Tmc428Math

To move RA in sidereal speed, we have to calculate the right settings for the TMC428 motor controller:

1. calculate closest upper and lower TMC428 parameter set for that speed, based on gear ratio of the mount and stepper motor resolution (fullsteps per rotation)

Parameters needed for speed calculation (based on TMC documentation):

Signal	Description	Range or Value
RPM:	revolution per minute
RPS:	revolution per second
fsf[Hz]:	fullstep frequency
µsf[Hz]:	microstep-frequency
f_CLK:	clock frequency	16MHz
velocity:	TMC428 speed parameter	0..2047
pulse_div:	divider for velocity, The higher the value, the less is the maximum velocity	0..13
µsrs:	microstep-resolution (µ-step per fullstep = 2^µsrs)	0..7 [7 is mapped to 6 by the TMC428]
fs_ro:	fullsteps per rotation for a given stepper	for example: 100 for an “Escap P530”

Formulas:

µsf[Hz] = f_CLK * velocity / (2^pulse_div * 2048 * 32)

fsf[Hz] = µsf[Hz] / 2^µsrs

fsf[Hz] = f_CLK * velocity / (2^pulse_div * 2048 * 32 * 2^µsrs )

RPM = fsf[Hz] * 60 / fs_ro

RPS = fsf[Hz] / fs_ro

RPM = f_CLK * velocity * 60 / (2^pulse_div * 2048 * 32 * 2^µsrs * fs_ro)

RPS = f_CLK * velocity / (2^pulse_div * 2048 * 32 * 2^µsrs * fs_ro)

=>

velocity = fsf[Hz] * 2^µsrs * 2^pulse_div * 2048 * 32 / f_CLK

velocity = RPM * fs_ro * 2^µsrs * 2^pulse_div * 2048 * 32 / (f_CLK * 60)

velocity = RPS * fs_ro * 2^µsrs * 2^pulse_div * 2048 * 32 / f_CLK

remark!: The slowest possible speed will be:

fsf_min (f_CLK=16MHz): 0.000466 and RPM_min (@fs_ro=100): 0.000279

Example:
For a EQ6 mount with Escap P530 motors with 1:12 secondary gear:

- primary mount gear ratio (RA gear teeth) : 180 => 2° mount movement at the sky / turn
- secondary (motor) gear ratio: 12

180 * 12 => 0.1667° mount movement at the sky per stepper motor turn

- sidereal day: 86164.1 seconds

sidereal_turn_rate: 360° / 86164.1s = 0.004178°/s
Exact sidereal motor RPS_req: 0.004178°/s / 0.1667° = 0.025068 required for this mount, motor and gear values.

The TMC428Calculator.exe proposed these values as a close match:

f_CLK = 16MHz
velocity = 168
pulse_div = 8
2^µsrs = 64
fs_ro = 100

=> RPS_lower: 0.025034

and with:

f_CLK = 16MHz
velocity = 169
pulse_div = 8
2^µsrs = 64
fs_ro = 100

=> RPS_upper: 0.025183

But the same speeds will also be obtained with:

f_CLK = 16MHz
velocity = 1344
pulse_div = 11
2^µsrs = 64
fs_ro = 100

=> RPS_lower: 0.025034

and with:

f_CLK = 16MHz
velocity = 1352
pulse_div = 11
2^µsrs = 64
fs_ro = 100

=> RPS_upper: 0.025183

Therefore the highest possible velocity and pulse_div values will give the closest match to the RPS of 0.025068 necessary in this example:

f_CLK = 16MHz
velocity = 1345
pulse_div = 11
2^µsrs = 64
fs_ro = 100

=> RPS_lower: 0.025052 ( equals ~ 0.1 % error)

and with:

f_CLK = 16MHz
velocity = 1346
pulse_div = 11
2^µsrs = 64
fs_ro = 100

=> RPS_upper: 0.025071

We have now to switch periodically between RPS_lower and RPS_upper around RPS_req.
The (scaled) difference between RPS_lower and RPS_req. equals 16 and between RPS_upper and RPS_req. equals 3.
=>
We get our exact average RPS_req by driving the motor for 3 seconds with RPS_lower and 16 seconds with RPS_upper and so on…
(Of course quartz oscillator errors (drift, absolute and temperature variations) still remain, but these errors are much smaller in the range of 0.01 percent)