1

Sanjiban Choudhury

PID and Pure Pursuit Control

TAs: Matthew Rockett, Gilwoo Lee, Matt Schmittle

The Control API

2

Input OutputControl

Law

The Control API

2

Input Output

x(⌧), y(⌧), ✓(⌧), v(⌧)

1. Reference path Control Law

The Control API

2

Input Output

x(⌧), y(⌧), ✓(⌧), v(⌧)

1. Reference path 2. Current state

2

4x

y

✓

3

5

Control Law

The Control API

2

Input Output

x(⌧), y(⌧), ✓(⌧), v(⌧)

1. Reference path 2. Current state

2

4x

y

✓

3

5

Control action

�

Control Law

Steps to designing a controller

3

1. Get a reference path / trajectory to track

2. Pick a point on the reference

3. Compute error to reference point

4. Compute control law to minimize error

Rough idea of what happens across timesteps

4

Robot is trying to track a desired state on the reference path

(Take an action to drive down error between desired and current state)

Rough idea of what happens across timesteps

4

Robot is trying to track a desired state on the reference path

(Take an action to drive down error between desired and current state)

Rough idea of what happens across timesteps

4

Robot is trying to track a desired state on the reference path

(Take an action to drive down error between desired and current state)

Rough idea of what happens across timesteps

4

Robot is trying to track a desired state on the reference path

(Take an action to drive down error between desired and current state)

Rough idea of what happens across timesteps

4

Robot is trying to track a desired state on the reference path

(Take an action to drive down error between desired and current state)

Rough idea of what happens across timesteps

4

Robot is trying to track a desired state on the reference path

(Take an action to drive down error between desired and current state)

Steps to designing a controller

5

1. Get a reference path / trajectory to track

2. Pick a point on the reference

3. Compute error to reference point

4. Compute control law to minimize error

Step 1: Get a reference path

6

x(⌧), y(⌧), ✓(⌧), v(⌧)

2

4x

y

✓

3

5

Step 2: Pick a reference (desired) state

7

E.g. Pick nearest state / pick state L distance ahead

2

4xref

yref

✓ref

3

52

4x

y

✓

3

5

Step 3: Compute error to this state

8

Error is simply the state of the car expressed in the frame of the reference (desired) state

2

4x

y

✓

3

5

2

4eatect✓e

3

5

Step 3: Compute error to this state

9

2

4xref

yref

✓ref

3

52

4x

y

✓

3

5

Step 3: Compute error to this state

9

2

4xref

yref

✓ref

3

52

4x

y

✓

3

5

Step 3: Compute error to this state

9

2

4xref

yref

✓ref

3

52

4x

y

✓

3

5

Step 3: Compute error to this state

9

2

4xref

yref

✓ref

3

52

4x

y

✓

3

5

Step 3: Compute error to this state

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Step 3: Compute error to this state

10

Step 3: Compute error to this state

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A

Ae =

x

y

��

xref

yref

�Position in frame A

Step 3: Compute error to this state

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AB

We want position in frame B

Be =BA R Ae

(rotation of A w.r.t B)

Step 3: Compute error to this state

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AB

We want position in frame B

Be =BA R Ae

(rotation of A w.r.t B)

= R(�✓ref )

✓x

y

��xref

yref

�◆

(rotation of A w.r.t B)

Step 3: Compute error to this state

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AB

We want position in frame B

Be =BA R Aee =

cos(✓ref ) sin(✓ref )

� sin(✓ref ) cos(✓ref )

�✓x

y

��

xref

yref

�◆eatect

�=

Step 3: Compute error to this state

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AB

We heading in frame B

✓e = ✓ � ✓ref

Step 3: Compute error to this state

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ect = � sin(✓ref )(x� xref ) + cos(✓ref )(y � yref )

eat = cos(✓ref )(x� xref ) + sin(✓ref )(y � yref )

{ eat {

ect

✓e = ✓ � ✓ref

✓e

(Along-track)

(Cross-track)

(Heading)

Some things to note

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{ eat {

ect

✓e

1. We will only control steering angle; speed set to reference speed

2. Hence, no real control on along-track error. Ignore for now.

3. Some control laws will only minimize cross-track error, others both heading and cross-track error.

Step 4: Compute control law

17

errorcontrol

Compute control action based on instantaneous error

Different laws have different trade-offs, make different assumptions,

look at different errors

u = K(e)

Different control laws

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1. PID control

2. Pure-pursuit control

3. Lyapunov control

4. LQR

5. MPC

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Proportional–integral–derivative (PID) controller

Used widely in industrial control from 1900s Do not try this

with PID!!!Regulate temp, press, speed etc

PID control overview

20

ectu

Select a control law that tries to drive error to zero (and keep it there)

PID control overview

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ectu

Select a control law that tries to drive error to zero (and keep it there)

Proportional Integral Derivative

u = �✓Kpect +Ki

Zect(t)dt+Kdect

◆

(current) (past) (future)

Some intuition …

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Proportional Integral Derivative

u = �✓Kpect +Ki

Zect(t)dt+Kdect

◆

(current) (past) (future)

Proportional - get rid of the current error!

Integral - if I am accumulating error, try harder!

Derivative - if I am going to overshoot, slow down!

Proportional control

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ect

u = �Kpect(Gain)

Proportional control

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ect

u = �Kpect

ect < 0, u > 0

(Gain)

Proportional control

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ect

u = �Kpect

ect < 0, u > 0

ect > 0, u < 0

(Gain)

The proportional gain matters!

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ect

What happens when gain is low?

The proportional gain matters!

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ect

What happens when gain is low?

The proportional gain matters!

23

ect

What happens when gain is low?

What happens when gain is high?

The proportional gain matters!

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ect

What happens when gain is low?

What happens when gain is high?

Proportional term

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What happens when gain is too high?

ect

� umax

Kp

Proportional term

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What happens when gain is too high?

ect

� umax

Kp

Proportional derivative control

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ect

u = � (Kpect +Kd ˙ect)

Proportional derivative control

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ect

u = � (Kpect +Kd ˙ect)

Proportional derivative control

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ect

u = � (Kpect +Kd ˙ect)

ect << 0, ect ⇡ 0, u >> 0

Proportional derivative control

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ect

u = � (Kpect +Kd ˙ect)

ect << 0, ect ⇡ 0, u >> 0

ect < 0, ect > 0, u ⇡ 0

Proportional derivative control

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ect

u = � (Kpect +Kd ˙ect)

ect << 0, ect ⇡ 0, u >> 0

ect < 0, ect > 0, u ⇡ 0

ect ⇡ 0, ect > 0, u < 0

How do you evaluate the derivative term?

26

How do you evaluate the derivative term?

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Terrible way: Numerically differentiate error. Why is this a bad idea?

How do you evaluate the derivative term?

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Terrible way: Numerically differentiate error. Why is this a bad idea?

Smart way: Analytically compute the derivative of the cross track error

How do you evaluate the derivative term?

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Terrible way: Numerically differentiate error. Why is this a bad idea?

Smart way: Analytically compute the derivative of the cross track error

ect = � sin(✓ref )(x� xref ) + cos(✓ref )(y � yref )

How do you evaluate the derivative term?

26

Terrible way: Numerically differentiate error. Why is this a bad idea?

Smart way: Analytically compute the derivative of the cross track error

ect = � sin(✓ref )x+ cos(✓ref )y

= � sin(✓ref )V cos(✓) + cos(✓ref )V sin(✓)

= V sin(✓ � ✓ref ) = V sin(✓e)

ect = � sin(✓ref )(x� xref ) + cos(✓ref )(y � yref )

How do you evaluate the derivative term?

26

Terrible way: Numerically differentiate error. Why is this a bad idea?

Smart way: Analytically compute the derivative of the cross track error

ect = � sin(✓ref )x+ cos(✓ref )y

= � sin(✓ref )V cos(✓) + cos(✓ref )V sin(✓)

= V sin(✓ � ✓ref ) = V sin(✓e)

ect = � sin(✓ref )(x� xref ) + cos(✓ref )(y � yref )

u = � (Kpect +KdV sin ✓e)New control law! Penalize error in cross track and in heading

Proportional integral control

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ect

u = �✓Kpect +Ki

Zect(t)dt

◆

Proportional integral control

27

ect

u = �✓Kpect +Ki

Zect(t)dt

◆

Only Proportional cannot overcome wind!

Proportional integral control

27

ect

u = �✓Kpect +Ki

Zect(t)dt

◆

Only Proportional cannot overcome wind!

Zect(t)dt � 0, u � 0

Different control laws

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1. PID control

2. Pure-pursuit control

3. Lyapunov control

4. LQR

5. MPC

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Pure Pursuit Control

 Aerial combat in which aircraft pursues another aircraft by pointing its nose directly towards it

Similar to carrot on a stick!

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Key Idea:

The car is always moving in a circular arc

31

L

Consider a reference at a lookahead distance

Problem: Can we solve for a steering angle that guarantees that the car will pass through the reference?

����

x

y

��xref

yref

����� = L

Solution: Compute a circular arc

32

We can always solve for a arc that passes through a lookahead point

Note: As the car moves forward, the point keeps moving

Pure pursuit: Keep chasing looakahead

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1. Find a lookahead and compute arc

2. Move along the arc

3. Go to step 1

Pure pursuit: Keep chasing looakahead

33

1. Find a lookahead and compute arc

2. Move along the arc

3. Go to step 1

Pure pursuit: Keep chasing looakahead

33

1. Find a lookahead and compute arc

2. Move along the arc

3. Go to step 1

Pure pursuit: Keep chasing looakahead

33

1. Find a lookahead and compute arc

2. Move along the arc

3. Go to step 1

Pure pursuit: Keep chasing looakahead

33

1. Find a lookahead and compute arc

2. Move along the arc

3. Go to step 1

Pure pursuit: Keep chasing looakahead

33

1. Find a lookahead and compute arc

2. Move along the arc

3. Go to step 1

Pure pursuit: Keep chasing looakahead

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1. Find a lookahead and compute arc

2. Move along the arc

3. Go to step 1

Control law derivation: Solve for arc

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R

Control law derivation: Solve for arc

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R

↵

L

2

↵

Control law derivation: Solve for arc

34

R

↵

L

2

↵

↵ = tan�1

✓yref � y

xref � x

◆� ✓

Control law derivation: Solve for arc

34

R

↵

L

2

↵

sin↵ =L

2R

↵ = tan�1

✓yref � y

xref � x

◆� ✓

Control law derivation: Solve for arc

34

R

↵

L

2

↵

sin↵ =L

2R

R =L

2 sin↵

↵ = tan�1

✓yref � y

xref � x

◆� ✓

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u = tan�1

✓2B sin↵

L

◆

Control law derivation

R

↵

L

2

↵

✓ =V

Btanu

w =V

R=

2V sin↵

L✓ = ! =

Question: How do I choose L?

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Question: How do I choose L?

36

Question: How do I choose L?

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