Dynamics and Control - uni-goettingen.de€¦ · Control Actuator dynamics. Control Actuator...
Transcript of Dynamics and Control - uni-goettingen.de€¦ · Control Actuator dynamics. Control Actuator...
Dynamics and Control
Modeling Sensing
Actuation
Motion Planning
AIControl
Kinematics
Dynamic
Modeling
Kinematics
Control
Sensing
ActuationDynamic
Today!Today!
Motion Planning
AI
q=(q1 , q2 ⋯ , q p−1 , qp)
Review Kinematics Generalized coordinates
T (q)
J (q)
Homogeneous Transform
JacobianV=J (q) q̇
Pi=T i , j(q)P j
Conservation of momentum
Linear momentum Angular momentum
p⃗=m v⃗ L⃗= r⃗× p⃗
F⃗=d p⃗dt
=m a⃗
τ⃗=d L⃗dt
=I ⃗̇ω=I α⃗
L⃗= r⃗×m v⃗
L⃗= r⃗×(m ω⃗× r⃗ )
v⃗=ω⃗× r⃗
ω⃗⊥ r⃗
L⃗=mr2 ω⃗
L⃗=I ω⃗
Moment of Inertia(scalar value)
m
Newton's second law
Moment of Inertia
L⃗= r⃗×m v⃗
L⃗= r⃗×(m ω⃗× r⃗ )
v⃗=ω⃗× r⃗
ω⃗⊥ r⃗
L⃗=mr2 ω⃗
L⃗=I ω⃗
Moment of Inertia(scalar value)
m
I=mr2 τ⃗=I α⃗
I=∑mi r i2
I=∫ρ(r)r2 dV I=mL2
12
example
Moment of Inertia
Moment of Inertia(Tensor)
I s=sT I s s: axis of rotationSpecific axis: (scalar)
Transform:(matrix)
R: Rotation matrix
Dynamics One dimensional system
Kinetic Energy
Newton's second law
Y
m
Dynamics One dimensional system
Potential Energy
Newton's second law
m
Y
Dynamics One dimensional system
Lagrangian
Dynamics One dimensional system
Lagrangian
=f (q1 , q2 , ... , qn)
Calculate
Dynamics n-dimensional system
Translational Rotational
Dynamics n-dimensional system
[ vnωn ]=J n θ̇=J n q̇
6×n
[ v4ω4 ]=J 4 θ̇=J 4 q̇
vi = J vi q̇
ωi = Jωi q̇3×n
mi : mass of link i
Dynamics n-dimensional system
vi = J vi q̇
ωi = Jωi q̇
D (q)
RotationalTranslational
Inertia matrix
Dynamics n-dimensional system
Dynamics n-dimensional system
Dynamics
quadratic Independent ofq̇
n-dimensional system
Dynamics n-dimensional system
Dynamics
InertialCoriolis and centrifugal
gravity Input torque
n-dimensional system
Dynamics example
Dynamics example
Control System
Control System
Control Open-loop control
Control Open-loop control
Control Output feedback
Sensors should be precise
Control Closed-loop control
Control Closed-loop control
Control Closed-loop control
Control Closed-loop control
How to design the controller?
Control Closed-loop control
How to design the controller?
Non-linear system
Approaches
Linear model, nonlinearity as a disturbance
Non-linear model
Control
Actuator(DC motor)Control signal
Actuator dynamics
Actuator dynamicsControl
Actuator dynamicsControl
Actuator dynamicsControl
Actuator dynamicsControl
Control
Actuator(DC motor)Control signal
Actuator dynamics
Control
Actuator(DC motor)Control signal
Actuator dynamics
Set-point trackingControl
Set-point trackingControl
Set-point trackingControl
Motor inertia + link inertia other link inertia + coriolis + centrifugal + gravity
PD controlControl
Control
Constant d
PD control
PD control performanceControl
Control PD control performance
Control
Reduce steady state error
Increase Kp
PID
PD control performance
Control PID control
Multivariable controlControl
PD controllerControl
Steady state error
Gravitational termsShould be known!
Computed torque controlControl
Computed torque controlControl
Computed torque controlControl
Steady state error Unmodeled dynamics (friction)
Computed torque controlControl
Dynamic is assumed to be known!
Unmodeled dynamics
Uncertainty in parameters
Friction
Joint and link elasticity
estimation
Robust and Adaptive control
Modeling Sensing
Actuation
Control
Kinematics
Dynamic
Questions?