Marcos Duarte

Let's estimate the joint force and moments of force at the lower limb during locomotion using the inverse dynamics approach.

We will model the lower limbs at the right side as composed by three rigid bodies (foot, leg, and thigh) articulated by three hinge joints (ankle, knee, and hip) and perform a two-dimensional analysis.

The free body diagrams of the lower limbs are:

The calculation above is tedious, error prone, useless, and probably it's wrong.

To make some use of it, we can clearly see that forces act on far segments, which are not directly in contact with these forces. In fact, this is true for all stuff happening on a segment: note that $\mathbf{F_1}$ and $\mathbf{M_1}$ are present in the expression for $\mathbf{F_3}$ and $\mathbf{M_3}$ and that the acceleration of segment 1 matters for the calculations of segment 3.

Instead, we can use the power of computer programming (like this one right now!) and solve these equations recursevely hence they have the same pattern.

We could write a function that it would have as inputs the body-segment parameters, the kinematic data, and the distal joint force and moment of force and output the proximal joint force and moment of force.

Then, we would call this function for each segment, starting with the segment that has a free extremity or that has the force and moment of force measured by some instrument (i,e, use a force plate for the foot-ground interface).

This function would be called in the following manner:

```
Fp, Mp = invdyn2d(rcm, rd, rp, acm, alfa, mass, Icm, Fd, Md)
```

So, let's write such function:

In [ ]:

```
# %load ./../functions/invdyn2d.py
"""Two-dimensional inverse-dynamics calculations of one segment."""
__author__ = 'Marcos Duarte, https://github.com/demotu/BMC'
__version__ = 'invdyn2d.py v.2 2015/11/13'
def invdyn2d(rcm, rd, rp, acm, alpha, mass, Icm, Fd, Md):
"""Two-dimensional inverse-dynamics calculations of one segment
Parameters
----------
rcm : array_like [x,y]
center of mass position (y is vertical)
rd : array_like [x,y]
distal joint position
rp : array_like [x,y]
proximal joint position
acm : array_like [x,y]
center of mass acceleration
alpha : array_like [x,y]
segment angular acceleration
mass : number
mass of the segment
Icm : number
rotational inertia around the center of mass of the segment
Fd : array_like [x,y]
force on the distal joint of the segment
Md : array_like [x,y]
moment of force on the distal joint of the segment
Returns
-------
Fp : array_like [x,y]
force on the proximal joint of the segment (y is vertical)
Mp : array_like [x,y]
moment of force on the proximal joint of the segment
Notes
-----
To use this function recursevely, the outputs [Fp, Mp] must be inputed as
[-Fp, -Mp] on the next call to represent [Fd, Md] on the distal joint of the
next segment (action-reaction).
This code was inspired by a similar code written by Ton van den Bogert [1]_.
See this notebook [2]_.
References
----------
.. [1] http://isbweb.org/data/invdyn/index.html
.. [2] http://nbviewer.ipython.org/github/demotu/BMC/blob/master/notebooks/GaitAnalysis2D.ipynb
"""
from numpy import cross
g = 9.80665 # m/s2, standard acceleration of free fall (ISO 80000-3:2006)
# Force and moment of force on the proximal joint
Fp = mass*acm - Fd - [0, -g*mass]
Mp = Icm*alpha - Md - cross(rd-rcm, Fd) - cross(rp-rcm, Fp)
return Fp, Mp
```

The calculations are implemented in only two lines of code at the end, the first part of the code is the help on how to use the function. The help is long because it's supposed to be helpful :), see the style guide for NumPy/SciPy documentation.

The real problem is to measure or estimate the experimental variables: the body-segment parameters, the ground reaction forces, and the kinematics of each segment. Fo such, it is necessary some expensive equipments, but they are typical of a biomechanics laboratory.

Let's work with some data of kinematic position of the segments and ground reaction forces in order to compute the joint forces and moments of force.

The data we will work are in fact from a computer simulation of running created by Ton van den Bogert. The nice thing about these data is that as a simulation, the true joint forces and moments of force are known and we will be able to compare our estimation with these true values.

All the data can be downloaded from a page at the ISB website:

In [2]:

```
from IPython.display import IFrame
IFrame('http://isbweb.org/data/invdyn/index.html', width='100%', height=400)
```

Out[2]:

In [3]:

```
# import the necessary libraries
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from IPython.display import HTML, display
import sys
sys.path.insert(1, r'./../functions')
```

In [4]:

```
# load file with ground reaction force data
grf = np.loadtxt('./../data/all.frc') # [Fx, Fy, COPx]
# load file with kinematic data
kin = np.loadtxt('./../data/all.kin') # [Hip(x,y), knee(x,y), ankle(x,y), toe(x,y)]
freq = 10000
time = np.linspace(0, grf.shape[0]/freq, grf.shape[0])
```

In [5]:

```
# this is simulated data with no noise, filtering doesn't matter
if False:
# filter data
from scipy.signal import butter, filtfilt
# Butterworth filter
b, a = butter(2, (100/(freq/2)))
for col in np.arange(grf.shape[1]-1):
grf[:, col] = filtfilt(b, a, grf[:, col])
b, a = butter(2, (100/(freq/2)))
for col in np.arange(kin.shape[1]):
kin[:, col] = filtfilt(b, a, kin[:, col])
```

In [6]:

```
# heel strike occurs at sample 3001
time = time[3001 - int(freq/40):-int(freq/20)]
grf, kin = grf[3001 - int(freq/40):-int(freq/20), :], kin[3001 - int(freq/40):-int(freq/20), :]
```

In [7]:

```
# plot data
hfig, hax = plt.subplots(2, 2, sharex = True, squeeze=True, figsize=(12, 6))
hax[0, 0].plot(time, grf[:, [0, 1]], linewidth=2, alpha=0.3)
hax[0, 0].legend(('Fx','Fy'), frameon=False)
hax[0, 0].set_ylabel('Force [N]')
hax[0, 1].plot(time, grf[:, 2], linewidth=2, alpha=0.3)
hax[0, 1].legend(['COPx'], frameon=False)
hax[0, 1].set_ylabel('Amplitude [m]')
hax[1, 0].plot(time, kin[:, 0::2], linewidth=2, alpha=0.3)
hax[1, 0].legend(('Hip x','Knee x','Ankle x','Toe x'), frameon=False)
hax[1, 0].set_ylabel('Amplitude [m]')
hax[1, 1].plot(time, kin[:, 1::2], linewidth=2, alpha=0.3)
hax[1, 1].legend(('Hip y','Knee y','Ankle y','Toe y'), frameon=False)
hax[1, 1].set_ylabel('Amplitude [m]')
hax[1, 0].set_xlabel('Time [s]'), hax[1, 1].set_xlabel('Time [s]')
plt.show()
```

In [8]:

```
# body-segment parameters [thigh, shank, foot]
mass = [6.85, 2.86, 1.00] # mass [kg]
Icm = [0.145361267, 0.042996389, 0.0200] # rotational inertia [kg.m2]
cmpr = [0.4323725, 0.4334975, 0.0] # CM [m] wrt. prox. joint [frac. of segment length]
```

In [9]:

```
# Kinematic data
# center of mass position of the thigh, shank, foot segments
rcm = np.hstack((kin[:, (0, 1)] + cmpr[0]*(kin[:, (2, 3)] - kin[:, (0, 1)]),
kin[:, (2, 3)] + cmpr[1]*(kin[:, (4, 5)] - kin[:, (2, 3)]),
kin[:, (4, 5)] + cmpr[2]*(kin[:, (6, 7)] - kin[:, (4, 5)])))
# center of mass linear acceleration of the thigh, shank, foot segments
acm = np.diff(rcm, n=2, axis=0)*freq*freq
acm = np.vstack((acm, acm[-1, :], acm[-1, :]))
# thigh, shank, foot segment angle
ang = np.vstack((np.arctan2(kin[:, 1] - kin[:, 3], kin[:, 0] - kin[:, 2]),
np.arctan2(kin[:, 3] - kin[:, 5], kin[:, 2] - kin[:, 4]),
np.arctan2(kin[:, 5] - kin[:, 7], kin[:, 4] - kin[:, 6]))).T
# hip, knee, and ankle joint angles
angj = np.vstack((-(ang[:, 0]-ang[:, 1]), np.unwrap(ang[:, 1] - ang[:, 2] + np.pi/2))).T*180/np.pi
# thigh, shank, foot segment angular acceleration
aang = np.diff(ang, n=2, axis=0)*freq*freq
aang = np.vstack((aang, aang[-1, :], aang[-1, :]))
```

In [10]:

```
# plot hip, knee, and ankle joint angles
hfig, (hax1, hax2) = plt.subplots(2, 1, sharex = True, squeeze=True, figsize=(10, 6))
hax1.plot(time, angj[:, 0], linewidth=2, label='Knee')
hax1.legend(frameon=False, loc='upper left'), hax1.grid()
hax2.plot(time, angj[:, 1], linewidth=2, label='Ankle')
hax2.legend(frameon=False, loc='upper left'), hax2.grid()
hax1.set_ylabel('Joint angle $[^o]$')
hax2.set_ylabel('Joint angle $[^o]$')
hax2.set_xlabel('Time [s]');
```

In [11]:

```
# inverse dynamics
# invdyn2d(rcm, rd, rp, acm, alpha, mass, Icm, Fd, Md)
from invdyn2d import invdyn2d
# ankle
[Fa, Ma] = invdyn2d(rcm[:, (4, 5)], grf[:, (2, 2)]*[1, 0], kin[:, (4, 5)], acm[:, (4, 5)], \
aang[:, 2], mass[2], Icm[2], grf[:, (0, 1)], 0)
# knee
[Fk, Mk] = invdyn2d(rcm[:,(2, 3)], kin[:, (4, 5)], kin[:, (2, 3)], acm[:, (2, 3)], \
aang[:,1], mass[1], Icm[1], -Fa, -Ma)
# hip
[Fh, Mh] = invdyn2d(rcm[:, (0, 1)], kin[:, (2, 3)], kin[:, (0, 1)], acm[:, (0, 1)], \
aang[:, 0], mass[0], Icm[0], -Fk, -Mk)
# magnitude of the calculated hip, knee, and ankle resultant joint force
Fam = np.sqrt(np.sum(np.abs(Fa)**2, axis=-1))
Fkm = np.sqrt(np.sum(np.abs(Fk)**2, axis=-1))
Fhm = np.sqrt(np.sum(np.abs(Fh)**2, axis=-1))
```

In [13]:

```
# load file with true joint forces and moments of force
forces = np.loadtxt('./../data/all.fmg') # [Hip, knee, ankle]
moments = np.loadtxt('./../data/all.mom') # [Hip, knee, ankle]
#heel strike occurs at sample 3001
forces, moments = forces[3001-int(freq/40):-int(freq/20), :], moments[3001-int(freq/40):-int(freq/20), :]
```

Let's plot these data but because later we will need to plot similar plots, let's create a function for the plot to avoid repetition of code:

In [14]:

```
def plotdata(time, Fh, Fk, Fa, Mh, Mk, Ma, forces, moments, stitle):
# plot hip, knee, and ankle moments of force
hfig, hax = plt.subplots(3, 2, sharex = True, squeeze=True, figsize=(12, 8))
# forces
hax[0, 0].plot(time, Fh, label='invdyn'), hax[0, 0].set_title('Hip')
hax[1, 0].plot(time, Fk), hax[1, 0].set_title('Knee')
hax[2, 0].plot(time, Fa), hax[2, 0].set_title('Ankle')
hax[1, 0].set_ylabel('Joint force [N]')
hax[2, 0].set_xlabel('Time [s]')
# moments of force
hax[0, 1].plot(time, Mh), hax[0, 1].set_title('Hip')
hax[1, 1].plot(time, Mk), hax[1, 1].set_title('Knee')
hax[2, 1].plot(time, Ma), hax[2, 1].set_title('Ankle')
hax[1, 1].set_ylabel('Moment of Force [Nm]')
hax[2, 1].set_xlabel('Time [s]')
# true joint forces and moments of force
hax[0, 0].plot(time, forces[:, 0], 'r--', label='True'), hax[0, 0].legend(frameon=False)
hax[1, 0].plot(time, forces[:, 1], 'r--')
hax[2, 0].plot(time, forces[:, 2], 'r--')
hax[0, 1].plot(time, moments[:, 0], 'r--')
hax[1, 1].plot(time, moments[:, 1], 'r--')
hax[2, 1].plot(time, moments[:, 2], 'r--')
plt.suptitle(stitle, fontsize=20)
for x in hax.flat:
x.locator_params(nbins=5); x.grid()
plotdata(time, Fhm, Fkm, Fam, Mh, Mk, Ma, forces, moments,
'Inverse dynamics: estimated versus true values')
```

The results are very similar; only a small part of the moments of force is different because of some noise.

Let's see what happens with the joint forces and moments of force when we neglect the contribution of some terms in the inverse dynamics analysis of these data.

Consider the case where the segment acceleration is neglected:

In [15]:

```
# ankle
[Fast, Mast] = invdyn2d(rcm[:, (4, 5)], grf[:, (2, 2)]*[1, 0], kin[:,(4, 5)], acm[:, (4, 5)]*0, \
aang[:,2]*0, mass[2], Icm[2], grf[:, (0, 1)], 0)
# knee
[Fkst, Mkst] = invdyn2d(rcm[:, (2, 3)], kin[:,(4, 5)], kin[:, (2, 3)], acm[:, (2, 3)]*0, \
aang[:, 1]*0, mass[1], Icm[1], -Fast, -Mast)
# hip
[Fhst, Mhst] = invdyn2d(rcm[:, (0, 1)], kin[:,(2, 3)], kin[:, (0, 1)], acm[:, (0, 1)]*0, \
aang[:,0]*0, mass[0], Icm[0], -Fkst, -Mkst)
# magnitude of the calculated hip, knee, and ankle resultant joint force
Fastm = np.sqrt(np.sum(np.abs(Fast)**2, axis=-1))
Fkstm = np.sqrt(np.sum(np.abs(Fkst)**2, axis=-1))
Fhstm = np.sqrt(np.sum(np.abs(Fhst)**2, axis=-1))
plotdata(time, Fhstm, Fkstm, Fastm, Mhst, Mkst, Mast, forces, moments,
'Inverse dynamics: quasis-static approach versus true values')
```

This is not a pure static analysis because part of the ground reaction forces still reflects the body accelerations (were the body completely static, the ground reaction force should be equal to the body weight in magnitude).

Consider the case where besides the acceleration, the body-segment parameters are also neglected.

This means that the joint loads are due only to the ground reaction forces (which implicitly include contributions due to the acceleration and the body-segment weights).

In [16]:

```
# ankle
[Fagrf, Magrf] = invdyn2d(rcm[:, (4, 5)], grf[:, (2, 2)]*[1, 0], kin[:, (4, 5)], acm[:, (4, 5)]*0, \
aang[:, 2]*0, 0, 0, grf[:,(0,1)], 0)
# knee
[Fkgrf, Mkgrf] = invdyn2d(rcm[:, (2, 3)], kin[:, (4, 5)], kin[:,(2,3)], acm[:, (2, 3)]*0, \
aang[:, 1]*0, 0, 0, -Fagrf, -Magrf)
# hip
[Fhgrf, Mhgrf] = invdyn2d(rcm[:, (0, 1)], kin[:, (2, 3)], kin[:, (0, 1)], acm[:, (0, 1)]*0, \
aang[:, 0]*0, 0, 0, -Fkgrf, -Mkgrf)
# magnitude of the calculated hip, knee, and ankle resultant joint force
Fagrfm = np.sqrt(np.sum(np.abs(Fagrf)**2, axis=-1))
Fkgrfm = np.sqrt(np.sum(np.abs(Fkgrf)**2, axis=-1))
Fhgrfm = np.sqrt(np.sum(np.abs(Fhgrf)**2, axis=-1))
plotdata(time, Fhgrfm, Fkgrfm, Fagrfm, Mhgrf, Mkgrf, Magrf, forces, moments,
'Inverse dynamics: ground-reaction-force approach versus true values')
```

Neglecting all the accelerations and the weight of the segments means that the only external force that actuates on the system is the ground reaction force, which although is only actuating at the foot-ground interface it will be transmited to the other segments through the joint forces. Because of that, the joint forces on the ankle, knee, and hip will simply be minus the ground reaction force. Note that the forces shown above for the three joints are the same and equal to:

$$\sqrt{GRF_x^2+GRF_y^2}$$

These simplications also mean that the moments of force could have been simply calculated as the cross product between the vector position of the the COP in relation to the joint and the GRF vector:

$$ \mathbf{M_{a}} = -cross(\mathbf{COP}-\mathbf{r_{a}},\mathbf{GRF}) $$

$$ \mathbf{M_{k}} = -cross(\mathbf{COP}-\mathbf{r_{k}},\mathbf{GRF}) $$

$$ \mathbf{M_{h}} = -cross(\mathbf{COP}-\mathbf{r_{h}},\mathbf{GRF}) $$

Where $\mathbf{r_{i}}\;$ is the position vector of joint i.

Let's calculate the variables in this way:

In [17]:

```
Fhgrfm2 = Fkgrfm2 = Fagrfm2 = np.sqrt(np.sum(np.abs(-grf[:, (0, 1)])**2, axis=-1))
Magrf2 = -np.cross(grf[:, (2, 2)]*[1, 0]-kin[:, (4, 5)], grf[:, (0, 1)])
Mkgrf2 = -np.cross(grf[:, (2, 2)]*[1, 0]-kin[:, (2, 3)], grf[:, (0, 1)])
Mhgrf2 = -np.cross(grf[:, (2, 2)]*[1, 0]-kin[:, (0, 1)], grf[:, (0, 1)])
plotdata(time, Fhgrfm2, Fkgrfm2, Fagrfm2, Mhgrf2, Mkgrf2, Magrf2, forces, moments,
'Inverse dynamics: ground-reaction-force approach versus true values II')
```

For these data set of 'running' (remember this is simulated data), in the estimation of the forces and moments of force at the hip, knee, and ankle joints in a two-dimensional analysis, to not consider the segment acceleration and/or the mass of the segments had no effect on the ankle variables, a small effect on the knee, and a large effect on the hip.

This is not surprising; during the support phase, ankle and knee have small movements and the mass of the segments only start to have a significant contribution for more proximal and heavy segments such as the thigh.

Don't get disappointed thinking that all this work for drawing the complete FBDs and their correspondent equations was a waste of time.

Nowadays, the state of the art and the demand for higher accuracy in biomechanics is such that such simplifications are usually not accepted.

- Winter DA (2009) Biomechanics and motor control of human movement. 4 ed. Hoboken, EUA: Wiley.
- Zatsiorsky VM (2202) Kinetics of human motion. Champaign, IL: Human Kinetics.