Technical Summary: Human as an Actuator Dynamic Model Identification

Problem Formulation

The paper addresses the challenge of characterizing human pilot response as a dynamic actuator within a closed-loop control system. When a human interacts with a physical vehicle (e.g., an aircraft or space vehicle), their reaction time and response characteristics introduce dynamics that affect the overall system stability and performance. This is critical for safety analyses, such as computing "H-V" (Height-Velocity) diagrams for helicopters, where the size of unsafe flight regimes depends directly on the pilot's reaction capabilities.

Existing literature has largely relied on low-order, linear, time-invariant models identified in the frequency domain, typically using single-input, single-output (SISO) assumptions. A significant limitation in previous approaches (e.g., [5]) is that they often fit pilot models to the pilot's raw commands (e.g., joystick deflection) rather than the resulting vehicle response. The authors argue that a pilot model should be evaluated based on its ability to achieve a desired vehicle state (e.g., altitude tracking) rather than simply replicating the input commands.

Methodology

The authors propose a time-domain method for estimating human pilot dynamic models using constrained trajectory optimization. The core of the methodology involves the following steps:

1. System Modeling:

   • Vehicle Dynamics: Modeled as $\dot{x}_v = f_v(x_v, u_v)$, where $x_v$ is the vehicle state and $u_v$ is the control input.
   • Pilot Dynamics: Modeled as a dynamic system $\dot{x}_p = f_p(x_p, u_p; \theta)$ and $y_p = h_p(x_p; \theta)$, where $\theta$ represents the unknown parameters to be estimated.
   • Joint System: The pilot and vehicle are coupled into a joint state vector $x = [x_v, x_p]^T$. The pilot's input is defined as the error between the desired target $\hat{y}$ and the current vehicle output $y_v$.

2. Data Collection:

   • Data is collected via a flight simulator where a pilot maneuvers a vehicle toward a visual marker representing a target position.
   • The system records the vehicle output $y_v(t)$ and the pilot's joystick commands over a finite horizon $[0, T]$ at discrete time steps.
   • The method does not require direct observation of the internal system states ($x$); instead, it uses an implicit formulation.

3. Optimization Framework:

   • The identification problem is formulated as a constrained optimization problem to find the parameter vector $\theta$ and the state trajectory $X$ that minimize the error between the measured outputs and the model-predicted outputs.
   • Objective Function: Minimize the squared $L_2$ norm of the difference between the measured output error ($\delta Y$) and the model output error ($H$), assuming Gaussian noise.
   • Constraints:
     • Dynamics Consistency: The state trajectory must satisfy the joint system dynamics $\dot{x} = f(x; \hat{y}, \theta)$. This is enforced using a discrete differentiation matrix $D$ to approximate the time derivative.
     • Physical/Task Constraints: Inequality constraints ($g(X, \theta)$) are applied to ensure physical realism, such as limiting joystick displacement and ensuring the stability of the pilot model (e.g., via Routh-Hurwitz criteria).
   • Multi-Trial Aggregation: The framework allows for the simultaneous estimation of a single, common model $\theta$ across multiple experimental trials ($M$), accounting for natural variations in human performance.

4. Implementation:

   • The optimization is solved using the Non-Linear Programming (NLP) solver IPOPT in MATLAB.
   • Automatic differentiation is used to compute the constraint Jacobian and Hessian.

Experimental Results

The proposed method was validated using a quadcopter drone simulation scenario:

• Task: The pilot was tasked with moving a quadcopter from a hover to a rest position at a specific target distance ($\hat{y}$).
• Vehicle Model: A 6-state model focusing on the pitch axis dynamics, including propeller thrust and rotational speeds.
• Pilot Model: A 3rd-order, single-input, single-output linear system defined by six parameters ($\theta \in \mathbb{R}^6$). Stability constraints were enforced to ensure the model remained Hurwitz.
• Data: Three independent runs were recorded for a 10m target and three runs for a 20m target.
• Outcome: A single model was estimated that successfully described the data from all six runs.
  • The estimated model parameters yielded a stable system.
  • Figure 5 in the paper demonstrates that the estimated model's response (red and yellow lines) converges to the goal positions, closely matching the empirical data (blue lines) for both target distances.
  • The results indicate that the model is robust to natural variations in human control across different trials and target distances.

Key Contributions and Significance

The paper claims the following contributions:

1. Time-Domain Identification: Unlike previous works that rely on frequency-domain approaches, this method solves for the model entirely in the time domain using trajectory optimization.
2. Output-Centric Regularization: The method incorporates vehicle dynamics directly into the estimation process. Instead of fitting the model to pilot commands, it fits the model to the resulting vehicle response. This acts as a regularization, ensuring the identified model is capable of achieving the desired task (e.g., reaching a target altitude) rather than just mimicking input signals.
3. Multi-Trial Robustness: The formulation allows for the aggregation of multiple data collections to find a single, common dynamic model, effectively handling the natural variability inherent in human performance.
4. Generalizability: The approach is designed to be easily generalized to multi-input, multi-output (MIMO) and nonlinear pilot models, making it applicable to a broad range of flight operations.

The authors conclude that this method provides a robust framework for characterizing human pilots as actuators, which is essential for accurate dynamic analysis of piloted vehicles and safety-critical calculations like H-V diagrams. Future work is noted to include experiments with MIMO pilot scenarios.