## Abstract

Flying animals ranging in size from fruit flies to hummingbirds are nimble fliers with remarkable rotational manoeuvrability. The degrees of manoeuvrability among these animals, however, are noticeably diverse and do not simply follow scaling rules of flight dynamics or muscle power capacity. As all manoeuvres emerge from the complex interactions of neural, physiological and biomechanical processes of an animal's flight control system, these processes give rise to multiple limiting factors that dictate the maximal manoeuvrability attainable by an animal. Here using functional models of an animal's flight control system, we investigate the effects of three such limiting factors, including neural and biomechanical (from limited flapping frequency) delays and muscle mechanical power, for two insect species and two hummingbird species, undergoing roll, pitch and yaw rotations. The results show that for animals with similar degree of manoeuvrability, for example, fruit flies and hummingbirds, the underlying limiting factors are different, as the manoeuvrability of fruit flies is only limited by neural delays and that of hummingbirds could be limited by all three factors. In addition, the manoeuvrability also appears to be the highest about the roll axis as it requires the least muscle mechanical power and can tolerate the largest neural delays.

## 1. Introduction

For flying insects and hummingbirds, orchestrating a repertoire of controlled aerobatic manoeuvres is essential for their aerial survival [1–5] and sexual selection [6–10]. In particular, they are masters of low-speed manoeuvres that demand precisely coordinated body rotations, such as (banked) yaw turns [11–14], pitch turns [15,16], pitch–roll turns [1,10,17,18] and arcing turns [1,14,19]. Compared with human-engineered fliers with fixed and rotary wings, these miniature flying animals gain higher control authority over their whole body movements by leveraging the unsteady aerodynamics of flapping wings [20–22] using elaborate neural [23,24] and physiological systems [25–27].

The manoeuvrability of these animals is remarkably diverse and does not simply follow scaling rules of flight dynamics [18,28] or muscle power output [29,30]. For example, figure 1*a* shows the scaling of total body mass against the cube of wing length for four animal species (table 1). Having similar wing length and time scales of dynamics, recent studies show black-chinned hummingbirds [17] achieve significantly higher linear and rotational manoeuvrability than hawkmoths [15]. On the other hand, despite three-orders difference in wing length, the peak roll and pitch rates (*ω*_{m}, table 2) in escape manoeuvres of magnificent hummingbirds [17] are similar to those of fruit flies [4]. The complexities in evaluating the flight performance of these animals arise, at least partly, from the fact that the flight performance depends on the complex interactions of an animal's neural, physiological and biomechanical processes. Therefore, to gain a system-level understanding of the consequence of each process on the resulting flight performance, modelling of the functional principles of these processes is needed, based on which feedback control theory can be used for performance analysis [15,16,33–36].

In this paper, we aim to understand how neural, physiological and biomechanical processes in an animal's flight system affect its rotational manoeuvrability in low-speed flight. Rotational manoeuvrability is considered here because fast coordinated body rotations are crucial to orient the averaged aerodynamic force of an animal to the desired flight direction and hence are critical to the linear manoeuvrability as well [4,37]. Particularly, here we consider the maximal manoeuvrability and its potential limiting factors arising from different processes in an animal's flight system. We first analyse a major limiting factor from the neural process, which is the time delay in the sensorimotor transduction. This is because nervous systems of animals have limited computation and transmission speed, therefore both stability and response time of their flight system unavoidably suffer from the time delay between sensory input and motor activation. Such delays have been previously identified for fruit flies [33,34,38], hawkmoths [39,40] and hummingbirds [17] through model fitting to free-flight data or by measuring their response time. The effects of the delays on the stability of animal locomotor system are also analysed in control-theoretic framework for cockroach wall-following [41]. The second limiting factor, which arises from the physiological process, is the maximum mechanical power available from the flight muscles. The limiting effects of muscle mechanical power on load-lifting performance in short-burst flight have been studied extensively for a large group of flying animals [29,42–44]; however, its limiting effect on flight manoeuvrability has rarely been investigated. The last limiting factor considered here arises inherently from the flapping wing motion, as the rate of wing kinematics modulation is constrained by the rate of muscle contraction–relaxation or wingbeat frequency. While this biomechanical limitation introduces discrete characteristics into the flight system, it can be also approximated as a form of delay.

We focus on two insect species: fruit fly (*Drosophila melanogaster*) and hawkmoth (*Manduca sexta*); and two hummingbird species: magnificent hummingbird (*Eugenes fulgens*) and black-chinned hummingbird (*Archilochus alexandri*). Note that these species are chosen for two reasons: first they have been widely studied in the literature with flight data available for assessing their flight dynamics and manoeuvrability; and second we aim to compare hummingbirds with fruit flies because despite drastically different body sizes, they have similar rotational manoeuvrability, and, by contrast, we will compare black-chinned hummingbirds with hawkmoths because they have similar body sizes but considerably different body mass and rotational manoeuvrability.

We first develop a functional model of an animal's flight control system considering only single-axis body rotations, which is a parsimonious but sufficient model for analysing low-speed rotational manoeuvres. Then, we investigate the effects of neural and biomechanical delays on the achievable closed-loop manoeuvrability based on this model by leveraging the flight data available in the literature. In addition, the stability and muscle mechanical power of rotational manoeuvres are also evaluated to further understand the limitation factors of rotational manoeuvrability.

## 2. Material and methods

### 2.1. Overview

Here we first derive a simplified functional model of an animal's flight system. This model will allow us to analyse the rotational manoeuvrability from a control-theoretic perspective. The flight system of an animal encompasses closed-loop interactions of several subsystems in a feedback loop (figure 2*a*) including neural control system, wing musculoskeletal system, aerodynamics, body dynamics and sensory system. Note that here we will not consider any internal models an animal may use to predict sensory consequences due to self-generated motion [45]. Such an internal model, if exists, may partly compensate the adverse effects of neural processing delay and improve the flight performance [5]. To reflect the functional principles and limiting factors of flight performance, a mathematical model (figure 2*b*) is then derived. Specifically, we use four separate functional blocks to represent (i) the control principle embodied by the combined neural and wing biomechanical systems, (ii) lumped neural delays from sensory inputs to motor outputs, (iii) the biomechanical constraints due to flapping wing motion, and (iv) open-loop body dynamics.

### 2.2. Modelling flight control system

#### 2.2.1. Open-loop body dynamics

The open-loop body dynamics of an animal can be modelled approximately using rigid body dynamics with six degrees-of-freedom translational and rotational motion [46]. The body of the animal is subjected to time-varying inertial and aerodynamic forces from the flapping wings and gravity [46,47]. This model can be further simplified using averaging and linearization around a flight equilibrium, e.g. hover, which leads to a linear time-invariant system [46,48,49].

To assess the rotational manoeuvrability at low flight speed, modelling single-axis rotational dynamics (figure 2*c*) is sufficient for the following reasons. First, the inertial coupling of different axes can be neglected because the insects and hummingbirds considered here have relatively small body length-to-width ratio, which results in relative small difference in body moments of inertia (MoI) about different principal axes. This can be confirmed by calculating the coupling terms using the kinematic and morphological data from escape manoeuvres in fruit flies [4] and hummingbirds [17,18]. However this would not be a legitimate assumption for dragonflies with an elongated body. Second, aerodynamic couplings and the resulting unstable modes in linearized flight dynamics can be also ignored, which is due to (i) the time scale at maximal manoeuvrability will be significantly smaller than that of natural modes [46]; (ii) single-axis stability ensures the stability of the coupled dynamics [46]; (iii) we are only considering the rapid rotational manoeuvre started from hovering or at low-speed flight, and therefore linear velocities are small throughout the manoeuvre and do not affect significantly the aerodynamics. Based on these considerations, the following differential equation that governs the single-axis rotational dynamics is derived:
2.1where *I*, *b* and *θ* are moment of inertia, damping coefficient and rotational angle with respect to the axis under consideration. In this equation of motion, the total aerodynamic moment acting on the wings is separated into those due to passive damping, , and active changes of wing kinematics with respect to hovering, *T*_{ac}(*t*), i.e. changes in both wing trajectory and flapping frequency. Here, the open-loop time constant *τ*_{p} is defined as
2.2

This time scale, *τ*_{p}, plays an important role in determining the natural response time of the open-loop system. By taking the Laplace transform of equation (2.1), one can derive a transfer function from moment *T*_{ac} as input and rotational angle *θ* as output,
2.3where *s* is the complex frequency.

To estimate damping coefficient *b*, we use blade-element analysis combined with quasi-steady aerodynamic models, the details of which are provided in Cheng *et al.* [18] and are briefly recapitulated here. The aerodynamic model calculates the aerodynamic forces and moments acting on each blade of a wing resulting from three distinct aerodynamic mechanisms: delayed stall (or translational forces), rotational lift, and added mass [50,51]. We superimpose body roll, pitch and yaw angular velocities individually to the hovering wing kinematics, and calculate the corresponding resultant stroke-averaged counter-moments based on the aerodynamic model. The damping coefficient *b* is then obtained by performing linear regression of the stroke-averaged moments as a function of the superimposed angular velocity, which linearly increases from −1500 to 1500 degree s^{−1}. The wing kinematic patterns used for damping estimation are collected from the literature for hummingbirds [17], fruit fly [4], and hawkmoth [15]. In addition, the aerodynamic coefficients used in the aerodynamic model are collected from the literature for hummingbirds [18], fruit fly [50] and hawkmoth [52].

#### 2.2.2. Neuro-biomechanical controller and neural delays

The neuro-biomechanical controller represents the lumped functional principle in neural and biomechanical processes that compares the measurements from the sensory system with the desired body motion trajectory from central nervous system (CNS), and outputs the aerodynamic control moments. As suggested by a number of previous studies [15,34,38], such a functional principle can be described parsimoniously by a proportional–integral (PI) controller (figure 2*b*). Specifically, by fitting the observed body kinematics to one or several hypothesized control models, these studies show that PI controllers are able to properly capture and predict the dynamic behaviours of these animals. This paper extends the work in [38] to include separate proportional and integral delays, i.e.
2.4where *K*_{P} and *K*_{I} are proportional and integral control gains, respectively, and are desired reference body angle and angular velocity, *T*_{p} and *T*_{i} are lumped delays in angular velocity and angle feedbacks respectively. Note that in many previous studies, proportional and velocity delays are not distinguished [16,38,41,53], i.e. *T*_{p} = *T*_{i} = *T*_{d}, where *T*_{d} is a general representation of the lumped delay. Neural delays arise from the time required for neural computation and transmission, and may occur anywhere in the sensory, neural control and motor systems. For example, it has been estimated previously that the total visual delay from visual stimuli to motor response in fruit flies is about 60 ms [4] and total mechanosensory delay from mechanical perturbation perceived by haltere to corrective motor response is about 10 ms [16,38]. Here we will further distinguish the delays in velocity and angle feedbacks. Note that an underlying assumption here is that body angular velocity *θ* is measured directly by an animal (e.g. through haltere in flies) and body angle *θ* is obtained through integration over time. Since it may take extra neural circuits and hence processing time to operate this integration, the integral delay, *T*_{i}, is expected to be greater than the proportional delay, *T*_{p}. Neural delays are considered as the primary limiting factor of the flight performance because they will reduce the stability margin and the bandwidth of a control system [16,33,54]. Estimations for the delays from various experiments are collected for hummingbirds [18], fruit fly [16,34,38] and hawkmoth [16], and the data are summarized in table 3 by . Taking the Laplace transform of both sides one can further obtain
2.5

Although the assumption that *T*_{p} = *T*_{i} = *T*_{d} represents an approximation to a more accurate model where *T*_{p} and *T*_{i} are different, it arguably captures the major effect of delay in the animal's feedback system and allows us to compare it with that from the biomechanical delay in a simpler fashion. Such a simplification also allows us to separate the delay from the PI controller as a separate transfer function, and equation (2.5) can be simplified to
2.6where
2.7
2.8

and the controller time constant *τ*_{c} is defined as
2.9

#### 2.2.3. Biomechanical constraints of flapping wings

Since insects and hummingbirds flap their wings in reciprocal manner with a finite frequency or muscle contraction–relaxation rate, the control input to the flight system, if seen as the muscle action potential, is generated approximately in a discrete manner, and therefore animals are unable to adjust their wing motion continuously. In this regard, the amount of the discrete control inputs during a wing stroke can be either one or two, and both cases are considered in this study. For insects and hummingbirds with synchronous flight muscles (e.g. pectoralis the supracoracoideus of hummingbirds) [55], we assume that downstroke and upstroke muscles can be each controlled once during a wingbeat cycle, and therefore in total wing motion can be controlled twice within a wingbeat cycle. For insects with asynchronous power muscles, such as flies, previous studies suggest that their synchronous steering muscles can be controlled only once within a wingbeat cycle (for example, first basalar muscle of blowfly fires a single phase-lock spike during each wingbeat) [56]. To model the effect of this biomechanical constraint, we approximate the flight system as a discrete control system. Specifically, we assume that the aerodynamic moment created by the flapping wings is maintained at a constant value within a (half) wingbeat cycle until the next cycle, which can be modelled as a sampling and zero-order-hold (ZOH) mechanism [57] (figure 2*b*). Note that although this is an approximation of the actual time-varying aerodynamic moment, it nevertheless captures the primary effect of limited control rate on the flight control due to flapping wings. Consequently, the closed-loop system becomes a sampled-data system, where both discrete and continuous signals appear [58]: the neural controller receives delayed continuous feedback information from sensory system and produces continuous control moment, which is then sampled discretely at every (or half) wing stroke (the sampling frequency is equal to (or twice) wingbeat frequency) and maintained as constant (ZOH) throughout a (or half) wingbeat cycle.

The effects of the biomechanical constraint in the sampled-data system can be studied by converting the flight system into a discrete one. The purpose of this conversion is to obtain the characteristic equation, which largely determines system behaviour. Consider the assumptions used to derive equations (2.6)–(2.8), if delays in angular velocity and position were not distinguished, the functional blocks of neural controller and delay can be separated (electronic supplementary material, figure S1*a*(i)). We further rearrange the sequence of the functional blocks (electronic supplementary material, figure S1*a*(ii)) to facilitate the derivation of a discrete equivalent of the initial sampled-data system (electronic supplementary material, figure S1*a*(iii)). It needs to be emphasized that this conversion maintains the characteristic equation although it results in different closed-loop systems. Finally, the discrete equivalent (or the loop gain of the discrete closed-loop system) can be derived as [58]
2.10where represents the z-transform. And the characteristic equation becomes 1 + *P*(*z*) = 0. The z-transform implicitly introduces to the system a sampling period, *T*_{s}, which is determined by the wingbeat frequency, *f*_{w} (table 2), as follows:
2.11where *n* represents the number of control input in one wingbeat cycle. As discussed above, for animals with synchronous flight muscles, such as hummingbirds and hawkmoths, *n* = 2; and for fruit flies with asynchronous flight muscles, *n* = 1.

A further simplification can be also made to only capture the signal-delaying nature of the biomechanical constraint, as one can show that a signal is delayed by half of the sampling period when passing through the sampling and ZOH [58]. As a result, an additional delay *T*_{b}, referred to as biomechanical delay, will be added to the flight system,
2.12

This simplification results in a flight system shown as electronic supplementary material, figure S1*b*.

#### 2.2.4. Nondimensionalization and closed-loop characteristic equation

In this section, based on the mathematical models derived above for each subsystems and with different assumptions corresponding to the biomechanical constraints and neural delays, two dimensionless closed-loop models are derived for an animal's flight control system. Both dimensionless models will be used to analyse the effects of biomechanical constraints and neural delays on the closed-loop time constant and stability, the results of which are then redimensionalized for different animals.

The first model (electronic supplementary material, figure S1*a*(i)) assumes that proportional and velocity feedback have the same amount of delay and the neural-biomechanical controller can be thus described by equations (2.6)–(2.8). It also uses sampling and ZOH to model the biomechanical constraint. Therefore, the characteristic equation of such a discrete system is 1 + *P*(*z*) = 0 (*P*(*z*) is the loop gain of the discrete equivalent system shown in equation (2.10)). In this model, there are a total of seven dimensional quantities, i.e. *s*, *I*, *b*, *K*_{I}, *τ*_{c}, *T*_{d}, *T*_{s}, and they reduce to five dimensionless quantities:
2.13where is dimensionless complex frequency, and are dimensionless controller gain and time constant, and and are dimensionless neural and biomechanical delays, respectively. Then, the dimensionless loop gain can be written as
2.14where and denotes nearest integer rounding operator. The characteristic equation becomes
2.15

The second model (electronic supplementary material, figure S1*b*), distinguishes neural delays in the angular velocity and angle feedbacks, and approximates the biomechanical constraint with a delay, *T*_{b}, of half of the sampling period *T*_{s}. The characteristic equation of this model can be obtained by substituting the delayed neural-biomechanical controller of equation (2.4), i.e. *T*(*t* − *T*_{b}), into the open-loop dynamics equation (2.1), and taking the Laplace transform of both sides of the resulting closed-loop equation of motion. The transfer function from reference angle *θ*_{r} as input and body angle *θ* as output is
2.16

Similar to the first model, we can reduce the eight dimensional quantities in second model to six dimensionless ones: 2.17and the transfer function in dimensionless form is 2.18

The characteristic equation is given by 2.19where the dimensionless loop gain is 2.20

### 2.3. Assessing closed-loop performance

In this paper we aim to understand rotational manoeuvrability and its limiting factors by evaluating the closed-loop time constant of an animal's closed-loop flight system (§2.3.1). While the closed-loop time constant indicates how quickly the response reaches a desired angular velocity, it does not provide any measure of system's stability. For example, a second-order system may have fast response but poor stability if the phase margin is low [57]. Therefore, here we will also evaluate the stability margin that measures the degree of stability (§2.3.2).

#### 2.3.1. Closed-loop time constant

The closed-loop time constant *τ*_{CL}, which indicates how quickly the animal can reach a desired angular velocity, can be obtained by analysing the characteristic equations (2.15) (first model) and (2.19) (second model), with smaller *τ*_{CL}, faster the response. In the following, we introduce our analysis using the first model as an example, and the analysis based on the second model follows the same method.

The closed-loop time constant *τ*_{CL} of the system can be obtained from the real component of the dominant pole(s), which are root(s) of the characteristic equation with the smallest magnitude of real component. From equation (2.15), the dimensionless () depends on four dimensionless system parameters (). Here, we aim to find the best combination of dimensionless controller parameters and so that the corresponding is the smallest for every combination of and , i.e. . The above estimation of assumes that the animal uses an optimal PI feedback controller that minimizes the closed-loop time constant . This assumption will lead to conservative estimations of the smallest allowable delays in the flight control system (see Results). The characteristic equation (equation (2.15)) is a transcendental equation because of the exponential terms introduced by neural delays, and it has infinite number of roots [59]. A third-order Padé approximation [57] of neural delays is therefore used to approximate these exponential terms, and render the number of roots finite for every combination of , , and , the roots of the characteristic equation are then found by evaluating the root locus using Matlab (Mathworks, Natick, MA). In sum, as a function of and is given by
2.21where *p*^{d} and *p _{i}* (

*i*= 1 :

*N*,

*N*is the number of roots) are the dominant root and roots of the characteristic equation respectively, and Re( ) represents the real part of the root. Dimensional can be restored from as defined by equation (2.13) by multiplying the corresponding open-loop time constant

*τ*

_{p}, i.e. . Note that we can also have

*τ*

_{CL}as a function of

*f*

_{w}and

*T*

_{d}, i.e. , as

*T*

_{s}and

*f*

_{w}are related by equation (2.11). The results based on characteristic equation of equation (2.19) (second model) can be dimensionalized similarly to obtain

*τ*

_{CL}as a function of angular velocity and position delays, i.e.

*τ*

_{CL}(

*T*

_{p},

*T*

_{i}).

In addition, there are also manoeuvring kinematics data available in the literature from which we can obtain the actual achieved by an animal in fast manoeuvres (e.g. hummingbirds [17,18], fruit flies [16,34,38] and hawkmoths [15]). Specifically, to estimate the from kinematics data, we first assume that flying animals perform rotational manoeuvres by following a step change in angular velocity. Then we measure the closed-loop time constant by calculating the time period spent by an animal to accelerate or decelerate to 63% peak velocity.

#### 2.3.2. Stability margin

Stability margin is a direct measure of the degree of stability or robust stability, which indicates how close the closed-loop system is to the stability boundary [57], i.e. how much variations can occur in the flight system before it becomes unstable. It is calculated here to evaluate how neural delays affect the closed-loop stability of the system in addition to the closed-loop response time, which is measured by *τ*_{CL}. Here we calculate stability margin, *s*_{m}, which is defined as the closest distance of a Nyquist plot to the stability boundary, i.e. (−1, 0) point on the complex plane [54],
2.22

For every , the stability margin, , is obtained by plotting Bode plot of and finding the minimum in the magnitude plot. A larger value of stability margin indicates a higher degree of stability and a more robust flight system.

### 2.4. Estimation of muscle mechanical power

Fast manoeuvres may impose additional demand on muscle mechanical power compared to that in steady flight, which also needs to be considered as a limiting factor of manoeuvrability. Changes of muscle power mainly arise from the changes in both wing kinematic pattern and frequency; and presumably, the latter contributes the most, since aerodynamic power is proportional to the third power of wing frequency [18]. The estimation of muscle mechanical power involves two steps, which are described in the following (also see electronic supplementary material, figure S2).

In the first step we estimate the power *P*_{h} when flapping frequency equals to that of hovering but wing kinematic pattern are varied. This involves generating a mapping from the changes of wing kinematic pattern (Δ*θ*) to the corresponding muscle mechanical power , where *f*_{wh} is the wingbeat frequency in hovering. Specifically, two types of changes in wing kinematic pattern are considered. The first creates pitch moment by increasing the bilateral symmetric stroke angle at the end of downstroke while maintaining other kinematic features the same as those of hovering [16]. It is assumed that the stroke angle can be increased until wingtips come into contact at the end of downstroke, based on which it is calculated that the maximum changes of stroke angle for magnificent and black-chinned hummingbirds, fruit flies and hawkmoths are 20°, 13°, 21° and 60°, respectively (data sources described in table 1). The second kinematic change creates roll and yaw moments by modulating the bilateral difference of angle of attack (AoA) [38,60]. It is estimated that the difference of AoA for magnificent, black-chinned hummingbirds [17] and fruit flies [60] are 20°, 15°, and 10°, respectively. The difference of AoA for hawkmoth is set to be equal to the largest of the above three animals, i.e. 20°. Next, based on above manoeuvring wing kinematic patterns, we can obtain the mapping and the corresponding moment by estimating the aerodynamic forces, moments and power using blade-element analysis and a quasi-steady aerodynamic model [18]. We also assume perfect muscle elastic energy storage, and therefore the calculation represents the minimum power requirement [18].

In the second step, for every pair , the response of the dimensionless closed-loop model to a step angular velocity reference input with a magnitude equal to nondimensionalized *ω*_{m} (table 2) is simulated in Matlab Simulink. Then the corresponding output (i.e. time history of dimensional aerodynamic moment *T*_{ac}(*t*)) from the dimensionless PI controller is recorded and dimensionalized, i.e. (according to equation (2.17)), from which the averaged dimensional moment *T*_{m} is calculated for each animal. The controller gains used here are those corresponding to the smallest closed-loop time constants (see §2.3.1). By matching *T*_{m} with *T*_{r} (from the mapping generated above), the corresponding power *P* can be obtained from *P*_{h}. If *T*_{m} is greater than the maximum *T*_{r}, increase in wingbeat frequency, i.e. , is then necessary to provide sufficient manoeuvring moment, in addition to changing wing kinematic pattern. Since power is proportional to 1.5th power of moment, the final aerodynamic power can be estimated as follows:
2.23

## 3. Results

### 3.1. Single axis open-loop dynamics

Tables 1⇑–3 summarize the morphological parameters (table 1), estimated damping coefficients *b* and open-loop time constant *τ*_{p} (table 2) for animals studied here. Strongly dependent on the body size, the damping of magnificent hummingbirds is about seven orders higher than that of fruit flies, while those of black-chinned hummingbirds and hawkmoths are comparable. It is noted that damping differs significantly among axes of rotation, as *b* is larger about roll and yaw axes than that about pitch axis for every animal species.

Interestingly, despite the differences in size and wingbeat frequency, as well as several orders of differences in damping coefficient *b* and the moment of inertia *I*, the interspecific differences of estimated open-loop dynamics (*τ*_{p}) are relatively small among the four animal species studied. For example, fruit fly and magnificent hummingbird are the smallest and the largest species studied here; their *I* and *b* differ in the order of seven, while *τ*_{p} only have differences up to the order of approximately 10^{1}. This suggests that open-loop dynamics are only weakly dependent on the size, and the differences between magnificent hummingbird and fruit fly are not as significant as indicated by their size. In addition, the open-loop dynamics of these two species are faster than those of black-chinned hummingbird and hawkmoth, which again does not simply follow an isometric scaling law. In addition, *τ*_{p} is the smallest in roll, larger in yaw and the largest in pitch (figure 1*b*), as *I* is smallest in roll and approximately equal between pitch and yaw. This implies that the roll has the fastest open-loop dynamics, yaw slower and pitch the slowest, and this pattern is consistent interspecifically.

### 3.2. Effects of biomechanical delays

Contour plots of closed-loop time constants *τ*_{CL} as functions of wingbeat frequency *f*_{w} and neural delay *T*_{d} are shown in figure 3 for pitch, roll and yaw rotations of magnificent hummingbird and fruit fly. The results for black-chinned hummingbird and hawkmoth are not qualitatively different and are provided in the supplementary materials (electronic supplementary material, figure S3); the difference between these two animals will be discussed later (see Discussion). Contours that correspond to the closed-loop time constants measured for each species from experiments (, table 3) and a reference closed-loop time constant (close to the median of all ) are also plotted. Note that figure 3 shows the lowest *τ*_{CL} achievable theoretically assuming that the animal has used an ‘optimal’ PI feedback controller to minimize the closed-loop response time (equation (2.21)). Therefore, given a known closed-loop time constant (e.g. and ), one can estimate the allowable delay to achieve this performance.

The results show that the biomechanical constraint limits *τ*_{CL} when wingbeat frequency *f*_{w} is relatively low, but has negligible effect when *f*_{w} is high. With the measured wingbeat frequencies, it follows that its effect on the manoeuvrability is negligible for fruit flies (*f*_{w} = 189 Hz for hovering, table 1) but not for magnificent hummingbirds (*f*_{w} = 25 Hz for hovering); therefore hummingbirds could improve their manoeuvrability by elevating their wingbeat frequency (figure 3*a–c*), while the same does not apply for fruit flies. Interestingly, this result seems to be congruent with the observation that hummingbirds increase their wingbeat frequency about 50% in manoeuvring while fruit only increase it about 7.5% (table 2).

### 3.3. Effects of neural delay

The results show that neural delays heavily constrain the response time (a measure of the manoeuvrability), as *τ*_{CL} in all cases increase monotonically as neural delays increase. Therefore, as fast response demands small neural delays, and with the ‘optimal’ PI controller assumption, we can have a conservative estimation of the allowable neural delay *T*_{a} that achieves a particular value of *τ*_{CL}, such as the measured . Specifically, to achieve the observed rotational manoeuvrability, one can find a contour in figure 3 that specifies a boundary, the ordinates of which indicate the largest neural delay allowable in an animal's flight sensing control system. In other words, the allowable neural delays have to locate in the region below this boundary (i.e. ‘performance region’). For example, as summarized in table 3, the smallest allowable delays *T*_{a} (*T*_{p} = *T*_{i}) for measured among roll, pitch and yaw rotation are 6 ms and 18 ms for fruit flies and magnificent hummingbirds respectively, both of which are in good agreement with previous estimation from experimental data [16,17]. Note that fruit flies have relatively small neural delay because of the fast sensorimotor transduction achieved by the strong connection between sensory cells innervating campaniform sensilla at its base with the steering motoneurons of the wings [61].

Furthermore, the results show that the allowable delay *T*_{a} differs considerably among roll, pitch and yaw (table 3). For example, to achieve the observed closed-loop time constants (), a fruit fly needs to have a delay less than 6 ms for roll rotation and less than 12 ms for pitch rotation; this is simply because the observed of roll is considerably smaller than that of pitch (table 3). In other words, fruit flies usually have faster roll closed-loop dynamics and therefore need faster sensorimotor transduction than pitch.

However, assuming that roll, pitch and yaw have the same closed-loop time constants as specified by = 30 ms, then *T*_{a} is largest in roll and approximately equal in pitch and yaw (table 3). This suggests that roll has the least demand on the speed of sensorimotor transduction, and it is the easiest one to attain small closed-loop response time. Interestingly, this is consistent with the observation that is the smallest for roll, since given the same neural delay for roll, pitch and yaw, the achievable *τ*_{CL} would be the smallest for roll.

However, in contrast to the inter-axis differences, the allowable neural delays *T*_{a} for the same reference are comparable between magnificent hummingbirds and fruit flies for all the axes of rotation. For example, *T*_{a} for fruit fly are 48 ms, 17 ms, and 21 ms in roll, pitch and yaw, which are similar to 37 ms, 16 ms, and 18 ms of magnificent hummingbirds. The allowable delays for black-chinned hummingbird and hawkmoth are smaller in general (table 3). This result seems to be consistent with that *τ*_{p} of the former two animals are less than those of the latter, suggesting a dependence of *T*_{a} on *τ*_{p}, rather than on the size.

### 3.4. Effects of angular position and velocity delays

We investigate further the effects of the neural delays by distinguishing the delays in angular velocity and position feedback. Figure 4 shows the contour plots of *τ*_{CL} as functions of velocity and position delays for magnificent hummingbird and fruit fly (results for black-chinned hummingbird and hawkmoth are shown in electronic supplementary material, figure S4). The result shows that *T*_{i} and *T*_{p} constrain the response time in a coupled and nonlinear fashion. It is seen that *τ*_{CL} increases monotonically with the velocity delay only when position delay is less than 20 ms; and *τ*_{CL} increases monotonically with the position delay only when velocity delay is less than 10 ms. However, from figure 4, their effects on *τ*_{CL} are rather unequal. For example, for pitch and yaw, which are less damped than roll, *τ*_{CL} increases slightly faster with velocity delay than with position delay, while the roll shows the opposite. Furthermore, the effect of velocity delay on roll is mitigated by the large roll damping and therefore is relatively small compared with pitch and yaw.

Notably, due to the strong nonlinear dependence of *τ*_{CL} on the two delays, compared with the results assuming equal *T*_{p} and *T*_{i} (figure 3), the performance region is extended into regions of larger neural delays. Specifically, the performance region includes points with larger neural delays than that indicated at the location of the white asterisks, which indicate the maximal *T*_{a} in single-delay case. For example, for pitch and yaw rotations of both magnificent hummingbird and fruit fly, the lowest closed-loop time constant occurs when position delay is approximately two times higher than the velocity delay (figure 4).

The result also shows notable changes of allowable delays among different axes. For example, fruit fly allows maximal 12 ms position delay when velocity delay is 7 ms in roll, while 40 ms position delay when velocity delay is 20 ms in pitch. When the manoeuvre is specified by , roll rotation can tolerate the largest neural delays, and pitch and yaw are approximately the same. For example, hummingbird allows 43 ms of velocity delay in roll, while about 30 ms in pitch and yaw. Consistent with the investigation based on single delay, the interspecific differences of allowable delays are insignificant compared to *I* and *b* (tables 1 and 2). For example, the allowable *T*_{p} and *T*_{i} of magnificent hummingbird and fruit fly are remarkably close for all three axes of rotation.

### 3.5. Stability margins

We further examine how neural delays affect the degree of the stability of an animal's closed-loop flight control system by calculating the stability margin *s*_{m} of each species corresponding to all the cases studied above, the contour plots of which are shown in figure 5. As the results show interspecific and inter-axis similarities, we will not distinguish the species as well as roll, pitch and yaw.

The closed-loop system remains stable when both position and velocity delays vary from 0 to 60 ms. Interestingly, when position delay is less than two times higher than the velocity delay, a good *s*_{m} (e.g. >0.5 [54]) can be achieved, while there is a sharp drop in *s*_{m} when position delay is more than two times higher than the velocity delay (figure 5). Considering the performance region, the putative biological constraint that *T*_{i} ≥ *T*_{p} and the results of stability margin, it can be inferred that if the position and velocity delays within an animal's flight control system satisfy *T*_{p} ≤ *T*_{i} ≤ 2*T*_{p} and located in the performance region, both high degree of manoeuvrability and stability can be achieved.

### 3.6. Muscle mechanical power

Figure 6 shows the averaged muscle-mass-specific power of magnificent hummingbirds and fruit flies required to perform roll pitch and yaw body rotation with the smallest *τ*_{CL} possible using PI controller, as position and velocity delays of the system vary (the results for black-chinned hummingbirds and hawkmoths are shown in electronic supplementary material, figure S6). Only those within the performance region are plotted. The effects of position and velocity delays on the mass-specific power are highly nonlinear and are consistent interspecifically. For example, when velocity delay is small, required muscle power increases substantially as position delay increases; however, the muscle power appears to be less dependent on pure velocity delays. In the performance region when *T*_{p} ≤ *T*_{i} ≤ 2*T*_{p}, the required muscle powers are relatively low, and are comparable to the highest values reported in the literature (e.g. hummingbirds [18] and fruit flies [64]).

Despite that the effects of neural delays on muscle power required are consistent interspecifically and among axes of rotation, the magnitude of required muscle power varies substantially among the four species and the axes of rotation. For example, both magnificent and black-chinned hummingbirds require much less power to roll than to pitch or yaw, fruit flies require less power to roll and yaw than to pitch, while hawkmoths require substantially more power to roll than to pitch and yaw. Notably, these results seem to be consistent with the commonly performed manoeuvres observed in these animals; for example, hawkmoths often perform pitch [15] and yaw [65] manoeuvres while roll less frequently (except for direct lateral manoeuvres [66]); on the other hand, hummingbirds and fruit flies often engage in coupled pitch and roll manoeuvres, especially when maximal performance is needed, such as in escape manoeuvres [4,17], and they tend to yaw only in slower voluntary manoeuvres [14,67].

Most importantly, although our results show that for the four animal species studied here (figure 1*a*), body size and MoI only weakly affect the response time of open-loop dynamics and the closed-loop performance (§§3.1–3.4), they have strong effect on the required muscle power, and therefore resulting in significant interspecific differences. For example, for fruit flies, the power required for roll and yaw manoeuvres is almost unchanged compared with that in hovering (approx. 59 W kg^{−1}, figure 6*e*,*f*); however, for magnificent hummingbird, the required power for pitch and yaw manoeuvring (figure 6*a*,*c*) can be as high as three times that during hover (185 W kg^{−1}) (similar for black-chinned hummingbird, electronic supplementary material, figure S6). On the other hand, hawkmoth is different from both hummingbirds and fruit flies, as its roll manoeuvring requires a significant increase in muscle power (electronic supplementary material, figure S6*e*) but pitch and yaw manoeuvring only require a mild increase (electronic supplementary material, figure S6*d* and *f*); again this seems to be congruent with the observation that hawkmoths barely roll actively.

## 4. Discussion

### 4.1. Inter-axis differences of open-loop dynamics and closed-loop performance

For all four species, the time constants of open-loop dynamics (*τ*_{p}, table 2) are significantly different among roll, pitch and yaw axes (smallest for roll, followed by yaw and largest for pitch). Since *τ*_{p} is the critical parameter that determines the dimensionless neural delays in the models (equations (2.13) and (2.17)), it follows that the effects of neural delays on the resulting closed-loop time constants (table 3) are also significantly different.

Specifically, the inter-axis differences of the allowable delay are investigated by setting equal closed-loop time constant (table 3). Results show that roll has the highest passive stability and therefore is able to tolerate the largest neural delays and therefore to alleviate the demand on the sensorimotor control [67]. However, pitch and yaw are more heavily constrained by neural delays due to slower open-loop dynamics. Moreover, roll manoeuvre also requires the least muscle power increase from hovering flight (except for hawkmoth, see §3.6), therefore from both control and muscle power perspectives, roll has the highest manoeuvrability. This seems to be congruent with the observation that rolling is often used [15] in the manoeuvres when maximal performance is more likely needed, such as in the escape manoeuvres in fruit flies [4,14] and hummingbirds [17].

Despite having high manoeuvrability, due to its small MoI, roll does not necessarily have high stability when subjected to external perturbations. In fact, previous results suggest that roll is the least stable axis for bumblebees [68,69] and hummingbirds [70,71] when flying in unsteady wakes. Interestingly, different from other species, hawkmoth has relatively large roll MoI (table 1) and roll damping (table 2), suggesting it is more stable under perturbation, which seems to be consistent with the recent experimental results [72]. Future studies may be needed to elucidate the determining factors of roll stability and potentially the trade-off between manoeuvrability and the stability.

### 4.2. Interspecific similarities of open-loop dynamics and closed-loop performance

Despite several orders of difference in body and wing size (figure 1*a*), the interspecific differences in open-loop and closed-loop time constants *τ*_{p} and (tables 2 and 3), as well as in the allowable neural delay *T*_{a} (table 3), are relatively small among the four animals studied here. This is somewhat surprising, since scaling theory [18,28] suggests that smaller species should have faster open-loop dynamics and manoeuvre more easily than larger species. Therefore, next we aim to further understand the underlying reason for the weak dependence of *τ*_{p} on the size and for the interspecific dynamic similarities (note that *τ*_{p} is the critical parameter in determining both the open-loop dynamics and the resulting closed-loop performance). Since *τ*_{p} = *I*/*b*, we first estimate the scaling of *b* based on the quasi-steady aerodynamic model [18,67],
4.1where , *ρ*, *R*, , , *Φ*, and *f*_{w} are a dimensionless constant, air density, wing span, mean wing chord, dimensionless radius of third moment of wing area, stroke amplitude and flapping frequency, respectively. The is approximately proportional to *R* assuming constant wing aspect ratio, i.e. . In addition, since , and *Φ* have relatively low variations interspecifically, they are also assumed to be constants. Then *τ*_{p} scales according to
4.2where *γ* = *m*/*R*^{3} (table 1) reflects the scaling of body mass relative to wing length, with a large value indicating higher body density or larger body size relative to wing size. Seen from figure 1*a*, where log_{10} *m* is plotted against log_{10} *R*^{3}, logarithmic values of *γ* are indicated by the intercepts of parallel straight lines of a unit slope with vertical log_{10} *m*-axis, i.e.
4.3Therefore, higher the line, higher the value of *γ*, and according to equation (4.2), greater the *τ*_{p}, or slower the open-loop dynamics.

According to equation (4.2), larger animals can reduce the time constants of their open-loop dynamics and therefore achieve comparable rotational manoeuvrability to smaller animals, by (i) reducing their body density to reduce *γ*, (ii) reducing their body-to-wing size ratio to reduce *γ*, or (iii) temporarily increasing wingbeat frequency during manoeuvring. Fruit fly has the largest *γ* and the hawkmoth has the smallest, while black-chinned hummingbird has larger *γ* than magnificent hummingbird. This shows that fruit fly, although being the smallest or has the highest wingbeat frequency (note that wingbeat frequency scaled inversely with body size [73]), has relatively slow body dynamics due to relatively large body size (or small wing size). On the other hand, hawkmoth and hummingbirds can achieve relatively fast body dynamics due to relatively small body density. To further illustrate the effect of *γ*, we calculate the (table 2) of the two hummingbird species and hawkmoth assuming they are scaled to fruit fly isometrically (or having the same *γ* as that of fruit fly), therefore . Since the fruit fly has the largest *γ* among the four, is greater than the corresponding *τ*_{p} for all the other three animal species. Particularly, the differences between and *τ*_{p} are most significant for hawkmoths and magnificent hummingbirds because they have the two smallest *γ*.

Note that the above discussion is carried out from control perspective only, and does not consider the muscle power required, which also depends on the actual angular velocity of the manoeuvre (table 2). Our estimation of the required muscle power shows that it is strongly size dependent and therefore larger animals, such as hummingbirds, need to increase their muscle power substantially to achieve similar angular velocity to those of fruit flies. Consequently, performance of burst flight using anaerobic metabolism [74] could be vital for hummingbirds, especially when they (both magnificent and black-chinned hummingbirds) have achieved higher peak body angular velocities than those of fruit flies in escape manoeuvres (table 2). Insects, however, have no anaerobic capacity, and this seems to suggest that if they cannot elevate their aerobic muscle power sufficiently, larger insects should have lower manoeuvrability in general. It seems that this speculation is somewhat consistent with the observation that the maximal pitch rate achieved by the hawkmoth is much lower than that of black-chinned hummingbirds (table 2).

### 4.3. Effect of unequal angular position and velocity delays

As delays usually have adverse effects on the stability and performance of a dynamic system [54,75], our results show that a proper combination of position (*T*_{i}) and velocity (*T*_{p}) delays, even with larger magnitude of individual delays, could potentially improve the response time, degree of stability and the required muscle power for performing rotational manoeuvres. Similar conclusions regarding the stability of a dynamic system with multi-delays can be also found in the control theory literature; for example, Sipahi *et al*. [59] shows that there exists a proper combination of two delays that could stabilize an otherwise unstable system. In particular, stability may become independent of the values of two delays if their ratio is a certain constant. Interestingly, the combination of *T*_{i} and *T*_{p} that improves the response time (figure 4) and ensures good stability margin (figure 5) and muscle power consumption (figure 6) does seem to be congruent with the putative biological constraint *T*_{i} ≥ *T*_{p} (assuming the angular velocity is measured directly, e.g. through haltere reafferent in fruit flies, and angular position is estimated through integration, the delays in position feedback should be higher than that of in velocity feedback). Also, it is reasonable to speculate that *T*_{i} cannot be significantly higher than *T*_{p}, and should be smaller than a certain value, although to authors' best knowledge, no direct measurements are available to make quantitative comparison of these two sources of delays.

### 4.4. Comparing hummingbirds and fruit flies

Despite the remarkably similar manoeuvrability between fruit flies and hummingbirds, the manoeuvrability of fruit flies is limited by neural delays while that of hummingbirds might be limited by both muscle power capacity and neural delay. For example, to achieve the measured peak roll rate, fruit flies need to have a neural delay less than 7 ms (figures 3*e* and 4*e*), which is remarkably close to those reported for haltere [34], the fastest sensor of flies to measure angular rate [56,76]. However, the reported neural delays of hummingbirds (table 3) are considerably lower than the estimated allowable delay of roll but similar to those of pitch, which indicates that their roll manoeuvrability is not limited by the neural delay, but not for the pitch manoeuvrability. For hummingbirds, especially larger magnificent hummingbirds, it is also reported that they change their wing trajectory and frequency drastically during manoeuvres [17], and significantly increase the power consumption of their flight muscles. In fact, a muscle mass-specific power as high as 881 W kg^{−1} is reported for magnificent hummingbirds, significantly higher than those reported in the literature for a variety of flying animal species [42,43]; although it is unknown whether this approaches the highest anaerobic power that hummingbirds are able to achieve, it is reasonable to speculate that the manoeuvrability of hummingbirds could be limited by muscle power capacity. The muscle power increase required for fruit flies in manoeuvring, however, is almost negligible for roll and yaw, and also small for pitch (figure 6), therefore it can be concluded unequivocally that muscle power capacity is not a limiting factor for rotational manoeuvrability of fruit flies.

## Data accessibility

The datasets and analysis code that support this article are available at Dryad: http://dx.doi.org/10.5061/dryad.5t0g4 [77].

## Authors' contributions

P.L. contributed to theoretical formulation, model simulation, data collection and analysis and preparation of the manuscript; B.C. conceived, coordinated and designed the study and helped in the preparation of the manuscript. All authors gave final approval for publication.

## Competing interests

The authors declare no competing interests.

## Funding

Funding was provided by the National Science Foundation (NSF CMMI 1554429 B.C).

## Acknowledgements

We would like to thank Tyson L. Hedrick and Noah J. Cowan for their valuable contributions to this work, including modelling, data collection and manuscript writing.

## Footnotes

Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.3805816.

- Received January 29, 2017.
- Accepted June 5, 2017.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.