Shear Waves and PID Drift

In AP Physics C and Calculus, we treat surfaces as flat, forces as instant vectors, and materials as perfectly rigid bodies.

Then you build a physical robot and discover that mechanics is an ongoing war against perfect calculus.

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📉 The Theoretical PID Loop

A Proportional-Integral-Derivative (PID) loop calculates an error value $e(t)$ as the difference between a desired setpoint and a measured process variable:

$u(t)=K_{p}e(t)+K_i{\int }_0^{t}e(\tau )d\tau +K_d\frac{de(t)}{dt}$

On paper, this produces a textbook critically-damped step response. The robot moves to position $x=2.0\text{m}$ and stops gracefully.

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🚜 What Physics Actually Does

In the physical world:

1. Mechanical Flex: The aluminum frame flexes under acceleration, introducing high-frequency shear waves that vibrate the IMU accelerometer.
2. Wheel Slip: The optical encoder records 100 revolutions, but the rubber tire slipped on smooth concrete for 12 of those revolutions.
3. Backlash & Friction: Static friction ($f_s$) holds the motor stuck until $u(t)$ ramps high enough, causing the motor to suddenly jerk forward past the target position.

[PID Output: Smooth Step] ───> [Real Frame: Shear Waves + Slip] ───> [Sensor Drift + Chaotic Noise]

Calculus gives you the smooth theoretical curve. Physical friction and mechanical flex will always append a chaotic $+C$ to your differential equations. 🤖