ARMWING STRUCTURES FOR AERIAL ROBOTS

Abstract

Robotic wings for an aerial drone include a plurality of armwing structures, each comprising a plurality of rigid members connected together by flexible living hinges in a single monolithic structure. Wing membranes are supported by the armwing structures. A drive mechanism is connected to the armwing structures for articulating the armwing structures. A motor is connected to the drive mechanism for actuating the drive mechanism to move the armwing structures through a series of wingbeats wherein the armwing structures expand in a downstroke and retract in an upstroke to move the wing membranes in a flapping motion.

Claims

1. Robotic wings for an aerial drone, comprising: a plurality of armwing structures, each comprising a plurality of rigid members connected together by flexible living hinges in a single monolithic structure; wing membranes supported by the armwing structures; a drive mechanism connected to the armwing structures for articulating the armwing structures; and a motor connected to the drive mechanism for actuating the drive mechanism to move the armwing structures through a series of wingbeats wherein the armwing structures expand in a downstroke and retract in an upstroke to move the wing membranes in a flapping motion.

2. The robotic wings of claim 1, wherein the rigid members and flexible living hinges comprise different materials, and wherein the armwing structures are formed in an additive manufacturing process.

3. The robotic wings of claim 2, wherein the additive manufacturing process uses a Polyjet 3-D printer.

4. The robotic wings of claim 1, wherein the armwing structures include four-bar linkage mechanisms.

5. The robotic wings of claim 1, wherein each armwing structure includes an upper section with a radius four-bar linkage mechanism connected to the drive mechanism and a lower section with a humerus four-bar linkage mechanism connected to the drive mechanism.

6. The robotic wings of claim 1, wherein each flexible living hinge comprises a joint with a notch design.

7. The robotic wings of claim 1, wherein the drive mechanism comprises a gear and crank mechanism operably coupled to the motor and to the plurality of armwing structures.

8. The robotic wings of claim 1, wherein the armwing structures comprise two sets of crank and four-bar mechanisms articulating linkages representing the radius and the humerus bones, respectively, in a bat's arm.

9. The robotic wings of claim 8, wherein the two sets of crank and four-bar mechanisms are off-plane and parallel to each other.

10. The robotic wings of claim 9, wherein gears of the drive mechanism that drive the two sets of crank and four-bar mechanisms are located in a midpoint of housing to enable a symmetric wing assembly.

11. The robotic wings of claim 1, wherein the motor comprises a single brushless motor.

12. The robotic wings of claim 1, wherein the aerial robot comprises a micro-aerial vehicle.

13. The robotic wings of claim 1, wherein the wing membranes each comprise a flexible printed circuit board.

14. The robotic wings of claim 13, wherein the printed circuit board includes computer processing units and sensors.

15. The robotic wings of claim 14, wherein the sensors perform flow measurement or wing movement measurement.

16. Armwing structures for an aerial robot, each comprising a plurality of rigid members connected together by flexible living hinges in a single monolithic structure, wherein each armwing structure is configured to support a wing membrane, and each armwing structure is configured to be articulated by a drive mechanism driven by a motor such that the armwing structures are moved through a series of wingbeats during which the armwing structures expand in a downstroke and retract in an upstroke to move the wing membranes in a flapping motion.

17. The armwing structures of claim 16, wherein the rigid members and flexible living hinges comprise different materials, and wherein the armwing structures are formed in an additive manufacturing process.

18. The armwing structures of claim 16, wherein the armwing structures include four-bar linkage mechanisms.

19. The armwing structures of claim 16, wherein each armwing structure includes an upper section with a radius four-bar linkage mechanism connected to the drive mechanism and a lower section with a humerus four-bar linkage mechanism connected to the drive mechanism.

20. The armwing structures of claim 16, wherein the armwing structures comprise two sets of crank and four-bar mechanisms articulating linkages representing the radius and the humerus bones, respectively, in a bat's arm.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0016] FIG. 1 is a set of time-lapsed images showing articulation of an exemplary armwing structure for an aerial robot in accordance with one or more embodiments.

[0017] FIGS. 2A, 2B, and 2C show perspective, front, and side views of the armwing structure in accordance with one or more embodiments.

[0018] FIGS. 3A and 3B show a set of links and hinges in an exemplary wing linkage mechanism in accordance with one or more embodiments.

[0019] FIG. 4 shows the dimension parameters for non-straight links of the exemplary wing linkage mechanism in accordance with one or more embodiments.

[0020] FIGS. 5A and 5B are charts showing unoptimized and optimized armwing shoulder and elbow angles, respectively, within a single wingbeat of the exemplary wing linkage mechanism in accordance with one or more embodiments.

[0021] FIGS. 6A and 6B show finite element analysis of the compliant mechanism under static torques acting on the crank arms. The strain values are shown at the right side of the charts.

[0022] FIGS. 7A, 7B, and 7C illustrate armwing elbow joint and wingtip trajectories generated by varying one of the design parameters.

[0023] FIG. 8 illustrates exemplary wing membranes connected to the armwing structure in accordance with one or more embodiments.

[0024] FIGS. 9, 10, and 11 contain Tables I, II, and III, respectively.

DETAILED DESCRIPTION

[0025] Various embodiments disclosed herein extend our prior contributions [21]—[24] by offering kinetic sculpture designs that can capture bat dynamically versatile wing conformations. The disclosed structures comprise rigid and flexible materials that are monolithically fabricated using novel computer-aided fabrication methods and additive manufacturing technology (e.g., PolyJet 3D printing). Like its predecessor, this armwing structure articulation is also designed to expand and retract within a single wingbeat through a series of crank and four-bar mechanisms as it is actuated by a single brushless DC motor. The use of a monolithic rigid and flexible armwing structure in a flying robot is novel and impactful for flapping robot design as this structure is capable of mimicking the range of motion and flexibility of an actual bat armwing. This mechanism design assumes a planar flapping motion and only articulates the wing plunging and extension-retraction gaits. Other modes such as supination-pronation, sweeping motion, and 3D flapping gait are also possible.

[0026] Various embodiments disclosed herein relate to an aerial robot having a novel bio-inspired monolithic bat armwing structure with both flexible and rigid materials. The armwing structure is designed to expand and retract during the wing flapping motion to maximize the net lift produced by the wings. FIGS. 2A, 2B, and 2C show perspective, front, and side views of an exemplary armwing structure 100 in accordance with one or more embodiments. The armwing structure 100 comprises a plurality of rigid members 102 connected together by flexible living hinges 104 in a single monolithic structure.

[0027] FIG. 1 is a set of time-lapsed images showing articulation of the armwing structure in accordance with one or more embodiments. The first two images (1 and 2) show the wing downstroke/expansion motion, while the second two images (3 and 4) show the wing upstroke/retraction motion.

[0028] The following set of design criteria can be used in developing a robotic wing structure that can mimic the speed and flexibility of a natural bat wing: (a) a mechanical structure that mimics as many meaningful degrees-of-freedom (DoF) as possible from the natural bat wing, (b) a robust and flexible wing structure that facilitates control through morphological computation, and (c) a small, lightweight, and compact mechanism. Meaningful DoFs include the plunging motion along with the wing extension/retraction, where the control is facilitated by either changing the wing morphology or by directly articulating the armwing kinetic sculpture.

[0029] A bat wing has up to 34 DoF and unparalleled flexibility [30], which is not feasible to replicate using a rigid mechanical structure in a small and compact form factor. By using flexible joints to form a compliant structure, we can mimic some of the natural bat wing's flexibility and the important DoFs for flapping flight packaged in a very compact mechanical structure. The multi-material printing capability of PolyJet 3D printers allows us to fabricate a monolithic wing structure composed of rigid and flexible materials, which is shown in FIG. 2.

[0030] The wing structure is articulated using a series of cranks and four-bar linkage mechanisms as shown in FIG. 3. This mechanism is preferably actuated with only a single motor, therefore the wing expansion/retraction is a slave to the flapping motion as it is actively actuated by the motor. In one exemplary embodiment, in order to achieve a small and lightweight structure, we constrain our design such that the driving mechanism is contained in a capsule-shaped tube of 50 mm diameter and 75 mm long. The total wingspan of the robot is approximately 300 mm wide which follows a similar form factor as our previous work [23], and we aim for a robot that weighs approximately 20 grams.

[0031] Flexible Hinge Design

[0032] The flexible joints are part of the wing's compliant mechanism. Several design considerations affect the hinge stiffness and robustness. There are several design variations for a compliant joint as outlined in [33], where they vary in size, off-axis stiffness, axis drift, stress concentration, and range of motion. In order to satisfy our design target of a small and lightweight aerial robot, we choose to use the simple planar notch design as shown in FIG. 2. This notch design has a disadvantage in the form of very low off-axis stiffness, particularly the torsional stiffness. The PolyJet flexible materials, in the case of Stratasys PolyJet 3D printers, come in Shore hardness scale range of 30 A to 85 A, which is formed by mixing the rigid Vero White and the flexible Agilus Black [34]. A softer and more flexible material has better compliance, which means that it can safely deform to counteract unexpected forces. However, it will also has worse resistance to torsion and off-axis perturbations.

[0033] The planar four-bar linkage mechanism shown in FIGS. 2A-2C assumes that the structure does not deform in the off-plane directions, so the low off-axis stiffness can be a significant issue. This problem can be addressed by using a larger cross-sectional area or by reinforcing the hinge with a flexible support structure post-fabrication to increase the off-axis stiffness of the hinge. The larger cross-sectional area increases the durability of the hinge, but also increase the overall hinge stiffness and weight which is a design trade-off.

[0034] One exemplary embodiment includes a combination of 1.3 mm and 2 mm hinge thickness with the flexible materials as shown in Table I (FIG. 9), where the material property values are taken from [34]. We use a shaft thickness of 3 mm and depth of 5 mm in the disclosed link designs. The deep shaft design increases the off-axis stiffness of the armwing structure and the durability of the hinge. The three material hardness in Table I capture a wide range of stiffness, which can be a reference point for other design iterations. Each armwing prototypes we fabricated has the same hinge thickness and material within an individual wing. Conversely, it is possible to use varying hinge design and properties depending on each joint's stress and flexibility requirements which we can investigate in our future work.

[0035] Driving Mechanism Design

[0036] The armwing driving mechanism can be separated into two sets of crank and four-bar mechanisms, as shown in FIGS. 3A and 3B, where they articulate the linkages representing the radius and the humerus bones, respectively, in a bat's arm.

[0037] As shown in FIGS. 3A-3B, the wing linkage mechanism comprises 12 links (L.sub.i, i∈{1, . . . , 12}) and 17 joints/living hinges (J.sub.k, k∈{1, . . . , 17}). The crank mechanisms' drive links L.sub.3 and L.sub.8 actuate the shoulder and elbow joints J.sub.7 and J.sub.8, respectively.

[0038] Both crank mechanisms operate at the same frequency but with a different phase Δϕ, which articulates the desired wing extension and retraction during a specific timing within a wingbeat. The monolithic wing structure has 8 links and 11 hinges per wing while the gears and crank mechanism add 4 links and 6 revolute joints per wing, which results in a grand total of 12 links and 17 joints per wing. The mechanism is designed by assuming that the flexible hinges act like an ideal axial joint that follows the parallel linkage mechanism design principles.

[0039] In the exemplary embodiment, the humerus and radius links have a length of 50 mm and 90 mm, respectively, which is based on the conformation of the Rousettus aegyptiacus [35]. This bat flies under a flapping rate of approximately 10 Hz, which we emulate. Due to space constraints, the four-bar mechanisms are placed off-plane and parallel from each other. The gears that drive the four-bar mechanisms are placed in the midpoint of the body so that we can implement a symmetric wing assembly. This way, each side of the wing can utilize the same wing structure and the mechanisms can be connected using a spur gear or other means of power transmission. This configuration results in a horizontally-symmetric but off-plane wing skeletal structure. However, this is not problematic because the wing membranes can be attached in a symmetric fashion.

[0040] Monolithically Fabricated Bat Armwing Structure

[0041] The 3D printed monolithic bat armwing structure, which in the exemplary embodiment, weighs 7 grams, can be seen in FIG. 1 as it is being articulated from the crank arms. This flexible armwing conformation is capable of achieving the desired wing expansion and retraction. We tested several combinations of hinge thickness and materials as listed in Table I. As expected, there is a large variation in the stiffness and ease of articulation for each wing variant. We found out that an armwing with durometer scale of 50 A is simply too soft to be used in the wing articulation and it has very little off-plane and torsional stiffness even in the thicker hinge design. An armwing with the durometer scale of 85 A and 2 mm hinge thickness is too stiff and relatively brittle. However, a wing with 85 A durometer and 1.3 mm hinge thickness has excellent off-plane and torsional stiffness, which can be beneficial as long as we can provide the torque to actuate the stiff armwing. In the end, we conclude that the wings with 1.3 mm hinge thickness and either 70 A or 85 A durometer have the best overall result in the wing articulation and stiffness.

[0042] The mechanism shown in FIGS. 3A-3B was developed by simulating the rigid body linkages in Solidworks Motion Study. This methodology is suffice to discover the initial configuration that works well. However, in order to get the best result, a design optimization framework can be used to search for the best armwing conformation that follows the desired wing gait we specified.

[0043] FIG. 8 illustrates exemplary wing membranes 106 connected to the armwing structures 100 in accordance with one or more embodiments. In this exemplary embodiment, the wing membranes 106 each comprise a flexible printed circuit board (PCB). The PCB includes processing units and sensors. The sensors can be used for various purposes including, e.g., for flow measurement and wing movement measurement.

[0044] Wing Conformation Design Optimization

[0045] The wing mechanism is composed of rigid links and flexible hinges that can be modeled as rigid body linkages with linear and rotational stiffness at the joints, as outlined in [36]. However, fully modeling the flexible joints is very difficult considering the complexity of our design. Therefore, we designed the mechanism and performed our analysis assuming a rigid parallel linkage mechanism. The following outlines the rigid body kinematic formulation and the wing morphology optimization problem to follow a specific flapping trajectory.

[0046] Kinematic Formulation

[0047] Assuming rigid body kinematics, the armwing mechanism has two DoF per wing, which are represented by the two crank arms of the wing (links L.sub.3 and L.sub.8). Since the crank gears are coupled, the assembled wing mechanism only has one DoF, which means that the full system states can be solved from the driving gear angle if the value is known.

[0048] Referring to FIGS. 3A-3B, let θ.sub.i be the angle of joint i with respect to the horizontal axis and p.sub.i=[p.sub.ix, p.sub.iy].sup.T be the position of joint i. Table II (FIG. 10) lists the wing design parameters (q), where lj represents the length components of link Lj. The dimension parameters for non-straight links are shown in FIG. 4. The actuation phase difference Δϕ defines the relation between two crank gear angles θ.sub.1=ωt+ϕ.sub.0 and θ.sub.9=θ.sub.1+Δϕ, where ω is the flapping frequency.

[0049] Given the humerus mechanism driving gear angle θ.sub.1, the system states can be solved sequentially as follows:

[0050] (1) Solve the humerus mechanism: Given θ.sub.1, solve the four-bar linkages (J.sub.1, J.sub.2, J.sub.3, J.sub.4) for p.sub.5(θ.sub.4), then solve the next four-bar linkages (J.sub.4, J.sub.5, J.sub.6, J.sub.7) for p.sub.8(θ.sub.7).

[0051] (2) Solve the radius mechanism: Calculate θ.sub.9=θ.sub.1+Δϕ, then solve for the four-bar linkages (J.sub.9, J.sub.10, J.sub.11, J.sub.12) for p.sub.13(θ.sub.12), then solve the next four-bar linkages (J.sub.12, J.sub.13, J.sub.15, J.sub.14) for p.sub.16(θ.sub.14). Finally, solve the last three-bar linkage (J.sub.8, J.sub.16, J.sub.17) for p.sub.17(θ.sub.8).

[0052] The four-bar and three-bar linkages listed above can be solved by using a root-finding algorithm. For example, given θ.sub.1, the solution to the four-bar linkages (J.sub.1, J.sub.2, J.sub.3, J.sub.4) can be found by solving the constraint equation

[00001]$\begin{array}{cc}{h}_c\left({\theta }_4\right)=.Math.p_2\left({\theta }_1\right)-p_3\left({\theta }_4\right).Math.-l_2=0p_2\left({\theta }_1\right)=l_1\left[\begin{array}{c}\cos \left({\theta }_1\right)\\ \sin \left({\theta }_1\right)\end{array}\right],p_3\left({\theta }_4\right)=-l_{3a}\left[\begin{array}{c}\cos \left({\theta }_4\right)\\ \sin \left({\theta }_4\right)\end{array}\right],& \left(1\right)\end{array}$ [0053] for θ.sub.4 which can be used to calculate p.sub.5(θ.sub.4). The remaining linkages can be solved in a similar fashion.

[0054] The angles that are biologically meaningful in this wing articulation are the shoulder and elbow angles, θs and θe, respectively, where θs represents the upstroke/downstroke motion and θe represents the retraction/expansion motion. We can then formulate a solver equation such that given the wing design parameters and the drive gear angle θ.sub.1∈[0, 2π]+ϕ.sub.0, solve for θs and θe.

θ.sub.s=θ.sub.7+α.sub.5, θ.sub.e=θ.sub.8−θ.sub.s+π

θ.sub.s,θ.sub.e.sup.T=f.sub.m(q,θ.sub.1).  (2)

[0055] Note that the solution of θe depends on θs but not the other way around.

[0056] The ideal desired flapping motion includes the following properties: (1) the wing extends and retracts during downstroke and upstroke, respectively, (2) the wing is already partially expanded before the downstroke motion begins. The desired trajectories {circumflex over (θ)}s and {circumflex over (θ)}e can be seen in FIG. 5B, which are defined as the following sinusoidal functions

[00002]$\begin{array}{cc}{\hat{\theta }}_s=35\mathrm{°sin}\left(\varphi \right)-10°{\hat{\theta }}_e=-0.5{\tan }^{-1}\left(\frac{-0.5\sin \left(\varphi +2\pi /3\right)}{1+0.5\cos \left(\varphi +2\pi /3\right)}\right)45°+120°,& \left(3\right)\end{array}$

where ϕ∈[0, 2π). θe is a skewed sinusoidal function which allows the wing to expands faster than the retraction and have a full wingspan in the middle of the downstroke.

[0057] The design optimization will solve for some of the mechanism design parameters q which is listed in Table II, using our initial mechanism design in Solidworks for the initial q. There are 38 parameters in the design space of this armwing and we constrain some of these parameters to fit our design criterion and reduce the search space of the optimizer. In order to have a symmetric gait between the left and right wing, the drive gears are centered (p.sub.1x=p.sub.9x=0) and the crank arm maximum horizontal length must be aligned with the body y axis (p.sub.4x=I.sub.3a, p.sub.12x=I.sub.8a). Additionally, we fix the values for the following parameters: p.sub.1y=15 mm, p.sub.9y=−15 mm, I.sub.h=50 mm, and I.sub.r=90 mm. This leaves us with 30 design parameters to optimize.

[0058] Considering the large design space of this wing structure, solving for all 30 parameters at the same time is not practical due to the large computational time and search space. The radius mechanism follows a trajectory in relation to the humerus mechanism to articulate the appropriate elbow angle. Therefore, we can separately optimize the humerus and radius mechanisms, starting from the humerus mechanism. The humerus and radius mechanisms have 13 and 17 design parameters, respectively.

[0059] The optimization problem can be formulated as

[00003]$\begin{array}{cc}\underset{q}{\min }\left(y^{T}y\right)/N\mathrm{subject}\mathrm{to}:\begin{array}{cc}{q}_{\min }\le q\le q_{\max },& f_c\le 0,\end{array}& \left(4\right)\end{array}$

where the cost function is the mean squared value of y, which is the difference between target vs. the simulated trajectory, N is the data size, q is the parameter to optimize, q.sub.min and q.sub.max are the parameter bounds, and f.sub.c is the constraint function. We used the interior-point method as the optimization algorithm in Matlab which has successfully found a solution that matches the target trajectory well.

[0060] 1) Humerus Mechanism Optimization: The humerus mechanism is optimized using the cost function y={circumflex over (θ)}s−θs where θs is the trajectory vector gained by solving (2) for θs,k given the input angle θ.sub.1,k=2πk/N+ϕ.sub.0, k={1, . . . , N}. We then optimize the following 13 parameters

q.sub.H=[l.sub.1,l.sub.2,l.sub.3a,l.sub.3b,l.sub.3c,l.sub.4,l.sub.5a,l.sub.5b,α.sub.5,p.sub.4y,p.sub.7x,p.sub.7y,ϕ.sub.0],  (5)

subject to the following constraints: (1) the body-fixed joint positions (p.sub.4 and p.sub.7) are within the robot's 50 mm diameter cylindrical body, (2) the linkages do not intersect or block each other, and (3) the length constraints for the linkages to prevent singularity in the four-bar mechanism. For example, the constraint equation (l.sub.1+|p.sub.4−p.sub.1|)−0.8 (l.sub.2+l.sub.3a)<0 constrains the linkage lengths to prevent singularity in the four-bar mechanism (J.sub.1, J.sub.2, J.sub.3, J.sub.4). We use a similar constraint for the other four-bar mechanisms.

[0061] 2) Radius Mechanism Optimization: Once the humerus parameters has been optimized, we can then optimize the remaining 17 parameters for the radius mechanism

q.sub.R=[l.sub.6,l.sub.7,l.sub.8a,l.sub.8b,l.sub.8c,l.sub.9,l.sub.10a,l.sub.10b,l.sub.10c,l.sub.10d,l.sub.11,l.sub.12a,l.sub.12b,p.sub.12y,p.sub.14x,p.sub.14y,Δϕ],  (6)

[0062] subject to similar constraints and follow the same procedures as the humerus optimization problem. The cost function calculates the trajectory error y={circumflex over (θ)}e−θe.

[0063] Optimization Results and Discussion

[0064] FIGS. 5A and 5B show the simulation of the shoulder and elbow angles for one flapping cycle using the rigid body kinematic formulation shown in (2). The unoptimized design trajectories, shown in FIG. 5A as a range of θs∈[−20°, 54° ] and θe∈[70°, 228° ], which have the characteristics of the ideal flapping motion that is discussed in Section III-B. However, the maximum elbow angle of 228° raises some concerns since that indicates a nontrivial joint hyperextension which might have an adverse effect on the armwing structure.

[0065] The design optimization has successfully found the design parameters q shown in Table II, which have 15.8% average difference compared to the initial values and closely follow the target trajectories as shown in FIG. 5B, with R2 values of 0.997 and 0.920 for θs and θe respectively which indicate a good fit. The optimized trajectory has motion range of θs∈[−43°, 27° ] and elbow angle range of θe, ∈[74°,174° ], which does not have the joint hyperextension present in the unoptimized design. The elbow joint and wingtip trajectories of the optimized design can be seen in FIG. 7. There should be no extreme hinge bending angles compared to its resting position to avoid breakage. Joints J.sub.8, J.sub.16, and J.sub.17, which are all connected to the radius link L.sub.12 have a maximum bending angles of 80° to 90° that bend significantly more towards one side than the other. Joint 11 has a maximum bending angle of 65° while the other joints have the maximum bending angle of less than 45°.

[0066] Structural And Sensitivity Analysis

[0067] The following discusses the structural and sensitivity analysis done on the optimized armwing structure. The structural analysis was done by using Solidworks Simulation FEA to simulate the flexible material bending as the armwing is articulated. Then a sensitivity analysis was done to show which design parameters have the most impact to the flapping gait and how the trajectories change with these parameters.

[0068] FIGS. 6A and 6B show the strain of the armwing structure during some of the key moments, which is simulated using the Solidworks Simulation FEA under static torques acting on the crank arms. We use the material property of FLX9870 (Table I), which has the density of 1.15 g/cm3 and average tensile strength of 5 MPa [34]. The simulation is done with the hyperelastic Mooney-Rivlin model using the material constants of a rubber shown in [37], with a Poisson's ratio of 0.4999, and first and second material constants of 0.3339 MPa and −0.337 kPa respectively. The key moments we considered are the beginning of the downstroke and upstroke motion where the mechanism starts to change its direction. As shown in FIGS. 6A-6B, the maximum strain during these two motions are 43% and 30% respectively. Since FLX9870 has an elongation at break of 120-140%, we have an adequate margin to deal with unforeseen strains, which might arise due to unexpected torsional and off-plane perturbations. A different hinge design can be considered to further reduce the stress/strain concentration, but our space constraints significantly limit our design options.

[0069] The sensitivity analysis is done to determine how much the parameters in q affect the θs and θe trajectories. Let A and M be the peak-to-peak amplitude and mean of the joint trajectories respectively, and Δθ be the phase difference between the peaks of θs and θe. The rate of change of Ae, As, Me, Me, and Δθ are evaluated about the optimized parameters q listed in Table II.

[0070] Table III (FIG. 11) shows the Jacobian of the A, M, and Δθ vs q.sub.H and q.sub.R. The parameters have the following units: mm for lengths and positions, and rad for the angles. The most sensitive parameter is l.sub.1 which affects the amplitudes As and Ae the most, and also significantly affects Δθ as well. The shoulder mean trajectory Ms are mostly affected by l.sub.4, l.sub.3a, l.sub.3b, l.sub.2, and p.sub.7x. The elbow trajectory is mostly affected by the parameters l.sub.6 and l.sub.7. The angle parameters as significantly affects Ms, Me, and Ae while Lϕ affects Ae, but they have a different unit than the length parameters so it is difficult to directly compare their effects. In conclusion, the length parameters of the links closer to the driving gears are among the most sensitive parameters in our mechanism.

[0071] FIGS. 7A, 7B, and 7C show how the elbow joint and wingtip trajectory varies with some of the design parameters, particularly l.sub.1, l.sub.4, and l.sub.9. l.sub.1 is chosen as it is the most sensitive parameter in the design while l.sub.4 and l.sub.9 are simple links that are relatively sensitive to change in the shoulder and elbow joint angle trajectories. l.sub.4 and l.sub.9 are also chosen to promote our control framework where we adjust the flapping gait by changing the wing conformation. These links are simple straight links and we might be able change the design to support variable length linkages on these links. The mechanism design and implementation of this idea can be a part of our future work. The values tested in FIGS. 7A-7C are adjusted by ±2.5%, up to ±10% from their original values in Table II. The green and red lines represent an increase and decrease compared to the original parameter value respectively. As shown in FIG. 7C, the elbow joint trajectory is unaffected by a change in l.sub.9, which is also shown in Table III. This indicates that we can adjust θe independently from θs, which supports our idea of applying control through morphology manipulation.

[0072] For reference, videos depicting exemplary robot wing motion can be viewed at https://www.youtube.com/watch?v=X_UhhCvFC_Q and https://www.youtube.com/watch?v=9nWx4rhUtm0.

[0073] Having thus described several illustrative embodiments, it is to be appreciated that various alterations, modifications, and improvements will readily occur to those skilled in the art. Such alterations, modifications, and improvements are intended to form a part of this disclosure, and are intended to be within the spirit and scope of this disclosure. While some examples presented herein involve specific combinations of functions or structural elements, it should be understood that those functions and elements may be combined in other ways according to the present disclosure to accomplish the same or different objectives. In particular, acts, elements, and features discussed in connection with one embodiment are not intended to be excluded from similar or other roles in other embodiments. Additionally, elements and components described herein may be further divided into additional components or joined together to form fewer components for performing the same functions. Accordingly, the foregoing description and attached drawings are by way of example only, and are not intended to be limiting.

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