Dynamics and Control

Modeling Sensing

Actuation

Motion Planning

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Kinematics

Dynamic

Modeling

Kinematics

Control

Sensing

ActuationDynamic

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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

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?