Speed Adjustment Using PID Control for TTC

Overview

In autonomous driving and Advanced Driver Assistance Systems (ADAS), Time-to-Collision (TTC) is used to adjust vehicle speed dynamically. Instead of braking abruptly when TTC drops below a threshold, a Proportional-Integral-Derivative (PID) controller can smoothly adjust speed based on how TTC changes over time.

This approach provides gradual deceleration and acceleration, improving both safety and driving comfort.

1️⃣ Why Use PID Control for Speed Adjustment?

A fixed TTC threshold (e.g., reducing speed when TTC < 1.0s) may cause:

  • Harsh speed reductions, leading to inefficient driving.

  • Oscillations, where speed fluctuates between slowing down and speeding up.

  • Overcorrections, making the vehicle feel unstable.

A PID controller allows:

✅ Gradual speed adjustments based on TTC deviation.

✅ Predictive control, slowing down in advance if TTC is decreasing rapidly.

✅ Smoother driving, eliminating unnecessary fluctuations in speed.

2️⃣ PID Control Formula for Speed Adjustment

The error is defined as the difference between desired and actual TTC:

\[e(t) = TTC_{\text{desired}} - TTC_{\text{actual}}\]

The speed adjustment output is calculated using:

\[V_{\text{adjustment}} = K_p e(t) + K_i \int e(t) dt + K_d \frac{de(t)}{dt}\]

Where:

  • \(V_{\text{adjustment}}\) = Speed adjustment (positive = accelerate, negative = decelerate).

  • \(K_p\) = Proportional gain (reacts to the TTC error).

  • \(K_i\) = Integral gain (corrects long-term speed errors).

  • \(K_d\) = Derivative gain (predicts sudden changes in TTC).

  • \(e(t)\) = TTC error.

3️⃣ How the Derivative Term Helps

The derivative term predicts future risks by monitoring the rate of change of TTC.

If TTC is dropping rapidly → Reduce speed aggressively to prevent a collision.

If TTC is stable → Hold current speed (avoid unnecessary slowing).

If TTC is increasing → Smoothly increase speed to improve efficiency.

Example:

  • Obstacle detected at a distance → Slowly reduce speed (TTC decreases gradually).

  • Sudden obstacle appears → Rapid speed reduction (TTC drops sharply).

  • Obstacle moves away → Smooth acceleration to return to normal speed.

4️⃣ Example: PID-Based Speed Control

Scenario: Adjusting Speed Based on TTC

  • Desired TTC = 2.0s

  • Actual TTC = 1.0s

  • Proportional Gain = \(K_p\) = 0.5

  • Integral Gain = \(K_i\) = 0.1

  • Derivative Gain = \(K_d\) = 0.4

  • TTC is dropping at 0.3s per second

Step 1: Compute the Error

\[e(t) = 2.0 - 1.0 = 1.0\]

Step 2: Compute Speed Adjustment

\[V_{\text{adjustment}} = (0.5 \times 1.0) + (0.1 \times \int 1.0 dt) + (0.4 \times 0.3)\]

If the accumulated integral error over time is 2.0, then:

\[V_{\text{adjustment}} = (0.5 \times 1.0) + (0.1 \times 2.0) + (0.4 \times 0.3)\]

Thus, the vehicle will reduce speed smoothly by 0.82 m/s, instead of braking immediately.

5️⃣ Considerations & Tuning

🛠 Tuning Kd Carefully

  • Kd too high → Reacts too aggressively to minor TTC changes (unstable speed control).

  • Kd too low → Delayed reaction to sudden changes (slower response).

🚗 Tuning Strategy

  1. Start with P-control only to see basic responsiveness.

  2. Add I-control to correct steady-state speed errors.

  3. Introduce D-control to anticipate sudden obstacles and prevent overcorrections.

6️⃣ Summary

✅ PID-based TTC speed control prevents abrupt slowdowns.

✅ The derivative term predicts risk before it happens.

✅ Results in smoother, more efficient speed adjustments.