Research Article
International Journal of Advanced Robotic Systems November-December 2016: 1–17 Design and motion analysis of a novel ª The Author(s) 2016 DOI: 10.1177/1729881416678139 coupled mechanism based on a regular arx.sagepub.com triangular bipyramid
Huifang Gao, Jingfang Liu and Yueqing Yu
Abstract Traditional methods and theories on synthesizing parallel mechanisms are not applicable to related researches on hybrid mechanisms, thus hampering the design of innovative coupled mechanisms. Polyhedrons with attractive appearance and particular geometrical construction provide many choices for coupled inventions. A novel mechanism with one trans- lational degree of freedom based on a regular triangular bipyramid is proposed in this article. First, the basic equivalent geometrical model is spliced with new-designed components substituting for vertexes and edges by revolution joints (R-pairs) only. The expected motion for the basic coupled model can be achieved by adding links to modify the constraint sets and arrange spatial allocation of an elementary loop based on the screw theory. Then, the mobility of one branch is calculated to investigate the movability of the novel structure, and a Denavit–Hartenberg (D-H) model with properties of symmetry is implemented to investigate the inverse kinematic analysis. Furthermore, a numerical example is given to verify the correctness of analysis results and related motion simulation is conducted to illustrate the potential application of the proposed novel system as an executing manipulator for mobile robots.
Keywords Regular triangular bipyramid, substitutive components, adding chains, reciprocal translational motion
Date received: 27 July 2016; accepted: 13 October 2016
Topic: Service Robotics Topic Editor: Marco Ceccarelli Associate Editor: Yan Li
Introduction several areas of mathematics, the use of polyhedron as a novel insight has been extended to chemistry, architecture, The need of synthesizing new mechanisms for innovation, 8 mechanism, and other scientific fields. The octahedral especially with symmetric structures, makes researchers to invention associated with the so-called Jitterbug motion investigate particular structures. In recent years, the coupled provided by Fuller in 1948 can be regarded as the first mechanisms with the advantages of the serial and parallel inspired product by polyhedrons. In recent decades, poly- mechanisms have been a hot topic. The new type of coupled hedrons have inspired many innovative coupled mechanisms that is characterized by coupled chains, which are always connected for complex multiloop or netting struc- tures between the base and the moving plane, is different from that of the serial and parallel constructions. However, the College of Mechanical Engineering and Applied Electronics Technology, complicated coupling chains bring about the impossibility Beijing University of Technology, Beijing, China to apply generalized methods and theories on parallel struc- tures1–7 to the construction of novel coupling mechanisms. Corresponding author: Jingfang Liu, Beijing University of Technology, No. 100 Pingleyuan, Polyhedral structures, interesting for their geometrical Chaoyang District, Beijing 100124, China. characteristics, are naturally occurring. Appeared first in Email: [email protected]
Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage). 2 International Journal of Advanced Robotic Systems mechanisms. Hoberman invented the Hoberman switch- pitch ball, radial expansion/retraction truss structure, and the folding covering panels by connecting different types of polygonal linkages for radial or reversible motion.9–11 Many types of rolling robots by combining a set of pris- matic pairs, revolution pairs, and parallelogram scaling units based on spatial tetrahedral framework have been presented in literature.12,13 Lalibert´e and Gosselin14 pro- posed a mechanical arrangement based on a single type of component for the construction of polyhedrons. Wei et al.15,16 applied a spatial eight-bar linkage to synthesize a series of deployable mechanisms based on different poly- hedrons. Ding et al.17 proposed a novel prism deployable mechanism by polyhedral linkages, which had potential application in aerospace field. Some other polyhedral struc- Figure 1. Sketch of a regular triangular bipyramid. tures,18,19 such as the antenna of Mir Space Station designed to be network construction for flexibility and transformation, on the regular triangular bipyramid in this article. The com- have already been used in recent years. Cai et al.20 studied a monness between the polyhedra and coupled structures is planar radial expanding linkage with generalized angulated used to establish fundamental framework for type synth- elements. For extensive range and superior performance of esis. Our work is scheduled as follows. polyhedral coupled mechanisms, the researchers also had In this article, a novel coupled mechanism based on regular conducted many theoretical studies on type synthesis triangular bipyramid is presented. The number of merging approaches and mobility analysis of the focused systems. edges at every vertex in a regular triangular bipyramid is not In 2004, Dai et al.21 investigated a scalable mechanism and identical. By designing three substitutive components for essen- analyzed its mobility by decomposing the mechanism into tial elements and adding reasonable links, based on the screw some elementary parallel mechanisms with single degree of theory, the basic coupled model is obtained in section ‘‘Model- freedom (DOF) based on the screw theory. You22 came up ing of the novel coupled mechanism.’’ Then, the mobility and with the efficiency of the productive application in satellite inverse kinematic analysis are conducted to identify motion and solar panels and shelters considering the motion character- symmetry properties. A numerical example is given to verify istics of the Hoberman sphere in 2007. In 2012, Kipper and the analytical results and corresponding simulations are imple- So¨ylemez23 introduced a family of highly overconstrained mented and demonstrate the validity and feasibility of the pro- spatial linkages using the Jitterbug motion introduced by posed mechanism in section ‘‘Numerical validation and Fuller. Based on the screw theory, Liu et al.24 proposed a simulation analysis.’’ Conclusions are drawn in the last section. new mobility method for coupled mechanisms by analyzing the motion of the node parts relative to a fixed base and converting the mechanism into a simple equivalent parallel Modeling of the novel coupled mechanism 25,26 structure. Then, Liu et al. improved the method for mul- The goal of this section is to establish the basic coupled tiloop coupling mechanisms by altering a loop linkage into model based on a regular triangular bipyramid. The poly- generalized limbs and operated conveniently. hedral solid bounded with six identical triangles is selected As mentioned earlier, polyhedrons characterized by par- for the construction of fundamental framework. First, by ticular geometry and charming symmetry provide scholars spatial arrangement and geometrical features of the poly- the possibility of establishing fundamental framework for hedral sketch, based on which three types of substitute inventing polyhedral coupled mechanisms. The studies on components are designed and assembled for the original type synthesis of innovative coupled mechanisms with basic model, the relationship of essential elements is ela- polyhedral structures have drawn researchers’ attention. borated. Then, to obtain a moveable basic coupled model, The invented polyhedral structures with coupling connec- the unit loop corresponding to a face in a regular triangular tions usually perform excellently. Possessing special output bipyramid is extracted and its mobility is modified by add- motion, polyhedral coupled mechanisms can transform into ing links based on the screw theory. At last, the whole suitable configurations conveniently to satisfy different spatial coupled model is built. operational requirements. However, the existing coupled mechanisms are not plentiful and related studies have not been developed overall. As thus, it is significant to invent- Construction of fundamental framework ing various new polyhedral coupled mechanisms for According to the connecting condition of each vertex, poly- increasing the variety of mechanisms and expanding appli- hedrons can be classified to the three types8: the regular cation widely. To enrich the family of the new mechanisms, polyhedra, the semiregular polyhedra, and some special we present a novel polyhedral coupled mechanism based irregular polyhedra. The selected regular triangular Gao et al. 3 bipyramid for type synthesis shown in Figure 1 is a kind of semiregular polyhedra. To clarify the synthesizing method, the geometrical property of a regular triangular bipyramid is first introduced. Edges and vertexes contained in a polyhedron are essen- tial elements and play an important role in modeling. It is known that the numbers of essential elements in a polyhe- dral solid satisfy the following Euler formula V þ F E ¼ 2(1) where V is the number of vertexes, F is the number of faces, and E is the number of edges. As shown in Figure 1, a regular triangular bipyramid is composed of six regular triangles. Not all merging condi- tions at every vertex are the same. Assume V3 is the number of the vertexes connecting three regular triangles and V4 is that of four regular triangles. Hence, the relationship Figure 2. Construction of fundamental framework. between all essential elements of a regular triangular bipyr- amid is obtained as Establishment of the basic equivalent geometrical V ¼ 2; V ¼ 3 ; F ¼ 6 ; E ¼ 9(2) 3 4 model Based on the spatial arrangement and relationship Considering the diversity and complexity of coupled struc- between all essential elements of the regular triangular tures, the novel mechanism under investigation is synthesized bipyramid, the fundamental framework can be constructed just with R-pairs. Based on the geometrical connections of by determining the types of all components and connec- essential elements in a regular triangular bipyramid, the rule tions of branches in the following steps: of replacing the vertex merging n edges with one n-gon part is (1) Marking all vertexes of a regular triangular taken to design three kinds of substitutive parts. bipyramid (1) Each edge in Figure 2 is replaced by a binary link Equation (2) explains the relationships between all with two parallel R-pairs at both ends, as shown in essential elements in a regular triangular bipyramid and Figure 3(a). (2) Both A and E vertexes are represented by regular meets equation (1). The two vertexes (V3 ¼ 2) merging three edges are labeled as A and E; the other three triangle substitutions with three R-pairs, as shown in Figure 3(b). vertexes (V4 ¼ 3) connecting four edges are labeled as B, C,andD. (3) Square nodes replace B, C,andD vertexes, as shown in Figure 3(c). (2) Selecting components as the base and moving plat- form for the basic model Then, the basic equivalent geometrical model shown in Figure 4 is spliced with two regular triangle parts, that is, D1 Considering the symmetry of the fundamental frame- and M1, three square nodes, that is, N1, N2,andN3, and nine work, choose A and E, located symmetrically with identical binary links, that is, Li (i ¼ 1–9). The three symmetric lines 0 0 connections, as the moving platform and the base, O Ni (i ¼ 1, 2, 3) intersects at the central point O of the model 0 respectively. and coincides with the three perpendicular bisectors O ni (i ¼ 0 1, 2, 3) of L5, L6,andL4. The three symmetric axes O Ni define (3) Determining coupled node parts and connecting aplane , which is the symmetric plane of the model and chains signed with red-dashed boxes. The line g, connecting the cen- ters of M1 and D1, is also a symmetric line and perpendicular All branches can be confirmed when the base and to the plane of symmetry . Passing through the symmetric the end effector are selected. Except two edges linked 0 line g and O Ni (i ¼ 1, 2, 3), respectively, three symmetric to A and E directly, every two of B, C,andD vertexes planes, that is, plane 1, plane 2, and plane 3, are determined share a common edge, defined as coupled chain. And correspondingly and are all revealed with blue-dashed boxes. then B, C,andD vertexes are selected as the three node parts. As mentioned earlier, the fundamental framework based Construction of the movable elementary loop on regular triangular bipyramid is established and shown in Since the basic equivalent model in Figure 4 connected Figure 2. with substitutive components can be viewed as 4 International Journal of Advanced Robotic Systems
Figure 3. Substitutive parts: (a) binary link; (b) regular triangle part; (c) square node part.
Table 1. Reciprocal condition for constraint screws of a known screw system S/ .
Demanded constraint No screws for S/ Reciprocal conditions r r 1 Line vector S/ 1 S/ 1 must be perpendicular to all S/ 2 and intersect with all S/ 1 r r 2 Couple S/ 2 S/ 2 must be perpendicular to all S/ 1
L5-N3-L3-D1 indicated in Figure 5 as an example, the group matches up with the triangular face EBC in Figure 2. The constraint screw system of the original loop in Fig- ure 5 is analyzed first. Taking D1 as the base and N2 as the end effector, the loop can be decomposed to two parallel branches, that is, branch 1 labeled as D1-R2-R21-N2- and branch 2 labeled as D1-R3-R31-N3-R32-R23-N2-, respec- tively. The motion of N2 is constrained by the aforemen- Figure 4. Basic equivalent geometrical model. tioned two parallel branches and corresponding constraints are marked by red arrows. Constraints of branch 1 contain r r two constraint couples, S/ 11 and S/ 12, and two constraint r r r r forces, S/ 13 and S/ 14. The direction of S/ 11 and S/ 12 is perpen- r dicular to R21-axis, one couple S/ 13 is along link L2 and the r other one S/ 14 is parallel to R21-axis. Constraints of branch 2 r r include a constraint force S/ 21 and a constraint couple S/ 22. r The direction of S/ 21 is along the line defined by two inter- section points, one determined by the R32- and R31-axes, r and the other by R23- and R3-axes. The coupleS/ 22 is normal to the plane defined by R32- and R31-axes. Therefore, the original loop is unmovable. To construct the movable loop with desired mobility, the constraint sets of branches are modified by adding chains for spatial rearrangement based on reciprocity and relativ- ity of screws.27 The reciprocal condition of constraint screws for a known screw system is summarized in Table Figure 5. Original loop. 1, where S/ i (i ¼ 1, 2) represents the motion screws in a r given screw system S/ and S/ i (i ¼ 1, 2) indicates the reci- composition of six identical loops, the primary issue is to procal constraint screws in S/ . Here, the condition i ¼ 1is ensure the movability of a loop for the possibility of obtain- the line vector and i ¼ 2 is the couple. ing a movable model. For the constituent triangular faces, The novel coupled model is expected to be an expand- each loop included in the regular triangular bipyramid able and retractable mechanism with translational motion. framework is usually constrained. Hence, the mobility of Because of the only use of R-pairs, motion screw systems one loop should be analyzed first. Taking loop D1-L2-N2- of the modified branches include line vectors merely. To Gao et al. 5
Figure 6. Modified loop. Figure 7. Structure and motion flow of the novel basic coupled model. avoid local or inactive DOFs for stability and effective mobility, the rule of adding the minimum components sym- metrically is followed. As shown in Figure 6, an additional joint R5 is introduced by adding a component shown in r Figure 3(a) to L5 in branch 2 and the constraint force S/ 21 can be eliminated correspondingly. The additional link and midpoint of the R5-axis are located in plane . To maintain the symmetry of the entire model, the additional R4 is introduced by adding a link to L4 and R6 by adding a link to L6. The rearranged link group denoted as D1-R2-R21-N2- R23-R5-R32-N3-R31-R3-D1 in Figure 6 is extracted as the movable elementary loop and used to splice the novel basic coupled model. Figure 8. Sketch diagram of three coupled branches. Mobility analysis of the novel coupled Structure description for the novel basic coupled mechanism model Mobility of a novel basic coupled mechanism should be As shown in Figure 7, the novel coupled model consists of analyzed primarily for further recognition. The improved two regular triangular components, that is, D1 and M1, and method with independent motion shunting is valid for three square nodes, that is, N1, N2, and N3, which are con- mobility analysis and output motion characteristics of the 24 nected by 12 binary links with 21 R-pairs. Ri (i ¼ 1, 2, 3) complicated multiloop coupled mechanisms. In this sec- are the R-pairs located on D1, and those fixed on Nj are tion, the obtained coupled system is disassembled into three labeled as Rj1–Rj4 (j ¼ 1, 2, 3). The three R-pairs for M1 are identical branches according to the method of motion R4k (k ¼ 1, 2, 3) and the remainder three mid-joints are Rq shunting first. Then, the motions of the nodes are analyzed (q ¼ 4,5,6), respectively. V ¼ 21, F ¼ 11, and E ¼ 30 can and expressed as simple generalized kinematic limbs. be calculated for the coupled model, which satisfy equation Finally, the complicated basic coupled model can be con- (1) and demonstrate the reasonability of the novel verted to an equivalent parallel mechanism. The mobility mechanism. analysis of the proposed novel mechanism is illustrated in The new loop N1-R12-R6-R33-N3-R32-R5-R23-N2-R22- detail as follows. R4-R13-N1 is located exactly in the symmetric plane and the three symmetric lines O0N (i ¼ 1, 2, 3) pass through i Decomposition of the novel basic coupled model corresponding axes of mid-joints Rq (q ¼ 4,5,6), respec- tively. Hence, the symmetric relations for original model in Based on the motion shunting measurements, the designed Figure 4 also satisfy the modified coupled model in Figure 7. coupled mechanism referred in Figure 7 is decomposed The initial configuration for the following mobility analysis into three identical branches i,wherei ¼ 1, 2, 3. Each is represented by the mechanism configuration at any time branch i has a node Ni, and the connection from D1 to node of the moving process in Figure 7. Ni is Ci. The node Ni connects three subchains labeled Ci1, 6 International Journal of Advanced Robotic Systems
Figure 9. The three subchains of the branch 2: (a) subchain C21; (b) subchain C22; (c) subchain C23.
C2 mobility analysis. The screw system of the chain C21 for the initial configuration in coordinate frame Oc- XcYcZc is S/ 21 ¼ð010; 000Þ (3) S/ 2 ¼ð010; d1 0 f1 Þ
where d1 and f1 represent two elements of dual unit of S/ 2. The two symbols are just related to the axis location of joints and do not depend on the mobility analysis of the mechanism. The subscripts of S/ 2 and S/ 21 identify with that of the corresponding R-pairs. Then, the constraint screw system is obtained as 8 1r >S/ 21 ¼ð010; 000Þ <> S/ 2r ¼ð000; 100Þ 21 (4) >S/ 3r 000; 001 :> 21 ¼ð Þ Figure 10. Coordinate systems for mobility analysis. 4r S/ 21 ¼ðf1 0 d1; 000Þ 1r 4r which indicates two constraint forces of S/ 21 and S/ 21 and Ci2,andCi3, respectively, so Ci is expressed as Ci ¼ 2r 3r 1r 4r two constraint couples of S/ 21 and S/ 21. S/ 21 and S/ 21 can limit Ci1[Ci2[Ci3. In addition, two R-pairs Ri4 and R4i are two translations along the R21-axis and the connecting link located between Ni and M1, and any branch i can be 2r 3r L2, respectively. S/ 21 and S/ 21 restrict the rotation of N2 expressed as a chain of C -R -R . In Figure 7, the motion i i4 4i around Xc-axis and Zc-axis. flow of every branch is marked with red arrows and the In the same coordinate system, the screw system of C22 corresponding sketch diagram is displayed in Figure 8. is expressed as Taking C2 as an example, Figure 9(a) to (c) shows the three 8 >S/ a c 0; d e f subchains C21, C22, and C23 enclosed in green dashed lines. > 1 ¼ð 1 1 2 2 2 Þ > <>S/ 11 ¼ða1 c1 0; 00f3 Þ >S/ 13 ¼ð001; d4 e4 0 Þ (5) > Mobility analysis of the mechanism >S/ 4 ¼ð001; d5 e5 0 Þ :> As illustrated in Figure 9, the three parallel subchains in the S/ 22 ¼ð001; d6 00Þ initial configuration can be indicated as C21 (R2-R21-), C22 The corresponding constraint screw system will be (R1-R11-R13-R4-R22-), and C23 (R3-R31-R32-R5-R23-). C22 and C23 are located symmetrically with plane 2, as shown 1r S/ ¼ð000; c1 a1 0 Þ (6) in Figure 7. To analyze the constraints of each subchain, a 22 local coordinate system Oc-XcYcZc in Figure 10 is built, Equation (6) defines a constraint couple limiting the where Yc-axis is along the direction of the R21-axis and the rotation of N2 around the normal line of the plane defined plane OcXcZc coincides with the symmetrical plane 2 of by R11- and R13-axes. branch 2 aforementioned. Suppose that the midpoint of the The screw system of chain C23 connecting with the R11-axis is A1 (a1, c1,0), the direction vector of R31-axis can same structure as chain C22 can be obtained be denoted as A2 (a1,-c1,0). symmetrically Gao et al. 7
8 > >S/ 3 ¼ða1 c1 0; d2 e2 f2 Þ > <>S/ 31 ¼ða1 c1 0; 00 f3 Þ >S/ 32 ¼ð001; d4 e4 0 Þ (7) > >S/ 5 ¼ð001; d5 e5 0 Þ :> S/ 23 ¼ð001; d6 00Þ then the corresponding constraint screw system is
1r S/ 23 ¼ð000; c1 a1 0 Þ (8) Figure 11. Equivalent parallel mechanism. By solving the reciprocal screw system of the union set of equations (4), (6), and (8), we obtain expressed in O-XYZ as g 8 S/ 2 ¼ð000; d1 0 f1 Þ (9) 1r >S/1 ¼ð010; 000Þ > which demonstrates the secondary reciprocal screw > 2r ; >S/1 ¼ð000 100Þ system of the symmetrical double loop and illus- > 3r >S/ ¼ð000; 001Þ trates an independent translation of N relative to > 1 2 > 1r ; the base. 2 2 2 2 Mobility analysis of branch 2. As shown in equation (9), the > 3r >S/2 ¼ð000; p3 q3 r3 Þ motion of C can be expressed as a prismatic pair P , there- > 2 2 >S/1r ¼ð110; 00r Þ fore, branch 2 can be equivalent to a generalized kinematic > 3 4 > 2r >S/ ¼ð000; p5 q5 r5 Þ chain P2-R24-R42 and the motion screw system in Oc- :> 3 3r XcYcZc is S/3 ¼ð000; p6 q6 r6 Þ 8 > 24 24 S/ ¼ð000; 001Þ (13) S/ 42 ¼ð010; d42 0 f42 Þ Equation (13) shows that M1 has one translation along Z- then the constraint screw system is axis, indicating the reciprocating motion of the moving plane. 0 1 010; 000 B C S/ r ¼ @ 000; 100A (11) Mobility analysis of the equivalent parallel mechanism. Based 2 on the mobility analysis of the end described in the previ- 000; 001 ous section, the novel coupled mechanism can be converted r to the equivalent parallel mechanism shown in Figure 11, where S/ 2 represents a constraint force limiting the transla- tion of M1 along Yc-axis, and two constraint couples which means the mobility of the coupled model can be restricting the rotation of M1 around the Xc- and Zc-axes, calculated by that of the equivalent parallel structure. respectively. It is well known that each generalized branch provides two constraint couples and a constraint force. There is a Mobility analysis of the end effector M1. To describe the common constraint ( ¼ 1) among these constraint couples overall constraints acting on M1, the global coordinate and the order of the mechanism is five (d ¼ 6 1 ¼ 5); the system O-XYZ showninFigure10isestablished,where rank of the three linearly dependent constraint forces is the origin O coincides with the projection of the geome- two. There are two parallel redundant constraints overall, trical center O0 onthebaseplaneandtheX-axis is that is, v ¼ 2, and the number of local DOF is zero, that is, perpendicular to R1-axis. Z-axis is vertical to the plane ¼ 0. According to the modified Gru¨bler-Kutzbach for- 27 of D1, pointing up to M1,andY-axis is along the direc- mula, the DOF of the terminal mechanism (M), which is tion of R1-axis. equal to that of the end effector (M0), can be solved Based on the identical constructions of the three Xg branches and the symmetry planes shown in Figure 7, con- M ¼ M0 ¼ dðn g 1Þþ fi þ v straint sets from branch 1 and branch 3 exerted to M1 are i¼1 identical to that of the branch 2 expressed in equation (11). ¼ 5ð8 9 1Þþ9 þ 2 0 ¼ 1 (14) Hence, on the whole, the end effector will be restricted by three constraint forces and six constraint couples. The established local and global coordinate systems The constraint screw system of all branches can be shown in Figure 10 are valid to analyze the mobility for 8 International Journal of Advanced Robotic Systems
Establishment of coordinate systems Based on the three symmetric planes, that is, plane 1, plane 2, and plane 3 as shown in Figure 7, only the branch 2 shown in Figure 12 is taken as example to perform the kinematics analysis in the initial configura- tion. To display the coordinate frames clearly, the branch 2 shown in Figure 12 is taken as example to perform the kinematics analysis in the initial configura- tion. The coordinate system O-XYZ,whichisfixedon D1 and built for the aforementioned mobility analysis, is the reference system. The midpoint of each Ri-axis on D1 is marked as Bi (i ¼ 1, 2, 3) correspondingly. The side B2B3 is perpendicular to the X-axis and parallels to the Y-axis. The origin of moving coordinate P-XpYpZp is attached to the center of M1,whereXp-axis is parallel to X-axis and Zp-axis coincides with Z-axis. The midpoint of each R4j-axis on M1 is labeled as Ej (j ¼ 1, 2, 3). The Figure 12. The coordinate systems of branch 2. line connecting the origins of O and P passes through the center O0. The point P locatedontheendeffectoris the unchanged relationship of the relative motion and con- selected as the reference point and its spatial position straint characteristics of the three symmetrical branches coordinate is expressed as (0, 0, Hz)inO-XYZ for one throughout the moving process. Moreover, it is proved translational DOF. that the mobility analysis of the coupled mechanism is Each subchain C2i (i ¼ 1, 2, 3) for branch 2 in Figure 9 full cycle. can be viewed as a linkage composed of a series of R-pairs. Axes Zij (i ¼ 1, 2, 3; j ¼ 0 ...7) of each joint marked with Kinematics analysis of the novel coupled red dashed lines in Figure 12 represent the jth joint of the mechanism ith subchain for the branch 2, where corresponding coordi- nate systems located at joints are set to describe the spatial To investigate more advantageous characteristics of the position transformation of adjacent links. Considering that designed structure, a kinematic module is established for the axis of R6 is located in the symmetric plane 2 and passes solving the inverse kinematic solution based on the geo- 0 through the O N2, the kinematical analysis of the subchain metrical construction and symmetrical properties men- C22 can be deduced by that of C22. Then, the D-H models of tioned in section ‘‘Modeling of the novel coupled C21 and C23 are established as shown in Figure 13(a) and mechanism.’’ The work conducted in this section pro- (b), respectively. vides the basis for future work, such as dynamic analysis. As far as a coupled mechanism concerned, the inverse kinematic analysis looks for the variable values of the Kinematic mathematic model for C21 actuator joints located on the base with given position and The position of N2 is determined by the comprehensive orientation of the end effector. To describe the relationship effects of C21, C22, and C23, hence, the kinematic equations of the position for every connecting rod, the D-H, which is for different subchains with equal parameters at N2 should frequently used for studying parallel mechanisms, is be obtained. adopted to define the geometrical parameters and joint By means of the 4 4 homogeneous matrix tool that variables in sequence. Note that the posture of the node has describes the geometrical rotation and transformation rela- the same parameters for every chain, thus symmetry prop- tionship between two directly connecting bars, the trans- erties allow to simplify the calculation and kinematics formation matrix for subchain C21 from (i-1)th bar to ith bar analysis. is defined as
2 3 cos i sin i 0 ai 1;i 6 7 6 sin cos cos cos sin d sin 7 6 i i 1;i i i 1;i i 1;i i i 1;i 7 Ti 1;i ¼ 4 5 (15) sin i sin i 1;i cos i sin i 1;i cos i 1;i dicos i 1;i 0001 Gao et al. 9
Figure 13. Coordinate systems attached to joins: (a) subchain C21; (b) subchain C23.
Table 2. D-H parameters for chain C21. transformation of C21 from the base to the end effector is shown in Figure 14(a) and that of C23 is shown in Figure i 0 1 2345(P) 14(b). i 1,i 0 90 000 90 Hence, the corresponding transformation matrix for ai 1,i (mm) 0 a01 a12 a23 a34 a45 chain C21 labeled T 1 can be expressed as i 60 1 2 3 4 5 di (mm) 0 0 0 0 0 0 T 1 ¼ T 0T 01T 12T 23T 34 T 45 (16)
where T 1 can be obtained by the position relationship where is the torsion angle about X -axis from Z - between the O-XYZ and P-XpYpZp i 1,i i 1 i 1 2 3 axis to Zi-axis, ai 1,i is the translational distance along 100 0 6 7 Xi 1-axis from Zi 1-axis to Zi-axis for the length of the 6 010 07 6 7 (i 1)th bar, di is the offset distance along Zi-axis between T 1 ¼ 4 5 (17) 001Hz Xi 1-axis and Xi-axis, i is the joint angle around the Zi-axis from Xi 1-axis to Xi-axis. 000 1 Referring to Figure 13(a), Table 2 lists the D-H parameters and the D-H transformation matrixes in equation (16) can for C21 with the established coordinate systems. The be written with parameters in Table 2 as
2 pffiffiffi 3 2 3 1=2 3=200 cos 1 sin 1 0 a01 6 pffiffiffi 7 6 7 6 3=21=2007 6 00107 6 7 6 7 T 0 ¼ 4 5 T 01 ¼ 4 5 0010 sin 1 cos 1 00 0001 0001 2 3 2 3 cos 2 sin 2 0 a12 cos 3 sin 3 0 a23 6 7 6 7 6 sin 2 cos 2 007 6 sin 3 cos 3 007 T 12 ¼ 6 7 T 23 ¼ 6 7 4 00105 4 00105 0001 0001 10 International Journal of Advanced Robotic Systems
Figure 14. Geometrical transformation relationship with D-H parameters: (a) chain C21; (b) chain C23.
2 3 2 pffiffiffi 3 cos 4 sin 4 0 a34 1=2 3=20a45 6 7 6 7 6 sin 4 cos 4 007 6 01=0107 T 34 ¼ 6 7 T 45 ¼ 6 pffiffiffi 7 (18) 4 00105 4 3=21=2005 0001 0001
8 > 1 þ 2 ¼ 90 Then, the transformation for C21 is converted to the > > equivalent kinematical equation. Considering the geome- > þ ¼ 90 > 3 4 trical dimensions of designed components, following rela- < tionships can be obtained sinð 1 þ 2 þ 3 þ 4Þ¼0 (20) > 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > ð3=4Þ ð1=4Þcosð 1 þ 2 þ 3 þ 4Þ¼1 > pffiffiffi > > h ¼ð2=3Þ ðr Þ2 ðr =2Þ2 ¼ð 3=3Þr > cos þ sin ¼ 0 > 1 1 1 1 :> 1 3 > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi <> 2 2 p sin 1 cos 3 ¼ðh2 HzÞ=L h2 ¼ð1=2Þ ðr2Þ þðr2Þ ¼ð 2=2Þr2 (19) > a ¼ a ¼ h Note that i assumes a positive or negative sign depend- > 01 45 1 > ing on the configuration of the chain. With regard to the > a23 ¼ h2 :> inverse kinematic analysis, the actuator variable 1 and a12 ¼ a34 ¼ L other joint angles i can be acquired once Hz for the initial where h is the length of the connecting line between position of the moving plane is given, which yields 1 8 the origin O and the point B2,andh2 is equal to half of > ¼ arcsin½ðh H Þ=2L > 1 2 z the distance of parallel axes of Z22 and Z23.Thesym- < 2 ¼ 90 1 bol L indicating the length of each connecting bar and > (21) > 3 ¼ 2 rj (j ¼ 1, 2) meaning the radius of axis hole for R-pairs :> of regular triangle parts and square nodes, respec- 4 ¼ 1 tively, are design parameters of all substitutive com- Finally, the variables of the actuator and other joints in ponents that will be given for the simulation model in C are solved. section ‘‘Numerical validation and simulation 21 analysis.’’ By substituting equation (19) into equation (18), the Kinematic mathematical model for C23 position of the end effector for C21 can be accom- plished by equating both sides of equation (16) and The inverse kinematic analysis for C23 can be carried simplified as out with the same reference system and moving Gao et al. 11
system shown in Figure 12. To distinguish the D-H meaning for C23 are used to express the transforma- parameters of C21, different symbols with the same tion matrix as
2 3 cos i sin i 0 bi 1;i 6 7 6 sin cos cos cos sin d sin 7 23 6 i i 1;i i i 1;i i 1;i i i 1;i 7 Ti 1;i ¼ 4 5 (22) sin i sin i 1;i cos i sin i 1;i cos i 1;i di cos i 1;i 0001
where i is the joint angle around the Zi-axis from X 1-axis The relative transformation matrix for C23 labeled as T 2 to Xi-axis, i 1,i is the torsion angle about Xi 1-axis from can be obtained Zi 1-axis to Zi-axis, bi 1,i is the translational distance along 23 23 23 23 23 23 23 23 23 T 2 ¼ T T T T T T T T T (23) Xi 1-axis from Zi 1-axis to Zi-axis for the length of the 0 01 12 23 34 45 56 67 7p (i 1)th bar. where T 2 can be expressed based on the position relation- The corresponding geometrical transformation of all ship. Similarly, the transformation matrices in equation joints for C23 has shown in Figure 14(b) using parameters (23) are written, respectively, as in Table 3.
2 pffiffiffi 3 2 3 1=2 3=200 cos 1 sin 1 0 b01 6 pffiffiffi 7 6 7 6 3=21=2007 6 00 107 23 6 7 23 6 7 T 0 ¼ 4 5 T 01 ¼ 4 5 0010 sin 1 cos 1 00 0001 0001 2 3 2 3 cos 2 sin 2 0 b12 cos 3 sin 3 00 6 7 6 7 6 sin cos 007 6 00 1 d 7 23 6 2 2 7 23 6 3 7 T 12 ¼ 4 5 T 23 ¼ 4 5 001d2 sin 3 cos 3 00 0001 0001 2 3 2 3 cos 4 sin 4 0 b34 cos 5 sin 5 0 b45 6 7 6 7 6 sin cos 007 6 sin cos 007 23 6 4 4 7 23 6 5 5 7 T 34 ¼ 4 5 T 45 ¼ 4 5 0010 001d5 0001 0001 2 3 2 3 cos 6 sin 6 00 cos 7 sin 7 0 b67 6 7 6 7 6 00 1 d 7 6 sin cos 007 23 6 6 7 23 6 7 7 7 T 56 ¼ 4 5 T 67 ¼ 4 5 sin 6 cos 6 00 0010 0001 0001
2 pffiffiffi 3 1=2 3=20b7p 6 7 6 00107 T 23 ¼ 6 pffiffiffi 7 (24) 7p 4 3=21=2005 0001 12 International Journal of Advanced Robotic Systems
Table 3. D-H parameters for the chain C23. the geometrical relations of D-H parameters for C23 are 8 i 012345678(P) <> b01 ¼ b7p ¼ h1 090 090 0090 0 90 b12 ¼ b67 ¼ L (25) i 1,i :> bi 1,i(mm) 0 b01 b12 0 b34 b45 0 b67 b7p d2 ¼ d3 ¼ d5 ¼ d6 ¼ h2 i 60 1 2 3 4 5 6 7 120 d (mm) 0 0 d d 0 d d 00 i 2 3 5 6 where h1, h2, and L have the same meaning as in equation (19). Based on the constraint conditions of laws of sine and Table 4. Parameters of substitutive parts. cosine, by substituting equation (25) into equation (24) and expanding equation (23) yield Components Regular triangle Square Binary link 8 > 1 þ 2 ¼ 180 Parameter symbol l1 r1 l2 r2 l3 r3 > > Size (mm) 9 1 11 1 20 1 > 3 þ 4 þ 5 ¼ 120 > > L þ½ðh2=2Þ= sin 3 2½Lcos 6 þð2=3Þh1 þð1=2Þh2cot 3 > ¼ > sinð =2Þ sin <> 4 4 Table 5. D-H parameters for chain C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 21. L 2L2 þ 2L2cos > ¼ 4 > i 012345(P) > sinð30 3Þ sin 4 > > i 1,i 0 90 000 90 > 6 þ 7 ¼ 0 > ai 1,i (mm) 0 5.20 20 15.56 20 5.20 > > 1 ¼ 1 i 60 31.49 –58.51 –58.51 –31.49 120 : L sin þ h2 þ L sin ¼ Hz di (mm) 0 0 0 0 0 0 1 6 (26)
Table 6. D-H parameters for chain C23. i 012345 678(P)
i 1,i 090 090 00900 90 bi 1,i (mm) 0 5.20 20 0 20 20 0 20 5.20 i 60 31.49 148.51 69.73 –79.46 –110.27 31.49 31.49 120 di (mm) 0 0 7.78 7.78 0 7.78 7.78 0 0
X 50 Y Z 40
30
20 Displacement (mm) Displacement 10
0 0 1 2 3 4 Time (s)
Figure 15. Simulation model. Figure 16. Motion simulation of M1 along X-, Y-, and Z-axis. Gao et al. 13
20 N1 N1 N N2 20 2 N3 N3 10 10
0 0 axis (mm) axis (mm) X Y –10 –10 –20 –20 0 1 2 3 4 0 1 2 3 4 Time (s) Time (s) (a) (b)
N1 N2 N3 25
20 axis (mm) Z
15
0 1 2 3 4 Time (s) (c)
Figure 17. Motion simulations along the three axes for nodes N1, N2, and N3: (a) X-axis; (b) Y-axis; (c) Z-axis.
Then, the variable 1 and other parameters for inverse 60 kinematics analysis of C22 can be implemented conveni- N1 ently by symmetry; for space limitations, further details are 50 N2 not given. Solving equation (26), the variables for C23 are N3 determined when the initial position is known 40 M1 0 axis (mm) 1 ¼ 1 Z 30 B B B 2 ¼ 180 þ 1 B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 20 B 2 B ¼ arcsin 1= ðh1=h2Þ þ1 cos arctanðh2=h1Þ B 3 6 10 B 50 B 4 ¼ 60 2 3 B 0 30 B 10 20 B 5 ¼ 3 180 0 B –50 –10 Y axis (mm) X @ 6 ¼ arcsin½ðHz h2 þ L sin 1Þ=L axis (mm)
7 ¼ 6 (27) Figure 18. Spatial displacement variations for components. Theoretical investigation of the inverse kinematic anal- Numerical validation and simulation ysis of the presented coupled mechanism is carried out with analysis the algebraic method. The variables of joint angles, i and j, can be determined if the position for the end device is To verify the theoretical kinematics analysis, a numerical given. In the next section, a numerical simulation is per- example is presented first. Then, a simulation model is formed to validate the correctness of the mobility and kine- evaluated to identify the synthesized manipulator and asso- matics analysis. ciated theoretical motion characteristics. 14 International Journal of Advanced Robotic Systems
Figure 19. Different configurations: (a) retractable configuration; (b) expandable configuration.
Numerical example θ 1 The dimensions of the parts composing the novel mechan- θ ism are listed in Table 4, where the symbol l (i ¼ 1, 2, 3) is 2 i –20 θ the edge length of corresponding components and rj (j ¼ 1, 3 2, 3) is the radius of axis hole for R-pairs. θ –40 4 The initial position of the moving plane is set to (0, 0, 36.45) in the overall coordinate system O-XYZ. Based on Angle (°)Angle equations (21) and (27), the corresponding D-H parameters –60 for C21 and C23 are shown in Tables 5 and 6, respectively. The inverse kinematic results listed in Tables 5 and 6 –80 reveal the symmetry of the initial configuration. Positive sign indicates counterclockwise rotation of each joint angle 0 1 2 3 4 and negative sign stands for clockwise rotation. Then, sub- Time (s) stituting the parameters into equations (16) and (23), the Figure 20. Joint angles of chain C . transformation matrixes T 1 and T 2 can be expressed as 21 2 3 2 3 1 000 100 0:0162 6 7 6 7 the software SolidWorks 2013 is shown in Figure 15. The 6 01 00 7 6 0100:0105 7 identical design parameters for the given numerical exam- T 1 ¼ 6 7 T 2 ¼ 6 7 4 00136:4581 5 4 00136:4445 5 ple listed in Table 4 are used for the following simulation. 00 01 0001 An actuator of rotational motor at 10 rpm marked with a (28) red rotation arrow is mounted on R3-axis to realize one translation of reciprocal motion for end. The position var- which indicate the relationship between the moving plane iations of moving plane and nodes are determined by the and the end effector. It is obvious that the inverse kinematic overall coordinate system O-XYZ mentioned in Figure 12. analysis including coupled chains by algebraic method is feasible. The main source of the errors in equation (28) is rounding. Simulation of displacement variation. The simulation of dis- placement variations of moving plane and three nodes is implemented by means of the point P and the three centers Motion simulation for kinematic analysis of N1, N2, and N3 in the global system O-XYZ. The motion To investigate the motion characteristics of the new simulations of the four measuring points along the three mechanism, the displacement variations of end and nodes, axes are carried out running the motor for 4 s. The corre- and joint angles of connecting links are simulated with a sponding range of the displacement of the end is from given rotational actuator. Motion properties are illustrated 30.0416 mm to 51.3406 mm, divided into 17 steps each clearly by the curves of simulation results. Then, the cor- lasting 0.25 s. rectness of theoretical motion analysis and symmetrical To show the mobility characteristics of M1, time–dis- property are further verified. placement curves of the end along the three axes are plotted in Figure 16. Simulation model and driving condition. The three- Figure 16 shows that the translation along X-axis (black dimensional model for simulation analysis established by solid line) remains unchanged with time, as Y-axis (red Gao et al. 15
φ 1 φ 3 150 φ 2 150 φ φ 4 100 6 100 φ 5 φ 7 50 50 0 Angle (°)Angle 0 (°)Angle –50 –50 –100
–100 –150 0 1 2 3 4 0 1 2 3 4 Time (s) Time (s) (a) (b)
Figure 21. Joint angles of the chain C23: (a) comparison of 1, 2, 6, and 7; (b) comparison of 3, 4, and 5.
θ 1 θ 100 3 150 θ θ 2 4 φ φ 100 1 50 6 φ φ 2 7 50 0 Angle (°)Angle 0 (°)Angle –50 –50 –100 –100 0 1 2 3 4 0 1 2 3 4 Time (s) Time (s) (a) (b)
Figure 22. Comparison of joint angles of C21 and C23: (a) 1, 1, 2, and 2; (b) 3, 4, 6, and 7. dashed line) does, and the translation of Z-axis (blue dashed retractable configurations of moving plane relative to the line) rises gentle. Therefore, the moving plane just has base are shown in Figure 19. translational motion along Z-axis, so the feasibility of the novel coupled mechanism and the correctness of theoretical Simulation for kinematic verification. In this part, the theore- mobility analysis in section ‘‘Mobility analysis of the novel tical inverse kinematical results are verified with the simu- coupled mechanism’’ are demonstrated. lation model. First, using the same displacement interval To highlight the symmetrical motion characteristic of (from 30.0416 mm to 51.3406 mm, time length of 4 s), the N , N , and N , the displacement of each node along the 1 2 3 value of each joint angle (i ¼ 1, 2, 3, 4) in C can be same axis is plotted in a graph simultaneously for i 21 obtained by simulation. Compared with simulated results, comparison. according to Figure 20, the relationship among these angles The comparison of the plots in Figure 17 highlights that can be illustrated in a Cartesian plane coordinate where the the displacement variation of N is confined in the plane 1 variation of (cyan solid line) and (black dashed line) XOZ, which also represents the plane 1 shown in Figure 7. 1 4 overlap and decrease. On the contrary, the angles of (red The variations of the three nodes along Z-axis keep the 2 thick dot dashed line) and (blue dashed line) overlap but same during the whole simulation. Figure 18 shows that 3 have positive trend, thus the sum of angles and or N , N , and N exhibit good characteristics of radial motion 1 2 3 1 2 3 and is always a constant value for negative 90 . tendency directing forward to geometrical center O0 with 4 The joint angles for C are simulated with the same associated reciprocal translation of M along Z-axis. 23 1 period of 4 s. Figure 21 shows the relationship among Basedontheaforementioned simulations, the novel obtained joint angles, and the particular relationship of coupled mechanism synthesized symmetrically is proved joint angles between C and C is shown in Figure 22. to be a deployable mechanism. The expandable and 21 23 16 International Journal of Advanced Robotic Systems
From Figures 21 and 22, the following relationships for authors are grateful to the project (No. 51475015) supported by simulated joint angles are established NSFC, the open project (No. 310825151130) of National Engi- neering Laboratory for Highway Maintenance Equipment j 1j¼j 6j¼j 7jj 1jþj 2j¼ 180 j 3jþj 5j¼ 180 (Chang’an University).
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