The field of bioinspired flapping-wing micro-air vehicles (FWMAVs) has seen major advancements in the last decade.
Researchers have achieved controlled flight on tethered [1], [2] and untethered [3]- [6] FWMAVs at the centimeter scale. They have integrated sensors [7]- [10] and implemented robots with a wide range of actuators including piezo bending actuators, mini DC motors [11]- [13], soft DEA actuators [14], and electromagnetic coils [15]. Others have developed autonomous control algorithms that (given sufficient knowledge of the state of the robot) can achieve not just stable hovering, but also impressive feats of agility [16]. The design, fabrication, and control tools now exist to design novel FWMAVs capable of flight.
However, the performance of such robots still lags behind that of their insect muses. The agility and versatility of insects like flies, bees, and dragonflies is unmatched by any FWMAV at similar scales. Untethered FWMAVs at the centimeter scale must be supplied with extremely highpower energy sources-lasers [17] or high-wattage light sources [3]-while insects are efficient enough to sustain flight over long distances during foraging and migration [18], [19]. Additionally, FWMAVs are often much more delicate {jelynch, ngravish}@eng.ucsd.edu. Jeff Gau is with the 2 Bioengineering Graduate Program and Simon Sponberg is with the 3 School of Physics and Biological Scienes, both at Georgia Institute of Technology, Atlanta, GA. Email {jeff.gau, sponberg}@gatech.edu Fig. 1. a) Insects such as moths use synchronous actuation. This is characterized by a periodic signal from an internal source with rate ω wingbeat . b) Bumblebees are an example of an insect using asynchronous actuation. This is characterized by feedback, where the rate of the mechanical system ω mech interacts with the rate of the feedback ω f eedback to produce the wingbeat.
than insects, constructed of 100-micron-thin carbon fiber, thin polymer sheets, and brittle piezoelectric materials. The dynamics of flight are sensitive to changes in mechanical properties (wing geometry, inertia, etc.), and yet insects are able to continue to fly despite damage caused by the environment or other animals [20], [21]. There is still much for us to learn about how insects achieve their impressive flight performance and translate these into advances in robotics.
Flapping wing insects can be classified into one of two actuation strategies: synchronous and asynchronous (Fig. 1). Insects such as moths generate wingbeats through a periodic signal generated by the nervous system that is "synchronous" with the wingbeat, while insects such as bees rely on a strain-dependent response of the muscle to generate selfexcited wingbeats whose frequency is higher than signals from the nervous system and therefore "asynchronous" from the neural signals. Asynchronous muscle actuation is thought to provide several distinct advantages to flying insects, including high wingbeat frequency (wbf), adaptive behavior after wing damage [22], separation of power and control, and improved efficiency [23]- [25]. To the authors knowledge, all previous actuation of flapping wing robots have relied on synchronous actuation strategies. We hypothesize that asynchronous actuation methods provide adaptive behaviors that could be beneficial for flapping wing robots.
A key specialization in asynchronous muscle is a phenomenon called delayed stretch activation (dSA), wherein, after an activated muscle is stretched, its tension will continue to increase, reaching a peak that is delayed in time w.r. the wings with no direct input from the nervous system. Biologists have studied asynchronous muscle in isolation by carefully removing the muscles from the insect thorax and applying techniques adapted from materials science such as tension increases in response to stretching [26]- [28].
Their results have been used to characterize asynchronous muscle as an active material and compare muscle behavior across species [25] and between lines of transgenic flies [29]. However, biologists have been typically interested in the elusive bio-molecular dynamics from which the dSA phenomenon arises. To enable asynchronous actuation in FWMAVs, it is necessary to characterize the system level behavior of asynchronous flight: the dynamical interactions between asynchronous muscle, the elastic thorax [30] and the complex aerodynamic forces on the wing [31]. One way to tease out the complexities of such integrated biological systems is to use robophysical methods [32], [33]. By building a robotic model of the biological system where we may control individual system parameters, we may be able to more clearly understand the relationships between system parameters and observed behavior.
In this manuscript, we seek to establish principles of asynchronous actuation for flapping wing robots and to demonstrate some unique properties of dSA actuation for adaptive and resilient flapping wing dynamics. We begin by deriving the dynamical equations for dSA and show that it is dynamically similar to a second-order low-pass filter on the strain rate of the muscle. We then integrate the dSA feedback law with the nonlinear equations of motion of a "spring-wing" system with aerodynamic drag and elastic energy storage [34]. Using the equations of motion, we derive conditions for the existence of stable limit cycles and use a combination of simulation and robophysical model experiments as validation.
We then present further experiments in the robophysical model that show that an asynchronously-actuated robot has a compelling ability to rapidly adapt wingbeats and respond to collisions with no direct control. Lastly, we implement dSA feedback in an insect-scale flapping wing as a proof of concept towards creating a full asynchronous FWMAV.
The first step towards integrating dSA into a robotic flapping system is to express the observed behavior of asynchrononus muscle as a function of the state of the system. The time-dependent force response (f step ) of the muscle to a step change in its length (see Fig. 2a) has been parameterized as the sum of three exponents [35]:
Each term corresponds to a phase of the response: An extremely fast decay (r 2 >> wbf), a slower rise (r 3 ≈ wbf), and a very slow decay (r 4 << wbf). The constant c represents the passive stiffness of the muscle. This 7-term model is fitted to stretch-and-hold data collected from insect muscle fibers and used to evaluate the behavior of the muscle.
To ease the complexity of analyzing the behavior of a system with dSA forcing, we focus on the Phase 3 and 4 dynamics, the slow rise and slower decay. We choose to focus on the r 3 term in particular because it has been shown to vary linearly with wbf across a range of insects [25]. The very fast dSA dynamics (Phase 2) are effectively damped out by the spring-mass-damper dynamics of an elastic flapping wing. Setting K 3 = K 4 = 1, c = 1, and defining the ratio of the slower rates, κ = r 4 /r 3 , we can write:
Since r 4 > r 3 , κ ∈ [0, 1], and typically is in the range of 0.01 -0.3 in insects [25]. By tuning r 3 and κ and adjusting the peak amplitude via a muscle "strength" term, µ, we can match the shape of the asynchronous muscle response (Fig. 2b).
The expression in Eq. 2 is the response of the system to a step in the muscle strain, which is equivalent to an impulse in the strain rate. This implies that Eq. 2 is the impulse response of the muscle, given strain rate as the input. We can express the forcing function as a convolution of the impulse response with the strain rate,
where g(t) = -e -r3t + e -κr3t . In the Laplace domain, convolution is a multiplication rather than an integration, i.e. F dSA (s) = G(s)V (s). Taking the Laplace transform of the response function g(t) as defined in Eq. 2, we get the transfer function G(s) which transforms the velocity feedback input to the dSA forcing output:
where we've defined 3 parameters, α 1 , α 2 , and α 3 for convenience. The dSA phenomenon, therefore, is qualitatively similar to a second-order low-pass filter on the velocity, fed back to the muscle as a force command.
The Laplace representation of dSA allows us to express the dynamics of asynchronous actuation as an ODE. In the Laplace domain, the force output is the product of the transfer function and the velocity input:
We can distribute the denominator of G(s) and take the inverse Laplace transform:
We can then connect the dSA actuation dynamics to the nonlinear "spring-wing" equation that is commonly used [34], [36], [37] to describe flapping systems with internal elasticity. In terms of inertia I, stiffness k, drag torque term Γ, and the applied torque τ applied , the spring-wing equation takes the form:
The asynchronous muscle dynamics dictate the torque applied to the system and are scaled by a gain coefficient, µ (units: Nm rad -1 ), so we may write the combined equations of motion:
Solving these equations simultaneously gives the trajectory of the flapping wing, θ(t). The prevalence of asynchronous insects that employ dSA actuation suggests that this system can produce stable limit cycle oscillations from the balance between quadratic aerodynamic damping and strain-rate dependent muscle actuation. However, this is not guaranteed; it is possible that this simplified model lacks some feature that is critical to creating stable oscillations. We must evaluate the dynamics of this system to understand the conditions that lead to stable oscillations as well as learn if there are combinations of feedback and mechanical parameters which produce more exotic dynamical behaviors (exponential growth, chaos, etc.) that should be avoided in a robot implementation.
The stationary state, θ, θ, f dSA , ḟdSA = 0, is a fixed point of the asynchronous dynamical system (Eq. 9). We now seek to understand if this fixed point is stable or unstable. We ask the following question: if there is a small perturbation to the closed-loop system with dSA feedback, will oscillations tend to decay back to the origin or will they grow?
The nonlinear system described in Eqs. 9 can be redefined with a new state vector σ = [θ, θ, We next linearize about the point σ = 0, constructing the Jacobian and the linear dynamics about the stationary state:
The growth or decay of perturbations from the stationary state are determined by the eigenvalues of the Jacobian. The characteristic equation for the linearized system is
with four eigenvalues λ j = a j + iω j . The sign of the real part of the largest eigenvalue dictates whether a perturbation away from the stationary point will tend to decay (stable) or grow (unstable). Understanding the conditions on the stability boundary will enable us to choose relevant feedback parameters to induce oscillations.
We can determine the boundary between decaying and growing solutions by setting the real part of the eigenvalue to zero, e.g. λ * = iω, where ω is the frequency of oscillation. Plugging in for λ * and separating the real and imaginary parts, we get two equations:
We are specifically interested in how the parameters µ and r 3 influence the onset of asynchronous oscillations due to the instability of the stationary point. We define ω 2 n = k/I as the natural frequency of the system and μ = µ/I.
The first equation is quadratic in ω 2 . Solving and using the definitions of α 1 , α 2 and α 3 , we get two solutions: ω 2 = κr 2 3 and ω 2 = ω 2 n . Plugging each into the second equation, we get a pair of equations:
Eq. 15 gives the trivial conditions (zero feedback gain = no oscillations), but Eq. 14 defines a relationship between the strength of the dSA feedback and its rate parameter, plotted in Fig. 3 (red line). The two equations define 4 quadrants in the r 3 -µ plane where perturbations tend to either grow or decay. In the current work, we'll focus just on the two regions that exist for µ > 0.
In the previous section we demonstrated that, for certain dSA parameters, the stationary state is unstable and oscillations will grow. In a system with nonlinear damping (i.e. quadratic aerodynamic damping) a limit cycle may form where the system oscillates such that the energy input from the muscle over one period exactly balances the energy dissipated by the environment. Unlike linear damping, dissipation from quadratic damping is amplitude-dependent, resulting in a stable amplitude and frequency that depends on system and aerodynamic In order to control flapping oscillations, we need to understand how the amplitude and frequency of oscillations vary with the system parameters. We can study the behavior of the system via numerical simulation since the nonlinear system presents challenges to deriving analytical solutions.
We simulated the nonlinear equations of motion from Eq. 10 in Matlab (R2021a, Mathworks) using the ode45 solver. We chose mechanical parameters (k, I, Γ) that matched those of a robophysical experimental system that we will use section III. We selected 0.8 as an arbitrary value for κ and the results are independent of this choice. We computed intercepts of Eq. 14,
and defined ranges of μ and r 3 from 0 to 3 times the intercept value. We ran simulations at each configuration and computed the amplitude and frequency of oscillations, which were typically sinusoidal. The results are shown in Fig. 3. Our simulations confirm that the stability boundary in Eq. 14 does divide the plane into oscillatory and non-oscillatory regions. Additionally, there is a clear trend that shows that increasing μ leads to an increase in flapping amplitude and a decrease in flapping frequency. The change in frequency is predicted by linear eigenvalue analysis (Fig. 3b, inset). However, when amplitudes get very high (high μ), nonlinear drag effects become more significant, causing the lines of constant frequency to deviate from the linear predictions. This discrepancy underlines the importance of considering the inherent nonlinearity of the system.
It is important to note that many of the configurations shown in Fig. 3 are simply impractical. At high μ, low r 3 , we see oscillation amplitudes well above 360 degrees, whereas a hinge on a flapping robot would be expected to have a maximum angle of only ∼90 degrees. This limitation, in addition to limits on the torque and max displacement of a potential actuator, means that in practice, dSA feedback will need to have a relatively small μ.
In this section we describe experiments on a dynamicallyscaled robotic flapping wing, a so-called robophysical system since it employs robotics and feedback and is coupled to real environmental physics. The system has well-characterized mechanical parameters and is easily modified thanks to its modular design, enabling a range of tests that would be more difficult at a smaller scale.
The robophysical system we use in this study (Fig. 4) was adapted from a similar system described in [34]. It consists of an elastic element (a molded silicone torsion spring), a main shaft supported by a thrust bearing and radial air bearings, an optical rotary encoder (4096 CPR, US Digital), and a rigid, fixed-pitch acrylic wing in water. The inertia can be changed by fixing one of a set of inertia plates to the main shaft. Data collection and control of the system is done via a DAQ (PCIe 6323, NI) and Simulink Desktop Real-Time (SLDRT, Mathworks), which enables hardware-in-the-loop control at a rate of 1000 samples/s.
We use a brushless DC motor (D6374 150KV, ODrive Robotics) and a motor driver to capable of closed-loop torque control at 10 kHz. The angular position of the wing is used as the input to a SLDRT model that implements the dSA transfer function (Eq. 4), multiplies the output by the strength, µ, and sends a torque command to the motor via USB, as shown in Fig. 4a. The direct torque control method eliminates the need to explicitly integrate the motor dynamics into the asynchronous ODE.
We tested the effect of changing the value of µ in the robotic model while holding r 3 constant. We chose a value of r 3 = 35s -1 = 2.4ω n , and κ = 0.5, which places this configuration on the side of the stability boundary that should mean that it oscillates as long as µ > 0. However, choosing a small value for µ, we find that an initial perturbation actually tends to decay back to zero (Fig. 5a-i). We don't model friction in our simulations, but we can see that as μ → 0 in Fig. 3a, the oscillation amplitude approaches zero. When friction is present in the system, arbitrarily small amplitudes are not possible, so the oscillation decays -dSA isn't strong enough to overcome friction. Increasing µ leads to a "borderline" case where the system oscillates for a few periods before decaying again (Fig. 5a-ii). When µ finally crosses the threshold, stable oscillations result. We observe a linear relationship between µ and amplitude for µ > 0.05 (4899 deg/Nm, R 2 > 0.99), as well as a subtle decrease in oscillation frequency with increasing µ (Fig. 5, b andc)
Beyond the control of flapping amplitude and frequency, we are able to examine novel behaviors of dSA flapping systems via the robophysical model. We observed that the asynchronous system was able to naturally adapt to changes in its mechanical properties. Additionally, the system features an extremely fast response to collisions with environmental obstacles, reducing the potential for serious damage to the wings or wing transmission.
1) Adaptation to changing mechanical parameters: Figure 6a shows the results of an experiment where additional mass is suddenly added to an inertia plate. One might expect that the addition of extra inertia would cause the amplitude Fig. 6. The asynchronous spring-wing adapts to changes in its mechanical system properties. a) When extra mass is added to the inertia plate on the large-scale robot model, the system transitions to a new amplitude and frequency. b) When we varied the inertia over a large range, we saw that increasing inertia decreases the frequency and increases the amplitude of the wingbeat to decrease, as the motor now needs to move more mass. However, we see that as soon as the inertia of the system changes at t = t * , the asynchronous flapper adjusts to the new loads on the system, actually increasing in amplitude and decreasing frequency. It adapts to the new system properties.
Keeping the same values of r 3 and µ, we measured amplitude and frequency of oscillation for 4 different inertias (a roughly 3-fold range). Figure 6b shows that this trend continues, suggesting that the product of amplitude and frequency remains roughly constant.
A robot with an adaptive control scheme like this is able to respond to changes to its mechanical properties automatically. This is qualitatively similar to the adaptive oscillators explored for legged-locomotion, in which robots adjust gait and frequency when loads are added [38]. Damage to a wing or accumulation of debris may cause changes in wing inertia that would seriously impact the performance of a synchronously driven robot whose frequency is dictated by the resonance curve of the transmission and wings [39]. An asynchronously-driven robot, on the other hand, would simply adapt to a new frequency and amplitude that may still enable it to fly. As FWMAVs move from safe laboratory conditions to the more unpredictable world-at-large, adaptation to new situations will be ever more critical.
2) Fast response to collisions with the environment: Another inevitable consequence of operating in unstructured environments is collisions. The brittle actuator materials and delicate microstructures that make up typical FWMAVs make it all the more important to avoid or mediate damage from collisions. We wanted to investigate the response of the asynchronous system to a collision with a rigid object in the environment.
We fixed an inertia plate fitted with vertical posts to the robotic model and set the system to oscillate. At t = t * , we interrupted the motion of the system by causing a post to collide with an obstacle. We observed that the system stopped almost immediately, well within a single period (Fig. 7). Very shortly after the angular velocity is reduced to zero, the dSA feedback also goes to zero, causing the actuator to In the synchronous forcing case, represented by the lightcolored trace in Fig. 7, the actuator would be oblivious to the collision and continues to apply torque to the wing after it has already stopped, potentially causing damage to the wing structure. The low-level feedback inherent to asynchronous actuation enables the system to respond immediately, reducing the potential for damage. In addition, as soon as the system is perturbed again, after it is clear of the obstacle, it resumes flapping at the same amplitude and frequency. The asynchronous system naturally avoids damage and does it within a single oscillation period, with no need for an explicit command to stop actuation. It may serve as a sort of distributed control, offloading some need for the flight controller to respond to environmental disturbances.
As a proof-of-concept demonstration, we implemented dSA feedback on an insect-scale robotic wing. The wing apparatus, consisting of a thin polymer wing (15mm x 5mm x 0.1mm) supported by a carbon fiber frame, a PZT bimorph bending actuator, and transmission, is based on the design from [1] (see Fig. 8a). Here, we used a single wing supported by acrylic brackets instead of a carbon fiber airframe. We aligned a fiber optic displacement sensor (D21, Philtec) with the tip of the actuator to track the actuator displacement. Oscillations were induced by an aerodynamic perturbation provided by a toy vortex ring gun (Zero Blaster, zerotoys.com). We recorded high speed video of the system from the top down as the vortex crossed the wing. Frames from the video can be seen in Fig. 8b.
The output from the displacement sensor was fed into an DAQ (PCIe 6343, NI) and used as input to the same SLDRT model that implemented the dSA feedback law described in Section III. To close the loop via dSA feedback, we took the derivative of the displacement to get velocity and fed the velocity into the dSA transfer function 4 with r 3 = 225 Hz (∼ 3ω n ). The output was converted to voltage and fed through an amplifier to the PZT control signal, inducing bending in the actuator. As with the large-scale robotic model, we slowly increased µ until stable oscillations were observed. Fig. 8 shows the result of two tests: one with µ too small to overcome friction, and one large enough to induce oscillations. Larger values of µ may have provided larger amplitudes, but we used this minimum µ value in our testing to avoid overloading the actuator and the robot.
In this paper we derive and study the first dynamical system representation of asynchronous wingbeat actuation in flapping wing robots and insects. The dSA feedback control law that we have described here is a novel method of achieving flapping in robots. Asynchronous actuation in current FWMAVs is simple to implement and requires only 1) a state estimate via strain gauge, encoder, gyroscope, or other sensor and 2) knowledge of the internal dynamics of the actuator. The method can be applied to a wide range of actuators using relatively simple analog hardware or digital logic. The resulting system naturally oscillates while powered and can be controlled by adjusting the feedback parameters -or by changing the mechanical properties (e.g. wing inertia) of the robot.
While our implementation of asynchronous actuation relied on actuators, sensors, and a feedback loop, the dSA response of insect flight muscle is "material" property of the muscle. Thus asynchronous wingbeats emerge from the lowest level of mechanical feedback within asynchronous insects. Recent efforts to incorporate strain-based sensing withing piezoelectric actuators [40], [41] show promise for incorporating the sensing component of dSA into mobile micro-robots. However, future work to engineer low-level sensing and actuation feedback properties into active materials and circuits will be of great interest for future FWMAVs.
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