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:
- Upper Command Control System – Generates acceleration commands based on sensor data.
- 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:
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:
Thus, the state-space representation of the system is:
where:
The calibrated control function is:
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:
- Bayesian Optimization (BO) – Efficient global search for optimal delay parameters (\(\eta_a, \eta_b\)).
- 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:
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):
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
- Data Input – Load collected vehicle trajectory data.
- Parameter Adjustment – Optimize both delay and control parameters using BO + SPSA.
- Validation – Compare calibrated model outputs with real-world observations.
- 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.