Model Calibration

Open-PAV provides tools for calibrating various physical vehicle dynamics models to improve the accuracy of PAV behavior simulations.

Three-Stage Car-Following Model

The Adaptive Cruise Control (ACC) system in PAV consists of two subsystems:

  1. Upper Command Control System – Generates acceleration commands based on sensor data.
  2. Lower Motion Response System – Regulates acceleration in response to commands.

To account for real-world system delays, the Three-Stage Car-Following Model incorporates: - Stage 1: Sensor Perception Delay (\(\eta_{a,1}\)) – The time required for the vehicle to detect and process leading vehicle data. - Stage 2: Control Computation Delay (\(\eta_{a,2}\)) – The time taken to compute acceleration commands. - Stage 3: Vehicle Response Lag (\(\eta_b\)) – The mechanical delay in executing the acceleration.

Mathematical Formulation

At any given time \( t \), let: - \( p_i(t) \), \( v_i(t) \), and \( a_i(t) \) be the position, velocity, and acceleration of the following vehicle. - \( p_{i+1}(t) \), \( v_{i+1}(t) \), and \( a_{i+1}(t) \) be the position, velocity, and acceleration of the preceding vehicle. - \( s_i(t) = (p_i(t), v_i(t), a_i(t)) \) represent the state of the following vehicle.

The command acceleration at time \( t_0 + \eta_{a,1} + \eta_{a,2} \) is computed as:

\[ u_i(t_0 + \eta_{a,1} + \eta_{a,2}) = g(v_i(t_0), v_{i+1}(t_0), p_i(t_0), p_{i+1}(t_0); \theta_C) \]

where: - \( g(\cdot) \) is a control law function. - \( \theta_C \) represents the control parameters to be calibrated.

In Stage 3, due to the response lag \( \eta_b \), the acceleration is modeled as a first-order system:

\[ \eta_b \frac{d a_i(t)}{dt} + a_i(t) = u_i(t) \]

Thus, the state-space representation of the system is:

\[ \dot{s}_i(t) = A_i s_i(t) + B_i u_i(t) \]

where:

\[ A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & -\frac{1}{\eta_b} \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 0 \\ \frac{1}{\eta_b} \end{bmatrix} \]

The calibrated control function is:

\[ u_i(t) = g(v_i(t - \eta_a), v_{i+1}(t - \eta_a), p_i(t - \eta_a), p_{i+1}(t - \eta_a); \theta_C) \]

where \( \eta_a = \eta_{a,1} + \eta_{a,2} \) represents the total control delay.

Hybrid Parameter Calibration Method

To calibrate model parameters, Open-PAV implements a hybrid optimization algorithm that combines:

  1. Bayesian Optimization (BO) – Efficient global search for optimal delay parameters (\(\eta_a, \eta_b\)).
  2. Simultaneous Perturbation Stochastic Approximation (SPSA) – Fast local optimization for control gains (\(\theta_C\)).

Mathematical Formulation of Calibration

The calibration problem is defined as an optimization problem:

\[ \min_{\theta} f(\theta, D^{m}, D^{r}) \]

where: - \( \theta \) is the parameter set to be calibrated. - \( D^m \) is the model-predicted vehicle states. - \( D^r \) is the real-world vehicle data.

The objective function \( f(\cdot) \) is based on the Root Mean Square Error (RMSE):

\[ f(\theta, D^{m}, D^{r}) = \sqrt{\frac{1}{N} \sum_{j=1}^{N} \left( a_{i,m}(t_j | \theta) - a_{i,r}(t_j) \right)^2 } \]

where: - \( a_{i,m}(t_j | \theta) \) is the model-predicted acceleration. - \( a_{i,r}(t_j) \) is the real observed acceleration at time \( t_j \).

Calibration Process

  1. Data Input – Load collected vehicle trajectory data.
  2. Parameter Adjustment – Optimize both delay and control parameters using BO + SPSA.
  3. Validation – Compare calibrated model outputs with real-world observations.
  4. Export – Save parameters in a compatible format for SUMO, VISSIM, or other simulators.

Supported Models

  • Linear Models – Suitable for basic simulations.
  • IDM Models – Ideal for SUMO simulations.
  • Wiedemann-99 – Compatible with VISSIM.
  • Machine Learning-Based Models – For advanced simulations.