Peter L. WangRobotics and Automation Laboratory,

University of California,

Irvine, CA 92697

e-mail: wangpl1@uci.edu

Haithem E. TahaFlight Dynamics Laboratory,

University of California,

Irvine, CA 92697

e-mail: hetaha@uci.edu

J. Michael McCarthyRobotics and Automation Laboratory,

Department of Mechanical and

Aerospace Engineering,

University of California,

Irvine, CA 92697

e-mail: jmmccart@uci.edu

Synthesis of a Flapping WingMechanism Using a ConstrainedSpatial RRR Serial ChainThis paper designs a one degree-of-freedom (1DOF) spatial flapping wing mechanismfor a hovering micro-air vehicle by constraining a spatial RRR serial chain using two SSdyads. The desired wing movement defines the dimensions and joint trajectories of theRRR spatial chain. Seven configurations of the chain are selected to define seven preci-sion points that are used to compute SS chains that control the swing and pitch jointangles. The result is a spatial RRR-2SS flapping wing mechanism that transforms theactuator rotation into control of wing swing and pitch necessary for hovering flight of amicro-air vehicle. [DOI: 10.1115/1.4038529]

1 Introduction

Flapping-wing micro-air vehicles are designed to fly like birdsor insects that exploit unconventional aerodynamic mechanismsto generate high lift at low speeds and to stabilize their bodies inflight [1]. While most micro-air vehicles such as the Nano-Hummingbird [2] rely on passive positioning of the wing pitch,Yan et al. [3] show that coordinated control of the pitch and wingmovement improves the aerodynamics of a micro-air vehicle.They demonstrate that increased aerodynamic performance can beachieved by coordinating the swing and pitch control functions. Inthis paper, we present a design methodology that yields a onedegree-of-freedom (1DOF) mechanism that coordinates these twofunctions.

This methodology begins with a designer specified spatial serialchain formed by three revolute, or hinged, joints denoted RRRchain—R denotes a revolute joint. The input that drives the basejoint h1 and output wing swing and pitch angles, h2 and h3, are thesecond two joints. The RRR spatial chain is constrained introduc-ing two SS dyads—S denotes a spherical or ball joint—to connectthe second and third links to the ground frame, Fig. 1. The result-ing one degree-of-freedom spatial RRR-2SS linkage coordinatesthe wing swing and pitch to the input.

In what follows, we present the design methodology for thisspatial six-bar linkage and demonstrate its use in the design of aflapping wing mechanism for a hovering micro-air vehicle.

2 Literature Review

A linkage that provides a prescribed function for an outputangle for a given input angle is called a function generator. Har-tenberg and Denavit [4] and Suh [5] provide the kinematic theoryfor the synthesis of the single loop spatial RSSR function genera-tors. Mazzotti et al. [6] presents the dimensional synthesis of anRSSR mechanism through optimization. Recent work on singleloop function generators has been presented by Cervantes-Sanchez et al. [7,8].

Our flapping wing mechanism must have two output angles foreach given input angle, one for swing and one for pitch. Thisrequires a design theory for coordinating three angles in two-loopspatial linkage system. Sandor et al. [9] and Chiang [10] presentdesign methodologies for two-loop spatial chains, but instead ofcoordinating joint angles they guide an end-effector along a speci-fied trajectory. Similarly, Chung [11] constructs a two-loop spatialmechanism that supports an end-effector to draw a specified curve

rather than coordinate joint angles. Cervantes-Sanchez et al. [12]analyze a spatial linkage with two loops; however, it does nothave the RRR spatial serial chain that characterizes our system.Hauenstein et al. [13] focus on the design of spatial RRR serialchains to guide an end-effector along a desired trajectory, how-ever, this mechanism has three degrees-of-freedom.

This is the first presentation of a synthesis theory for a RRR-2SS spatial linkage to coordinate the joint angles of the RRR spa-tial chain. This can be viewed as a spatial version of the synthesisof planar six-bar linkages by adding RR constraints to a planarRRR serial [14]. The synthesis of SS dyads that provide the con-straining links for the spatial six-bar linkage is presented by Inno-centi [15] and McCarthy and Soh [16].

3 The Spatial RRR Chain

The design of the RRR-2SS spatial six-bar linkage begins withthe selection of a spatial RRR serial chain. The Denavit Harten-berg frames are shown in Fig. 1 and the DH parameters are shownin Table 1, where hi, i¼ 1, 2, 3, are joint variables and di, i¼ 1, 2,3, ai, i¼ 1, 2, and ai, i¼ 1, 2, are specified by the designer.

Let Li, i¼ 1, 2, 3, be the coordinate frames with the z-axis alongthe ith joint and the x-axis along the common normal to the next

Fig. 1 The spatial RRR-2SS linkage constructed by constrain-ing a spatial RRR serial chain using two SS dyads that connectthe second and third links to the ground frame

Manuscript received May 25, 2017; final manuscript received October 27, 2017;published online December 20, 2017. Assoc. Editor: Robert J. Wood.

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joint axis. The position of these links relative to the ground frameis defined by the kinematics equations

T1 ¼ Zðh1; d1ÞT2 ¼ Zðh1; d1ÞXða1; a1ÞZðh2; d2ÞT3 ¼ Zðh1; d1ÞXða1; a1ÞZðh2; d2ÞXða2; a2ÞZðh3; d3Þ

(1)

where Z(hi, di) and X(ai, ai) are the 4� 4 homogeneoustransforms

Zðhi; diÞ ¼

cos hi �sin hi 0 0

sin hi cos hi 0 0

0 0 1 di

0 0 0 1

266664

377775

Xðai; aiÞ ¼

1 0 0 ai

0 cos ai �sin ai 0

0 sin ai cos ai 0

0 0 0 1

266664

377775; i ¼ 1; 2; 3

(2)

These kinematics equations are used in the design procedure.

4 SS Constraint Synthesis

Let the desired movement of the RRR serial chain be definedby the joint trajectories h1(t), h2(t), and h3(t), which are knownfunctions of a parameter t. In order to design a one degree-of-freedom system that approximates this movement, select sevenprecision points from these joint trajectories and denote them asqi¼ (h1j, h2j, h3j), j¼ 1,…, 7. Two SS constraints can be calcu-lated that ensure the system reaches these seven precision points.

In order to constrain the RRR spatial chain to one degree-of-freedom, we introduce an SS dyad BC that connects L2 to theground frame F, and another SS dyad EF that connects L3 to theground frame.

Let Bj denote the coordinates of the moving pivot attached toL2 measured in F, when the RRR chain is in the configurationdefined by qj. Similarly, let E

j be the coordinates of the movingpivot attached to L3 for each of the precision positions qj. Intro-ducing the relative displacements

R1j ¼ T2ðqjÞT2ðq1Þ�1

and S1j ¼ T3ðqjÞT3ðq1Þ�1

(3)

we have

Bj ¼ R1jB1 and Ej ¼ S1jE

1 (4)

The coordinates for the SS dyad BC must satisfy the constraintequations

ðR1jB1 � CÞ � ðR1jB

1 � CÞ ¼ h2; j ¼ 1;…; 7 (5)

where

B1 ¼ ðx; y; zÞ; C ¼ ðu; v;wÞ (6)

Similarly, the coordinates for the SS dyad EF must satisfy

ðS1jE1 � FÞ � ðS1jE

1 � FÞ ¼ k2; j ¼ 1;…; 7 (7)

where

E1 ¼ ðm; n; oÞ; F ¼ ðp; q; rÞ (8)

where h and k are the lengths of BC and EF, respectively.Both Eqs. (5) and (7) can be simplified by subtracting the first

equation in the set from the remaining six equations. This cancelsthe constants h2 and k2 as well as the squared terms for all 12coordinates of BC and EF. The result is the two sets of designequations

Aj : ðR1jB1 � CÞ � ðR1jB

1 � CÞ � ðB1 � CÞ � ðB1 � CÞ ¼ 0;

j ¼ 2;…; 7 (9)

and

Bj : ðS1jE1 � FÞ � ðS1jE

1 � FÞ � ðE1 � FÞ � ðE1 � FÞ ¼ 0;

j ¼ 2;…; 7 (10)

The equations Aj and Bj are each bilinear in their respective sixunknown coordinates for BC and EF. They can be solved inde-pendently to determine as many as 20 SS dyads, which is as manyas 400 pairs of SS dyads that guide the RRR serial chain throughthe seven precision points, qj, j¼ 1,…, 7 [15,16].

5 Flapping Wing Mechanism

In order to design the flapping wing mechanism, we start withthe RRR serial chain defined in Table 2. These values were chosenby the designer to fit the workspace and packaging requirements.The RRR chain matches the scale of a hummingbird as seen inAerovironment’s Nano Hummingbird [2] and the motor is ori-ented such that it fits within the body of the bird.

This serial chain will be installed in the micro-air vehicle bymounting link L1 to the body so the ground link F is rotated by amotor. This is shown in Fig. 2 where the S-joints C and F aremounted to interconnected cranks that simultaneously drive wingswing, link AD, and wing pitch, link DE. The joint trajectories forthe RRR chain that move this system as recommended by Yanet al. [3] are giving by

h1 ¼ t

h2 ¼p2� p

3cos t

h3 ¼p2� p

3sin t

(11)

as shown in Fig. 3. Seven precision positions selected from thesetrajectory curves are given in Table 3. The precision points areshifted slightly in the design process, the difference between theselected precision points and the desired function values is shownin Table 4, and plotted in Fig. 4.

Substitute the precision points into the design equations Aj; j ¼2;…; 7 for the link BC. The result is the following set of designequations:

Table 1 Denavit–Hartenberg table for the RRR serial chain. hi,i 5 1, 2, 3 are joint variables. The remaining parameters areselected by the designer.

Link i hi di ai ai

1 h1 d1 a1 a1

2 h2 d2 a2 a2

3 h3 d3 — —

Table 2 Denavit–Hartenberg table for the RRR serial chain ofthe flapping wing mechanism

Joint hi di (cm) ai (deg) ai (cm)

1 h1 7.07 45 �3.002 h2 �5.00 90 �1.003 h3 2.00 — —

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A2 : 1:36uxþ 1:87uyþ 0:32uz� 0:39u� 1:78vxþ 1:29vy

þ 0:56vz� 4:68v� 0:64wxþ 0:11wyþ 0:11wz� 2:67w

þ 5:15xþ 1:15yþ 1:15zþ 7:30 ¼ 0

A3 : 3:43uxþ 0:99uy� 0:99uzþ 10:39u� 0:21vxþ 3:55vy

þ 1:25vz� 3:5v� 1:38wxþ 0:79wyþ 0:79wz� 9:74w

þ 14:52x� 1:44y� 1:44zþ 53:74 ¼ 0

A4 : 1:79ux� 0:91uy� 1:77uzþ 6:46uþ 1:52vxþ 3:07vy

� 0:73vzþ 12:34v� 1:28wxþ 1:43wyþ 1:43wz� 13:92w

þ 17:6x� 6:24y� 6:24zþ 96:94 ¼ 0

A5 : 0:77ux� 1:45uy� 0:63uz� 4:72uþ 0:88vxþ 2:03vy

� 1:79vzþ 12:88v� 1:31wxþ 1:38wyþ 1:38wz� 13:68w

þ 17:53x� 5:83y� 5:83zþ 93:85 ¼ 0

A6 : 0:88ux� 1:63uyþ 0:3uz� 7:03uþ 0:98vxþ 1:06vy

� 1:47vzþ 7:54v� 1:34wxþ 0:67wyþ 0:67wz� 8:78w

þ 13:5x� 0:76y� 0:76zþ 45:84 ¼ 0

A7 : 0:36ux� 1:05uyþ 0:44uz� 4:08uþ 0:99vxþ 0:3vy

� 0:37vzþ 1:11v� 0:57wxþ 0:09wyþ 0:09wz� 2:32w

þ 4:56xþ 1:11yþ 1:11zþ 5:83 ¼ 0 (12)

The solution of these equations yields four real sets of coordinatesfor B1¼ (x, y, z) and C¼ (u, v, w), listed in Table 5.

Substitute the precision points into the design equations Bi; j ¼2;…; 7 for the link EF. The result is the following set of designequations:

Fig. 3 The wing swing and wing pitch functions recommendedby Yan et al. [3]

Table 3 Seven precision points (radians) selected from thejoint trajectories of the RRR spatial chain to design the SSdyads

Point qj h1 h2 h3

1 0. 0.52 1.572 0.88 0.99 0.813 1.72 1.88 0.564 2.69 2.53 1.045 3.56 2.48 2.036 4.44 1.76 2.587 5.44 0.94 2.32

Table 4 The difference Dhi between the resulting input, swingand pitch angles, and the desired function values at each of theprecision points (radians)

Point qj Dh1 Dh2 Dh3

1 0.0 0.0 0.02 �0.014 0.073 0.0543 �0.072 0.079 0.0084 0.002 0.020 �0.0755 �0.027 �0.030 0.0016 �0.044 �0.041 �0.0087 0.051 0.023 �0.074

Fig. 4 The precision points selected for the synthesis of theswing and pitch linkages. The precision points are shiftedslightly from the required curves in the design process.

Table 5 Real-valued solutions to design equations Aj for BC

u v w x y z

1 40.08 �50.05 298.50 �4.94 7.53 2.252 0.00 0.00 234.83 �3.00 93.91 �86.833 �1.54 0.06 2.65 �4.47 0.53 5.994 0.00 0.00 3.70 �3.00 9.29 �2.22

Fig. 2 The link OA of the RRR-2SS linkage is held fixed so theinterconnected cranks supporting the joints C and F simultane-ously drive the links AD and DE

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B2 : 1:96mp� 1:94mqþ 0:5mr þ 4:41mþ 1:79npþ 2:19nq

þ 0:87nr � 0:66nþ 0:89op� 0:46oqþ 0:27or þ 2:36o

þ 0:38p� 4:87q� 1:23r þ 6:36 ¼ 0

B3 : �0:0072npþ 3:95mp� 0:38mqþ 0:2mr þ 10:71m

þ 2:92nqþ 1:78nr � 8:61nþ 0:43opþ 1:73oqþ 1:1or

þ 2:62oþ 11:05p� 3:72q� 7:72r þ 48:91 ¼ 0

B4 : 2:19mpþ 1:91mq� 0:57mr þ 16:58m� 0:4npþ 2:59nq

þ 1:87nr � 9:74n� 1:95opþ 0:06oqþ 1:57or � 3:58o

þ 6:97pþ 12:83q� 13:01r þ 95:64 ¼ 0

B5 : 1:15mpþ 0:38mq� 1:77mr þ 16:75m� 1:81npþ 1:78nq

þ 0:82nr � 3:4nþ 0:04op� 1:95oqþ 1:56or � 8:91o

� 4:23pþ 12:24q� 14:27r þ 92:86 ¼ 0

B6 : 2:47mpþ 0:71mq� 1:81mr þ 9:85m� 1:39npþ 0:83nq

� 0:82nr þ 3:13nþ 1:35op� 1:46oqþ 1:78or � 7:62o

� 5:02pþ 7:19q� 9:38r þ 41:21 ¼ 0

B7 : 1:36mpþ 1:51mq� 1:14mr þ 3:98m� 1:23npþ 0:75nq

� 0:96nr þ 2:09nþ 1:44op� 0:4oqþ 0:67or � 0:35o

� 2:82pþ 1:77q� 3:05r þ 5:09 ¼ 0 (13)

The solution to these equations yields four sets of coordinatesfor E¼ (m, n, o) and F¼ (p, q, r) listed in Table 6.

The two sets of four solutions to the design equations can becombined to define 16 candidate linkages that are analyzed to ver-ify performance.

6 Analysis of RRR-2SS Mechanism

The flapping wing mechanism consists of two loops: (i) oneformed by OABC that forms an RRSS closed chain and (ii) theother formed by OADEF is an RRRSS closed chain. Once thetwo SS dyads are determined, then the coordinates of B1 C and E1

F are known, as are the lengths h ¼ jB1Cj and k ¼ jE1Fj. Themovement of the system is determined by specifying the inputangle h1 and solving the loop constraint equations for h2 and h3.

The constraint equation for the loop OABC is given by

ðR1qðDh1;Dh2ÞB1 � CÞ � ðR1qðDh1;Dh2ÞB1 � CÞ ¼ h2 (14)

where

R1qðDh1;Dh2Þ ¼ T2ðh1; h2ÞT2ðh11; h21Þ�1(15)

and Dh1¼ h1� h11 and Dh2¼ h2� h21. Expand this equation to obtain

AðDh1Þcos Dh2 þ BðDh1Þsin Dh2 ¼ CðDh1Þ (16)

where the coefficients are given as listed in Appendix equation(A1). The solution to this equation is given by

Dh2 ¼ arctanB

A6arccos

CffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2 þ B2p (17)

The constraint equation for the loop OADEF is given by

ðS1qðDh1;Dh2;Dh3ÞE1 � FÞ � ðS1qðDh1;Dh2;Dh3ÞE1 � FÞ ¼ k2

(18)

where

S1qðDh1;Dh2;Dh3Þ ¼ T3ðh1; h2; h3ÞT3ðh11:h21; h31Þ�1(19)

and Dh3¼ h3� h31. Expand this equation to obtain

DðDh1;Dh2Þcos Dh3 þ EðDh1;Dh2Þsin Dh3 ¼ FðDh1;Dh2Þ (20)

where the coefficients are given in Appendix equation (A2). Thesolution to this equation is

Dh3 ¼ arctanE

D6arccos

FffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2 þ E2p (21)

For each value of h1, Eq. (17) yields two values for h2, whichwe denote as hþ2 and h�2 . Similarly, Eq. (21) yields two values hþ3and h�3 . Therefore, we obtain four joint trajectories

q1 ¼ ðh1; hþ2 ; h

þ3 Þ; q2 ¼ ðh1; h

þ2 ; h

�3 Þ;

q3 ¼ ðh1; h�2 ; h

þ3 Þ; and q4 ¼ ðh1; h

�2 ; h

�3 Þ

(22)

Each of these trajectories is compared to the required task preci-sion points to verify the performance of the RRR-2SS mechanism.

7 Analysis of the Flapping Wing Mechanism

For each of the 16 combinations of solutions for BC and EF,substitute the coordinates into Eqs. (17) and (21) to evaluate themovement of the candidate design for the Flapping Wing Mecha-nism. Of these 16 candidates, only the combinations of solutions 1and 3 for BC and solution 3 for EF yielded linkages that movedsmoothly through all of the precision points.

The candidate selected for this mechanism combines solution 3for BC and solution 3 for EF and is listed in Table 7. Solution 1for BC was eliminated because the length of BC would be over300 cm for a micro-air vehicle that is to have wings on the orderof 30 cm in length

The ability of the flapping wing mechanism to drive the desiredswing and pitch trajectories is demonstrated in Fig. 5.

A geometric model of the flapping wing mechanism is shownin Fig. 6. The wing swing and the wing pitch are driven by anactuator and a gear train system that connect cranks at the top andbottom of the mechanism. Figure 7 is a view from the oppositeside that shows that the two cranks move together to simultane-ously drive the swing and pitch of the wing. A rear view of themechanism shows the perpendicular wing swing and wing pitchaxes at the center of the device.

8 Design Process

The synthesis of the Flapping Wing Mechanism followed theprocess is shown in Fig. 8. It consists of three primary steps thatare repeated as specified by the designer: (i) values for the task arerandomly selected from within the tolerance zones around the pre-cision points specified by the designer; (ii) the design equationsformulated and solved using these task values to identify candi-date designs; then (iii) each design is analyzed to evaluate its per-formance. Successful designs are saved. The tolerance zones werefor this problem 65 deg (60.087 rad) around the precision points

Table 6 Real-valued solutions to design equations Bj for EF

p q r m n o

1 2.80 �3.09 26.22 �4.94 4.76 4.152 �1.68 �0.48 3.65 �3.50 �41.90 �26.313 3.02 �1.06 �6.87 �2.05 3.39 2.534 �3.15 2.43 �10.00 �4.63 4.19 4.25

Table 7 The links selected for the flapping wing mechanism

BC u v w x y z

�1.54 0.06 2.65 �4.47 0.53 5.99

EF p q r m n o

3.02 �1.06 �6.87 �2.05 3.39 2.53

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given in Table 8. The process was iterated 1100 times to obtain 91successful designs.

The successful designs then were ranked by the ratio of thelengths of the longest to shortest links

j ¼ longest link

shortest link(23)

Smaller values of this parameter are considered more preferablefor packaging, and designs with j> 10 were eliminated. Theremaining designs were sorted by the RMS error of their swingand pitch curves compared to the desired swing and pitch curves.

Of the 91 successful designs, 18 designs had a link length ratioof less than 10. The design with the lowest RMS error wasselected with j¼ 8.200 and an RMS error of 0.159. The SS dyadsof the selected design are shown in Table 7.

This design uses a motor to drive links OC and OF, which areconnected by a simple gear train, to drive them at the same veloc-ity as seen in Fig. 7. The wing is attached to link DE, as shown inFig. 6. The pitch of the wing is controlled by link EF, which con-nects the lower gear and the wing, as shown in Fig. 6. Link ABD,shown in Fig. 6, controls the swing of the wing and connects thewing, the structural frame, and link BC. Link BC, shown inFig. 9, connects the upper gear and link ABD.

Fig. 5 The movement of the swing control with link BC andpitch control with link EF

Fig. 6 Geometric model of the RRR-2SS flapping wing mecha-nism. The wing is attached to link DE. The pitch of the wing iscontrolled by link EF, which connects the lower gear and thewing. Link ABD controls the swing of the wing and connectsthe wing, the structural frame, and link BC.

Fig. 7 A motor drives links OC and OF at the same velocity viaa simple gear train. Links OC and OF control swing and pitch,respectively.

Fig. 8 The process used to find successful solutions to thedesign equations involves adjustment of the precision pointswithin user defined tolerance zones. Iteration of this procedureproduces a large number of design candidates.

Table 8 Seven precision points (radians) which define the cen-ter of the search zones

Point qj h1 h2 h3

1 0.00 0.52 1.572 0.90 0.92 0.753 1.80 1.80 0.554 2.69 2.51 1.125 3.59 2.51 2.036 4.49 1.80 2.597 5.39 0.92 2.39

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9 Comparison With Existing Micro-Air Vehicles

The Aerovironment Nano Hummingbird [2] uses a four-barlinkage and cable driven to produce the wing swing movementand relies on flexure of the wing due to aerodynamic drag to pro-duce the pitch. Seshadri et al. [17] and Conn et al. [18] use a pairof phased four-bar linkages to control swing and pitch for each

wing. Plecnik and McCarthy [19] use four six-bar linkages to con-trol the four joints of a serial chain that models the wing gait of ablack-billed magpie.

The flapping wing mechanism presented here provides controlof both swing and pitch with a six-bar linkage for each wing. Thishas many fewer parts than a pair of phased four-bar linkages andis only slightly more complicated than Aerovironments wingmechanism. A prototype is being developed to test its perform-ance, see Figs. 10 and 11.

10 Conclusion

This paper presents a design process for an RRR-2SS spatialmechanism that controls both the swing and the pitch of the wing of amicro-air vehicle, in order to improve the aerodynamics of hoveringflight. The mechanism is obtained by introducing two SS constraininglinks to a designer-specified RRR spatial chain. The result is a flap-ping wing mechanism that achieves seven precision points alongspecified swing and pitch trajectories. The results are demonstratedwith the design of a new flapping wing mechanism.

Acknowledgment

The assistance of Benjamin Liu in preparation of the geometricmodels is gratefully acknowledged.

Funding Data

� Division of Civil, Mechanical and Manufacturing Innovation(Grant No. 1636017).

Appendix: Coefficients for the Analysis Equations

The coefficients for Eq. (16) are listed here

AðDh1Þ ¼ 2a1xþ 2d1y sin a1 � 2ux cos Dh1

þ 2uy cos a1 sin Dh1 � 2vx sin Dh1

� 2vy cos a1 cos Dh1 � 2wy sin a1

BðDh1Þ ¼ �2a1yþ 2d1x sin a1 þ 2ux cos a1 sin Dh1

þ 2uy cos Dh1 � 2vx cos a1 cos Dh1

þ 2vy sin Dh1 � 2wx sin a1

and

CðDh1Þ ¼ v2 þ w2 þ x2 þ y2 þ z2 þ d22 þ u2 � 2d1wþ 2d2z

þ d21 þ a2

1 � b2 � 2uz sin a1 sin Dh1

þ 2 cos a1ðd1 � wÞðd2 þ zÞ � 2a1v sin Dh1

þ cos Dh1ð2v sin a1ðd2 þ zÞ � 2a1uÞ� 2d2u sin a1 sin Dh1 (A1)

The coefficients for Eq. (20) are listed here

DðDh1;Dh2Þ¼2a1mcosDh2þ2a2m�2a1ncosa2 sinDh2

þ2d1msina1 sinDh2þ2d2nsina2

þ2d1nsina1 cosa2 cosDh2þ2d1nsina2 cosa1

þ2mpcosa1 sinDh1 sinDh2�2mpcosDh1 cosDh2

�2mqcosa1 sinDh2 cosDh1

�2mqsinDh1 cosDh2�2mrsina1 sinDh2

�2npsina1 sina2 sinDh1

þ2npcosa1 cosa2 sinDh1 cosDh2

þ2npcosa2 sinDh2 cosDh1

�2nqcosa1 cosa2 cosDh1 cosDh2

þ2nqsina1 sina2 cosDh1þ2nqcosa2 sinDh1 sinDh2

�2nrsina1 cosa2 cosDh2�2nrsina2 cosa1

Fig. 9 The geometric model with the wings attached showingthe rear of the model

Fig. 10 Front view of the physical model without the wings

Fig. 11 The wing swing mechanism of the physical model

011005-6 / Vol. 10, FEBRUARY 2018 Transactions of the ASME

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EðDh1;Dh2Þ¼�2a1mcosa2 sinDh2�2a1ncosDh2�2a2n

þ2d1msina1 cosa2 cosDh2þ2d2msina2

þ2d1msina2 cosa1�2d1nsina1 sinDh2

�2mpsina1 sina2 sinDh1þ2mpcosa2 sinDh2 cosDh1

þ2mpcosa1 cosa2 sinDh1 cosDh2

�2mqcosa1 cosa2 cosDh1 cosDh2þ2mqsina1

�sina2 cosDh1

þ2mqcosa2 sinDh1 sinDh2�2mrsina1 cosa2 cosDh2

�2mrsina2 cosa1�2npcosa1 sinDh1 sinDh2

þ2npcosDh1 cosDh2þ2nqcosa1 sinDh2 cosDh1

þ2nqsinDh1 cosDh2þ2nrsina1 sinDh2

and

FðDh1;Dh2Þ¼2a2d1 sina1 sinDh2þ2a1d3 sina2 sinDh2

þ2cosa1ða2psinDh1 sinDh2�a2qsinDh2 cosDh1

þsina2ðd3þoÞcosDh2ðqcosDh1�psinDh1Þþcosa2ðd3þoÞðd1�rÞ�d2rþd1d2Þþ2a1a2 cosDh2þ2a1osina2 sinDh2�2a1pcosDh1

�2a2pcosDh1 cosDh2�2a1qsinDh1

� 2a2qsinDh1 cosDh2

�2a2rsina1 sinDh2þa21þa2

2�b2

�2d1d3 sina1 sina2 cosDh2�2d1osina1 sina2 cosDh2

þ2cosa2ðd3þoÞðd2�psina1 sinDh1

þ qsina1 cosDh1Þþ2d3o�2d2psina1 sinDh1

�2d3psina2 sinDh2 cosDh1

�2d3qsina2 sinDh1 sinDh2þ2d2qsina1 cosDh1

þ2d3rsina1 sina2 cosDh2�2opsina2 sinDh2 cosDh1

�2oqsina2 sinDh1 sinDh2þ2orsina1 sina2 cosDh2

þp2þq2þr2�2d1rþd21þd2

2þd23þm2þn2þo2

(A2)

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Journal of Mechanisms and Robotics FEBRUARY 2018, Vol. 10 / 011005-7

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