Vehicle Dynamics & Control - 05 Kinematic bicycle model

Kinematic Bicycle Model Derivation

Introduction to the Kinematic Bicycle Model

• The kinematic bicycle model is a simplified version of the kinematic four-wheeler model, crucial in vehicle dynamics and control.

• In this model, the two rear wheels and two front wheels are combined into one imaginary rear wheel and one imaginary front wheel.

Structure of the Kinematic Bicycle Model

• The instantaneous center of rotation for both models remains identical, ensuring similar behavior under kinematic assumptions (zero slip angle).

• A significant advantage of the bicycle model is its single front wheel steering angle compared to two angles in the four-wheel model.

Equations of Motion

• The derivation begins with defining distances: L_F (distance from reference point C to front wheel) and L_R (to rear wheel), where L_F + L_R = L, representing the wheelbase.

• The steering angle Delta is defined as the difference between the vehicle's heading angle and that of the front wheel.

Velocity Vectors and Slip Angle

• At any moment, velocity vectors at both wheels are perpendicular to their connection line with point O.

• The side slip angle beta, which depends on reference point C's position, can equal Delta or be zero based on chosen reference points.

Vehicle Dynamics Description

• Vehicle motion is described using inertial coordinates x, y, and yaw angle psi.

• To derive differential equations for vehicle motion over time T, constant absolute velocity is assumed; changes in coordinates relate directly to components of velocity vector V.

Angular Velocity and Geometry Considerations

• Change in yaw angle dotpsi equals angular velocity around point O (Omega), expressed as V/R, where R is radius from O to C.

• Finding expressions for side slip angle beta and radius R requires geometric analysis involving triangles formed by steering angles.

Trigonometric Relationships in Motion

• Using trigonometry, relationships between lengths related to steering angles are established; specifically, tangent functions help express these relationships mathematically.

Kinematic Bicycle Model: Deriving Key Equations

Derivation of the Tangent of Beta

• The tangent of beta is expressed as the ratio of lengths L_R and R . By substituting the expression for R , we derive that:

• tan(beta) = L_R/L_F + L_R cdot tan(Delta).

• Taking the arctangent of both sides provides an explicit formula for beta in terms of variables from the kinematic bicycle model.

Radius Calculation

• To find the radius R , we analyze a small triangle where:

• The cosine of beta relates to R_textbar and R.

• We derive that:

• 1/R = 1/R_textbar cdot cos(beta).

Summary of Motion Equations

• The equations of motion for the kinematic bicycle model are established through three nonlinear differential equations, which describe vehicle states (X, Y, and psi):

• The steering angle ( Delta) serves as a single input.

• The side slip angle ( beta) is derived from a static trigonometric transformation based on vehicle reference point location.

Consideration of Velocity

• A final note addresses the assumption regarding constant velocity ( V):

• In many scenarios, this velocity can be treated as a second input to the kinematic bicycle model.