# Exerpts from Chapter 8, Appendix B of Manual:  HEMMER  EXAMPLE  10:

For this appendix, there is a spring between the load inertia and a fixed  point and there is no compensation for the current loop or position loop.

This appendix was prepared long after the the main part of Chapter 8. In the past, compensation of the current loop required for brushless electric motors was considered to be necessary. This appendix refutes that assumption and it represents a major break-through in servo design technique.

It is noted that a servo with an uncompensated current loop is much easier to impliment because the need for 3 compensators, one for each of the 3 phases required by the brushless AC servomotor, is avoided. Each of the 3 circuits need only have a DC gain, instead of a complex operational amplifier circuit.

The pictorial diagram for this is similar to Figure 1-7, except a brushless AC electric motor with a gear train is used to drive the load instead of a hydraulic cylinder and yoke.

Figure 8B1 – Example of a drive containing a spring between the load inertia and a fixed point.

Figure 8B2 – Block Diagram: Current Loop, Gv

Note: Ranges are 0 to 18 degrees for load position,

0 +/- 750 deg./sec. for load velocity, and

0+/- 6.846 amps for current (thus +/- 21.36 lb.in. for motor torque )

Figure 8B3 – Block Diagram: Velocity & Position Loops

The transfer function for the amplifier for the current loop is:

Kac

Gac =    -------------------------------------

( Tn 2 S 2 + 2 Zn Tn S +1 )

where: Kac = 250 vac / vdc

Tn = T9 = .000016 sec., Zn = Z9 = 0.5    , Wn = 62,832 rad./sec,

fn = 10 khz  = switching frequency in PWM power stage.

The above values were specified in input file V-FR-I-10B.TXT for the frequency response program.

Inspection of the notes below Table 8B2-2 indicates that the closed loop transfer function for the current loop is:

Kv  (Txx S + 1)

Gv =  -------------------------------------------------------------------------

(Tx 2 S2 + 2 Zx Tx S +1) (Tn 2 S 2 + 2 Zn Tn S +1)

where: Tn = T9 = 16 usec. and  Zn = Z9 = 0.5

Tx = Txx = 41 usec. and Zx = 0.8091

Loop gain, Kolc =  Kf1 Kf2 Hoc = Ka Kie Hoc = 250 x 0.1471 x 1.111 = 40.857

1/ Hoc             1 / 1.111                   0.90009

Kcur = ----------------- = --------------------- =  ---------------- = 0.878585 = 0.88 amps ac / vdc

1 + 1/ Kolc        1 + 1 / 40.857        1.0244756

Kv = R x Kcur = 6.8 x 0.88 = 5.984 vac / vdc

TABLE 8B1: HEMMER PROGRAM DT: Drive Train Analysis

DATA FILE: V-DT-10B.TXT           TIME: 12:32:57             DATE: 01-01-2007

The load is driven by a Brushless Servomotor at ultimate temperature.

The power amplifier contains a current control loop.

TABLE 8B1-1: Electric Motor Characteristics for a

Kollmorgan Model B-102-A Brushless AC Servomotor

SYMBOL  VALUE         UNITS                            DESCRIPTION

Em     169.8         volts                    max. motor terminal voltage

Nm     7500          rpm                      max. motor speed

Tmc    7.44          lb.in.                   max. continuous motor torque

Tmp    21.36         lb.in.                   peak motor torque

Kti    3.12          lb.in./amp               motor torque sensitivity

Kbe    0.0213        volts/rpm                motor back emf constant

R      6.8           ohms                     cold resistance, motor + source

L      30            mh                       inductance of motor

Jm     0.000274000   lb.in.sec.^2             motor inertia

Tfm    0.36          lb.in.                   static friction in motor

Dmv    0.000172      lb.in./rad./sec          viscous damping at motor shaft

Ct     1.06          -----                    Tmc,cold / Tmc,hot

Kt     0.458824      lb.in. / volt            motor torque or force gain

Dm     0.093496      lb.in./rad./sec          motor damping if no currrent loop

Te     0.0022        sec.                     electric motor time constant0.5L/R

Kv     5.984         volts/volt               current loop static gain (Reference)

TABLE 8B1-2: GENERAL CHARACTERISTICS

SYMBOL  VALUE         UNITS                            DESCRIPTION"

Ji     0             lb.in.sec.^2             intermediate inertia

JL     0.843         lb.in.sec^2              load inertia or mass

Nga    0.96          -----                    efficiency of converter (a)

Ngb    1             -----                    efficiency of converter (b)

Ng     0.960         -----                    overall conversion efficiency

Acl    143.4         rad./sec./sec.           => cont. acceleration limit

Apl    425.2         rad./sec./sec.           => peak acceleration limit

TABLE 8B1-3: TRANSFER FUNCTIONS

Case 2: Spring between the load inertia and a fixed or movable point.

TL = JL x A + DL x V + KL x P + TLs  (reference)

Gu = Ku ( Te S + 1 )

Where: Ku = 0.0521 lb.in./lb.in.

Te = 0.0000 sec. ( Te = 0 when there is a current loop. )

Gb = 1/Kgb

Kpt

Gpt = -----------------------------

( T5^2 S^2 + 2 Z5 T5 S + 1 )

Kpt = 1/K = 4.80000000 rad./lb.in.

T5 = 0.108870 sec.

Z5 = 0.0038

Guf = Kuf S ( Tuf S + 1 )

Where: Kuf = 0.003440 lb.in./(in./sec.)).

Tuf = 14.3565 sec."

Kie ( T5^2 S^2 + 2 Z5 T5 S + 1 )

Gie = ---------------------------------------------

( T1'S + 1 ) ( T2'S + 1 ) ( T3'S + 1 )

Where: Kie = 0.1471 amps/volt.

T1' = 0.423118 sec.

T2' = 0.025442 sec.

T3' = 0.002429 sec.

TABLE 8B2:  HEMMER PROGRAM  FR:  Frequency Response

Current Control Loop

DATA FILE: V-FR-I-10B.TXT    TIME: 17:51:53     DATE: 12-30-2006

The loop gain is 40.86 sec.^0

The phase margin is 72.42 deg.

The unity gain crossover frequency is 19368.68 rad./sec.

There is no resonant peak in the frequency range from

TABLE 8B2-1 - INPUT PARAMETERS FOR TRANSFER FUNCTIONS:

INPUT TRANSFER FUNCTION, Gi:

Ki = 1 volts / volt , Ni = 0                  (* denotes a lead.)

* Ti = 0,* Tj = 0,* Tq = 0,* Tk = 0,* Zk = 1

T15 = 0, T16 = 0, T17 = 0

T18 = 0, Z18 = 1, T19 = 0, Z19 = 1

FEEDBACK TRANSFER FUNCTION, H:

Ho = 1.111 volts / volt , Nh = 0

* Td = 0,* Tf = 0,* Th = 0,* Zh = 1, T10 = 0

T11 = 0, T12 = 0, T13 = 0,T14 = 0, Z14 = 1

OUTPUT TRANSFER FUNCTION, Go: Delay = 0sec.

Ko = 1,* Taa = 0,* Tbb = 0,* Zbb = 1

T22 = 0, T23 = 0, Z23 = 1

FORWARD LOOP TRANSFER FUNCTIONS, Gf1 AND Gf2:

Kf1 = 250, Nf1 = 0, * Ta = 0, * Tb = 0                     Ga

* Ty = 0 * Tg = 0, * Zg = 1, T7 = 0, T8 = 0

T9 = 0.000016, Z9 = 0.5, Tn = 0, Zn = 1

Tna = 0, Tnb = 0, Tv = 0

Tr = 0, Zr = 1, * Tm = 0, * Tp = 0

Kf2 = 0.1471, Nf2 = 0, * Tc = 0.10887, * Zc = 0.0038       Gie

T1 = 0.423118, T2 =0.025442, T3 = 0.002429, T4 = 0

T5 = 0, Z5 = 0, T6 = 0, Z6 = 0

Delay = 0sec. The excitation of the servo is a sine wave of fixed amplitude.

TABLE 8B2-2 - MAGNITUDE & PHASE SHIFT VS. FREQUENCY

FOR OPEN & CLOSED LOOP OPERATION.

w         20 LOG GH       A            M          20 LOG M       B

rad/sec         db          deg.          ---          db          deg.

1.00     31.40       -24.48         0.88        -1.12        -0.62

3.00     27.05       -56.39         0.88        -1.13        -2.07

10.00      4.55        85.07         0.75        -2.54        29.31

30.00     27.82        52.81         0.88        -1.13         1.81

100.00     32.11         9.03         0.88        -1.12         0.22

300.00     31.13       -28.45         0.88        -1.12        -0.74

1000.00     24.67       -66.16         0.88        -1.13        -2.99

3000.00     15.74       -84.15         0.87        -1.17        -9.08

10000.00      5.46       -96.73         0.84        -1.56       -29.48

30000.00     -3.34      -121.09         0.70        -3.07       -79.14

100000.00    -21.63      -224.02         0.08       -22.03      -227.52

300000.00    -51.26      -257.63         0.00       -52.16      -257.78

1000000.00    -82.80      -266.38         0.00       -83.71      -266.39

9.19    -22.46        -0.01         0.06       -24.01        -0.01

The approximate closed loop transfer function, Gcl = G' GiGo/H'

where G' has a static gain of 1 and contains the following time

constants and damping ratios of the Input Parameter Table:

,,,,,,,T9, Z9,,,,,,,,,.,,,,,,,,,,,,..

G' also contains: Txx = Tx = 0.000041 sec. and Zx =  0.8090

Note: Symbols in the frequency response plots are defined below:

GH is the ratio of output/input for the open loop, in db.

A is the phase shift of the output sine wave from the input sine wave for GH, in deg

M is the ratio of output/input for the closed loop, in db.

B is the phase shift of the output sine wave from the input sine wave for M, in deg

Mu is the ratio of output/input for the disturbance response, q/Tu, in db.

Wco is the unity gain crossover frequency for which GH = 0 db.

PM is the phase margin. PM = 180 + A at W = Wco .

Figure 8B4 : Open & Closed Loop Frequency Responses for

the Current Loop with No Compensation for

Frequencies  from 100 to 100,000 rad./sec.

TABLE 8B3:  HEMMER PROGRAM  FR:  Frequency Response

Velocity Control Loop

DATA FILE: V-FR-V-10B.TXT    TIME: 16:17:24     DATE: 12-30-2006

The loop gain is 250.50 sec.^0

The phase margin is 84.24 deg.

The unity gain crossover frequency is 2510.47 rad./sec.

There is no resonant peak in the frequency range from

TABLE 8B3-1 - INPUT PARAMETERS FOR TRANSFER FUNCTIONS:

AMPLIFIER, Ga:

Ka = 190 Volts/sec./volt         (* denotes a lead.)

N1 = 1

* Tg = 0.10887 sec.

* Zg = 0.0076

T7 = 0.1 sec.

CURRENT LOOP, Gv:

Kv = 5.984 volts/volt

Tn = 0.000041 sec.

Zn = 0.8091

Tr = 0.000016 sec.

Zr = 0.5

* Tp = 0.000041 sec.

MOTOR TORQUE/VOLTAGE RATIO, Gt :

Kt = 0.459 lb.in./volt

MOTOR POSITION / TORQUE OR FORCE RATIO, Gpt:

Np = 0 and N2 = -1.

T5 = 0.10887 sec.

Z5 = 0.0038

LOAD POSITION / INTERMEDIATE SHAFT POSITION RATIO, Gb:

FORWARD LOOP & OUTPUT TRANSFER FUNCTIONS, G and Go :

G = Ga Gv Kt Gpt and Go = 1 / Kga.

INPUT TRANSFER FUNCTION, Gi:

Ki = 2 volts / rad., Ni = 0

* Ti = 0,* Tj = 0,* Tq = 0,* Tk = 0,* Zk = 1

T15 = 0, T16 = 0, T17 = 0

T18 = 0, Z18 = 1, T19 = 0, Z19 = 1

FEEDBACK TRANSFER FUNCTION, H:

Ho = 0.1 volts / rad., Nh = 0

* Td = 0,* Tf = 0,* Th = 0,* Zh = 1,T10 = 0

T11 = 0,T12 = 0,T13 = 0,T14 = 0,Z14 =1

This is a Type 0 servo with a # 2 drive that controls VELOCITY

Input parameters not listed elsewhere: Ku =0.0521, Te = 0,

Kg = 20, Ng = 0.96, Kga = 20, Kgb = 1, FB1 = 1, FB2 = 0,

KL = 80, Tdi = 0, SPRING = 2, Cu = 0, Tdo = 0.

TABLE 8B3-2: MAGNITUDE & PHASE SHIFT VS. FREQUENCY

FOR OPEN & CLOSED LOOP OPERATION.

w         20 LOG GH       A            M          20 LOG M       B

rad/sec         db          deg.          ---          db          deg.

1.00     47.93        -5.66         1.00        -0.03        -0.02

3.00     47.60       -16.55         1.00        -0.03        -0.07

10.00     44.99       -47.57         1.00        -0.03        -0.24

30.00     37.98       -71.78         1.00        -0.03        -0.68

100.00     27.93       -84.57         1.00        -0.04        -2.28

300.00     18.43       -88.81         0.99        -0.08        -6.81

1000.00      7.98       -91.80         0.94        -0.55       -21.99

3000.00     -1.53       -96.98         0.68        -3.29       -54.17

10000.00    -11.78      -115.55         0.28       -11.04      -100.89

30000.00    -22.97      -175.49         0.08       -22.34      -175.15

100000.00    -51.18      -305.21         0.00       -51.20        54.66

300000.00    -90.42      -344.82         0.00       -90.42        15.18

1000000.00   -132.43      -355.54         0.00      -132.43         4.46

9.19     51.34       -42.59         1.00        -0.02        -0.10

The approximate closed loop transfer function, Gcl = G' GiGo/H'

where G' has a static gain of 1 and contains the following time

constants and damping ratios of the Input Parameter Table:

,,,Tn, Zn,Tr, Zr,,,,,,,,,,,,.Tp,,,,,,,,,,,,..

G' also contains:

Txx = Tx = 0.000375 sec. and Zx =  0.7293

Figure 8B5 : Open & Closed Loop Frequency Responses for

the Velocity Loop with Conventional Compensation

for Frequencies  from 100 to 100,000 rad./sec.

TABLE 8B4:  HEMMER PROGRAM  FR:  Frequency Response

Position Control Loop – Two Loop Analysis

DATA FILE: V-FR-P-10B.TXT    TIME: 17:32:41     DATE: 12-30-2006

The loop gain is 747.02 sec.^-1

The phase margin is 73.82 deg.

The unity gain crossover frequency is 723.68 rad./sec.

There is no resonant peak in the frequency range from

TABLE 8B4-1: INPUT PARAMETERS FOR TRANSFER FUNCTIONS:

INNER LOOP CONSTANTS:

AMPLIFIER, Ga = servoamplifier for the inner velocity loop.

Ka = 190 Volts/sec./volt                       (* denotes a lead.)

N1 = 1

* Tg = 0.10887 sec.

* Zg = 0.0076

T7 = 0.1 sec.

CURRENT LOOP, Gv:

Kv = 5.984 volts/volt

Tn = 0.000041 sec.

Zn = 0.8091

Tr = 0.000016 sec.

Zr = 0.5

* Tp = 0.000041 sec.

MOTOR TORQUE/VOLTAGE RATIO, Gt :

Kt = 0.459 lb.in./volt

MOTOR POSITION / TORQUE OR FORCE RATIO, Gpt:

Np = 0 and N2 = 0.

T5 = 0.10887 sec.

Z5 = 0.0038

LOAD POSITION / INTERMEDIATE SHAFT POSITION RATIO, Gb:

FORWARD LOOP & OUTPUT TRANSFER FUNCTIONS, G and Go :

G = Ga Gv Kt Gpt and Go = 1 / Kga.

INPUT TRANSFER FUNCTION, Gi

Gi = servoamplifier for the outer position loop.

Ki = 100 volts / rad., Ni = 0

* Ti = 0,* Tj = 0,* Tq = 0,* Tk = 0,* Zk = 1

T15 = 0, T16 = 0, T17 = 0

T18 = 0, Z18 = 1, T19 = 0, Z19 = 1

FEEDBACK TRANSFER FUNCTION, H:

Ho = 0.1 volts / rad., Nh = 1

* Td = 0,* Tf = 0,* Th = 0,* Zh = 1,T10 = 0

T11 = 0,T12 = 0,T13 = 0,T14 = 0,Z14 =1

OUTER LOOP CONSTANTS: FB2 = 3 and Go2 = Gb.

Ho2 = 15, * Tu = 0, * Tw = 0, * Zw = 1

T20 = 0, T21 = 0, Z21 = 1

Ki2 = 15, T26 = 0

Ko2 = 1

This is a Type 1 servo with a # 2 drive that controls POSITION.

Input parameters not listed elsewhere: Ku =0.0521, Te = 0,

Kg = 20, Ng = 0.96, Kga = 20, Kgb = 1, FB1 = 1, FB2 = 3,

KL = 80, Tdi = 0, SPRING = 2, Cu = 0, Tdo = 0.

TABLE 8B4-2: MAGNITUDE & PHASE SHIFT VS. FREQUENCY

FOR OPEN & CLOSED LOOP OPERATION.

w         20 LOG GH       A        20 LOG G2H2      A2       20 LOG M2        B2

rad/sec         db          deg.          db          deg.          db          deg.

1.00     47.93        -5.66        57.47       -90.02         0.00        -0.08

3.00     47.60       -16.55        47.92       -90.07         0.00        -0.23

10.00     44.99       -47.57        37.47       -90.24         0.00        -0.77

30.00     37.98       -71.78        27.92       -90.68         0.00        -2.30

100.00     27.93       -84.57        17.46       -92.28        -0.03        -7.66

300.00     18.43       -88.81         7.88       -96.81        -0.28       -22.84

1000.00      7.98       -91.80        -3.05      -111.99        -2.91       -70.42

3000.00     -1.53       -96.98       -15.33      -144.17       -14.09      -137.53

10000.00    -11.78      -115.55       -33.54      -190.89       -33.36      -191.12

30000.00    -22.97      -175.49       -54.38      -265.15       -54.38      -265.26

100000.00    -51.18      -305.21       -93.70       -35.34       -93.70       -35.34

300000.00    -90.42      -344.82      -142.46       -74.82      -142.46       -74.82

1000000.00   -132.43      -355.54      -194.93       -85.54      -194.93       -85.54

9.19     51.34       -42.59        38.22       -90.10         0.00        -0.70

Figure 8B6 : Open & Closed Loop Frequency Responses for

the Position Loop with No Compensation

for Frequencies  from 0.1 to 100 rad./sec.

Note: The disturbance response, Mu = 0 at W = 0. This is in agreement with the simulation and the Servo Error Analysis, shown in Figure 8B9 and Table 8B7, respectively.

Figure 8B7 : Open & Closed Loop Frequency Responses for the Position Loop

with No Compensation for a  2 loop analysis for Frequencies  from

100 to 100,000 rad./sec. ( GH and A are for the inner loop.)

Note: The remaining pages in this appendix reflect a gain reduction of 4 to 1 in the velocity loop. This was necessary because the maximum sampling frequency for digital controllers is typically only 4 khz or a sampling time of 250 usec for velocity loops. The original gain would have required a sampling frequency of 16 khz or more. The performance of the position loop is only slightly reduced, since the following error is increased by only 1.17 %.

Fig. 8B8a:  Left side of the simulation block diagram for Example EX10B

Note:  See TABLE 8B1,HEMMER PROGRAM DT: Drive Train Analysis to find values for

the diagram

Shown below is the Z Transform that corresponds to the Laplace Transform for the servoamplifier in the velocity loop, Ga, when the sampling time is 250 usec. This yields the same transient responses for velocity and position when it is substituted for the Laplace Transform for use in a digital controller.

Fig. 8B8b: Right side of the simulation block diagram for Example EX10B

Note:  See TABLE 8B1,HEMMER PROGRAM DT: Drive Train Analysis to find values for

the diagram

Figure 8B9:  Simulation of Transient Responses to a 15 deg. Step Command & a 100 lb.in.

Step Disturbance Applied Simultaneously to the Position Loop using Digital Compensation

The current and position loops have no compensation, other than proper amplifier gain settings.

The Velocity Loop is compensated with one PID control and one lag, which can be implimented with Conventional Compensation. ( Has 1 integrator, 2 leads, and 1 lag.) This compensation can be accomplished with analog circuits or by a sampled data digital compensation system using a sampling time of 250 usec; The transient responses are identical to those resulting from the use of the Laplace transform transfer function for the servoamplifier in the simulation program, except there is less oscillation of voltage and torque.

Curve # 1, red  = A’, Velocity,  deg./sec./ 50

( +/- 2,500 rpm limit at motor results in  +/- 750 deg./sec. limit at load)

Curve # 2, blue  = A, Position, deg.                       ( +/- 18 deg. limit )

Curve # 3, violet  = Tm, Motor Torque,  lb.in. / 2

( +/- 21.36 lb.in. peak limit allowed for at least 2 seconds )

Curve #4, green = Emc, Motor Voltage, vac / 20  ( +/- 200 vac limit )

TABLE 8B5:  HEMMER PROGRAM  SE:  Servo Error Analysis

Position Control Loop

DATA FILE: V-SE-P-10B.TXT    TIME: 08:51:36    DATE: 01-24-2007

This is a 2 loop, Type 1 servo with a # 2 drive that controls position.

There is a spring between the final inertia & a fixed or movable point.

The steady state error, excluding the feedback transducer error,

is 0.000000 deg.. This includes a proportional error of

0.000000 deg. at command, r = 15 deg.

The following error is 0.948 deg. at command velocity dr/dt = 700 deg./sec.

The output stiffness is infinite.

TABLE 1 - DRIVE TRAIN CHARACTERISTICS

SYMBOL     VALUE       UNITS                       DESCRIPTION

Kv        5.984      volt/volts           Static Gain of Gv

Kt        0.459      lb.in./volt          Motor torque or force gain

Nga       0.960      --------             Efficiency of convertor (a)

Ng        0.960      --------             Overall conversion efficiency

Pba       0          deg.                 backlash of Kga

Cont.     P          --------             controlled variable

Feed.     2          --------             outermost feedback location

Aoe       1.748      volts                max. static error at Gv input

Pe        0.000      deg.                 uncontrolled load static error

Kpse      0.0000000  deg./deg.            proportional static error constant

Ese1      0.00000    deg.                 static error, excluding any

proportional error

TABLE 2 - SERVO ELEMENT CHARACTERISTICS

SYMBOL     VALUE       UNITS                       DESCRIPTION

Kil       62.625     1/sec^0              loop gain of innermost loop

Eae       0          u Volts              amplifier input drift for Ga

Eie       0          u Volts              amplifier input drift for Gi

Ka        47.5       volts/sec^1/vdc      gain of amplifier Ga

N1        1          --------             degree of S in the denominator of Ga

Ho        0.1        vdc/rad./sec^1       gain of feedback element H

Nh        1          --------             degree of S in the numerator of H

Ki        100        vdc/vdc              gain of input element Gi

Ni        0          --------             degree of S in the denominator of Gi

Ko        0.05       -----                gain of output element Go

Ho2       15         vdc/rad.             gain of feedback element Ho2

Ki2       15         vdc/rad.             gain of input element Gi2

DATA FILE: V-SE-P-10B.TXT              TIME: 09:15:48              DATE: 01-24-2007

The following error is 0.937 deg. at command velocity dr/dt = 700 deg./sec. or 1.17 % less than above value where the gain is 4 times less.

Kil       250.5      1/sec^0              loop gain of innermost loop

Ka        190        volts/sec^1/vdc      gain of amplifier Ga

Figure 8B10: Simulation of Transient Responses to a 700 deg./sec. Ramp Command & a 100 lb.in. Step Disturbance Applied Simultaneously to the Position Loop without feed-forward corrections  ( Step Size = 10 usec.)

Curve # 1, black  =  Command Position, degrees  ( The rate of change is 700 deg./sec.)

Curve # 2, red  = Controlled Position, degrees.             ( The +/- 18 deg. limit is ignored. )

Curve # 3, blue  = Following Error, degrees x 10

The peak following error is 2 deg. at  .005 sec. and the steady state following error is 0.965 deg. after .025 sec. This is only1.8 % greater than the 0.948 deg. following error predicted by the Hemmer Servo Error Program.

Figure 8B11: Following Errors, degrees x 100,  vs time, seconds,  for a Simulation

of  Transient Responses to a 700 deg./sec. Ramp Command & a 100 lb.in. Step

Disturbance Applied Simultaneously to the Position Loop ( Step Size = 1 usec.)

Curve # 1, blue  =  No feed-forward corrections are used.

The Steady State Error = - 0.934 deg.

Curve # 2, red  = A 2 x  Velocity feed-forward correction is used.

The Steady State Error = + .018 deg.

Curve # 3, black  =  Both 2 x Velocity & .02 x Acceleration feed-forward corrections are used.

The Steady State Error = - 0.113 deg.

Figure 8B12: Simulation of the Following Error, degrees x 100,  vs time, seconds,

when a Sinusoidal Command with a 15 deg. amplitude and a frequency of 1 hz & a 100 lb.in.

Step Disturbance are Applied Simultaneously to the Position Loop ( Step Size = 1 usec.)

Curve # 1, blue   =  No feed-forward corrections are employed. The peak error =  0.137 deg.

This is only 0.9 % of the 15 deg. amplitude.

Curve # 2, red  =   A 2 x Velocity feed-forward correction is employed. The peak error =  0.023

deg.  This is 6 times better than Curve #1.

Curve # 3, black  =  Both 2 x Velocity and .02 x Acceleration feed-forward corrections are

employed. The peak error =  0.025 deg. ( This is slightly worse than Curve #2. )