Evolutionary Robotic Approaches in Primate Gait Analysis

Abstract

Understanding how primates move is particularly challenging because many of the experimentation techniques that would normally be available are unsuitable for ethical and conservation reasons. We therefore need to develop techniques that can maximize the data available from minimally intrusive experimentation. One approach for achieving this is to use evolutionary robotic techniques to build a musculoskeletal simulation and generate movement patterns that optimize some global parameter such as economy or performance, or to match existing kinematic data. If the simulation has a sufficiently high biofidelity and can match experimentally measured performance criteria then we can use it to predict aspects of locomotor mechanics that would otherwise be impossible to measure. This approach is particularly valuable when studying fossil primates because it can be based entirely on morphology and can generate movements spontaneously. A major question in human evolution is the origin of bipedal running and the role of elastic energy storage. By using an evolutionary robotics model of humanoid running we can show that elastic storage is required for efficient, high-performance running. Elasticity allows both energy recovery to minimize total energy cost and also power amplification to allow high performance. The most important elastic energy store on the human hind limb is the Achilles tendon: a feature that is at best weakly expressed among the African great apes. By running simulations both with and without this structure we can demonstrate its importance, and we suggest that identification of the presence or otherwise of this tendon—perhaps by calcaneal morphology or Sharpey’s fibers—is essential for identifying when and where in the fossil record human style running originated.

Appendix

This section describes the muscle model used. All equations are given in Mathematica format so that they can simply be copied into a notebook and solved as required. The muscle model consists of the standard Minetti and Alexander contraction model (Minetti and Alexander 1997) that represents concentric and eccentric contractions by the following equations (rearranged from the originals so that they incorporate linear rather than rotational quantities and altering the sign of the contraction direction so that a negative velocity represents shortening):Eccentric (vce > 0)

$$ eq1: = fce = = alpha\,f0\left( {1.8 + \left( {0.8\,k\left( {vce - 1.\,vmax} \right)} \right)/7.56\,vce + k\,vmax} \right) $$

Concentric (vce ≤ 0)

$$ eq2: = fce = = \left( {alpha\,f0\,k\left( {vce + vmax} \right)} \right)/\left( { - vce + k\,vmax} \right) $$

where

fce :

contractile element force (N)

alpha :

activation (0–1)

k :

shape constant (generally 0.17)

vce :

contractile element velocity (m/s)

vmax :

maximum shortening velocity (m/s)

f0 :

isometric force (N)

Serial and parallel springs are then added in the Hill style and are implemented using the following equations:

$$ eq3: = fce + fpe = = fse $$

$$ eq4: = fse = = ese\left( {lse - sse} \right) $$

$$ eq5: = fpe = = epe\left( {lpe - spe} \right) $$

where

fce :

contractile force (N)

lpe :

contractile and parallel length (m)

spe :

slack length parallel element (m)

epe :

elastic constant parallel element (N/m)

fpe :

parallel element force (N)

lse :

serial length (m)

sse :

slack length serial element (m)

ese :

elastic constant serial element (N/m)

fse :

serial element force (N)

Two other equations can be derived based on the total length of the muscle-tendon unit and the previous model state:

$$ eq6: = len = = lpe + lse $$

$$ eq7: = vce = = \left( {lpe - lastlpe} \right)/timeIncrement $$

where

len :

total length of system (m)

lastlpe :

length of the parallel element at the last time step (m)

timeIncrement :

integration time step of the simulator (s)

For the concentric case the C code can now be generated using the Mathematica command:

$$ CForm\left[ {FullSimplify\left[ {Solve\left[ {\left\{ {eq2,eq3,eq4,eq5,eq6,eq7} \right\},\left\{ {lpe,lse,fpe,fse,fce,vce} \right\}} \right]} \right]} \right] $$

Similarly for the eccentric case:

$$ CForm\left[ {FullSimplify\left[ {Solve\left[ {\left\{ {eq1,eq3,eq4,eq5,eq6,eq7} \right\},\left\{ {lpe,lse,fpe,fse,fce,vce} \right\}} \right]} \right]} \right] $$

These results need to be checked for validity because the solutions are not unique. However, there is a unique solution when range criteria are checked. There are also special cases when springs are slack. If both parallel and serial springs are slack then the model reverts back to the non-spring version. If just the parallel spring is slack then Eq. 5 changes and there are different solutions:

$$ eq5b {:} = fpe = = 0 $$

The concentric case C code is now generated by:

$$ CForm\left[ {FullSimplify\left[ {Solve\left[ {\left\{ {eq2,eq3,eq4,eq5b,eq6,eq7} \right\},\{ lpe,lse,fpe,fse,fce,vce\} } \right]} \right]} \right] $$

The eccentric case C code is now generated by:

$$ CForm\left[ {FullSimplify\left[ {Solve\left[ {\left\{ {eq1,eq3,eq4,eq5b,eq6,eq7} \right\},\{ lpe,lse,fpe,fse,fce,vce\} } \right]} \right]} \right] $$