KINEMATICS - Biomechanics and Motor Control of Human Movement - page 77

3.5.2 Joint Angles
Each joint has a convention for describing its magnitude and polarity. For
example, when the knee is fully extended, it is described as 0 ◦ flexion, and
when the leg moves in a posterior direction relative to the thigh, the knee is
said to be in flexion. In terms of the absolute angles described previously,

knee angle = θ k = θ 21 − θ 43

If θ 21 > θ 43 , the knee is flexed; if θ 21 < θ 43 , the knee is extended.
The convention for the ankle is slightly different in that 90 ◦ between the leg
and the foot is boundary between plantarflexion and dorsiflexion. Therefore,

90 ◦

ankle angle = θ a = θ 43 − θ 65 +

If θ a is positive, the foot is plantarflexed; if θ a is negative, the foot is
dorsiflexed.

3.5.3 Velocities — Linear and Angular
As was seen in Section 3.4.3, there can be severe problems associated with
the determination of velocity and acceleration information. For the reasons
outlined, we will assume that the raw displacement data have been suitably
smoothed by digital filtering and we have a set of smoothed coordinates and
angles to operate upon. To calculate the velocity from displacement data, all
that is needed is to take the finite difference. For example, to determine the
velocity in the x direction, we calculate x /t , where x = x i + 1 − x i , and

t is the time between adjacent samples x i + 1 and x i .
The velocity calculated this way does not represent the velocity at either of
the sample times. Rather, it represents the velocity of a point in time halfway
between the two samples. This can result in errors later on when we try to
relate the velocity-derived information to displacement data, and both results
do not occur at the same point in time. A way around this problem is to
calculate the velocity and accelerations on the basis of 2 t rather than t .
Thus, the velocity at the i th sample is:

x i + 1 − x i − 1

2 t

Vx i = m/s (3.15)

Note that the velocity is at a point halfway between the two samples, as
depicted in Figure 3.24. The assumption is that the line joining x i − 1 to x i + 1
has the same slope as the line drawn tangent to the curve at x i .
For angular velocities, the formula is the same except that we use angular
data rather than displacement data in Equation (3.14); the angular acceleration