1. Dynamics

2. Motion with Regard to Mass Particle Dynamics Mass concentrated in point Newton’s Equation Governs Motion f = M x

3. Rigid Body Dynamics Two equations govern motion: Newton’s Equation for Translations Euler’s Equation for Rotational Motion where I is the interial tensor that describes the distribution of mass F= M x  = I 

4. Dynamics of Links Axis i+1 Axis i FiFi f i+1 Newton’s Equation

5. Dynamics of Links Axis i+1 Axis i TiTi f i+1 Euler’s Equation ii  i+1

6. Bodies in space Conservation of Momentum A body in motion remains in motion Conservation of Angular Momentum 0 = I   The relationship between angular momentum  and orientation is tricky 0 = M x

7. Making Them Move In the real world, we do not directly control the kinematic properties of object. We indirectly control position, velocity, and acceleration by exerting forces and torques Ground f Current position Desired position

8. Controllers What force should we apply to move the box to the destination? Ground f Current position Desired position

9. Proportional Control A control law, function, or algorithm for computing forces (or torques). Force is proportional to distance to goal: F = K p ( x d – x) Workhorse of robotics and animation

10. Problem with Proportional Control Overshoot Goal x x time xdxd

11. Solution: Damping Proportional Derivative Controller F = K p ( x d – x) – K v x virtual friction x x time xdxd

12. Problem: How Much Damping? Too little damping leads to overshoot Too much damping leads to sluggishness x time xdxd x xdxd

13. Critical Damping Constrain the constants such that: K v 2 - 4K p = 0 Just right: No overshoot Fastest possible approach (given gain K p )

14. There’s always something else What about considering other forces (such as gravity)? current position desired position G F? The PD controller will converge to a point where K p (x d -x) = G F= M x - G

15. PID Control Proportional Integral Derivative Control F = K p ( x d – x) – K v x + K i  ( x d – x) dt