In the earlier section on polar coordinates, we showed how linear and angular motion are related. When a rigid body undergoes angular rotation, it is possible to calculate linear velocity and acceleration from the known angular velocity and acceleration. Consider the
r
Now, attach to the body a 2-D reference frame with its normal axis, is at right angles to the curvature of the path, whereas the tangential axis is at a tangent
the body, this reference frame will rotate with the body so that the normal and tangential axes change their orientations within the GCS. The angular motion of the body is described by its angular velocity (") and angular acceleration (&) within the GCS. Note that such a system could represent a human body segment such as the thigh, knee joint, and r the length of the thigh.
-ence system can be calculated from the equation vt = r", where vt is the tangential velocity (i.e., in the direction of the tangential axis). Note that the normal velocity (vn) is zero, because the length r is a constant (because points
FROM THE SCIENTIFIC LITERATURE
Pezzack, J.C., R.W. Norman, and D.A. Winter. 1977. An assessment of derivative determining techniques used for motion analysis. Journal of Biomechanics 10:377-82.
This paper uses both direct kinematic measures of angle and angular acceleration and indirect kinematics of markers to assess two methods of smoothing digitized marker kinemat-ics, particularly angular acceleration. First, data from an aluminum arm that could be rotated only in the horizontal from above while simultaneously measuring its angular position with a potentiometer (see appendix C) and its linear acceleration with a uniaxial accelerometer. The aluminum arm was manually moved in several different ways. The accelerometer was mounted so that it measured transverse acceleration (a ), which was then converted to angular acceleration by dividing by the distance from the arm’s center of rotation to the accelerometer (& = at . Next, the
arm were computed. Then, the angular acceleration of the arm was computed in three different ways—without data smoothing, after smoothing with Chebyshev least-squares
the three methods of computing angular acceleration.
First, the unsmoothed accelerations were very noisy, although the waveform, on average, did follow the signal derived from the direct measure of acceleration. Second,
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1.20 1.60 2.00 2.40 2.80
Time (s)
1.20 1.60 2.00 2.40 2.80
Time (s)
1.20 1.60 2.00 2.40 2.80
Time (s)
1.20 1.60 2.00 2.40 2.80
Theta (rad)
Theta vs time Alpha vs time
Alpha vs time Alpha vs time
Raw film data
Figure 1.25 (a) Angular displacement of a humanly moved aluminum arm. (b) Angular acceleration of the arm after differentiation without data smoothing. (c) After least-squares curve fitting and differentiation. (d) After digital filtering and differentiation.
Journal of Biomechanics
placement signal did not follow the accelerometer signal polynomial to even a relatively simple human motion was not suitable. Third, after two time derivatives, the digitally measured by the accelerometer. Some attenuation of the signal occurred at the peaks, but these could have been corrected by increasing the cutoff frequency. See the fol-lowing section and chapter 12 for more details on the use
The importance of this paper cannot be overestimated.
In addition to successfully reducing the effects of high-data from one of their trials so that other researchers could evaluate their data-smoothing techniques (Lanshammer shown to have acceptable smoothing capabilities. Wood used optimally regularized Fourier series to appropriately smooth human motion data.
also undergo tangential acceleration, computed from the equation at = r&. This acceleration will be nonzero if the body increases or decreases its angular velocity. If the body rotates at a constant angular speed, the angular and tangential accelerations will be zero.
Curiously, although the normal velocity is zero for circular motions, there must always be a nonzero normal acceleration (an) if the body is traveling in a circular path. The normal acceleration is calculated as an = r"2. This acceleration, also called centripetal acceleration, is caused by the continual change of direction of the point -for more detailed consideration of these accelerations.
The normal and tangential velocities and accelerations rotating body but can be converted to the GCS using knowledge of the body’s angular orientation and simple
# is the angle formed by the tangential velocity and the right-horizontal, which is parallel to the X-axis in the GCS.
The vector can be resolved into the components vx and vy, corresponding to the principal directions of the GCS, using the equations vx = vt cos # and vy = vt sin #. Similar transformations can be applied to the accelerations an and at to express them in the GCS, as well.
Electrogoniometers
goniometer is a manual device for measuring joint a). It is essentially a protractor with
rotates to measure angles. To measure joint angles electronically during motion, electrogoniometers are used. Typically, they are much less expensive than imag-ing systems and allow data to be collected and viewed immediately. Unfortunately, these devices do encumber movement because various electronics must be attached to the subject and most systems require cables to be run to a data collection system.
The most common type of electrogoniometer uses a potentiometer constant voltage is applied across its terminals, and a wiper that turns with the potentiometer taps off voltage in an amount proportional to the amount of the turn (see
c
to one segment of the joint and the other to the adjacent potentiometer to rotate and therefore change its output -ers, polarized-light photography, strain gauges, and
act as pure hinges; any translational motion of the joint creates an erroneous angular rotation of the electrogoni-ometer. In principle, designing self-aligning mechanisms
b
electrogoniometer system developed by Hannah and
E5144/Robertson/fig1.26a-b/414858,415067/alw/r1-pulled
Figure 1.26 (a) Angular motion (!, ", and &) of a rigid body produces linear motion of points attached to it.
Point P is located at a distance r from fixed point Q. As the body rotates through the angle !, P scribes an arc such that its (x, y) coordinates are constantly changing. (b) LCS attached at P allows conversion from angular motion of the body to linear motion of the point P. The tangential axis (T) of the LCS describes a tangent to the arc scribed by P, whereas the orthogonal axis N is directed toward the axis of rotation point Q. Angle ! is the angle between the tangential axis and the right-horizontal and can be used to convert the tangential velocity (vt
= r!) into its x and y components.
triaxial angular motion of the ankle, knee, and hip of this system was clinically valuable for some patients, it is not as useful for examining people with severe
dis-they only measure joint angles. This prevents them from recording absolute motion of the segments with respect to a Newtonian frame of reference (in other words, a GCS), which is necessary for performing inverse dynamics analyses (see chapter 5). However, this type of system is useful in clinical environments where immediate joint kinematic information is required.
When selecting or building an electrogoniometer, use a sensing element that is continually differentiable.
For example, as the wiper of most inexpensive potenti-ometers moves, it jumps from one loop of a coil of wire to another to vary its resistance. This contaminates the output position data with discontinuity spikes that dis-rupt the calculation of derivatives for obtaining angular velocity or acceleration. To eliminate these discontinuous steps, more expensive potentiometers have a continuous strip of conductive material. There is usually a break at one spot along the strip, and this portion must be placed
c).
transmits less light as the bend in the cable increases, allowing continuous measurement of the amount of joint -to quantify the degree of bending in a steel wire—and For example, a transducer that crosses the elbow can simultaneously measure elbow flexion and forearm supination. Yet another solution is to use a polarized-light goniometer, which uses two sensors that are sensitive to
-tions of the sensors are determined by the orientation of each sensor within the polarized light plane.
Calibration of electrogoniometers is relatively straightforward. If the electrogoniometer design allows, it can be attached directly to a manual goniometer.
Moving the manual goniometer from one known posi-tion to another while recording data from the electrogo-niometer will provide a voltage measure equivalent to an actual angular displacement. From these
measure -trogoniometer cannot be directly attached to a manual goniometer, a similar procedure can be used while the electrogoniometer is attached to the joint of interest.
E5144/Robertson/fig1.27a-c/414857,415084,415085/alw/r2-pulled Potentiometer
Four-bar link
Wiper
Wiper
Battery or DC power
Output signal
a b
c
Figure 1.27 (a) Manual goniometer, (b) electrogoniometer, and (c) schematic of a potentiometer and elec-trogoniometer circuit.