Concept 03: Accumulation, Area & Numerical Integration

▶ Interactive Demo: Numerical Integration Visualizer

Open the interactive demo below to compare Euler rectangles vs. Trapezoidal slices and see how Trapezoidal integration drastically cuts odometry drift.

1. The Real-World Problem: Where Did the Robot Go?

During the 15-second autonomous period, your robot’s wheel encoders measure velocity every 20 milliseconds (dt = 0.02s).

How does the robot calculate its total distance traveled from a sequence of velocity readings?

Distance is the Accumulation of Speed over Time:

• In each small slice dt, distance traveled is speed · dt.
• Total distance is the Area Under the Velocity Curve!

2. Solving It in Code (Java & WPILib)

First-Principles Java: Trapezoidal Integration (Dead Reckoning)

// Accumulate robot distance over time steps
double totalPosition = 0.0;
double dt = 0.020; // 20ms control loop

double[] velocityStream = {0.0, 1.0, 2.0, 3.0, 3.0, 3.0, 2.0, 1.0, 0.0};

for (int i = 1; i < velocityStream.length; i++) {
    double vPrev = velocityStream[i - 1];
    double vCur = velocityStream[i];
    
    // Trapezoidal rule: Area = 0.5 * (vPrev + vCur) * dt
    double stepDistance = 0.5 * (vPrev + vCur) * dt;
    totalPosition += stepDistance;
}

System.out.printf("Integrated Odometer Distance: %.4f meters%n", totalPosition);

3. Review Checkpoints

Checkpoint 1

A robot drives at a constant speed of 2.5 m/s for 3.0 seconds. What is the area under its velocity curve?

Solution: Since speed is constant, the area is a simple rectangle: Area = width · height = (3.0 s) · (2.5 m/s) = 7.5 meters.

Checkpoint 2

Why does Trapezoidal integration produce zero error under constant acceleration?

Solution: Because under constant acceleration, velocity is a straight line (v = a * t). The area under a straight line is an exact trapezoid, which the trapezoid formula calculates perfectly!