Simple Ball Trajectory Estimation And Simulation

Code to estimate and simulate the trajectory of a ball undergoing projectile motion.

Trajectory is modeled using linear drag (see Wiki)

Trajectory Modelling

The estimator in misc/estimate_ball_trajectory.py assumes the ball follows a linear-drag model:

• Velocity: dv/dt = g - β v
• Position: dp/dt = v

For a time offset Δt = t - t₀, the closed-form solution becomes:

• p(t) = p₀ + A(Δt, β) v₀ + B(Δt, β) g
• v(t) = exp(-β Δt) v₀ + A(Δt, β) g
• A = (1 - exp(-β Δt)) / β
• B = Δt/β - (1 - exp(-β Δt)) / β²

When β → 0, the model smoothly reduces to the standard ballistic solution (A → Δt, B → 0.5 Δt²).

Trajectory Estimation

The fit_trajectory method of the BallTrajectoryEstimator class in misc/estimate_ball_trajectory.py fits the parameters p₀, v₀, and drag β using timestamped position measurements (Time, TX, TY, TZ). The implemented method is:

1. Data validation: requires at least N recent samples with finite values and sufficient upward Z velocity before fitting.
2. Solve for β = 0 baseline: computes a no-drag solution for comparison.
3. Optimize drag coefficient: performs a 1D search over β (bounded between beta_min and beta_max) with SciPy's minimize_scalar.
4. Variable projection for p₀ and v₀: for each candidate β, solves the linear system p = p₀ + A v₀ + B g via least squares to obtain the best p₀ and v₀.
5. Select best fit: keeps the (p₀, v₀, β) combination with the lowest sum of squared residuals and stores it for prediction.

Predicted positions use the stored parameters and the same closed-form equations to evaluate p(t) for any desired timestamps.

Misc:

• Included is BallTrajectoryEstimator_KalmanFilter in misc/estimate_ball_trajectory.py, that estimates trajectories using a Linear Kalman Filter and a gravity only model.