Estimating ankle muscle parameters
Developing a tendon dynamics included neuromechanical muscle model and
expanding the measurement protocols to improve ankle muscle parameter
estimation accuracy
By
Koen van de Poll
Student ID: 1369563
Date: 21-12-2015
Supervisors:
Prof. dr. F.C.T. van der Helm, supervisor TU Delft
Dr. Ir. J.H. de Groot, supervisor LUMC Leiden
Dr. Ir. E. de Vlugt, former supervisor
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Acknowledgements
I would like to thank both Jurriaan de Groot and Erwin de Vlugt for their support, supervision and
discussions during all phases of this research. I want to thank Frans van der Helm as well, for his
insights, providing some important last minute changes. My thanks as well to my friends and family
who both supported me morally and helped me by participating as test subjects in this study.
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Abstract
Background: Upper motor neuron diseases (UMND) are characterized by increased joint resistance
which origin can be neural, e.g. improper neural activation, non-neural, e.g. altered visco-elastic
properties or a combination. Current clinical methods such as the Ashworth scale do not allow for
unbiased quantification of the underlying tissue and neural (reflexive) properties. By means of
Instrumented identification techniques such as proposed by de Vlugt et al. promising results were
obtained in quantifying the muscle tissue characteristics. Yet essential active muscle characteristics
to guide treatment such as the optimal muscle length can currently not be estimated accurately. In
this study it is proposed to enable estimating the active muscle parameters by adding tendon
dynamics to the identification model and expanding the protocols used for patient.
Methods: Current neuromechanical ankle muscle model as developed by de Vlugt et al., is expanded
to include tendon dynamics and developed into a muscle-tendon ankle model. Optimal model input
regarding ankle position profile and muscle neural activity are determined through model
simulations on both muscle model and muscle-tendon model. Fifteen passive and active muscle
neuromechanical parameters are estimated. Modelling results are assessed on variance accounted
for (VAF), standard error of mean (SEM) and cross-covariance between passive and active muscle
parameters. From simulations, best performing ankle position profiles and neural activation (EMG)
profiles are selected and combined into new protocols and are tested during subject experiments.
Fourteen healthy subjects are measured over full ankle range of motion (ROM) with the Achilles
robotic manipulator, using the established passive ramp and hold protocol and the newly proposed
active protocols. Newly developed measurement protocols are validated for both the old
neuromechanical ankle muscle model and the newly developed neuromechanical ankle muscletendon model.
Results: During model simulations, highest VAF- and lowest SEM-values are found for the partially
passive, partially active EMG-task. Ankle position profiles rich in velocities and accelerations resulted
in lower SEM-values. Highest VAF and lowest SEM are found for the multiple holds profile and the
multi sine with holds profile.
Muscle model parameter estimation resulted in VAF-values for both multiple holds and multi sine
with holds measurement protocols of a 95% mean. SEM-values are in the range of 1-10 times the
SEM-values found during model simulations. Model validation resulted in mean VAF-value of 90.4%.
Muscle-tendon parameter estimation resulted in a mean VAF-value of 90% for the active multiple
holds protocol and 93% for the active multi sine with holds protocol. SEM-values low, with mean
values of 1 times the model simulation SEM-values. Adding the tendon strain factor or tendon toe
force to the optimization parameters increased the modelling fit with VAF-values of 94.3% and 91.7%
respectively.
Conclusion: The muscle model combined with the newly developed active multiple holds protocol
resulted in high VAF-values and similarly low SEM-values as compared to model simulation results.
Model validation resulted in high VAF-values indicating a good estimation of the modelled muscle
parameters. The newly developed muscle-tendon model performed equally good when including the
tendon strain factor as optimization parameter. Since both muscle and muscle-tendon models
perform equally but the muscle-tendon model is much slower in modelling the results and more
likely to land in a local minimum, the muscle model is preferred over the muscle-tendon model.
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Index
Abstract ....................................................................................................................................... 4
Introduction ................................................................................................................................. 8
Methods......................................................................................................................................11
Results ........................................................................................................................................28
Discussion ...................................................................................................................................50
Conclusion ...................................................................................................................................61
References ..................................................................................................................................62
Appendix A. Covariance matrix .....................................................................................................64
Appendix B. Restrictions tendon-model.........................................................................................68
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Introduction
Stroke, multiple scleroses (MS) or other motor neuron diseases (UMND) may cause people to suffer
from upper motor neuron lesions. UMND’s are characterized by increased joint stiffness; mechanical
resistance to movement [Wissel et al. 2010]. Changes in mechanical resistance to movement can be
attributed to changes in the muscles as depicted in the muscle model in Figure 1.
Figure 1. Schematic muscle model including the neural reflex, passive muscle characteristics and active muscle
characteristics. Changes in each of the included elements can be a cause for increased joint resistance. The first part is the
neural reflex, which is depicted as a variable gain resistor. The second part describes the passive muscle characteristics,
which is depicted as a resistor. The final part describes the active muscle characteristics and consists of the Hill type muscle
model. This Hill type model contains a parallel elastic component (PEC), a series elastic component (SEC) and a contractile
component (CC). [Lieber et al. 2004].
For current assessment of the severity of the condition, a physician performs a clinical test using the
Modified Ashworth Scale. During assessment, the physician will judge each muscle by moving the
limb between full flexion and extension. Depending on the amount of muscle tone felt by the
physician, the muscle is awarded a score between 0 and 4. The score 0 being no increase in muscle
tone and the score 4 having a muscle being very rigid.
Unfortunately it is difficult for the physician to distinguish between the different contributing
elements (i.e. the reflexive, passive and active muscle components). Further problems include the
fact that the results are subjective as they depend on the physician performing the test and the
limited reproducibility [Pandyan et al. 1999]. As different treatment methods are advisable for
different causes of the increased joint resistance and spasticity, it is of importance to find the exact
muscle characteristics.
One method to find the underlying muscle characteristics is through the use of biomechanical
measurements [de Vlugt et al. 2010]. Biomechanical measurements enable quantification of
characteristics of the muscle such as muscle slack length, stiffness, viscosity and neural activity. The
method of biomechanical measurements makes use of a robotic manipulator that can impose torque
or position perturbations on a person’s limb, for instance on the ankle. Measured data is entered in a
computerized muscle model, which holds all the relevant muscle parameters. By optimizing the
modelled torque with respect to the measured muscle torque, the model makes an estimation of the
subject’s muscle parameter characteristics.
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The robotic manipulator used in this study is the Achilles ankle manipulator located in the Leiden
University Medical Center. The neuromuscular optimization model that is developed in collaboration
between Leiden and Delft was first published in 2010 [de Vlugt et al. 2010]. The model includes the
mechanical and neural parameters of the joint system (Figure 1.) for description of joint resistance.
The computerized quantification method is already being used to estimate the ankle muscle
parameters in UMND inflicted patients [de Gooijer, 2013]. The first group of parameters, holding the
passive muscle characteristics, are currently possible to estimate with relatively small error values
(more on this in the methods section). The active muscle parameters still pose difficulties regarding
their estimation. During patient measurements, it was found that most patients show reflexes during
the faster movements. Due to reflexes, the active parameters where possible to estimate, even
though with low accuracy. However, when patients or healthy subjects are asked to actively exert
force with their ankles, the model is incapable of accurately estimating the active parameters. So
why is it that reflexive behaviour results in sufficient information to estimate the active parameters,
but voluntary activation results in worse active parameter estimations?
The inability of previous studies to accurately estimate the active neuromechanical parameters was
attributed to the lack of tendon dynamics [de Vlugt et al. 2010], [de Gooier, 2013], [M. de Jong,
2015]. One of the assumptions of the current model included an infinitely stiff tendon. For the
infinitely stiff tendon, all forces exerted by the muscle are fully transmitted to the limbs. For passive
movements, where the muscles are relaxed, the tendon tissue is much stiffer than the muscle
tissues. Because of the higher stiffness of the tendon, then assumption can be made that the tendon
is infinitely stiff for passive movements. For movements where the subject actively exerts forces, the
difference between muscle stiffness and tendon stiffness is less, thus for active movements the
assumption of an infinitely stiff tendon may no longer hold. Studies of Thelen [Thelen, 2003] show
that the tendon rather acts as a non-linear spring. The inclusion of the non-linear part of the tendon
force-length relation could explain the misfits in the modelling results of the current muscle model,
thus it is proposed that the neuromuscular model is expanded with the tendon mechanics.
The neuromuscular model has been expanded over the course of the last few years, giving a better
representation of the actual muscle. The expansions have led to the model becoming more complex
and containing more parameters that will need to be included in the optimization. The increased
number of parameters also increases the possibility of the model landing in a local minimum during
parameter optimization. In other words, the model did not have sufficient information to reliably
estimate all muscle parameter values. For the model to accurately estimate the neuromechanical
parameters, each parameter is required to have a clear influence on the measured torques. This is
called the richness of the measurement protocol. Previous studies [de Gooijer, 2013] have focussed
mainly on the passive neuromechanical parameters for which it was determined that a simple ramp
and hold position profile with a relaxation task was sufficient. A ramp and hold profile (Figure 2.)
does move the muscle through its full range of motion, but at a limited range of speeds. Figure 3.
shows the muscle length vs velocity mapping as the muscle moves through the ramp and hold. For
most of the muscle length, the corresponding velocities are limited to 0.3-0.4 L0/s. The velocities
from 0-0.3 L0/s are only hit for the beginning and end of the range of motion. The ideal position
profile fills as much muscle length-velocity relation as possible while still being measurable on a
limited time scale. Simulations are performed to find the protocol with the optimal amount of data,
such that all neuromechanical parameters can be accurately estimated.
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Ramp and hold protocol
ankle angle [rad]
1.5
1
0.5
0
0
2
4
6
8
10
time [s]
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14
16
18
Figure 2. The ramp and hold protocol. Full plantarflexion at ankle angle 0 rad, full dorsiflexion at ankle angle 1.5 rad.
Triceps surae muscle length vs velocity mapping
5
4
3
Velocity L0/s
2
1
0
-1
-2
-3
-4
-5
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Muscle length L/L0
1
1.1
1.2
Figure 3. Triceps surae muscle muscle length - velocity mapping as for the ramp and hold position profile.
Using the current measurement protocol of a ramp and hold movement profile with a relaxation
task, the passive muscle parameters can be accurately obtained. The active muscle parameters
however, can only roughly be obtained when reflexes arise during patient measurements. When
patients or healthy subjects are asked to actively exert forces, neither parameter types can be
obtained anymore [M. de Jong, 2015]. It is hypothesized in this study that the inability of the model
to estimate the muscle parameters can be attributed to the lack of tendon dynamics and insufficient
information in the measured data.
The goals of this study will be to accurately estimate both the passive and active neuromechanical
parameters. This is attempted by expanding the current neuromuscular model with the inclusion of
tendon mechanics. Simulations are then performed with both the old muscle model and the newly
developed muscle-tendon model to find the measurement protocol that yields rich information with
respect to the muscle mechanics. The results of the simulations will be assessed with three criteria,
the VAF, the SEM and a newly developed criterion. Ultimately, the developed protocols will be tested
in subject measurements to validate their effectiveness.
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Methods
The aforementioned order of model simulations, parameter estimations from the subject
measurements and model validation will be used through this study. The decision for measurement
protocols will be elaborated in the discussion section while the measurement protocols are already
applied for the subject measurements.
2.1 Instrumentation
Subjects are measured on the ankle using the Achilles® robotic manipulator. The Achilles (Figure 4.) is
a single axis manipulator developed by MOOG FCS Inc. (Nieuw Vennep, the Netherlands). Subjects
are seated on the chair (incorporating the Achilles) in such a manner that the to be measured leg is
aligned in the sagittal plane with the footplate of the Achilles. The foot is strapped to the footplate
using Velcro straps. There are three sizes of footplates, the best fit is selected for each subject,
ensuring that the ankle rotational axis is aligned with the rotational axis of the Achilles. The baseplate
of the footplate is to be placed under a 90 o angle as compared to the knee rotational axis. The
footplate is rotated around a horizontal axis using a servomotor (C40 actuator). Positive rotations are
defined as ankle dorsiflexion and negative rotations as ankle plantarflexion. Before measurements,
the hard limits of the Achilles are set. The ankle of the subject is moved in both dorsiflexion and
plantarflexion direction until the subject notes their limit of comfort. It is ensured that the found
limits are not transgressed by inserting pins around the rotational axis of the Achilles, blocking
movements outside the range of comfort. Secondarily, the Achilles has a safety program that moves
the footplate away from the hard limits if the Achilles either hits or comes close to the hard limits.
Finally, in case the subject feels it necessary to stop the measurement, they can do so by pressing a
safety button which they hold in their hands. Perturbations performed by the Achilles in this study
are all position perturbations.
Muscle activation of the tibialis anterior (TA), gastrocnemius lateralis (GL), gastrocnemius medialis
(GM) and soleus (SL) was measured by electromyography (EMG) using a TMSi recording system
(Twente Medical Systems international). EMG signals were sampled at 2048Hz. Potential difference
between the two measurement points on each muscle was taken after which the mean was
subtracted. The adjusted EMG signals are then filtered with a 3 rd order high pass Butterworth filter at
0.04Hz. Reaction torque and ankle angles were also sampled at 2048Hz. To avoid amplification of
noise due to differentiation, angle and force signals were low pass filtered at 20 Hz (Bessel filter).
Filtered EMG data, reaction torque and ankle angles were resampled with a sampling factor 8, giving
the sampled data a frequency of 256Hz. Angular velocity and acceleration were derived by single and
double differentiation of the sampled angle signal respectively.
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Figure 4. Achilles single axis ankle manipulator as developed by MOOG FCS Inc. (Nieuw Vennep, the Netherlands).
2.2 Neuromechanical model
A non-linear neuromechanical muscle model was used to estimate the ankle torque exerted by the
subject. The neuromechanical model optimizes the different parameters by minimizing the quadratic
difference (least squares non-linear) between the measured and the modelled torque. Input for the
neuromuscular model are the recorded EMG signals, the ankle angle and the derived ankle
movement velocity and acceleration (Figure 5). Two types of ankle muscle model are applied in this
study. First is the muscle model as developed by Erwin de Vlugt et al. [de Vlugt et al., 2010] and
expanded over the years [Gooijer et al., 2015]. This model was extended with the inclusion of tendon
dynamics.
Figure 5. Schematic representation of the parameter estimating process. The Achilles ankle manipulator exerts position
perturbations on the subjects ankle. The subject responds by exerting torque on the Achilles, muscle activity is recorded
through surface EMG. EMG data, position data and the derived ankle velocity and acceleration are fed to the muscle model,
which estimates the exerted torques. Measured and modelled torques are compared and minimized through a non-linear
least squares algorithm. Model parameters are assumed estimated for the minimum error.
2.3 Muscle model
The neuromechanical muscle model is as developed by de Vlugt et al [de Vlugt et al., 2010] with a
few alterations over the past years [Gooijer et al., 2015]. The model includes both a passive element
and an active (Hill-type) element for both dorsiflexion and plantarflexion muscle groups. For this
neuromechanical model the tendon is assumed infinitely stiff. The passive element represents the
resistance of the passive muscle tissues as a result of stretching the muscle beyond its slack length.
The active Hill-type element represents the active contraction of the muscle as a result of muscle
activation. Recorded ankle angles and EMG signals were used as inputs for the model. Forces exerted
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by the passive and active elements are determined for both muscle groups. Modelled torque is
determined by summed torque in dorsiflexion direction, torque in plantarflexion direction, torque
exerted by the weight of the foot and the inertial torque. Model parameters are estimated through
minimization of the quadratic difference (error function) of the modelled torques and recorded
torques. Parameter estimation and analysis were performed in Matlab (The Mathworks Inc., Natick
MA). 15 model parameters were estimated and are summarized in Table 1.
2.3.1 Modelling torque
Modelled ankle torque is described by:
𝑇𝑚𝑜𝑑 (𝑡) = 𝐼𝜃̈ (𝑡) + 𝑇𝑡𝑟𝑖 (𝑡) − 𝑇𝑡𝑖𝑏 (𝑡) + 𝑇𝑔𝑟𝑎𝑣 (𝜃)
(1.)
Where t is the independent time variable in [s], T mod the modelled ankle reaction torque in [Nm], I
the inertia of the foot plus footplate in [kg·m2], 𝜃̈(𝑡) the ankle angular acceleration in [rad/s2], Ttri the
torque generated by the plantarflexion muscles (GM, GL and SL) or triceps surae in [Nm], T ti b the
torque generated bt the dorsiflexion muscle (TA) in [Nm] and T gra v the torque due to gravity on the
foot and footplate [Nm].
Triceps and tibialis torques are obtained by adding the passive (elastic) and active (neural) muscle
forces, multiplied with their respective moment arms:
𝑇𝑡𝑟𝑖 (𝑡) = (𝐹𝑒𝑙𝑎𝑠,𝑡𝑟𝑖 + 𝐹𝑎𝑐𝑡 ,𝑡𝑟𝑖 ) ∙ 𝑟𝑎𝑐ℎ𝑖𝑙
(2.)
𝑇𝑡𝑖𝑏 (𝑡) = ( 𝐹𝑒𝑙𝑎𝑠,𝑡𝑖𝑏 + 𝐹𝑎𝑐𝑡,𝑡𝑖𝑏 ) ∙ 𝑟𝑡𝑖𝑏𝑖𝑎
(3.)
Where Fel a s are the passive (elastic) forces of the muscle generated by the connective tissues, Fa ct are
the active (neural) forces generated through muscle activation. The moment arms r for respectively
the achilles tendon and the tibialis tendon are dependent on the ankle joint angle [Maganaris, 1998],
[Maganaris, 1999]:
𝑟𝑎𝑐ℎ𝑖𝑙 (𝜃 ) = 0.0480 − 0.0104 ∙ 𝜃
(4.)
𝑟𝑡𝑖𝑏𝑖𝑎 (𝜃 ) = 0.0393 + 0.0171 ∙ 𝜃
(5.)
Where 𝜃 is the ankle rotational angle in [rad].
Inertia of the foot plus flootplate was modelled as a point mass, being the combined mass of the foot
m and the mass of the footplate mpl a te in [kg], determined at a distance la (fixed at 0.15 m) from the
center of rotation:
𝐼 = 𝑚 ∙ 𝑙𝑎2
(6.)
Torque due to gravity as used in equation 1. is determined by:
𝑇𝑔𝑟𝑎𝑣 (𝜃 ) = 𝑚 ∙ 𝑔 ∙ 𝑙𝑎 ∙ cos(𝜃 − 𝜃𝑓𝑔𝑛𝑑 )
(7.)
Where 𝜃𝑓𝑔𝑛𝑑 represents the angle of the foot with respect to the horizontal (ground) at zero degrees
ankle angle [rad] and g the gravitational acceleration (g = 9.81 m/s 2).
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2.3.2 Passive muscle properties
Elastic force delivered by the connective tissues is determined through:
𝐹𝑒𝑙𝑎𝑠,0,𝑡𝑟𝑖 (𝑥𝑡𝑟𝑖 ) = 𝑒 𝑘𝑡𝑟𝑖 ∙(𝑥𝑡𝑟𝑖 −𝑥0,𝑡𝑟𝑖 )
(8.)
Where ktri is the stiffness coefficient [1/m], xtri is the muscle length in [m] and x 0,tri is the muscle slack
length in [m], which is the length of the muscle where passive muscle stiffness will start to increase.
Tibialis elastic force and sub-equations are built up similarly. The current muscle length is determined
by:
𝑥𝑡𝑟𝑖 = 𝑙𝑚,𝑡𝑟𝑖 − 1.67 ∙ 𝑟𝑎𝑐ℎ𝑖𝑙 (𝜃 )
𝑥𝑡𝑖𝑏 = 𝑙𝑚,𝑡𝑖𝑏 − 1.56 ∙ 𝑟𝑡𝑖𝑏𝑖𝑎 (𝜃)
(9.)
Where lm,tri is the initial muscle length for the ankle under an ankle angle flexion of 0 rad. Initial
muscle lengths [Maganaris, 1998], [Maganaris, 1999] are set at lm,tri = 0.118 [m] and lm,ti b = 0.136 [m].
The initial muscle length here is an alteration on the muscle model as developed by de Vlugt et al.
[de Vlugt et al., 2010]. The study of [M. de Jong, 2015], showed relations between the estimated
passive and active muscle parameters. This could be explained from the use of optimal muscle length
l0, which was used instead of lm in the previous studies. The optimal muscle length is an active
neuromechanical parameter, which is the muscle length at which maximum muscle force can be
generated. Using l0 in determining the muscle length causes l0 to indirectly be used in the passive
muscle characteristics. De-coupling the muscle length from the optimal muscle length was attempted
by replacing l0 for lm.
When keeping the muscle length constant, the elastic muscle force will decrease in time. The
decrease in delivered force is attributed to muscle relaxation and has to be taken into account for the
passive muscle force. The muscle relaxation dynamics are given by:
𝐹𝑒𝑙𝑎𝑠,𝑡𝑟𝑖 (𝑠) =
𝜏 𝑟𝑒𝑙 ∙𝑠+1
𝜏 𝑟𝑒𝑙 ∙𝑠+1+𝑘𝑟𝑒𝑙
∙ 𝐹𝑒𝑙𝑎𝑠 ,0,𝑡𝑟𝑖(𝑠)
(10.)
Where 𝜏𝑟𝑒𝑙 is the relaxation time constant in [s] and 𝑘𝑟𝑒𝑙 the relaxation factor [-]. In the 2010 study
of de Vlugt et al. a viscous dynamics part was used to describe the relaxation dynamics. The viscous
part was replaced by relaxation dynamics which was tested in the study of [M. de Jong, 2015] and
found to be an accurate description of the muscle viscosity.
Since connective tissue cannot deliver force in a negative direction (i.e. can only pull, not push),
negative elastic forces were nullified:
𝐹𝑒𝑙𝑎𝑠,𝑡𝑟𝑖 = {
𝐹𝑒𝑙𝑎𝑠,𝑡𝑟𝑖 ;
0;
𝐹𝑒𝑙𝑎𝑠,𝑡𝑟𝑖 ≥ 0
𝐹𝑒𝑙𝑎𝑠,𝑡𝑟𝑖 < 0
(11.)
2.3.3 Active muscle properties
Active muscle forces as used in equations 2. and 3. is are consequences of neural muscle activation.
EMG signals are recorded for GM, GL, SL and TA. Muscle activation is estimated from the four
recorded EMG signals:
𝐸𝑡𝑖𝑏 (𝑡) = 𝑔1 ∙ 𝑒1 (𝑡)
(12.)
𝐸𝑡𝑟𝑖 (𝑡 ) = 𝑔2 ∙ 𝑒2 (𝑡) + 𝑔3 ∙ 𝑒3 (𝑡 ) + 𝑔4 ∙ 𝑒4 (𝑡)
(13.)
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Where e1, e2, e3 and e4 are the filtered EMG signals respectively for the TA, GL, SL and GM in [V], E ti b
and Etri are the neural muscle activities for the tibialis and triceps surae in [N] and g 1, g2, g3 and g4 are
the weighting factors for their respective EMG signals in [N/V].
The neural activity is then passed through a linear second order filter describing the muscle
activation process. The same filter is used for both tibialis and triceps surae neural activities.
𝑎𝑡𝑟𝑖 =
𝜔 02
2
𝑠 +2∙𝛽∙𝜔 0 ∙𝑠+𝜔 02
∙ 𝐸𝑡𝑟𝑖 (𝑠)
(14.)
Where 𝑎𝑡𝑟𝑖 is the active state of the triceps surae, 𝜔0 = 2𝜋 ∙ 𝑓0 is the cut-off frequency of the
activation filter, and s is the Laplace operator denoting the first time derivative. The relative damping
𝛽 was set to one (critically damped).
From the active state a Hill-type muscle model was used to determine the muscle force using the
force-length and force-velocity characteristics of a muscle.
𝐹𝑎𝑐𝑡,𝑡𝑟𝑖 (𝑡) = 𝑓𝑙 (𝑥𝑡𝑟𝑖 ) ∙ 𝑓𝑣 (𝑥̇ 𝑡𝑟𝑖 ) ∙ 𝑎𝑡𝑟𝑖
(15.)
The correspong force-length and force-velocity characteristics are described by the following
exponentials [Thelen, 2003]:
𝑓𝑙 (𝑥𝑡𝑟𝑖 ) = 𝑒
2
( 𝑥𝑡𝑟𝑖−𝑙0,𝑡𝑟𝑖 )
𝑤𝑓𝑙𝑡𝑟𝑖
−
𝑥̇ 𝑡𝑟𝑖 +𝑣𝑚𝑎𝑥,𝑡𝑟𝑖
𝑥̇𝑡𝑟𝑖
−𝑣𝑚𝑎𝑥,𝑡𝑟𝑖
𝑚𝑣𝑠ℎ
𝑓𝑣 (𝑥̇ 𝑡𝑟𝑖 ) = {
1−
(16.)
;
(1+𝑚𝑣𝑠ℎ ∙𝑚𝑣𝑠ℎ𝑙 ) ∙( 𝑓𝑒𝑐𝑐 −1 ) ∙𝑥̇ 𝑡𝑟𝑖
𝑚𝑣𝑠ℎ ∙𝑚𝑣𝑠ℎ𝑙 ∙𝑣𝑚𝑎𝑥,𝑡𝑟𝑖 +𝑥̇ 𝑡𝑟𝑖
𝑥̇ 𝑡𝑟𝑖 < 0
(17.)
;
𝑥̇ 𝑡𝑟𝑖 ≥ 0
Where mvs h (=0.25), mvs hl (=0.5) and wfl are shaping factors. wfl being determined from
2
𝑤𝑓𝑙 = 𝑐𝑓 ∙ 𝑙0 where cf (=0.1)is another shaping factor. The maximum eccentric force is 𝑓𝑒𝑐𝑐 which is
1.5 times the isometric force, the isometric force was normalized to 1 since scaling of the force was
determined by the EMG-weighting factors. Ultimately there is the maximum shortening velocity
𝑣𝑚𝑎𝑥 [m/s] which is eight times the optimal muscle length per second.
All undefined parameters in the above mentioned equations are neuromechanical parameters
estimated by the model and are listed in table 1. With respect to the 2010 ankle model [de Vlugt et
al., 2010] some parameters changed. For the estimated mass, the mass of the footplate is already
determined and only the foot mass is estimated. Where in the previous model only the
characteristics of the triceps surae muscles were estimated, this model includes the tibialis anterior,
thus also the stiffness coefficient, muscle slack length and optimal muscle length for the tibialis
anterior. Muscle relaxation is currently calculated from muscle relaxation dynamics as opposed to
viscosity dynamics. Integrated EMG signals are replaced with filtered signals. Optimal muscle length
was added as an independent parameter. A relative damping coefficient was included for
determining the calcium activation dynamics.
15
Ankle model 2010 [de Vlugt et al, 2010]
#
parameter
1
m
Mass (ankle + footplate)
2
ktri
Stiffness coefficient
3
x0,tri
4
e1
5
e2
6
e3
7
e4
8
9
10
f
btri
F0
Muscle length shift
IEMG weighting factor for
the tibialis anterior
IEMG weighting factor for
the gastrocnemius
lateralis
IEMG weighting factor for
the soleus
IEMG weighting factor for
the gastrocnemius
medialis
Cut-off frequency
activation filter
Viscosity coefficient
Muscle force shift
units
kg
1/m
m
N/V
N/V
N/V
N/V
Hz
Ankle model 2015
#
parameter
1
m
Mass ankle
2
ktri
Stiffness coefficient triceps
surae
3
ktib
Stiffness coefficient tibialis
anterior
4
x0,tri
Muscle slack length triceps
surae
5
x0,tib
Muscle slack length tibialis
anterior
6
𝜏𝑟𝑒𝑙
Relaxation time constant
7
𝑘𝑟𝑒𝑙
Relaxation factor
8
g1
fEMG weighting factor for
the tibialis anterior
9
g2
fEMG weighting factor for
the gastrocnemius
lateralis
10
g3
fEMG weighting factor for
the soleus
11
g4
fEMG weighting factor for
the gastrocnemius
medialis
12
𝑙 0,𝑡𝑟𝑖
Optimal muscle length
triceps surae
13
𝑙 0,𝑡𝑖𝑏
Optimal muscle length
tibialis anterior
14
f0
Cut-off frequency
activation filter
15
β
Relative demping
coefficient activation filter
units
kg
1/m
1/m
m
m
s
N/V
N/V
N/V
N/V
m
m
Hz
N.s/m
N.s/m
N
Table 1. Neuromechanical muscle parameters estimated by optimization of the ankle muscle model. Left part encompassed
the model parameters as for the model developed by de Vlugt et al. [2010]. The right part encompasses the current ankle
model as per 2015. Above fat line are the passive model parameters, below fat line are the active model parameters.
2.4 Muscle-tendon model
During passive tasks, the tendon is relatively much stiffer than the passive muscle tissues. However,
during active torque exertion, this might no longer be the case. Taking into account the possible
effects of tendon dynamics on the active muscle neuromechanical parameters , this study proposes a
new tendon included ankle model.
Tendon dynamics included in the muscle-tendon model are as proposed by Thelen [Thelen, 2003]. In
contrast to the muscle model, the muscle-tendon model determines the exerted forces on the joint
from the tendon force-length characteristic. One difficulty however, is that it is now no longer
possible to directly determine the muscle length from the joint angle. In the muscle model, the
tendon is assumed infinitely stiff and only the muscle length depends on the ankle angle. In the
muscle-tendon model the ankle angle influences both muscle length and tendon length and it is not
directly able to determine both lengths. To enable determining both muscle and tendon length, the
assumption has to be made that the difference in muscle length for one sample time step is small
enough that the correct new tendon length can be determined. Using an iterative process, both
16
muscle length and tendon length can be determined, resulting in the exerted tendon force. The
equations for the muscle-tendon model will use [n] for the current parameter values and [n-1] for
the previous parameter values.
Tendon force is determined as proposed by Thelen [Thelen, 2003].
𝐹𝑡𝑜𝑒
∙ (𝑒
𝑙𝑡 −𝑙𝑡,0
𝑙𝑡,𝑡𝑜𝑒 −𝑙𝑡,0
𝑘𝑡𝑜𝑒 ∙
− 1) ;
𝐹𝑡𝑒𝑛𝑑𝑜𝑛 (𝑙𝑡 ) = {𝑒 𝑘𝑡𝑜𝑒 −1
𝑘𝑙𝑖𝑛 ∙ (𝑙𝑡 − 𝑙𝑡,𝑡𝑜𝑒 ) + 𝐹𝑡𝑜𝑒 ;
𝑙𝑡,𝑡𝑜𝑒 ≥ 𝑙𝑡 > 𝑙𝑡,0
(18.)
𝑙𝑡 > 𝑙𝑡,𝑡𝑜𝑒
Where lt is the tendon length [m], Ftoe is the tendon force [N] where the transition from non-linear to
linear behaviour occurs, ktoe is an exponential shape factor [-], lt,0 is the tendon slack length [m] which
is the length where the tendon starts exerting force, l t,toe is the tendon length [m] where transition
from non-linear to linear behaviour occurs and kl i n is a linear scale factor [-].
𝐹𝑡𝑜𝑒 = 𝐹̅𝑡𝑜𝑒 ∙ 𝐹𝑡 ,𝑚𝑎𝑥
(19.)
Where 𝐹̅𝑡𝑜𝑒 is the normalized tendon force [-] where the transition from non-linear to linear
behaviour occurs and 𝐹𝑡,𝑚𝑎𝑥 the maximum exertable tendon force [N] for the respective tendon.
𝑙𝑡,𝑡𝑜𝑒 = 𝑙𝑡,0 +
𝜀0 ∙𝐹̅𝑡𝑜𝑒∙𝑒𝑘𝑡𝑜𝑒 ∙𝑘𝑡𝑜𝑒
𝑘
𝑡𝑜𝑒
(
𝑒
∙ 1−𝐹̅𝑡𝑜𝑒 +𝐹̅𝑡𝑜𝑒∙𝑘𝑡𝑜𝑒)+𝐹̅𝑡𝑜𝑒 −1
(20.)
Where 𝜀0 is the tendon strain [-] due to maximum isometric force.
𝑘𝑙𝑖𝑛 =
𝐹𝑡,𝑚𝑎𝑥
𝜀0
̅
𝑘𝑡𝑜𝑒
𝐹 ∙𝑒
∙𝑘
∙ (1 − 𝐹̅𝑡𝑜𝑒 + 𝑡𝑜𝑒𝑒𝑘𝑡𝑜𝑒−1 𝑡𝑜𝑒)
(21.)
Equations 18. to 21. use five predefined parameter values (table 2.) for both the triceps surae and
the tibialis anterior as from Thelen [Thelen, 2003].
Tendon dynamics
parameter
𝑙 𝑡,0
𝜀0
𝐹𝑡,𝑚𝑎𝑥
𝐹̅𝑡𝑜𝑒
𝑘𝑡𝑜𝑒
Triceps surae
Tibialis anterior
0.415 [m]
0.04 [-]
8000 [N]
0.3 [-]
4 [-]
0.216 [m]
0.04 [-]
1400 [N]
0.3 [-]
4 [-]
Table 2. Predefined tendon dynamics parameter values.
During the iterative process the following steps are taken. Steps are similar for both triceps surae and
tibialis anterior, both using their respective muscle and tendon lengths. First the tendon length is
determined from the current total muscle-tendon length and the previous muscle length.
𝑙𝑡 [𝑛] = 𝑙𝑚𝑡 [𝑛] − 𝑥[𝑛 − 1]
(22.)
Where lmt is the combined muscle-tendon length. Muscle-tendon length is given by the initial muscletendon length combined with the ankle rotation length change.
𝑙𝑚𝑡,𝑡𝑟𝑖 = 𝑙𝑚𝑡,𝑖𝑛𝑖𝑡,𝑡𝑟𝑖 − 1.67 ∙ 𝑟𝑎𝑐ℎ𝑖𝑙
(23.)
𝑙𝑚𝑡,𝑡𝑖𝑏 = 𝑙𝑚𝑡,𝑖𝑛𝑖𝑡,𝑡𝑖𝑏 − 1.56 ∙ 𝑟𝑡𝑖𝑏𝑖𝑎
(24.)
17
Where the initial muscle-tendon lengths are given by adding the initial muscle lengths (equation 9.)
and the tendon slack lengths, which gives 𝑙𝑚𝑡,𝑖𝑛𝑖𝑡,𝑡𝑟𝑖 = 0.467 m and 𝑙𝑚𝑡 ,𝑖𝑛𝑖𝑡,𝑡𝑖𝑏 = 0.291 m. For the first
iterative step muscle length x is still unknown. Assuming the trial starts with the subject fully relaxed,
the neural active state  assumed zero and the muscle length and tendon length can be solved from
the combined muscle-tendon length and the tendon dynamics.
From the current tendon length, the current tendon force can be determined using equations 18. to
21. The current passive (elastic) force exerted by the muscles can be determined as described in
equations 10. and 11. Since the force pulling on the tendon is equal to the combined passive and
active muscle forces, the current active muscle force can found by:
𝐹𝑚,𝑎𝑐𝑡 [𝑛] = 𝐹𝑡𝑒𝑛𝑑𝑜𝑛 [𝑛] − 𝐹𝑒𝑙𝑎𝑠 [𝑛]
(25.)
The current neural active state and the muscle force-length relation can be determined as through
equations 14. and 16. respectively. Using equation 15. the resulting force from the force-velocity
relation can be determined.
𝑓𝑣 ( 𝑥̇ ) =
𝐹𝑚,𝑎𝑐𝑡 (𝑡)
(26.)
𝑓𝑙 (𝑥)∙𝑎
Using the inverse equations of equation 17. the current muscle velocity can be derived. An extra limit
is added here since the muscle velocity cannot exceed the maximum muscle velocity.
𝑥̇ (𝑡, 𝜃 ) = 𝑣𝑚𝑎𝑥 ;
𝑓𝑣 ( 𝑥̇ ) > 𝑓𝑒𝑐𝑐
(27.)
Now ultimately the new muscle length can be determined from the previous muscle length and the
length change caused by the muscle velocity over one sample time step.
𝑥[𝑛 ] = 𝑥[𝑛 − 1] + 𝑥̇ [𝑛] ∙ 𝑑𝑡
(28.)
Where in the next iterative step the tendon force and thus the respective torques can be
determined.
𝑇𝑡𝑟𝑖 (𝑡) = (𝐹𝑡𝑒𝑛𝑑𝑜𝑛 ,𝑡𝑟𝑖 ) ∙ 𝑟𝑎𝑐ℎ𝑖𝑙
(29.)
𝑇𝑡𝑖𝑏 (𝑡) = (𝐹𝑡𝑒𝑛𝑑𝑜𝑛,𝑡𝑖𝑏 ) ∙ 𝑟𝑡𝑖𝑏𝑖𝑎
(30.)
Concluding with the modelled torque as given in equation 1.
18
2.5 Model simulations
To arrive at the ultimate position profile and muscle activation task a number of basic position
profiles will be chosen based on the pragmatic chosen profiles in thesis de Jong. The basic position
profiles are to follow a number of criteria. The position profile must move through the entire subjects
range of motion in order to find sufficient information for the force-length relation. The position
profile must have at least one period in time where slow or no movements are performed. The
relaxation parameters of the muscle are best estimated if the position profile includes a period of
slow movements that allow detectable muscle relaxation. Muscle velocities may not exceed the
maximum muscle elongation velocity of eight times the optimal muscle length [Thelen, 2003]. Finally,
for the muscle model to identify differences between position and velocity dependent forces, high
variations in combinations of velocities at different positions are desirable (i.e. many independent
combinations of x, ẋ, ẍ and EMG). The basic signals are adapted to increase the area that is spanned
in the muscle length – velocity mapping.
In this study an EMG control task is preferred over a torque control task. Delivered torque is an
output from the muscle, whereas muscle activation is an input. Furthermore, for torque control tasks
there is the risk of coupling muscle input and output directly, rather than through the muscle model
[Zajac, 1989]. Direct input – output relations can cause the estimated model output to overlap with
the measured output (high VAF values), but at wrongly estimated model parameters. For controlling
the EMG input, it is assumed unlikely to directly link inputs and output through any other means than
the muscle model.
Each protocol will be tested through simulations for a set of realistic muscle parameters for a healthy
subject, which the optimization model will try to accurately reobtain. Both muscle model and the
newly developed muscle-tendon model are used as models for the simulations. Results of the
simulations are judged for a set of criteria (elaborated in section 3.3).
2.5.1 Profiles
The position profiles for the protocol will be determined from alterations on a number of basic
profiles. These profiles are chosen as basics for their use in previous studies and/or for fulfilling the
position profile criteria.
Position profiles:
 Sinusoidal signal (0.20 Hz). (Figure 6A)
The sinusoidal profile was chosen for simplicity yet the ability to continuously move through the
full range of motion with a large range of velocities. This profile will be used to check if a simple
profile is sufficient for identifying the muscle parameters.
 Multi hold with variable speeds. (Figure 6B)
The multiple holds profile was investigated by [M. de Jong, 2015] and is included for
comparability to the previous study. The signal moves through the full range of motion but is
limited in the velocities the profile holds by itself.
 Multi sine. (Figure 6C)
The multi sine profile, created from three added sine waves, is included for the ability to move
through the full range of motion while including a high variety of velocities and accelerations.
19
The multi sine profile distinguishes itself from the sinusoidal profile in the diverse combinations
of positions and velocities. Note for the multi sine that a ramp and hold was included at the start.
The relaxation parameters of the muscle are best estimated for slow movements. Since the multi
sine did not include slow movements for longer durations of time, the ramp and hold was added.

Multi sine holds. (Figure 6D)
A multi holds like profile built from multiplying sine wavelets, combining the best of both other
signals. The multi sine holds profile includes slower movements at the start for estimation of the
relaxation parameters with increasing frequency for fast movements later on.

Scaled chirp. (Figure 6E)
An adaption on the multi sine holds profile. The scaled chirp moves from low frequencies at the
start of the signal to faster movements later on. The entire chirp is scaled with the first half of a
cosine wavelet to have holds at various muscle lengths. Turning points in position are more
evenly distributed than the multi sine holds profile, but might be more predictable for subjects
during experiments.
20
Multi hold protocol
1.5
1
1
ankle angle [rad]
ankle angle [rad]
Sine 0.20Hz protocol
1.5
0.5
0
0.5
0
5
10
15
time [s]
20
25
0
30
0
5
10
25
30
35
Figure 6B. Multi hold protocol. Full plantarflexion at
ankle angle 0 rad, full dorsiflexion at ankle angle 1.5 rad.
Multi sine holds protocol
Multi sine protocol
1.5
1.5
1
1
ankle angle [rad]
ankle angle [rad]
20
time [s]
Figure 6A. Sine wavelet of 0.20Hz. Full plantarflexion at
ankle angle 0 rad, full dorsiflexion at ankle angle 1.5 rad.
0.5
0.5
0
15
0
5
10
15
20
time [s]
25
30
35
40
Figure 6C. Multi sine protocol. Full plantarflexion at
ankle angle 0 rad, full dorsiflexion at ankle angle 1.5 rad.
0
0
5
10
15
20
time [s]
25
30
35
40
Figure 6D. Multi sine holds protocol. Full plantarflexion
at ankle angle 0 rad, full dorsiflexion at ankle angle 1.5
rad.
Scaled chirp protocol
ankle angle [rad]
1.5
1
0.5
0
0
5
10
15
20
time [s]
25
30
35
40
Figure 6E. Scaled chirp protocol. Full plantarflexion at
ankle angle 0 rad, full dorsiflexion at ankle angle 1.5 rad.
21
Modifications on the basic position profiles
All five basic position profiles move through the full range of motion. However, to add more velocity
and acceleration information a chirp signal is added to the five basic position profiles. The added
chirp is varied between simulations in amplitude and in rate of frequency increase. An example of a
chirp signal to be added to a basic position profile is shown in Figure 7. The added chirp signals
consists of a combination of the following alterations:

Four levels of chirp amplitude: 0 rad, 0.1 rad, 0.2 rad and 0.3 rad.

23 rates of increasing chirp frequency. Scale factor range from 0.05 to 0.6 in steps of 0.025.
The frequency scale factor changes the rate in which the frequency of the chirp signal
increases. Higher frequency scale factors cause faster movements as the duration of the
chirp continues.
Chirp signal to be added to position signal
0.4
0.3
chirp amplitude [rad]
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0
5
10
15
time [s]
20
25
30
Figure 7. Chirp signal created with an amplitude of 0.2 rad and a frequency increasing scale factor of 0.3.
An example of adding the chirp signal to a basic position profile is depicted in Figure 8. The chirp
signal is added to the multiple holds position profile, increasing the combinations of velocities and
accelerations on different positions in the range of motion. I.e. the position profile in Figure 8. is an
addition of the position profiles in Figure 6B and Figure 7.
22
Multiple holds with added chirp
Full dorsiflexion 1
0.8
0.6
Part of range of motion [-]
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
Full plantarflexion-1
0
5
10
15
20
25
30
35
Time [s]
Figure 8. Position profile of the multiple holds profile with addition of a 0.2 rad amplitude chirp signal.
EMG activation profile
The EMG activation profile is the amount of muscle activation exerted by the subjects during
measurement trials. Current studies focus mostly on passive tasks, where subjects are asked to keep
muscle activation to a minimum. To be able to accurately estimate the active muscle
neuromechanical parameters, a rich profile of muscle activation is required. To test the influence of
the muscle activation, two different EMG activation profiles will be used in parameter optimization
simulations.


Part passive part activation. (Figure 9A)
No activation is given during the first part of the trial. For the measurements in this case the
subject is asked to remain as relaxed as possible. After the low activation part the muscles will be
active alternating in plantarflexion and dorsiflexion direction. The relaxation part is for the
hypothesis that the relaxation parameters are better estimated during passive tasks.
Purely activation. (Figure 9B)
For the case that the passive muscle parameters can be accurately obtained even with the lack of
a passive part (low activation), an EMG activation profile where high muscle activity starts
immediately should give more information to estimate the active muscle parameters.
An alternating EMG activation profile is chosen such that a high variety of muscle activation is
achieved during the full position profile. The alternating EMG activation profile combined with varied
combinations of muscle lengths and velocities ensures rich input to the muscle model for parameter
optimization and reduces the correlation between activation and position input signals.
A random EMG activation profile could also achieve a varied combination of muscle activations and
muscle lengths and velocities. However, for subjects trials in which the subjects use random
activation profiles it is more likely that subjects make movements exceeding the muscle reflex
threshold, thus increasing the chance on reflexes. The muscle model is not thoroughly tested for
reflexes and reflex occurrence should be avoided if possible.
23
x 10
-4
TA
6
Sol
Gm
Gl
5
EMG [uV]
4
3
2
1
0
0
5
10
15
20
time [s]
25
30
35
Figure 9A. EMG activation profile combining a passive part (0 – 11s) and an active part alternating between the four ankle
muscles (11s – end). Tibialis anterior (blue), solus (green), gastrocnemius medialis (red) and gastrocnemius lateralis (cyan).
x 10
-4
6
TA
Sol
Gm
Gl
5
EMG [uV]
4
3
2
1
0
0
5
10
15
time [s]
20
25
Figure 9B. Full active torque exertion. Tibialis anterior (blue), solus (green), gastrocnemius medialis (red) and gastrocnemius
lateralis (cyan).
2.5.2 Assessment criteria
The simulated protocols are evaluated and assessed by three criteria. The variance accounted for
(VAF), the standard error of mean (SEM) and an inter-parameter dependency estimator (IPDE).
Complete model validity is assessed by the variance accounted for (VAF), which is a measure of the
model goodness of fit. The VAF is described with:
𝑉𝐴𝐹 = (1 −
∑( 𝑇𝑐 𝑚𝑜𝑑 −𝑇𝑐 𝑚𝑒𝑎𝑠 ) 2
∑( 𝑇𝑐 𝑚𝑒𝑎𝑠 )2
) ∙ 100%
(31.)
Where Tcmea s is the torque measured by the robotic manipulator and Tc mod is the estimated ankle
torque from the model over the time span used for parameterization. The model parameters are
optimized by minimization of the mean squared model error:
𝑒 = 𝑇𝑐𝑚𝑒𝑎𝑠 − 𝑇𝑐𝑚𝑜𝑑
(32.)
24
The error vector is used to calculate the covariance matrix:
𝐶𝑜𝑣𝑃 =
1
𝑁𝑒𝑟𝑟
∙(
𝐽𝑇 ∙𝐽
𝑁𝑒𝑟𝑟
)
−1
∙𝐸
(33.)
Where Nerr is the number of time samples used for the estimation of the parameters and J is the
Jacobian matrix. The Jacobian is an N errxNp matrix, with N p = 15 estimated parameters, and contains
the first derivatives of the final error to each parameter. From the covariance matrix CovP, the SEM
for each parameter can be calculated by taking the square root of the auto -covariance (diagonal
terms of CovP).
𝑆𝐸𝑀 (𝑖 ) = √|𝐶𝑜𝑣𝑃(𝑖, 𝑗)|
|
(𝑖 = 𝑗)
(34.)
The SEM-values are compared between the tests, where a lower SEM-value is considered better. A
low SEM means that the corresponding parameter has a substantial contribution to the total
generated ankle torque.
Newly suggested is an inter-parameter dependency estimator (IPDE), which corresponds to the
accumulated inter-parameter relations. As elaborated in Appendix A. the muscle model is not fully
linear in the parameters; some parameters might show similar error variances. For the optimal
estimation of the parameters though, there should still be strived for parameters that contribute
maximally to the criterion function and are as independent as possible. The relations between
parameters is found within the covariance matrix in the cross-covariances. The proposed estimator
to show the influence of the cross-covariances on the ability of the muscle model to accurately
estimate the muscle parameters is determined by
2
𝐼𝑃𝐷𝐸 = √∑ ∑(𝑥𝐶𝑜𝑣𝑖𝑗 )
|
(𝑗 > 𝑖 )
(35.)
or in other words the RMS of the cross-covariances. xCovi j is the cross-covariance between
parameters i and j. The condition of (j>i) gives that only the off-diagonal upper triangular part of the
covariance matrix is included. The RMS of the cross-covariances was chosen such that larger crosscovariances have more impact on the total IPDE value.
2.5.3 Influence of noise
Subject measurements suffer from noise on the measured signals, to resemble this in the simulations
5% noise was added to the simulated signals. The simulated signals being the four EMG signals of the
tibialis anterior, soleus, gastrocnemius medialis and gastrocnemius lateralis, and the simulated
torque level. Noise on the EMG signals was added to represent ambient noise from other devices and
from possible transducer noise in the body that was not removed by the grounding signal. Noise on
the simulated torque level represents measurement noise on the torque measured by the Achilles.
The noise level was 5% [Ferrier et al., 2013] of the mean of the respectively measured signals. Both
noise on the EMG signals and on the simulated torque was added after the simulation step to
represent a situation similar to real measurements. I.e. noise was added after the simulation step but
before the parameter optimization step.
25
The signal noise gives a closer representation of actual measurements and provides the possibility to
analyse its effect on the parameter estimation. To check the effect of input signal noise on the three
assessment criteria (VAF, SEM and IPDE) a series of consecutive simulations have been run. The five
basic position profiles are checked for the effects of adding noise (Figures 6A – 6E). The added chirp
to the position signals had an amplitude of 0.1rad and varied in chirp frequency scale factor from 0.3
to
0.6.
2.5.4 Patient simulation
To check the protocols for possible dependencies on the chosen simulated neuromechanical
parameter values, simulations have been performed using “patient”-like parameter values. Mass of
the foot has been reduced, muscle stiffness’s are increased and muscle slack length and optimal
muscle lengths are reduced to give a better representation of muscles from patients struck by
UMND’s.
2.6 Subjects
The study measured on 14 healthy subjects (10 male and 4 female) on the age range of 20-30.
Subjects had no history of neurological disorders or lower extremity injuries. All subjects signed
informed consent prior to the measurements. Measurements are performed on the right ankle.
Visual feedback is given for the EMG-task. Subjects have had the opportunity to rest if so desired.
2.7 validation method
The developed muscle-tendon model and the newly developed protocols will be validated in two
manners. First the measured data of the subject measurements will be applied through both the
muscle model and the muscle-tendon model. Both models estimate the muscle/tendon parameters
for each subject individually. The results on the parameter estimation can then be checked for
correctness using the three aforementioned criteria, being the VAF, SEM and parameter covariance.
Secondly, the resulting parameter values are validated by reproduction of the measured torque time
traces.
1. Variance Accounted For (VAF). Simulations on the measurement protocols have shown that
for VAF-values lower than 99.85% one or more of the estimated parameters differed more
than 10% of the actual parameter value. Thus ideally found VAF-values are 99.85% or higher.
2. Simulation normalized Standard Error of Mean (snSEM). The resulting SEM-values will be
compared to the mean SEM-values of the simulations, since it was found that the muscle
parameters were correctly estimated for those SEM and levels under realistic conditions. For
snSEM-values below 1, more emphasis has been put on that particular parameter than was
found during model simulations. For snSEM-values above 1, less emphasis has been put on
that particular parameter than was found during model simulations. A lower snSEM-value
does not necessarily correspond to a better estimated parameter. For lower snSEM-values
the correspond parameter has had more influence on the modelled torque level.
3. Parameter cross-covariance. The passive and active components of the muscle describe
different contributions to the final torque levels. It is expected that the influence of passive
parameters on active parameters and vice versa is kept to a minimum.
26
4. Model validation step. The resulting parameter values will be validated by reproduction of
the measured torque time traces. Estimated parameter values from one measurement set
will be used to predict the torque for two other measurement sets for validations using the
VAF. Next checked is the ability of the parameters estimated for the active multiple holds
protocol to model the torque levels on the passive ramp and hold protocol. In the past
problems were found here for active tasks since the force delivered was expected to be
around the non-linear toe-region. If the new active protocols it are able to the estimate the
active parameters, the torque on the ramp and hold should be predictable as well.
Structure of the models used for the parameter optimization step is clarified in Figure 10. For the
muscle model [1.0] it is expected that the model will be unable to accurately estimate the muscle
parameters, since this was a result of previous studies. It is hypothesized that the newly developed
muscle-tendon model [2.0] combined with the newly developed measurement protocols will be able
to accurately estimate all muscle parameters, resulting in high VAF-values and low SEM-values crosscovariances. If however, the muscle-tendon model [2.0] is not able to give more accurate parameter
estimations, two more possibilities will be checked.
Figure 10. Complexity of the muscle models used. Chart arranges the muscle models used for the parameter optimization
where each step down defines a specific alteration in the model.
2.7.1 Varying the tendon parameters
When the muscle-tendon model appears able estimate the muscle parameters without showing
signs of the model being over defined, it is possible to include on or more of the tendon parameters
in the model optimization. The muscle-tendon model includes five model parameters that can be
either set or included in the optimization (Table 2.). Investigating initial results it was found that the
tendon strain factor (𝜀0) henceforth named dlt and the tendon toe force (𝐹̅𝑡𝑜𝑒 ) henceforth named pt, show
most promise in improving the modelling results. The tendon strain factor (dlt) influences the tendon
compliancy. The tendon toe force (pt) determines the force level where the tendon transitions from a non linear force-length relation to a linear force-length relation.
27
Results
3.1 Simulation results
3.1.1 EMG profiles
Comparing the simulations with regard to the part passive part active EMG profile vs the purely
active EMG profile it was found that the VAF values did not differ noticeably. For each movement
profile, the VAF values remained roughly the same. SEM values were lower for the part passive part
active EMG profile for all simulations, where the amount of difference depended on the movement
profile simulated with. Similarly, IPDE values were lower for the part passive part active EMG profile
for all simulations performed.
3.1.2 Muscle model
VAF-values were similar for all simulation performed with the muscle model. All VAF-values were
found around 99.91%. SEM-values decrease when the chirp frequency scale factor increases for
constant chirp amplitude and basic position profile (Figure 11a). Most prominent decrease is in the
mass SEM. SEM-values of the relaxation parameters remain roughly steady for chirp frequency scale
factors of 0.25 and higher. Comparing the SEM-values for different levels of chirp amplitude at steady
chirp frequency scale factor and basic position profile (Figure 12a), a decrease in SEM-values were
found for higher chirp amplitudes for all muscle parameters except for the relaxation parameters,
which minimized at a chirp amplitude of 0.1 rad. Finally comparing the different basic position
profiles using a chirp amplitude of 0.2 rad and a chirp frequency scale factor of 0.5 (Figure 13).
Highest SEM-values are found for the sine basic profile. Lowest SEM-values are found in the three
basic profiles of the multi sine profile, the multi sine with holds profile and the scaled chirp profile,
where the difference lies in which parameters where found with lowest SEM-values.
The IPDE-values are found decreasing over an increasing chirp amplitude. For steady chirp amplitude
levels, the IPDE-values were found lowest at chirp frequency scale factors around 0.5 for all basic
position profiles (Figure 15a). Between the five different basic position profiles lowest IPDE-values
are found for the multiple holds profile and the multi sine with holds profile at 2.60 and 2.64
respectively. Other position profiles settled at IPDE-values of 2.70-2.80.
3.1.3 Muscle-tendon model
For successful simulations, VAF-values were found around 99.89%. The model was more likely to
arriving in a local minimum, thus incorrectly estimating the parameter values. Incorrect estimations
resulted in lower VAF-values and were found mostly for the high chirp amplitude of 0.3 rad. Low
VAF-values and bad estimations (more than 10% difference from simulated parameter values) were
found for almost all simulations using the multi sine position profile.
SEM-values showed a tendency of decreasing for an increasing frequency scale factor (Figure 11b),
although showing more variations when compared with the results of the muscle model. Especially
the SEM-value of the mass decreases for higher chirp frequencies. The relaxation parameters appear
more volatile. Looking at the effects of altering the chirp amplitude, the lowest SEM-values are found
for a chirp amplitude of 0.2 rad (Figure 12b). The results for chirp amplitudes of 0.3 rad were omitted
from the Figure as these were all bad parameter estimations with low VAF-values. Comparing the
basic position profiles, lowest SEM-values were found for the multiple holds profile, the multi sine
with holds profile and the scaled chirp profile (Figure 14). Figure 14 shows high SEM-values for the
28
relaxation parameters for the multi sine with holds profile and lower SEM-values for the scaled chirp
profile, yet the estimated parameter values differed more from the simulated parameter values for
the scaled chirp profile than the multi sine with holds profile.
Similar to the muscle model, the lowest IPDE values were found for frequency scale factors around
the 0.5 for all position profiles (Figure 15b), with exception of the multi sine profile, which had low
VAF-values. Likewise, the IPDE-values decreased for higher chirp amplitudes. Mean IPDE-values were
at similar levels (around 3.45) for the multiple holds profile, the multi sine with holds profile and the
scaled chirp profile. The IPDE-value for the sine basic position profile was found higher at 3.80.
SEM for the multiple holds protocol with chirp amplitude of 0.2 rad
2
1.8
m
k
1.6
ktib
1.4
x0tri
1.2
x0tib
1
taurel
SEM (mean normalized) [-]
a) Muscle model
tri
krel
0.8
0.6
0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
g1
g2
g3
0.7g4
l0
tri
l0tib
2.5
f0
bet
SEM (mean normalized) [-]
b) Muscle-tendon model
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
chirp frequency scale factor [-]
0.5
0.6
0.7
Figure 11. The influence on SEM values for increasing frequencies of the chirp added to the basic position profile of the
multiple holds protocol. A) Simulation using the muscle model, B) Simulation using the muscle-tendon model. SEM-values
are normalized over their mean respective SEM.
29
SEM for the multiple holds protocol using chirp frequency scale factor of 0.5
SEM (mean normalized) [-]
2.5
m
ktri
a) Muscle model
2
k
tib
x0tri
1.5
x0
1
k
0.5
0
0
0.1
0.2
0.3
rel
g1
g2
g3
g4
l0tri
l0
tib
f0
bet
2
SEM (mean normalized) [-]
tib
taurel
b) Muscle-tendon model
1.5
1
0.5
0
0
0.1
0.2
chirp amplitude [rad]
0.3
Figure 12. The influence on SEM values for increasing amplitudes of the chirp added to the basic position profile of the
multiple holds protocol. A) Simulation using the muscle model, B) Simulation using the muscle-tendon model. SEM-values
are normalized over their mean respective SEM.
30
SEM for the position profiles using chirp amplitude of 0.2 [rad] and chirp frequency scale factor of 0.5
2
m
ktri
k
tib
1.8
x0tri
x0
tib
taurel
1.6
k
normalized SEM [-]
rel
g1
g2
g3
g4
l0
1.4
tri
l0tib
f0
bet
1.2
1
0.8
sine
multiple holds
multi sine
potition profile
multi sine with holds
scaled chirp
Figure 13. Resulting SEM values for simulations on the five basic position profiles using an added chirp of 0.2 rad amplitude
and a chirp scale factor of 0.5. Simulation using the muscle model. SEM-values are normalized over their mean SEM.
SEM for the position profiles using chirp amplitude of 0.2 [rad] and chirp frequency scale factor of 0.5
3
m
ktri
k
tib
x0tri
2.5
x0
tib
taurel
k
rel
normalized SEM [-]
2
g1
g2
g3
g4
l0
tri
1.5
l0tib
f0
bet
1
0.5
0
sine
multiple holds
multi sine with holds
scaled chirp
potition profile
Figure 14. Resulting SEM values for simulations on the five basic position profiles using an added chirp of 0.2 rad amplitude
and a chirp scale factor of 0.5. Simulation using the muscle-tendon model. SEM-values are normalized over their mean SEM.
31
Inter-parameter dependency for multiple holds protocol with chirp amplitude of 0.2 rad
4.2
Muscle model
Muscle-tendon model
4
3.8
IPDE [-]
3.6
3.4
3.2
3
2.8
2.6
0
0.1
0.2
0.3
0.4
0.5
Chirp frequency scale factor [-]
0.6
0.7
Figure 15. The influence on IPDE values for increasing frequencies of the chirp added to the basic position profile of the
multiple holds protocol. A) Simulation using the muscle model, B) Simulation using the muscle-tendon model.
3.1.4 Influence of noise
Results of the noise simulations were similar for all five position profiles, examples are given for the
multiple holds position profile and the multi sine with holds position profile. Shown in Figure 16A the
addition of noise to the input signals has decreased the VAF from 100% to around 99.91%. The
random nature of the simulated noise affects the VAF at most with 0.006. On the scale of 100 the
0.006 effect can be neglected. The effects of the random nature of the noise on the SEM is shown in
Figure 16B. At most, the noise affects the SEM with 3.75%. Differences between SEM values lower
than 3.75% can thus be attributed to noise influence. Figure 16C and Figure 16D show the results of
the effects of noise on the IPDE, simulated with the multiple holds and the multi sine with holds
protocols respectively. Highest differences in the IPDE approach 0.02. When using the IPDE for
assessment, differences in IPDE lower than 0.02 can thus be attributed to noise influence.
3.1.5 Patient simulation
Simulation results were similar to those performed with healthy subjects neuromechanical
parameter values. High VAF values and lowest SEM and IPDE values were found for the multiple
holds, multi sine with holds and the scaled chirp protocols. The amplitude of the added chirp
remained optimal at 0.2 rad. The SEM values decreased for increasing frequency scale factors yet the
IPDE values were found lowest for frequency scale factors around 0.5. The difference in results
between the scaled chirp protocol and the two chosen protocols of multiple holds and multi sine
with holds was smaller (IPDE difference <0.2) than compared to the healthy simulations.
32
Noise influence on VAF
99.913
99.912
VAF [-]
99.911
99.91
99.909
99.908
99.907
0.6
0.575
0.55
0.525
0.5
0.475
0.45
0.425
0.4
0.375
0.35
0.3
0.325
99.906
frequency scale factor [-]
Figure 16A. The effects of the randomness of noise on the VAF. Determined through five successive simulations using the
th
th
multiple holds protocol. Red lines are the means, blue boxes the ranges from 25 to 75 percentile, the black ranges are
the outermost values minus outliers and the red plusses are outliers.
x 10
Noise influence on SEM in ktri
-4
8.3
SEM [-]
8.2
8.1
8
7.9
0.6
0.575
0.55
0.525
0.5
0.475
0.45
0.425
0.4
0.375
0.35
0.325
0.3
7.8
frequency scale factor [-]
Figure 16B. The effects of the randomness of noise on the SEM of the stiffness parameter ktri. Determined through five
th
th
successive simulations using the multiple holds protocol. Red lines are the means, blue boxes the ranges from 25 to 75
percentile, the black ranges are the outermost values minus outliers and the red plusses are outliers.
33
Noise influence on inter-parameter dependency
2.74
2.72
IPDE [-]
2.7
2.68
2.66
2.64
2.62
0.6
0.575
0.55
0.525
0.5
0.475
0.45
0.425
0.4
0.375
0.35
0.325
0.3
2.6
frequency scale factor [-]
Figure 16C. The effects of the randomness of noise on the IPDE. Determined through five successive simulations using the
th
th
multiple holds protocol. Red lines are the means, blue boxes the ranges from 25 to 75 percentile, the black ranges are
the outermost values minus outliers and the red plusses are outliers.
Noise influence on inter-parameter dependency
2.78
2.76
IPDE [-]
2.74
2.72
2.7
2.68
2.66
0.6
0.575
0.55
0.525
0.5
0.475
0.45
0.425
0.4
0.375
0.35
0.325
0.3
2.64
frequency scale factor [-]
Figure 16D. The effects of the randomness of noise on the IPDE. Determined through five successive simulations using the
th
th
multi sine with holds protocol. Red lines are the means, blue boxes the ranges from 25 to 75 percentile, the black ranges
are the outermost values minus outliers and the red plusses are outliers.
34
3.2 Subject experiments results
During the experiments, 14 subjects were measured using three measurement protocols. The passive
ramp and hold protocol, the active multiple holds protocol and the active multi sine with holds
protocol. Not every subject was able to complete the active multi sine with holds protocol. Table 3
lists the number of subjects that completed each protocol.
Table 3. Number of subjects that completed the measurement protocols.
Protocol
Number of subjects that
completed the protocol
14
14
7
Passive, ramp and hold [Protocol 1]
Active, multiple holds [Protocol 2]
Active, multi sine with holds [Protocol 3]
3.2.1 Muscle model
Starting with the results for parameter estimations using the muscle model [1.0]. The resulting VAFvalues for the three measurement protocols are shown in Figure 17. The muscle model combined
with the ramp and hold protocol are what is currently used for patient trials and will be useful for
comparison purposes. The multiple holds protocol and the multi sine with holds protocol are
investigated to compare later with the results of the muscle-tendon model [2.0].
VAF [%]
100
VAF, muscle model
100
b
a
95
95
90
90
85
85
80
80
75
Ramp and hold
75
Multiple holds
Multi sine with holds
Figure 17. Resulting VAF-values for using the muscle model [1.0] in estimating the muscle parameters. Depicted VAF-value
boxplots are respectively for the passive ramp and hold protocol (panel a.) and the active multiple holds protocol and multi
th
sine with holds protocol (panel b.). The red line in the box depicts the mean VAF-value. The blue box depicts the 25 th
percentiles and the black span the 75 -percentiles. Red crosses depict the outliers.
Variance accounted for
The passive ramp and hold protocol found similar results as found during studies in the ROBIN
project [de Gooijer, 2013]. VAF-values were consistently found around 99.9%. The active multiple
holds protocol yielded lower VAF-values of around 95% and are found to be at least 90%. The active
multi sine with holds protocol yielded VAF-values with a mean of 93%, with the 75 th percentile going
down to 85%.
35
EMG [uV]
x 10
4
Angle [rad]
Sol
Gm
Gl
5
10
15
20
25
30
5
10
15
20
25
30
25
30
1
0
-1
0
Torque [Nm]
TA
2
0
0
Torque [Nm]
Ramp and hold, muscle model
-5
20
10
0
Measured
-10
0
5
20
10
15
Tissue
Neural
Model
20
Inertial
Gravitational
10
0
-10
0
5
10
15
Time [s]
20
25
30
Figure 18. Modelling results [1.0] on the passive ramp and hold protocol. First panel shows the EMG levels for the four
measured muscles. Second panel shows the position profile for the measured protocol. Third panel shows both the torque
measured by the Achilles setup as the torque levels modelled by the muscle model. Fourth panel shows the contributions of
the individual muscle components to the modelled torque. VAF for this Figure was found at 99.95%
Modelled torques
Examples of results for the modelled torques by the muscle model [1.0] are shown in Figures 18, 19
and 20. The top plot of the Figures show the level of muscle activation by the subject during the
measurement. The second plot shows the position profile for the measured protocol. The third plot
depicts both the torque measured by the Achilles during the trial and the torque modelled by the
model. The fourth plot separates the modelled torque level and gives the respective contribution of
the torque delivered by the elastic tissues, the torque exerted through the active neural components,
the inertial torques and the torque exerted by gravitational effects.
36
EMG [uV]
4
x 10
TA
Angle [rad]
Gm
Gl
5
10
15
20
25
5
10
15
20
25
20
25
1
0
-1
0
Torque [Nm]
Sol
2
0
0
Torque [Nm]
Multiple holds, muscle model
-4
100
50
0
-50
-100
Measured
0
5
Model
10
15
100
50
0
-50
-100
Tissue
0
5
Neural
Inertial
10
15
Gravitational
20
25
Time [s]
Figure 19. Modelling results [1.0] on the active multiple holds protocol. First panel shows the EMG levels for the four
measured muscles. Second panel shows the position profile for the measured protocol. Third panel shows both the torque
measured by the Achilles setup as the torque levels modelled by the muscle model. Fourth panel shows the contributions of
the individual muscle components to the modelled torque. VAF for this Figure was found at 92.4%
The modelled torque (Figure 19. third panel) appears to follow the measured torque level quite
good. The VAF-value of this specific measurement was 92.4% but was characteristic for most
measurements. It can be seen that the modelled torque does not reach the measured torque’s peak
levels at certain moments. Active (neural) torque levels, as shown in the fourth panel of Figure 19,
are significantly higher than the passive (tissue) torque levels. For some subjects, active torque were
found close to 80 Nm where passive torque levels were found up to 10 Nm. Inertial effects are found
very low. Similar to the multiple holds protocol, the multi sine with holds protocol has parts where
the modelled torque does not reach the peak levels of the measured torques (Figure 20).
37
EMG [uV]
4
x 10
TA
Angle [rad]
Gm
Gl
5
10
15
20
25
30
35
40
5
10
15
20
25
30
35
40
30
35
40
1
0
-1
0
Torque [Nm]
Sol
2
0
0
Torque [Nm]
Multi sine with holds, muscle model
-4
100
50
0
-50
-100
Measured
0
5
10
15
Model
20
25
100
50
0
-50
-100
Tissue
0
5
10
Neural
15
Inertial
20
Time [s]
25
Gravitational
30
35
40
Figure 20. Modelling results [1.0] on the active multi sine with holds protocol. First panel shows the EMG levels for the four
measured muscles. Second panel shows the position profile for the measured protocol. Third panel shows both the torque
measured by the Achilles setup as the torque levels modelled by the muscle model. Fou rth panel shows the contributions of
the individual muscle components to the modelled torque. VAF for this Figure was found at 91.2%
Simulation normalized Standard error of mean
The snSEM-values on the results are shown in Figures 21 and 22 for respectively the multiple holds
protocol and the multi sine with holds protocol. The snSEM-values for the multiple holds protocol
were found distributed with large variations (Figure 21). With respect to the resulting SEM-values of
the model simulations, it was found that slightly more emphasis was put on the activation filter
parameters (f0 and beta), all other parameters attained higher snSEM-values. Especially the snSEMvalues of the muscle activation scale factors were found high (g1, g2, g3 and g4). The results f or the
snSEM-values are similar to the multiple holds protocol as well. Lower snSEM-values are found for
the activation filter parameters (f0 and beta) and all other parameters attained higher snSEM-values.
38
SEM multiple holds, muscle model
150
snSEM [-]
100
50
bet
f0
l0_tib
l0_tri
g4
g3
g2
g1
k_rel
tau_rel
x0_tib
x0_tri
k_tib
k_tri
m
0
Parameters
Figure 21. Resulting snSEM-values for using the muscle model [1.0] in estimating the muscle parameters on the multiple
holds protocol. SEM-values are normalized with respect to the mean values found during model simulations. The red line in
th
th
the box depicts the mean snSEM-value. The blue box depicts the 25 -percentiles and the black span the 75 -percentiles.
Red crosses depict the outliers.
SEM multi sine with holds, muscle model
3000
2500
snSEM [-]
2000
1500
1000
500
bet
f0
l0_tib
l0_tri
g4
g3
g2
g1
k_rel
tau_rel
x0_tib
x0_tri
k_tib
k_tri
m
0
Parameters
Figure 22. Resulting snSEM-values for using the muscle model [1.0] in estimating the muscle parameters on the multiple
holds protocol. SEM-values are normalized with respect to the mean values found during model simulations. The red line in
th
th
the box depicts the mean snSEM-value. The blue box depicts the 25 -percentiles and the black span the 75 -percentiles.
Red crosses depict the outliers.
39
3.2.2 Muscle-tendon model
Variance accounted for
Compared to the muscle model [1.0] (Figure 17), the muscle-tendon model [2.0] achieves slightly
lower VAF-values for the passive ramp and hold protocol. Mean VAF-values are found around 99.6%,
with all results above 99%. The active multiple holds protocol has found lower VAF-values with a
mean value of 90% and ranging down to 76%. The active multi sine with holds protocol finds VAFvalues with a mean of 93% and ranging down to 87%.
VAF [%]
100
VAF, muscle-tendon model
100
b
a
95
95
90
90
85
85
80
80
75
Ramp and hold
75
Multiple holds
Multi sine with holds
Figure 23. Resulting VAF-values for using the muscle-tendon model [2.0] in estimating the muscle parameters. Depicted
VAF-value boxplots are respectively for the passive ramp and hold protocol (panel a.) and the active multiple holds protocol
and multi sine with holds protocol (panel b.). The red line in the box depicts the mean VAF-value. The blue box depicts the
th
th
25 -percentiles and the black span the 75 -percentiles. Red crosses depict the outliers.
Modelled torques
Figure 24 depicts the results for a passive ramp and hold trial. The third panel shows that the
modelled torque is able to follow the measured torque levels (corresponding subject VAF at 99.9%).
The fourth panel shows a slight contribution of the active (neural) components during the ramp in
dorsiflexion direction. Passive (tissue) components account for almost all modelled torques. The
results for an active multiple holds protocol in Figure 25 show that the modelled torque is mostly
able to follow the measured torque.. As was found for the results from the muscle model [1.0],
modelled torque from the muscle-tendon model [2.0] is unable to reach the peak torque levels for
certain parts of the measurement. During the passive phase (first 7 seconds) of the measurement
contribution to the modelled torque comes from the passive components. Only after the active
phase starts can the contributions of the active components be found. Similar are the findings for the
active multi sine with holds protocol.
40
EMG [uV]
x 10
4
Angle [rad]
Sol
Gm
Gl
5
10
15
20
25
30
5
10
15
20
25
30
Measured
Model
15
20
25
30
1
0
-1
0
Torque [Nm]
TA
2
0
0
Torque [Nm]
Ramp and hold, muscle-tendon model
-5
20
10
0
-10
-20
0
5
10
20
10
0
-10
-20
Tissue
0
5
Neural
10
Inertial
15
Time [s]
Gravitational
20
25
30
Figure 24. Modelling results [2.0] on the passive ramp and hold protocol. First panel shows the EMG levels for the four
measured muscles. Second panel shows the position profile for the measured protocol. Third panel shows both the torque
measured by the Achilles setup as the torque levels modelled by the muscle model. Fourth panel shows the contr ibutions of
the individual muscle components to the modelled torque. VAF for this Figure was found at 99.9%
41
EMG [uV]
4
x 10
TA
Angle [rad]
Gm
Gl
5
10
15
20
25
5
10
15
20
25
20
25
1
0
-1
0
Torque [Nm]
Sol
2
0
0
Torque [Nm]
Multiple holds, muscle-tendon model
-4
100
50
0
-50
-100
Measured
0
5
Model
10
15
100
50
0
-50
-100
Tissue
0
5
Neural
Inertial
10
15
Gravitational
20
25
Time [s]
Figure 25. Modelling results [2.0] on the active multiple holds protocol. First panel shows the EMG levels for the four
measured muscles. Second panel shows the position profile for the measured protocol. Third panel shows both the torque
measured by the Achilles setup as the torque levels modelled by the muscle model. Fourth panel shows the contributions of
the individual muscle components to the modelled torque. VAF for this Figure was found at 88.2%
Simulation normalized Standard error of mean
The snSEM-values for the results on the multiple holds protocol (Figure 26) were found with mean
values close with respect to the model simulation results for most parameters. Mostly the mass, the
relaxation stiffness (k_rel) and the optimal triceps muscle length (l0_tri) were estimated with mean
SEM-values higher than the simulation modelled resulting SEM-values. All snSEM values appear a lot
lower with respect to the muscle model.
The snSEM-values for the results on the multi sine with holds protocol (Figure 27) are found similar
to the multiple holds protocol, but lower. The mean SEM-values for most parameters are found at
similar levels compared to the mean SEM-values of the model simulations. Only slightly higher values
are found for the mass, the tibialis slack length (x0_tib), the relaxation stiffness (k_rel) and the
optimal triceps muscle length (l0_tri).
42
SEM multiple holds, muscle-tendon model
20
18
16
snSEM [-]
14
12
10
8
6
4
2
bet
f0
l0_tib
l0_tri
g4
g3
g2
g1
k_rel
tau_rel
x0_tib
x0_tri
k_tib
k_tri
m
0
Parameters
Figure 26. Resulting snSEM-values for using the muscle-tendon model [2.0] in estimating the muscle parameters on the
multiple holds protocol. SEM-values are normalized with respect to the mean values found during model simulations. The
th
th
red line in the box depicts the mean snSEM-value. The blue box depicts the 25 -percentiles and the black span the 75 percentiles. Red crosses depict the outliers.
SEM multi sine with holds, muscle-tendon model
20
18
16
snSEM [-]
14
12
10
8
6
4
2
bet
f0
l0_tib
l0_tri
g4
g3
g2
g1
k_rel
tau_rel
x0_tib
x0_tri
k_tib
k_tri
m
0
Parameters
Figure 27. Resulting snSEM-values for using the muscle-tendon model [2.0] in estimating the muscle parameters on the
multi sine with holds protocol. SEM-values are normalized with respect to the mean values found during model simulations.
th
The red line in the box depicts the mean snSEM-value. The blue box depicts the 25 -percentiles and the black span the
th
75 -percentiles. Red crosses depict the outliers.
43
Covariance matrix
The covariance matrix is shown in Figure 28 for the results of the parameter estimation given in
Figure 25. Covariances in the matrix are normalized to the corresponding auto-covariances.
Parameters used for determining the passive (elastic) muscle components are k_tri, k_tib, x0_tri,
x0_tib, tau_rel and k_rel. Parameters used for determining the active (neural) muscle components
are g1, g2, g3, g4, l0_tri, l0_tib, f0 and beta. The mass m is determined by the model, but contributes
to the inertial torque, rather than the active or passive torques. From Figure 28 it can be seen that
some high cross-covariances arise between the passive and active parameters (red box). Some higher
cross-covariances are found between the g1 tibialis activity scale factor and the mass. With similar
findings for the g4 gastrocnemius lateralis activity scale factor and both the triceps surae and tibialis
anterior passive parameters. Mainly the optimal muscle lengths of the tibialis anterior shows high
cross-covariances with the mass, the muscle slack lengths and muscle stiffness factors.
Figure 28. Covariance matrix for the multiple holds protocol, parameters estimated using the muscle -tendon model [2.0].
All covariances are normalized to the auto-covariance. Auto-covariance is shown on the diagonal, cross-covariance is shown
in the off-diagonal. The red box depicts the cross-covariance between the passive parameters and the active parameters.
44
3.2.3 Model validation
Resulting parameter values from the multiple holds protocol and the multi sine with holds protocol
are used to validate both the muscle model [1.0] and the muscle-tendon model [2.0]. The results
from the parameter optimization step are taken from the first measurement for each subject and
used to determine the modelled torque levels on the second and third measurement for each
subject. Modelled torque levels are then compared with the measured torque levels and used to
determine the VAF.
Figure 29 shows the resulting VAF-values for each measurement protocol, combined with the model
used to produce the torque level. The multiple holds protocol combined with the muscle model [1.0]
produced a mean VAF-value of 90.4% and included three outliers with a minimum VAF-value of
68.0%. The multi sine with holds protocol combined with the muscle model [1.0] yielded a lower
mean VAF-value of 83.1% with some results going down to 50%.
The multiple holds protocol combined with the muscle-tendon model [2.0] produced a mean VAFvalue of 82.6% with a number of low VAF-values going down to 59%. The multi sine with holds
protocol combined with the muscle-tendon model [2.0] gave similar results compared to the muscle
model [1.0] validation, with a mean VAF-value of 82.0% and the lowest VAF-value being 51%.
Model validation VAF
95
90
85
VAF [%]
80
75
70
65
60
55
50
Multiple holds
Multi sine with holds
Muscle model
Multiple holds
Multi sine with holds
Muscle-tendon model
Figure 29. Resulting VAF-values for the model validation step. Top label in the x-axis depicts the measurement protocol
used during subject trials. Bottom label in the x-axis depicts the muscle model used for obtaining the results and model
th
validation. The red line in the box depicts the mean VAF-value. The blue box depicts the 25 -percentiles and the black span
th
the 75 -percentiles. Red crosses depict the outliers.
45
Next checked is the ability of the parameters estimated for the active multiple holds protocol to
model the torque levels on the passive ramp and hold (Figure 30). The muscle model [1.0] has been
able to predict to measured torque levels with high VAF-values. Mean VAF-value was found at 94%.
The muscle-tendon model [2.0] was less able to accurately predict the ramp and hold torque levels,
having a mean value of 90% and a much higher spread. Results on the modelled torque levels
strongly varied per subject.
An example of the validation step can be seen in Figure 31. Torques found during the hold in plantar
flexion are found (absolutely) higher and torques found during the hold in dorsi flexion are found
lower compared to the measured torque. The modelled torque during the upward ramp shows a
close fit.
VAF model validation on Ramp and Hold
100
95
VAF [%]
90
85
80
75
70
65
Muscle model
Muscle-tendon model
Figure 30. Resulting VAF-values for the model validation step. Data from the active multiple holds protocol is used to
validate the passive ramp and hold protocol. The red line in the box depicts the mean VAF -value. The blue box depicts the
th
th
25 -percentiles and the black span the 75 -percentiles. Red crosses depict the outliers.
46
Muscle-tendon model validation
1.5
Angle [rad]
1
0.5
0
-0.5
-1
-1.5
0
5
10
15
20
25
30
20
Torque [Nm]
Measured
Model
10
0
-10
0
5
10
15
Time [s]
20
25
30
Figure 31. Validation results [1.0] on the passive ramp and hold protocol using the active multiple holds parameter results.
First panel shows the position profile for the measured protocol. Second panel shows both the torque measu red by the
Achilles setup as the torque levels modelled by the muscle model. VAF for this Figure was found at 94.3%
3.2.4 Setting tendon strain parameter as to be estimated parameter
Modelling results with the muscle-tendon model using the tendon strain parameter [2.1] resulted in
a mean VAF-value of 94.3% (Figure 32 left box). The snSEM-values are generally found higher when
compared to the results of the muscle-tendon model [2.0], with the exception of the muscle activity
scale factors, which are found very low. The low snSEM-values on the muscle activity scale factors
indicates a strong influence on the modelled torques. The mass snSEM is found highest at a mean
value of 5.9. Model validation resulted in a mean VAF-value of 88.5% with some of the lower values
ranging down to 70%.
3.2.5 Setting tendon toe force parameter as to be estimated parameter
Modelling results with the muscle-tendon model using the tendon toe force parameter [2.2] resulted
in a mean VAF-value of 91.7% (Figure 32 right box). The snSEM-values are found very similar
compared to the results of the muscle-tendon model [2.0], with again the highest snSEM-values on
the m, x0_tib, k_rel and l0_tri parameters. The snSEM-values on the muscle activity scale are again
found at low values and the mass snSEM is found highest. Model validation resulted in a mean VAFvalue of 85.4% with some of the lower values ranging down to 55%.
47
VAF muscle-tendon model with variable tendon parameter
100
VAF [%]
95
90
85
80
75
strain factor
toe force
Variable tendon parameter
Figure 32. Resulting VAF-values for using the muscle-tendon model with (left) variable tendon strain [2.1] and (right)
variable tendon toe force [2.2] in estimating the muscle parameters. The red line in the box depicts the mean VAF-value.
th
th
The blue box depicts the 25 -percentiles and the black span the 75 -percentiles. Red crosses depict the outliers.
SEM multiple holds, muscle-tendon model with variable dlt
20
18
16
snSEM [-]
14
12
10
8
6
4
2
dlt_tib
dlt_tri
bet
f0
l0_tib
l0_tri
g4
g3
g2
g1
k_rel
tau_rel
x0_tib
x0_tri
k_tib
k_tri
m
0
Parameters
Figure 33. Resulting snSEM-values for using the muscle-tendon model with variable dlt [2.1] in estimating the muscle
parameters on the multiple holds protocol. The red line in the box depicts the mean SEM-value. The blue box depicts the
th
th
25 -percentiles and the black span the 75 -percentiles. Red crosses depict the outliers.
48
SEM multiple holds, muscle-tendon model with variable pt
20
18
16
snSEM [-]
14
12
10
8
6
4
2
pt_tib
pt_tri
bet
f0
l0_tib
l0_tri
g4
g3
g2
g1
k_rel
tau_rel
x0_tib
x0_tri
k_tib
k_tri
m
0
Parameters
Figure 34. Resulting snSEM-values for using the muscle-tendon model with variable pt [2.2] in estimating the muscle
parameters on the multiple holds protocol. The red line in the box depicts the mean SEM-value. The blue box depicts the
th
th
25 -percentiles and the black span the 75 -percentiles. Red crosses depict the outliers.
Validation VAF, muscle-tendon model with variable tendon parameter
100
95
90
VAF [%]
85
80
75
70
65
60
55
50
strain factor
toe force
Variable tendon parameter
Figure 35. Resulting VAF-values for the model validation step. Model used is the muscle-tendon model with (left) variable
tendon strain [2.1] and (right) variable tendon toe force [2.2] on the active multiple holds protocol. The red line in the box
th
th
depicts the mean VAF-value. The blue box depicts the 25 -percentiles and the black span the 75 -percentiles. Red crosses
depict the outliers.
49
Discussion
The discussion will follow the structure of the results section. First the model simulations and the
decision of the measurement protocol is discussed, followed by the subject measurements and
model validation. For the subject measurements and model validation section, first the muscle model
and then the muscle-tendon model are discussed with regards to the VAF, snSEM, modelled torque,
parameter covariance and model validation. The muscle-tendon model will be further discussed for
the alterations of including the tendon strain parameter and the tendon toe force parameter to the
set of optimized parameters. Some disadvantageous modelling choices will be discussed as well.
4.1 Simulation
The goal of the model simulations was to determine the optimal measurement protocol, based on
the optimal position profile and the optimal EMG-task profile. Both muscle model and the newly
developed muscle-tendon model were applied during the model simulations. During the simulations
it was found that the simulation VAF should not decrease below 99.85%. For VAF values below
99.85% one or more of the estimated parameter values differed more than 10% from the simulated
parameter values. The muscle-tendon model gave bad results (below the 99.85% VAF threshold) for
high changes in muscle velocity, caused by a discrepancy in the iterative steps in the model. This
problem will be further elaborated in Appendix B.
It was found that higher chirp amplitudes and higher chirp frequencies of the added chirp signal
resulted in lower SEM-values on the estimated parameters. The EMG-task consisting of both a
passive and active part also resulted in higher VAF- and lower SEM-values. Based on the results for
both the muscle model and the muscle-tendon model, the chosen position profiles and the multiple
holds profile and the multi sine with holds profile, using a chirp amplitude of 0.2 rad and a chirp
frequency scale factor of 0.5.
4.1.1 EMG-task profile
Simulations using an EMG profile of a combined low activation part and an alternating activation part
and an EMG profile containing only alternating activation were compared. For all simulations, VAF
values did not appear to change. SEM values were lower for the part passive part active EMG profile.
IPDE values were lower for all simulations with the part passive part active EMG profile. Since the full
EMG activation profile performed worse in the fields of SEM and IPDE values, the chosen task will be
an EMG task including both a passive part and an EMG active part.
4.1.2 Muscle model
As the VAF-values did not differ between the different simulations, it will not be possible to draw
conclusions based on the VAF-values and focus will go to the SEM and IPDE-values. SEM-values were
found lower for higher chirp amplitudes, although the difference becomes lower for the highest
amplitudes. It should be noted that the relaxation parameters show increasing SEM-values after a
chirp amplitude of 0.1 rad. The increase in relaxation parameter SEM could be attributed to the
increasing speeds and accelerations in the position profiles, since relaxation depends on slower
movements. To avoid increasing the SEM on the relaxation parameters too much, chirp amplitude
should be chosen around 0.1 – 0.2 rad. As for the frequency scale factor. SEM-values decreased for
higher frequency scale factors, except for the relaxation parameters τ rel and krel , which remained
50
roughly the same for frequency scale factors of 0.2 and higher. Regarding the SEM-values, an as high
as possible frequency scale factor seems optimal. The scale factor of 0.6 is the highest possible for
these simulations, since higher frequencies would result in the muscle velocity approaching the
maximum muscle velocity. For the basic position profiles, lowest SEM-values were found in the multi
sine profile, the multi sine with holds profile and the scaled chirp profile.
The IPDE-values show similar characteristics as compared to the SEM-values. Higher chirp amplitudes
appear to result in lower IPDE-values, making it less likely that parameters will influence one another.
A chirp frequency scale factor of 0.5 appears to be optimal. The basic position profiles of the multiple
holds and the multi sine with holds profiles appear to give the lowest IPDE-values. As the lowest
SEM-values and IPDE-values were found in the multi sine with holds profile, this position profile with
an chirp amplitude of 0.2 rad and a chirp frequency scale factor of 0.5 appears optimal.
4.1.3 Muscle-tendon model
The multi sine position profile resulted in VAF-values below the 99.85% threshold for all simulations.
Same low VAF-values were found for the other position profiles when simulating with a chirp
amplitude of 0.3 rad. These simulations resulted in the estimated parameter values differing more
than 10% from the simulated parameter values and were omitted from the results. For the remaining
simulations, the SEM and IPDE-values will be examined.
The SEM-values show a tendency to decrease for higher frequency scale factors, although not as
apparent as for the muscle model. Most prominent decrease is in the mass parameter and least
effect is found in the relaxation parameters. A higher frequency scale factor appears optimal. A
higher chirp amplitude seems optimal, but since a chirp amplitude of 0.3 rad yielded bad estimations,
the amplitude of 0.2 rad appears best. Not apparent from Figure 14. the estimations for the
relaxation parameters were far off. The position profiles with lowest SEM-values were thus found in
the multiple holds profile and the multi sine with holds profile.
Like the muscle model, lowest IPDE-values are found for a chirp frequency scale factor around 0.5
and a chirp amplitude of 0.2 rad. Between the position profiles, the IPDE-values did not differ more
than 0.2 between the multiple holds profile, the multi sine with holds profile and the scaled chirp
profile, meaning that each of these three can be taken into consideration. When taking both SEMand IPDE-values into account, the multiple holds profile and the multi sine with holds profile appear
best, using a chirp frequency scale factor of 0.5 and a chirp amplitude of 0.2 rad.
4.1.4 Patient simulations
The chosen multiple holds and multi sine with holds protocols with the chosen chirp amplitude and
scale factor performed best for the patient neuromechanical parameter values as well. The effects of
muscle neuromechanical parameter values on the protocol optimality are assumed to be negligible.
4.1.5 Final protocols
Of the five basic position profile for the protocols, the multiple holds profile and the multi sine with
holds profile performed best for simulations with both the muscle model and the muscle-tendon
model. Muscle activation; active torque exertion, was found to give lower SEM and IPDE values for
the part passive activation task. The added chirp to the position profiles decreased the SEM and IPDE
values for the higher two chirp amplitudes (0.2 and 0.3 rad). The highest chirp amplitude gave
problems for the muscle-tendon model, since too high muscle velocities arose. SEM and IPDE values
were found lowest for chirp frequency factors of around 0.5. The suggested protocols are the
multiple holds and multi sine with holds position profiles, using an added chirp of amplitude 0.2 and
51
a chirp frequency factor of 0.5. The EMG activation task given to the subjects will be a part passive
part alternating activation task, were the first part will be passive and the second part alternating
activity between the plantarflexion and dorsiflexion muscles.
Due to the physical limits of the Achilles, some adaptions had to be made to the chosen movement
profiles. Achilles velocities of 1.5 rad/s and higher are likely to suffer from position overshoot [van
der Burg, 2013]. To remove Achilles velocities that result in overshoot, the movement profiles are
filtered with a one dimensional digital filter using discrete time filter coefficients.
Additionally to filtering the movement profiles, the amplitude of the added chirp signal to the
multiple holds profile had to be lowered from 0.2 rad to 0.1 rad. The limit on the Achilles velocity
also limits the range of achievable muscle velocities. Unfortunately, this can currently not be
avoided. The final movement profiles are shown in Figure 36 and Figure 37.
Multiple holds position profile adapted for Achilles
Ankle angle [rad]
1.5
1
0.5
0
0
5
10
15
Time [s]
20
25
30
Figure 36. Multiple holds position profile. Decreased chirp amplitude and filtered to lower Achilles movement velocities.
Note that due to filtering, the chirp amplitude decreases further as the chirp frequency increases.
52
Multi sine with holds position profile adapted for Achilles
Ankle angle [rad]
1.5
1
0.5
0
0
5
10
15
20
25
30
35
Time [s]
Figure 37. Multi sine with holds position profile. Filtered to lower Achilles movement velocities. Note that due to filtering,
the chirp amplitude decreases further as the chirp frequency increases.
Triceps surae muscle length vs velocity mapping
5
4
3
Velocity L0/s
2
1
0
-1
-2
-3
-4
-5
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Muscle length L/L0
1
1.1
1.2
Figure 38. Triceps surae muscle muscle length - velocity mapping as for the multi sine with holds position profile. Red dots
represent a muscle length with its corresponding velocity during simulation.
53
40
4.2 Subject measurements
The goal of this study was to accurately estimate both of the passive and active muscle parameters
by developing both a new muscle model which includes tendon dynamics and a new measurement
protocol which acquires richer data in the fields of position, velocity, acceleration and muscle
activation. Both the old muscle model from de Vlugt et al. [de Vlugt et al. 2010] and the newly
developed muscle-tendon model were used in combination with the newly developed measurement
protocols. From the results of parameter estimation with found VAF- and snSEM values and the
model validation step, can be concluded that the muscle model can estimate the measured torque
with reasonable accuracy. VAF-values were found high for comparison with the modelled torque and
the snSEM-values were low with the exception of the muscle activity scale factors. The multiple holds
protocol achieved higher VAF-values and lower snSEM-values than the multi sine with holds protocol.
VAF for the model validation step was high as well, indicating accurate estimations of the modelled
parameter values.
The muscle-tendon model with set tendon parameters does not improve the parameter estimations
with respect to the muscle model. While snSEM-values are found lower with respect to the musclemodel, VAF-values for the muscle-tendon model were found slightly lower for the multiple holds
protocols than the muscle model. The multi sine with holds protocol showed similar results between
the muscle model and the muscle-tendon model. Model validation also showed higher validation
VAF-values for the muscle model for both measurement protocols. Including tendon parameters in
the set of optimization parameters had positive effects on the modelling fits. For both the dlt and pt
parameters, the modelling VAF and validation VAF increased. The snSEM showed similar levels
between all tendon models, i.e. with and without variable tendon parameters. Ultimately, the muscle
model and the muscle-tendon model with variable tendon strain showed to be equally good in
estimating the model parameters. The decision on which model to use will thus be researcher and
situation dependent.
By adding tendon dynamics to the muscle model, it was expected to increase the accuracy of the
active muscle parameters. During active tasks, the muscle stiffness is closer to the tendon stiffness
than during passive tasks. The smaller difference in stiffness causes tendon dynamics to influence the
exerted forces and the assumption in the muscle model of an infinitely stiff tendon can no longer be
upheld. With tendon dynamics present in the model, it was expected that torque patterns during
muscle activation, which could not be followed by the model in the past, could now be achieved.
With expanding the measurement protocol to include more variations in muscle length, velocity,
acceleration and activation, it was expected to enable the muscle model to more easily differentiate
between the muscle parameters. The different muscle parameters are influenced by different
aspects of the model input. By including higher muscle velocities and accelerations and more
variations it was expected to increase the accuracy of all muscle parameters.
4.2.1 Muscle model
VAF-values for both multiple holds and multi sine with holds measurement protocols are found high
with a mean value of around 95%. The outliers that were found for the VAF-values were for subjects
that showed knee movements or incorrect alignment with the Achilles rotational axis, more on this in
the subjects discussion.
54
Most snSEM-values are found low in the range of 1 – 10 times the simulation SEM levels. Highest and
most distributed snSEM-values are found for the triceps surae activation scale factors (g2, g3 and g4).
This is expected as subjects are each differently build with respect to their muscles. One subject
might have a higher activation level in the medial gastrocnemius while another has higher activations
in the lateral gastrocnemius. The muscle activation filter parameters (f0 and beta) are found with
very low snSEM-values. This could be attributed to more muscle activation found for the subject
trials than for the model simulations. The snSEM-value for the mass is also found relatively low with a
mean value of twice the simulation SEM. From the parameter estimations however, it was found that
the estimated mass (ankle + footplate) was at its allowed limit (1.2 kg lower limit and 3k upper limit)
for almost all subjects. Due to this it is expected that the mass is incorrectly estimated. This could be
explained by the muscle accelerations. The mass parameter is estimated by having enough
information on the inertial torques. Since inertial torques were found relatively low with respect to
the passive (tissue) and active (neural) torques, the mass could not be accurately estimated.
Visualizations of the modelled vs measured torques show the modelled torque closely following the
measured torques for both new measurement protocols. It is mostly at peak levels for muscle
activation that the modelled torque does not reach the level of the measured torques. It is expected
that this faster reduction in delivered torque is due to the absence of the tendon dynamics. If tendon
dynamics are included, the opposing muscle first has to move the tendon through its non-linear toe
region. In the toe region, forces delivered are lower thus the opposing muscle is slower in exerting
torque, which results in the delivered torque reducing at a slower rate. If is found that the muscle
model is capable of separating the torques exerted by the passive (tissue) components and the active
(neural components). During the passive phase (first 7 seconds) of the measurement, all modelled
torques are attributed to the passive components.
The multiple holds measurement protocol in combination with the muscle model was able to predict
the measured torques levels for most subjects in the model validation step. A mean VAF-value of
90.4% was found. The lower VAF-values are again found for the subjects that showed ankle
misalignment or knee movements. The multi sine with holds protocol was less able to accurately
predict the measured torque levels. Similar to the VAF-values found for the modelling results, the
low VAF-values can be attributed to two subjects of the measured seven having a bad ankle
alignment. The higher velocities in the multi sine with holds protocol could possibly have induced
reflexive behaviour, although this study assumes reflexive and voluntary activation can be kept
together for determining full muscle activation.
4.2.2 Muscle-tendon model
The passive ramp and hold task modelled with the muscle-tendon model resulted in slightly lower
VAF-values than the modelling results with the muscle model. A mean VAF-value of 99.6% was
found. This indicates that the muscle-tendon model is able to estimate parameters on a similar scale
as compared to the muscle model. The VAF-values found for the active multiple holds protocol are
more widely distributed for the muscle-tendon model than for the muscle model. The mean VAFvalue is found around 90% and ranges between 75% and 93%. The VAF-values for the active multi
sine with holds protocol are higher and with a smaller span, the higher VAF could be attributed to the
smaller subjects group.
55
The snSEM-values are lower for almost all parameters for the muscle-tendon model when compared
with the muscle model results. Mean snSEM-values are mostly around 1 times the simulated SEM
results for both measurement protocols. Largest variations are found in the mass, tibialis slack
length, relaxation stiffness factor and optimal triceps muscle length. The high mass snSEM can be
attributed to low contributions of the inertial torques to the measured torque. Tibialis slack length
snSEM might be higher because the torques during the passive movement phase are mostly found in
plantar flexion direction. The higher snSEM of the relaxation stiffness factor can be attributed to the
tendon acting as a means of relaxation as well, decreasing the influence of the muscle relaxation. The
higher snSEM of the optimal triceps muscle length can be related to the maximal torque the Achilles
manipulator can handle. Torque levels may not rise above 80Nm (see limitations of research
discussion section). The triceps surae muscles are stronger than the tibialis anterior muscle, subjects
may have held back on delivering high torques with the triceps surae muscles. Lower exerted forces
from the muscle means that the muscle length was further from the optimal muscle length, thus the
model may have more difficulty in determining the triceps surae optimal muscle length.
Modelling results of the muscle-tendon model show similar results as compared to the muscle mode.
The modelled torque follows the measured torque pattern closely, but again has difficulty reaching
the peak torques. The inability to reach the peak torques is unfortunate, since the lack of reaching
the peak torque levels in the muscle model was attributed to the lack of tendon dynamics. It is likely
caused by setting the tendon parameters at set values. Including the tendon parameters in the set of
optimization parameters should increase the ability to reach the peak torques.
The covariance matrix shows dependencies between the estimated muscle parameters. It is
expected for some of the passive muscle parameters (k_tri, k_tib, x0_tri, x0_tib, tau_rel and k_rel) to
show higher cross-covariances within their group and likewise for the active muscle parameters (g1,
g2, g3, g4, l0_tri, l0_tib, f0 and beta), since parameters in the same group contribute to the same
part of the modelled torque. Therefore it is expected that the cross-covariance between passive and
active muscle parameters are found low. The mass can be found for passive measurements but
contributes to the inertial torque. High cross-covariances were found for the optimal muscle lengths
(l0_tri and l0_tib) with respect to their muscle stiffness factor (k_tri and k_tib) and their muscle slack
length (x0_tri and x0_tib). This could indicate some interaction between the passive and active
muscle components, although this was not visible from the active (neural) and passive (tissue)
modelled torques. The mass parameter is also found to have high cross-covariance with g1 and
l0_tib, indicating the bad estimations for the mass.
Model validation for the muscle-tendon model gives slightly lower VAF values when compared with
the muscle model. This indicates that parameter values estimated from one measurement set could
less accurately predict the measured torques of the other two measurement sets. Similar results are
found for model validation on the passive ramp and hold task, using estimated parameters from the
active multiple holds task. These findings show that the muscle-tendon model in this state is not as
capable in estimating the muscle parameters as the muscle model.
Including the tendon strain factor to the set of optimization parameters has positive effects on the
parameter estimations. Mean VAF of the modelling results has increased to about 94%. Compared to
the results of the muscle model, the mean VAF is at a similar level, although the span in VAF found is
larger for the muscle-tendon model with variable dlt. This can be attributed to the higher chance of
56
the muscle-tendon model of resulting in a local minimum. The snSEM-values are found at similar
levels for the muscle-tendon model with variable dlt when compared with the muscle-tendon model
with set tendon parameters, except for the muscle activation parameters which are all much lower.
The inclusion of the tendon strain factor dlt as variable parameter may have put more emphasis on
the effects of the muscle activation, lowering the snSEM-values. Model validation again shows similar
results when compared with the muscle-model. Mean VAF is at a similar level and the entire
validation VAF span is slightly higher, likely coming from the higher possibility of resulting in a local
minimum.
Including the tendon toe force to the set of optimization parameters also has positive effects on the
parameter estimations, although slightly lower than the effects of adding the tendon strain factor.
Modelling and validation VAF have increased when compared to the muscle-tendon model with set
tendon parameters, but not as much as found when using the tendon strain factor. The snSEMvalues also show similar yet slightly higher levels compared to the muscle-tendon model with
variable dlt. It can be concluded that adding the tendon strain factor to the parameter optimization
set yields better parameter estimations than using the tendon toe force parameter.
The total length of the muscle-tendon couple at zero degrees dorsiflexion is currently determined by
a zero angle length for the muscle and tendon. The muscle lengths at the zero position is determined
from the literature (Maganaris, 1999 for the Tibialis anterior and Maganaris, 1998 for the triceps
surae muscles). The tendon length at the zero position for no muscle activity is currently assumed at
the muscle slack length (from Thelen, 2003), as only the achilles tendon length in rest was found in
the literature [Rosso et al., 2012]. The assumption of the tendon slack length at the zero position
brings a certain phenomenon into the model. The tendon length is the tendon slack length plus or
minus a deviation depending on the current state of the muscle-tendon couple. The tendon force is
determined through the length difference with respect to the tendon slack length. In this scenario,
the slack length is taken from the total tendon length, which is the tendon slack length plus or minus
a deviation, causing only the tendon length change to determine the tendon force. In other words,
the value chosen for the tendon slack length does not matter in the model. In Figure 39 it can be
seen that the tendon force-length characteristic is the same when the tendon slack length is
subtracted. A longer tendon is expected to be more compliant. But since tendon stiffness depends on
the tendon specific parameters that are taken from the literature [Thelen, 2003], it is expected that
the phenomenon does not affect the results.
Tendon slack lengths are taken from [Thelen, 2003], which are found to be full tendon lengths from
attachment to the bone to the end. The full tendon length causes the tendons to be longer than the
length from bone attachment to the start of the muscle. The achilles tendon for example is 41.2cm in
its full length. Unfortunately it is not possible to let the tendon slack length be determined by the
model, as this would put two undetermined parameters in series, making them unobtainable. For
consistency with the study of Thelen and for the fact that the chosen tendon slack length does not
influence the tendon force, the tendon slack length as reported by Thelen is taken for this study.
57
Tendon force-length relation
10000
9000
8000
tendon force [N]
7000
6000
5000
4000
3000
2000
1000
0
0
0.1
0.2
0.3
tendon length [m]
0.4
0.5
Figure 39. Tendon force-length characteristic. Tendon slack length does not affect the tendon force-length relation after
slack length. Blue line depicts tendon force-length relation for lt0 = 0.3m and the red line for lt0 = 0.41m.
4.3 Implications for clinical use
The clinic is currently able to accurately estimate the passive muscle parameters with the passive
ramp and hold protocol. Accurate estimation of the active muscle parameters would be beneficial for
the patients. The optimal muscle lengths describe on which part of the active force-length
characteristic the patients are working in their daily lives. Altered optimal muscle lengths would give
reason to investigate if treatments like tendon extension are required to bring the patient to a more
favourable part on the active force-length characteristic.
With the active multiple holds protocol, both the muscle model and muscle-tendon model are able
to estimate the active muscle parameters to a reasonable degree. It has to be investigated if this
protocol is usable for patients suffering from a neural disease.
4.4 Limitations of this research
For both the muscle model and the muscle-tendon model it was found that the validation VAF was
lower for the active protocols than for the passive ramp and hold protocol. Since the validation VAF is
still high with mean VAF-values around 95%, it is thought that the slight difference in estimated
parameters does not result from overparametrization. Investigating the power-spectral density plots
of three succeeding measurements of the same subject showed varying power distributions (Figure
40). This might be the result of the subject responding differently in the actively exerted torques per
measurement.
58
Power spectral density of 3 repeated subject measurements
60
repetition 1
repetition 2
repetition 3
50
40
Power [dB]
30
20
10
0
-10
-20
-30
0
10
20
30
Frequency [Hz]
40
50
60
Figure 40. Power spectral density plot of three succeeding same subject measurements.
During this study, the limits of the Achilles ankle perturbator were met. The maximum torque the
motor of the Achilles can produce is 80Nm. One of the stronger aspects of the multiple holds
protocol and the multi sine with holds protocol are the fast movements. The Achilles is force
controlled, for a high acceleration a high torque has to be delivered. It was seen that for high torque,
with relatively light test subjects, the Achilles had a chance to overshoot the to be achieved position.
When this overshoot happened, the measurement protocol was no longer followed and instead a
steps movement was followed by the Achilles. For the stronger subjects it also happened that the
Achilles was giving torque in one direction and the subject in the other direction, thus increasing the
total torque on the Achilles over the limit of 80Nm. Whenever this happened the Achilles stopped
and the current measurement had to be redone.
During subject measurements it was found that some subjects had feet with higher ankle rotation
points. The higher rotation points caused a misalignment with the rotational axis of the Achilles,
likely influencing the torques delivered during the measurement. Another observation was on
subjects with very flexible joints. For the more flexible subjects, it was possible that they slightly
twisted their knee in order to exert more torque on the Achilles. This could cause other muscles that
are not measured to influence the measured torque level. Because the measured torque is partially
caused by muscles that are not included in the model, the model might not give accurate results.
For subject measurement close after one another, it happened a number of times that the EMG
measurement device did not send the measured EMG signals to the pc. This could only be solved by
unplugging all devices and waiting for a period of time usually longer than 30 minutes. The EMG
device not sending the measured signals could have to do with some previous signal values
remaining in the data acquisition device after the previous measurement.
59
The muscle-tendon model has to use a sampling rate at 1000Hz as opposed to the 125Hz of the
muscle model. The increased sampling rate is required to avoid a stiffness issue in the muscle-tendon
model. The muscle-tendon model finds the modelled torque iteratively. Because of the iterative
nature of the model it has the risk of a length discrepancy influencing the resulting torques. The
tendon length is determined from the previous muscle length and the total muscle-tendon length. If
the total muscle-tendon length changes more than the maximum muscle velocity can follow, the
remainder of the length change is attributed to the tendon. Small length changes in the tendon can
result in large force differences. By using a higher sampling rate this issue can be avoided, although it
increases the modelling time significantly.
This study makes use of the VAF- and SEM-values of the modelling results in order to assess the
results. Both the VAF and the SEM are found to have their weaknesses however. The VAF gives a high
value if the fit between the measured torque and the modelled torque is high. A possible torque
measurement is one where the torque is at a steady level for most of the measurement and one
peak in exerted torque is included. For this case, if the modelled torque would be a steady line at the
same height as the measured torque, omitting the peak torque, the found VAF-value would still be
reasonably high. If however, the modelled torque accurately follows the peak torque and is slightly
below the steady torque level for the remainder of the measurement, the VAF-value would be lower
than for the first case. I.e. The highest VAF-value might not give the most accurate description of the
estimated parameters.
The SEM is currently used in most studies to evaluate the accuracy of the estimated parameter,
where a lower SEM-value corresponds with a more accurate estimation. The SEM is found from the
inverse hessian and describes the influence of that specific parameter to the final solution. From
modelling simulations with too high velocities it was seen that the model could not accurately
reobtain the modelled parameters. The estimated parameters however, did receive very low SEMvalues. For a model that accurately describes the situation of the to be modelled system, all SEMvalues should be low, but one low SEM-value does not necessarily correspond with a correctly
estimated parameter.
4.5 Future research
Both muscle model and muscle-tendon model are able to follow most of the measured torque levels.
The muscle-tendon model with variable tendon strain performs equally compared to the muscle
model. This study has only investigated including one tendon parameter to the optimization set at a
time. Using combinations of multiple tendon parameters as to be optimized parameters could
further increase the fit of the model. Difficulties could lie in the higher chance of arriving in a local
minimum when using multiple tendon parameters.
The mass of the foot and footplate was found mostly at the optimization limits, thus suggesting bad
estimations of the mass parameter. For the future it is suggested to either increase the movement
accelerations to offer the model more information for estimating the mass or removing the mass
from the set of optimization parameters.
The VAF as currently used for assessing modelling results evaluates all parts of the torques equally.
Since some parts are of more significance to the results, it could be possible to add a weighting factor
to these parts. The weighting factors can place more emphasis on the significant parts focussing the
VAF.
60
Conclusion
The goal of this study was to accurately estimate of both passive and active muscle parameters by
developing both a new muscle model which includes tendon dynamics and a new measurement
protocol which acquires richer data in the fields of position, velocity, acceleration and muscle
activation. The muscle model as developed by de Vlugt et al. in its current state, combined with the
newly developed active multiple holds protocol resulted in high VAF-values and similarly low snSEMvalues as compared to model simulation results. Model validation resulted in high VAF-values
indicating a good estimation of the modelled muscle parameters. Peak torques could not yet be fully
achieved, but the high results give reason to use the model and protocol for estimating both the
passive and active muscle parameters. The newly developed muscle-tendon model is equally able to
result in high VAF and low snSEM values when using the tendon strain parameter as a variable
parameter. With one variable tendon parameter, the muscle-tendon model performs similarly with
respect to the muscle model. Using a combination of multiple tendon parameters as optimization
parameters might be able to find even better modelling fits, but at the risk of overparametrization.
Both muscle and muscle-tendon models perform equally but the muscle-tendon model is much
slower in modelling the results and more likely to land in a local minimum, currently providing reason
to choose the muscle model over the muscle-tendon model.
61
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63
Appendix A. Covariance matrix
The developed protocols will be tested for relations between the different neuromechanical
optimization parameters. One method of checking the relationship between parameters is through
their covariance matrix. It is a necessity to check the dependence between different parameters, for
whereas it is possible for some parameters to relate to each other, certain other combinations of
parameters should share no relation at all. For instance, it is possible that the two relaxation
parameters share some dependence since both are used to describe the muscle relaxation. The
muscle stiffness and the optimal muscle length however, should share no dependence at all since
both parameters describe different characteristics of the muscle. Hence if there appears to be a
relationship between two parameters for a certain protocol where there should be no relation at all,
that protocol should be discarded.
A.1 The covariance matrix
In the field of probability and statistics, the covariance matrix is a matrix whose element in the i, j
position is the covariance between the ith and jth elements of a random vector. In this study, the
covariance matrix depicts the covariance between two optimized neuromechanical parameters. After
optimization, each parameter has a vector which contains that parameters set of variance. From the
measured variance, the covariances are calculated as shown in Figure 41.
Figure 41, The covariance matrix. Note that this Figure shows the raw covariance matrix, the matrix used in this study will
be normalized in order to compare parameter dependence.
Where Xi represents the vector of variance of the ith parameter and μi is the expected value of vector
Xi .
𝜇 𝑖 = 𝐸 (𝑋𝑖 )
Now the covariance between two parameter variances X1 and X2 can be written as:
𝐶𝑜𝑣(𝑋, 𝑌) = 𝐸 [(𝑋1 − 𝜇 1)(𝑋2 − 𝜇 2)]
= 𝐸 [(𝑋1 𝑋2)] − 𝐸 [ 𝑋1 ]𝐸[ 𝑋2 ]
If parameters X1 and X2 were completely independent of one another, it would be possible to write
them as two separate functions:
64
Thus for two independent parameters the covariance becomes zero. If the covariance is anything
other than zero, a dependence exists between those two parameters.
In this study, the cross-covariances are normalized by division with RMS of the i th and jth autocovariances. In the normalized covariance matrix, all auto-covariances will have a normalized value of
1. The higher the cross-covariance between two neuromechanical parameters, the closer that value
will be to 1.
A.2 Linear and non-linear models
Linear and non-linear models differ in the linearity in the parameters. Linear models are linear in the
statistical sense such that all functions
𝑓 (𝒙, 𝛽 → ) = 𝛽0 + 𝛽1 ∙ 𝑥 + 𝛽2 ∙ 𝑥 2
𝑓 (𝒙, 𝛽 → ) = 𝛽0 + 𝛽1 ∙ 𝑙𝑛(𝑥 )
𝑓 (𝒙, 𝛽 → ) = 𝛽0 + 𝛽1 ∙ 𝑠𝑖𝑛 (𝑥 ) + 𝛽2 ∙ 𝑠𝑖𝑛 (2𝑥 ) + 𝛽3 ∙ 𝑠𝑖𝑛 (3𝑥 )
are linear because every parameter’s effect on the outcome can be written as a separate function.
An example of a non-linear model would be
𝑓 (𝒙, 𝛽 → ) = 𝛽0 + 𝛽0 𝛽1 ∙ 𝑥
since the input vector x is now related to two parameter values and the equation cannot be written
as adding 𝛽0 and 𝛽1 separately. Since 𝛽0 and 𝛽1 are both determined through x, it is possible for
them to share similar variances and have a high cross-covariance. The difference in cross-covariance
can be shown in the covariances matrices of a linear and non-linear model respectively.
Linear mass-spring-damper model
Parameter optimization for a linear mass-spring-damper model described by the equation
𝐹 (𝑃, 𝑥 ) = 𝑘 ∙ 𝑥 + 𝑏 ∙ 𝑥̇ + 𝑚 ∙ 𝑥̈
The normalized covariance matrix in Figure 42. shows cross-covariances close to zero. The non-zero
cross-covariance can be explained by incidental small similarities between the x-vector and its
derivatives.
65
Covariance Matrix
1
0.8
k
0.6
0.4
0.2
0
1
0.8
b
0.6
0.4
0.2
0
1
0.8
m
0.6
0.4
0.2
0
k
b
m
Figure 42. Normalized covariance matrix for a linear mass-spring-damper model. All cross-covariances approach zero.
Covariance Matrix
k
1
0.5
0
b
1
0.5
0
m
1
0.5
0
u
1
0.5
0
k
b
m
u
Figure 43. Normalized covariance matrix for a non-linear mass-spring-damper model. The cross-covariances of the nonrelated parameters approach zero, while the relation between parameters b and u can be seen in the high cross-covariance.
66
Non-linear mass-spring-damper model
Parameter optimization for a non-linear mass-spring-damper model described by the equation
𝐹 (𝑃, 𝑥 ) = 𝑘 ∙ 𝑥 + 𝑏 ∙ ( 𝑥̇ − 𝑢) + 𝑚 ∙ 𝑥̈
The normalized covariance matrix in Figure 43. shows cross-covariances approaching zero for the
independent parameters and shows a high cross-covariance between parameters b and u. The high
cross-covariance results from the model’s equation where both parameters are related to 𝑥̇ and
share similar variances. The model however, is still able to accurately estimate both parameters b
and u, since both are related to 𝑥̇ differently.
A.3 Relation muscle model inputs to covariance matrix
Both muscle model and muscle-tendon model are non-linear in parameters, thus it is possible for
high cross-covariances to arise between certain parameters. It is still possible to estimate these
parameters accurately, but this requires the model to be variant enough in its inputs (position, EMG
and derivatives). Inputs sharing similarities make it more difficult for the optimization to distinguish
different parameters. For a position input of 𝑥 = 𝑒 𝑡, the derivatives would result in
𝑥 = 𝑥̇ = 𝑥̈ = 𝑒 𝑡
which would make it impossible to distinguish the parameters in the linear mass-spring-damper
model. If parts of the model input share similar behaviour it already affects the ability to estimate the
model parameters, which is shown though high cross-covariances. The model input (position and
EMG) must be rich signals such that the effects of all parameters can be seen in the resulting torques,
which requires determining the optimal protocol.
67
Appendix B. Restrictions tendon-model
The muscle-tendon model includes both muscle and tendon mechanics, giving a closer
approximation of a true muscle than the previous model without tendon. The muscle-tendon model
also has its restrictions. In the tendonless model, the speed of muscle contraction/extension could be
determined by calculating the derivative of the muscle length. In the muscle-tendon model however,
only the length of muscle and tendon combined is known. To determine the respective muscle and
tendon lengths and speeds, a different approach was required.
The tendon-included model uses an iteration loop to determine the muscle and tendon lengths and
speeds. Known for each data point is the total muscle + tendon length. Following script has been
shortened to convey the actual process. The new muscle length is calculated through the following
steps:
1. Force exerted by the tendon is determined from previous tendon length, and which is
depending on the measured muscle-tendon length Lmt(t) and the state of the muscle length
Lm(t).
ft_tri(i) = tendon_l2f(lt_tri(i), par_tri);
2. Force exerted by the elastic muscle components is determined.
F_elas_tri_rel(i) = exp(k_tri * (lm_tri(i) - x0_tri(i)));
3. Muscle force is determined by subtracting the elastic force component from the tendon
force.
fm_tri(i) = ft_tri(i) - F_elas_tri(i);
4. Muscle speed is determined from the active force-length and force-speed characteristics.
fl_tri(i) = a_tri(i) * force_length_char;
fdl_tri(i) = fm_tri(i) / fl_tri(i);
dlm_tri(i) = function(fdl_tri(i));
5. The new muscle length Lm(t+dt) is then determined from the speed.
lm_tri(i+1) = lm_tri(i) + dlm_tri(i) * dtr;
6. In the next iteration the new tendon length is determined from the new total length
Lmt(t+dt) and the previously calculated muscle length.
lt_tri(i+1) = lmt_tri(i+1) - lm_tri(i+1);
Muscle contraction/extension speed has a maximum, set by
vmax_tri = 10 * l0_tri;
% factor 10 for young and factor 8 for old
If the muscle speed dlm_tri were to become larger than the maximum speed vmax_tri, the speed is
set to be the maximum value. In the iteration loop this speed restraint can pose a problem. If the
length change in the muscle-tendon couple lmt_tri is much larger than the maximum length change
that the muscle itself can undergo, the remaining length change will go to the tendon length (step 6).
68
Since the tendon force is determined from the tendon length, a large step in the tendon force-length
curve can move the tendon to exert high levels of force (Figure 44).
Tendon force-length characteristic
6000
Tendon force [N]
5000
4000
3000
2000
1000
0
0
0.1
0.2
0.3
Tendon length [m]
0.4
0.5
Figure 44. Tendon force-length characteristic. The tendon exerts no force until the tendon slack length has been surpassed.
Tendon force exertion starts non-linear then continues with a linear relation.
Triceps muscle rate of length change
0.5
0.4
0.3
muscle speed [m/s]
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
0
5
10
15
20
time [s]
25
30
35
40
Figure 45. The triceps muscle rate of length change. The muscle is oscillating due muscle active force exertion alternating
between on and off.
69
The problem lies in the lengthening muscle. While the one muscle is shortening, the other should
exert no active force and exist purely of the elastic force component; ft = F_elas. The discrepancy in
muscle-tendon total length and muscle length causes the lengthening muscle to exert force as well,
counteracting the shorting muscle. While the lengthening muscle should not actively exert force it is
now actively pulling against the shortening muscle, causing both muscles to oscillate between
actively pulling and non-active. An example of the oscillating speed is shown in Figure 45.
During simulations of the protocols, this restriction can be overcome by adding more damping to the
system. A higher damping by the active muscle components reduces the speed of the muscle,
keeping it below the maximum level. Since the actual muscle length elongation of humans cannot
move faster than the maximal muscle elongation speed, this restriction in the iteration loop should
only pose a problem during the protocol simulations and should not arise during clinical trials.
70