1. Lecture
Rigid Body Dynamics

2. Rigid Bodies Rigid bodies have both a position and orientation
Rigid bodies assume no object deformation
Rigid body motion is represented by 2 parameters
- center of mass
- orientation (rotation matrix)

3. Rigid Bodies Objects are defined in body space and transformed
by the position and orientation into world space

4. Linear Velocity The change of the center of mass over time
For a pure translation ( constant), all points
move with velocity

5. Angular Velocity The change in orientation over time
Encodes both the axis and speed of the rotation
direction encodes the axis
magnitude encodes the speed (rad/s)
But, how are and related?

6. Angular Velocity For a given vector
The columns of represent the transformed axes

7. Angular Velocity We can represent the cross product with a matrix
Therefore

8. Velocity of a Point Since a point can be represented at any time by
Total velocity can then be expressed as
Which can be rewritten as

9. Force We can apply forces to the object at any point
Total force on an object is simply
No information about where the forces are applied

10. Torque Torque describes the “rotational force”
Total torque on an object is simply
Tells us about the force distribution over the object

11. Linear Momentum Linear momentum of a particle is
Linear momentum of a rigid body is then
density
integration over the body

12. Linear Momentum Linear momentum can be simplified as follows
Assuming constant mass gives

13. Angular Momentum Angular momentum of a rigid body
inertia tensor
Taking the time derivative
Angular momentum is conserved for no torque

14. THANK YOU