Vehicle Dynamics & Control - 03 Review: Kinematics of a rigid body

Kinematic Bicycle Model: A Review of Rigid Body Kinematics

Introduction to Rigid Body Kinematics

• The discussion begins with a review of the kinematics of a rigid body, focusing on the position of point P in space represented by vector R.

• Position is defined using three coordinates (X(t), Y(t), Z(t)), indicating that point P moves through space along a trajectory dependent on time.

• The first two time derivatives of the position vector are introduced: velocity (RP dot) and acceleration (RP double dot). Velocity represents the change in position over time, while acceleration indicates changes in velocity.

Definition and Motion Description of Rigid Bodies

• A rigid body is defined as a collection of mass points that maintain fixed relative positions over time. This allows for more efficient motion description.

• Instead of describing every point's motion, we can describe one reference point's motion plus the relative motions of other points.

• The position vector for an arbitrary point P within the rigid body is expressed as the sum of the reference point C's position vector and its relative position to C.

Angular Velocity and Its Implications

• Due to rigidity, all points perform relative rotations about reference point C. This leads to defining angular velocity (Ω), which indicates rotation speed and axis direction.

• Angular velocity is independent of the choice of reference point; it remains constant across different frames within the rigid body.

• The velocity for any arbitrary point P can be calculated using both the reference point's velocity and angular velocity via cross products.

Degrees of Freedom in Rigid Body Motion

• A rigid body has six degrees of freedom: three positional coordinates for a reference point plus three angles representing orientation.

• Correspondingly, there are three velocities (linear and angular) associated with these degrees, allowing comprehensive movement analysis.

Instantaneous Center of Rotation

• Any chosen reference point can serve as a basis for analysis; however, there exists an instantaneous center where its velocity vector equals zero at any moment.

• This center signifies pure rotational movement around it. Understanding this concept aids in analyzing various mechanical systems effectively.

Example: Turning Wheels

What is the Instantaneous Center of Rotation?

Understanding the Instantaneous Center of Rotation

• The instantaneous center of rotation for a wheel is its center point, as it has a zero velocity vector in the XY coordinate system.

• To find the velocity vector at any point on the wheel, such as point P at the top, we use the formula Omega times R_oP , resulting in a vector perpendicular to both Omega and R_oP .

• The properties of cross products indicate that velocity vectors along a line from the instantaneous center will increase proportionally with distance from this center.

Analyzing Rolling Motion

• In cases where a wheel rolls without slipping on fixed ground, only one point (the contact point with the ground) has zero velocity in terms of XY coordinates.

• This contact point becomes the instantaneous center of rotation due to no slip conditions; all other points rotate around this contact point.

Velocity Vectors and Magnitudes

• For any rigid body point like P (center of wheel), its velocity vector points right with magnitude Omega R ; for another point like P', it can be calculated similarly.

• Each chosen reference point yields a velocity vector pointing right, proportional to its distance from the instantaneous center.

Conceptualizing Instantaneous Centers Over Time

• The first case maintains a constant center of rotation over time, while in rolling scenarios, this center changes continuously—hence termed "instantaneous."

• The set of all potential instantaneous centers for any rigid body is called "pole hold," while those in an XY coordinate system are referred to as "her pole hold."

Reflection on Pole Hold and Her Pole Hold