IEEE Robotics & Automation Magazine - March 2016 - 88

the performance of the selected primitives by computing
the normalized root-mean-square error and added components until this error became less than 4% between the reconstructed and actual
signals. Only three primThese devices may
itives were necessary to
account for approximatebenefit from a primitive
ly 98% of the data set
total variance.
based control in complex
These selected primitives are displayed in the
environments where
right panel of Figure 2(a)
along with the corredifferent locomotion
sponding weights to reconstruct the original
modes are requested.
data. Finally, the weights
for the different walking
cadences were simplified
by fitting their evolution using second-order polynomials.
Neural-Level Primitives
In this case, the objective was to find the minimum set of
components to reconstruct all possible muscle stimulations.
Hereafter, the selected musculoskeletal model is described.
Next, we explain how the muscle stimulations used to activate
the model were obtained from the data in Table 1. Finally, the
method to identify the target set of artificial motor primitives
is presented.
The musculoskeletal model was implemented based on
the one developed by [14], comprising seven virtual muscletendon units per leg. Each unit captures the following leg
muscle groups (Figure 3):
● hip flexor (HFL) muscle: monoarticular hip flexor
● gluteus (GLU) muscle: monoarticular hip extensor
● hamstring (HAM) muscle: biarticular knee flexor and hip
extensor
● vastus (VAS) muscle: monoarticular knee extensor
● gastrocnemius (GAS) muscle: biarticular ankle plantar flexor and knee flexor
● tibialis anterior (TA) muscle: monoarticular ankle dorsiflexor

soleus (SOL) muscle: monoarticular ankle plantar flexor.
These muscle-tendon units are modeled as Hill-type muscles
[Figure 3(a)]: the force Fm being generated by a muscle results
from the interaction between a series elastic element (SE), a
parallel element (PE), a buffer elasticity (BE) preventing the
muscle from collapsing, and the active contractile element
(CE) [14]. The force generated by the CE depends on l CE and
its first derivative v CE, i.e., the muscle length and contraction
velocity, respectively, capturing the force-length and force-velocity relationships of a biological muscle.
This model thus introduces a local force feedback in the
muscle-tendon unit based on muscle length and velocity. This
feedback, together with the serial and parallel elasticities, plays
a significant role to endow the muscle unit with an intrinsically
nonzero impedance and enhance its stability. Afterward, these
muscle forces are converted into muscle torques through geometrical relationships: x m = rm $ Fm, where rm corresponds to
the lever arm of the muscle attachment point [14]. Finally, the
muscle model computes the joint torques as a function of the
torque provided by each muscle:

●

= x HAM + x GLU - x HFL + x lH
= x VAS - x GAS - x HAM + x lK
x ANKLE = x SOL + x GAS - x TA + x lA,
x HIP

x KNEE

(2)

where x lH , x lK , and x lA are torques preventing the hip, knee,
and ankle, respectively, from reaching their physical limits
[14]. Numerical parameters of the muscle units and of the
skeleton geometry were taken from the ones in [14], pending
some adaptations to the subjects, anthropometry (see the
"Experimental Validation" section).
Neural primitives reconstruct muscle stimulations. Therefore, these target muscle stimulations have first to be computed from the available kinematic and dynamic data (Table
1) before computing the corresponding primitives. For this
purpose, an inverse model of the introduced musculoskeletal
model was created and scaled depending on the anthropometry reported for each data set. The outcome of this stimulation computation is displayed in the left panel of Figure 2(b).
To get a reduced set of primitives that could reconstruct all
these muscle stimulations, NNMF was performed. Because NNMF may converge to
local minima, the process was repeated 100
i
Iopt
times, and the one with the lowest residual
error was kept as the solution.
ICE
ISlack
As input to the NNMF process, there
PE
were seven muscle stimulations for seven
GLU
HFL
different conditions: the five different walkSE
HAM
VAS
CE
ing cadences and ascending and descending stairs. In coherence with the first
BE
GAS
decomposition, the first primitives were
SOL
IMTU
kept up to accounting for at least 95% of the
TA
variance and less than 4% of reconstruction
(a)
(b)
error. This led to six components, which
also accounted for around 98% of the variFigure 3. (a) The seven main muscle groups actuating one leg. (b) The Hill-type
ance [see the right panel of Figure 2(b)].
muscle. (Figure adapted from [14].)

IEEE ROBOTICS & AUTOMATION MAGAZINE

march 2016

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - March 2016