Lecture 17:Robot Control

Katie DC

Oct. 31, 2017

Mod. Rob. Ch 11

Admin

• Reflection 3 due Sunday 11/10 at midnight

• Will post Quiz 3 rubric soon

• Project:• Project Update 4 due Sunday 11/17 at midnight

• Will post final project rubric soon for presentation and final report

• Note that presentations are 12/3 and 12/5, right after fall break

• Will post extra credit video rubric soon

Historical Fun Fact

Meet Shakey the Robot:An Experiment in Robot Planning and Learning

Developed by Stanford Research Institute (SRI)

1. An operator types the command "push the block off the platform" at a computer console.

2. Shakey looks around, identifies a platform with a block on it, and locates a ramp in order to reach the platform.

3. Shakey then pushes the ramp over to the platform, rolls up the ramp onto the platform, and pushes the block off the platform.

4. Mission accomplished.

Control Paradigm

Error Dynamics

Feedback Control

PID Controllers

• Proportional 𝑢 = 𝑘𝑝𝑒

• Integral 𝑢 = 𝑘𝑖∫ 𝑒 𝜏 𝑑𝜏

• Derivative 𝑢 = 𝑘𝑑 ሶ𝑒

𝑘𝑝 UE

𝑘𝑖1

𝑠UE

𝑘𝑑𝑠 UE

U

Error Dynamics (1)

Error Dynamics (2)

Linear Error Dynamics (1)

Linear Error Dynamics (2)

Stability of Linear Systems

First Order Dynamics

Second Order Dynamics

Second Order Dynamics: Cases

• Overdamped: 𝜁 > 1• Roots 𝑠1 and 𝑠2 are distinct

• 𝜃𝑒 𝑡 = 𝑐1𝑒𝑠1𝑡 + 𝑐2𝑒

𝑠2𝑡

• Time constant is the less negative root

• Critically damped: 𝜁 = 1• Roots 𝑠1 and 𝑠2 are equal and real

• 𝜃𝑒 𝑡 = (𝑐1+𝑐2𝑡)𝑒−𝜔𝑛𝑡

• Time constant is given by 1/𝜔𝑛

• Underdamped: 𝜁 < 1• Roots are complex conjugates:

𝑠1,2 = −𝜁𝜔𝑛 ± 𝑗𝜔𝑛 1 − 𝜁2

• 𝜃𝑒 𝑡 = (𝑐1cos𝜔𝑑𝑡 + 𝑐2 sin𝜔𝑑𝑡)𝑒−𝜁𝜔𝑛𝑡

Simple Damped Spring System

ሷ𝑥 +𝑏

𝑚ሶ𝑥 +

𝑘

𝑚𝑥 = 𝑢

ሷ𝑥 + 2𝜉𝜔0 ሶ𝑥 + 𝜔02𝑥 = 𝑢

𝜉 damping ratio

𝜔0 natural frequency

𝑚 ሷ𝑥 + 𝑏 ሶ𝑥 + 𝑘𝑥 = 𝐹

Simple Damped Spring System

ℒ 𝑚 ሷ𝑥 + 𝑏 ሶ𝑥 + 𝑘𝑥 =

𝑚𝑠2𝑋 𝑠 + 𝑏𝑠𝑋 𝑠 + 𝑘𝑋 𝑠

Transfer Function:𝑋 𝑠

𝑈(𝑠)=

1

𝑚𝑠2 + 𝑏𝑠 + 𝑘

Poles:

𝑠 =−𝑏 ± 𝑏2 − 4𝑚𝑘

2𝑚

𝑚 ሷ𝑥 + 𝑏 ሶ𝑥 + 𝑘𝑥 = 𝑢

Undamped Case: 𝑏 = 0

Overdamped Case: 𝑏2 − 4𝑚𝑘 > 0

Note that when 𝑏2 − 4𝑚𝑘 = 0 the system is critically damped.

Underdamped Case: 𝑏2 − 4𝑚𝑘 < 0

Sorry for not having a video!

With Feedback Control