Synthesis of a Flapping Wing Mechanism Using a Constrained Spatial RRR Serial Chain

Wang, Peter L.; Taha, Haithem E.; Michael McCarthy, J.

doi:10.1115/1.4038529

This paper designs a one degree-of-freedom (1DOF) spatial flapping wing mechanism for a hovering micro-air vehicle by constraining a spatial RRR serial chain using two SS dyads. The desired wing movement defines the dimensions and joint trajectories of the RRR spatial chain. Seven configurations of the chain are selected to define seven precision points that are used to compute SS chains that control the swing and pitch joint angles. The result is a spatial RRR-2SS flapping wing mechanism that transforms the actuator rotation into control of wing swing and pitch necessary for hovering flight of a micro-air vehicle.

Introduction

Flapping-wing micro-air vehicles are designed to fly like birds or insects that exploit unconventional aerodynamic mechanisms to generate high lift at low speeds and to stabilize their bodies in flight [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 wing movement improves the aerodynamics of a micro-air vehicle. They demonstrate that increased aerodynamic performance can be achieved by coordinating the swing and pitch control functions. In this paper, we present a design methodology that yields a one degree-of-freedom (1DOF) mechanism that coordinates these two functions.

This methodology begins with a designer specified spatial serial chain formed by three revolute, or hinged, joints denoted RRR chain—R denotes a revolute joint. The input that drives the base joint θ₁ and output wing swing and pitch angles, θ₂ and θ₃, are the second two joints. The RRR spatial chain is constrained introducing two SS dyads—S denotes a spherical or ball joint—to connect the second and third links to the ground frame, Fig. 1. The resulting one degree-of-freedom spatial RRR-2SS linkage coordinates the wing swing and pitch to the input.

Fig. 1

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The spatial RRR-2SS linkage constructed by constraining a spatial RRR serial chain using two SS dyads that connect the second and third links to the ground frame

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

Literature Review

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

Our flapping wing mechanism must have two output angles for each given input angle, one for swing and one for pitch. This requires a design theory for coordinating three angles in two-loop spatial linkage system. Sandor et al. [9] and Chiang [10] present design methodologies for two-loop spatial chains, but instead of coordinating joint angles they guide an end-effector along a specified trajectory. Similarly, Chung [11] constructs a two-loop spatial mechanism 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 not have the RRR spatial serial chain that characterizes our system. Hauenstein et al. [13] focus on the design of spatial RRR serial chains to guide an end-effector along a desired trajectory, however, 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 spatial chain. This can be viewed as a spatial version of the synthesis of planar six-bar linkages by adding RR constraints to a planar RRR serial [14]. The synthesis of SS dyads that provide the constraining links for the spatial six-bar linkage is presented by Innocenti [15] and McCarthy and Soh [16].

The Spatial RRR Chain

The design of the RRR-2SS spatial six-bar linkage begins with the selection of a spatial RRR serial chain. The Denavit Hartenberg frames are shown in Fig. 1 and the DH parameters are shown in Table 1, where θ_i, i = 1, 2, 3, are joint variables and d_i, i = 1, 2, 3, α_i, i = 1, 2, and a_i, i = 1, 2, are specified by the designer.

Table 1

Denavit–Hartenberg table for the RRR serial chain. θ_i, i = 1, 2, 3 are joint variables. The remaining parameters are selected by the designer.

Link i	θ_i	d_i	α_i	a_i
1	θ₁	d₁	α₁	a₁
2	θ₂	d₂	α₂	a₂
3	θ₃	d₃	—	—

Let L_i, i = 1, 2, 3, be the coordinate frames with the z-axis along the ith joint and the x-axis along the common normal to the next joint axis. The position of these links relative to the ground frame is defined by the kinematics equations

\begin{matrix} T_{1} = Z (θ_{1}, d_{1}) \\ T_{2} = Z (θ_{1}, d_{1}) X (α_{1}, a_{1}) Z (θ_{2}, d_{2}) \\ T_{3} = Z (θ_{1}, d_{1}) X (α_{1}, a_{1}) Z (θ_{2}, d_{2}) X (α_{2}, a_{2}) Z (θ_{3}, d_{3}) \end{matrix}

(1)

where Z(θ_i, d_i) and X(α_i, a_i) are the 4 × 4 homogeneous transforms

\begin{matrix} Z (θ_{i}, d_{i}) = [\begin{matrix} cos θ_{i} & - sin θ_{i} & 0 & 0 \\ sin θ_{i} & cos θ_{i} & 0 & 0 \\ 0 & 0 & 1 & d_{i} \\ 0 & 0 & 0 & 1 \end{matrix}] \\ X (α_{i}, a_{i}) = [\begin{matrix} 1 & 0 & 0 & a_{i} \\ 0 & cos α_{i} & - sin α_{i} & 0 \\ 0 & sin α_{i} & cos α_{i} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], i = 1, 2, 3 \end{matrix}

(2)

These kinematics equations are used in the design procedure.

SS Constraint Synthesis

Let the desired movement of the RRR serial chain be defined by the joint trajectories θ₁(t), θ₂(t), and θ₃(t), which are known functions of a parameter t. In order to design a one degree-of-freedom system that approximates this movement, select seven precision points from these joint trajectories and denote them as q_i = (θ₁_j, θ₂_j, θ₃_j), j = 1,…, 7. Two SS constraints can be calculated 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 the ground frame F, and another SS dyad EF that connects L₃ to the ground frame.

Let B^j denote the coordinates of the moving pivot attached to L₂ measured in F, when the RRR chain is in the configuration defined by q_j. Similarly, let E^j be the coordinates of the moving pivot attached to L₃ for each of the precision positions q_j. Introducing the relative displacements

R_{1 j} = T_{2} (q_{j}) T_{2} {(q_{1})}^{- 1} and S_{1 j} = T_{3} (q_{j}) T_{3} {(q_{1})}^{- 1}

(3)

we have

B^{j} = R_{1 j} B^{1} and E^{j} = S_{1 j} E^{1}

(4)

The coordinates for the SS dyad BC must satisfy the constraint equations

(R_{1 j} B^{1} - C) \cdot (R_{1 j} B^{1} - C) = h^{2}, j = 1, \dots, 7

(5)

where

B^{1} = (x, y, z), C = (u, v, w)

(6)

Similarly, the coordinates for the SS dyad EF must satisfy

(S_{1 j} E^{1} - F) \cdot (S_{1 j} E^{1} - F) = k^{2}, j = 1, \dots, 7

(7)

where

E^{1} = (m, n, o), F = (p, q, r)

(8)

where h and k are the lengths of BC and EF, respectively.

Both Eqs. and can be simplified by subtracting the first equation in the set from the remaining six equations. This cancels the constants h² and k² as well as the squared terms for all 12 coordinates of BC and EF. The result is the two sets of design equations

\begin{matrix} A_{j} : (R_{1 j} B^{1} - C) \cdot (R_{1 j} B^{1} - C) - (B^{1} - C) \cdot (B^{1} - C) = 0, \\ j = 2, \dots, 7 \end{matrix}

(9)

and

\begin{matrix} B_{j} : (S_{1 j} E^{1} - F) \cdot (S_{1 j} E^{1} - F) - (E^{1} - F) \cdot (E^{1} - F) = 0, \\ j = 2, \dots, 7 \end{matrix}

(10)

The equations $A_{j}$ and $B_{j}$ are each bilinear in their respective six unknown coordinates for BC and EF. They can be solved independently to determine as many as 20 SS dyads, which is as many as 400 pairs of SS dyads that guide the RRR serial chain through the seven precision points, q_j, j = 1,…, 7 [15,16].

Flapping Wing Mechanism

In order to design the flapping wing mechanism, we start with the RRR serial chain defined in Table 2. These values were chosen by the designer to fit the workspace and packaging requirements. The RRR chain matches the scale of a hummingbird as seen in Aerovironment's Nano Hummingbird [2] and the motor is oriented such that it fits within the body of the bird.

Table 2

Denavit–Hartenberg table for the RRR serial chain of the flapping wing mechanism

Joint	θ_i	d_i (cm)	α_i (deg)	a_i (cm)
1	θ₁	7.07	45	−3.00
2	θ₂	−5.00	90	−1.00
3	θ₃	2.00	—	—

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

\begin{matrix} θ_{1} = t \\ θ_{2} = \frac{π}{2} - \frac{π}{3} cos t \\ θ_{3} = \frac{π}{2} - \frac{π}{3} sin t \end{matrix}

(11)

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

Fig. 2

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The link OA of the RRR-2SS linkage is held fixed so the interconnected cranks supporting the joints C and F simultaneously drive the links AD and DE

Fig. 3

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The wing swing and wing pitch functions recommended by Yan et al. [3]

Fig. 4

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The precision points selected for the synthesis of the swing and pitch linkages. The precision points are shifted slightly from the required curves in the design process.

Table 3

Seven precision points (radians) selected from the joint trajectories of the RRR spatial chain to design the SS dyads

Point q_j	θ₁	θ₂	θ₃
1	0.	0.52	1.57
2	0.88	0.99	0.81
3	1.72	1.88	0.56
4	2.69	2.53	1.04
5	3.56	2.48	2.03
6	4.44	1.76	2.58
7	5.44	0.94	2.32

Table 4

The difference Δθ_i between the resulting input, swing and pitch angles, and the desired function values at each of the precision points (radians)

Point q_j	Δθ₁	Δθ₂	Δθ₃
1	0.0	0.0	0.0
2	−0.014	0.073	0.054
3	−0.072	0.079	0.008
4	0.002	0.020	−0.075
5	−0.027	−0.030	0.001
6	−0.044	−0.041	−0.008
7	0.051	0.023	−0.074

Substitute the precision points into the design equations

A_{j}, j = 2, \dots, 7

for the link BC. The result is the following set of design equations:

\begin{matrix} A_{2} : 1.36 u x + 1.87 u y + 0.32 u z - 0.39 u - 1.78 v x + 1.29 v y \\ + 0.56 v z - 4.68 v - 0.64 w x + 0.11 w y + 0.11 w z - 2.67 w \\ + 5.15 x + 1.15 y + 1.15 z + 7.30 = 0 A_{3} : 3.43 u x + 0.99 u y - 0.99 u z + 10.39 u - 0.21 v x + 3.55 v y \\ + 1.25 v z - 3.5 v - 1.38 w x + 0.79 w y + 0.79 w z - 9.74 w \\ + 14.52 x - 1.44 y - 1.44 z + 53.74 = 0 A_{4} : 1.79 u x - 0.91 u y - 1.77 u z + 6.46 u + 1.52 v x + 3.07 v y \\ - 0.73 v z + 12.34 v - 1.28 w x + 1.43 w y + 1.43 w z - 13.92 w \\ + 17.6 x - 6.24 y - 6.24 z + 96.94 = 0 A_{5} : 0.77 u x - 1.45 u y - 0.63 u z - 4.72 u + 0.88 v x + 2.03 v y \\ - 1.79 v z + 12.88 v - 1.31 w x + 1.38 w y + 1.38 w z - 13.68 w \\ + 17.53 x - 5.83 y - 5.83 z + 93.85 = 0 A_{6} : 0.88 u x - 1.63 u y + 0.3 u z - 7.03 u + 0.98 v x + 1.06 v y \\ - 1.47 v z + 7.54 v - 1.34 w x + 0.67 w y + 0.67 w z - 8.78 w \\ + 13.5 x - 0.76 y - 0.76 z + 45.84 = 0 A_{7} : 0.36 u x - 1.05 u y + 0.44 u z - 4.08 u + 0.99 v x + 0.3 v y \\ - 0.37 v z + 1.11 v - 0.57 w x + 0.09 w y + 0.09 w z - 2.32 w \\ + 4.56 x + 1.11 y + 1.11 z + 5.83 = 0 \end{matrix}

(12)

The solution of these equations yields four real sets of coordinates for B¹ = (x, y, z) and C = (u, v, w), listed in Table 5.

Table 5

Real-valued solutions to design equations $A_{j}$ for BC

	u	v	w	x	y	z
1	40.08	−50.05	298.50	−4.94	7.53	2.25
2	0.00	0.00	234.83	−3.00	93.91	−86.83
3	−1.54	0.06	2.65	−4.47	0.53	5.99
4	0.00	0.00	3.70	−3.00	9.29	−2.22

Substitute the precision points into the design equations

B_{i}, j = 2, \dots, 7

for the link EF. The result is the following set of design equations:

\begin{matrix} B_{2} : 1.96 m p - 1.94 m q + 0.5 m r + 4.41 m + 1.79 n p + 2.19 n q \\ + 0.87 n r - 0.66 n + 0.89 o p - 0.46 o q + 0.27 o r + 2.36 o \\ + 0.38 p - 4.87 q - 1.23 r + 6.36 = 0 B_{3} : - 0.0072 n p + 3.95 m p - 0.38 m q + 0.2 m r + 10.71 m \\ + 2.92 n q + 1.78 n r - 8.61 n + 0.43 o p + 1.73 o q + 1.1 o r \\ + 2.62 o + 11.05 p - 3.72 q - 7.72 r + 48.91 = 0 B_{4} : 2.19 m p + 1.91 m q - 0.57 m r + 16.58 m - 0.4 n p + 2.59 n q \\ + 1.87 n r - 9.74 n - 1.95 o p + 0.06 o q + 1.57 o r - 3.58 o \\ + 6.97 p + 12.83 q - 13.01 r + 95.64 = 0 B_{5} : 1.15 m p + 0.38 m q - 1.77 m r + 16.75 m - 1.81 n p + 1.78 n q \\ + 0.82 n r - 3.4 n + 0.04 o p - 1.95 o q + 1.56 o r - 8.91 o \\ - 4.23 p + 12.24 q - 14.27 r + 92.86 = 0 B_{6} : 2.47 m p + 0.71 m q - 1.81 m r + 9.85 m - 1.39 n p + 0.83 n q \\ - 0.82 n r + 3.13 n + 1.35 o p - 1.46 o q + 1.78 o r - 7.62 o \\ - 5.02 p + 7.19 q - 9.38 r + 41.21 = 0 B_{7} : 1.36 m p + 1.51 m q - 1.14 m r + 3.98 m - 1.23 n p + 0.75 n q \\ - 0.96 n r + 2.09 n + 1.44 o p - 0.4 o q + 0.67 o r - 0.35 o \\ - 2.82 p + 1.77 q - 3.05 r + 5.09 = 0 \end{matrix}

(13)

The solution to these equations yields four sets of coordinates for E = (m, n, o) and F = (p, q, r) listed in Table 6.

Table 6

Real-valued solutions to design equations $B_{j}$ for EF

	p	q	r	m	n	o
1	2.80	−3.09	26.22	−4.94	4.76	4.15
2	−1.68	−0.48	3.65	−3.50	−41.90	−26.31
3	3.02	−1.06	−6.87	−2.05	3.39	2.53
4	−3.15	2.43	−10.00	−4.63	4.19	4.25

The two sets of four solutions to the design equations can be combined to define 16 candidate linkages that are analyzed to verify performance.

Analysis of RRR-2SS Mechanism

The flapping wing mechanism consists of two loops: (i) one formed by OABC that forms an RRSS closed chain and (ii) the other formed by OADEF is an RRRSS closed chain. Once the two SS dyads are determined, then the coordinates of B¹C and E¹F are known, as are the lengths $h = | B^{1} C |$ and $k = | E^{1} F |$ ⁠. The movement of the system is determined by specifying the input angle θ₁ and solving the loop constraint equations for θ₂ and θ₃.

The constraint equation for the loop OABC is given by

(R_{1 q} (Δ θ_{1}, Δ θ_{2}) B^{1} - C) \cdot (R_{1 q} (Δ θ_{1}, Δ θ_{2}) B^{1} - C) = h^{2}

(14)

where

R_{1 q} (Δ θ_{1}, Δ θ_{2}) = T_{2} (θ_{1}, θ_{2}) T_{2} {(θ_{11}, θ_{21})}^{- 1}

(15)

and Δθ₁ = θ₁ − θ₁₁ and Δθ₂ = θ₂ − θ₂₁. Expand this equation to obtain

A (Δ θ_{1}) cos Δ θ_{2} + B (Δ θ_{1}) sin Δ θ_{2} = C (Δ θ_{1})

(16)

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

Δ θ_{2} = \arctan \frac{B}{A} \pm \arccos \frac{C}{\sqrt{A^{2} + B^{2}}}

(17)

The constraint equation for the loop OADEF is given by

(S_{1 q} (Δ θ_{1}, Δ θ_{2}, Δ θ_{3}) E^{1} - F) \cdot (S_{1 q} (Δ θ_{1}, Δ θ_{2}, Δ θ_{3}) E^{1} - F) = k^{2}

(18)

where

S_{1 q} (Δ θ_{1}, Δ θ_{2}, Δ θ_{3}) = T_{3} (θ_{1}, θ_{2}, θ_{3}) T_{3} {(θ_{11} . θ_{21}, θ_{31})}^{- 1}

(19)

and Δθ₃ = θ₃ − θ₃₁. Expand this equation to obtain

D (Δ θ_{1}, Δ θ_{2}) cos Δ θ_{3} + E (Δ θ_{1}, Δ θ_{2}) sin Δ θ_{3} = F (Δ θ_{1}, Δ θ_{2})

(20)

where the coefficients are given in Appendix equation . The solution to this equation is

Δ θ_{3} = \arctan \frac{E}{D} \pm \arccos \frac{F}{\sqrt{D^{2} + E^{2}}}

(21)

For each value of θ₁, Eq. yields two values for θ₂, which we denote as

θ_{2}^{+}

and

θ_{2}^{-}

⁠. Similarly, Eq. yields two values

θ_{3}^{+}

and

θ_{3}^{-}

⁠. Therefore, we obtain four joint trajectories

\begin{matrix} q^{1} = (θ_{1}, θ_{2}^{+}, θ_{3}^{+}), q^{2} = (θ_{1}, θ_{2}^{+}, θ_{3}^{-}), \\ q^{3} = (θ_{1}, θ_{2}^{-}, θ_{3}^{+}), and q^{4} = (θ_{1}, θ_{2}^{-}, θ_{3}^{-}) \end{matrix}

(22)

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

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 the movement of the candidate design for the Flapping Wing Mechanism. Of these 16 candidates, only the combinations of solutions 1 and 3 for BC and solution 3 for EF yielded linkages that moved smoothly through all of the precision points.

The candidate selected for this mechanism combines solution 3 for BC and solution 3 for EF and is listed in Table 7. Solution 1 for BC was eliminated because the length of BC would be over 300 cm for a micro-air vehicle that is to have wings on the order of 30 cm in length

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

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

Fig. 5

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The movement of the swing control with link BC and pitch control with link EF

A geometric model of the flapping wing mechanism is shown in Fig. 6. The wing swing and the wing pitch are driven by an actuator and a gear train system that connect cranks at the top and bottom of the mechanism. Figure 7 is a view from the opposite side that shows that the two cranks move together to simultaneously drive the swing and pitch of the wing. A rear view of the mechanism shows the perpendicular wing swing and wing pitch axes at the center of the device.

Fig. 6

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Geometric model of the RRR-2SS flapping wing mechanism. The wing is attached to link DE. The pitch of the wing is controlled by link EF, which connects the lower gear and the wing. Link ABD controls the swing of the wing and connects the wing, the structural frame, and link BC.

Fig. 7

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

Design Process

The synthesis of the Flapping Wing Mechanism followed the process is shown in Fig. 8. It consists of three primary steps that are repeated as specified by the designer: (i) values for the task are randomly selected from within the tolerance zones around the precision points specified by the designer; (ii) the design equations formulated and solved using these task values to identify candidate designs; then (iii) each design is analyzed to evaluate its performance. Successful designs are saved. The tolerance zones were for this problem ±5 deg (±0.087 rad) around the precision points given in Table 8. The process was iterated 1100 times to obtain 91 successful designs.

Fig. 8

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The process used to find successful solutions to the design equations involves adjustment of the precision points within user defined tolerance zones. Iteration of this procedure produces a large number of design candidates.

Table 8

Seven precision points (radians) which define the center of the search zones

Point q_j	θ₁	θ₂	θ₃
1	0.00	0.52	1.57
2	0.90	0.92	0.75
3	1.80	1.80	0.55
4	2.69	2.51	1.12
5	3.59	2.51	2.03
6	4.49	1.80	2.59
7	5.39	0.92	2.39

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

κ = \frac{longest link}{shortest link}

(23)

Smaller values of this parameter are considered more preferable for packaging, and designs with κ > 10 were eliminated. The remaining designs were sorted by the RMS error of their swing and pitch curves compared to the desired swing and pitch curves.

Of the 91 successful designs, 18 designs had a link length ratio of less than 10. The design with the lowest RMS error was selected with κ = 8.200 and an RMS error of 0.159. The SS dyads of the selected design are shown in Table 7.

This design uses a motor to drive links OC and OF, which are connected by a simple gear train, to drive them at the same velocity as seen in Fig. 7. The wing is attached to link DE, as shown in Fig. 6. The pitch of the wing is controlled by link EF, which connects 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 the wing, the structural frame, and link BC. Link BC, shown in Fig. 9, connects the upper gear and link ABD.

Fig. 9

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The geometric model with the wings attached showing the rear of the model

Comparison With Existing Micro-Air Vehicles

The Aerovironment Nano Hummingbird [2] uses a four-bar linkage and cable driven to produce the wing swing movement and relies on flexure of the wing due to aerodynamic drag to produce the pitch. Seshadri et al. [17] and Conn et al. [18] use a pair of phased four-bar linkages to control swing and pitch for each wing. Plecnik and McCarthy [19] use four six-bar linkages to control the four joints of a serial chain that models the wing gait of a black-billed magpie.

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

Fig. 10

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Front view of the physical model without the wings

Fig. 11

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The wing swing mechanism of the physical model

Conclusion

This paper presents a design process for an RRR-2SS spatial mechanism that controls both the swing and the pitch of the wing of a micro-air vehicle, in order to improve the aerodynamics of hovering flight. The mechanism is obtained by introducing two SS constraining links to a designer-specified RRR spatial chain. The result is a flapping wing mechanism that achieves seven precision points along specified swing and pitch trajectories. The results are demonstrated with the design of a new flapping wing mechanism.

Acknowledgment

The assistance of Benjamin Liu in preparation of the geometric models 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. are listed here

\begin{matrix} A (Δ θ_{1}) = 2 a_{1} x + 2 d_{1} y sin α_{1} - 2 u x cos Δ θ_{1} \\ + 2 u y cos α_{1} sin Δ θ_{1} - 2 v x sin Δ θ_{1} \\ - 2 v y cos α_{1} cos Δ θ_{1} - 2 w y sin α_{1} \end{matrix}

\begin{matrix} B (Δ θ_{1}) = - 2 a_{1} y + 2 d_{1} x sin α_{1} + 2 u x cos α_{1} sin Δ θ_{1} \\ + 2 u y cos Δ θ_{1} - 2 v x cos α_{1} cos Δ θ_{1} \\ + 2 v y sin Δ θ_{1} - 2 w x sin α_{1} \end{matrix}

and

\begin{matrix} C (Δ θ_{1}) = v^{2} + w^{2} + x^{2} + y^{2} + z^{2} + d_{2}^{2} + u^{2} - 2 d_{1} w + 2 d_{2} z \\ + d_{1}^{2} + a_{1}^{2} - b^{2} - 2 u z sin α_{1} sin Δ θ_{1} \\ + 2 cos α_{1} (d_{1} - w) (d_{2} + z) - 2 a_{1} v sin Δ θ_{1} \\ + cos Δ θ_{1} (2 v sin α_{1} (d_{2} + z) - 2 a_{1} u) \\ - 2 d_{2} u sin α_{1} sin Δ θ_{1} \end{matrix}

(A1)

The coefficients for Eq. are listed here

\begin{matrix} D (Δ θ_{1}, Δ θ_{2}) = 2 a_{1} m cos Δ θ_{2} + 2 a_{2} m - 2 a_{1} n cos α_{2} sin Δ θ_{2} \\ + 2 d_{1} m sin α_{1} sin Δ θ_{2} + 2 d_{2} n sin α_{2} \\ + 2 d_{1} n sin α_{1} cos α_{2} cos Δ θ_{2} + 2 d_{1} n sin α_{2} cos α_{1} \\ + 2 m p cos α_{1} sin Δ θ_{1} sin Δ θ_{2} - 2 m p cos Δ θ_{1} cos Δ θ_{2} \\ - 2 m q cos α_{1} sin Δ θ_{2} cos Δ θ_{1} \\ - 2 m q sin Δ θ_{1} cos Δ θ_{2} - 2 m r sin α_{1} sin Δ θ_{2} \\ - 2 n p sin α_{1} sin α_{2} sin Δ θ_{1} \\ + 2 n p cos α_{1} cos α_{2} sin Δ θ_{1} cos Δ θ_{2} \\ + 2 n p cos α_{2} sin Δ θ_{2} cos Δ θ_{1} \\ - 2 n q cos α_{1} cos α_{2} cos Δ θ_{1} cos Δ θ_{2} \\ + 2 n q sin α_{1} sin α_{2} cos Δ θ_{1} + 2 n q cos α_{2} sin Δ θ_{1} sin Δ θ_{2} \\ - 2 n r sin α_{1} cos α_{2} cos Δ θ_{2} - 2 n r sin α_{2} cos α_{1} \end{matrix}

\begin{matrix} E (Δ θ_{1}, Δ θ_{2}) = - 2 a_{1} m cos α_{2} sin Δ θ_{2} - 2 a_{1} n cos Δ θ_{2} - 2 a_{2} n \\ + 2 d_{1} m sin α_{1} cos α_{2} cos Δ θ_{2} + 2 d_{2} m sin α_{2} \\ + 2 d_{1} m sin α_{2} cos α_{1} - 2 d_{1} n sin α_{1} sin Δ θ_{2} \\ - 2 m p sin α_{1} sin α_{2} sin Δ θ_{1} + 2 m p cos α_{2} sin Δ θ_{2} cos Δ θ_{1} \\ + 2 m p cos α_{1} cos α_{2} sin Δ θ_{1} cos Δ θ_{2} \\ - 2 m q cos α_{1} cos α_{2} cos Δ θ_{1} cos Δ θ_{2} + 2 m q sin α_{1} sin α_{2} cos Δ θ_{1} \\ + 2 m q cos α_{2} sin Δ θ_{1} sin Δ θ_{2} - 2 m r sin α_{1} cos α_{2} cos Δ θ_{2} \\ - 2 m r sin α_{2} cos α_{1} - 2 n p cos α_{1} sin Δ θ_{1} sin Δ θ_{2} \\ + 2 n p cos Δ θ_{1} cos Δ θ_{2} + 2 n q cos α_{1} sin Δ θ_{2} cos Δ θ_{1} \\ + 2 n q sin Δ θ_{1} cos Δ θ_{2} + 2 n r sin α_{1} sin Δ θ_{2} \end{matrix}

and

\begin{matrix} F (Δ θ_{1}, Δ θ_{2}) = 2 a_{2} d_{1} sin α_{1} sin Δ θ_{2} + 2 a_{1} d_{3} sin α_{2} sin Δ θ_{2} \\ + 2 cos α_{1} (a_{2} p sin Δ θ_{1} sin Δ θ_{2} - a_{2} q sin Δ θ_{2} cos Δ θ_{1} \\ + sin α_{2} (d_{3} + o) cos Δ θ_{2} (q cos Δ θ_{1} - p sin Δ θ_{1}) \\ + cos α_{2} (d_{3} + o) (d_{1} - r) - d_{2} r + d_{1} d_{2}) \\ + 2 a_{1} a_{2} cos Δ θ_{2} + 2 a_{1} o sin α_{2} sin Δ θ_{2} - 2 a_{1} p cos Δ θ_{1} \\ - 2 a_{2} p cos Δ θ_{1} cos Δ θ_{2} - 2 a_{1} q sin Δ θ_{1} - 2 a_{2} q sin Δ θ_{1} cos Δ θ_{2} \\ - 2 a_{2} r sin α_{1} sin Δ θ_{2} + a_{1}^{2} + a_{2}^{2} - b^{2} \\ - 2 d_{1} d_{3} sin α_{1} sin α_{2} cos Δ θ_{2} - 2 d_{1} o sin α_{1} sin α_{2} cos Δ θ_{2} \\ + 2 cos α_{2} (d_{3} + o) (d_{2} - p sin α_{1} sin Δ θ_{1} + q sin α_{1} cos Δ θ_{1}) \\ + 2 d_{3} o - 2 d_{2} p sin α_{1} sin Δ θ_{1} - 2 d_{3} p sin α_{2} sin Δ θ_{2} cos Δ θ_{1} \\ - 2 d_{3} q sin α_{2} sin Δ θ_{1} sin Δ θ_{2} + 2 d_{2} q sin α_{1} cos Δ θ_{1} \\ + 2 d_{3} r sin α_{1} sin α_{2} cos Δ θ_{2} - 2 o p sin α_{2} sin Δ θ_{2} cos Δ θ_{1} \\ - 2 o q sin α_{2} sin Δ θ_{1} sin Δ θ_{2} + 2 o r sin α_{1} sin α_{2} cos Δ θ_{2} \\ + p^{2} + q^{2} + r^{2} - 2 d_{1} r + d_{1}^{2} + d_{2}^{2} + d_{3}^{2} + m^{2} + n^{2} + o^{2} \end{matrix}

(A2)

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Synthesis of a Flapping Wing Mechanism Using a Constrained Spatial RRR Serial Chain

Introduction

Literature Review

The Spatial RRR Chain

SS Constraint Synthesis

Flapping Wing Mechanism

Analysis of RRR-2SS Mechanism

Analysis of the Flapping Wing Mechanism

Design Process

Comparison With Existing Micro-Air Vehicles

Conclusion

Acknowledgment

Funding Data

Appendix: Coefficients for the Analysis Equations

References

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Synthesis of a Flapping Wing Mechanism Using a Constrained Spatial RRR Serial Chain

Introduction

Literature Review

The Spatial RRR Chain

SS Constraint Synthesis

Flapping Wing Mechanism

Analysis of RRR-2SS Mechanism

Analysis of the Flapping Wing Mechanism

Design Process

Comparison With Existing Micro-Air Vehicles

Conclusion

Acknowledgment

Funding Data

Appendix: Coefficients for the Analysis Equations

References

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Cited By

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