Supplementary information:
Trunk orientation causes asymmetries in leg function in small bird terrestrial
locomotion
Emanuel Andrada1, Christian Rode1, Yefta Sutedja1, John Nyakatura23, Reinhard Blickhan1.
Institutions:
1
Science of Motion, Friedrich-Schiller University Jena, Seidelstraße 20, 07749 Jena, Germany.
2
Institut für Spezielle Zoologie und Evolutionsbiologie mit Phyletischem Museum, Friedrich-SchillerUniversität, D-07743 Jena, Germany.
3
Image Knowledge Gestaltung: In interdisciplinary laboratory and Institute of Biology, HumboldtUniversity, Philippstraße 13, D-11015 Berlin, Germany.
S1 Material and methods (extended)
Experimental data
Animals, x-ray and force data
We corrected for distortion (using MATLAB scripts available free of charge from www.xromm.org),
filtered the raw video data (e.g., gamma correction, contrast, sharpness) and, finally, digitized the
joints and other landmarks (Fig. 2A) using SimiMotion® software (SimiMotion Systems,
Unterschleißheim, Germany). To measure GRF, 6-DOF (degree of freedom) force-torque sensors (ATI
nano17®) were used as transducer elements. GRFs were collected at 1 kHz (NI USB-6229, National
Instruments®; custom software LabView 2009 National Instruments®) and force and X-ray data
synchronized electronically (post-trigger).
Center of mass mechanics
As described in (1, 2), we calculated the instantaneous position of the body’s CoM from kinematic and
cadaveric data (X-ray motion analysis data of the limbs, additional kinematic data of digitized head,
neck and torso landmarks, and each element’s CoM position).
%congruity: Gait transitions in birds are smooth (2, 4-6), but we defined walking as %Congruity < 50 %
and bouncing gaits as %Congruity > 50 % (for detailed information see (1)).
VPP in body and CoM coordinates
In the experimental data we estimate the VPP height where the horizontal spread of GRFs is minimal.
In our model the VPP is fixed to the trunk. In simulations the height at which the horizontal spread of
GRFs is minimal in CoM coordinates corresponds to the height of the body fixed VPP point (rvpp), see
Fig.S1.
Fig.S1. Simulations of grounded running and walking with the PVPP. Grounded running: k=100, C=4.1,
rVPP=0.04; walking: k=100, C=4.7, rVPP=0.065. Larger trunk oscillations make that a body fixed VPP
leads to an area where GRFs intersects when plotted in CoM coordinates.
Data analysis
Non-steady state trials with a horizontal speed deviation of more than 10 % between two midstance
events were discarded for later analysis. In most cases the successful trials were grounded runs or
walking trials. Aerial running was rarely used by the quail.
Simulations
Bipedal locomotion requires the regulation of hip torques to balance the trunk. In line with the
tradition of keeping models as simple as possible to understand principles of locomotion (7), we
extended the SLIP model by adding a trunk, controlling the hip torques such that the GRFs pointed to
the VPP, and accounted for asymmetric leg function by using a damper in parallel to a spring-like leg.
We call this model the Pronograde Virtual Pivot Point (PVPP) model.
PVPP model and geometry
Hip coordinates are given by
𝑥ℎ = 𝑥 − 𝑟ℎ ∙ 𝑠𝑖𝑛𝜃
𝑦ℎ = 𝑦 + 𝑟ℎ ∙ 𝑐𝑜𝑠𝜃
and touchdown occurs for
𝑦ℎ = 𝑙0 ∙ sin⁡(𝛼 − 𝜙0 )
with x and y are the CoM coordinates in x- and y-axes, θ the trunk angle measured from vertical, the
angle between the posterior leg and the ground, and 0 the aperture angle (Fig. 2).
Finally, the position of the VPP is obtained as:
𝑥𝑣𝑝𝑝 = 𝑥 + 𝑟𝑣𝑝𝑝 ∙ sin⁡(𝜃 − 𝜓0 )
𝑦𝑣𝑝𝑝 = 𝑦 + 𝑟𝑣𝑝𝑝 ∙ cos(𝜃 − 𝜓0 ).
The GRF of each leg is required to point towards the VPP at all times. This is achieved by an additional
force Ft acting at the tip of the leg perpendicular to the leg force Fa along the leg axis (Fig. 1). The GRF
of each leg is the sum of Fa and Ft. The magnitude of Fa is defined by Eq. 1, while Ft is defined by Ft =
Fa tanβ, with  being the angle between leg and VPP (Fig. 2A). The hip torque is given by Mh = Ft.l.
Equations of motion
The equations of motion are:
𝑚ẍ = 𝐹𝑥
𝑚ӱ = −𝑚𝑔 + 𝐹𝑦
𝐽𝜃̈ = 𝑟𝑉𝑃𝑃 (𝐹𝑥 cos(𝜃 − 𝜓0 ) − 𝐹𝑦 sin⁡(𝜃 − 𝜓0 ))
Where Fx and Fy are the sum of the horizontal and vertical components of the GRF of the legs, g is
gravitational acceleration, and x , y , and  are the horizontal, vertical, and angular CoM accelerations,
respectively.
Quail model parameters and parameter space
We constrained the scanned parameter space according to the experimental results: 1.5 Nsm1
≤ c ≤ 8 Nsm-1, c = 0.2 Nsm-1; 90° ≤  ≤ 140°, 0 = 2°; 0.01 m ≤ rVPP ≤ 0.1 m, rVPP = 0.005 m; CoM
initial height 0.07 m ≤ y0 ≤ 0.116 m, y0 = 0.002 m. Note that during locomotion, the trunk angle θ
oscillates about the value, i.e. θmean≈. Initial speed for both categories is close to the mean values
observed in the experiments (see results). Initial vertical speed is vy0 = 0 ms-1, trunk angular speed is
 = 0 rads-1, and the stance leg is oriented vertically. Initial trunk angle θ0 is set to 
Stability
A rigorous analysis of stability was not the aim of this paper. In accordance with literature (8), we
defined stability as the ability to cope with even undetected perturbations. A common way of assessing
gait stability is the steps-to-fall method (9). We adopted a similar approach and analyzed whether the
vertical amplitude of CoM movement was still decreasing in forward simulations after 100 completed
steps (stable solutions converge asymptotically to the periodic oscillation). If this was the case, the
simulation was deemed to be stable, though it might also be mathematical partially stable.
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