Dynamics and Control - uni-goettingen.de€¦ · Control Actuator dynamics. Control Actuator...

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Dynamics and Control

Transcript of Dynamics and Control - uni-goettingen.de€¦ · Control Actuator dynamics. Control Actuator...

Page 1: Dynamics and Control - uni-goettingen.de€¦ · Control Actuator dynamics. Control Actuator Control signal (DC motor) Actuator dynamics. Control Actuator Control signal (DC motor)

Dynamics and Control

Page 2: Dynamics and Control - uni-goettingen.de€¦ · Control Actuator dynamics. Control Actuator Control signal (DC motor) Actuator dynamics. Control Actuator Control signal (DC motor)

Modeling Sensing

Actuation

Motion Planning

AIControl

Kinematics

Dynamic

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Modeling

Kinematics

Control

Sensing

ActuationDynamic

Today!Today!

Motion Planning

AI

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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

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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

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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

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Moment of Inertia

Moment of Inertia(Tensor)

I s=sT I s s: axis of rotationSpecific axis: (scalar)

Transform:(matrix)

R: Rotation matrix

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Dynamics One dimensional system

Kinetic Energy

Newton's second law

Y

m

Page 9: Dynamics and Control - uni-goettingen.de€¦ · Control Actuator dynamics. Control Actuator Control signal (DC motor) Actuator dynamics. Control Actuator Control signal (DC motor)

Dynamics One dimensional system

Potential Energy

Newton's second law

m

Y

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Dynamics One dimensional system

Lagrangian

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Dynamics One dimensional system

Lagrangian

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=f (q1 , q2 , ... , qn)

Calculate

Dynamics n-dimensional system

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Translational Rotational

Dynamics n-dimensional system

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[ 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

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vi = J vi q̇

ωi = Jωi q̇

D (q)

RotationalTranslational

Inertia matrix

Dynamics n-dimensional system

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Dynamics n-dimensional system

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Dynamics

quadratic Independent ofq̇

n-dimensional system

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Dynamics n-dimensional system

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Dynamics

InertialCoriolis and centrifugal

gravity Input torque

n-dimensional system

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Dynamics example

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Dynamics example

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Control System

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Control System

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Control Open-loop control

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Control Open-loop control

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Control Output feedback

Sensors should be precise

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Control Closed-loop control

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Control Closed-loop control

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Control Closed-loop control

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Control Closed-loop control

How to design the controller?

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Control Closed-loop control

How to design the controller?

Non-linear system

Approaches

Linear model, nonlinearity as a disturbance

Non-linear model

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Control

Actuator(DC motor)Control signal

Actuator dynamics

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Actuator dynamicsControl

Page 34: Dynamics and Control - uni-goettingen.de€¦ · Control Actuator dynamics. Control Actuator Control signal (DC motor) Actuator dynamics. Control Actuator Control signal (DC motor)

Actuator dynamicsControl

Page 35: Dynamics and Control - uni-goettingen.de€¦ · Control Actuator dynamics. Control Actuator Control signal (DC motor) Actuator dynamics. Control Actuator Control signal (DC motor)

Actuator dynamicsControl

Page 36: Dynamics and Control - uni-goettingen.de€¦ · Control Actuator dynamics. Control Actuator Control signal (DC motor) Actuator dynamics. Control Actuator Control signal (DC motor)

Actuator dynamicsControl

Page 37: Dynamics and Control - uni-goettingen.de€¦ · Control Actuator dynamics. Control Actuator Control signal (DC motor) Actuator dynamics. Control Actuator Control signal (DC motor)

Control

Actuator(DC motor)Control signal

Actuator dynamics

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Control

Actuator(DC motor)Control signal

Actuator dynamics

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Set-point trackingControl

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Set-point trackingControl

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Set-point trackingControl

Motor inertia + link inertia other link inertia + coriolis + centrifugal + gravity

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PD controlControl

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Control

Constant d

PD control

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PD control performanceControl

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Control PD control performance

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Control

Reduce steady state error

Increase Kp

PID

PD control performance

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Control PID control

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Multivariable controlControl

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PD controllerControl

Steady state error

Gravitational termsShould be known!

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Computed torque controlControl

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Computed torque controlControl

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Computed torque controlControl

Steady state error Unmodeled dynamics (friction)

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Computed torque controlControl

Dynamic is assumed to be known!

Unmodeled dynamics

Uncertainty in parameters

Friction

Joint and link elasticity

estimation

Robust and Adaptive control

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Modeling Sensing

Actuation

Control

Kinematics

Dynamic

Questions?