Automated High-Speed Rail Jerk Analysis (Robotics & Computer Engineering) Solution

Problem Overview

We compute higher-order kinematic derivatives (velocity, acceleration, jerk) for the autonomous transit position profile:

on $t \in [0, 10]\text{ s}$.

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Step-by-Step Mathematical Solution

Step 1: Compute Successive Derivatives

1. Velocity $v(t) = s'(t)$:
   $$v(t) = 0.05(4t^3) - 0.4(3t^2) + 2.5(2t) = 0.2 t^3 - 1.2 t^2 + 5 t$$
2. Acceleration $a(t) = s''(t)$:
   $$a(t) = 0.2(3t^2) - 1.2(2t) + 5(1) = 0.6 t^2 - 2.4 t + 5$$
3. Jerk $j(t) = s'''(t)$:
   $$j(t) = 0.6(2t) - 2.4(1) + 0 = 1.2 t - 2.4$$

Step 2: Evaluate Jerk at Specific Timestamps

• At $t = 2\text{ s}$:
  $$j(2) = 1.2(2) - 2.4 = 2.4 - 2.4 = 0\text{ m/s}^3$$
• At $t = 6\text{ s}$:
  $$j(6) = 1.2(6) - 2.4 = 7.2 - 2.4 = 4.8\text{ m/s}^3$$

Step 3: Verify Passenger Comfort Constraint ($|j(t)| \le 1.5$)

The jerk function $j(t) = 1.2 t - 2.4$ is a linear function on $[0, 10]$:

• At $t = 0$: $j(0) = -2.4\text{ m/s}^3 \implies |j(0)| = 2.4 > 1.5$ (Fails constraint!)
• At $t = 10$: $j(10) = 1.2(10) - 2.4 = 9.6\text{ m/s}^3 \implies |j(10)| = 9.6 > 1.5$ (Fails constraint!)

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Engineering Conclusion

Although velocity and acceleration appear smooth, the third derivative (jerk) reaches $-2.4\text{ m/s}^3$ at startup and $9.6\text{ m/s}^3$ at $t = 10\text{ s}$, substantially exceeding the $1.5\text{ m/s}^3$ passenger comfort threshold. The controls engineer must redesign the position profile with a higher-order polynomial (quintic spline) to clamp jerk to acceptable bounds and eliminate passenger whiplash.