Inverse mechanics 2
Determination of joint forces and moments in two dimensions
WCR, modified from notes for the 3D case by James Richards
In the following, the sagittal plane is assumed to be the X-Z plane, with +Z pointing up, for
consistency with later notes. Moments are computed about the Y axis in the 2D case. The Y
axis is perpendicular to the X-Z (i.e. sagittal) plane, and points to the subject’s left, if +X points
forward.
0.
Gather needed data. This includes:
a. Fgrx(t), Fgrz(t) = ground reaction force; Xcop(t),Zcop(t)=location of center of pressure of
ground reaction force, in lab coordinates. The z component of c.o.p. is the height of
the forceplate surface, often defined as zero. This data is obtained from a force plate.
b. Xprox(t), Zprox(t), Xdist(t), Zdist(t) = coordinates of proximal and distal ends of segment,
in the lab frame. This data is obtained from the motion capture system.
c. Body mass, segment length L.
1.
Compute constants and coefficients, including:
a. m=segment mass, computed from body mass and the appropriate coefficient for that
segment.
b. Iyy=moment of inertia of the segment about its center of mass, computed from
segment mass and length, and the appropriate coefficient for that segment. If using
Table 4.1, Winter 4th ed.: Iyy=m(cL)2, where c = (radius of gyration about center of
mass)/(segment length). Example: c=0.302 for leg, according to Table 4.1. This may
also be thought of as the Iyy component of the segment inertia tensor.
c. b=distance ratio (proximal end to center of mass)/(segment length). Example:
b=0.433 for leg segment, according to Table 4.1. This will be used to estimate the
segment center of mass location.
2.
From the data above, compute the other data you will need, including:
a. Xcg(t), Zcg(t) = coordinates of the segment center of mass, using the proximal and
distal end coordinates plus coefficient b computed earlier.
Xcg(t) = Xprox(t) + b(Xdist(t)-Xprox(t))
Zcg(t) = Zprox(t) + b(Zdist(t)-Zprox(t))
b. aX(t), aZ(t) = segment center of mass acceleration, by the second time derivative of
the center of mass coordinates.
c. θY(t) = segment angle (about the Y axis), in the lab frame, computed from the
proximal and distal end coordinates. Use a 4-quadrant arctangent function:
θY(t) = atan2(X,Z) = atan2(Xdist(t)-Xprox(t), Zdist(t)-Zprox(t))
d. y(t) = segment angular acceleration, in the lab frame, by the second time derivative
of the segment angle. Watch out for incorrect huge angular accelerations which can
occur if angle has discontinuities (jumps between+-180, or between +-90 if angle was
computed with a 2-quadrant arctangent function).
3.
Determine X, Z coordinates of distal end relative to segment center of gravity:
DX(t) = Xdist(t)-Xcg(t)
DZ(t) = Zdist(t)-Zcg(t)
If the segment is in contact with the forceplate, then Xcop(t),Zcop(t) should be used in
place of Xdist(t),Zdist(t), when computing DX(t),DZ(t).
4.
Determine X, Z coordinates of proximal end relative to segment center of gravity:
PX(t) = Xprox(t)-Xcg(t)
PZ(t) = Zprox(t)-Zcg(t)
5.
Obtain or calculate force on distal end of segment: Fdx(t), Fdz(t). If the segment is in
contact with the ground, this will be the ground reaction force. Otherwise, the force at
the distal end of the segment is the negative of the force at the proximal end of the more
distal segment. Example: force at distal end of thigh = -force at proximal end of shank.
6.
Calculate X, Z forces on proximal end of segment:
Fpx(t) = max(t) - Fdx(t)
Fpz(t) = maz(t) - Fdz(t) - mg
7.
Calculate the moment about the segment center of gravity, due to proximal and distal
forces. In the 2D case, this is:
Mcgy(t) = - PXFpZ + PZFpX - DXFdZ + DZFdX
The fact that PX , FpZ, and other right-hand-side quantities are functions of time is not
shown, for clarity.
8:
Calculate the proximal joint moment with the equation below. Note that the distal
moment, Mdy, which appears on the right had side of the equation, is the negative of
the proximal moment from the distal segment. The segment touching the ground is
assumed to have zero distal moment about the Y axis.
Mpy(t) = - Mdy(t) - Mcgy(t) + Iyyy(t)