module 7 slides

School of Electrical and Computer Engineering

Control of Mobile Robots

Dr. Magnus Egerstedt Professor School of Electrical and Computer Engineering

Module 7 Putting It All Together

How make mobile robots move in effective, safe, predictable, and collaborative ways using modern control theory?

•  We need to understand when and how our models are relevant!

Lecture 7.1 – Approximations and Abstractions

Magnus Egerstedt, Control of Mobile Robots, Georgia Ins<tute of Technology 7.1.1

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cFW i > 0

kx� x

o

k = � and

d� := kx� xgk

kx� x

o

k = � andhuGTG, u

ccFW i > 0

ucFWucc

FW

kx� x

o

k < �

kx� xgk < d� and

uAO

kx� xgk �

u = 0

d� := kx� xgk

kx� x

o

k < �kx� x

o

k = � andhuGTG, u

ccFW i > 0

d� := kx� xgk

kx� x

o

k = � andhuGTG, u

cFW i > 0

d� := kx� xgk

huAO, uGTGi > 0

•  We need to understand when and how our models are relevant!

Lecture 7.1 – Approximations and Abstractions


“Slow and steady wins the race”

Don’t rush it when doing the quizzes…

•  Dynamics:

•  Sensors:

Main Assumptions Made So Far


x = u, x 2 <2

(d1,�1)

(d2,�2)

More or less ok…

Not even close to being reasonable!

•  Recall the unicycle model (e.g., for describing differential drive mobile robots)

What’s The Problem?


x = u, x 2 <2

�

(x, y)

8<

:

x = v cos�y = v sin�˙� = ⇥

•  How do we make a unicycle robot “act” like ?

Next Few Lectures…


x = u

•  We have a problem: Even with a simple robot model, the navigation architecture becomes rather involved

•  Would like to be able to reuse this while allowing for more realistic robot models

Lecture 7.2 – A Layered Architecture


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cFW i > 0

kx� x

o

k = � and

d� := kx� xgk

kx� x

o

k = � andhuGTG, u

ccFW i > 0

ucFWucc

FW

kx� x

o

k < �


uAO

kx� xgk �

u = 0

d� := kx� xgk

kx� x

o

k < �kx� x

o

k = � andhuGTG, u

ccFW i > 0

d� := kx� xgk

kx� x

o

k = � andhuGTG, u

cFW i > 0

d� := kx� xgk

huAO, uGTGi > 0

All Good Things Come in Threes


•  Standard navigation systems are typically decoupled along three different levels of abstraction: –  Strategic Level: Where to go (high-level, long-term)? –  Operational Level: Where to go (low-level, short-term)? –  Tactical Level: How to go there?

All Good Things Come in Threes


•  Or slightly less militaristic: –  High-Level Planning: Where should the (intermediary)

goal points be? –  Low-Level Planning: Which “direction” to move in-

between goal points? –  Execution: How make the robot move in those

directions?

Not in this course!

Use the navigation architecture!

Control design with reference signal!

High-Level Planning


•  There are many AI methods (e.g., Dijkstra, Dynamic Programming, A*, D*, RRT…) for doing this!

Low-Level Planning


•  We already know how to do this! Assume that

and get to work! •  The “output” is a desired direction (and magnitude) of

travel

x = u, x 2 <2

Execution-Level


•  This is where we make the unicycle (or any other mobile robot) act like a simpler system over which we are performing the low-level planning

The Architecture


PLAN

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cFW i > 0

kx� x

o

k = � and

d� := kx� xgk

kx� x

o

k = � andhuGTG, u

ccFW i > 0

ucFWucc

FW

kx� x

o

k < �


uAO

kx� xgk �

u = 0

d� := kx� xgk

kx� x

o

k < �kx� x

o

k = � andhuGTG, u

ccFW i > 0

d� := kx� xgk

kx� x

o

k = � andhuGTG, u

cFW i > 0

d� := kx� xgk

huAO, uGTGi > 0

TRACK

Intermediary Waypoints

Reference trajectory

Actual trajectory

x = f(x, u), u = g(x, r)

•  Next time: Let’s do this for the differential-drive mobile robot

•  How should we design the tracker when the robot is a differential-drive mobile robot?

Lecture 7.3 – Differential-Drive Trackers


PLAN TRACK



Actual trajectory

Recap: The Model

7.3.2

L

R

vrv`

�

(x, y)

8<

:

x = v cos�y = v sin�˙� = ⇥

8>>>><

>>>>:

x =

R2 (vr + v�) cos�

y =

R2 (vr + v�) sin�

˙� =

RL (vr � v�)

vr =2v + �L

2R

v` =2v � �L

2R

Magnus Egerstedt, Control of Mobile Robots, Georgia Ins<tute of Technology

•  How drive the robot in a specific direction?


Recap: Dealing With Angles

e = �d � �, ⇥ = PID(e)

(x, y)

�d

8<

:

x = v cos�y = v sin�˙� = ⇥

PID(e) = KP e(t) +KI

Z t

0e(�)d� +KD e(t)

•  Let the output from the planner be


Adding In The Speed Component

(x, y)

u =

u1

u2

�(x = u)

�d = atan

✓u2

u1

◆8<

:

x = v cos�y = v sin�˙� = ⇥

u?

v = kuk ) v =qu21 + u2

2

px2

+ y2 =

qv2 cos2 �+ v2 sin2 � = v

The Complete Differential-Drive Architecture


PLAN

uGTGhuGTG, u

cFW i > 0

kx� x

o

k = � and

d� := kx� xgk

kx� x

o

k = � andhuGTG, u

ccFW i > 0

ucFWucc

FW

kx� x

o

k < �


uAO

kx� xgk �

u = 0

d� := kx� xgk

kx� x

o

k < �kx� x

o

k = � andhuGTG, u

ccFW i > 0

d� := kx� xgk

kx� x

o

k = � andhuGTG, u

cFW i > 0

d� := kx� xgk

huAO, uGTGi > 0

TRACK


(u1, u2) (vr, v�)

vr =2v + �L

2R

v` =2v � �L

2R

�d = atan

✓u2

u1

◆

v =qu21 + u2

2

! = PID(�d � �)

•  We can use a layered architecture for making differential drive robots act like

•  Key idea: Plan using the simple dynamics, then track using some clever controller (PID?)

•  Today: We can be even more clever!

Lecture 7.4 – A Clever Trick


x = u

•  What if we ignored the orientation and picked a different point on the robot as the point we care about?

Transforming the Unicycle


�

(x, y)

8<

:

x = v cos�y = v sin�˙� = ⇥

New point

⇢x = x+ ⇥ cos�y = y + ⇥ sin�

New Dynamics


8<

:

x = v cos�y = v sin�˙� = ⇥

⇢x = x+ ⇥ cos�y = y + ⇥ sin�

˙x = x� ⇥� sin�

˙y = y + ⇥ ˙� cos�

= v cos�� ⇤⇥ sin�

= v sin�+ ⇤⇥ cos�

New Inputs


•  Let’s assume that we can control the new point directly

˙x = u1,

˙y = u2

˙x = v cos�� ⇤⇥ sin� = u1˙y = v sin�+ ⇤⇥ cos� = u2

cos� � sin�sin� cos�

� v⇤⇥

�=

u1

u2

�

R(�) 1 00 ⇥

� v�

�

New Inputs


•  Let’s assume that we can control the new point directly

˙x = u1,

˙y = u2

R(�)

1 00 ⇤

� v⇥

�=

u1

u2

�

v⇥

�=

1 00 1

`

�R(��)

u1

u2

�

What’s The Point?


•  Before:

(u1, u2)PLAN

uGTGhuGTG, u

cFW i > 0

kx� x

o

k = � and

d� := kx� xgk

kx� x

o

k = � andhuGTG, u

ccFW i > 0

ucFWucc

FW

kx� x

o

k < �


uAO

kx� xgk �

u = 0

d� := kx� xgk

kx� x

o

k < �kx� x

o

k = � andhuGTG, u

ccFW i > 0

d� := kx� xgk

kx� x

o

k = � andhuGTG, u

cFW i > 0

d� := kx� xgk

huAO, uGTGi > 0

TRACK (v,�)

e = �d � �, ⇥ = PID(e)

v = kuk

What’s The Point?


•  Now:

(u1, u2)PLAN

uGTGhuGTG, u

cFW i > 0

kx� x

o

k = � and

d� := kx� xgk

kx� x

o

k = � andhuGTG, u

ccFW i > 0

ucFWucc

FW

kx� x

o

k < �


uAO

kx� xgk �

u = 0

d� := kx� xgk

kx� x

o

k < �kx� x

o

k = � andhuGTG, u

ccFW i > 0

d� := kx� xgk

kx� x

o

k = � andhuGTG, u

cFW i > 0

d� := kx� xgk

huAO, uGTGi > 0

TRANSFORM (v,�)

v⇥

�=

1 00 1

`

�R(��)

u1

u2

�

•  Last time: It is indeed possible to make differential drive mobile robots “act” like –  If we are willing to ignore orientation –  And accept a small offset error

•  Today: Does this generalize to other types of robots? And, what other types are there?

Lecture 7.5 – Other Robot Classes


x = u

•  There are lots and lots of different types of robotic systems •  We cannot cover them all. Instead, we will focus on what they

have in common:

Other Models


UNICYLE

x = v cos�y = v sin�˙� = ⇥

States: position and orientation Inputs: Angular and transl. velocities


have in common:

Other Models


CAR-LIKE ROBOT

States: position, orientation, steering angle Inputs: Transl. vel. and steering angular vel.

x = v cos(�+ ⇥)y = v sin(�+ ⇥)˙� =

v� sin(⇥)

˙⇥ = u


have in common:

Other Models


SEGWAY ROBOT

States: position, orientation, tilt angle, and velocities Inputs: Wheel torques

base: unicycle pendulum:

3

(mw +mb)v �mbd co

s⇥ ¨⇥+mbd s

i

n⇥(

˙⇥2+

˙⇤2)

= � 1

R(�L + �R)

(

(

3

L2+

1

2

R2)

mw +

mbd2s

i

n

2 ⇥+

I2)¨⇤ +

mbd2s

i

n

⇥ cos⇥˙⇤ ˙⇥ =

L

R(

�L � �R)

mbd cos⇥ ˙v + (

�mbd2 � I3)¨⇥+

mbd2s

i

n

⇥ c

o

s

⇥ ˙⇥2+

mbgd sin⇥ =

�L +

�R


have in common:

Other Models


FIXED-WING AIRCRAFT

States: position, orientation, altitude Inputs: Transl., angular, vertical vel.

x = v cos(�)y = v sin(�)˙� = ⇥z = u


have in common:

Other Models


UNDERWATER GLIDER

States: position, orientation, altitude Inputs: Transl., angular, vertical vel.

x = v cos(�)y = v sin(�)˙� = ⇥z = u

•  Everything (almost) involves POSE = position and heading! •  Everything (almost) with pose is almost a unicycle! •  So we can (almost) use what we have already done and then

make the actual model class fit the unicycle – Just add a layer –  Next lecture: Do this for the car robot!

Punchline


•  A lot of times we actually need constraints, which unfortunately make it harder to control the robots (not in this class)

Adding Constraints


UNICYLE x = v cos�y = v sin�˙� = ⇥

x = v cos(�)y = v sin(�)˙� = ⇥v = 1, ⇥ 2 [�1, 1]

x = v cos(�)y = v sin(�)˙� = ⇥|v| = 1, ⇥ ⇥ [�1, 1]

DUBINS

REEDS-SHEPP

•  Humanoids •  Snakes •  Mobile manipulators

•  Next time: Cars (when it is ok…)

When is POSE Not Reasonable?


•  Claims: –  Pose (position and heading) is central –  Other “pose-based” models can be made to look like a

unicycle •  Today: Car-like robots

Lecture 7.6 – Car-Like Robots


•  What’s different about the car is that it has four wheels. •  Only the front wheels turn, which means that the steering

wheel angle (“=“ front wheel angle) becomes important

Car Kinematics


(x, y)`

�

x = v cos(⇥+ ⇤)y = v sin(⇥+ ⇤)˙⇥ =

v� sin(⇤)

˙⇤ = �

(x, y)�

states: position heading steering angle

v�

inputs: speed angular steering velocity

•  How do we make this act like a unicycle? •  Assume a unicycle is driving along a

circular arc

Curvature Control


x = v cos(⇥+ ⇤)y = v sin(⇥+ ⇤)˙⇥ =

v� sin(⇤)

˙⇤ = �

⇢

↵

�

↵ = �� ⇡

2

x = x0 + ⇥ cos(�)

= x0 + � sin(⇥)

x = ⇤� cos(⇥)= v cos(�)

� =v

⇥� =⇥

v

•  Let’s redo this for the car

Curvature Control


⇢

↵

�+

↵ = �+ � ⇡

2

x = x0 + � sin(⇥+ ⇤)

x = �( ˙⇥+

˙⇤) cos(⇥+ ⇤)

= 0� =v

⇤sin(⇥)

= v cos(�+ ⇥)

⇢ =`

sin( )

=sin( )

`

Lining Up The Curvatures


UNICYCLE CAR

� =⇥

v =

sin( )

`

sin(�) =⇥⇤

v) �d = arcsin

✓⇥⇤

v

◆

But we can actually stay with sinus instead of dealing with arcsin!

An Almost P-Regulator


= �

� = C(⇥d � ⇥) � = C

✓⇤⌅

v� sin(⇥)

◆

Summing It Up


(u1, u2)PLAN TRANSFORM

(v,�)

TRACK (v,�)

� = C

✓⇤⌅

v� sin(⇥)

◆

•  Believe it or not – there are lots of things not covered in this course!

Lecture 7.7 – To Probe Further


Nonlinear and Optimal Control


x = f(x, u)

minu

Z T

0L(x(t), u(t))dt+�(x(T ))

Machine Learning


V �(x0) =1X

k=0

�kc(xk,⇥(xk))

V �(x) = minu

{c(x, u) + �V �(f(x, u))}

Perception and Mapping


High-Level AI


•  Not only are there things not covered in the class, there are lots of things we don’t know yet!

To Probe Further


•  That’s it folks!

•  Ambition with the course: –  Learn how to make mobile robots move in effective and

safe ways using modern control theory –  Appreciate the value of systematic thinking/design –  Bridge the theory-practice gap –  Have fun and spark further investigations

Lecture 7.8 – In Conclusion


•  Without a model, we cannot say much about how the system will behave: –  Need models to predict behavior forward in time –  Need models to be able to derive control laws in a

systematic manner •  The model should be rich enough to be relevant yet simple

enough to be useful •  Bananas vs. Non-bananas

High-Level Punchline #1 – The Model


•  Given a model, feedback control should be used to make the system behave the way we want it to (if possible)

•  Stability, Tracking, Robustness •  State feedback and observers

High-Level Punchline #2 – Feedback


y

x

x = Ax+Bu

y = Cx

u

˙x = Ax+Bu+ L(y � Cx)

u = �Kx

•  Plan for simple systems, execute on the “real” system

High-Level Punchline #3 – Architectures


PLAN

uGTGhuGTG, u

cFW i > 0

kx� x

o

k = � and

d� := kx� xgk

kx� x

o

k = � andhuGTG, u

ccFW i > 0

ucFWucc

FW

kx� x

o

k < �


uAO

kx� xgk �

u = 0

d� := kx� xgk

kx� x

o

k < �kx� x

o

k = � andhuGTG, u

ccFW i > 0

d� := kx� xgk

kx� x

o

k = � andhuGTG, u

cFW i > 0

d� := kx� xgk

huAO, uGTGi > 0

TRACK



Actual trajectory

x = f(x, u), u = g(x, r)

•  Three abstraction layers •  Differential-drive robots

•  Don’t take my word for it •  Experiment and tweak •  The field is certainly not done yet

High-Level Punchline #4 – Whatever…


THANKS!


Greg Droge

JP de la Croix

Amy LaViers

Smriti Chopra

Rowland O’flaherty

Zak Costello

All of you!

module 7 slides

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