Quasi-steady state aerodynamics of the cheetah tail

The cheetah's hunting success has been attributed to both its speed and agility during acceleration and turning, with tail movements often being associated with these manoeuvres (Wilson, et al., 2013). During pursuit of prey the tail is swung from one side of the body to the other (in the roll axis) for turn initiation at high-speed (P. Hudson, PhD Thesis, Royal Veterinary College, London, 2011) and upwards or downwards (in the sagittal plane) during rapid acceleration and braking manoeuvres (Fig. 1A; Movie 1). These movements have the potential to exert inertial forces and torques on the body (Wilson et al., 2013) similar to those observed in lizard tails (Libby et al., 2012). The cheetah tail is commonly referred to as functioning as a rudder (used to change the heading of body) or a counterweight (used for balance) (Eaton, 1972; Thompson, 1998). Actuated tails on cheetah-inspired robots provide inertial compensation, stabilising the platforms when making aggressive manoeuvres (Patel and Braae, 2013, 2014). Measurements of cheetah tails however, indicate that the tail is relatively light, and, as most of its mass is located near the base (P. Hudson, PhD Thesis, Royal Veterinary College, London, 2011), the relevant moment of inertia is modest. Aerodynamic effects of the cheetah tail during rapid manoeuvres have not been investigated and our hypothesis is that the long, furry tail generates aerodynamic forces that contribute to the angular impulse (especially at high speeds), thereby assisting manoeuvrability. In an aerodynamic study of tarsiers, effects of tails were found to be negligible because of low airspeed (Peters, 1985) but in cockroach-inspired robots with sail-like tails they were significant and allowed rapid turns (Kohut et al., 2013).

In this paper, we report on measurements of the aerodynamic force of two distinct sections of the tail (mid-section and tip) in a series of wind tunnel tests. Using this data, we ran dynamic simulations of a rigid tail executing swinging motions to estimate the contribution of aerodynamic forces on the body and draw conclusions from the results.

The effective area was determined for each tail so as to minimise the weighted (via propagation of the wind speed and force measurement accuracies as independent statistical quantities) mean square error in Eqn (1) with fixed C=1.2. After accounting for the skin thickness, the effective fur lengths were ℓ_f=8.8 mm and ℓ_f=11.3 mm for Tails 1 and 2, respectively. These lengths are less than the lengths of the guard hairs (ranging from 22 mm to 39 mm over both tails; see Fig. S1). The effective area in the mid-section of both tails is nearly double (1.8 times) the frontal area of the tail without fur, allowing an increase in aerodynamic forces without significant mass penalty. The final drag coefficient data with effective areas determined above are depicted in Fig. 1B.

The cheetah tail tip effective areas (calculated as above) resulted in fur lengths of 11.9 mm and 8 mm for Tails 1 and 2, respectively and the aerodynamic coefficient data is depicted in Fig. 1E. Again, these fur lengths are plausible based on the measured guard fur (ranging from 31 mm to 50 mm). During the wind tunnel tests, we observed that the very long fur on the tail tip tended to collapse at higher airspeeds (>20 m/s) but this deformation of the tip fur was not observed in video footage of cheetahs performing rapid manoeuvres. This implies that the tail may potentially exert even greater forces on the body than we have estimated.

The simulation results are summarised in Fig. 2B. As can be seen, aerodynamic effects make a contribution to the angular impulse, in the pitch, roll and yaw cases (respectively 26%, 12.6% and 28% of the total 30 m/s). Initially the effect of the tail is inertial, but as the inertial effects decrease, the total torque is maintained by an increase in the aerodynamic torque (Fig. 2C). Tail motion in the pitch axis has been attributed to both increasing pitch stability and lowering the peak forces on the front legs (Wilson et al., 2013). In addition, by yawing the tail in the aerial phase the cheetah can redirect its body heading and by rolling the tail it can bank the body appropriately into a turn (P. Hudson, PhD Thesis, Royal Veterinary College, London, 2011) or counteract the destabilising effect of the centrifugal force (Patel and Braae, 2013). In all cases, the aerodynamic force also has a braking effect.

We have not considered non-linear, time varying and distributed aerodynamic effects. Nevertheless, our first order, quasi-steady state results clearly support the hypothesis that aerodynamic effects of the cheetah's long, furry tail contribute to the angular impulse that can be applied to the body, especially at higher speeds. Both inertial and aerodynamic effects must therefore be considered in modelling the use of the cheetah tail for manoeuvring tasks. Our results further support the observations that the cheetah tail can be used as a ‘rudder’ to contribute to fast change of heading, and as a ‘stabiliser’ during rapid acceleration and turning.

The use of the cheetah tails complied with the University of Cape Town Science Faculty Ethics policy. Tails from two euthanized cheetahs were obtained from the National Zoological Gardens of South Africa. The cheetah tails were removed from the intervertebral disc between S3 and C1, weighed (Boeco BBL61, 0.1 g) and measured. The vertebrae of Tail 1 were individually measured to calculate the moment of inertia about the base (via Steiner's theorem). Our measurements are compared to previous results in Table 1.

We measured the drag of a section of a furry cylinder normal to the airflow, for varying wind speeds up to 30 m/s. This is comparable to the average hunting speed of ∼20 m/s (Wilson et al., 2013) executing a tail flick of ∼17 rad/s at a length of 0.7 m (P. Hudson, PhD Thesis, Royal Veterinary College, London, 2011). A 300-mm long piece of 32-mm diameter PVC pipe (from the average diameter of Tail 1 of ∼31 mm, Table S2), was covered with cheetah tail pelt from the middle section of each tail. To minimise end effects, dummy fur-covered cylinders were fixed above and below the test section as illustrated in Fig. 1C. The fur was compressed using Vernier callipers to estimate skin-fur boundary diameter. A third set of measurements was made with pelt from Tail 1 but using a 50-mm diameter pipe, giving a higher Reynolds number at the same airspeed.

The tip of the tail is bushy, typically experiences the largest speeds, has the largest lever arm, and complicated span-wise or end-effects. Based on the approximate geometry of the last ten caudal vertebrae of Tail 1 (Tables S1 and S2), a truncated cone (Fig. 1G) of length 238 mm with initial diameter 18 mm and tip diameter 5 mm was constructed from wood and covered with pelt. The tail tip test measured the force normal to the tail tip at varying angles of inclination (0° to 60°) and wind speeds (up to 30 m/s) (Fig. 1F). We assumed that the tip is extended by the fur length (Fig. 1G), and neglected span-wise effects of the tapered cone and tip end. Although these are approximations for a very complex underlying problem, the morphometric tail tips have comparable geometry and scale to actual tails and we therefore expect the force data to be recovered in calculations for other tail tips.

The motion of any animal in its natural environment is very complex. From observations of captive cheetahs at the Cheetah Outreach (Somerset West, South Africa) and from wildlife documentaries (Table S3), we abstracted the following simplified motion primitives to aid understanding; pitching in the sagittal plane, yawing in coronal plane, and rolling in the transverse plane. We developed rigid tail models using Euler–Lagrange dynamics for each of the three motions (see Fig. 2A). The assumption that the tail is rigid is a reasonable starting point as it appears that the tail is often held rigidly when swinging (P. Hudson, PhD Thesis, Royal Veterinary College, London, 2011). For the pitch and yaw case it is assumed that the tail is attached at some distance, L_b, from the centre of mass (CoM) of the body. For each of the models, the velocity of the tail is calculated using geometry.

We thank Cheetah Outreach for allowing us to film their exercise runs, The Ann van Dyk Cheetah Centre for use of the tails, Charlie Turnley and Javaad Patel for technical support, and Prof Sandra Corr for her support.