 2 Journal of Robotics 7 &e six components x , y , x , y , x , y represented the six linear equations in equation (22). Furthermore, 1 1 2 2 3 3 withnaturalcoordinatesontheplaneαβaresolvedoutfrom x , y , x , y , x , y are linear expressed with τ ,τ ,τ : 1 1 2 2 3 3 1 2 3 k +2C k −2C k + k τ − k τ 1 4 1 2 2 1 1 1 2 x � − , 2k k +2C k −2C k + k τ − k τ 1 5 1 3 2 1 1 1 3 y � − , 2k k +2C k +2C k + k τ1 −2k τ1 − k τ k + 2C k − 2C k + k τ1 − k τ 0 4 0 1 2 0 2 0 2 1 4 1 2 2 1 1 2 x � − − , 2k k 2k 0 2 2 (26) −1 − 2C − 2C k − τ + τ k +2C k − 2C k + k τ − k τ 5 6 2 1 3 1 5 1 3 2 1 1 1 3 y � − − , 2k 2k 2 2 k + 2C k − 2C k + k τ1 − k τ −1 − 2C − 2C k − τ + τ 1 4 1 2 2 1 1 2 5 6 2 1 3 x � −2C − − , 3 6 2k 2k 2 2 C 1 C τ τ τ k +2C k −2C k + k τ − k τ 1 4 1 1 2 1 5 1 3 2 1 1 1 3 y � − + + + + − − . k 2k k k 2k 2k 2k 0 2 2 0 2 2 2 After we substitute the expressions of However, this kind of output singularity is impossible if we x , y , x , y , x , y inequation(26)intotheJacobian J in consider the early stage of structural design. 1 1 2 2 3 3 2 equation (25), then J becomes a new 12 ×12 dimensional expression with 6 variables z , z , z ,τ ,τ ,τ and 6DOF 1 2 3 1 2 3 4.2.2. :e Second Category parallel robot’s structure parameters, r , r ,θ ,θ . a b a b (1) If z � z � 0 and s � 0, then det[J ] � 0 1 2 10 2 (2) If z � z � 0 and s � 0, then det[J ] � 0 2 3 1 2 4.2. Output Singularity Trajectory Equations Represented by (3) If z � z � 0 and s � 0, then det[J ] � 0 1 3 4 2 Natural Coordinates. τ ,τ ,τ are the coupling variables of 1 2 3 positionandorientation;therefore,themobileplatformisat In this case, there is a singular surface in τ ,τ ,τ space. 1 2 3 singular configuration only when the determinant of Jaco- bian J is zero while the singular configuration is expressed 4.2.3. :e :ird Category when the position and orientation are coupled. &e singular trajectory equation of coupled pose is written as (1) If z � 0 and s � s � s � s � 0, then det[J ] � 0 1 4 7 9 10 2 detJ (τ, z) � 0, (27) (2) If z � 0 and s � s � s � s � 0, then det[J ] � 0 2 1 5 8 10 2 T (3) If z � 0 and s � s � s � s � 0, then det[J ] � 0 3 1 2 3 4 2 where remember z � z , z , z . 1 2 3 &e specific form of Jacobian J determinant after ex- In this case, in τ ,τ ,τ space, the intersections of four 2 1 2 3 pansion is expressed as follows: curved surfaces are singular points, lines, and surfaces. 3 2 2 3 2 detJ � s z + s z z + s z z + s z + s z z + s z z z 2 1 1 2 1 2 3 1 2 4 2 5 1 3 6 1 2 3 5. Example 2 2 2 3 + s z z + s z z + s z z + s z , 7 2 3 8 1 3 9 2 3 10 3 5.1.PreliminaryDefinitions. Taketheplanarplatform6-UPS (28) parallel robot for example, as shown in Figure 2, where the geometric structure parameters are θ � 0.28618πrad, where s is polynomial combinations of different degrees of θ � 0.046988πrad, r � 0.849864m, r � 0.849864m. &e b a b τ ,τ ,τ , namely, s (τ). Furthermore, s is any combination 1 2 3 i i six pairs of vertices P of the mobile and base platforms are polynomials with the highest degree to cubic orders of circularly and symmetrically arranged on a plane circle, as τ ,τ ,τ . 1 2 3 shown in Figure 3. If det[J ] � 0 is true, then three major categories in- cludingsevensecondarykindsofsingularconfigurationsare obtained from equation (28) as follows: 5.2. Output Singularities. Substituting the structural pa- rameters into det[J ], the numerical expression of the singular trajectory is obtained as follows: 4.2.1. :e First Category. det[J ] � 0 when z � z � z � 0 2 1 2 3 detJ � f z , z , z ,τ ,τ ,τ . (29) while the mobile platform and the base are coplanar. 2 1 2 3 1 2 3 A 8 Journal of Robotics (P ) (S) τ2 2θ (P ) r 2 a 2 O (P ) a 1 (P) τ3 (U) b O 6 x 2 2θ 5 B Figure2:Diagram of 6-UPS planar parallel robot. &ere are two platforms and six legs. &e passive joints A connecting the moving platform are spherical hinge S. &e passive joints B i 4 τ1 connecting the base are universal joint U. &e active joints Pi driving the legs are prismatic joint P fixed on each leg. O αβc is therelativemovingframe. O xyzistheabsolutestaticframe.P , 1 Figure4:Case1ofclassIIoutputsingulartrajectories. τ ,τ ,τ are 1 2 3 P , P are selected as natural coordinates. O and O are the 2 3 the coupling variables of position and orientation. a b circle center of the two platforms. r , r are circumradii of the a b base and the moving platform, respectively. θ ,θ are center a b semiangles corresponding to the short side of the platform, respectively. τ2 β (y) B3 A3 B4 A2 τ3 B2 A4 α (x) A5 B1 0 τ1 B5 A1 Figure5:Case2ofclassIIoutputsingulartrajectories.τ ,τ ,τ are B6 A6 1 2 3 the coupling variables of position and orientation. Figure 3: Diagram of vertices arrangement for 6-UPS planar parallel robot. A are the vertices of the moving platform. B are i i 5.2.1. :e First Category. It is obviously impossible. . the vertices of the base. &e circles in the dashed line are the circumscribed circle of the moving and base platforms, respectively. 5.2.2. :e Second Category (1) If z � z � 0 and s � 0, then det[J ] � 0. &e 1 2 10 2 spatial distribution of singular trajectories is shown In the following, we study the output singular types of in Figure 4. the mechanism and the spatial distribution of singular trajectories. If det[J ] � 0 is true, three kinds of output (2) If z � z � 0 and s � 0, then det[J ] � 0. &e 2 3 1 2 singularities obtained in Section 4.2 are analyzed as spatial distribution of singular trajectories is shown follows. in Figure 5. Journal of Robotics 9 τ2 τ3 τ3 4 τ2 τ1 τ1 Figure6:Case3ofclassIIoutputsingulartrajectories.τ ,τ ,τ are Figure 8: Case 2 of class III output singular trajectories. τ ,τ ,τ 1 2 3 1 2 3 the coupling variables of position and orientation. are the coupling variables of position and orientation. τ2 τ3 0 τ3 4 6 2 τ2 τ1 6 Figure 7: Case 1 of class III output singular trajectories. τ ,τ ,τ τ1 6 1 2 3 are the coupling variables of position and orientation. Figure 9: Case 3 of class III output singular trajectories. τ ,τ ,τ 1 2 3 are the coupling variables of position and orientation. (3) If z � z � 0 and s � 0, then det[J ] � 0. &e 1 3 4 2 spatial distribution of singular trajectories is shown in Figure 6. (2) If z � 0 and s � s � s � s � 0, then det[J ] � 0. 2 1 5 8 10 2 &e spatial distribution of singular trajectories is According to the6-UPS structure,we knowthat z ≠0. As shown in Figure 8. a result, the two cases, (1) and (3), are impossible. &erefore, only case (2) exists in this kind of output singularities. What is (3) If z � 0 and s � s � s � s � 0, then det[J ] � 0. 3 1 2 3 4 2 more, there is a three-dimensional curved surface, s � 0, of &e spatial distribution of singular trajectories is singular trajectory with six-dimensional parameters for cou- shown in Figure 9. pled position and orientation in space τ ,τ ,τ . 1 2 3 Accordingto the 6-UPSstructure, we know that z ≠0. As aresult,case(1)isimpossible.&erefore,onlycases(2)and(3) 5.2.3. :e :ird Category existinthiskindofoutputsingularities.Whatismore,thereare (1) If z � 0 and s � s � s � s � 0, then det[J ] � 0. three-dimensional singular trajectories of points, lines, and 1 4 7 9 10 2 &e spatial distribution of singular trajectories is surfaces,whicharetheintersectionsofthefourcurvedsurfaces, shown in Figure 7. s � s � s � s � 0 and s � s � s � s � 0, respectively, 1 5 8 10 1 2 3 4 10 Journal of Robotics with six-dimensional parameters for coupled position and Acknowledgments orientation in space τ ,τ ,τ . 1 2 3 &e authors disclosed receipt of the following financial support for the research, authorship, and/or publication of 6. Conclusions and Future Work this article. &is research was supported by the National Natural Science Foundation of China (Grant no. &enaturalcoordinatesmethodisusedtorepresentposition 51975277). &e authors gratefully acknowledge these PandorientationR.&en,asetofkinematicmodelsandtwo support agencies. sets of constraint models are represented as quadratic equations.&esemodelscontainquadraticterms,first-order References terms,andconstantterms,inwhichthehighestordertermis quadratic. &us, the simplest and nonlinear kinematic [1] K. H. Hunt, Kinematic Geometry of Mechanisms, Claredon models and constraint models for a kind of planar platform Press, Oxford, UK, 1978. type 6DOF parallel robot are established. &e Jacobian [2] L. Yan-Wen, Z. Huang, and F. 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Journal
Journal of Robotics – Hindawi Publishing Corporation
Published: Jun 10, 2021