Characterization of a Series Viscous Actuator For
Use in Rehabilitative Robotics
by
Nicholas Eric Wiltsie
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
OF TECHNOLOcjY
Bachelor of Science in Mechanical Engineering
O THL
JUN 3 0 2010
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
LIBRARIES
June 2010
© Massachusetts Institute of Technology 2010. All rights reserved.
Author .............................................
Department of Mechanical Engineering
May 10, 2010
Certified by............
Neville Hogan
Professor of Mechanical Engineering
Thesis Supervisor
Accepted by .......
ohn H. Lienhard V
C'oillis rofessor of Mechanical Engineering
Chairman, Undergraduate Thesis Committee
2
Characterization of a Series Viscous Actuator For Use in
Rehabilitative Robotics
by
Nicholas Eric Wiltsie
Submitted to the Department of Mechanical Engineering
on May 10, 2010, in partial fulfillment of the
requirements for the degree of
Bachelor of Science in Mechanical Engineering
Abstract
In this thesis, a series viscous actuator, composed of a viscous rotary damper connected to the output of a geared electric motor, is proposed as a novel solution to the
high actuator impedances encountered when using such a motor. High impedances
can be undesirable when interacting with the environment or animal subjects as unexpected forces may damage already-compromised joints. A state space model for the
series viscous actuator was derived and verified by measurements upon a physical system. Physical design considerations for implementing a series viscous actuator, such
as the appropriate gearing ratio and damper selection, are discussed and supported
using the derived model.
Thesis Supervisor: Neville Hogan
Title: Professor of Mechanical Engineering
Acknowledgments
The author would like to thank Yun-Seong Song and Neville Hogan for their invaluable
help in completing this project.
6
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.1
SVA State Equations . . . . . . . . . . . . . . . . . . . . . . .
8
2.2
System Dynamics . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.3
SVA Impedance . . . . . . . . . . . . . . . . . . . . . . . . . .
13
Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.1
SVA Design . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.2
Damper Characterization . . . . . . . . . . . . . . . . . . . . .
14
3.3
System Dynamics Verification . . . . . . . . . . . . . . . . . .
15
3.4
SVA Impedance Measurement . . . . . . . . . . . . . . . . . .
16
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
4.1
SVA Design . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
4.2
Damper Characterization . . . . . . . . . . . . . . . . . . . . .
18
4.3
System Verification . . . . . . . . . . . . . . . . . . . . . . . .
19
4.4
SVA Impedance Measurement . . . . . . . . . . . . . . . . . .
20
5
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
3
4
1
Introduction
The need for low-impedance actuators when interacting with humans or animals
has recently been emerging.
Particularly in the field of rehabilitative robotics, a
high output-impedance actuator can be dangerous if it interferes with the subject's
natural movement; due to inadequate muscle tone, a joint such as the shoulder may
be particularly vulnerable to unexpected resistance. However, most actuators with
high enough torque output usually suffer from high output impedance. For example,
the gearbox necessary to transform the high-speed low-torque output of an ordinary
DC motor into a low-speed high-torque output increases the motor inertia as the
square of the gear ratio, in addition to introducing significant friction. These sum to
a high-impedance output that is difficult to back-drive.
Previous work on this problem has resulted in the development of series elastic
actuators (SEA) consisting of an elastic spring in series with the high-impedance
actuator. In exchange for altered dynamics and increased complexity, the output force
is delivered through the low-impedance spring. However, SEAs have limited travel and
displacement-dependent impedance. As an alternative, series viscous actuators (SVA)
have been proposed. Previous work on this project has resulted in an experimental
setup consisting of a brushed DC motor in series with a gearbox and a viscous damper,
with the far end of the damper connecting to a static torque gauge. A computer with
data acquisition setup commands the motor, reads and logs the outcome. This system
was characterized using linear models, and the expected torque outputs resulting from
ramp and step voltage inputs were compared with measured results on the physical
system.
2
2.1
Background
SVA State Equations
The physical SVA system is designed as shown in Figure (1), and the equivalent
impedance model is shown in Figure (2); it consists of a DC motor in series with a
SVA
Vin
Motor
B1
Gearbox
B2
Figure 1: System diagram of a loaded SVA
SVA
Motor
R
L
0
Gearbox
Figure 2: Equivalent impedance diagram of a loaded SVA
gearbox connects to a load through a rotary damper, B1. The motor is modeled as
a resistance R in series with an electrical-to-mechanical transformer. The motor and
gearbox have rotary inertias Jm and Jg, respectively. The load is modeled as a rotary
intertia J in series with a second rotary damper, B2, fixed to the ground.
The derivation of the system model begins with the constitutive equations
r. = KT -1
Vemf = KT
w,
ET = Jo
T
= B(Aw)
(1)
(2)
(3)
(4)
Equations (1) and (2) relate the electrical inputs of the motor to the mechanical
outputs through the motor constant KT, while (3) and (4) are the characteristic
equations of a rotating mass and a rotary damper.
The angular velocities of the gearbox output, wi, and the load, W2 , are used as
the two state variables of the system, with the input voltage Vm as the driving input.
The state equations are obtained via manipulations of the free body diagrams of the
following subsystems:
TI
O-X
---
-NT
WBia
Figure 3: Subsystems of loaded SVA
The first state equation is the torque balance of the fourth subsystem,
JLL
2 =
B1(w2 -
-B22-
W1)
(5)
The other state equation requires the inclusion of the motor dynamics,
Vn = Vemf + RI = KTwm + RTm
Kt
(6)
as well as the relationship between the internal variables on either side of the gearbox,
T2
Wm
~
1
W1
=
9W2
Combining these with the torque balance of the remaining two subsystems and elim-
inating the internal variables
and Wm reveals the final state equation:
Ti, T2, Tm,
Tm -
ri = JmAJm
T2
Tm-
9
72 = grm
B1(wi
T2 =
Jgu)1
2
Jm9 C)i
-
- w 2 ) = Jdi
+ B1(wi
Jmg 2Ci91 = JcAi
g7m -
rm
T2-
Jmgw1
=
--
=
+
JgCA1
Jmg)1N
+ B 1 (wi - W2 )
B1
+ B(1 -
9
'rm
KT
Vin -
R
C')1(Jg
+
gK
R
g 2 Jm)
R
KT 2
Vin
-
W1
2
w
WM
R
Vin -
=
1
W2)
9
Kr
=-
w2 )
-
= Jmgc 1 +
B1(wi -
W2)
JLA1
g
B1
+ B,(Wi g
(gKW) 2
gKT
R
R
W2)
(7)
Converting the two state equations (5) and (7) into the canonical c = Ax + Bu
state space form and making the substitutions of Jd
=
Jg+g 2 Jm and Beff = (gKT) 2 /R
results in
X=
W2
Vin
-B1-Beff
B
Ja
Jd
Bi
J,
-BJ-BJ
J,
gK
JdR
0
]
1
2.2
System Dynamics
With the system model thus derived, the dynamics of any variable of the system ("r")
can be derived in the Laplace domain using the state space equations
(8)
r = Cx
C(sI - A)-'Bu
As the potentially desired results of these calculations are the dynamics of the gearbox
output and the load shaft, as well as the torque between the SVA and the load, the
transfer functions can be found as
Wi = [1 O]x
W2 = [0 1]x
r = [B2 - B 2 |X
in
Un
W2
Vin
T
in
(B 1 + B 2 + sJ,)gKT/R
s (JdJl)+ s(Ji(Bi + Bef) + Jd(B1 + B 2 ))+ (B 1 B 2 + (B1 + B 2 )Beff)
(10)
BlgKT/R
S ( JdJI) + s(J(Bi + Beff) + Jd(B1 + B 2 )) + (B 1 B 2 + (B1 + B 2 )Be55)
(11)
(B 2 + Jls)BlgKT/R
s (JdJl)+ s(J(Bi + Be55) + Jd(B1 + B 2 ))+ (B 1 B 2 + (B 1 + B 2 )Beff)
(12)
2
_
2
2
Using the final value theorem, commonly expressed as
lim f(t) = lim sF(s)u(s)
t-++oo
s40
(13)
the steady state values of these variables in response to a step voltage input are
wi(t = oo)
=
W2 (t = oo)
=
(B 1 + B 2 )gKT/R
B1 B 2 + (B1 + B 2 )Beff
BlgKT/R
B1 B 2 + (B1 + B 2 )Beff
(14)
(15)
B1 B 2 g K / R
-r(t= oo) =
B
29tR(16)
B 1 B 2 + (B 1 + B 2 )Beff
As the torque on either side of the damper is the same, the power efficiency of the
SVA as compared to the geared motor is exactly the ratio of the output shaft speed
to the SVA speed:
17
2.3
=
TW2
TWi
(t = oo) =-
W2
W
(t =
B
00)=
B1
B
+
(17)
B2
SVA Impedance
The rotary impedance of a system is defined as the relationship between applied
torque and angular velocity. The torque transmitted from the SVA is at all times
defined by Equation (4), and the dynamics of the angular velocities were derived in
the previous section. Assuming that the SVA input Vm was shorted, simulating a
command of 0 volts, Equation (7) can be rearranged into
JdcAti
+ (B 1 + Beff)wi = Biw2
(18)
Translated into the Laplace domain, this becomes
B1
W1- =,
Be55(19)
s Jd+ B
W2
_1
(19)
From this, the SVA impedance can be found as
= B 1 (w 2 - W 1 )
S
ZSVA
-
_____(20)_
W2
ZsvA =
sB1Jd
W
+ B1Beff
B
f(21)
(20)
3
Experimental Design
3.1
SVA Design
In order to evaluate the characteristics of the system and compare them to the
expected dynamics found in Equation (11), three experiments were designed: one
to measure the steady-state response of a loaded SVA system, one to measure the
impedance of the back-driven SVA, and one to accurately characterize the behavior
of the dampers used in the experiment.
The SVA was designed on a modular platform, allowing multiple loads to be
attached to the output shaft and different measurements to be taken.
The SVA
itself consists of a Maxon A-max 114813 brushed DC motor in series with a Maxon
110483 64:1 spur gearbox mounted horizontally on an ABS platform. A rigid shaft
and adapter plate couple the gearbox output shaft and the body of an Ace G2 rotary
damper. The damper rotor acts as the output of the SVA. An Accu-Coder 15T-02-SA
rotary encoder records the angle subtended by the gearbox shaft.
Three grades of Ace G2 dampers were used in the SVA, with torque codes of
200, 600, and 101. These dampers are referred to as "low," "medium," and "high"
respectively, and are noted as such in the subscript of the damper labels; for example,
the 200-grade damper used in the SVA is referred to as B1L while the 101-grade used
as the load is referred to as B2H.
3.2
Damper Characterization
SVA
Encoder
+
Motor Gearbox
Bi Torque Gage
Figure 4: System model for measuring individual damper characteristics
To control for variability in the manufacturing process, the same dampers were
used in each experiment.
To accurately characterize the relationship between the
torque and speed across these specific dampers, a Futek TFF325 torque gage was
attached between the SVA output and a rigid surface. A computer controller sent
a stair voltage waveform of 0, 4, 7, and 10 volts through an Apex PA16A power
amplifier circuit to drive the motor, producing a sequence of steady-state angular
velocities across the damper currently mounted on the adapter plate. The turn angles
measured by the encoder and the voltage produced by the torque gage were logged
to a computer with a 2 kHz sampling rate.
The numerical derivative of the shaft angle as a function of time was computed
from the recorded data to find the angular velocity of the damper. The torque voltage
signal was initially very noisy, and was cleaned using a median filter with a window
of 20 following samples. The voltage change induced by each successive applied speed
was then converted to a torque value using a table of calibration values provided by
the manufacturer.
A least-squares linear fit was performed on the collected values to obtain a single
characteristic ratio between torque and angular velocity for each damper.
3.3
System Dynamics Verification
To measure the steady state response of a loaded SVA system, a load consisting
of a rigid shaft and a second damper was coupled to the output of the SVA, as
shown in Figure (1). The casing of the load damper was held stationary, producing
a load impedance proportional to the output shaft speed. A second rotary encoder
measured the turn angle of the output shaft. A step input of 10 volts was applied
to the motor for several seconds to allow any transient response to die away, leaving
only the steady-state angular velocities of the gearbox and output shaft. The turn
angles of the gearbox shaft and the output shaft were logged using a computer and
differentiated to produce angular velocities as a function of input voltage. The ratio
between these was computed and compared to the expected value derived from the
measurements of the dampers and Equation (17).
3.4
SVA Impedance Measurement
Figure 5: Setup to measure impedance of SVA.
To measure the impedance of the SVA, a second motor and gearbox set were
coupled to the SVA damper as shown in Figure (5).
The terminals of the SVA
motor were shorted to simulate a command input of 0 volts. The second motor was
connected to a BK Precision 1672 bench-top power supply and driven with 5, 10, 15,
20, 25, and 30 volts. At each step, an Amprobe 37XR-A multimeter recorded the
current passing through the second motor while the encoder recorded the turn angle
of the shaft. This experiment was repeated with each of the three SVA dampers.
To measure the static frictional torque within the second motor, the current
through the motor was decreased until the output shaft stopped turning. This threshold current at this operating point was recorded.
The torque applied to the SVA system was found from the recorded current as
shown in Equation (1). The angular velocity of the load shaft was found from the
encoder data as shown in Section (3.2).
The impedance of the SVA system as a
function of damper type and shaft speed was found as the ratio of the applied torque
and the angular velocity, as given in Equation (20).
4
4.1
Results
SVA Design
Table 1: System constants found from component datasheets.
g
KT
R
837/13
0.049 N-m/A
120
Relationship Between Torque and rpm
73'F (23'Q
in m (ion)
200
2A
160
120
1.6
so
0.8
40
Figure 6: Damper torque-speed curves as given by manufacturer. The 200, 600, and
101 grades were used in this experiment.
4.2
Damper Characterization
Table 2: Damping coefficients for SVA and load dampers as a function of applied
voltage (mN.m.s/rad).
4V
7V
10V
B1H
7.90
5.38
4.45
BiM
4.67
3.51
2.94
B1L
1.54
1.17
1.05
B2H
7.60
6.22
5.00
B2M
4.22
3.56
2.90
B2L
0.95
1.32
0.89
Rotor angular velocity (RPM)
Figure 7: Torque-speed curve for B1H damper with least-square fit.
Table 3: Least-squares fit damping coefficients from Table (2) (mN-m-s/rad).
B1H
4.73
B1M
3.14
B1L
1.10
B2H
5.40
B2M 3.11
B2L
4.3
1.01
System Verification
Table 4: Measured w2/wi for SVA damper B1 and load damper B2.
B2H
B2M
B2L
B1H
0.462
0.619
0.817
B1M
0.271
0.553
0.744
B1L
0.037
0.021
0.209
Table 5: Expected W2/wi derived from Table (3) and Equation (17).
B2H
B2M
B2L
B1H
0.467
0.603
0.824
B1M
0.368
0.502
0.757
B1L
0.169
0.261
0.521
Table 6: Ratio of measured u 2 and expected W2 .
4.4
B2H
B2M
B2L
B1H
0.989
1.027
0.992
BlM
0.736
1.102
0.983
B1L
0.219
0.080
0.056
SVA Impedance Measurement
_
SVAH Meastred
*
SVAM Measured
SAL Measured
20 |-
_B... SVAH Expected
- SVAM Expected
-
151
SVAL Expected
10 .
-D--------
01
1
2
3
4
5
6
7
8
9
Shalt Speed (radis)
Figure 8: SVA impedance as a function of speed and damper type.
5 -
4.--
-
--
- -
4.5 .-
0
SYAM Measured
0-
SYA Measured
L
SVAH Expected
--
-
SYAM Expected
4-
E
-5--
a)
C
3
- - - --- - - - -
8
SVAL Expected
---
- - - - e--
- - - - -o0-
-
2.5 C
2
0.5
1
2
3
4
5
6
7
8
Shaft Speed (radis)
Figure 9: SVA impedance as a function of speed and damper type corrected for static
friction torque.
5
Discussion
The damper characterization measurement produced results within a few percent of
the manufacturer's data sheet. It is important to note that the expected range of
operation should be considered when linearizing the damper, as the decreasing slope
of the damper's measured torque-speed curve shown in Figure (7) cannot be perfectly
modeled by a straight line.
The entries in the upper right of Table (6) reveal the that a medium or heavy
SVA damper driving a load of equal or lesser damping closely matches the expected
response from the model. A medium SVA damper driving a heavy load is poorly
approximated by the model, and a light SVA damper behaves completely differently
than expected. This is likely due to the non-modeling of dry or Coulomb friction
within the system; friction has a significantly greater effect in low-torque, low-speed
configurations as it is a larger percentage of the overall torque.
It was observed during the SVA impedance measurements that the gearbox output
shaft did not rotate for anything except a 30 volt excitation and the heaviest SVA
damper.
From the point of view of the SVA damper, it acted as a fixed shaft,
simplifying the system impedance to simply that of the damper itself. This result
clearly follows from Equation (21) - the motor was shorted, reducing R to the internal
winding resistance of the motor. Inserting the system values from Table (1) into this
equation reveals an expected steady-state SVA impedance of the SVA within 1% of
that of the damper itself.
As a check whether this high motor impedance was caused by a high electromechanical impedance or merely static friction within the SVA, the experiment was
repeated with the short between the SVA motor terminals removed. It was observed
that the gearbox shaft began to spin under lighter loads, indicating that a significant
reduction of the SVA impedance.
The derivation of Equation (21) neglected any
loss mechanisms other than the motor resistance, so it predicts that the effective
steady-state impedance will drop to zero as R goes to infinity. While this is clearly a
deficiency of the model, the trend of reduced impedance is correctly predicted.
The primary unmodeled loss mechanism is is static friction within the SVA, most
likely within the motor gearbox. The gearbox used in these experiments made use
of spur gears, which are traditionally less efficient than other gearbox configurations.
The effect of static friction can clearly be seen in the SVA impedance measurements
displayed in Figure (8). The expected steady-state impedance is independent of the
output shaft speed, but the measured impedances of three SVA configurations at low
speeds range from 5 to 20 times too large. The measured impedances asymptotically
approach the expected values, dropping off like the inverse of the shaft speed. This
behavior can be explained by a constant frictional torque resisting the shaft motion.
Figure (9) shows the same data with the measured static torque of the gearbox
subtracted, and shows the greatly decreased dependence of impedance upon angular
velocity.
Taking all of this into consideration, the general design procedure for an SVA
should begin with an appropriate model of the system to be driven, including the
load damping, the desired torques and operating speeds, and the desired SVA output
impedance. Equation (21) and Figure (9) indicate that the overall system impedance
is effectively the impedance of the SVA damper, so the damper should be chosen to
match this desired output impedance. Once this value is determined, the necessary
motor power can be chosen by back-calculation from the efficiency of the SVA given in
Equation (17). The appropriate gear ratio and motor constant KT can then be chosen
from the desired steady-state torque or angular velocity values given in Equations (15)
and (16).
The deviations from the expected behavior resulting from very low choices of SVA
dampers shown in Table (6) result from mis-modeling of the useful torque within
the system. This problem may be addressed by either improving the quality of the
gearbox or carefully modeling the damper and using encoders to measure wi and
W2.
By increasing the gearbox quality, the losses to static friction can be largely
reduced. This will result in a transition similar to that from Figure (8) to Figure (9)
and bring the low-speed impedance SVA impedance closer to the expected value.
Harmonic drives are one such way to reduce the gearbox friction. Alternatively, by
characterizing the damper well and measuring the speed on either side, the torque
delivered to the load can be calculated and controlled using a feedback loop. Allowing
such low output impedances will allow lighter, smaller, and cheaper motors augmented
with high-ratio gearboxes to meet the desired specifications.
6
Conclusion
The series viscous actuator configuration was found to be an acceptable means of
reducing the effective impedance of a high-impedance actuator. The dynamic model
derived for this research accurately predicted the physical response and of the system
in configurations where the SVA damping matched or exceeding the damping of the
driven load. The largest discrepancies between the model and the physical system
concerned static and dynamic friction, which were particularly prevalent in low-speed
and low-torque configurations.
Future work should focus on adapting the the adaptation of this system for use
in rehabilitative robotics. As the conditions that produce significant deviations from
the model coincide with the optimal conditions for use of an SVA - low speed and
low output impedance - methods to reduce the losses to static friction should be
investigated. Completely novel solutions may produce the best result - for example,
the motor itself might be used as the vane of the damper in order to minimize the
space usage of the SVA.