Flight Dynamics and Control of Flapping-wing Mavs a Review

one. Introduction

The recent survey papers on flapping flying highlight the intense attention that bio-inspired flying is receiving in the aerospace and robotics research community. Each of these survey papers addressed a particular broad sub-area of flight. Shyy and co-authors [i] focused on aerodynamics and structures; the papers by Girard [2] and Nayfeh [3] provide a consummate review from a dynamics and control perspective; books by Azuma [4] and Mueller [5] form a consummate tutorial on flapping flying. Dark-brown [6] reviews the flapping flight of birds from the perspective of flight performance, with detailed observations on the flapping wing kinematics.

The same survey papers primarily concentrate on insect flight. This preponderance reflects the considerable piece of work washed past aeronautical every bit well as the robotics customs at large toward the evolution of engineered insect flight [7-12]. Ane scientifically challenging aspect of insect flight is the unsteady and nonlinear aerodynamics. It is known that the aerodynamics of insect flight also involves several unconventional circulatory also equally non-circulatory mechanisms [xiii, xiv]. However, since the flapping frequencies are far college than those of the flight dynamic modes, it suffices to model the aerodynamics via quasi-steady approximations for the purpose of stability analysis and control [9, 13, xiv]. In dissimilarity, the aerodynamics of bird and bat flight are relatively easier to model analytically. However, since the flapping frequency is similar to the natural frequency of several modes of the airframe (eastward.thousand., see [15]), the resulting flapping flight dynamics are much more complex than their insect-calibration counterparts.

Evolution of bird-scale flapping flying has led to interesting results and advances in the flight mechanics and command of non-flapping flight likewise, under the broad umbrella of wing articulation, morphing wing technologies, and bio-inspired maneuvers. In [16,17], we developed an articulated wing shipping which employed the (symmetric and asymmetric) wing dihedral for both longitudinal and lateral command. We besides flying-tested the technology, which was the event of a first-principle reappraisal of flying mechanics of non-stock-still-wing aircraft [18]. Leylek and Costello [xix] performed a parametric study and stability analysis of a similar aircraft concept which uses a combination of active and passive articulation. Cuji and Garcia [20] analyzed the force distribution on morphing aircraft wings which modify shape to yield variable span-wise dihedral. They focused on turning flight and demonstrated that asymmetric wings produce a reduced load gene for every value of the turn rate. Obradovic and Subbarao [21,22] computed the power requirements for wing morphing nether dynamic loading from maneuvers and identified cases nether which morphing is more efficient than traditional control mechanisms. Nonflapping aircraft with stock-still as well as articulated wings have been used extensively to study the perching maneuver [eighteen,23-27,52], which is unique to birds. The perching maneuver

has the potential to be adjusted by bird-scale micro aerial vehicles (MAVs) to significantly improve their portfolio of maneuvering and mission capabilities.

The nowadays review newspaper focuses on bird-scale flight, and complements the aforementioned papers. Information technology is worth noting that although bird-similar aircraft, such as the Festo SmartBird and the Aerovironment Hummingbird, accept been developed in the recent years (run into Fig. 1), and non-flapping aircraft take drawn considerable inspiration from bird flight, the academic literature on the flapping flight of birds is sparse compared to insect flight. 1 event is that there are very few results on the stability and control of flapping flight of birds. This paper attempts to consolidate the existing results in a tutorial-like framework.

In this paper, we review flapping flight of birds from a flight mechanics and command perspective. We review the get-go principles of flapping flight, and present results on stability and control from the literature. In Section ii, we derive the equations of motion of a flapping wing aircraft. In Section three, we review flapping wing kinematics. Stability and control of flapping flight are discussed in Sections 5 and 6. Ii case studies, the Festo SmartBird and a robotic bat testbed developed past the authors, are presented in Section 7.

2. Equations of Motion

In this section, we state the equations of motion for a rigid flapping wing aircraft. The reader is referred to [17] for a complete derivation, and to [xvi] for a derivation of the equations of motion of an aircraft with flexible flapping wings. The equations presented in this department have been borrowed from [17].

   2.1 Fly Kinematics

Two approaches are commonly used to model the kinematics of a flapping wing. The first and more mutual approach starts by identifying the stroke plane of the fly, which is divers by the fly root and the 2 extreme positions of the wing tip during a flapping bicycle. The stroke plane is visually convenient, and its orientation correlates well with maneuvers. For example, the stroke plane is almost horizontal in hovering flight, and almost vertical in forward flight. A drawback of using the stroke planes for modeling is that the resulting kinematics are relatively more cumbersome to codify.

The 2d approach is more convenient and models the wing motion equally a limerick of standard Euler rotations. Wing motion represented by Euler rotations is not hard to visualize. In fact, it directly describes the physical facets of flapping movement, viz., pb-lag, flapping (up-and-downward beating), and twist.

Let the matrix T _FG denote the rotation matrix which transforms the components of a vector from the K frame to F, where the frames F and 1000 are capricious. The frame R is the frame based at the right wing root. Information technology is related to the B frame via a sweep rotation ψ_R at the wing root, followed past dihedral rotation Ø_R and a twist rotation θ_R near the y axis. Let R_one, R₂, R₃ denote the trunk-to-wing frame rotation matrices for wing rotations most the root hinge respective to lead-lag (ψ_R ), dihedral (Ø_R ) and incidence (θ_R ), respectively. Therefore,

The following rotation matrix connects the right-wing root frame to the trunk frame:

A like matrix T _BL (δ_L,θ_L ) tin can be derived for the left wing.

   2.2 Local Velocity and Force Adding

Without whatever loss of generality, consider the right wing of an aircraft, with (semi) span b/two and chord c(y), where y denotes the spanwise location. Let V _∞=[u v w]^T denote the body axis wind velocity of the shipping. Permit ∞_B =[p q r] denote the body axis angular velocity of the fuselage.

The angular velocity perceived at a spanwise strip at a distance y forth the bridge is given by

and the local velocity at that strip on the right wing is

where r_ac is the position vector of the aerodynamic middle of the station given past

and ten_{air conditioning} is the chordwise location of the aerodynamic heart with respect to the mid-chord. The local aerodynamic strength at the station is given by the vector sum of the lift and the drag, with components calculated in the trunk frame:

where

Detailed expressions for C_L and C_D are given in Sec. 2.4. The local aerodynamic moment at the station is given past

The total aerodynamic strength and moment are obtained by integrating the above expressions, performed in practice by using strip theory [28].

   2.3 Equations of Motion

In the following equations, given a vector p=[p_ane, p₂, p₃ ], define the cross product matrix operator

Let r_cg denote the position vector of the centre of gravity (CG) of the shipping, while r_{cg, R} , and r_{cg, L} denote the position vectors of the CG of the right and left wings, respectively. The translational equations of motion are given by the following vector expression [17]:

and the CG variation is given past

This CG variation could play an of import role in cases where the wing weight is substantial and where the CG position is used as a control variable, as in [10]. The CG variation is rarely used by birds, only can exist used past insects and insect-size aircraft. Information technology is as well used for decision-making underwater vehicles.

Therefore, the equations of rotational movement are given by [17]

where

and

In the above equations, J_{R, R} and J_{L, 50} announce the moments of inertia of the correct and left wings, respectively, in their respective local coordinate frames based at the wing root.

The kinematic equations relate the angular velocity of the aircraft to the rates of change of the Euler angles:

The equations which chronicle the position of the aircraft to its translational velocity are essentially decoupled from the flying dynamics, and are given by

Finally, the flight path angle (γ) and the wind axis heading angle (χ) in equation (xviii) are defined as follows:

   2.4 Aerodynamic Models

The aerodynamic model presented by DeLaurier [28] is one of the most widely used aerodynamic models in the flapping flight literature. Information technology incorporates the unsteady added mass result, delayed stall equally well every bit downwash due to a finite fly. The model is in the class of explicit analytical formulae for computing the forces and moments at every span-wise station on the fly, and the blade element theory is used for computing the net forces and moments. The model is limited past its use of a linear C_L-α relationship, which restricts its use to large, irksome flapping ornithopters.

Goman and Khrabrov [29] presented a model for an aquiver airfoil that is applicable to high α flight. Information technology incorporates a nonlinear C_L-α human relationship, valid at post-stall angles of set on, and delayed stall is modelled as arising due to chord-wise movement of the flow separation signal on the upper surface of the wing. They take a similar nonlinear model for computing the quarter-chord pitching moment. Still, the model is obtained for airfoils rather than finite wings. Bommanahal and Goman [thirty] presented a loftier allegiance model based on Volterra series for oscillating rigid airfoils which tin can also be applied to flapping wings.

Another popular model used in the literature is the finite state model of Peters and co-authors [31,32], which is motivated by and improves upon the classic model of Theodorsen [33]. In item, instead of using Theodorsen's function, the finite land approximation yields a closed-form analytical model.

Goman and Khrabrov's model offers at least two advantages over the other existing models. First, the model is cast in the class of a unmarried ordinary differential equation (ODE) and 2 algebraic equations, one each for lfte and the quarter chord pitching moment. The land variable for the ODE corresponds, physically, to the chordwise location of menstruum separation on the airfoil. Therefore, the model is quite easy to implement every bit part of a numerical routine. 2nd, the model is inherently nonlinear and applicable to poststall weather.

The following equation describes the motion of the separation signal for unsteady catamenia conditions

where τ_one is the relaxation time constant, τ₂ captures the time delay effects due to the flow, and five₀ is an expression for the nominal position of the separation betoken. These three parameters are identified experimentally or using CFD. The coefficients of lift and quarter-chord moment are then given by

At that place is, unfortunately, no uncomplicated expression for the sectional drag coefficient. Assuming laminar menstruation on the wing, the exclusive drag coefficient tin exist written as

where A_R is the aspect ratio of the wing,

is the chordwise Reynolds number, and eastward is Oswald'southward efficiency factor. A refined model for calculating elevate, incorporating dynamic stall, may be found in DeLaurier [28]. Notation that inertial contributions from the motion of the surrounding air need to be added to the forces computed using the above coefficients.

3. Flight Mechanics of Flapping

The kinematics of flapping are different in forrad flight and hover. In forward flight, the fly primarily flaps and twists, and the lead-lag movement, if any, is strictly for the purpose of control. On the other hand, while hovering, the atomic number 82-lag motion is as important as the other 2 degrees of liberty. In this section, we separately consider simple theoretical models of forrard flight and hovering. The purpose of this modelling is to understand the stage relations betwixt the three degrees of freedom, and decide ways to choose the amplitude and bias value of each degree of freedom.

   3.1 Model

In this department, we consider a rigid fly. Since the phase relations between the unlike degrees of freedom are independent of the spanwise location on a rigid wing, we consider a single representative spanwise location. Without loss of generality, suppose that the angle of set on of the aircraft (defined with respect to the fuselage reference line) is goose egg. And then, the local velocity vector V on the right wing at a altitude b from the root is given by

not counting the effect of θ, the wing twist. The oscillatory motility of the fly is given by

so that

Information technology is worth noting that sinusoidal functions in the higher up expression can be generated using nonlinear oscillators, such as the central pattern generator (CPG) networks described in Sec. 6. And so, the local angle of attack is given by

Assuming linear a erodynamics and ignoring the added mass effect, the cycle-averaged values of lift and thrust are given past

where c is the chord length.

   three.2 Forward Flight

In frontwards flight, we set ψ_a =0. Thus, from (23), nosotros tin can write

Substitution into (27) yields the following expression for bike-averaged values of lift:

We deduce that the boilerplate value of elevator at a given V_∞ depends only on two variables: the bias value of wing incidence angle, θ_c , and the peak flapping speed of the fly, given by Ø_aω. Interestingly, the stage difference ξ_？ does non change the cycle averaged value of elevator, which is a consequence of choosing a linear aerodynamic model.

The role of the phase divergence ξ_？ becomes apparent when one computes the thrust produced during a flapping wheel. Since the thrust is proportional to ∫ V²α (t)cos(ωt)dt (from (27)), we compute : α(t)cos(ωt)

It follows that thrust is maximised when nosotros cull ξ_？ = π/ii, i.e., when

The wheel-averaged value of thrust is given by

The in a higher place results demonstrate that both thrust and lift increase for a given V_∞ with increasing

The higher up results besides suggest an interesting point: it is possible, at least in principle, to produce thrust without using pitch oscillations, i.east., by setting θ_a = 0. Alternately, at least within the limits of linear aerodynamics, θ_a tin exist increased to obtain greater thrust, while the pick of the bias parameter θ_a can be dictated by elevator requirements alone.

Remark: The term

is usually referred to as reduced frequency, and b is replaced in the standard definition of reduced frequency by c, the chord length. The term

is a scaled version of the Strouhal number, and it is a measure of whether the flow is dominated past viscosity and vortex shedding (Strouhal number ？ane) or fast quasi-steady motion (Strouhal number ？10^-4).

   three.3 Hovering Flight

In hovering flying, V_∞ ？ 0, so that (26) becomes

Clearly, in order for α to exist finite and the bike averaged lift in (27) to be positive, we need cos(ωt + ξ_ψ ) = ？cos(ωt), and then that the stage departure between lead-lag and plunging motions is given past ξ_ψ = π. The choice of θ _o and θ_a can be made on the basis of lift and thrust requirements, respectively, every bit in the case of forrad flight. Note, however, that the cycle averaged drag will not exist naught and hence a non-naught wheel averaged value of thrust is required to maintain the hover.

Remark: The lead-lag motion is a secondary motility in frontwards flight. The stage relationship obtained here for hovering is indeed used in forward flight too, e.g., in the CPG-based scheme in [34].

   3.4 Force Product during Fast Flying and Hovering

Although the preceding word in this section derived conditions under which positive lift and thrust can be generated in a flapping cycle, it did not specifically highlight the distribution of forces in a given wheel. A typical flapping cycle consists of two strokes: a downstroke where the wing flaps downward in forward flight (or frontward in hovering flight), and an upstroke. Figure 2, taken from [vi], shows the typical flapping cycle of a pigeon in forward flight. Sketches (A - C) bear witness the downstroke, while (D - E) show the upstroke. The wing produces both elevator and thrust predominantly in the forward downstroke. During the upstroke, the wing nonetheless produces some lift, but little or no thrust. Annotation the bent outer segment in Sketch D: this is a event of a degree of passivity in its hinging at the root, i.e., where it is fastened to the inner wing. This folding of the wing reduces the drag produced during the upstroke. For a modest role of the upstroke, the fly tip does provide a modest amount of propulsive force, presumably due to a delayed reversal of motion as compared to the inner wing.

Figure 3 shows the downstroke and upstroke in slow flight (which is not exactly hovering, just a close analog). In boring flight, the office of upstroke and downstroke are reversed. A bulk of lift and thrust are obtained from the upstroke [six]. On the other hand, the downstroke yields some lift, but no meaning propulsive force. In particular, the propulsive force during the upstroke comes from the rapid, almost instantaneous, pronation and extension of the wing shown in Sketch C of Fig. 3(b) [6].

This give-and-take serves to illustrate a limitation of the discussion in the previous department where the kinematics of hovering were modelled using a start-principls approach. The effects of the rapid wing "pic" are about incommunicable to capture in that framework, but it provides a bulk of the propulsive strength and therefore cannot be ignored in force and moment calculations. Such phenomena correspond a challenge even to the general aerodynamic modelling of flapping flight.

iv. Effect of Nonlinear Aerodynamics and Fly Flexibility

   4.1 Aerodynamics

There are several important nonlinear effects that affect the aerodynamics of flapping wings. Broadly, their influence depends strongly on the Reynolds number, Re, i.e., on the size and speed of shipping. The contributions themselves can be split into two sets: (1) those that change the circulatory lift, such as by deforming the C_L-α curve, and (2) non-circulatory terms which are generated past inertial furnishings.

The aerodynamics at moderate to high Reynolds numbers (Re>10⁴) are dominated primarily by traditional circulatory mechanisms of lift and thrust generation. The other significant contributor is the added mass effect, which may contribute up to twenty% of the internet aerodynamic strength on the shipping, depending on the weight of the shipping.

The delayed stall effect primarily leads to flapping-phasedependent hysteresis in the C_L-α curve, causing the value of optimum stage difference between pitch and flapping (derived in Sec. 3.2) to shift from 90deg. In fact, the optimum value stated in the literature is approximately 95deg, and this value is equally influenced by the structural flexibility of the fly.

The aerodynamics at depression R_e are strongly driven by unconventional mechanisms, notably wake capture and delayed stall, the latter caused by the stabilization of leading edge vortices on the wing. These furnishings together contribute nearly 30% of the internet lift [ix]. In addition, insects (which are the stereotypical representatives of low R_{due east} flight) are known to use unconventional inertial mechanisms such equally clap-and-fling which brand utilise of the added mass effect for producing lift and thrust [1].

The effectiveness of the unconventional mechanisms listed in a higher place is primarily a issue of the rapid flapping of insect wings. Whereas birds and bats (high R_e fliers) typically crush their wings at frequencies of roughly 5-10Hz, insect wings are known to beat at frequencies running from 100Hz to 250Hz. At such flapping frequencies, although the aerodynamics themselves are highly nonlinear, the flying dynamics and control themselves are unaffected by the transient backdrop of the aerodynamics and depend entirely on the cycleaveraged values of the aerodynamic forces and moments. On the other other hand, quasi-steady approximations of aerodynamic forces and moments lucifer poorly with actual values in case of birds and bats.

   4.2 Effect of Wing Flexibility on Force Product

Wing flexibility affects the efficiency of flapping flying in 3 ways, by changing (a) the local air current speed, (b) the local angle of set on, and (c) the phase relations between twisting, flapping and lead-lag. To overcome the detrimental furnishings of this altered stage relationship, a different phase relation from that of a rigid wing must be commanded at the wing root [35].

A comprehensive experimental study on the consequence of flexibility on flapping wing propulsion was performed by Heathcote, Gursul, and co-authors [36,37]. They considered three wings: inflexible, flexible, and highly flexible. For spanwise flexibility, their results showed that a moderate degree of flexibility offers a considerable comeback over a rigid wing, but a highly flexible wing shows a considerable deterioration in performance. They signal out a shut correspondence between the Strouhal number (measured as a function of mid-span amplitude), and force production and efficiency. The propulsive efficiency, in particular, peaks for Strouhal number ？ 0.i-0.two. At college Strouhal numbers, a moderately flexible wing shows a marked improvement in propulsive efficiency.

For chordwise flexibility, they observed that although the thrust produced by the wing increases with increasing flexibility, so does elevate. Thus, a moderate corporeality of flexibility is still the optimal configuration.

Flexibility plays another role in flapping flying, namely reducing the sensitivity of the fuselage to gusts [38] and periodic disturbances from flapping. Passive flexible joints are known to help in flow control and delaying wing stall. Equally shown in Sec. three, they also assist help the wing to generate elevator and thrust through unconventional mechanisms.

v. Stability

There are very few results describing a formal stability analysis of flapping flight. A probable cause for this paucity is a conventionalities that the stability of an aircraft in flapping flight can be related to that in gliding flying, under the assumption that flapping frequencies typically exceed the natural modal frequencies of the airframe [39]. This is occasionally used to justify a quasi-steady modelling of flapping flight aerodynamics. It is instructive, therefore, to review the stability of bird-sized aircraft in gliding flight.

   5.1 Stability of Gliding Flying

Birds lack a vertical tail, which could potentially render them inherently unstable in yaw, depending on the relative location of the center of gravity and the wings. It has been argued by Taylor and Thomas [38] and Sachs [40-42] that birds are laterally-directionally stable despite the absence of a vertical tail.

The stability of birds comes from iii sources: (i) elevate, (2) lift, and (3) pendulum effect. Taylor and Thomas [38] showed that drag and pendulum effect are the dominant contributors to stability. The fly itself, according to them, is sufficient to provide longitudinal stability provided it is located behind the center of gravity. Sachs [41,42] derived belittling approximations to the standard flight dynamic modes (short period, screw, Dutch coil), and showed that the wings are indeed sufficient to provide even lateral-dynamic stability. The stability is largely a result of a favorable placement of the CG with respect to the wing.

In contrast with the arguments in the aforementioned references, Paranjape, Chung, and Selig [17] argued that birds would nearly probable be laterally-directionally unstable under routing flying weather. The nature of the instability, arising from the Dutch roll mode, depends on the fly dihedral angle. For big dihedral angles, the Dutch coil mode is indeed stabilized, only such big dihedral angles are rarely used during gliding flight in the midst of soaring or cruising.

Wing flexibility is believed to play a role in stabilizing the airframe, reducing its sensitivity to gusts, and in improving the performance. It was shown by Paranjape and coauthors [xvi] that flexibility does non necessarily bring well-nigh a significant improvement in the operation, and can in fact degrade certain metrics such as the coordinated (nada sideslip) turn rate by reducing the trim speed for a given tail setting. Moreover, unless the fly is highly flexible, at that place is no qualitative difference in the stability of rigid and flexible wings. Therefore, it is condom to conclude that wing flexibility helps in making the wing and the shipping lighter, improves the efficiency of passive mechanisms, and even aids flow control, simply does not, by itself, improve the traditional flight mechanic operation metrics and stability.

In [43], the authors used an approach identical to [17], but replaced the aerodynamic model with a loftier-elevate model. They demonstrated that the lateral-directional dynamics tin be stabilized by drag. In fact, increasing the drag coefficient alone can stabilize the dynamics completely [44].

   five.ii Stability of Flapping Flying

The stability of the airframe during flapping flight has been as much a matter of contention as that of gliding flight. Taylor and Thomas [39] argued that flapping wing aircraft are stable longitudinally also as laterally-directionally, although they lack a vertical tail. The stability in pitch is largely a outcome of the horizontal tail, but is also a event of the flapping kinematics [45]. Mwongera and Lowenberg [45] argued that forces arising from circulatory mechanisms tend to be stabilizing, while those that ascend from translational mechanisms (such as the unsteady added mass event) practice not contribute to stability. Consequently, they ended that flyers such as birds tend to be stable, while insects do non tend to be stable. The survey of the stability of insect flight in [2] complements this observation. A study of the modal structure of longitudinal insect flight dynamics past Leonard [46] showed that the instability in insect flight arises primarily from a slow mode.

Flapping motion gives rise to limit cycles rather than equilibria in the country-parameter space. Stability assay of limit cycles is performed past calculating the Floquet multipliers of the linearized dynamics most the limit cycles (much like the eigenvalues of the linearized dynamics well-nigh equilibria) [47].

Bifurcation assay is one of the near sophisticated and generic methods for analysing the global stability of nonlinear systems. Numerical continuation methods are used to compute the steady states (equilibria and the limit cycles) of the arrangement, together with the respective eigenvalues or Floquet multipliers. Bifurcation and continuation methods have been used widely to predict instabilities in flight dynamics, aircraft structures, and integrated aircraftstructure- propulsion systems [48].

The first awarding of bifurcation methods to flapping flight was reported recently by Mwongera and Lowenberg [45]. They considered an MAV consisting of two wings, each with a span of 10cm, and a fuselage, but no tail. They used continuation and bifurcation methods to report the stability of the longitudinal flapping dynamics for unlike flight atmospheric condition as well as for varying the longitudinal position of the wings.

Interestingly enough, Mwongera and Lowenberg's study concluded that the longitudinal stability depends primarily on the flapping frequency of the wing, with secondary dependence on the longitudinal position of the fly. The latter observation is in stark contrast to the conventional understanding that placing the fly behind the CG ensures pitch stability and vice-versa. In this detail case, it was seen that the lead-lag motion of the wing supplied the necessary stabilizing moments. Moreover, the observed instabilities were largely benign. Figure 4, reproduced from [45], shows the flapping limit cycle amplitudes, together with their stability, every bit a function of the flapping frequency and longitudinal position of the wing. The unstable regions in Fig. 4 are obtained via flow-doubling or Neimark-Sacker bifurcations, which give rise to quasi-periodic beliefs [47].

Dielt and Garcia [49] reported a stability analysis of the longitudinal dynamics of a bird-sized ornithopter with a fly bridge of 72cm. They observed unstable longitudinal dynamics, where the instability was divergent, and the respective eigenvector afflicted all longitudinal states more or less uniformly. The unstable manner was fast (compared to the deadening unstable modes in the prior references in this section). Additionally, a stable phugoid-like irksome mode was also detected, along with a fast stable mode.

One could ponder about the possibility of a correlation between the stability of flapping and gliding flight of an shipping under identical weather condition (flying speed and angle of attack). In that location is no conclusive evidence to suggest any correlation. The most obvious analog is flutter: a fly whose plunging and twist dynamics are themselves stable in isolation can still undergo flutter due to adverse phase relationships betwixt plunging and twisting. Morever, from the work of Mwongera and Lowenberg [45], it appears that an airframe that is unstable in gliding could be rendered stable due to flapping. Whereas the lack of a correlation does not appear surprising, it strikes at the root of the rationale behind quasi-steady modelling of flapping flight for stability analysis. Quasi-steady modelling may not work for stability prediction because it leaves no room for instability induced by adverse phase relations betwixt the different elements of the flapping flight dynamics, since it implicitly assumes a stably beating fly interacting with an approximately static fuselage.

Moreover, quasi-steady modelling of the aerodynamics is likely to yield erroneous estimates even of the performance, because medium and big sized birds flap their wings at frequencies which are comparable to the natural frequencies of the air frame. This is one of the reasons why unsteady aerodynamic modelling of flapping wings is essential for analysing bird flight.

6. Flight Control

In this department, we review contempo work on control of flapping flight. Specifically, we focus on ii aspects of control: the pick of control inputs and the choice of command methods.

   half dozen.one Control of Gliding Flying

Control of gliding flight appears at first sight to be no different than the control of conventional fixed fly aircraft. However, at that place are some crucial differences: (1) birds lack a vertical tail and a rudder, and (2) the control system in birds is overactuated. In fact, most birds can exert at least 8 control inputs: three degrees of freedom on each wing and two on the horizontal tail (rotations about the in-plane axes). Moreover, birds can control the deflections of their fly leading edge and trailing edge feathers, likewise equally feathers on elevation of the fly surface. Together, the feathers play the roles of ailerons, trailing edge flaps, leading edge slats and wingtop spoilers on conventional shipping. Thus, strictly speaking, the problem of matching the desired control input to the advisable command surfaces represents a trouble in control allocation in over-actuated systems [fifty]. To the best of our knowledg, the literature is devoid of reports wherein this arroyo has been applied bird-scale MAVs.

In lodge to judge the capabilities and limitations of command inputs bachelor to birds, it is occasionally instructive to consider their opposite-engineered instances in the MAV literature. For instance, Abdulrahim et al. [51] optimized the wing twist actuation for a flexible membrane-like wing for achieving a rapid ringlet rate. Paranjape, Chung and coauthors [16, 17] developed an MAV concept which uses the wing dihedral for longitudinal as well as lateral-directional control. The concept shed additional insight into the roles of quarter chord pitching moment and abaft edge flaps in yaw command, and was flying tested successfully [xviii]. The MAV adult past the authors has been shown in Fig. five, while Fig. 6 shows a perching maneuver performed past the MAV using articulated wing-based control. The Festo SmartBird (cf. Sec. 7) uses a two-degree of freedom horizontal tail for pitch and yaw control, and the fly dihedral is varied symmetrically for controlling the flying path.

Gliding is of import in birds considering information technology helps to conserve energy in flight. It allows birds to extract energy from the surrounding air flow to increase their endurance, a process known as dynamic soaring [53]. Even without the possibility of dynamic soaring, which requires specific wind atmospheric condition, it was shown by Sachs [54,55] that switching between flapping and gliding flight can in fact yield a much improved performance, even in terms of the flight speed, over optimized steady state flapping flying.

In one case the control inputs are chosen, the control problem involving stabilization and tracking tin can be solved by any of a vast number of well-established methods, although methods such as adaptive command [56] or dynamic inversion [18,57] may exist required to address problems arising from nonlinearities from an unconventional pick of control inputs. Occasionally, if the wing is highly flexible, a command approach which incorporates wing deformation may need to employed to stabilize the elastic dynamics of the fly and

ensure that information technology produces the desired forcefulness and moment [66].

   vi.2 Choice of Control Inputs for Flapping Flight

The modelling in Section 3 shows that there is a broad variety of possible control inputs for flapping. They are listed in Table 2, together with their primary effectiveness. Nosotros also indicate sources in the literature where they have been employed.

There is clearly a considerable diversity in the choice of command inputs. Chung [34] and Bhatia [58] used the kinematics of lead-lag motion of the two wings to command the motion. In fact, Bhatia [58] demonstrated that, for hovering, LQR command of the lead-lag motion alone is more robust than LQR using larger sets of control parameters. This is to be expected since atomic number 82-lag motion is the primary wing motion during hovering flight. Chung and Dorothy considered frontward and turning flight in [34], and although their choice of control variable was motivated just by physical intuition, it was seen to exist equally effective. In add-on to atomic number 82-lag control, they leveraged wing beating frequency, the stage deviation between flapping and pitch, and flying mode switching to accomplish multiple tasks, including altitude/ velocity regulation and smooth turning.

The SmartBird developed by Festo (cf. Sec.) used a V-tail for pitch as well as lateral-directional control. Therefore, despite flapping-based propulsion, the three- axis control of the SmartBird was essentially identical to that of a stock-still fly aircraft.

Hedrick and Biewener [59] observed the turning flight of cockatoos and cockatiels, which differ considerably in size and speed. They observed that both birds used asymmetry in the flapping and feathering amplitudes for roll and yaw control. This can be explained along the lines of dihedralbased yaw command mechanism proposed in [17]. Asymmetric feathering yields direct ringlet control, but very lilliputian yaw control. On the other mitt, disproportionate flapping (i.east., asymmetric dihedral) provides direct yaw command, merely very little roll command. Therefore, feathering and flapping act as independent roll and yaw control mechanisms, respectively. The reader is referred to Orlowski and Girard [2] for a like table of command inputs found in the liteature on insect flying.

In contrast to birds, whose wings are structurally more or less undeformed, bats deform and slant their wings significantly in flight, as demonstrated past Breuer and coauthors [threescore]. Their wings are cambered and fully stretched during downstroke, and folded inward during the upstroke. This helps to reduce the drag, and particularly since the upstroke contributes no thrust either. Birds are known to fold their wings to maneuver rapidly, such as to perform butt rolls, merely not systematically in a single stroke as in the case of bats. The main reason is that bat wings are fabricated of

skin, which acts like a flexible, malleable membrane, while feathers that make up bird wings are more or less rigid. This feature may contribute to a bat'south ability to consummate a 180deg turn in approximately three wingbeats [threescore].

For turns, birds as well as bats bank considerably, turning the lift vector inward [half dozen,61]. However, banking is not solely responsible for turning. D?az and Swartz [62] estimated that, for bats, at most lxx% of the required turning forcefulness was due to banking. The remaining portion was the effect of a crabbed turn - irresolute yaw orientation during upstroke and flying direction in the subsequent downstroke [62]. In contrast, studies of turns performed past the Parajape et al. [16,17] showed that when asymmetric dihedral (or flapping angle) is used in gliding turns, the body depository financial institution bending is considerably smaller (less than 20deg) even for large plow rates.

   6.three Control Methods for Flapping Flying

The survey papers by Orlowski and Girard [2], and by Taha, Hajj and Nayfeh [3] give a comprehensive review of control methods employed commonly for flapping flight aircraft. In this paper, we review 2 control approaches, each of which sheds lite on a fundamental aspect of decision-making flapping flight. The first approach is based around cardinal pattern generators (CPGs), and uses synchronization properties of coupled oscillators [34]. The second approach is based on linear quadratic regulator (LQR) control, and indicates the importance of specific control effectors and on the relevance of specific state variables for feedback.

Chung and Dorothy proposed a CPG-based controller, leveraging the backdrop of the symmetric Hopf oscillator [34]. The key thought was to produce shine signals for multiple motions while assuasive for nifty flexibility in toplevel controller blueprint. Such a coupled oscillator network could easily incorporate frequency, amplitude, and phase difference modulation. They used all iii types of control logic - frequency for velocity control, aamplitude for yaw control, and stage difference for roll and pitch command. Such a CPG network could also reproduce intra-wingbeat frequency controllers like the split-cycle [11] without requiring any analytic solutions [63]. Effigy 7 shows the schematic of a CPG array for the 2 wings. The start plot shows a nominal, symmetric configuration for forrard flight. The nominal phase differences were derived in Sec. 3. The second plot shows a configuration where the phase difference betwixt the lead-lag motion of the 2 wings is used as a control input, with the plunging motion of the two wings retained in sync. This is not a unique choice of control inputs, but serves to illustrate how the stage differences and symmetry-breaking tin can be honed for control. A block diagram showing an implementation of CPG-based control is shown in Fig. 8.

Bhatia and co-authors [58] presented an LQR-based controller for hovering flying in the presence of gusts. Their metric for evaluating controllers was the maximum speed of a transient gust that the controller could withstand. They designed an LQR controller and systematically scaled the penalty functions and varied the choice of control inputs, while evaluating the maximum tolerable gust speed. They concluded that controlling the aamplitude and bias of leadlag motion is not just sufficient just too the most constructive way of achieving tolerance to gusts. They also demonstrated, at least for their detail model, that it is necessary to feed back athwart positions and angular rates for increased gust tolerance, while translational position and velocity feedback play a comparatively insignificant role. The final version of their controller (obtained after the parameter study) yielded satisfactory tolerance to longitudinal gusts whose speeds matched the tip speeds of the wing, and to lateral gusts with a speed equal to a 3rd of the tip speed.

vii. Mechanical Implementation

In this section, we will describe two examples of mechanical implementation of bird-scale flapping flying - the Festo SmartBird and a robotic bat testbed adult at the Academy of Illinois at Urbana-Champaign (UIUC). The purpose is to consider practical blueprint issues that arise in the implementation of the aforementioned ideas, which were presented largely from a theoretical standpoint, too as solutions used in practice.

   7.1 Festo SmartBird

The SmartBird, designed by Festo, is probably the commencement successful flapping wing remote-controlled aircraft which mimics some relevant characteristics of avian flight (in this case, a sea gull). The aircraft incorporated several technologies, and most details are unpublished. We will summarize some relevant design features, gathered from the production brochure, and relate them to the theoretical results presented in the previous section.

Each wing of SmartBird has two segments. The flapping motion of the outer segment is non synchronized actively with that of the inner segment, but is instead coupled to the inner segment passively. The twisting motion of the wing, withal, is controlled actively for optimizing the lift and thrust produced during a flapping bicycle. No additional lifting devices are used. Interestingly, the fly is designed to be rigid in torsion despite its size, although reasons for this blueprint choice are unknown.

The inboard segment primarily generates lift, while the outboard segment provides thrust. This separation of roles is also seen in large birds such as sea gulls and swans. The SmartBird utilizes a horizontal tail with two degrees of freedom: information technology can deflect about the transverse axis for pitch control, and nearly the longitudinal axis of the aircraft for yaw control. Roll control is achieved by controlling the torsion motility of the 2 wing.

   7.2 Robotic Bat

The RoboBat was developed to investigate the effectiveness of dissimilar control strategies on forces and moments [27,64,65]. RoboBat incorporates six degrees of freedom (flapping, lead-lag, and pitch for each fly), which would be synchronized and controlled via a CPG network. Each wing is driven by a single DC motor, while the phase departure between the different degrees of freedom is controlled past servo motors. Figure 9 shows the Robotbat testbed mounted

in a air current tunnel for early testing, as well as the current south experimental setup where information technology is mounted on a rotating 3-DOF pendulum.

In order to test airtight-loop CPG control, information technology was placed on a compound pendulum, restricting the arrangement to longitudinal modes but. The Quanser-built encoder interface integrated in the pendulum provided orientation and velocity feedback. Phase differences between flapping and lead-lag proved to exist effective in stabilization and control [27]. An example trajectory for pitch control is shown in Fig.10. However, the organisation was not equally sensitive to command input equally indicated by prior simulations, as the compound pendulum arrangement increased the pitch moment of inertia.

8. Conclusions and Future Work

In this paper, we surveyed the literature on flapping flight of birds and bird-like airplanes from a flight mechanics perspective, in a tutorial-similar setting. Stability and control of flapping flight were addressed with insightful case studies from the literature. Open problems in flapping flight incover both stability and control. In particular, very little is understood about lateral-directional stability of birds and bird-scale aircraft in the flapping phase. Quasi-steady aerodynamic modelling, which forms the cornerstone of a considerable torso of work on the modelling and analysis of flapping flight, presents a strong possibility of erroneous stability and control results in bird-calibration flapping flying due to a close lucifer between the typical flapping frequencies and the natural frequencies of the flying dynamics of the air frame. Flapping fly aircraft, dissimilar typical stock-still wing aircraft, are over-actuated, which presents as yet largely unsolved problems in command allocation. Despite these shortcomings in our noesis of flapping flight, there are some instances of flapping wing MAVs being developed and flown successfully by the academia too every bit the industry. A deeper understanding of stability, coupled with sophisticated schemes to optimally uilizse the multitude of control inputs, will significantly heighten the performance and maneuverability of flapping wing aircraft in the hereafter.

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Source: http://oak.go.kr/central/journallist/journaldetail.do?article_seq=11296

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