Telescope Axes Servo Design

EOST Servo Design Specification

ER-80

Gary J. McGrath
November 27, 1995

The purpose of this document is to record the current status of the servo systems that control pointing and instrument rotation for the E. O. Smith Telescope. The design parameters may change as the servos are optimized for tracking performance. This document will be updated as the designs evolve.

The Azimuth, Altitude, Right and Left Instrument Rotator servos all share a similar design architecture.

The command position (THETAcmd) is computed from the equation of a cubic spline by the servo CPU. The coefficients of the spline are calculated by the supervisory CPU from trajectories that are generated by the TCC (Telescope Control Computer). This gives the servo a smooth position command trajectory.

The PID compensation is performed digitally using the following difference equations:

x1(k) = Kp u(k) + Ki x2(k) + [Kd (u(k) - u(k-1))]/T
x2(k) = x2(k-1) + T u(k)

Where x1(k) is the output, and u(k) is the input.

These difference equations are derived from the transfer function in the z-domain (see "Digital Implementation of PID Compensation"):

X1(z) = [Kp + (Ki T z) / (z-1) + Kd(z-1) / (T z)] U(z)

The output of the PID compensator is then fed into a series of digital notch filters. Up to four filters can be cascaded in series via software. These filters are used to reduce the amplitude of mechanical resonances in the open-loop frequency response. The difference equation for each filters is:

y(k T) = [ b0 x(k T) + b1 x((k-1)T) + b2 x((k-2)T) - a1 y(k-1)T) - a2 y((k-2) T) ] / a0

Which comes from the transfer function in the z-domain (see "Derivation and Simulation of Second-Order Digital Notch Filter Using Prewarping"):

The output of the notch filters is sent through an 18 bit D/A converter. The output of the D/A must be low-pass filtered to smooth out the quantized signal. A single real pole analog RC filter with unity gain buffer is used.

The values of R and C are selected to produce a bandwidth of half the sampling rate (or slightly lower).

f(Hz) = 1/(2 PI R C)

The output of the low-pass filter drives the motor current amplifier. Kpa is the gain of the amplifier (Amps/Volt).

The amplifier supplies current to the drive motor. The motor current is proportional to its output torque. Kt is the motor torque factor (lb-ft / Amp). The azimuth axis has two motors and two amplifiers which are driven in parallel.

The drive motor(s) apply torque to a drive ring through a friction drive. N is the drive ratio.

The azimuth and altitude axes use Farrand encoders to measure position. These encoders are directly driven by the axis and produce sin and cos waves that are resolved into 2^16 counts per sector. There are 512 sectors per revolution. The computer reads the binary output of the resolver and converts the position to degrees by multiplying by the scalefactor 1.0728836E-9 (deg/count). The sin and cos waves are band-limited to 500 Hz. by a single pole low pass filter in the resolver circuit. This filter eliminates aliasing which would otherwise occur in the digitized signal.

The instrument rotators use Heidenhain friction driven incremental encoders. These optical encoders generate sin and cos waves and have an internal resolver. The band-limiting internal filter cuts off at 25 KHz and cannot be modified. The encoder outputs 18000 x 4 counts per revolution of the encoder and has a drive ratio of approximately 24. The encoder output is multiplied by the nominal scalefactor 2.083333E-4 to get the axis position in degrees.

The encoder is sampled every T seconds and is subtracted from the command signal in the computer to get the error signal. This error signal is the input to the digital PID compensator.

The servo can be modeled by the following block diagram:

System Parameters

                      Azimuth     Elevation    Right Inst.    Left Inst.
Drive Ring Dia.       4 ft.       4 ft.        2 ft.          2 ft.
Motor Capstan Dia.    1.5 in.     1.5 in.      0.75           0.75 in.
Encoder Capstan Dia.  direct      direct       1 in.          1 in.
Encoder Counts/rev    512x2^16    512x2^16     18000x4        18000x4
Encoder Turns Ratio   1           1            24             24
Kpa (A/V)             0.5         0.5          0.27           0.27
Kt  (lb-ft/A)         2.07        2.07         0.40           0.40
sample rate (Hz)      1000        1000         1000           1000          
low pass cuttoff (Hz) 500         500          500            500
J (lb-ft-sec^2)       2400        1800         1.375          1.375
N                     32          32           32             32

PID Constants

                      Azimuth     Elevation    Right Inst.    Left Inst.
fz (Hz)              
Kp                   
Ki
Kd

Notch Filter Constants