Unit
- Earth-fixed frame: An inertial reference frame (X_e, Y_e, Z_e) used to describe absolute position and orientation. The Z_e axis points downward, following the right-hand rule. This frame is essential for global path tracking and inertial navigation.
- Vehicle-fixed frame: Attached to the vehicle's center of mass, with X_v forward, Y_v to the right, and Z_v downward. All dynamic equations are typically expressed in this frame to simplify force and moment calculations.
- Yaw, pitch, roll: Rotations around Z_v (yaw), Y_v (pitch), and X_v (roll) define the vehicle's orientation. These Euler angles are used to relate the vehicle-fixed frame to the earth-fixed frame through rotation matrices.
- Transformation matrices: Converting vectors from one frame to another requires sequential rotations. For small angles, approximations linearize the transformation, which is common in handling analysis.
- Center of gravity (CG): The point where the vehicle's mass is concentrated. All external forces and moments produce accelerations relative to the CG. Its height and longitudinal position critically affect weight transfer during acceleration and braking.
- Axes and sign conventions: Standard SAE J670 defines vehicle dynamics terminology. Positive yaw is clockwise when viewed from above, positive roll is left side down, and positive pitch is nose up.
- Degrees of freedom: A full vehicle model typically includes 6 DOF (three translations and three rotations). Simplified models (e.g., bicycle model) reduce to 3 DOF (longitudinal, lateral, yaw) for handling studies.
- Application in simulation: All multibody dynamics software (e.g., CarSim, Adams/Car) relies on these frames. Engineers must carefully define initial conditions and outputs in the correct reference frame to avoid errors.
- Tire contact patch frame: A local frame at each wheel contact point, with X along the wheel heading, Y lateral, and Z normal to the road surface. This frame is used to compute tire forces based on slip angles and slip ratios. It rotates with steering angle and suspension motion, making it essential for accurate traction and braking models.
- Aerodynamic reference frame: For high speed vehicles, aerodynamic forces and moments are often expressed in a wind axis frame aligned with the relative wind vector. This frame simplifies lift and drag calculations by separating longitudinal and lateral components. Transformations between vehicle and wind frames depend on sideslip angle and angle of attack.
System Architecture
Pattern: Coordinate Transformation Pipeline
Components:
- Inertial sensor: measures acceleration and angular velocity in the vehicle frame.
- Navigation algorithm: transforms measurements to earth frame using strapdown integration.
- State estimator: fuses sensor data with vehicle model to produce orientation and position.
Code Example
Python
import numpy as np
def rotation_matrix_yaw(yaw):
c = np.cos(yaw)
s = np.sin(yaw)
return np.array([[c, -s, 0],
[s, c, 0],
[0, 0, 1]])
# Transform velocity from vehicle to earth frame
v_vehicle = np.array([10.0, 0.0, 0.0]) # m/s forward
yaw_angle = np.radians(15)
R = rotation_matrix_yaw(yaw_angle)
v_earth = R @ v_vehicle
print('Earth-frame velocity:', v_earth)The function creates a yaw rotation matrix. The example transforms a forward velocity of 10 m/s by 15 degrees of yaw, giving components in the earth frame. This is the basis for dead-reckoning navigation.
Quick Revision
- Earth-fixed and vehicle-fixed frames are essential for describing motion.
- Yaw, pitch, roll Euler angles define orientation between frames.
- Transformation matrices convert vectors between frames using rotation sequences.
- Center of gravity location governs weight transfer and dynamic response.
- SAE J670 standardizes sign conventions and terminology.