A.A. 2006/2007

Roboticafor Computer Engineering students

Marcello RestelliDipartimento di Elettronica e InformazionePolitecnico di Milanoemail: restelli@elet.polimi.ittel: 02-2399-3470 Mobile Robotics

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From Industrial Robotics to Mobile Robotics

Industrial roboticscomplex robotssimple worldmanipulation in known environment

Mobile roboticssimple robotscomplex worldnavigation in dynamic environments

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Robot Dynamics vs World Dynamics

In industrial robotics we need to model the robot functioning

this may be hard according to the simplifications adopted

In mobile robotics we need to model the environment

this may be impossible and depends from

possible events external sensors

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Development of Mobile Robotics

The first autonomous robot was Shakey (1969)1970s: JPL Lunar rover: planetary explorationlate 1970s: CART followed a line on the road1980s: Automated Guided Vehicles (AGV) used in factories, based on magnetic and optic guides1994: Dante II. a six-legged robot explored a volcano1997: the Sojourner rover explored Mars1997: Honda presented the wonderful humanoid P31997: the RoboCup competitions began2001: Sony presented the humanoid Q-rio

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Autonomous Land Vehicle (ALV)

We focus our attention on ALVvehicle that autonomously move on a plane surfacemay be equipped with robotic arms

we will focus on the navigation problemThree basic questions

Where am I?Where am I going?How do I get there?

To answer these questions the robot has tohave a model of the environmentperceive and analyze the environmentfind its position within the environmentplan and execute the movement

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Industrial Robotics vs Mobile Robotics

There are several differences between industrial robotics and mobile robotics

A manipulator, typically, is an open kinematic chain, while a wheeled robot is a closed multiple chain, since it has at least two wheels on the ground. Also legged robots are closed multiple chains, but it opens when legs are risenManipulator joints have only 1 DOF, while wheels may have 2 (or even 3) DOFsIn manipulators all the joints are actuated with the aim of moving the end-effector, while mobile robots may have passive wheelsTo control the trajectory of the end-effector we need to control position, speed, and acceleration of each joint, while for mobile robots we need to control each DOF of each wheel

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Applications of Mobile Robotics

Indoor Structured worlds

transportation: industry and servicescustomer support: museums and shopsresearch, entertainment, toyscleaning of large buildingssurveillance buildings

Outdoor EnvironmentsUnstructured worlds

spaceforestdeminingfire fightingagricultureconstructionminingsewage tubesunderwatermilitaryair

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DOF for Mobile Robots

If a robot has an actuator for each DOF, then all DOF are controllableMobile robots often cannot control all the DOF related to their positionLet us consider the example of a car

A car has 3 DOFOnly two of them can be controlledSome motions cannot be doneThe two available DOF can get to any position and orientation in 2D

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Holonomicity

A robot is holonomic if the number of controllable DOF is equal to the number of DOF of the robotA robot is non-holonomic if the number of the controllable DOF is smaller than the number of DOF of the robotA robot is redundant if the number of the controllable DOF is larger than the number of DOF of the robotExamples

a car is non-holonomica human arm (7 DOF) is a holonomic, redundant system

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Effectors in Mobile Robotics

In mobile robotics effectors are mainly used for locomotion

legswalking, crawling, climbing, jumping, hopping

wheelsrolling

armsswinging, crawling, climbing

flippersswimming

and many others taken from biological examples

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Locomotion

Concepts found in naturedifficult to imitate technically

Biped walking mechanism

not far from real rollingrolling of a polygonsmall steps -> circle

Fully rotating joints are not developed in nature

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Characterization of Locomotion Concept

Locomotionphysical interaction between the vehicle and its environment

Locomotion is concerned with interaction forces and the mechanisms and actuators to generate themThe most important issues in locomotion are

stabilitycharacteristics of contacttype of environment

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Mobile Legged Robots

The fewer the legs the more complicated becomes the locomotion

3 legs are required for static stabilityDuring walking some legs are lifted

the stability may be loosedFor static walking at least 6 legs are required

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Number of Joints of Each Leg

A minimum of two DOF are required to move a leg forward

a lift and a swing motionsliding free motion in more than only one direction is not possible

Three DOF for each leg in most casesFourth DOF for the ankle joint

might improve walkingadditional DOF increase the complexity of the design and especially of the locomotion control

Each leg may be considered as a manipulator

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Gaits

The gait is characterized as the sequence of lift and release events of the individual legs

it depends on the number of legsthe number of possible events N for a walking machine with k legs is N=(2k-1)!

For a biped (k=2) the number of possible events is 6For a robot with 6 legs N=39,916,800

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Most Obvious Gaits

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Examples of Walking Robots

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Walking or Rolling?

Number of actuatorsStructural complexityControl expenseEnergy efficient

terrain (flat ground, soft ground, climbing...)Movement of the involved masses

walking/running includes up and down movements of COGsome extra losses

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Wheeled Mobile Robots (WMR)

A WMR is a robot capable of locomotion solely through the actuation of wheel assemblies mounted on the robot and in contact with the surface. A wheel assembly is a device which provides or allows

motion between its mount and surface on which it is intended to a single point of rolling contact

Most robots use wheels for locomotionsimplicity of controlstability

If so, why don't animals have wheels?some do!!! some bacteria have wheel-like structurehowever, legs are more prevalent in nature

Most robots have three or four wheels (recently robots with two wheels have been produced)

three wheels are sufficient to guarantee stabilitywith more than 3 wheels a flexible suspension is neededselection of wheels depends on the application

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Wheels Types

Standard wheel: two DOFrotation around the motorized wheel axlerotation around the contact point

Castor wheel: three DOFrotation around the wheels axlerotation around the contact pointrotation around the castor axle

Swedish wheel: three DOFrotation around the motorized wheel axlerotation around the contact pointrotation around the rollers

Ball or spherical wheel: three DOFsuspension technically not solved

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Stability

Robots need to be stable to get their job doneStability can be

Static: the robot can stand still without falling overis achieved thorugh the mechanical design of the robot

Dynamic: the body must actively balance or move to remain stable

is achieved through control

For stability the Center Of Gravity (COG) of the body needs to be above the polygon support

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Different Arrangement of Wheels

Two wheels

Three wheels

Four Wheels

Six Wheels

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Characteristics of Wheeled Robots and Vehicles

Stability of the vehicle can be guaranteed with 3 wheelscenter of gravity must fall inside the triangle formed by the ground contact points of the wheels

Stability is improved by 4 and more wheelsthese arrangements are hyper-static and require a flexible suspension system

Bigger wheels allow to overcome higher obstaclesbut they require higher torque or reductions in the gear box

Most arrangements are non-holonomicrequire high control effort

Combining actuation and steering on one wheel makes the design complex and adds additional errors for odometry

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Examples of Wheeled Robots

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Motion Control

To control the movement of a wheeled robot we needkinematic/dynamic model of the robotmodel of the interaction between the wheel and the grounddefinition of the required motion (speed and position control)control law that satisfies the requirements

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Mobile Robot Kinematics

Aimdescription of mechanical behavior of the robotsimilar to manipulator kinematicsmobile robots can move unbound w.r.t. the environment

we cannot measure the robot's positionposition must be integrated over timelead to inaccuracy in the position estimation

To understand mobile robot motion we need to understand wheel constraints

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Kinematics Model

Goalgiven the geometric parameters of the robot, wheel speeds, steering angles, and steering speeds we want to establish the robot speedForward kinematics

Inverse kinematics

why not using the following formulation?

=[ xy ]= f 1, ,n ,1, ,m , 1, , m[1 n 1 m 1 m ]T= f −1 x , y ,

=[ xy ]= f 1, ,n ,1, ,m

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Representing Robot Position

Fix an initial frameinitial frame: {X

I,Y

I}

robot frame: {XR,Y

R}

robot position:

mapping between the two frames

=[ xy ]R=R I=R [ xy ]R =[ cos sin 0

−sin cos 00 0 1 ]

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Wheel Kinematic Constraints: Assumptions

Movement on a horizontal planePoint contact of the wheelsWheels not deformablePure rolling (v=0 at contact point)No slipping, skidding or slidingNo friction for rotation around contact pointAll the steer axes are orthogonal to the groundWheels connected by rigid frame (chassis)

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Kinematic Constraints: Fixed Standard Wheel

Pure rolling constraint

Lateral movement constraint

[sin −cos −lcos ]R I−r =0

[cos sin l sin ]R I=0

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Kinematic Constraints: Steered Standard Wheel

Pure rolling constraint

Lateral movement constraint

[sin −cos −lcos ]R I−r =0

[cos sin l sin ]R I=0

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Kinematic Constraints: Castor Wheel

Pure rolling constraint

Lateral movement constraint

[sin −cos −lcos ]R I−r =0

[cos sin dl sin ]R Id =0

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Kinematic Constraints: Swedish Wheel

Pure rolling constraint

Lateral movement constraint

[sin −cos −lcos ]R I−r cos=0

[cos sin l sin ]R I−r sin −r sw sw=0

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Kinematic Constraints: Spherical Wheel

Pure rolling constraint

Lateral movement constraint

[sin −cos −lcos ]R I−r =0

[cos sin sin ]R I=0

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Robot Kinematic Constraints

Given a robot with M wheelseach wheel imposes zero or more constraintsonly fixed and steerable and standard wheels impose constraints

What is the maneuverability of a robot considering a combination of different wheels?Suppose we have a total of N = Nf + Ns standard wheels

Rolling

Lateral movement

J 1sR IJ 2=0

t = [ f t s t ]

N fN s×1

J 1s= [ J 1fJ 1ss]

N fN s×3J 2=diag r 1r N

C 1sR I=0 C 1s= [ C1fC 1s s]N fN s×3

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Mobile Robot Maneuverability

The maneuverability of a mobile robot is the combination

of the mobility available based on the sliding constraintsplus additional freedom contributed by the steering

Three wheels are sufficient for static stabilityadditional wheels need to be synchronizedthis is also the case for some arrangements with three wheels

It can be derived using the equation seen beforeDegree of mobility δ

m

Degree of steerability δs

Robots maneuverability δM

= δm

+ δs

Two robots with same δM are not necessary equal

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Instantaneous Center of Rotation

Imposing the absence of lateral movement is the same as requiring the existence of the Instantaneous Center of Rotation (ICR)The ICR is the point of intersection between the wheel axles

For any robot with δM = 2 the ICR is always constrained to lie on a line For any robot with δM = 3 the ICR is not constrained an can be set to any point on the plane

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Three-Wheel Configurations

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Mobile Robot Workspace

Workspacehow the vehicle is able to move between different configurations in its workspace?

DOF: Degrees of freedomrobots ability to achieve various poses

DDOF: Differentiable degrees of freedomrobots ability to achieve various path

DDOF ≤ δM ≤ DOF

Holonomic robotsa holonomic kinematic constraint can be expressed as an explicit function of position variables onlya non-holonomic constraint requires a different relationship, such as the derivative of a position variableFixed and steered wheels impose non-holonomic constr.

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Differential Drive Robots

It is the easiest configuration2 actuated wheels on same axle1 passive wheel

Inputdesired velocities

Outputwheel velocities

when vr = vl , R goes to infinity -> straight movement

when vr = -vl , R is zero, the robot rotates on itself

R L2=vR

R− L2=v L

=

v r−v LL

R= L2

vRv LvR−v L

=

v r−vLL

v=vrvL2

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Kinematics of Differential Drive Robots

Forward kinematicsgiven the wheel speeds find the robot position

Inverse kinematicsfind the speeds to reach a given destinationthe problem has infinite solutionsthe equations for the constraints in the velocities cannot be integrated in a constraint for the positionparticular cases

when the wheel velocities are the same the robot moves straight

x t =∫ v t cos t dt=12∫ vL t v R t cost d t

y t =∫ v t sin t dt=12∫ v L t vR t sin t d t

t =∫ t dt=1L∫ v R t −vL t dt

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Synchronous Drive Robots

It is a robot with simple kinematicsthree actuated steering wheelsall the wheels receive the same actuationtwo motors

one to make the wheels runone to make the wheels steer

the wheels have always the same directionrotations are around the robot center

it is possible to directly control θ

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Kinematics of Synchronous Drive Robots

Forward kinematicsrotation around the center is equal to the angular speed ωthe movement speed is equal to the linear velocity vCIR is always at infinity

when the wheels steer CIR direction changes

Inverse kinematicswe use the special case

when v = 0 and the angular speed is ω during the interval δt, the robot turns of an angle equal to ω*δtwhen ω = 0 and the linear velocity is v then the robot moves forward for v*δt

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Kinematic Control

The objective of a kinematic controller is to follow a trajectory described by its position and/or velocity profiles as function of timeMotion control is not straightforward since mobile robots may be non-holonomic systems

some solutions have been proposedopen loop controlfeedback controlkinematic position control

Most controllers do not consider the dynamics of the system

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Open Loop Control

The trajectory is divided in motion parts of clearly defined shape

straight linesarcs of a circle

Control problempre-compute a smooth trajectory

Disadvantagesnot easy to pre-compute a feasible trajectorylimitations to robot velocities and accelerationsdoes not adapt to dynamical changestrajectory are not smooth

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Feedback Control

Find a control matrix K (if it exists), with kij = k(t,e)

such that the control of v(t) e ω(t)

drives the error e to 0

K=[k11 k12 k 13k21 k 22 k 23]

[ v t t ]=K⋅e=K⋅[ xy ]limt∞

e t =0

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Kinematic Position Control

The kinematic of a differential drive mobile robot described in the initial frame {xI, yI, θ} is given by

where the first two components are the linear velocity in the direction of xi and yI

let α be the angle between the xR axis and the vector connecting

the center of the axle of the wheels with the final position

[ xy ]=[cos 0

sin 00 1 ][ v]

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Kinematic Position Control

Transform the coordinates into polar coordinates with its origin at goal position

System in the new coordinates

= x2 y2

=−arctan x y =−

[]=[−cos 0

sin

−1

0 −1][v] for ∈−2 ,2 ]

[ ]=[−cos 0

−sin

−1

0 −1][ v] for ∈− ,−2 ]∪2 ,]

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Kinematic Position Control

It can be shown that with

the feedback controlled system

will drive the robot to the final positionThe control signal v has always constant sign

the direction of the movement is kept positive or negative during movementparking maneuver is performed always in the most natural way and without ever inverting its motion

v=k=k k

[ ]=[ −k cos−k alpa−kk sin

−k −k ]

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Non-Holonomic Systems

Differential equations are not integrable to the final positionThe measure of the traveled distance of each wheel is not sufficient to compute the final position of the robotWe need to know also how this movement was executed as a function of time

non-holonomic

v t =∂ s∂ t=∂ x∂ t

cos∂ y∂ t

sin

d s=d x cosd y sin

d s=∂ s∂ x

d x∂ s∂ y

d y∂ s∂d

∂ s∂ x=cos ; ∂ s

∂ y=sin ; ∂ s

∂=0

∂2s∂ x ∂

=−sin≠0= ∂2 s

∂∂ x