Position, Velocity, and Acceleration Sensors
69
Position, Velocity, and Acceleration Sensors
Analog Position Sensing Methods
Several types of analog sensors are used to measure position, but the most common are
1) potentiometers, and
2) linear variable differential transformers (LVDT's).
Analog sensor systems are frequently characterized by relatively low cost, high resolution,
and simple construction. However, linearity is often a problem, and interfacing to digital
computers requires additional hardware.
Potentiometers
Potentiometers are constructed of electrical resistance elements in either linear or
rotary form. Rotary potentiometers are available in continuous turn, one turn, or multiple
turn configurations. In all types of potentiometers a mechanical motion of the wiper changes
the output voltage in proportion to the wiper displacement.
E
X
+
= out
Several factors are important in determining the
X max E in
relationship between position input and voltage output for a
potentiometer. The output voltage depends on nonlinearities
Xmax
such as zero offset and "ripple". Potentiometer
manufacturers specify resistance "linearity" as a percentage.
Ein
Standard potentiometers have linearities from +1% to +5%,
+
but values from +0.1% to +0.25% are available. Linearity is
X
best expressed as a band centered about the ideal voltageEout
position relationship. Note carefully that the linearity
specified by the manufacturer does not consider zero offsets
which are frequently caused by signal wire resistance, wiper
Figure PVA-1. Potentiometer
resistance, and
amplifier offsets.
Modern potentiometers use conductive
plastics as the resistance element. The
conductive plastic is characterized by infinite
resolution, low electrical noise, and long life. A
wide variety of nominal resistance values are
available with resistance tolerances from +5% to
+20%.
Displacement sensors based on
potentiometers are frequently used to measure
relatively long distances. A thin wire rope
wrapped around a capstan provides a linear
position to rotary displacement conversion.
Special care must be exercised to prevent “slack”
Figure PVA-2. Long-Distance Position in the wire from adversely affecting the position
measurement.
Position, Velocity, and Acceleration Sensors
70
Linear Variable Differential Transformers
Linear variable differential transformers (LVDT's) use the relationship between a movable
ferrous core and a magnetic field to convert mechanical position to an electrical signal. The
LVDT consists of three windings, the primary and two secondaries, surrounding a hollow
core and a separate ferrous core. A high-frequency alternating-current (AC) signal is passed
through the primary windings. Voltages are induced in the secondary windings by this
primary current. The two secondary winding outputs are connected in series opposition so
that the difference in their voltages is output. The voltage induced in each secondary
depends on the position of the ferrous core. As the core moves towards secondary winding
1, voltage V1 increases while V2 in secondary winding 2 decreases. The net effect is an
increase in the output voltage Vout = V1-V2. Conversely, Vout decreases as the core moves
towards secondary winding 2.
Figure PVA-3. LVDT
The relationship between the core position and the AC output voltage is very nearly linear
over the middle 50-60% of the full travel. Linearity for typical LVDT's ranges from +0.1%
to +0.5% of the full scale voltage in this region. Saturation occurs as the core leaves the
vicinity of either secondary winding. The linear region for LVDT's depends on core and
winding lengths and ranges from less than +0.050 inch to more than +4 inches.
The rugged physical construction of the typical LVDT is advantageous in industrial
and automation applications. Other advantages of include
1) simple construction,
2) availability of several ranges of operation,
3) negligible friction or actuating force,
4) non-critical alignment of core and windings,
5) infinite output resolution, and
Position, Velocity, and Acceleration Sensors
71
6) nearly linear relationship between voltage and position,
A disadvantage to LVDT use is the AC output signal, which is difficult to interface directly
to DC control systems. To overcome this problem, LVDT's with integral demodulator
electronics to transform the AC output signal to a DC voltage are available. Integral solid
state oscillators are also available to allow DC input voltages. These two circuits can be
combined to give a convenient LVDT package with DC input and output voltages.
Digital Position Sensing Methods - Encoders
Encoders are a type of displacement sensor which give a direct digital output. Two broad
classes of encoders based on the type of output information supplied are
1. absolute output, and
2. incremental output.
The types of encoders will be discussed before the methods of implementation.
Absolute Encoders
The absolute encoder gives a finite number of unique bit patterns spread uniformly over 1
revolution. An illustrative example of a 3 bit absolute encoder disk is shown in Figure PVA4. Since there are 3 output lines (or bits) and each line can be either "dark" or "light", there
are 23 = 16 unique bit patterns. Therefore the resolution of this particular encoder is
⎛ 360 degrees ⎞ ⎛ 8 patterns ⎞ 45 degrees
⎜
⎟ /⎜
⎟=
⎝ revolution ⎠ ⎝ revolution ⎠
pattern
If we arbitrarily assign the position shown in Figure 3.2 as 0 degrees, there are 8 possible bit
patterns correspond to angular position as shown in Table PVA-1.
0ο
Table PVA-1. Absolute Encoder Bit Patterns
Angular Position
Binary bit pattern
000
22.5 ± 22.5°
001
67.5 ± 22.5°
010
112.5 ± 22.5°
011
157.5 ± 22.5°
100
202.5 ± 22.5°
101
247.5 ± 22.5°
110
292.5 ± 22.5°
111
337.5 ± 22.5°
315ο
45ο
270ο
90ο
225ο
135ο
Figure PVA-4: Absolute Encoder
Note that the most significant bit corresponds to the innermost track, while the least
significant bit corresponds to the outermost track. Of course a 3 bit absolute encoder does
not have enough resolution to be used in most industrial applications, but the concepts
presented above can be readily extended to 8, 10 or 12 bit encoders. The unique bit pattern
corresponding to a given angular position of the absolute encoder is its biggest advantage
over the incremental types. If power is temporarily lost to an absolute encoder, the correct
Position, Velocity, and Acceleration Sensors
72
bit pattern will be present when power is restored. However, each bit of this encoder requires
a separate signal wire, which can lead to a large number of signal wires for a robot with five
or more degrees of freedom. Also, the physical size of the encoder increases as the number
of bits becomes greater since more area is required to differentiate between "dark" and
"light" areas.
Incremental Encoders
The incremental encoder uses a single mask pattern as
B
A
shown in Figure PVA-5. Two stationary sensors (either
optical or brush) are mounted such that one is half
blocked while the other is in the middle of the "light"
area. If the incremental encoder disk is rotated clockwise
, the sensors will produce a output pattern similar to that
shown in Figure PVA-6. The number of positive peaks
in either output voltage depends soley on the number of
"light" areas on the encoder disk. Incremental encoders
are frequently specified by the number of "counts" per
revolution, which corresponds to the number of "light"
(or "dark") areas on the disk. Values such as 100, 150,
200, 360, or up to 2000 counts are common, where
Figure PVA-5. Incremental
encoders with the larger count values are more difficult
Encoder
to manufacture and cost more.
The typical incremental encoder has only four or
five connecting wires, two for input and either two or
three for output. The inputs are the supply voltage and ground for the light sources, while
the outputs are the two detector voltages V1 and V2, and possibly a separate output signal
ground. Note that with only two output signals (Channel A, Channel B) and an infinite
number of possible angular positions for each output pattern, there is no one-to-one
Channel A
"dark"
"light"
0
45
90
135
180
225
270
315
360
Channel B
"dark"
"light"
0
45
90
135 180 225 270 315 360
relationship between the output signals and the absolute position of the incremental encoder.
However, if the encoder
is moved
fromIncremental
an initial position
and
number of output pulses is
Figure
PVA-6.
Encoder
Bitthe
Patterns
counted, then the angular distance from the initial position can be determined (if the number
Position, Velocity, and Acceleration Sensors
73
of counts per revolution is known). The term "incremental" comes from this operating
feature that the distance is measured incrementally from some initial position. Some other
type of sensor must be used (limit switch, proximity switch, or potentiometer) to determine
the “home” position that the incremental encoder will count from.
Note in Figure PVA-6 that Channel A "lags" behind Channel B by 1/4 of a cycle (or
90 “electrical” degrees) if the disk is rotated in the clockwise direction. If the direction of
rotation is reversed, then Channel B would "lag" behind Channel A. By determining which
channel “lag” the other, the direction of rotation can be established. A "positive" rotation
can be defined to be when Channel A lags Channel B, therefore a "negative" rotation occurs
when Channel B lags Channel A. If we initialize our pulse counter to any value at the initial
angular position, then we add pulse counts to the counter value when traveling in the positive
direction. Conversely we subtract pulse counts from the counter value when traveling in the
negative direction. This adding or subtracting of pulse counts can be handled electronically
by an up/down counter which uses the lead/lag relationship of Channels A and B.
There is another feature of incremental encoder outputs that can be used to effectively
increase the encoder count by a factor of 4. If we examine closely the first 90 degrees of
“mechanical” rotation, we note that there are four distinct bit patterns:
Mechanical Angle
0° to 22.5°
22.5° to 45°
45° to 67.5°
67.5° to 90°
Channel A
“light”
“light”
“dark”
“dark”
Channel B
“light”
“dark”
“dark”
“light”
Special electronic hardware has been developed to determine these four additional patterns.
Manufacturers frequently refer to this as “quadrature" output since it effectively increases the
encoder count by a factor of four.
One of the most important factors involved in using incremental encoders is the
maximum size of the count value. If a 16 bit value is used to store the encoder count, then
there are 216 = 65,536 different possible counts ranging from 0 to 65,535. If signed binary is
used then the possible counts range from -32,767 to +32,768. An encoder with 1000 counts
per revolution could turn at most approximately 32.7 revolutions in either direction without
losing the true encoder count. For this reason, incremental encoders are frequently mounted
on the shaft of a motor prior to any gear reductions. This allows a fairly low resolution
encoder (100 to 200 counts) to be reasonably accurate if a large gear reduction ratio is used.
However, accuracy can be lost if there is any backlash, windup, or hysteresis in the gear
reduction system. Almost all motor mounted incremental encoder systems suffer reduced
accuracy and repeatability for this reason.
Incremental encoders provide only relative position information, i.e. position relative
to the place where the counter was initialized. Some external means must be provided to
initialize the encoder counters at the same absolute position each time if overall system
accuracy and repeatability are to be maintained. Some industrial control systems use a set of
Position, Velocity, and Acceleration Sensors
74
limit switches which indicate a "home" or initial position. After powering the actuators, each
is slowly moved until it trips the limit switch corresponding to that actuators's initial position.
When all the joints have reached their initial positions, then all encoder counters can be
initialized. In other industrial systems a potentiometer (or other absolute position measuring
device) is mounted to each axis along with the incremental encoder. The potentiometer is
used to bring the axis into the home position only, where the encoder counters are
initialized. During the actual axis motion operation only the encoders are used and the extra
potentiometers are ignored. Unfortunately, both of these methods of encoder initialization
suffer the same limitations on accuracy and repeatability as using limit switches or
potentiometers for the actual joint measurements.
Another important drawback to incremental encoder systems is the lack of position
information upon power-up after a shutdown. Even if the counter value is maintained during
the power-off time (by a battery backup system), the motion axis could slip during this time
and the counter would not "know" it. Some control systems also power both the encoders
and the counters during a power outage to overcome this deficiency.
Optical Encoding
+V
+V
Optical sensing of the encoder position
removes many of the problems associated
R2
with brush encoders. As shown in Figure
R1
PVA-7, light sources are placed on one side
Vout
of the encoder disk while light detectors are
placed on the other side. Disks are typically
LED
formed from glass with a photoetched
"mask" creating the encoder pattern. The
"light" areas of the disk are essentially
transparent so that light from the appropriate
Phototransistor
source strikes the detector in these regions.
The "dark" areas are essentially opaque such
that no light reaches the detectors in these
Figure PVA-7. Optical Encoder
regions. In many optical encoders the light
source is a light-emitting- diode (LED),
while the detectors are phototransistors or photocells. A typical electrical schematic for an
LED - phototransistor pair is shown in Figure PVA-7. Resistor R1 is placed in the LED
circuit to limit the source current to a typical value of 10-100 milliamps. The light input to
the phototransistor acts much like the base current of a conventional NPN transistor. As the
phototransistor is exposed to more light, additional current is drawn from the +V supply.
This current is passed thru the terminating resistor R2, which produces an output voltage,
Vout. This output voltage can then be compared to a threshold value to determine if a "light"
or "dark" area is present. Light-emitting-diodes and phototransistors that operate in the
infrared light spectrum are frequently used to eliminate possible contamination of the sensed
output by ambient light. The non-contact nature of the sensing device also makes the optical
encoder insensitive to most vibration and wear is almost non-existent.
Analog Velocity Sensing Methods
Position, Velocity, and Acceleration Sensors
75
There are two common types of analog sensors used to measure velocity:
• Linear Velocity Transducers – LVT, and
• DC tachometers.
Both methods used the properties of magnetic
Linear Motion of Magnet, X
fields and wire coils to produce DC voltages
proportional to either linear (LVT) or angular
(tachomter) velocity. Figure PVA-8 shows the
operating principle for an LVT. A magnetized
rod is moved through a coil of wire as shown.
Coil of
The magnetic field induces a voltage in the coil
Wire
of wire that is proportional to the linear velocity.
Vout ∝
A DC tachometer works in a similar fashion to
the LVT, except
• the magnet is fixed (“stator”)
• a coil of wire rotates inside the magnet,
which
Figure PVA-8. Linear Velocity
• produces a voltage proportional to the
Transducer
angular velocity.
Note that a DC motor works similarly to a DC tachometer, but
• voltage/current is input to the wire coil, and
• velocity/torque is output from the motor.
dX
dt
Digital Velocity Sensing Methods
Digital velocity sensing methods are all timer-based methods. The definition of linear
velocity is given by
dX ΔX
v = lim
≈
Δt
dt → 0 dt
The basic idea behind all timer based methods is to fix either ΔX or Δt, then measure the
other to determine velocity. Each of these two different methods for determining require the
accurate measurement of time. This is typically done by simply counting the number of
“off” to “on” transitions of a “clock” signal at a known frequency (think of a square wave
function generator). Figure PVA-9 shows an example of a “clock” signal operating at 1000
Hz. Any time interval between two events (T1 and T2) can be determined with a resolution
of 0.001 sec ( = 1/1000) by counting the number of pulses that have occurred (Δt = 0.011 sec
in Figure PVA-9). Note that there is always some uncertainty in this process because the
exact amount of time between T1 and the first “count” (and between T1 and the last “count”)
is unknown.
Position, Velocity, and Acceleration Sensors
76
“Clock” at 1000 Hz
1
2
3
4
5
6
7
8
T1
9
10 11
T2
Figure PVA-9. Counting “Clock” Signals to Determine T2-T1
“Instantaneous” Velocity Timer Method
“lobe”
In this method only one counter is used to count events
counting
between any two events. An example problem for
sensor
measuring “instantaneous” angular velocity is described
below. The rotating wheel of Figure PVA-10 has 8 “lobes”
that can be counted as they pass a fixed point (sensing of the
lobes is typically done optically like Figure PVA-7 or by
8 “lobes”on rotating wheel
proximity sensing techniques). The timing counter begins
Figure PVA-10. Angular
when the first “lobe” passes the sensor. The timing counter
Velocity Measurement
stops when the next “lobe” passes the sensor (on a low-tohigh transition). The final count of the timing counter can
then be used to determine the instantaneous velocity between the two lobes.
“Clock” at 100,000 Hz
0
1
2
3
4
5
6
7
8
…..
234 235 ……..
Timing counter
“Lobe” sensor output
start
stop
Figure PVA-11. Instantaneous Velocity Timer Method
For the example shown above in Figure PVA-11 the average angular velocity during the 1
second period would be
1 / 8rev 100,000 clocks
ω=
*
k clocks
sec
1 / 8rev
100,000 clocks 53.2 rev
ω=
*
~ 3190 RPM
≈
235 clocks
sec
sec
“Average” Velocity Timer Method
In this method a counter is used to count events for a fixed time interval. The fixed time
interval is set by also counting a known clock signal for a fixed number of counts. An
example problem for measuring angular velocity is described below. The same rotating
Position, Velocity, and Acceleration Sensors
77
wheel of Figure PVA-10 with 8 “lobes” is used again for this example. As shown below in
Figure PVA-12, an arbitrarily selected timing interval of 1 second is used, therefore the
timing counter determines when 1000 of the 1 kHz clock signals have occurred. The lobe
counter starts and stops counting (low to high transitions) at the same times as the timing
counter.
“Clock” at 1000 Hz
0
1
2
3
4
5
6
7
8
…..
997 998 999 1000
Timing counter
T1
1
2
3
T2
….. 420
421
“Lobe” counter
Figure PVA-12. Average Velocity Timer Method
The final count of the lobe counter can then be used to determine the average velocity during
the 1 second period. For the example shown above in Figure PVA-11 the average angular
velocity during the 1 second period would be
N lobes 1 rev
421 lobes 1 rev 60 sec
ω=
*
→ ω=
*
*
~ 3160 RPM
1 sec
8 lobes
1 sec
8 lobes 1 min
Figure PVA-13 below shows examples of handheld “phototachs” that use the average
velocity timer method to measure the angular velocity of a rotating shaft.
Figure PVA-12. Average Velocity Timer Method