Recent progress in the development of a gravity-sensor-based instrument for determining the elevation angle of DSN antennas is described. The benefits of such a system include the capability to locate the Gravity Referenced Elevation Encoder (GREE) directly on the primary reflector (thus bypassing structural flexure and deformation error sources), anticipated lower maintenance costs compared to the present gimbal encoders, direct replaceability, or supplementation of the present gimbal encoders and the utilization of off-the-shelf components to construct the GREE. This article includes a description of the nominal GREE design. Test results on a laboratory breadboard model are given. Rigid-body dynamics of the GREE are derived and the simulated performance in response to measured antenna vibrations is given

Goddard, R. E.

English

NASA Technical Reports Server

N93-29592
TDA Progress Report 42-113 May 15,1993
, j/
//,
Gravity Referenced Elevation Encoder Development
R. E. Goddard
Guidance and Control Section
Recent progress in the development of a gravity-sensor-based instrument for
determining the elevation angle of DSN antennas is described. The benefits of such
a system include the capability to locate the Gravity Referenced Elevation Encoder
(GREE) directly on the primary reflector (thus bypassing structural flexure and
deformation error sources), anticipated lower maintenance costs compared to the
present gimbal encoders, direct replaceability or supplementation of the present
gimbal encoders and the utilization of off-the-shelf components to construct the
GREE. This article includes a description of the nominal GREE design. Test results
on a laboratory breadboard model are given. Rigid-body dynamics of the GREE are
derived and the simulated performance in response to measured antenna vibrations
is given.
I. Introduction
Large structure ground-based antennas are subject to
pointing errors which prevent achieving the full designed
performance of the antennas [1-3]. Major error sources
are gravity sag, thermal gradients and hysteresis of tile
primary reflector and its supporting structure. Therefore,
the mechanical boresight of the primary reflector is not
directly observable from the angle encoders on the axes'
gimbals because of the large amount of structural sepa-
ration. As an alternative to gimbal measurements, an in-
strument capable of sensing the Earth's gravity vector may
be placed directly on the primary reflector to bypass the
structural error sources. Knowledge of the gravity vector
in antenna coordinates defines the elevation angle. In Sec-
tion II, a Gravity Referenced Elevation Encoder (GREE)
is described and the results of recently completed labora-
tory proof-of-concept tests of noise and long term drift are
given. The rigid body dynamics of the GREE are derived
in Section III and the nominal controller design is given.
Simulated performance of the GREE in a DSN disturbance
environment is detailed in Section IV.
II. Gravity Referenced Elevation Encoder
The GREE is constructed of all off-the-shelf compo-
nents. Figure 1 is a pictorial of the GREE. The major
components are the brushless DC motor, optical encoder,
and accelerometer. The accelerometer is mounted via an
adapter block onto the optical encoder rotor shaft. The
opposite end of this shaft is rotated by the motor until
the accelerometer output is nulled. Tile angular rotation
required to null the accelerometer is read on the optical en-
coder. This angle is the elevation angle of the GREE itself.
Knowledge of the calibration matrix between the GREE
and the primary reflector mechanical boresight yields the
antenna elevation angle. A block diagram of the GREE is
shown in Fig. 2 and the laboratory unit is shown in Fig. 3.
It is anticipated that thermal stabilization will be utilized
in the field deployable unit to prevent temperature residual
bias drift of the accelerometer. IIowever, the laboratory
model has neither heaters nor thermal insulation.
Long term stability of the sensor bias determines the
time between recalibrations. Test results (Fig. 4) over 27
63
https://ntrs.nasa.gov/search.jsp?R=19930020403 2020-03-17T04:29:46+00:00Z
days indicate a 3a of 0.8 mdegs after detrending. A trend
of 0.2 mdeg is apparent in the data. The suspected cause
is cyclic microtilting of the laboratory pier. The test was
limited to an accelerometer on hand; however a superior
version of the accelerometer, available at a modest cost in-
crease, is expected to reduce the long term drift by 50 per-
cent per manufacturer's specifications. Further, the drift
may not be linear, which means that extrapolation of the
test results is not applicable. Although the results are en-
couraging, longer duration tests should be completed to
characterize the bias drift. A Hewlett-Packard spectrum
analyzer recorded the short term noise spectrum. The rms
noise is 0.32 mdeg at a 3.63-Hz sampling resolution and an
accelerometer bandwidth of 250 Hz.
III. Rigid Body Dynamics
In this section, the GREE rotational dynamics are de-
rived. The accelerometer is a torque-balanced pendulous
mass sensor as shown in Fig. 5. The torque v is developed
by internal control electronics to maintain the angular po-
sition of the mass in accelerometer body coordinates. The
amount of torque required is v = m g 1 sin 0 and the out-
put is zero when the accelerometer sensitive axis is aligned
with gravity.
The GREE dynamics are derived utilizing a two degree-
of-freedom planar model. One degree of freedom repre-
sents the rotation axis of the optical encoder and the sec-
ond degree of freedom is the accelerometer rotation axis.
Figure 6 details the dynamic model where for simplicity
the misalignments are assumed zero, i.e., the calibration
matrix is the identity transformation. The following quan-
tities are defined:
01 =
zi, Yi----
ZO, YO -=
COMi =
kl =
01 =
02 =
fxi, fYi =
g --
_=
elevation angle
rotor shaft angle with respect to gravity
location of center of mass of member i, i =
1,2
rotor axis of rotation
center of mass of member i, i = 1, 2
distance of accelerometer axis from rotor
COM1
accelerometer pendulous mass angle with re-
spect to accelerometer case
accelerometer pendulous mass angle with re-
spect to gravity
resultant coupling forces on rotational
axisi, i=1,2
gravitational constant
COMi offset from rotational axis i, i = 1,2
inertia of member i about COMI, i = 1, 2
The axis of rotation of the rotational member con-
sisting of the optical encoder shaft, accelerometer, and
actuator rotor (see Fig. 1) is given by the coordinates
x0,Y0. The kinematics of the system defined in Fig. 6
are (si = sin0_,c_ = cosOi):
Zl = ZO -_-11Cl Z2 = Zl + klcl + 12c2
Yl = YO "4- llSl Y2 = Yl + klsl + 12s2
Twice time differentiating the above equations yields:
(1)
The translational dynamics are
64
ml_l =/x, - f_, + mlg "lib =/y, -/y, /
im2x2 = fz, + m2g --2_)2 = fy_
(2)
Solving for the coupling forces (fx,, fy,) in terms of the translational accelerations and substituting out these accel-
erations via Eq. (1) yields
(3)
The rotational dynamics may now be written and the coupling forces substituted out to arrive at the dynamics in
rotational coordinates. The results are
[I1.+121 (.1-4-m2)-4-211kl.2-l-k_Yn2] 01 + [(tl--l-kl)t21Tt2cos(02-/91)]_2 }
- [(11 + kl)12m2sin(/92-/91)10_ + llsl ('1 + "2)g + klslm2g
= "4-llSl ('1 "4" )7/2) _0 -- llCl (_"nl -_- "2) Y0 + klSlm2xo - klClm2_lo -4- T1 -- 7*2
(4a)
[(/1 "4- kl)12"2 c0s(/92 -/91)]01 + [(/1 + kl)12--2 sin(02 -/91)] #x2
+ [12 + "2/22]_2 + 12"2s2g = +12"2s2x0 --12rn2c2ijo + T2
(4b)
In the dynamics, the accelerations (xo,_)o) represent
composite motion of the GREE due to antenna vibrations
(_, y_) and a jitter on the elevation angle (g,g). This is
expressed as
Xo "" --to(sin c g + cos _ _2)X 0 (5a)
i)o = Y_ -/o(cos _ k' - sin e _2) (5b)
where t0 is the off-axis mounting of the GREE. Table 1
lists the parametric values utilized for the GREE. The val-
ues are obtained from actual measurements of the GREE
components, vendor specifications, or calculations. The
parameter l0 represents the assumed off-axis mounting of
the GREE on the antenna; and the parameter ll repre-
sents the assumed center-of-mass imbalance on the optical
encoder rotor. These two quantities are installation and
fabrication constraints, respectively.
The linearized GREE model is shown in Fig. 7. The
actuator is modeled in the frequency domain as a simple
first-order torque generator:
,.-,..0>
. = (61F(s+7)
65
where kr, /%, r and ! are the torque constant, back elec-
tromotive force (EMF) constant, coil resistance, and coil
inductance, respectively, and s is the complex frequency.
The accelerometer model is a second-order critically damp-
ed system with an appropriate scale factor (SF). The lin-
earized dynamics of the rotor are derived utilizing/2 <</1 :
e _ k,
r= ( - + =) (r)
A generalized model of bearing friction [4], Fig. 8, de-
scribes the restoring torque as a function of rotation. This
represents the resistance to rotation exhibited by the opti-
cal encoder bearing action. The model is characterized by
coulomb torque (T¢), the spatial time constant (re), and
the material factor (i). All of the curves in Fig. 8 have
equal slope (To\re) at the origin of the torque-rotation
curve and therefore an exponential approximation may be
utilized. An analytical torque-rotation curve with hystere-
sis is shown in Fig. 9. This model is valid for/_ > 0 and
/_ < 0. The expression for this model utilized in the sub-
sequent simulations is
T(/3) = sgn(_)Tc + [T(0) - sgn(/})T¢] [e -(_/,°)] (8)
where T(0) is the starting torque and /3 is the angular
change from the starting (last reversal) angle. Since a
linearized model is required for the controller design, a
spring constant is defined by
AT = _-_A/3 = A/3 - kA/3 (9)
Table 2 lists values for the GREE components.
The linearized dynamics (Fig. 7) are utilized to com-
pute a controller. The calculations are considerably sim-
plified if higher order modes are neglected. Referring to
Table 2, the motor and accelerometer dynamics are widely
separated from the plant dynamics. Also combining the
back EMF and friction loops into the plant dynamics yields
the reduced-order block diagram in Fig. 10. A classical
proportional, integral, and derivative (PID) controller is
b-1
H=bls+bo+-- (10)
$
The rotor angle 01 is the error signal. If, for example,
01(w) = 0 for all w, then/3 = e, and the system is error
free. The controller H must keep 01 small. From Fig. 10,
the closed-loop error transfer function is
81
m
sa +
k_8
[2_w,_ + krSFkrbl ]s2_b [krSFkrb° +w2nJs w [kTS? _b-1]
(11)
For simplicity let the closed loop response be three first
order responses possessing break frequencies Wl,W2, and
wa. The controller solution is
(12)
The PID controller places a zero at the origin in the
error transfer function, Eq. (11). For the analytical mod-
els used, a constant input will result in zero error. This is
desirable since the nominal trajectory of a celestial track
has a fundamental period of 1/(86,400) Hz. The error fre-
quency response can be made flat over a particular band.
For example, let wl = 2_r(0.01) and w2 = w3 = 27r(500)
rad/sec. The resulting Bode diagram for the error 0](w),
Fig. 11, is flat from 0.01 to 500 Hz with a minimum error
attenuation of-80 dB.
IV. Nonlinear Simulations
The nonlinear equations derived in Section IlI are uti-
lized to simulate the GREE response to DSN antenna
66
vibrations. The nonlinear friction model (Fig. 9) is in-
cluded. Measured vibration data on DSS 14, as given in
Fig. 12, are the input. Also included are the effects of
angular jitter on the elevation angle g(t) at a 20-Hz rate
and 0.01-rad amplitude. A fourth-order Runge-Kutta in-
tegration at a step size of 10 msec computed the GREE
response given in Fig. 13. The expected error is less than
1 mdeg under these conditions.
V. Summary
The present state of development of an inertial instru-
ment for DSN antennas has been described. A breadboard
model GREE tested in the laboratory demonstrated ac-
ceptable noise and long-term drift characteristics. Rigid
body analytical models and simulations have been devel-
oped for design and analysis. A nominal controller design
maintained the simulated elevation angle error to less than
1 mdeg when driven by measured 70-m antenna vibrations.
The performance of a GREE under actual antenna vi-
brations can be determined by a field test. Because the
GREE will retrofit both mechanically and electrically onto
existing antennas, a performance comparison study with
the existing gimbal encoders can be designed. The recently
completed laboratory demonstration unit can be upgraded
for the field test by the addition of heaters, thermal insu-
lation, mechanical mounting hardware and an electrical
interface. Prior to the field testing, the analytical model
needs to be upgraded to incorporate the dominant flexible
modes and the controller upgraded to maintain the desired
performance in the specific antenna environment.
References
[1] C. N. Guiar, F. L. Lansing, and R. Riggs, "Antenna Pointing Systematic Error
Model Derivations," The Telecommunications and Data Acquisition Progress Re-
port 42-88, vol. October-December 1986, Jet Propulsion Laboratory, Pasadena,
California, pp. 36-46, February 15, 1987.
[2] K. Abichandani, "Unified Analysis for Antenna Pointing and Structural Error
Part 1. Review," The Telecommunications and Data Acquisition Progress Report
_2-73, vol. January-March 1983, Jet Propulsion Laboratory, Pasadena, Califor-
nia, pp. 40-59, April 15, 1983.
[3] S. Katow, J. J. Cucchissi, K. L. Chuang, R. Levy, J. L. Lansing, F. W. Stoller,
and F. Menninger, "Structural Design Options for the New 34 Meter Beam Wave-
guide Antenna," The Telecommunications and Data Acquisition Progress Report
42-88, vol. October-December 1986, Jet Propulsion Laboratory, Pasadena, Cal-
ifornia, pp. 183-186, February 15, 1987.
[4] P. R. Dahl, "Solid Friction Damping of Mechanical Vibrations," AIAA Journal,
vol. 14, no. 12, pp. 1675-1688, December 1976.
67
Table 1. Mass properties for simulation.
Pazazneter Value
Inertia, kg-m 2
Mass, kg
Distance to center of mass, m
Distance to accelerometer rotation axis, m
Distance to elevation axis, m
Vibration scale factor, kg-m
Rotor transfer function, (kg-m2) -1
Rotor transfer function, rad/sec
I1 = 1.28 x 10 -2
mz = 1.02x 10 °
Iz = 5.00 x 10 -s
kl =1.00x10 -2
Io =1.00x 100
kx = 5.09 X 10 -3
kr =7.78X 101
tao = 1.97 x 10 °
I2 = 5.65 X 10 -4
m2 = 1.20 X 10 -4
12 = 8,89 X 10 -3
Table 2. Component mechanical and electflcal properties.
Component Value
Brushless dc motor
Torque constant, newton-m/ampere
Inductance, henries
Resistance, ohms
Voltage constant, volts/rad/sec
Half-power frequency, rad/sec
Accelerometer
Scale factor, volts/rad
Break frequency, Hz
Optical encoder friction
Coulomb torque, newton-m
Spatial constant, rad
Spring constant, newton-m/rad
k_- = 1.91 X 10 -1
1=4.32X 10 -2
r = 2.40 X 101
k,, = 1.91 x 10 -1
p3 = 5.55 x 102
SF= 2.87x 102
Pl,P2 = 2.00 x 103
Tc = 5.30 x 10 -3
rc = 4.00 x 10 -4
k = 1.32 x 10 z
68
ELEVATION ANGLE
, ,NTE FAO , OONN C O I - ¢'_.__E OOD  OPT'OAL
-12r--h_:_N _ . O._E
GREE I I I ANGLE
III1_1 I_ _ II_NAL[t"_ _M_Z -_w_ _.OL_,_; C_L_.O_,S_S O)
. tOON  OLL liJ,,,, ._.CT.O.C_ _L.OT.ON,OSI"
L OPTICAL ENCODER Fig. 2. GREE block diagram.
Fig. 1. GREE components.
i
Fig. 3. Laboratory demonstration unit.
69
04
03
c_
4_
1D
E
02
rr
rr
I,U
01
0.8
0.7
0.6
0.5
-o_0.4
E
_ 0.3
0
er
_" 0.2
0.1
0
--0.1
-0.2
(a)
a i r 1 J I _ 1 J
50 100 150 200 250
FREQUENCY, Hz
I l I I i t I I
(b)
f--
i i
50 100 150 200 250 300 350 400 450
TIME, hr
Fig. 4. Accelerometsr-messursd laboratory data: (a) noise spec-
trum and (b) 27-day drift teal (Subatrand QA-2000 eccelerometar
bias stability lest, June 22 to July 17).
ROTATION
AXIS
NULL ANGLE/ _W, " , _J) \
RELATIVE
POSITION
SENSORS
Fig. 5. Accelerometer principle of operation.
g
)
Fig. 6. Dynamic model of GREE.
7O
ANTENNA
VIBRATIONS
_OM
! OFFSET
.._+f
FRICTION
CONSTANT
QUANTIZATION NOISE
ELEVATION ANGLE -- -- OUTPUT
I I ROTOR I I
_, IANGLEIO,I I p,p_SF [_
r _ / i. _____- I
ROTOR DYNA_ ACCELEROMETER /
I BACKEMF /
k_ _ I CONTROLLER 4
/ s + r F'_k,,.j / - MOTOR H ACCELEROMETER
i VOLTAGE I I OUTPUT VOLTAGE
DC MOTOR DYNAMICS
Fig. 7. Llnearized dynamics for controller design.
0.5
I I
-0.5o
1/2 1 2
0 1/4
t I I I I
0.5 1.0 1.5 2.0 2.5
T c
Fig. 8. Generalized force-displacement curve of solid friction
damping,
TORQUE (newton-m)
Tc
-T_
p (rad)
Fig. 9. Torque-rotation curve for linear controller design.
71
ANTENNAVIBRATIONS
r OFFSET_
ELEVATION ANGLE (c)
1
FRICTION
ONSTANT
s2 + (2 r o_n) s + o)2
ROTOR DYNAMICS
QUANTIZATION NOISE
j -OUTPUT
ROTOR ANGLE
(el)
i= SF t--
ACCELEROMETER
i + i I'  ELE O  TE kt VOLTAGE CONTROLLER OUTPUT VOLTAGEII IIr H
DC MOTOR DYNAMICS
Rg. 10. PID controller and reduced-order GREE model (_N = 32.2 red/sec; _ = 0.002).
-75
-80
nn
"_ -85
<: -90
-95
-100
' ' ' 'I ' ' ''I ' ' ' 'I ' ' ''I ' ' ' 'I ' ' ''
, , ,,I , , ,,I n _ =_I , , ,,I , , ,+I , , ,,
100
5O
o
10 --50--
-100 -
-150
10-3
' ''t ' ' ''l ' ' ''1 ' ' ''
(b) _
I , I il i i = ,I i I I lJ I i ill i J JJJ i k i
10-2 10-1 100 101 102 103
FREQUENCY, Hz
Rg. 11. Error frequency response: (a) error magnitude (small eleva-
tion changes) and (b) error phase (small elevaUon changes).
72
8
6_
?
2i°
-2
-4
0.5 1.0 1.5 2.0 2.5 3.0 3.5
TIME, secx 10 4
Fig. 12. Measured DSS-14 (70-m) antenna vibration data for
simulation.
0,5
c_
0_
"o 0E
¢-
o
rr -0.5
or-
w
-1.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
TIME, sec
Fig. 13. Simulated GREE response to DSS-14 disturbance environment.
I I I I I I I I I
I I 1 I i I A I i
1.0
73


Gravity referenced elevation encoder development

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