Trajectory Simulation Skill

Physics Model

Objects follow projectile motion:

x(t) = x0 + vx * t
y(t) = y0 + vy * t  
z(t) = z0 + vz * t - 0.5 * g * t²

Landing time found by solving z(t) = catch_height using quadratic formula.

Camera Model

ToF camera at origin pointing +z (up). FOV is a cone:

# Point visible if:
# 1. Above camera (dz > 0)
# 2. Within range (distance < max_range)
# 3. Within cone (angle_from_vertical < fov/2)

horizontal_dist = sqrt(dx² + dy²)
angle = atan2(horizontal_dist, dz)
in_fov = angle < (fov_degrees / 2)

Prediction Algorithm

1. •Collect N visible points from camera
2. •Fit parabola to z-coordinates: z = a*t² + b*t + c
3. •Fit lines to x, y coordinates: x = mx*t + bx
4. •Solve quadratic for landing time
5. •Evaluate x(t), y(t) at landing time

Use np.linalg.lstsq for fitting - handles noise well.

Key Parameters

Testing Predictions

Always test with:

1. •Perfect data (no noise) - should be <1cm error
2. •Noisy data - should be <10cm error for 80%+ throws
3. •Edge cases: steep angles, fast throws, late FOV entry

# Quick accuracy check
python scripts/run_benchmark.py 100

Common Bugs

Negative landing time: Object already past catch plane. Check discriminant and filter for future times only.

NaN in prediction: Division by zero when trajectory is nearly vertical. Add abs(a) < 1e-10 check.

High error on angled throws: Usually FOV issue - object enters late, fewer frames to fit. Increase MIN_FRAMES or accept lower confidence.

Inconsistent results: Forgot to set random seed. Use np.random.seed(N) for reproducibility.

Extending the Predictor

To add ML predictor later:

1. •Same interface: predict_landing(List[TrajectoryPoint]) -> Prediction
2. •Train on synthetic data first
3. •Export training pairs: (observed_points, actual_landing)
4. •Start with MLP, skip LSTM (trajectories are short)