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extras/doc/ramp.md

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+# Ramp generator
+
+# Introduction
+
+The ramp generator increases the speed linearly with the acceleration.
+If e.g. a=1000 steps/s², then after 1s the speed is 1000 steps/s.
+
+The full ramp consists of three phases: acceleration+coasting+deceleration.
+If the number of steps are less than the required steps for acceleration and deceleration, then there will not be any coasting phase.
+
+For illustration here an ascii chart of the ramp
+
+```
+Speed
+    |       -------------------
+    |      /                   \
+    |     /                     \
+    |    /                       \
+    0---/                         \------
+        |<->|<--------------->|<->|
+          Ta        Tc          Td
+    -----------------------------------> Time
+
+Ta ... acceleration time.
+Tc ... coasting time.
+Td ... deceleration time.
+```
+
+The total ramp time is Ta + Tc + Td = T 
+
+Due to acceleration = deceleration, the related times Ta and Td are same.
+$$
+T=2*Ta+Tc
+$$
+
+The steps executed in coasting are simply:
+$$
+steps_{coasting} = v * Tc
+$$
+
+The steps executed during acceleration and deceleration are:
+$$
+steps_{acceleration} = 0.5 * a * Ta²
+$$
+
+The total steps are:
+$$
+\begin{align}
+steps &= steps_{acceleration} + steps_{coasting} + steps_{deceleration} \\
+      &= 2 * steps_{acceleration} + steps_{coasting}                    \\
+      &= a * Ta² + v * Tc                                               \\
+      &= a * Ta² + (a * Ta) * Tc                                        \\
+      &= a * Ta * (Ta + Tc)                                             \\
+      &= a * Ta * (T - Ta)
+\end{align}
+$$
+This can be rearranged to calculate the required acceleration to perform `steps` during the total ramp time T and acceleration time Ta:
+$$
+a = \frac{steps}{ Ta * (T - Ta) }
+$$
+
+# Calculation example
+
+The stepper motor should perform 32000 steps in 10s.
+
+The required acceleration and speed for different acceleration times are calculated as this:
+$$
+\begin{align}
+Ta = 1s => a &= \frac{32000}{1 * 9}  steps/s² = 3556 steps/s² \\
+           v &= a*Ta = 3556 steps/s => 281us/step \\
+Ta = 2s => a &= \frac{32000}{2 * 8} steps/s² = 2000 steps/s² \\
+           v &= a*Ta = 4000 steps/s => 250us/step \\
+Ta = 3s => a &= \frac{32000}{3 * 7} steps/s² = 1524 steps/s² \\
+           v &= a*Ta = 4571 steps/s => 218us/step \\
+Ta = 4s => a &= \frac{32000}{4 * 6} steps/s² = 1333 steps/s² \\
+           v &= a*Ta = 5333 steps/s => 188us/step \\
+Ta = 5s => a &= \frac{32000}{5 * 5} steps/s² = 1280 steps/s² \\
+           v &= a*Ta = 6400 steps/s => 156us/step
+\end{align}
+$$
+
+As commands for the StepperDemo:
+```
+	M1 A3556 V281 R32000
+	M1 A2000 V250 R32000
+	M1 A1524 V218 R32000
+	M1 A1333 V188 R32000
+	M1 A1280 V156 R32000
+```
+
+# Special case
+
+The minimum command time for a bundle of steps is `MIN_CMD_TICKS`. The ramp generator need to make sure,
+that any command issued to the command queue meets this requirement.
+
+With high acceleration this can get problematic coming from or going to stand still.
+
+The required condition to meet this requirement is:
+$$
+    steps * \frac{1}{v} \ge T_{CMD}
+$$
+
+From stand still the speed after `MIN_CMD_TICKS`is:
+$$
+    v_{start} = a * T_{CMD}
+$$
+
+So the related condition for steps is:
+$$
+\begin{align}
+    steps * \frac{1}{a * T_{CMD}} &\ge T_{CMD}  \\
+    steps &\ge a * T_{CMD}^2 
+\end{align}
+$$
+
+Based on this there can be deducted two problems:
+
+1. Any move command with less steps cannot be fulfilled with this acceleration
+   
+    => Remedy: Run these steps at maximum allowed speed based on `MIN_CMD_TICKS`
+
+2. While ramping down the last issued command may violate this condition
+
+    => Remedy: Run these steps at maximum allowed speed based on `MIN_CMD_TICKS`.
+       The problem is to distinguish this case from a legitimate overshoot situation.