Cinematica dei Robot Mobili su Ruote

Corso di Robotica

Prof. Davide Brugali

Università degli Studi di Bergamo

Cinematica dei

Robot Mobili su Ruote

Riferimenti bibliografici

2

Roland SIEGWART, Illah R. NOURBAKHSH

Introduction to Autonomous Mobile Robots

Capitolo 2.3 «Wheeled Mobile Robots»

Robotica - UNIBG - Prof. Brugali

Wheeled Mobile Robots (WMR)

3 Robotica - UNIBG - Prof. Brugali

Wheeled Mobile Robots (WMR)

Kinematics – study of the mathematics of motion without considering the forces that affect the motion. Deals with the geometric relationships that govern the system Deals with the relationship between control parameters and the

behavior of a system.

Dynamics – study of motion in which these forces are modeled Deals with the relationship between force and motions.


Wheels

Lateral slip

Rolling motion


1. The robot is built from rigid mechanisms.

2. No slip occurs in the orthogonal direction of rolling (non-slipping).

3. No translational slip occurs between the wheel and the floor (pure rolling).

4. The robot contains at most one steering link per wheel.

5. All steering axes are perpendicular to the floor.

Non-slipping and pure rolling

Assumptions

Idealized Rolling Wheel


Robot wheel parameters

For low velocities, rolling is a reasonable wheel model.

This is the model that will be considered in the kinematics models of wheeled mobile robots (WMR)

Wheel parameters:

r = wheel radius

v = wheel linear velocity

w = wheel angular velocity

t = steering velocity


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Wheel Types

Fixed wheel Centered orientable wheel

Off-centered orientable wheel

(Castor wheel) Swedish wheel:omnidirectional

property


Smooth motion

Risk of slipping

Some times use roller-ball to make balance

Bi-wheel type robot

Omnidirectional robot

Caterpillar type robot

Exact straight motion

Robust to slipping

Inexact modeling of turning

Free motion

Complex structure

Weakness of the frame

Examples of WMR


Mobile Robot Locomotion

Instantaneous center of rotation (ICR) or Instantaneous center of curvature (ICC) A cross point of all axes of the wheels


Non-holonomic constraint

So what does that mean?

Your robot can move in some directions (forward

and backward), but not others (sideward).

The robot can instantly

move forward and backward,

but can not move sideward

Parallel parking,

Series of maneuvers

A non-holonomic constraint is a constraint on the

feasible velocities of a body


v : Linear velocity of the robot

: Angular velocity of the robot

Control input

12

R = curvature radius

V = R *

Twist {

V

Differential Drive

Relazione tra le velocità delle ruote (VL e VR)

e la velocità del robot (TWIST)


13

V

RVL

R )2

( LVL

R )2

(

Differential Drive


14

V

Differential Drive

Straight motion

R = Infinity VR = VL

Rotational motion

R = 0 VR = -VL


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V

Differential Drive


V

Twist { Velocità ruote {

Tricycle

Three wheels and odometers on the two rear wheels

Steering and power are provided through the front wheel

control variables:

steering direction α(t)

angular velocity of steering wheel ws(t)

The ICC must lie on

the line that passes

through, and is

perpendicular to, the

fixed rear wheels


Tricycle


Linear velocity of

steering wheel

Tricycle

Kinematics model in the world frame

---Posture kinematics model


Car Drive (Ackerman Steering)

Used in motor vehicles, the inside front wheel is rotated slightly sharper than the outside wheel (reduces tire slippage).

Ackerman steering provides a fairly accurate dead-reckoning solution while supporting traction and ground clearance.

Generally the method of choice for outdoor autonomous vehicles.

R

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where

d = lateral wheel separation

l = longitudinal wheel separation

i = relative steering angle of inside wheel

o = relative steering angle of outside wheel

R=distance between ICC to centerline of the vehicle


Carrello


Synchronous Drive

In a synchronous drive robot (synchronous drive) each wheel is

capable of being driven and steered.


Synchronous Drive

All the wheels turn in unison

All of the three wheels point in the same direction and turn at the same rate This is typically achieved through the use of a complex collection of belts that

physically link the wheels together

Two independent motors, one rolls all wheels forward, one rotate them for turning

The vehicle controls the direction in which the wheels point and the rate at which they roll

Because all the wheels remain parallel the synchro drive always rotate about the center of the robot

The synchro drive robot has the ability to control the orientation θ of their pose directly.


Omidirectional

Swedish Wheel


Odometry for Differential Drive Rovers


25

V

Differential Drive

Straight motion

R = Infinity VR = VL

Rotational motion

R = 0 VR = -VL


3 1 0 2

3 1 0 2

Velocity

Profile

Basic Motion Control

26

: Radius of rotation

: Length of path

: Angle of rotation


Differential Drive: odometria

x

x

y

y θ

dD

dttttrtx LR cos2

1

dttttrty LR sin2

1

dtttrdD LR 2

1

dtttL

rt LR



Esempio : velocità costanti

RR t LL t

tL

rt LR

t

L

rLtx LR

LR

LR

sin

2

t

L

rLty LR

LR

LR

cos

2



1 kkLk ttrDL

1 kkRk ttrDR

kk

kk

kDLDR

DLDRLr

2

Distanze percorse dalle due ruote nell’intervallo di tempo tk – tk-1

Raggio di curvatura del robot nell’intervallo di tempo tk – tk-1

L

DLDR kk

kk

1

kkkkk rxx sinsin 11

kkkkk ryy coscos 11

Posizione del robot all’istante tk


Effector Noise: Odometry, Dead Reckoning

Odometry and dead reckoning:

Position update is based on proprioceptive sensors

Odometry: wheel sensors only

Dead reckoning: also heading sensors

The movement of the robot, sensed with wheel encoders and/or heading

sensors is integrated to the position.

Pros: Straight forward, easy

Cons: Errors are integrated -> unbound

Using additional heading sensors (e.g. gyroscope) might help to reduce the

cumulated errors, but the main problems remain the same.


Imprecisione dell’odometria

Nr. posizionamenti = 35 ;

Dati di scostamento : Media = 11 gradi ; Deviazione standard = 5.47 gradi


Odometry: Error sources

deterministic non-deterministic

(systematic) (non-systematic)

deterministic errors can be eliminated by proper calibration of the system. non-deterministic errors have to be described by error models and will always

leading to uncertain position estimate.

Major Error Sources: Limited resolution during integration (time increments, measurement resolution

…) Misalignment of the wheels (deterministic) Unequal wheel diameter (deterministic) Variation in the contact point of the wheel Unequal floor contact (slipping, not planar …) …


Odometry: Classification of Integration Errors

Range error: integrated path length (distance) of the robots movement

sum of the wheel movements

Turn error: similar to range error, but for turns

difference of the wheel motions

Drift error: difference in the error of the wheels leads to an error in the

robots angular orientation

Over long periods of time, turn and drift errors

far outweigh range errors!

Consider moving forward on a straight line along the x axis. The error in the y-

position introduced by a move of d meters will have a component of dsinD, which

can be quite large as the angular error D grows.



1 kkLk ttrDL

1 kkRk ttrDR

kk

kk

kDLDR

DLDRLr

2

L

DLDR kk

kk

1

kkkkk rxx sinsin 11

kkkkk ryy coscos 11

Posizione del robot all’istante tk


Odometry: Growth of Pose uncertainty for Straight Line Movement

Note: Errors perpendicular to the direction of movement are growing

much faster!


Odometry: Growth of Pose uncertainty for Movement on a Circle

Note: Errors ellipse does not remain perpendicular to the direction of

movement!


Riduzione degli errori non sistematici

Utilizzo di ruote ausiliarie non

motrici


Cinematica dei Robot Mobili su Ruote

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