docs: coralator configuration space

docs/coralator-tuning.md

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+# Coralator Tuning
+
+The coralator essentially has three degrees of freedom.
+Those are height (from the elevator), arm angle, and wrist angle.
+Not all of these can be adjusted independently - certain configurations can result in the robot colliding with itself.
+We can model the various valid configurations to create what's called a configuration space.
+To move our end effector from one position to another, we must ensure that we stay within the confines of the configuration space.
+
+To define this configuration space, we must consider the bounds of each mechanism.
+We'll start with the individual components - the arm and the wrist.
+Let's say the arm has a lower limit of -10 degrees and an upper limit of 90 degrees, while the wrist can rotate a full 360 degrees.
+The arm is constrained by the drivetran and the elevator, while the wrist isn't constrained.
+However, the wrist does have a constraint, but it's dependent on the arms position.
+We can use the following diagram to determine the relative angle bettween the arm and the wrist:
+
+![angle diagram](./imgs/angle-diagram.jpg)
+
+Doing some trig, we find that the wrist's relative angle to the arm can be found with the following equation:
+
+$$
+\theta_r = 180-\theta_a-\theta_w
+$$
+
+Let's say this angle, $\theta_r$, is constrained to the bound \[-120,160\] degrees.
+If we graph all of these constraints, this is the result:
+
+![configuration space](./imgs/configuration-space.png)
+
+In this graph, the x-axis represents the wrist angle while the y-axis represents the arm angle.
+The white space represents all possible valid configurations.
+When performing path planning for the coralator, we must ensure that it's path never drifts outside of this configuration space.
+
+For more details on condiguration spaces and motion planning, check out chapter 16 of [Controls Engineering in FRC](https://file.tavsys.net/control/controls-engineering-in-frc.pdf).
+
+## Appendix
+
+**Wrist Angle Diagram:**
+
+![wrist angle diagram](./imgs/wrist-diagram.jpg)
+
+**Arm Angle Diagram:**
+
+![arm angle diagram](./imgs/arm-diagram.jpg)