Tail Control Enhances Gliding in Arboreal Lizards: An Integrative Study Using a 3D Geometric Model and Numerical Simulation: Complimentary/Contributed Paper

Abstract

The ability to glide through an arboreal habitat has been acquired by several mammals, amphibians, snakes, lizards, and even invertebrates. Lizards of the genus Draco possess specialized morphological structures for gliding, including a patagium, throat lappets, and modified hindlimbs. Despite being among the most specialized reptilian gliders, it is currently unknown how Draco is able to maneuver effectively during flight. Here, we present a new computational method for characterizing the role of tail control on Draco glide distance and stability. We first modeled Draco flight dynamics as a function of gravitational, lift, and drag forces. Lift and drag estimates were derived from wind tunnel experiments of 3D printed models based on photos of Draco during gliding. Initial modeling leveraged the known mass and planar surface area of the Draco to estimate lift and drag coefficients. We developed a simplified, 3D simulation for Draco gliding, calculating longitudinal and lateral position and a pitch angle of the lizard with respect to a cartesian coordinate frame. We used PID control to model the lizards’ tail adjustment to maintain an angle of attack. Our model suggests an active tail improves both glide distance and stability in Draco. These results provide insight toward the biomechanics of Draco; however, future in vivo studies are needed to provide a complete picture for gliding mechanics of this genus. Our approach enables the replication and modification of existing gliders to better understand their performance and mechanics. This can be applied to extinct species, but also as a way of exploring the biomimetic potential of different morphological features.

Introduction

Arboreal habitats are inundated with a complex 3D array of branches and surfaces. In addition to the structural variation in the physical parts of the trees, there are also gaps of variable sizes. Although some animals can simply avoid these gaps and move exclusively on trees, many animals will execute maneuvers to cross these gaps. In these cases, being able to bridge these gaps while avoiding injury is crucial for survival in these habitats. Whereas some animals simply fall from an arboreal surface in a controlled way (e.g., Yanoviak et al. 2005; Higham et al. 2017), many vertebrates have evolved specialized morphology and behavior to cross the gaps. These strategies can include reaching, swinging, leaping, and gliding (Günther et al. 1991; Demes et al. 1995; Hunt et al. 1996; Byrnes and Jayne 2012; Graham and Socha 2020). Of these strategies, gliding is likely the most effective for crossing large gaps.

To glide effectively from one place to another, animals often exhibit morphological, physiological, and behavioral specializations. Squirrel and lizard gliders use patagia, and frogs use enlarged feet and webbings to increase lift (Gupta 1966; Emerson and Koehl 1990; Russell et al. 2001). Snakes undulate in mid-air to dynamically adjust their aerodynamics (Socha 2002; Sochaet al. 2005; Yeaton et al. 2020). These biomechanical phenomena can be challenging to study, but they provide important insight into the evolution of gliders, as well as the development of biomimetic robots (Yanoviak et al. 2005; Shin et al. 2018).

Among reptiles, several groups have modified morphological features that assist with directed descents (Dudley et al. 2007). For example, geckos from Southeast Asia, including Ptychozoon, Luperosaurus, and Cosymbotus, have fully webbed hands and feet, along with dorsoventrally flattened tails and flaps (or folds) of skin along the lateral edges of the body (Russell 1979; Brown et al. 2000; Russell and Dijkstra 2001). Lizards of the agamid genus Draco are some of the most advanced reptilian gliders (Herre 1958; Mcguire and Dudley 2011). A modified patagium, as well as throat lappets and modified hindlimbs help Draco maximize their aerodynamic lift during gliding (Colbert 1967; Russell and Dijkstra 2001; Mcguire 2003). Their musculoskeletal anatomy, including elongate ribs and modified muscles, is also specialized to modulate the position of the patagium (Colbert 1967).

Like many other Southeast Asian and Indian vertebrates, Draco likely developed their specialized gliding morphology as a result of dipterocarp forest development, which is characterized by large trees and tall, open canopies (Heinicke et al. 2012). Now, Draco is composed of over 45 species, which range significantly in both size and gliding performance (Mcguire and Heang 2001). It is thought that the ability to control trajectory during aerial descent is a major driver in the evolution of gliding flight (Smith 1952; Yanoviak et al. 2005). Despite research on comparative glide distance and motion planning in Draco, the mechanism through which Draco controls its aerial trajectory is largely unknown (Mcguire 2003; Mcguire and Dudley 2005; Khandelwal and Hedrick 2020).

Recent research has shown that the tail can play a significant role in lizard locomotory stability and reorientation (Gillis et al. 2009; Libby et al. 2012; Gillis and Higham 2016). Specifically, geckos are able to self-right during free-fall and adjust their yaw during gliding with the tail alone (Jusufi et al. 2008, 2010). In addition, other lizards within the family Agamidae use the tail to ensure pitch stability during jumping (Libby et al. 2012). Similarly, flying squirrels of the genus Glaucomys are thought to use the tail to stabilize their pitch (Dudley et al. 2007; Hayssen 2008). However, tail function during gliding in Draco, arguably the most proficient gliding reptile, has not been examined. Prior studies on Draco have relied on observations of glide behavior in videos, and they show significant pitch and yaw movement of the tail during landing and maneuvering in gliding (Fig. 1).

Fig. 1.

Draco shows significant tail movement during gliding (Image from J. M. Dehling on creative commons).

The goals of this study are to determine (1) the aerodynamic properties of Draco’s gliding morphology and (2) the role of the tail in pitch stability and glide performance. A three degree-of-freedom model of Draco gliding dynamics is presented. We organize our methods into three distinct processes: (1) construction of a glide dynamics model, (2) construction of a 3D geometric model and parameter estimation, and (3) numerical simulation. This methodology was successful in determining aerodynamic parameters of a 3D geometric Draco model, and characterizing the effect of tail control on Draco flight stability and glide distance. We hypothesized that an active tail will improve Draco glide stability and distance.

Materials and methods

Museum specimens

We quantified the relative size of the tail and body of two species of Draco from the collection of Lee Grismer at La Sierra University. A digital scale was used to measure mass and photographs (dorsal view with limbs in the plane of the body) were taken of each specimen for additional measurements. ImageJ was used to quantify lengths and centroid. The latter was quantified by first outlining the lizard using the freehand tool. The centroid was then the average of the x and y coordinates of all of the pixels in the selection. Finally, the center of mass was determined by balancing the stiff (preserved) lizard on a thin rod. The fore-aft location that resulted in a balanced lizard was noted and then measured. Total body (with tail) masses of D. fimbriatus, and D. maculatus were 18.5 and 5.9 g, respectively. The tail for the two species represented 6.5% and 8.5% of total body mass, respectively.

The snout–vent length (SVL) and tail length of D. fimbriatus was 10.50 and 18.38 cm, respectively. The centroid (without tail) was 6.95 cm from the tip of the snout (66% of SVL). The SVL and tail length of D. maculatus was 7.69 and 12.02 cm, respectively. The centroid (without tail) was 5.02 cm from the tip of the snout (65% of SVL). In contrast, the center of mass was approximately 54% of SVL (from the tip of the snout) in all species.

Mathematical model of motion

A pitching dynamics model was developed to characterize the effects of lift, drag, and weight forces on the lizard’s motion. The aerodynamic model was similar to that of Yeaton et al. (2017), however, the lift and drag coefficients were experimentally determined as opposed to extracted from a characteristic model. The lizard is modeled as two rigid rods, connected via a rotational joint located at the end of each rod. The tail was designed to pivot about the ventral end of the lizard’s torso, adjusting its center of mass and inertia. In Jusufi et al. (2010), aerial reorientation of geckos is modeled using the conservation of momentum. In this work, the model is generated using Newtonian dynamics of the system, that is, the net forces and torques (e.g., from lift, and drag) on the lizard are balanced to compute acceleration, velocity, and position/orientation states.

Using the free-body diagram in Fig. 2, we defined horizontal, vertical, and rotational equations of motion:

$$m_{\mathrm{tot}}\ddot{x}=-L\sin \left(\theta -\alpha \right)\mathrm{ }-D\cos \left(\theta -\alpha \right).$$

(1)

$$m_{\mathrm{tot}}\ddot{y}=-Mg\mathrm{ }-mg\mathrm{ }-D\sin \left(\theta -\alpha \right)+L\cos \left(\theta -\alpha \right).$$

(2)

$$I\ddot{\theta }=-r_{1}D\sin \left(\alpha \right)\mathrm{ }-r_{1}L\cos \left(\alpha \right)+r_{2}mg\cos \left(\theta \right)+I_{\mathrm{tail}}{\alpha }_{\mathrm{tail}}.$$

(3)

where $m_{\mathrm{tot}}$ is the total mass, $g$ is the gravitational acceleration, $M$ and $m$ are the body and tail masses, $L$ and $D$ are the lift and drag forces, and $\theta$ and$\mathrm{ }\alpha \mathrm{ }$are the angles of attack and pitch. $r_1$ and $r_2$ were defined as the distance from the body’s center-of-mass to the center of pressure, and the distance from the tail’s center-of-mass to the center-of-pressure. $I_{\mathrm{tail}}$ was the inertia of the tail, and ${\alpha }_{\mathrm{tail}}$ was the angular acceleration of the tail. We assumed the moment of inertia of both the lizard’s body and tail as costs and modeled them as solid rods. By definition, lift is the aerodynamic force normal to the direction of motion, and drag is that parallel to the direction of the motion in still air. Pitch is the angle of the lizard relative to the horizontal plane, and the angle of attack is the angle of the lizard relative to its direction of motion.

Fig. 2.

The free-body-diagram describing the pitching dynamics of Draco. Here, Mbodyg and mtailg are the weight forces from the body and tail, L and D are the lift and drag forces, and θ and α are the angle-of-attack and pitch. v represents the velocity of lizard, which is antiparallel to the direction of airflow relative to the lizard.

3D geometric model

To determine the lift and drag of Draco, a 3D geometric model was developed in Blender based on photographs of Draco dussumieri mid-flight (Dehling 2017). The 3D geometric model was designed to reflect the geometry of Draco during the equilibrium phase of the glide, as opposed to the landing or takeoff phases (Fig. 3). Photographs from the gliding phase were prioritized as Draco employs landing and takeoff specific mechanisms during their respective phases as seen in Fig. 4 (Khandelwal and Hedrick 2020).

Fig. 3.

The Draco 3D geometric model (A) was developed in Blender based on Draco gliding morphology (B). (Image from J. M. Dehling on creative commons).

Fig. 4.

A 3D geometric model was developed based on photos of the equilibrium phase of Draco gliding (B). Draco also adjusts its gliding morphology during the takeoff phase (A) and the landing phase (C). (Images from J. M. Dehling on creative commons).

Wind tunnel experimentation

To determine aerodynamic characteristics, experiments were run on the 3D geometric Draco model in a custom-built Harvey Mudd College wind tunnel designed for teaching and research. This wind tunnel was a closed circuit and has a rectangular cross-section with a height and width of 31 cm. The working distance was 60 cm and the wind tunnel was completely horizontal with respect to the ground. The 3D geometric model was affixed to an linear variable differential transformer (LVDT) transducer facing into the direction of fluid flow. The 3D geometric model was rotated to provide different angles of attack. The LVDT measured forces normal and parallel to the wind’s direction, which correlated with lift and drag measurements, respectively. Airspeed was measured with a pitot tube using a Dwyer Series 475 Mark III Digital Manometer, and the angle of attack was measured using basic trigonometry based on the configuration described in Fig. 5.

Fig. 5.

Wind tunnel experiments were performed to determine lift and drag coefficients of Draco. The 3D geometric model was affixed to an LVDT transducer facing into the direction of fluid flow. The 3D geometric model was rotated to provide different angles of attack. The LVDT measured forces normal and parallel to the wind’s direction, which correlated with lift and drag measurements, respectively. Airspeed was measured with a pitot tube using a Dwyer Series 475 Mark III Digital Manometer.

Lift and drag were measured at airspeeds ranging from 3 to 34 m/s at angles of attack between −15° and 35°. These airspeeds produced Reynolds numbers in the range of 13,000–220,000. Reynolds numbers tested at higher speeds significantly exceeded values reported for both Draco maculatus (Re ∼ 20,300) and Draco fimbriatus (Re ∼ 36,000). Although these speeds are unrealistic for Draco to attain, Reynolds numbers between ${10}^3$ and $4\mathrm{*}{10}^5$ (for bluff bodies, and higher for streamlined bodies) generally produce similar aerodynamic coefficients (Hoerner 1965; Vogel 1994). Lift and drag coefficients were calculated as a function of angle of attack based on the following equations:

$${\mathrm{ }C}_D=\frac{2D}{\rho v^{2}A}\mathrm{ }.$$

(4)

$$C_L=\frac{2L}{\rho v^{2}A}.$$

(5)

in which $D$ and $L$ are the drag and lift, $C_D$ and $C_L$ are the drag and lift coefficients, $\rho$ is the density, $v$ is the velocity of the air, and $A$ is the reference area of the lizard. The reference area was defined as the frontal portion of the 3D geometric model directly exposed to the oncoming fluid.

Numerical simulation and tail control

We sought to understand the role of tail activity on Draco glide stability. To do this a reduced-order gliding simulation was developed in Matlab based on our pitching dynamics model (Fig. 2) and experimentally determined lift and drag coefficients. Glide paths were simulated from a starting height of 5 m, and glide distance was computed based on the horizontal distance a lizard had traveled upon reaching a height of 0 m. The lizard was initially horizontal (a pitch angle of 0) with no angular velocity, and with a horizontal velocity of 2.5 m/s. Each timestep was 0.001 s.

This technique has previously been used to evaluate non-equilibrium gliding in vertebrate gliders and characterize gliding squirrel flight (Yeaton et al. 2017; Zhao et al. 2019). Reduced order simulations have also proven useful to understand the use of tail as an inertial appendage for self-righting (Jusufi et al. 2010). However, this approach has not been utilized to study Draco flight. Moreover, the work presented here extends such work to include a feedback control system that tracks tail angle as a function of a desired angle of attack.

The relative location of the center-of-pressure of an aerofoil is an important factor in flight stability. In typical aerofoils with limited control structures, when the centroid is aft to the center-of-mass, an aerofoil is often aerodynamically unstable (Anderson 2001). Aerofoils with a more fore centroid are often easier to control due to self-stabilizing aerodynamic forces. Based on measurements from museum specimens, the center-of-pressure likely lies very close to (within ∼0.01 m of) the center-of-mass. This means that behavior likely plays a role in flight stability, as small changes in Draco posture during gliding could change the relative locations of the center-of-pressure, and center-of-mass. Tail activity is one way Draco could shift its center-of-mass during gliding.

In the simulation, the tail was set to adjust the angle between −90° and 90° with respect to the lizard’s body length about a pivot on the ventral end of the lizard (Fig. 6). This resulted in corresponding changes in the location of the tail’s center of mass. Tail angle was modulated using a proportional–integral–derivative (PID) controller designed to track the angle of attack. PID controllers are standard in many control systems. The controller accepts the desired output (in this case an angle of attack) and modulates an input to minimize the difference between the actual and desired output based on PID responses:

$${\text{e}=\alpha }_{\mathrm{desired}}\mathrm{ }-\alpha .$$

(6)

$${\theta }_{\mathrm{tail}}\mathrm{ }={{\theta }_{\mathrm{tail}}+k}_{p}e+k_d\dot{e}{+k}_i\int e\mathrm{ }\mathrm{d}t$$

(7)

in which $k_p$⁠, $k_d$⁠, and $k_i\mathrm{ }$are the control coefficients, $\alpha$ is the angle of attack, and $e$ is the error.$\mathrm{ }{\theta }_{\mathrm{tail}}$ was the tail angle. An error was defined as the difference between the desired angle of attack and the actual angle of attack at the previous timestep. Maximum angular velocity of the tail was approximated based on data from aerial reorientation in geckos (Jusufi et al. 2010).

Fig. 6.

Tail angle (θtail) was controlled to track a desired angle-of-attack. The distance (r2) of the tail center-of-mass (mtail) changes relative to body center-of-mass (Mbody) when the tail angle changes.

To characterize the role of tail activity on glide performance, simulations on models with an active tail (using the PID controller), a rigid tail, and no tail were performed. In addition, the effect of different relative tail and mass lengths on glide distance was examined. To better understand the aerodynamics of Draco morphology, simulations were also performed to determine the effect of angle of attack on glide distance.

Results

Lift and drag coefficients

The wind tunnel experiments (see the “Wind tunnel experimentation” section) produced the following lift and drag curves (Fig. 7):

$$C_D=0.000008{\alpha }^2+0.0002\alpha +0.004\mathrm{ }{R}^2=0.9868$$

(8)

$$C_L=-0.00006{\alpha }^2+0.0029\alpha +0.0669\mathrm{ }{\mathrm{ }R}^2=0.9789$$

(9)

in which $\alpha$ represents the 3D geometric model’s angle of attack.

Fig. 7.

Lift and drag coefficients were measured in a wind tunnel at speeds ranging from 3 to 30 m/s as a function of angle of attack. The drag and the lift coefficients were estimated as a quadratics function relative to angle of attack (A). Panel B depicts the lift coefficient (C_L) to drag coefficient (C_D) ratio as a function of angle of attack. Effect of Reynolds number (Re) on lift coefficient (C) and drag coefficient (D) at an angle of attack of 16.4° are also depicted.

Effectiveness of tail controller

PID control coefficients were determined based on an exhaustive search to minimize the mean-squared error of angle of attack in the last 2 s of a glide.

Draco maculatus had a mean-squared error in angle of attack of $0.163$ square radians. Draco fimbriatus had a mean-squared error in angle of attack of 0.271 square radians over the last 2 s of its glide (Fig. 8).

Fig. 8.

Draco maculatus (A) and D. fimbriatus (B) angle-of-attack control was simulated using an active tail. Angle of attack (α) is depicted in red, and lizard pitch (θ) is depicted in blue. The tail angle was adjusted to track a desired angle of attack. Draco maculatus had a mean-squared error in the angle of attack of 0.163. Draco fimbriatus had a mean-squared error in the angle of attack of 0.271 square radians over the last 2 s of its glide.

Active versus rigid versus no tail

Both D. maculatus and D. fimbriatus glided farthest with the active tail, than the rigid tail, and finally with no tail. Draco maculatus glided 8.29 m with active tail control, 3.64 m with a rigid tail, and 2.43 m with no tail (Fig. 9). Draco fimbriatus glided 4.16 m with active tail control, 2.93 m with a rigid tail, and 2.40 m with no tail (Figs. 10 and 11).

Fig. 9.

To characterize the role of tail activity on glide performance in Draco, simulations on models with an active tail (using the PID controller), a rigid tail, and no tail were performed. From a height of 5 m, D. maculatus glided 8.29 m with active tail control, 3.64 m with a rigid tail, and 2.43 m with no tail.

Fig. 10.

To characterize the role of tail activity on glide performance in Draco, simulations on models with an active tail (using the PID controller), a rigid tail, and no tail were performed. From a height of 5 m, D. fimbriatus glided 4.16 m with active tail control, 2.93 m with a rigid tail, and 2.40 m with no tail.

Fig. 11.

We compared glide distance of D. fimbriatus and D. maculatus during glide simulations with an active tail. From a height of 5 m, D. fimbriatus glided 4.16 and D. maculatus glided 8.29 m.

Discussion

Gliding is the use of control features to direct an animal horizontally during aerial descent (Socha et al. 2015). Among lizards, some geckos and agamids have the ability to glide, and this is facilitated by morphological specializations of the body, feet, and tail (Young et al. 2002). Our integrative study examined gliding performance in two species of Draco that vary in gliding ability. We found that glide distances in our simulation parallel those found by Mcguire and Dudley (2005) when they studied gliding performance in a number of Draco species. In that study, Draco was coaxed to fly between poles 9.3 m apart from a 6 m tall pole, which was referred to as a “comfortable” gliding distance. Lizards lost between 2.6 and 6.2 m over the course of the 9.3-m glide.

• Our simulation produced a glide distance of 4.16 m for D. fimbriatus and 8.29 m for D. maculatus from a starting height of 5 m. These distances fall slightly short of the maximum glide distances in the same species during live tests, however, these simulations only accounted for the tail as a control structure. Adjustment of the camber of the patagium and other control strategies could further improve actual glide distance in Draco. As a result, we conclude that the tail is effective in improving glide distance in Draco, but other control strategies may also be used to enhance performance.

• Given the similarity between the previous empirical measurements and our simulations, our approach could be used to predict maximum glide distance. This has great utility for studies that are limited by enclosure size during laboratory experiments, and for studies involving species for which empirical data are difficult to obtain. These simulations might also serve as a measure of maximum performance, which can then be compared with the actual performance employed during gliding in nature. This is especially important given that lizards often underperform under natural conditions (e.g., Irschick and Losos 1998).

Role of the tail

Tails are critical for numerous ecologically-relevant behaviors such as prey capture, climbing, running, jumping, maneuvering, and maintaining stability (Gillis et al. 2009; Jusufi et al. 2010; Gillis and Higham 2016; M. F. Vollin and T. E. Higham, submitted for publication). The tail function is often studied by capitalizing on the innate ability of most lizards to self-amputate their tail (autotomy). However, this is not always possible since not all lizards are capable of autotomy. Our geometric model, wind tunnel experiments, and numerical simulation revealed how active tails can enhance gliding performance. This approach may be used as a surrogate for tail autotomy studies, especially for those species lacking that ability. Given that the role of tails during aerial locomotion is poorly understood, especially for gliding reptiles, our study is a potential aid for understanding tail function.

Tails are used to enhance aerial behavior in several vertebrates. Examples include gliding in squirrels and righting and mid-air turning in geckos (Jusufi et al. 2010), among others. The ability to control aerial descent is thought to be associated with the evolution of gliding and flapping flight (Yanoviak et al. 2005). Our simulation demonstrated that D. maculatus can significantly improve glide distance using tail control. However, while tail control slightly improved the gliding distance of D. fimbriatus, it did not effectively stabilize the angle of attack in this simulation. Based on past work, D. fimbriatus is a relatively poor glider due to its high mass and consequent wing-loading (Mcguire 2003). As a result, it makes sense to conclude that this species may also struggle to maneuver and stabilize during gliding.

Limitations

It should be noted that the 3D geometric model does not accurately represent certain aspects of Draco morphology. The 3D geometric printed model is rigid and smooth. However, Draco has rough scales which may influence boundary layer aerodynamics on the patagium (Vogel 1994). In addition, Draco has been hypothesized to adjust their camber during flight (Mcguire and Dudley 2005; Dehling 2017), whereas the 3D geometric model has a fixed camber.

Our simulation is limited in that it only takes into account the tail as a control structure for gliding. Other studies have suggested that there is significant camber adjustment during gliding (Mcguire and Dudley 2005; Dehling 2017). Draco also likely uses their forelimbs to manipulate the patagium, which could influence aerial maneuverability (Dehling 2017). Draco also possesses modified hindlimbs to increase lift, which could be used for flight control. Future in vivo studies are needed to mechanistically study the roles of these control structures during Draco gliding.

In addition, this work did not consider the aerodynamics associated with tail movement. Also, the tail and body were modeled as solid rods. Future simulations should more accurately measure the inertia of Draco body and tail.

Conclusion

In this study, we used computational methods to quantify gliding flight control in Draco using a custom-developed 3D geometric model, wind tunnel experiments, and numerical simulation. We determined that tail control can be used to improve glide distance and stability in both D. maculatus and D. fimbriatus. We also corroborated past results showing that species with higher wing-loading (D. fimbriatus) have shorter maximum glide distances (Mcguire 2003).

In the future, we believe this methodology could be useful for studying the control of gliding flight, the role of autotomy on locomotion in lizards, and the evolution of gliding in reptiles. In regards to the latter, it is unclear how much control early gliders had over their tail, but we assume that both gliding ability and tail control likely evolved in parallel. Applying our approach to reconstructed extinct specimens would provide insight into the gliding ability of these species.

Conflicts of interest

The authors declare no conflicts of interest.

Data availability

Data and simulation code for this article is available at https://github.com/jadenvc/Active-tails-enhance-gliding-in-arboreal-lizards.