robotics 01peeqw - ladispe

ROBOTICS

01PEEQW

Basilio Bona

DAUIN – Politecnico di Torino

Mobile & Service Robotics

Kinematics

Fundamental problems in mobile robotics

� Locomotion: how the robot moves in the environment

� Perception: how the robot perceives the environment

� Mapping: how to build the map of the environment

� Localization: where is the robot wrt the map

� Representation: how the robot organizes the knowledge about

the environment

� Path planning/action planning: what the robot shall do to go

from here to there; what are the actions to be performed to

complete a specified task

� Supervision and control: how are the command to actuators

generated to perform simple or complex tasks. How to

generate tasks

3ROBOTICS 01PEEQW - 20152016Basilio Bona - DAUIN - PoliTo

Kinematics

� robot kinematics depends on the adopted locomotion stuctures

� Almost all mobile robots are underactuated, i.e., they have less

actuators (and the relative control signals) that is less than the

number of degrees of freedom of the structure

� Various wheels arrangements exist

� Each one has a different kinematic model

� We will study the following structures

� Differential drive robots

� Bicycle-like robots

� And a brief introduction to quadcopters

Basilio Bona - DAUIN - PoliTo 4ROBOTICS 01PEEQW - 20152016

Examples of underactuated structures

� A car has two actuators, thrust and steer, and moves in a 2D

space (3dof)

� A fixed-wing aircraft has four actuators, forward thrust,

ailerons, elevator and rudder, and moves in a 3D space (6dof)

� A helicopter has four actuators: thrust vector magnitude and

direction from the main rotor, and yaw moment from the tail

rotor, and moves in a 3D space (6dof)

� A quadcopter has four actuators, the four rotor thrusts , and

moves in a 3D space (6dof)

� A ship has three actuators: two propellers on parallel axes and

one rudder, and moves in a 2D space (3dof)


Instantaneous Curvature Centre (ICC)


ICC

If an ICC exists, each wheel rotates without slippage

The wheel axes meet in a common point called ICC

ICC


?

wheels slip = skid steering

If an ICC does not exist, the wheel motion occurs with slippage

Kinematics of a wheeled robot (aka rover)

� the steering angle is the angle between the velocity vector

and the steered wheel direction

� is equivalent to the angle between the normal to the velocity

vector and the steered wheel rotation axis

� the robot pose is the position and the orientation (with respect

to some give inertial axis)

( )

0

( )

cos sin 0

sin cos 0

0 0 1m

t x y θ

θ θ

θ θ

= − =

p

R

T

( )tβ

Basilio Bona - DAUIN - PoliTo ROBOTICS 01PEEQW - 20152016 8

βθ

Kinematics


d

mω

α

αδ

rϕ=v ɺ

Wω

mi

mj

Wi

Wj

0R

mR

x

y

WR

Kinematics – Fixed wheel


The kinematic model

represents a rover frame

with a generic active non-

steering wheel, located at

a given position, with a

local orientation

The kinematic equations

describe the relations and

constraints between the

wheel angular velocity and

the angular velocity of the

frame

( )0x yv v=v

T

( )0W Wx Wyω ω=ω

T

( )0 0m mz

ω=ω

T

Linear velocity of the wheel at the contact point with the plane

Angular velocity of the wheel

Angular velocity of the robot

,α δ

( )cos sin 0d dα α=dT

given constants

Kinematic constraints


Hypothesis

m

W

r

θ

ϕ

ϕ

=

=

=v

ɺ

ɺ

ɺ

ω

ω

Constraints:To avoid slippage the tangential velocity at the wheel contact point, due to the rover rotation around its center, must be equal to the advance velocity of the wheel

In the wheel reference frame we have

( )( )( )

TT

T

T

0 0 1

0 1 0

1 0 0

m

W

r

θ

ϕ

ϕ

=

=

=v

ɺ

ɺ

ɺ

ω

ω

Differential Drive (DD) Rover


Left wheel, active, non steering

Passive support(castor, omniwheel)

Right wheel, active, non steering

Differential Drive Rover


mi

mj

mk

Left wheel

2

ℓ r

Right wheel

2

ℓ

Differential Drive Kinematics


( )tθɺ

( )tv

( )tvℓ

( )rtv

ICC

0R

mR



( )tvℓ

( )rtv

( )tθɺ

( )tv

ICC

0R



vℓ

rv

v

ICC

iC

iρ

Consider the generic time instant

ℓ



mR

before

after

vℓ

rv

v

vℓ

rv

v

iC

r>v v

ℓ

r>v v

ℓ

Turn left

Turn right

mR

tδθ

θ



ks

A

B

AB = t��

2kδθ

ikC

kδθ

rks

ksℓ



ikC

1k+t

kθ

kδθ

0i

0j

0

rkt

ikc

ik rkρ− j

ik rkρ j

0R

kθ

kδθ

kv

kv


Considering the sampled time equations and assuming

constant values during the time interval


( / 2)

( / 2)k k rk

k k k

v

v

ω ρ

ω ρ

+ =

− =ℓ

ℓ

ℓ

ICCsin

cosxk k k k

k

yk k k k

C x

C y

ρ θ

ρ θ

− = = +

;

2( )

2 ( )

rk k rk k

k k

rk k

k

k rk k

k

k rk k

v v s s

v vv

v v v

v v

ω δθ

ρω

− −= =

+=

+= =

−

ℓ ℓ

ℓ

ℓ

ℓ

ℓ ℓ

ℓ



1

d

dk k k k kT

tω θ ω θ θ δθ+= ⇒ −≃ ≐

1

1

1

cos( ) sin( ) 0

sin( ) cos( ) 0

0 0 1

k k k k xk xk

k k k k yk yk

k k k

x x C C

y y C C

δθ δθ

δθ δθ

θ θ δθ

+

+

+

− − = − +

Approximation when sampling period is small

0 0

0

( ) cos ( ) ( ) d ; ( ) sin ( ) ( ) d ;

( ) ( ) d

t t

t

x t v y t v

t

θ τ τ τ θ τ τ τ

θ ω τ τ

= =

=

∫ ∫

∫

(A)



1

1

1

cos

sink k k k

k k k k

k k k

x x v T

y y v T

T

θ

θ

θ θ ω

+

+

+

= += +

= +

EULER APPROXIMATION



1

1

1

1cos

2

1sin

2

k k k k k

k k k k k

k k k

x x v T T

y y v T T

T

θ ω

θ ω

θ θ ω

+

+

+

= + + = + +

= +

RUNGE-KUTTA APPROXIMATION



EXACT INTEGRATION

( )

( )

1 1

1 1

1

sin sin

cos cos

k

k k k k

k

k

k k k k

k

k k k

x xv

y yv

T

ωθ θ

ωθ θ

θ θ ω

+ +

+ +

+

= + −

= − −

= +

(B)

(A) and (B) are the same

Path Planning


Path Planning


initial pose

final pose

initial pose

final pose

obstaclesfree space

2D

Path Planning


3D

F

I

Path Planning

A robot path from pose A to

pose B can always be

decomposed into a series

of arcs of different radius

� arcs of zero radius

represent turn-in-place

maneuvers

� arcs of infinite radius

represents straight line

maneuvers


Path Planning

Driving a vehicle to a goal (no orientation specified)


θ−

(0, 0)

,f fx y

(0, )ρ−

AR

Arelative to frame R

Path Planning

Driving a vehicle to a goal (orientation specified)


1θ−

(0, 0)

1 1,

f fx y

1(0, )ρ−

AR

2θ

2(0, )

yr

BR

2 2,

f fx y

Arelative to frame R

BR

Path Planning


Unicycle

Unicycle is the simplest example of a wheeled robot.

It consists of a single actuated steering wheel

ω is the steering velocity


mR

0R

x

yθ

rvcos 0

sin 0

0 1

x

y v

θ

θ ω

θ

= +

ɺ

ɺ

ɺ

control commands

(underactuated)

x y θ

T

states/coordinates

Unicycle motion


Unicycle-like robot (polar coordinates)


x

θ

δ

rR

rv

ρα

y

2 2

( , )

( , )

x y

y x

y x

ρ

α θ

δ θ α

= +

= −

= + =

atan2

atan2

θ′

Bicycle-like robot


x

y

rR

0R

θ

L

φv

cos

sin

tan

x v

y v

v

L

θ

θ

θ φ

=

=

=

ɺ

ɺ

ɺ

NON HOLONOMIC CONSTRAINT

sin cos 0x yθ θ− =ɺ ɺ

Lθɺ

fy

fx

steering angle

orientation

Bicycle equations 1


1’)

2)

1)

Bicycle equations 2


Bicycle equations 3


Bicycle equations 5


Bicycle vs unicycle motion


Quadcopters


Quadcopter model


ℓ

4F

BR

3N

Bk

Bi

Bj

4N

2F

2N

3F

1N

1F

Quadcopter equations


Odometry

Odometry is the estimation of the successive robot poses based on

the wheel motion. Wheel angles are measured and used for

pose computation

Odometric errors increase with the distance covered, and are due to

many causes, e.g.

� Imperfect knowledge of the wheels geometry

� Unknown contact points: in the ideal DD robot there are two

geometric contact points between wheels and ground. In real

robots the wheels are several centimeters wide and the actual

contact points are undefined

� Slippage of the wheel wrt the terrain

Errors can be compensated sensing the environment around the

robot and comparing it with known data (maps, etc.)


Odometry Errors


Map

Desired pathEstimated path

based only on odometry

Visual odometry

Visual odometry is the estimation of the robot poses based only on

the acquisition of successive images of the environment

Stereo visual odometry uses two cameras to estimate the ego-

motion of a camera by tracking features in a video sequence.

If the camera system is calibrated then the full 3D Euclidean

geometry can be reconstructed

Using a single camera, the reconstruction is up to an unknown scale

factor

A RGB-D (Kinect or similar) camera can be used


Other important problems that must be addressed


1. Mapping: how to build environment maps (geometrical vs

semantic maps)

2. Localization: where am I in the map?

3. Simultaneous Localization and Mapping (SLAM): build a map

and at the same time localize the robot in the map while

moving

4. Path planning: the definition of an “optimal” geometric path

in the environment from A to B, and avoid obstacles

Path Planning – A map is given

� Path planning algorithm: it computes which route to take, from

the initial to the final pose, based on the current internal

representation of the terrain possibly considering a “cost

function” (minimum time, minimum energy, maximum comfort,

etc.)

� Complete path generation: the path is created as a sequence of

successive moves from the initial pose to the final one

� Motion planning: is the execution of the above defined

theoretical route; it translates the plan from the internal

representation to the physical movement of the wheels

� Next move selection: at each step, the algorithm must decide

which way to move next. An efficient dynamic implementation

is required


Path Planning


Introduction to AI Robotics (MIT Press) Robin Murphy 2000

Navigation� Where am I going? Mission

planning

� What’s the best way there? Path planning

� Where have I been? Map making

� Where am I? Localization

MissionPlanner

Carto-grapher

BehaviorsBehaviors

BehaviorsBehaviors

deliberative

reactive

How am I going to getthere?

Path Planning


A

BFixed

obstacles

Mobile obstacles

Acceleration constraints

Velocity constraints

Path Planning


0i

0j

0k

0k =

1k =

2k =

k N=

Which commands shall I give to the rover?

0R

Path Planning


Initial pose

Final pose

Path A

Path B

A

B

Motion Planning


Initial pose

Final pose

Move 1: rotation

Move 3:rotation

A

B

Move 2: go straight

Non-holonomic constraints


� Non-holonomic constraints limit the possible incremental

movements in the configuration space of the robot

� Robots with differential drive move on a circular trajectory and

cannot move sideways

� Omni-wheel robots can move sideways

Holonomic vs. Non-Holonomic

� Non-holonomic constraints reduce the control space with respect

to the current configuration (e.g., moving sideways is impossible)

� Holonomic constraints reduce the configuration space

Basilio Bona - DAUIN - PoliTo ROBOTICS 01PEEQW - 20152016 61

62ROBOTICS 01PEEQW - 20152016

Academic year 2010/11 by Prof. Alessandro De Luca

Basilio Bona - DAUIN - PoliTo

robotics 01peeqw - ladispe

Documents