DeKA: The Deterministic Kaczmarz Algorithm

A Python library implementing the Deterministic Kaczmarz Algorithm with Greedy Selection and Smoothing for online parameter estimation, optimized for quadrotor inertial parameter estimation.

Installation

From Source

git clone https://github.com/A2R-Lab/DeKA.git
cd DeKA
pip install -e .

Quick Start

from deka import OnlineParamEst, visualize_trajectory_with_est

# Create parameter estimation instance
estimator = OnlineParamEst()

# Run simulation
x_history, u_history, theta_history, theta_hat_history = (
    estimator.simulate_quadrotor_hover_with_DEKA(
        NSIM=300 # 300 simulation steps
    )
)

# Visualize results
visualize_trajectory_with_est(
    x_history,  
    u_history,
    theta_history, 
    theta_hat_history,
    "Parameter Estimation Results"
)

Features

• Online Parameter Estimation: Real-time parameter estimation for dynamic systems with support for:

  • Sudden parameter changes (e.g., payload pickup)
  • Gradual parameter drift
  • Measurement noise handling

• Multiple Estimation Methods:

  • Deterministic Kaczmarz Algorithm (DeKA)
  • Recursive Least Squares (RLS)
  • Extended Kalman Filter (EKF)

• Quadrotor Dynamics: Built-in support for quadrotor parameter estimation with:

  • Full 6-DOF dynamics
  • Real-time control
  • Hover and trajectory tracking

• Visualization Tools: Comprehensive plotting utilities

Performance

Based on empirical testing:

• Convergence Speed: DeKA achieves ~7x faster convergence compared to base Kaczmarz methods
• Estimation Accuracy: DeKA has 45% lower estimation error than start-of-the-art variants of RLS and KF
• Parameter Tracking: Superior tracking of sudden parameter changes (see examples)

API Documentation

DeKA Parameters

DEKA(
    num_params: int,
    damping: float = 0.1,            # Update step damping (recommended: 0.1-1.0)
    regularization: float = 1e-6,    # Numerical stability term
    smoothing_factor: float = 0.9,   # Exponential smoothing (0: none, 1: max)
    tol: float = 1e-3                # Convergence tolerance
)

Visualization

The package includes three main visualization functions:

• visualize_trajectory(): Plot position, inertia parameters, and controls
• visualize_trajectory_with_est(): Compare true vs estimated parameters
• visualize_all_parameter(): Detailed parameter tracking visualization

Command Line Interface

python -m deka.param_est --robot quadrotor --task hover --method DEKA

Options:

• --robot: quadrotor
• --task: hover, tracking
• --method: DEKA, RLS, EKF
• --update_type: immediate_update, late_update, never_update

Contributing

Contributions are welcome! Please feel free to submit a Pull Request.

Citation

If you use this code in your research, please cite:

@article{deka2025,
    title={DeKA: The Deterministic Kaczmarz Algorithm with Greedy Selection and Smoothing for Online Inertial Parameter Estimation},
    author={Chris Sha, Anupam Bhakta, Justin Jiang, Kevin Qiu, Brian Plancher},
    journal={arXiv preprint},
    year={2025}
}

Contact

For questions and support:

• GitHub Issues: Create an issue