Kinematic Analysis of a Flapping-wing Micro-aerial-vehicle with Watt Straight-line Linkage

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1 Journal of Applied Science and Engineering, Vol. 18, No. 4, pp (2015) DOI: /jase Kinematic Analysis of a Flapping-wing Micro-aerial-vehicle with Watt Straight-line Linkage Chao-Hwa Liu* and Chien-Kai Chen Department of Mechanical and Electro-mechanical Engineering, Tamkang University, Tamsui, Taiwan 251, R.O.C. Abstract In this study kinematic analysis of a particular flapping wing MAV is performed to check the symmetry of the two lapping wings. In this MAV symmetry is generated by a Watt straight line mechanism. After appended by two more links to provide a continuous input, and it becomes a Stephenson type III six-bar linkage. Together with the two wings the vehicle has 10 links and 13 joints. Since a Watt four-bar linkage can only generate approximate straight lines, the deviation from an exact straight line causes phase lags of the two wings. The goal of this study is to determine the phase lags. To achieve this goal a forward kinematic analysis of the Stephenson III linkage is performed, which refers to the procedures that may determine the position, velocity, and acceleration of the MAV. Among these procedures, position analysis involves equations that are highly nonlinear and deserves special attention. The authors developed two solution techniques for the forward position analysis of Stephenson III mechanisms: an analytic procedure which leads to closed-from solutions; and a numerical technique to obtain approximate solutions. We use the numerical technique to perform kinematic analysis because solutions obtained by the two methods agree almost exactly but the numerical method is much faster. We analyzed the MAV with the same dimension as the real model, and found it to have very good symmetry with negligible phase lags between the two wings. Key Words: Micro-aerial-vehicle, Forward Position Analysis, Watt Straight-line Linkage, Stephenson Type III six-bar Linkage 1. Introduction Recently many research efforts have been made on design and construction of Micro Aerial Vehicles (MAV). Insect-like MAVs generally have two flapping wings, and various mechanisms to drive the wings have been suggested and tested. For example, Yang [1] utilized a fourbar crank-rocker linkage in his flapping wing device. Galinski and Zbikowski [2] developed a mechanism in *Corresponding author. chaohwa@mail.tku.edu.tw This paper is the extension from the authors technical abstract presented in the 1 st International Conference on Biomimetics And Ornithopters (ICBAO-2015), held by Tamkang University, Tamsui, Taiwan, during June 28 30, which a double rocker linkage is driven by a crank rocker mechanism. Galinski and Zbikowski [3] made use of a double spherical Scotch yoke mechanism to generate desired motion. The mechanism developed by McIntosh et al. [4] includes planar four-bar linkages, spatial cam mechanisms, and slotted arms. Zhang et al. [5] used a mechanism with a spatial single crank double rocker mechanism. Yang, et al. [6] demonstrated the design, fabrication, and performance test of a 20 cm-span MAV, which has a flapping angle up to 100. Recently, a model of a flapping wing MAV is proposed [7], the basic structure of which consists of a Stephenson type III six-bar mechanism that includes a Watt four-bar linkage, and two wing mechanisms. Symmetry of the two wings is due to the straight line motion generated by the Watt four-bar link-

2 356 Chao-Hwa Liu and Chien-Kai Chen age. However, a Watt four bar linkage can only generate an approximate straight-line, the purpose of this study, therefore, is to determine the difference, or the phase lag, between the two wings. Phase lags include differences of the two wings in position, velocity, and acceleration. Kinematic analysis must be performed to determine position, velocity, and acceleration of the mechanism. The paper is organized as follows. The construction of the mechanism is introduced in the next section; then the procedure for position analysis is discussed in section 3. Velocity and acceleration analysis procedures are explained in section 4. Analysis results are shown in section 5, followed by the conclusion in section Construction of the Mechanism Figure 1 is an illustrative diagram for the flapping wing MAV proposed by Cheng [7]. The mechanism has 10 links and 13 pin joints. Links,,,and constitute a Watt four-bar mechanism that makes joint C moving along an approximate straight line. A Stephenson type III six-bar linkage is formed by adding links and to the Watt straight line mechanism. The reason for adding these two links is obvious, since not a single link in the Watt mechanism is able to make a full revolution. With two links added, input to this mechanism is the continuous rotation of link generated by a motor. Finally, links and links are the right wing and the left wing mechanisms, and flapping of both wings are produced by motion of joint C. Symmetric motion of the two wings is achieved if joint C moves along a perfect straight line along the vertical direction, which can only be generated by more complex mechanisms [8]. In the current design, the Watt four-bar linkage is used to reduce number of links and total weight, the cost of using this simple linkage is that this mechanism can only produce approximate straight lines. In the subsequent analysis, we determine the phase lag of the two links when link rotates with a constant angular velocity. 3. Position Analysis 3.1 Stephenson Type III Six-bar Linkage Position analysis of the Stephenson type III six-bar linkage is performed first, from which the position of joint C in Figure 1, as a function of position of the input link, may be determined. Previous study includes Jawad [9] who obtained algebraic equations that govern the position of this linkage, and he solved the equations numerically. Watanabe and Funabashe [10,11] obtained solutions for positions of various Stephenson six-bar linkages by solving six order algebraic equations. Watanabe et al. [12] extended this study to 23 planar linkages with a similar construction. Since these solution techniques involve equations with page-long coefficients, in this study we developed different methods to obtain solutions, as introduced below. Figure 2 shows a general Stephenson type III six-bar linkage driven by link. When dimension of all links are known, the purpose is to determine position variables 3, 4, 5,and 6, for a given input angle 2.The Stephenson III linkage is composed of a four-bar linkage that includes links, and a two-bar chain that contains links. Note that the separation of this sixbar linkage this way has been suggested by Watanabe and Figure 1. An illustrative diagram for a flapping wing MAV. Figure 2. A Stephenson type III six-bar linkage.

3 Kinematic Analysis of a Flapping-wing Micro-aerial-vehicle with Watt Straight-line Linkage 357 Funabashe [10,11]. Yet they used this idea in determining limiting positions of the linkage, not in determining positions in general. In the following, we will discuss two techniques based upon this separation, one leads to closed-form solutions and the other is numerical in nature. In the first method, the point D in Figure 2 is considered as the intersection of two curves. The first curve is the circle of radius L 3, whose center G may be determined from input angle 2. The second curve, called coupler curve, is the curve generated by D as the four-bar linkage is in motion. Figure 3 illustrates a portion of the coupler curve for an arbitrary four-bar linkage. Using the notations used in Figure 3, the coupler curve is a sixth degree polynomial whose equation is [13]. (1) Closed-form solutions for position analysis of a Stephenson III six-bar linkage are obtained by locating intersections of the two curves, and they are found by MATLAB in this study. Each real solution so obtained is a position of the joint D in Figure 2. As long as the location of D is known, 3, 4, 5, and 6 may be calculated by geometric relations. In the second method, we choose link (or link ) Figure 3. A part of the coupler curve generated by the point D on the coupler of a four-bar linkage. as the driving link to the four-bar linkage BCDEF in Figure 2. For each given value of 5, values of 4, 6, and the corresponding positions of the point D may be calculated using analytical expressions that may be found in a textbook such as [8]. Then, for this particular value of 5, the distance between D and the known position of G is calculated. The purpose now is to determine the values of 5 so that the distance DG equals to L 3. The procedure used in this study to determine these values is as follows. We first incrementally search for intervals of 5 within which the value (DG L 3 ) changes sign, then the bisection method is used in these intervals to locate values of 5 which make (DG L 3 ) nearly zero. As a check of these two procedures, we analyzed the Stephenson III six-bar linkage in Figure 2 with the nondimensioned lengths: a =5,b = 13, c =8,d =4,e = 6.5, f =3,L 2 =1,L 3 =6,L 5 =1,L 6 = 3, and input angle 2 = 300, Convergence criterion for numerical procedure is that convergence is achieved when DG L 3 < Two solutions are found for this problem and they are given in Table 1. One may observe that results obtained from the two procedures agree up to at least 7 digits after the decimal point. Since the numerical procedure is faster, the following results for position analysis are calculated by this procedure. 3.2 Wing Mechanisms Figure 4 shows the Stephenson III six-bar linkage in the flapping wing MAV. As a value of input angle 2 is given, positions of all other links may be found by the method just been discussed, and from these positions the coordinates of C, namely C x and C y, can be calculated. When the position of C is found, joint D of the right wing mechanism (Figure 1) can be located since it is the intersection of the following two circles, the circle centers at Table 1. Comparison of results obtained using two procedures; differences are underlined Closed-form Numerical 3 1 st sol nd sol st sol nd sol st sol nd sol st sol nd sol

4 358 C. H. Liu and Chien-Kai Chen C with a radius L 4, and the circle centers at E whose radius is L 5. In general, there are two intersections, as shown in Figures 5 and 6. The first solution shown in Figure 5 is given by where R is the known distance between C and E,and (2) (5) Note that, generally the two solutions do not intersect. Once the right wing is assembled in either of the two ways, it remains the same configuration unless it is disconnected and reassembled. The left wing also has two solutions, given by [13] (6) (3) The second solution as shown in Figure 6 contains the following two angles (4) and (7) (8) (9) 4. Velocity and Acceleration Analysis Velocity analysis is performed after the position analysis, that is, when all link positions have been found. The order for velocity analysis is similar to that for position analysis; the Stephenson III six-bar linkage must be treated first, since both wings are driven by it. Figure 4. The Stephenson III six-bar linkage in the MAV. 4.1 Stephenson Type III Six-bar Linkage The Stephenson III linkage illustrated by Figure 4 has the following two closed loops: Figure 5. The first solution for the position analysis of the right wing mechanism. Figure 6. The second solution for the position analysis of the right wing mechanism.

5 Kinematic Analysis of a Flapping-wing Micro-aerial-vehicle with Watt Straight-line Linkage 359 AB + BC + CG = AI + IG (10) (20) HF + FG = HI + IG (11) Upon differentiation, we may obtain The x and y components of equation (10) are (21) (12) where (13) (22a) Similarly vector equation (11) has the following two components (14) (15) Differentiating these four equations once with respect to time, one may obtain where the matrix M 1 is given by (16) (22b) Angular velocities 4 and 5 may be found from equation (21). Finally, the loop equations for the left wing is (see Figure 8) AB + BC + CK = AJ + JK (23) Following the same procedure as before, one may obtain [13] in which (24) (25) (17) With positions of all links known, for a given input velocity 2 one may solve equation (16) for unknown velocities 3, 8, 9 and Wing Mechanisms Referring to Figure 7, one may notice the vector loop closure equation for the right wing AB + BC + CD = AE + ED (18) The x and y components of this equation are (19) Figure 7. Notations of the right wing mechanism for velocity and acceleration analysis.

6 360 Chao-Hwa Liu and Chien-Kai Chen From these equations the velocities 6 and 7 can be obtained. Unknown accelerations may be calculated when positions, velocities, and driving accelerations 2 are known. Equations for calculating accelerations are obtained by differentiating velocity equations (16), (21), and (24). For the Stephenson III six-bar linkage, the equation is and (29) 5. Results and Discussions where (26) (27) After accelerations 3, 8, 9,and 10 are obtained from equation (26), accelerations for the right wing and the left wing may be found from (28) The dimension for the MAV, as illustrated by Figure 1, is as follows [7]: AB = L 2 = 2 mm, BC = L 3 = 16 mm, CD = L 4 = 6.5 mm, CK = L 6 = 6.5 mm, DE = L 5 = 3 mm, JK = L 7 = 3 mm, GI = L 8 = 10 mm, HF = L 9 = 10 mm, GF = L 10 = 4 mm, and GC = FC = 2 mm. These lengths are used in the subsequent analysis. Figure 9 shows the particular configuration obtained from position analysis when 2 = 0. Angular position 5 of the right wing is shown in Figure 10. Note that the angle 5 defined in Figure 7 is not the flapping angle. The flapping angle generally refers to the angle by which the wing extends outward from the body (see Figure 1). The two angles always differ by a constant. The angular stroke of the wing, also called flapping range, is the maximum difference in 5. For this mechanism its value is To check the symmetry between the two wings, we compare the angle 5 in Figure 7, to the angle 7 in Figure 8; also we compare and the angle 4 in Figure 7 to the angle 6 in Figure 8. The differences between these angles are shown in Figure 11. The maximum difference in this figure is between 5 and 7, its value is , and it occurs when 2 =90. The coupler curve generated by the point Figure 8. Notations of the left wing mechanism for velocity and acceleration analysis. Figure 9. Configuration of the flapping wing MAV when 2 =0.

7 Kinematic Analysis of a Flapping-wing Micro-aerial-vehicle with Watt Straight-line Linkage C on the Watt linkage approximates a vertical straight line. The position q2 = 90 corresponds to one extreme position for joint C where it begins to deviate further from a straight line, causing a larger degree of asymmetry of the two wings at this position. Results for velocity and acceleration analysis are obtained with the constant input velocity w2 = 1 rad/s. The difference in angular velocity between link and link is shown in Figure 12. The maximum value in this figure is rad/s and it occurs when q2 = 113. In Figure 13 we show the velocity difference between link and link. Its maximum value is rad/s, which occurs when q2 = 114. The difference in acceleration between link and link is shown in Figure 14. The maximum difference is rad/s2, which occurs at the position q2 = 90. Finally, the difference in acceleration between link and link is shown in Figure 15. The maximum difference occurs Figure 10. Angular position of link. Figure 11. Phase lag of the two wings. 361 when q2 = 100, whose value is rad/s2. 6. Conclusions In this study we show that a Stephenson III six-bar Figure 12. Velocity difference between link and. Figure 13. Velocity difference between link and link. Figure 14. Acceleration difference between link and link.

8 362 Chao-Hwa Liu and Chien-Kai Chen Figure 15. Acceleration difference between link and link. linkage may be viewed as two separate mechanisms in forward position analysis. Based on this separation, two techniques for position analysis of this linkage are suggested. While one technique may lead to closed-form solution without any iterative procedure, the numerical technique is much faster and converges to nearly the same value obtained by using closed-form solution. The numerical technique is used for position analysis of a flapping wing MAV whose basic structure is Stephenson III six-bar linkage. We find the two wings of this MAV are highly symmetric and phase lags of the two wings are insignificant. Acknowledgements The authors gratefully acknowledge that this study was supported by the National Science Council of ROC under Grant NSC E MY3. References [1] Yang, L. J., U.S. Patent 8,033,499B2 (2011). [2] Galinski, C. and Zbikowski, R., Materials Challenges in the Design of an Insect-like Flapping Wing Mechanism Based on a Four-bar Linkage, Materials and Design, Vol. 28, pp (2007). doi: /j. matdes [3] Galinski, C. and Zbikowski, R., Insect-like Flapping Wing Mechanism Based on a Double Spherical Scotch Yoke, Journal of the Royal Society Interface,Vol.2, No. 3, pp (2005). doi: /rsif [4] McIntosh, S., Agrawal, S. and Khan, Z., Design of a Mechanism for Biaxial Rotation of a Wing for a Hovering Vehicle, IEEE/ASME Transactions on Mechatronics, Vol. 11, No. 2, pp (2006). doi: /TMECH [5] Zhang, T., Zhou, C., Zhang, X. and Wang, C., Design, Analysis, Optimization and Fabrication of a Flapping Wing MAV, Proceedings of 2011 International Conference on Mechatronic Science, Electric Engineering and Computer, Jilin, China, pp (2011). doi: /MEC [6] Yang, L. J., Esakki, B., Chandrasekhar, U., Huang, K.-C. and Cheng, C.-M., Practical Flapping Mechanisms for 20 cm-span Micro Air Vehicles, International Journal of Micro Air Vehicles, Vol. 7, No. 2, pp (2015). doi: / [7] Cheng, C.-M., Preliminary Study of Hummingbirdlike Hover Mechanisms, Master Thesis, Tamkang University, Damsui Dist., New Taipei, City, Taiwan (2013). [8] Norton, R. L., Design of Machinery: an Introduction to the Synthesis and Analysis of Mechanisms and Machines, 5 th ed., McGraw-Hill, New York (2012). [9] Jawad, S. N. M., Displacement Analysis of Planar Stephenson Mechanism, Proceedings of the International Computers in Engineering Conference, Vol. 2, pp (1984). [10] Watanabe, K. and Funabashi, H., Kinematic Analysis of Stephenson Six-link Mechanisms, (1st Report), Bulletin of the JSME, Vol. 27, No. 234, pp (1984). doi: /jsme [11] Watanabe, K. and Funabashi, H., Kinematic Analysis of Stephenson Six-link Mechanisms, (2 nd Report), Bulletin of the JSME, Vol. 27, No. 234, pp (1984). doi: /jsme [12] Watanabe, K., Zong, Y. H. and Kawai, Y., Displacement Analysis of Complex Six Link Mechanisms of the Stephenson-type, JSME International Journal, Vol. 30, No. 261, pp (1987). doi: / jsme [13] Chen, C.-K., Kinematic and Error Analysis of a Hummingbird-like Micro-aerial-vehicle, Master Thesis, Tamkang University, Damsui Dist., New Taipei, City, Taiwan (2014). Manuscript Received: Sep. 9, 2015 Accepted: Oct. 23, 2015