IMA Preprint Series # 2323

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A MATRIX FORMULATION OF THE NEWTON DYNAMICS FOR THE FREE FLIGHT OF AN INSECT By Sheng Xu IMA Preprint Series # 2323 ( June 21 ) INSTITUTE FOR MATHEMATICS AND ITS

1 A MATRIX FORMULATION OF THE NEWTON DYNAMICS FOR THE FREE FLIGHT OF AN INSECT By Sheng Xu IMA Preprint Series # 2323 ( June 21 ) INSTITUTE FOR MATHEMATICS AND ITS APPLICATIONS UNIVERSITY OF MINNESOTA 4 Lind Ha 27 Church Street S.E. Minneapois, Minnesota Phone: Fax: URL:

2 A matrix formuation of the Neton dynamics for the free fight of an insect Sheng Xu Department of Mathematics, Southern Methodist University Daas, TX , USA Astract An insect fies and maneuvers y couping its Neton dynamics and aerodynamics. In this note, e estaish a matrix formuation of the Neton dynamics for the free fight of an insect. The fight can e driven y to means. One is to prescrie the kinematics of each ing reative to the ody, and the other is to prescrie the torque exerted y the ody on each ing. Our formuation unifies the to and is very easy to use. It is usefu for studying the staiity and maneuveraiity in insect fight. Nomencature α τ Ω Π c f Tait-Bryan anges eteen to reference frames. Fuid or interna torque Anguar veocity of an insect ody or ing reative to the a frame. Anguar veocity of an insect ing reative to the ody. Musce torque activey exerted on an insect ing y the ody. Fuid, gravitationa, or intena force e-mai:sxu@smu.edu, epage:facuty.smu.edu/sxu 1

3 m x E H I K R Mass of an insect, an insect ody, or ing Dispacement of an insect ody, ing, or ody-ing hinge. 3 3 indentity matrix. Ske-symmetric matrix formed from the coordinates of a ody-ing hinge. Moment of inertia matrix of an insect ody or ing. Matrix reating anguar veocity ith rate of change of Tait-Bryan anges. Matrix of transformation eteen to reference frames. 1 Introduction A fying insect is a mutiody system consisting of a ody and mutipe ings. The physica connections eteen its ody and ings are physica constrains in the system. To study the staiity and maneuveraiity of free fight, e need to coupe the Neton dynamics and aerodynamics of this mutiody system. Specificay, the Neton dynamics needs the aerodynamic force and torque from the aerodynamics, and the aerodynamics needs the veocity and acceeration from the Neton dynamics. In this note, e estaish a formuation of the Neton dynamics for the free fight of an insect. Genera formuations to rite genera codes are avaiae for the Neton dynamics of mutiody systems [1]. Equations specific for the Neton dynamics for insect fight ere aso presented in [2, 3]. Hoever, ho to impement these formuations and equations in a simpe and straightforard manner is not aays cear to the readers. The ojective of this note is to provide a neat and concise formuation hich is very easy to use. In particuar, e estaish a matrix formuation y choosing suitae reference frames and unknon variaes Sheng Xu JAM

4 and manipuating dynamica equations and physica constraints. The formuation unifies to cases. In one case, the kinematics of each ing reative to the ody is prescried. In the other, the torque exerted y the ody on each ing is prescried. This formuation is neat, concise, and simpe. It oud e usefu in studying insect fight or aquatic anima simming. In particuar, it is usefu for couping the Neton dynamics and aerodynamics of insect fight in CFD or for anayzing insect fight ith reduced fuid force and torque modes. In Section 2, e prepare the notations for the presentation of the formuation. In Section 3, e present the compete formuation. In Section 4, e give detaied derivations of the formuation. In Section 5, e sho some tests on the formuation. In Appendix, e ist the formuas for cacuating some transformation matrices. The information given in Sections 2 and 3 and Appendix is sufficient for just impementing the formuation. 2 Preparations 2.1 Reference frames A fying insect has a ody and mutipe ings. Each ing is connected to the ody at a hinge. Mutipe reference frames (coordinate systems) shon in Fig. 1 are used to descrie the positions and orientations of the ody and ings. They are x = (x, y, z ), the static a frame used for the motion of the hoe insect (the system); x = (x, y, z ), the ody frame attached to the insect ody such that the origin B is the center of mass (CM) of the ody, and x, y, and z are the principe axes of the Sheng Xu JAM

5 2 z x y y z x B H y z W x O 1 Figure 1: Mutipe reference frames (coordinate systems) for a free-fying insect. moments of inertia I B of the ody; x = (x, y, z ), a ing frame attached to each insect ing such that the origin W is the CM of the ing, and x, y, and z are the principe axes of the moments of inertia I W of the ing. Hereafter, e use oercase superscripts,, and to specify the a frame, the ody frame, and a ing frame, respectivey, capita suscripts B, W, H to specify the ody, a ing, and a hinge, respectivey, and one dot and to dots atop a symo to denote first and second order time derivatives, respectivey. For exampe, x W denotes the transationa acceeration of the CM of a ing in the a frame. The coordinates of a ody-ing hinge in the ody frame are denoted as x H. For any vector p, e can rite x H p as x H p = H p, (1) Sheng Xu JAM

6 here the ske-symmetric matrix H is H = z H y H z H x H y H x H. (2) The coordinates of a ody-ing hinge in the ing frame are denoted as x H. Simiary, e can rite x H p as x H p = H p. (3) Let x e the dispacement and q e the veocity, anguar veocity, force, or torque. Then x and u transform eteen different reference frames as foos. x = R ( x x B ), q = R q, (4) x = R ( x x W ), q = R q, (5) x = R ( x x W ), q = R q, (6) here R, R, and R are the a-to-ody, ody-to-ing, and a-to-ing transformation matrices, respectivey. They are orthogona matrices and are given in Appendix. 2.2 Anguar veocity We denote the anguar veocity of the ody reative to the a frame as Ω B and the anguar veocity of a ing reative to the ody as Π W. Then the anguar veocity of the ing reative to the a frame is Ω W = Ω B + Π W. (7) Sheng Xu JAM

7 In the ing frame, it reads Ω W = Ω B + Π W = R Ω B + Π W. (8) 2.3 Force and torque We denote the aerodynamic force and torque on an insect ing as f fw and τ fw, respectivey. In the a frame, they can e ritten as [5] f fw = f W + A fw Ω W + m fw x W, (9) τ fw = τ W + τ 1W + A τw Ω W, (1) here f W and τ W are proportiona to the fuid viscosity, τ 1W depends ony on the shape, position, and the anguar veocity of the ing, the matrices A fw and A τw depends ony on the shape and position of the ing, and m fw is the mass of the fuid dispaced y the ing. The transationa and anguar acceerations of the ing are x W and Ω W, respectivey, and their contriutions to the aerodynamic force and torque are separated in the aove formuas. Pease refer to [4, 5] for the derivation of the formuas. Simiary, the aerodynamic force and torque on the insect ody can e ritten as f fb = f B + A fb Ω B + m fb x B, (11) τ fb = τ B + τ 1B + A τb Ω B. (12) We use f gw, f gb, and f gs to denote the gravitationa force (due to eight and uoyancy) Sheng Xu JAM

8 on a ing, the ody, and the insect respectivey. In the a frame, they are given y f gw = (m W m fw ) g, (13) f gb = (m B m fb ) g, (14) f gb = f gb + W f gw, (15) here m W and m B are the mass of the ing and the ody, respectivey, and g is the gravationa acceeration. 3 Matrix formuation In this section, We present the compete matrix formuation of the Neton dynamics for the free fight of an insect. We unify to cases in our formuation. In the first case, the kinematics of each ing reative to the ody is prescried, and Π W (the anguar veocity of the ing reative to the ody) is thus a knon function of the time. In the second case, the torque exerted y the ody on each ing (denoted as c W ) is prescried in the ody frame. To unify the to cases into the formuation, e introduce a W = δ i2 Π W + δ i1 c W, W = δ i1 Π W + δ i2 c W, (16) here δ ij is the Kronecker deta. The first and second cases aove correspond to i = 1 and i = 2, respectivey. Hence, a W is unknon, W is knon, and e have Π W = δ i2 a W + δ i1 W, c W = δ i1 a W + δ i2 W. (17) We choose x B and Ω B as unknon variaes for the ody B and x W, Ω W, f fw, and a W as unknon variaes for a ing W. If there is ony one ing, the matrix equation for the Sheng Xu JAM

9 unknon variaes reads C 11 C 12 C 13 W C 15 C 22 C 23 W C 25 C 26 C 33 W C34 W C35 W C36 W C 41 C 42 C 43 C 44 W C 52 W C 54 C 56 C 63 W C64 W C65 x B Ω B x W Ω W f fw a W = d 1 d 2 d 3 W d 4 W d 5 W d 6 W, (18) here, ith E denoting the 3 3 identity matrix, the entries in the coefficient matrix and the right hand side are C 11 = (m B m fb )E, C 12 = A fb R T, C 13 W = m W E, C 15 = E, C 22 = I B R A τb R T, C23 W = m W H R, C 25 = H R, C 26 = δ i1 E, C 33 W = m W H R, C 34 W = I W R A τw R T, C35 W = H R, C 36 W = δ i1r, C 41 = E, C 42 = R T H, C 43 = E, C 44 W = R T H, C 52 W = R, C 54 = E, C 56 = δ i2 E, C 63 W = m fw E, C 64 W = A fw R T, C65 = E, Sheng Xu JAM

10 and d 1 = f B + f gs, d 2 = Ω B (I B Ω B ) + R ( τ B + τ 1B) + W ( δ i2 W + H R f gw ), d 3 W = δ i2 R W Ω W (I W Ω W ) + R ( τ W + τ 1W ) H R f gw, d 4 W = RT ( Ω B 2 2 x H ( Ω B x H ) Ω B ) + RT ( Ω W 2 2 x H ( Ω W x H ) Ω W ), d 5 W = δ i1 W R ((R T Π W ) Ω B ), d 6 W = f W. The coefficient matrix ony depends on the shape, position, and mass distriution of the ing and the ody. Besides the shape, position, and mass distriution, the right hand side aso depends on the kinematic. The position and kinamatics of the ody and the ing is fuy descried y = [ x B, x B, α, Ω B, x W, x W, α, Π W ] T, (19) here α is the vector formed y Tait-Bryan anges eteen the a frame and the ody frame, and and α eteen the a frame and the ing frame. Definitions of these anges are given in Appendix. We can numericay integrate in time the fooing first order differentia equations d [ dt = x B, x B, K 1 B Ω B, Ω B, x W, x W, K 1 W Π W, ] T Π W, (2) here the matrices K B and K W are given in Appendix. The dynamic entries at the right hand side of Eq. (2) are otained y soving Eq. (18). Mutipe ings can e incuded easiy. For exampe, if there are to ings W 1 and W 2, Sheng Xu JAM

11 then e have C 1 C 12 2 C2 21 C x W 2 Ω W 2 f fw 2 = d 1 d 3 W 2 d 4 W 2 d 5 W 2 d 6 W 2, (21) a W2 here C 1, 1, and d 1 are the coefficient matrix, unknon vector, and right hand side of Eq. (18) for the first ing W 1, and the su-matrices C 21 the second ing W 2 and are given y C 41 C 42 C2 21 = CW 52 2 C 12 2 = 4 Derivation C 13 W 2 C 15 C 23 W 2 C 25 C 26, C 22 2 = 2, C22 2, and C12 2 are associated ith C 33 W 2 C 34 W 2 C 35 W 2 C 36 W 2 C 43 C 44 W 2 C 54 C 56 C 63 W 2 C 64 W 2 C 65.. (22) First, e introduce a fe reations hich i e used ater in our derivation. Sheng Xu JAM

12 Let the vector q e veocity, anguar veocity, force, or torque. We have Ṙ T R q = Ω B q, (23) Ṙ T R q = Ω W q, (24) Ṙ T R q = Π W q. (25) Using Eqs. (23) and (24), e have Ṙ T Ω B = (ṘT R )(R T Ω B ) = (ṘT R ) Ω B = Ω B Ω B =, (26) Ṙ T Ω W = (ṘT R )(R T Ω W ) = (ṘT R ) Ω W = Ω W Ω W =. (27) Let β and γ e any vectors and R e a 3 3 orthogona matrix, e have (R β) (R γ) = R( β γ). (28) We can convert a cross product to a matrix-vector product as β γ = A γ, (29) here the ske-symmetric matrix A is defined using the three components of β as in Eq. (2). 4.1 Geometric and kinematic reations The ody and a ing of an insect are connected at a hinge, denoted as H in Fig. 1. By Eqs. (4) and (6), e have x H x B = RT x H, (3) x H x W = RT x H. (31) Sheng Xu JAM

13 The geometric reation for the ody-ing constraint is otained y sutracting the aove to equations. It reads x W x B = RT x H RT x H. (32) The coordinates of the hinge, denoted as x H in the ody frame and x H in the ing frame, are time-independent if oth the ody and the ing are rigid. Using Eqs. (23), (24), (28), and (29), e can otain the first order time derivative of Eq. (32) as the fooing x W x B = ṘT x H ṘT x H = ṘT R R T x H ṘT R R T x H = Ω B (R T x H) Ω W (R T x H) = (R T Ω B ) (RT x H ) (RT Ω W ) (RT x H ) = R T ( Ω B x H ) RT ( Ω W x H ). (33) Then e can otain the second order time derivative as x W x B = ṘT ( Ω B x H) ṘT ( Ω W x H) +R T ( Ω B x H ) RT ( Ω W x H ) = d 4 W RT H Ω B + RT H Ω W, (34) Sheng Xu JAM

14 here d 4 W is defined and computed as d 4 W = ṘT ( Ω B x H) ṘT ( Ω W x H) = ṘT R R T ( x H Ω B ) + ṘT R R T ( x H Ω W ) = Ω B (RT ( x H Ω B )) + Ω W (RT ( x H Ω W )) = (R T Ω B) (R T ( x H Ω B)) + (R T Ω W ) (R T ( x H Ω W )) = R T ( Ω B ( x H Ω B )) + RT ( Ω W ( x H Ω W )) = R T ( Ω B 2 2 x H ( Ω B x H ) Ω B ) + RT ( Ω W 2 2 x H ( Ω W x H ) Ω W ). (35) Eq. (34) cooresponds to the fourth equation in Eq. (18) after it is ritten as CW 41 x B + CW 42 Ω B + CW 43 x W + CW 44 Ω W + f fw + a W = d 4 W. (36) As R R T = E, e have ṘR T = RṘT. Using Eq. (25), e then otain Ṙ Ω B = ṘR T R Ω B = R ṘT R Ω B = R Π W Ω B = R ((R T Π W ) Ω B ). (37) With the susitution of Eq. (17), the first order time derivative of Eq. (8) is therefore given y Ω W = Ṙ Ω B + R Ω B + Π W = R ((R T Π W ) Ω B ) + R Ω B + Π W = R ((R T Π W ) Ω B ) + R Ω B + δ i2 a W + δ i1 W, (38) hich cooresponds to the fifth equation in Eq. (18) after it is ritten as x B + CW 52 Ω B + x W + CW 54 Ω W + f fw + CW a 56 W = d 5 W. (39) Sheng Xu JAM

15 4.2 Transationa dynamics In the a frame, the Neton equation for the CM of the hoe free-fying insect reads m S x S = f gs + f fb + W f fw, (4) here m S is the mass of the insect, x S is the coordinates of its CM in the a frame, f gs is the gravitationa force on the insect, f fb is the fuid force on its ody, and f fw is the fuid force on its one ing. We have m S x S = m B x B + W m W x W. (41) Using Eqs. (9) and (26), e can rite f fb as f fb = f B + A fb Ω B + m x fb B = f B + A fb (ṘT Ω B + R T Ω B) + m fb x B = f B + A fbr T Ω B + m x fb B. (42) After sustituting Eqs. (41) and (42) into Eq. (4), e otain m B x B + W m W x W = f gs + f B + A fb R T Ω B + m fb x B + W f fw, (43) hich corresponds to the first equation in Eq. (18) after it is ritten as C 11 x B + C 12 Ω B + W ( CW 13 x W + Ω W + CW 15 f ) fw + a W = d 1. (44) Using Eqs. (9) and (27), e can rite f fw as f fw = f W + A fw Ω W + m fw x W = f W + A fw (ṘT Ω W + R T Ω W ) + m fw x W = f W + A fw R T Ω W + m x fw W, (45) Sheng Xu JAM

16 hich corresponds to the sixth equation in Eq. (18) after it is ritten as x B + Ω B + CW 63 x W + C64 W Ω W + C65 W ffw + a W = d 6 W. (46) In the a frame, the Neton equation for the CM of a ing reads m W x W = f gw + f fw + f W, (47) here f W is the force exerted y the ody on the ing at the hinge. By Neton s third a, the force exerted y this ing on the ody at the hinge is thereforce given y f B = f W = f gw + f fw m W x W. (48) 4.3 Rotationa dynamics In the ody frame, the Euer equation for the ody reads I B Ω B + Ω B (I B Ω B ) = τ fb + W τ B, (49) here the diagona matrix I B is the moments of inertia of the ody, τ fb is the fuid torque on the ody, and τ B is the torque on the ody received from a hinge. Using Eqs. (1) and (26), e have τ fb = R τ fb = R ( τ B + τ 1B + A τb Ω B ) = R ( τ B + τ 1B + A τb R T Ω B). (5) Using Eq. (48), e have τ B = c B + x H f B = c B + H f B = c B + H R f B, = c W + H R ( f gw + f fw m x W W ), (51) Sheng Xu JAM

17 here c B = c W and c W is the musce torque activey exerted y the ody on this ing. After sustituting Eqs. (5), (51), (17) into Eq. (49), e have I B Ω B + Ω B (I BΩ B ) = R ( τ B + τ 1B + A τbr T Ω B ) + ( (δ i1 a W + δ i2 W ) + H R ( f gw + f ) fw m W x W ), (52) W hich cooresponds the second equation in Eq. (18) after it is ritten as x B + C22 Ω B + W ( CW 23 x W + Ω W + CW 25 f ) fw + C26 W a W = d 2. (53) In the ing frame, the Euer equation for a ing reads I W Ω W + Ω W (I W Ω W ) = τ fw + τ W, (54) here the diagona matrix I W is the moments of inertia of the ing, τ fw is the fuid torque on the ing, and τ W is the torque on the ing received from the hinge. Using Eqs. (1) and (27), e have τ fw = R τ fw = R ( τ W + τ 1W + A τw Ω W ) = R ( τ W + τ 1W + A τw R T Ω W ). (55) Using Eq. (48), e have τ W = c W + x H f W = c W + H f W = R c W + H R f W = R c W + H R (m W x W f gw f fw ). (56) After sustituting Eqs. (55), (56), and (17) into Eq. (54), e have I W Ω W + Ω W (I W Ω W ) = R ( τ W + τ 1W + A τw R T Ω W ) + R (δ i1 a W + δ i2 W ) + H R (m W x W f gw f fw ), (57) Sheng Xu JAM

18 hich cooresponds the third equation in Eq. (18) after it is ritten as x B + Ω B + CW 33 x W + C34 W Ω W + C35 W ffw + C36 W a W = d 3 W. (58) 5 Tests We perform to simpe tests on our formuation for an insect mode ith to ings. The test code in MATLAB is posted on the author s epage. One thing can aays e checked in the tests is ho e Eq. (32) (the constraint eteen the ody and ings) is satisfied hen Eq. (2) is numericay integrated to update the position and orientation of oth the ody and ings. In rea appications, e shoud update ony the orientation of the ings from numerica integration, and otain the position of the ings directy from Eq. (32). Otherise, numerica errors may reak the constraint. Note that the initia conditions for integrating Eq. (2) must satisfy Eqs. (32) and (33). In the first test, e examine the conservation of anguar momentum of a torque free insect mode. In this mode, the to identica ings are fixed reative to the ody ith refective symmetry through the x z pane, and the CM of the mode fas on the x axis. If the initia anguar veocity of the mode has ony the roing component, then the mode keeps roing aout the roing x axis ith the initia rate. The simuation verifies this conservation, as shon in Fig. 2. The constraint reation, Eq. (32), is aso e satisfied in the simuation, as shon in Fig. 2. In the second test, e examine Neton s second a for an insect mode suject to an externa force hich has three non-zero components in the a frame. The to ings fap ith respect to the ody under the action of prescried ody-to-ing torque. The CM of Sheng Xu JAM

19 1.5 Body anguar veocity in the ody frame 3 Wing coordinates in the a frame 1 Ω Bx Ω By Ω Bz 1 2 ODE sover constraint reation t t Figure 2:.3 System CM coordinates in the a frame.5 Wing coordinates in the a frame.2 ODE sover anaytica ODE sover constraint reation t t Figure 3: the mode foos a knon trajectory according to Neton s second a, hich is verified in Fig. 3. The constraint reation, Eq. (32), is e satisfied again in this simuation, as shon in Fig. 3. Appendix To rite the expicit forms of the transformation matrices, e use the fooing Tait-Bryan anges (anges of ro, pitch, and ya) to descrie the orientation of one reference frame Sheng Xu JAM

20 reative to another. φ, θ, and ψ : the ro (around the x axis), pitch (around the y axis), and ya (around the z axis) anges of the ody reative to the a frame, respectivey; φ, θ, and ψ : the ro (around the x axis), pitch (around the y axis), and ya (around the z axis) anges of a ing reative to the ody, respectivey. Appying the rotations in the order of ya, pitch, and ro, e have R = R ro R = R ro R pitch R ya, (59) R pitch Rya, (6) R = R R, (61) here 1 1 R ro = cos φ sin φ, R ro = cos φ sin φ, (62) sin φ cos φ sin φ cos φ cos θ sin θ R pitch = 1, R pitch = sin θ cos θ cos ψ sin ψ R ya = sin ψ cos ψ 1, R ya = cos θ sin θ 1 sin θ cos θ, (63) cos ψ sin ψ sin ψ cos ψ. (64) 1 Sheng Xu JAM

21 The anguar veocity Ω B in the ody frame and the anguar veocity Π W in the ing frame are Ω B = Π W = φ φ + R ro + R ro θ θ + R ro + R ro R pitch Rpitch ψ = K B α, (65) ψ = K W α, (66) here 1 sin θ K B = cos φ sin φ cos θ, α = sin φ cos φ cos θ φ θ 1 sin θ K W = cos φ sin φ cos θ, α = sin φ cos φ cos θ ψ, (67) φ θ ψ. (68) Note that the initia conditions for integrating Eq. (2) must satisfy Eq. (66) hen the ing kinematics (the Tait-Bryan anges of the ing reative to the ody) is prescried. If the Tait-Bryan anges of the ing reative to the ody are prescried, then y Eq. (66), Π W can e cacuated from Π W = K W α + K W α, (69) Sheng Xu JAM

22 here K W is K W = φ sin φ cos φ cos θ + θ cos θ sin φ sin θ. cos φ sin φ cos θ cos φ sin θ (7) References [1] R. E. Roerson and R. Schertassek, Dynamics of mutiody systems, Springer-Verag (1998) [2] G. Geert, P. Gameier and J. Evers, Equations of motion for fapping fight, AIAA Paper, AIAA [3] M. Sun, J. Wang and Y. Xiong, Dynamic fight staiity of hovering insects, Acta Mech. Sin. 23 (27), [4] Sheng Xu, Singuar forces in the immersed interface method for rigid ojects in 3D, App. Math. Lett., 22 (29), [5] Sheng Xu, The fuid force and torque on a rigid oject in an incompressie viscous fo, App. Math. Lett., sumitted (29) Sheng Xu JAM

23 List of Figure Captions 1. Figure 1: Left: Conservation of the roing anguar momentum of an insect mode; Right: Comparison of the three coordinates of a ing eteen the simuation and the exact formua. 2. Figure 2: Left: Comparison of the CM trajectory of the insect mode eteen the numerica simuation and the exact formua; Right: Comparison of the three coordinates of a ing eteen the numerica simuation and the exact formua. Sheng Xu JAM

Implemental Formulation of Newton Dynamics for Free Insect Flight

Implemental Formulation of Newton Dynamics for Free Insect Flight Impementa Formuation of Neton Dynamics for Free Insect Fight Sheng Xu Department of Mathematics, Southern Methodist University, Daas, TX 75275-156, USA Astract A free-fying insect fies and maneuvers y