Online parameter estimation for the Crazyflie quadcopter through an EM algorithm

Technical Summary: Online Parameter Estimation for the Crazyflie Quadcopter through an EM Algorithm

Problem Statement
Quadcopters are increasingly utilized in diverse fields such as disaster rescue, agriculture, and aerial photography due to their low cost and reliability. However, effective model-based control relies on accurate knowledge of system parameters, such as mass and inertia matrices. Experimental measurement of these parameters is often laborious, requires specific hardware, and may yield inaccurate results. Furthermore, parameters can change dynamically during flight (e.g., when carrying payloads). Additionally, real-world flight environments introduce uncertain factors like wind and sensor inaccuracies, necessitating robust state estimation. This paper addresses the challenge of estimating quadcopter parameters directly from flight data in the presence of random noise, comparing both offline and online approaches.

Methodology
The paper proposes a framework integrating stochastic modeling, state estimation, and parameter learning:

1. System Modeling: The quadcopter dynamics are modeled using Stochastic Differential Equations (SDEs) to account for random process noise (e.g., wind, thrust fluctuations) and measurement noise. The state vector includes position, velocity, Euler angles, and angular velocity.
2. State Estimation: An Extended Kalman Filter (EKF) is employed to estimate the system state from noisy sensor observations. The EKF linearizes the nonlinear system dynamics to filter out noise and recover missing state information (such as velocity and Euler angles) when sensors provide partial data.
3. Control: A Linear Quadratic Gaussian (LQG) controller is implemented to track reference trajectories. The controller utilizes the state estimates from the EKF and calculates the required thrust and torque based on the current parameter estimates.
4. Parameter Estimation (EM Algorithm): The core contribution is the application of the Expectation Maximization (EM) algorithm to estimate unknown parameters (mass and inertia matrix).
   • E-Step: Uses the EKF for forward recursion and a Rauch-Tung-Striebel (RTS) smoother for backward recursion to compute the posterior distribution of the hidden states given the observations and current parameter estimates.
   • M-Step: Maximizes the expected log-likelihood function to update the parameter estimates (mass and inertia).
5. Simulation Scenarios: Two simulation models were developed:
   • Offline Estimation: Parameters are estimated after a complete flight trajectory is recorded in a database.
   • Online Estimation: The EM module is embedded within the flight loop. It iteratively updates parameters using only the most recent 800 observations to limit computational load, allowing the system to adapt in real-time.
6. Sensor Configurations: Three sensor schemes were tested:
   • Sensor A: Observes position, acceleration, and angular velocity (requires EKF for state recovery).
   • Sensor B: Full-state observation (position, velocity, Euler angles, angular velocity).
   • Sensor C: Partial observation (position, velocity, angular velocity; lacks Euler angles).

Key Results
The study presents simulation results based on 20 cycles for both offline and online scenarios:

• State Estimation: The EKF successfully filtered noise, achieving a mean position error of $1.02 \times 10^{-2}$ m and a mean Euler angle error of $1.04 \times 10^{-2}$ rad, outperforming benchmarks cited from previous literature.
• Offline Parameter Estimation:
  • Using EKF state estimates or full-state observations, the estimated mass converged closely to the true value (0.18 kg), with ranges of $(0.1797, 0.1802)$ kg and $(0.1788, 0.1801)$ kg, respectively.
  • Inertia matrix components ($I_{xx}, I_{yy}, I_{zz}$) also converged near true values.
  • Partial state observations (lacking Euler angles) resulted in less accurate mass estimates, converging to a range of $(0.1896, 0.196)$ kg, which is 5.33% to 8.89% higher than the true mass. The authors attribute this to the dependency of mass estimation on Euler angles in the derived update formula.
• Online Parameter Estimation:
  • The online method successfully estimated parameters in real-time but exhibited a larger convergence range compared to offline methods.
  • The mass convergence range for online EKF-based estimation was approximately 2.2 times larger than the offline range.
  • Inertia matrix convergence ranges were significantly wider (2.6 to 6 times larger) in the online setting, particularly for partial observations.
  • The authors note that the online module's limitation to 800 observations and its interaction with the controller contribute to the reduced precision compared to offline processing of the full trajectory.

Significance and Claims
The paper claims to demonstrate that the Expectation Maximization algorithm is effective for learning quadcopter parameters directly from flight data, eliminating the need for laborious physical measurements. The study highlights that while online parameter estimation is feasible and allows for adaptive control, it currently offers a slightly larger range of convergence values (lower precision) compared to offline estimation. This difference is attributed to the limited data window used in online processing and the impact of real-time parameter updates on the control loop. The work establishes a foundation for adaptive control in uncertain environments but acknowledges that the results are currently limited to simulation. The authors suggest that future work should involve real-world flight verification, removal of small-angle assumptions, and the exploration of alternative estimation algorithms.