16 August 2016
of change of velocity (speed), so we need a circuit which
converts the rate of change to velocity. Likewise, velocity
is the rate of change of distance, so we need a similar
circuit to convert this rate of change of distance to distance
There is a mathematical process called integration
which will do the job. Basically, integration takes a rate
signal and adds it up over time to determine a result. This is
like a gasoline pump meter at the local service station. The
gasoline is pumped at a certain rate (gallons per minute)
and the meter “accumulates” this flow rate over the time
you are pumping to determine the number of gallons
you have pumped. An electronic integrator is built using
an operational amplifier (op-amp) with a capacitor in the
feedback loop instead of a resistor.
To obtain distance traveled from acceleration, we need
a double integrator as shown in Figure 13. The position
output of Figure 13 represents the total distance traveled
over the time interval of concern, and the velocity output
represents the speed at the present time. The outputs
from the accelerometer sensor and the double integrator
circuit are analog (continuous) signals, so to be used by
a microcontroller or PC these signals must be sent to an
analog-to-digital converter (ADC) to create the digital
representation of speed and position.
By starting the distance calculation from a known
position, we can determine our location after a period
of time. It’s kind of like the old pirate’s map instruction:
Starting at the coconut tree, walk 10 paces due east.
Thus, we have developed a means of determining
distance traveled for a vehicle using a one-dimensional
(1-D) motion mode of operation. However, in the real
world, we need two-dimensional (2-D) motion information
for surface vehicles, and three-dimensional (3-D) motion
for aerial vehicles. This is where things start to get really
complicated really fast.
In 1-D motion, the orientation of the acceleration
never changes. It is either forward or backward but never
side-to-side or up-down, so it is very easy to keep the
accelerometer’s sensing axis aligned with the direction of
travel. In 2-D and 3-D motion, we need a means of keeping
the accelerometer sensors aligned with their original
orientations or the distances traveled are meaningless.
From our pirate’s map example: Now, take 10 paces due
south and climb 10 feet up the palm tree. If I had not
specified the “orienting” directions, we would never be
able to find the buried treasure.
Figure 14 shows the 3-D Cartesian coordinate system
(or reference frame) in which X represents side-to-side
motion; Y represents backward and forward motion; and
Z represents up and down motion of our vehicle. A simple
solution is to mount three accelerometers on our vehicle,
n FIGURE 12. Capacitive MEMS Accelerometer Operating
n FIGURE 14. Three-Dimensional Acceleration.
x represents side-to-side motion
y represents backward and forward motion
z represents up and down motion
n FIGURE 13. Calculating Position Using an Accelerometer.
∫f(x) dx means to integrate the f(x) [y-axis function values
related to the x-axis values - basically finds the area under
the f(x) curve along the x-axis]