6 
	
  
Supplementary Figures S1-S3 and Table S7: Model analysis and results 
 
	
  
Figure S1.  Force-posture relations for the actuated spring-mass-damper model with various and arbitrary actuator 
motions.  Here, the model parameters and touch-down conditions have been held constant, and arbitrary actuator 
motions applied. This demonstrates a wide range of possible force-length relations with the mathematical model.  
The arrangement of the actuator in series with the spring and damper decouples posture from force, allowing for 
forces that deviate significantly from a Hooke’s law relation. The specific force-length trajectory of the simulation 
results arises from minimal-work optimisation.  
The Journal of Experimental Biology | Supplementary Material
7 
	
  
 
Figure S2.  Example work-optimal solutions for the mathematical model satisfying the level-running ostrich gait 
boundary conditions (touchdown conditions of current and subsequent step).  By changing either the model stiffness 
or damping coefficient by a factor of two, different work-optimal solutions emerge from the control optimization, 
which all satisfy the boundary conditions (i.e. the problem is not over-constrained).  The modelling methods allow 
for freedom in take-off conditions, such that the model solutions could yield longer or shorter flight phases that 
satisfy the touch-down conditions for the subsequent step. Thus, the modelling approach can yield solutions with 
gait parameters and GRFs that deviate substantially from observed data. To make choice of stiffness and damping 
parameters non-arbitrary, we choose the model parameters for which a work-optimal control matched the data best 
(Fig. S3). However, the set of solutions from which these parameters were chosen (e.g., Fig. S3) were all work-
optimal for their respective parameter values, and were not constrained to fit the bird data. Consequently, the 
modelling approach could have failed to fit the data, potentially refuting the work-minimising hypotheses.  
 
The Journal of Experimental Biology | Supplementary Material
8 
	
  
 
Figure S3: A typical example of a parameter-fitting surface for the reduced order model of avian running: 
The results of our search for the best fitting parameters to the simple model with minimal actuation (Fig. 6A), 
visualised as a fitting landscape. All solutions shown on the surface are work-optimal for their respective parameter 
values. In this example, computed using ostrich data, the surface shows a characteristic ‘trough’ of parameter fits 
that emerge when searching for knorm and cnorm that best fit bird data. The red ‘trough’ line connects the best fits for 
each value of cnorm. Parameters are normalised as described Table S7, and mean-squared error is computed between 
model and mean-measured GRF. While some regions of this fitting landscape clearly performed better than others, 
there was often a large set of solutions that performed similarly well.  Given the non-unique nature of the parameter 
fits, we do not make scientific claims about the functional significance of the fit set of parameters.  Nonetheless, we 
did find a relatively narrow range of damping ratios (a standard measure of decay in oscillating systems) resulting in 
fits consistent with bird running data (Table S7). We report this as a successful result for the general model, which 
yielded good match between bird and model GRF, given a two-parameter fit (MSE: quail: 0.0103, pheasant: 0.0280, 
guinea fowl: 0.0032, turkey: 0.0086, ostrich: 0.0063, calculated by force error normalised to body weight). 
 
 
The Journal of Experimental Biology | Supplementary Material
1 
	
  
Supplementary Tables S1-S6: Statistical results from experimental data 
 
 
Dependent Variable 
Step Type 
Species 
Species X StepType 
θTD 
20.60 
3.47 
4.49 
HTD 
41.85 
0.07 
2.77 
αTO 
15.59 
1.69 
4.60 
ΔEP 
31.25 
0.27 
3.55 
ΔEK 
7.51 
0.78 
3.24 
ΔECoM 
8.64 
0.42 
2.97 
Fmax 
5.06 
6.07 
2.69 
*Bolding indicates a statistically significant result 
Table S1: ANOVA F-statistic results for 5 species, including ostriches, testing for effects of step type and species in 
0.1Lleg obstacle terrain (see Methods). Degrees of freedom are as follows: step type = 3; species = 4; species x step 
type = 12; αTO total = 743; all other variables total = 790. The F-statistic for the effect of step type on leg posture 
(θTD, HTD) and change in potential energy (ΔEp), which are most indicative of obstacle negotiation strategy, are 
much larger than the corresponding F-statistics for species (F < 1) and species x step type (F < 5). This reflects a 
uniform obstacle negotiation strategy across species the species studied here (see posthoc comparisons in Tables S2-
S3 for further detail). All species used a consistent balance of ‘vaulting’ and ‘crouching’ strategies (Figs, 1 and 2). 
 
Step 1 
Step 2 
θTD (degrees) 
HTD 
αTO (degrees) 
ΔEP 
ΔEK 
ΔECoM 
Fmax 
Level 
Step -1 
-0.80 
0.015 
2.28 
0.041 
0.053 
0.093 
0.01 
Level 
Step 0 
2.57 
-0.044 
-0.36 
0.015 
-0.005 
0.010 
-0.09 
Level 
Step 1 
-2.41 
0.056 
-2.27 
-0.051 
0.065 
0.014 
0.06 
Step -1 
Step 0 
3.37 
-0.059 
-2.63 
-0.026 
-0.057 
-0.084 
-0.10 
Step -1 
Step 1 
-1.61 
0.041 
-4.55 
-0.092 
0.013 
-0.079 
0.05 
Step 0 
Step 1 
-4.98 
0.099 
-1.91 
-0.066 
0.070 
0.005 
0.15 
*Bolding indicates significant difference based on Bonferroni threshold of 0.0083, for 6 possible step type pairwise comparisons within level and 
0.1 Lleg obstacle height. 
Table S2: Post hoc results on the ANOVA using pairwise mean differences between step types (column 2 - column 
1), in normalised units.  
	
  
	
  
The Journal of Experimental Biology | Supplementary Material
2 
	
  
Species 1 
Species 2 
θTD (degrees) 
HTD 
αTO (degrees) 
ΔEP 
ΔEK 
ΔECoM 
Fmax 
Step -1 
 
 
 
 
 
 
 
 
Quail 
Pheasant 
-6.89 
-0.020 
-3.73 
0.016 
0.117 
0.133 
-0.51 
Quail 
Guinea fowl 
-5.26 
-0.294 
-1.05 
0.021 
0.021 
0.042 
-0.22 
Quail 
Turkey 
-4.34 
-0.037 
-2.75 
-0.025 
0.009 
-0.017 
-0.14 
Quail 
Ostrich 
-3.48 
-0.126 
0.36 
0.027 
0.030 
0.056 
-0.08 
Pheasant 
Guinea fowl 
1.63 
-0.010 
2.68 
0.005 
-0.096 
-0.092 
0.30 
Pheasant 
Turkey 
2.55 
-0.018 
-0.98 
-0.042 
-0.109 
-0.150 
0.38 
Pheasant 
Ostrich 
3.41 
-0.107 
4.10 
0.010 
-0.087 
-0.077 
0.43 
Guinea fowl 
Turkey 
0.91 
-0.008 
-1.70 
-0.046 
-0.012 
-0.058 
0.08 
Guinea fowl 
Ostrich 
1.77 
-0.097 
1.41 
0.006 
0.009 
0.015 
0.14 
Turkey 
Ostrich 
0.86 
-0.089 
3.12 
0.052 
0.021 
0.073 
0.06 
Step 0 
 
 
 
 
 
 
 
 
Quail 
Pheasant 
-4.81 
-0.032 
1.71 
0.014 
0.159 
0.174 
-0.35 
Quail 
Guinea fowl 
-2.95 
-0.029 
4.57 
0.045 
0.095 
0.141 
-0.05 
Quail 
Turkey 
-3.36 
-0.014 
3.73 
0.013 
-0.030 
-0.017 
0.09 
Quail 
Ostrich 
-1.47 
-0.058 
-0.36 
0.013 
0.061 
0.074 
-0.22 
Pheasant 
Guinea fowl 
1.86 
0.004 
2.85 
0.031 
-0.064 
-0.033 
0.30 
Pheasant 
Turkey 
1.45 
0.019 
2.01 
-0.002 
-0.189 
-0.191 
0.45 
Pheasant 
Ostrich 
3.34 
-0.025 
-2.08 
-0.001 
-0.098 
-0.099 
0.13 
Guinea fowl 
Turkey 
-0.41 
0.015 
-0.84 
-0.033 
-0.125 
-0.158 
0.14 
Guinea fowl 
Ostrich 
1.48 
-0.029 
-4.93 
-0.032 
-0.034 
-0.067 
-0.16 
Turkey 
Ostrich 
1.89 
-0.044 
-4.09 
0.000 
0.091 
0.091 
-0.31 
Step +1 
 
 
 
 
 
 
 
 
Quail 
Pheasant 
-6.46 
0.027 
-3.99 
-0.067 
0.123 
0.056 
-0.12 
Quail 
Guinea fowl 
-7.77 
0.024 
-1.53 
-0.038 
0.069 
0.031 
-0.01 
Quail 
Turkey 
-2.18 
-0.023 
0.93 
0.015 
-0.068 
-0.053 
0.00 
Quail 
Ostrich 
-2.94 
-0.024 
-1.61 
0.006 
0.024 
0.031 
-0.01 
Pheasant 
Guinea fowl 
-1.31 
-0.003 
2.46 
0.028 
-0.053 
-0.025 
0.12 
Pheasant 
Turkey 
4.28 
-0.050 
4.92 
0.081 
-0.190 
-0.109 
0.13 
Pheasant 
Ostrich 
3.52 
-0.051 
2.38 
0.073 
-0.098 
-0.026 
0.11 
Guinea fowl 
Turkey 
5.59 
-0.047 
2.46 
0.053 
-0.137 
-0.084 
0.01 
Guinea fowl 
Ostrich 
4.83 
-0.048 
-0.08 
0.044 
-0.045 
-0.001 
-0.00 
Turkey 
Ostrich 
-0.76 
-0.001 
-2.54 
-0.009 
0.092 
0.083 
-0.01 
*Bolding indicates significant difference based on Bonferroni threshold of 0.005, for 10 possible species pairwise comparisons within each step 
category. 
Table S3: Post hoc pairwise mean differences between species (column 2 - column 1), in normalised units. 
The Journal of Experimental Biology | Supplementary Material
3 
	
  
Dependent Variable 
Step Type 
Species 
Obstacle Height 
Obstacle Height X Step 
Type 
Species X Step Type 
θTD 
663.92 
6.37 
9.99 
105.26 
1.13 
HTD 
1421.61 
1.32 
8.20 
223.14 
0.98 
αTO 
584.10 
9.00 
1.60 
115.29 
6.53 
ΔEP 
1402.79 
2.33 
7.13 
267.98 
3.01 
ΔEK 
78.83 
4.57 
5.37 
13.99 
2.08 
ΔECoM 
217.03 
2.49 
9.39 
34.40 
3.58 
Fmax 
114.41 
16.33 
9.79 
19.11 
2.51 
*Bolding indicates a statistically significant result 
Table S4: ANOVA F-statistic results for galliform birds, with obstacle heights from 0.1-0.5Lleg (see Methods). 
Degrees of freedom are as follows: step type = 2; species = 3; obstacle height = 5; obstacle height x step type = 10; 
species x step type = 6; αTO total = 2360; all other variables total = 2522. Most of the variance in the model is 
explained by step type and the interaction of obstacle height and step type, reflecting a consistent obstacle 
negotiation strategy across species. The F-statistics for the effects of step type and obstacle height on leg posture 
(θTD, HTD) and potential energy (ΔEp), which are most indicative of obstacle negotiation strategy, are much larger 
than the corresponding F-statistics for the effects of species. We did not observe a significant shift in obstacle 
negotiation strategy with body size between small and large birds (see Supplementary Table S6).  
 
 
The Journal of Experimental Biology | Supplementary Material
4 
	
  
 
Terrain 
θTD (degrees) 
HTD 
αTO (degrees) 
ΔEP 
ΔEK 
ΔECoM 
Fmax 
Step -1 
 
 
 
 
 
 
 
ObsH=0.1 
-1.02 
0.027 
1.91 
0.037 
0.048 
0.085 
0.02 
ObsH=0.2 
0.16 
0.005 
3.98 
0.085 
0.034 
0.119 
0.14 
ObsH=0.3 
-1.68 
0.016 
5.99 
0.141 
0.066 
0.207 
0.26 
ObsH=0.4 
0.45 
0.001 
9.88 
0.242 
0.041 
0.283 
0.35 
ObsH=0.5 
-1.16 
0.011 
11.22 
0.293 
-0.048 
0.245 
0.30 
Step 0 
 
 
 
 
 
 
 
ObsH=0.1 
2.78 
-0.036 
0.33 
0.011 
-0.042 
-0.031 
0.02 
ObsH=0.2 
7.20 
-0.101 
-0.07 
0.018 
-0.039 
-0.020 
-0.08 
ObsH=0.3 
8.16 
-0.122 
-0.69 
0.001 
-0.051 
-0.050 
-0.08 
ObsH=0.4 
10.43 
-0.148 
-2.75 
-0.014 
-0.044 
-0.058 
-0.14 
ObsH=0.5 
9.09 
-0.156 
-3.99 
-0.042 
-0.040 
-0.082 
-0.21 
Step +1 
 
 
 
 
 
 
 
ObsH=0.1 
-1.79 
0.051 
-1.60 
-0.043 
0.037 
-0.006 
0.06 
ObsH=0.2 
-2.95 
0.096 
-4.24 
-0.090 
0.098 
0.008 
0.05 
ObsH=0.3 
-5.02 
0.133 
-4.44 
-0.113 
0.090 
-0.023 
0.14 
ObsH=0.4 
-6.54 
0.181 
-8.63 
-0.164 
0.121 
-0.043 
0.13 
ObsH=0.5 
-11.08 
0.222 
-9.02 
-0.187 
0.146 
-0.041 
0.10 
*Bolding indicates a significant difference based on Bonferroni threshold of 0.0033, for 15 possible obstacle pairwise comparisons within each 
step category 
Table S5: Post hoc pairwise mean differences (Obs- Level) in normalised units, for obstacle heights by step type 
across galliform birds. 
 
The Journal of Experimental Biology | Supplementary Material
5 
	
  
Species 1 
Species 2 
θTD (degrees) 
HTD 
αTO (degrees) 
ΔEP 
ΔEK 
ΔECoM 
Fmax 
Step -1 
 
 
 
 
 
 
 
 
Quail 
Pheasant 
-- 
-- 
-1.91 
0.015 
-- 
0.060 
-0.28 
Quail 
Guinea fowl 
-- 
-- 
-0.63 
-0.000 
-- 
0.021 
-0.07 
Quail 
Turkey 
-- 
-- 
-0.31 
0.011 
-- 
0.013 
-0.01 
Pheasant 
Guinea fowl 
-- 
-- 
1.28 
-0.015 
-- 
-0.039 
0.21 
Pheasant 
Turkey 
-- 
-- 
1.60 
-0.004 
-- 
-0.047 
0.27 
Guinea fowl 
Turkey 
-- 
-- 
0.32 
0.011 
-- 
-0.008 
0.06 
Step 0 
 
 
 
 
 
 
 
 
Quail 
Pheasant 
-- 
-- 
0.14 
-0.012 
-- 
-0.037 
-0.26 
Quail 
Guinea fowl 
-- 
-- 
1.33 
-0.010 
-- 
-0.013 
-0.05 
Quail 
Turkey 
-- 
-- 
0.64 
-0.018 
-- 
-0.062 
0.06 
Pheasant 
Guinea fowl 
-- 
-- 
1.19 
0.003 
-- 
0.024 
0.21 
Pheasant 
Turkey 
-- 
-- 
0.50 
-0.005 
-- 
-0.025 
0.31 
Guinea fowl 
Turkey 
-- 
-- 
-0.69 
-0.008 
-- 
-0.049 
0.11 
Step +1 
 
 
 
 
 
 
 
 
Quail 
Pheasant 
-- 
-- 
-1.97 
0.000 
-- 
0.005 
-0.32 
Quail 
Guinea fowl 
-- 
-- 
-1.73 
-0.016 
-- 
-0.001 
-0.03 
Quail 
Turkey 
-- 
-- 
-1.89 
-0.013 
-- 
-0.019 
-0.04 
Pheasant 
Guinea fowl 
-- 
-- 
0.24 
-0.016 
-- 
-0.006 
0.29 
Pheasant 
Turkey 
-- 
-- 
0.08 
-0.013 
-- 
-0.024 
0.28 
Guinea fowl 
Turkey 
-- 
-- 
0.16 
0.003 
-- 
-0.019 
-0.01 
*Bolding indicates significant difference based on Bonferroni threshold of 0.0083, for 6 possible species pairwise comparisons within each step category. 
Table S6: Post hoc pairwise mean differences between galliform species (column 2-column 1) from ANOVA 
(Table S4). Notably, pairwise differences in leg posture (θTD, HTD) and change in potential energy (ΔEp), which are 
most indicative of obstacle negotiation strategy, do not significantly differ between species.	
  
 
 
 
The Journal of Experimental Biology | Supplementary Material
9 
	
  
 
 
Species 
 
Quail 
Pheasant 
Guinea fowl 
Turkey 
Ostrich 
Fitted Parameters 
 
 
 
 
 
Spring stiffness 
(knorm=k*Lleg /(m*g)) 
8.0 
11 
15 
10 
12 
Damping coefficient  
 
0.10 
0.20 
0.40 
0.20 
0.40 
Computed Property 
 
 
 
 
 
Damping ratio 
 
0.018 
0.020 
0.052 
0.032 
0.058 
Optimal trajectory performance 
 
 
 
 
 
Mean-squared error 
0.0102 
0.0266 
0.0032 
0.0081 
0.0063 
Net unsigned work 
(Joules / (m*g*Lleg)) 
0.3149 
0.1708 
0.0992 
0.0761 
0.0081 
Normalising Parameters 
 
 
 
 
 
m (kg) 
0.200 
1.02 
1.48 
2.96 
116 
Lleg (m) 
0.117 
0.201 
0.228 
0.287 
0.974 
g (m/s2) 
9.81 
Table S7: Normalised results of trajectory optimisation applied to the actuated model (Fig. 6A), resulting in the 
reported fits to bird GRF (Fig. 6B) and leg length trajectories (Fig. 6C).  Bird size spanned over a 500-fold mass 
range, but the damping ratio remained with a factor of 3.27 across species.  Average masses reported in this table 
differ somewhat from those reported in main text because here the mass averaging was weighted by number of level 
step samples, not by individual birds. Given the non-unique nature of the parameter fits (Fig. S3), we do not make 
scientific claims about the functional significance of any one particular set of parameters.  Nonetheless, a relatively 
narrow range of damping ratios results in fits consistent with bird running data.   
 
 
The Journal of Experimental Biology | Supplementary Material

Don't break a leg: running birds from quail to ostrich prioritise leg safety and economy on uneven terrain.
Birn-Jeffery, Aleksandra V,Hubicki, Christian M,Blum, Yvonne,Renjewski, Daniel,Hurst, Jonathan W,Daley, Monica A