Download Dynamics

Dynamics
Motion with Regard to Mass
Particle Dynamics
Mass concentrated in point
Newton’s Equation Governs Motion
f=Mx
Rigid Body Dynamics
Two equations govern motion:
Newton’s Equation for Translations
F= M x
Euler’s Equation for Rotational Motion
T=Iw+wIw
where I is the interial tensor that
describes the distribution of mass
Dynamics of Links
Axis i+1
Newton’s Equation
Axis i
fi+1
Fi
fi+1
Dynamics of Links
Axis i+1
Euler’s Equation
Axis i
ti+1
fi+1
ti
fi+1
Ti
Bodies in space
Conservation of Momentum
0=Mx
A body in motion remains in motion
Conservation of Angular Momentum
0=Iw
The relationship between angular momentum
and orientation is tricky
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
Current position
f
Ground
Desired position
Controllers
 What force should we apply to move the box to
the destination?
Current position
f
Ground
Desired position
Proportional Control
 A control law, function, or algorithm for
computing forces (or torques).
 Force is proportional to distance to goal:
F = Kp ( xd – x)
Workhorse of robotics and animation
Problem with Proportional Control
 Overshoot Goal
xd
x
x
time
Solution: Damping
 Proportional Derivative Controller
F = Kp ( xd – x) – Kv x
virtual friction
xd
x
x
time
Problem: How Much Damping?
 Too little damping leads to overshoot
x
xd
time
 Too much damping leads to sluggishness
xd
x
time
Critical Damping
 Constrain the constants such that:
Kv 2 - 4Kp = 0
Just right:
No overshoot
Fastest possible approach (given gain Kp)
There’s always something else
 What about considering other forces (such as
gravity)?
desired position
F= M x - G
G
The PD controller will
converge to a point where
Kp(xd-x) = G
current position
F?
PID Control
 Proportional Integral Derivative Control
F = Kp ( xd – x) – Kv x + Ki  ( xd – x) dt