RoboticsMobility: wheels and whegs

Dipartimento di Elettronica Informazione e Bioingegneria

@ G. Gini 2015

G. Gini

Wheeled Mobile robots (WMR)

A robot capable of locomotion on a surface 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 whichit is intended to have a single point of rolling contact.

(Muir and Newman, 1986)

AGVAutomated Guided VehicleALVAutonomous Land Vehicle

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• 1. only rigid parts.• 2. each wheel may have a 1 link for steering.• 3. steering axes are orthogonal to soil.• 4. pure rolling of the wheel about its axis (x axis)• 5. no translation of the wheel

• This model will be used in the kinematic model

Basic assumptions

y

roll

z motion

x

y

Wheel parameters:r radiusv linear velocityw angular velocity

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fixedCannot move in directions

perpendicular to the wheel plane

orientable centered

x

y

caster wheelomnidirectional

composed or Svedish

wheel

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WMR

• Easy model• Sensible to terrain irregularities

• 2 wheels

• Omnidirectional

• tracks

• For outside terrains• Imprecisions in

movements, in particular rotations

• Free movement• Complex structure

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Instantaneous Center of Curvature or RotationICC, ICR, CIR = point where the wheel axes intersect

ICC

Cannot move

ICC (or CIR – ICR)

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Representing a pose

xb

yb

θ

xm, ym is in the mobile system

xb, yb is in the base system

xmym

ICC

The rotation θ is about the z axis

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• DIRECT KINEMATICSGiven the control parameters (wheels velocities, etc) and a time of movement t, find the pose (x, y, θ) reached by the robot

• INVERSE KINEMATICSGiven the final pose (x, y, θ) find the controlparameters to move the robot there in a given time t

WMR Kinematics

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Direct and inverse kinematics

Direct kinematics: integrating

. . . x(t), y(t), θ(t)

Inverse kinematics

wheels parameters

wheels parameters

robot pose:

x(t), y(t), θ(t) control variables:

R angular velocity ω(t) linear velocity ν(t)

control variables

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2 wheels – differential drive

A robot for RoboCup

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pose

v : linear velocity of the robotω : angular velocity of the robot

Control input

Differential drive - 1

2 variables independently controlledVr – velocity right wheel

Vl - velocity left wheel

Construction: 2 wheels on the same axis2 motors

-one for wheel3rd passive supporting wheel

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differential drive - 2The wheels L and R follow a circular pathwith ω angular velocityand different curvature radius

ω(R + L/2) = vRω(R - L/2) = vL

find ω given Vl and VrSolve for R and equate

ω = vR - vL /L

find Rsolve for ω and equatePARTICULAR CASES

Rotation in placeR = 0 VR = -VL

Linear movementR = infinite VR = VL

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The wheels move around ICC on a circumferenceistantaneous radius = R angular velocity = ω

Find ICC: ICC = (x +Rcos(θ+π/2), y + Rsin(θ+π/2) = (x - Rsin(θ), y + Rcos(θ)

ICC

R

ICC in differential drive

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ICC computation

ICC

R

P(t)

P(t+δt)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

tICCyICCx

tttt

yx

y

x

y

x

ωδθωδωδωδωδ

θICCICC

1000)cos()sin(0)sin()cos(

'''

Given pose(t), pose(t+dt) is: rotation at distance R fromICC with velocity ω. –translatee ICC in origin–rotate arounda ICC withω velocity–Translate back

Rot(z,wdt) Trasl(x_ICCx, y _ICCy,θ)+)+Trasl(ICCx, ICCy,ωdt)

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Differential Drive - equations

R

ICCω

(x,y)

y

l/2

θx

vl

v r

lvv

vvv

vvvvlR

lRvlRv

lr

lr

lr

rl

l

r

−=

+=

−+

=

−=+=

ω

ωω

2

)()(

2

)2/()2/(

]cos,sin[ θθ RyRxICC +−=

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Integrate to obtain position

VR

VL

2d

ICCR(t)

robot’s turning radius

ω(t)

ω = ( VR - VL ) / 2d

R = 2d ( VR + VL ) / ( VR - VL )

V = ωR = ( VR + VL ) / 2

Vx = V(t) cos(θ(t))

Vy = V(t) sin(θ(t))

with

x

y

x(t) = ∫ V(t) cos(θ(t)) dt

y(t) = ∫ V(t) sin(θ(t)) dt

θ(t) = ∫ ω(t) dt

Thus,

Position from velocity

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Direct kinematics - direct drive

CIR

R

P(t)

P(t+δt)

')]'()'([1)(

')]'(sin[)]'()'([21)(

')]'(cos[)]'()'([21)(

0

0

0

∫

∫

∫

−=

+=

+=

t

lr

t

lr

t

lr

dttvtvl

t

dtttvtvty

dtttvtvtx

θ

θ

θ

Using the control input')'()(

')]'(sin[)'()(

')]'(cos[)'()(

0

0

0

∫

∫

∫

=

=

=

t

t

t

dttt

dtttvty

dtttvtx

ωθ

θ

θ

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VR(t)

VL (t)

starting position final position

x

y Finding some solution is not hard, but finding the “best”solution is very difficult...

• quickest time• most energy efficient• smoothest velocity

profiles

VL (t)

tVL (t)

Key question: Given a desired position or velocity, what can we do to achieve it?

define “best”...

Inverse kinematics

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Inverse kinematics

The equations of the direct kinematics describe a constraint on the velocity of the robot that cannot be integrated into a positionalconstraint (non holonomic constraint)

How to find Vr and Vl to get to the wanted pose? underconstrained: infinite combinations of Vr e Vlthe robot moves on a circlepassing for (0,0) at time 0 and (x,y) at time t. Infinite solutionsNo independent control of θ

But the solution is straingthforwardfor a limited class of control functionsvr and vl

0,0,0

x,y,t

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Usual approach: decompose the problem and control only a few DOF at a time

VR(t)

VL (t)

starting position final position

x

y

Differential Drive

(1) turn so that the wheels are parallel to the line between the original and final position of the robot origin.

-VL (t) = VR (t) = Vmax

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Usual approach: decompose the problem and control only a few DOF at a time

VR(t)

VL (t)

starting position final position

x

y

Differential Drive

(1) turn so that the wheels are parallel to the line between the original and final position of the robot origin.

(2) drive straight until the robot’s origin coincides with the destination

-VL (t) = VR (t) = Vmax

VL (t) = VR (t) = Vmax

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Usual approach: decompose the problem and control only a few DOF at a time

VR(t)

VL (t)

starting position final position

x

y

Differential Drive

(1) turn so that the wheels are parallel to the line between the original and final position of the robot origin.

(2) drive straight until the robot’s origin coincides with the destination

(3) rotate again in order to achieve the desired final orientation

-VL (t) = VR (t) = Vmax

VL (t) = VR (t) = Vmax

-VL (t) = VR (t) = Vmax

VL (t)

tVR (t)

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equations

1. If VR different VL:x(t) = [vL + vR]/ [vR - vL]* L/2*sin [t(vR- vL)/L] y(t) = [vL + vR]/ [vR - vL]* L/2*cos [t(vR- vL)/L] + [vL + vR]/ [vR -

vL]* L/2θ(t) = t/l * (vR - vL )

2. If VR=VL, R is infinite, θ(t) constant – linear movementof the robot

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Synchronous drive

3 wheels actuated and steered, controlled together• a motor to roll all the wheels, a second motor to rotate them•The 3 wheels point in the same direction•It is possible to control directly θ. •The robot can rotate on site aboutthe center of the robot•Holonomic behavior

CONTROL VARIABLES:•v(t), ω(t)

x

yv(t)

θ

ICC is always at infinity

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synchronous drive kinematics - 1

• DK: rotation around the robot center at angularvelocity ω(t), translation at linear velocity v(t). ω and v are directly controlledICC is at infinite distanceSteering changes the direction of ICC

Particular cases:v(t) = 0, w(t) = w for Δt, the robot rotates in placev(t) = v, w(t) = 0 for Δt, the robot moves linearly

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Synchonous Drive kinematics - 2

Inverse kinematicsConsider that1. v(t) = 0 and ω(t) = ω for δt, then the robot rotates in place2. v(t) = v and ω(t) = 0 for δt, then the robot moves in the

direction it is pointing

θ

y

x

v(t)

ω( )t

')'()(

')]'(sin[)'()(

')]'(cos[)'()(

0

0

0

∫

∫

∫

=

=

=

t

t

t

dttt

dtttvty

dtttvtx

ωθ

θ

θ

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Usual approach: decompose the problem and control only a few DOF at a time

V(t)

ω(t)

starting position

final position

x

y

Synchro Drive

(1) turn so that the wheels are parallel to the line between the original and final position of the robot origin.

(2) drive straight until the robot’s origin coincides with the destination

(3) rotate again in order to achieve the desired final orientation

ω(t) = ωmax

V(t) = Vmax

ω(t) = ωmax

Inverse kinematics synchro drive

Similar to the special cases of differential drive

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Other steered wheels

Tricicle (2 passive wheels, one actuated and steerable) as in AGV.

Robots that do not use eitherdifferential or synchronous drive have both wheels that can besteered and fixed wheels

Bicycle (1 fixed and 1 stearable)

ICC

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• Ackermann Steering

Car-like (2 fixed and 2 steerable)

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Other robot vehiclesFor outdoor environments, more degrees of freedom and

more complex wheels

Omnidirectional: easier to control

Sojourner Truth4 steerable wheels + 2 fixed

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Soil mechanism

When a wheel is driven, the driving force turns the wheel and isopposed by the frictional force between wheel and ground.

If driving force > frictional force the wheel may slipThe frictional force limits the rate of acceleration of the vehicle.On soft ground, wheels and legs sink. If a wheel moves forward, it

compacts the soil in front of it using energy. A driven leg has to compact the soil too, but has the advantage that it happens only in the point of contact, not on all the path

If frictional force is low, a wheel may not be able to generate enough force to compact the soil and will spin. In contrast a leg can move over a surface with low friction.

friction

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Tracked vehicles

• Kinematics as in differential drive

v of inside and outside tracksr diameter of the wheel• vo = rω0

• vi = rωi

It is necessary a model of the slippingTracks eliminate most of the sinking

problems• iRobot, 4 tracks

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tracks

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Degrees of Freedom

• “Maneuverability” is equivalent to the vehicle’s degree of freedom (DOF) – a body in 2D – characterize the positions that ca be reached in space

• Examples:• Bicycle: DOF=3• Omni Drive: DOF=3

• The robot’s independently achievable velocities= differentiable degrees of freedom (DDOF)

Examples:• Differential drive: DDOF = 2, DOF 3• Omni Drive: DDOF=3; DOF=3

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Degrees of Freedom -> Holonomy

• DOF degrees of freedom:Robots ability to achieve various poses

• DDOF differentiable degrees of freedom: Robots ability to achieve various path

HolonomicA 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 standard wheels impose non-holonomic constraints

DOFDDOF≤

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kinematic holonomic/nonholonomic constraintsKinematic constraint - a situation that reduces positions/velocities of the

mechanical system, so reducing the relative mobilityConstraints can be expressed as a set of equations/disequations of

position and velocity of the points in the system:

A position constraint is called holonomic (position) if in the equation thereis no dependence on the velocity.

is non holonomic ( mobility) if it constraints also velocity

This equation cannot be integrated (Holos in Greek = integer)

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Example holonomic constraints

rolling cylinder6 coordinates, x, y, z, r, θ, φ5 constraints:

z=0, since it rolls on the planer = constantφ = 0, the plane faces are orthogonal to the planespace s = (x2 + y2)1/2 – it replaces 2 coordinates with 1s’ = aθ’ if the cylinder rolls without slipping- in holonomic form it isds= adθ so s-s0= a(θ−θ0), a relation between coordinates so holonomic

only 1 degree of mobility (6 – 5), s or θ.

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Example non holonomic constraint

A thin disk rolling on an horizontal plane• Six coordinates as before• Holonomic constraints:

Z=0R=a=constantBut it can spin about both θ (roll) and, φ (turn) so it isnon holonomic

5 degrees of freedom• Since the disk can turn (change φ) it can roll from one

place to another by different routes and so the distance in the plane space s = (x2 + y2)1/2 is not related to the angle θ, as it was in the cylinder, and so this constraint does notlead to a definite condition on the coordinates.

φ

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Restrict DOF or velocity

Each configuration of a system is univocally determinedby n independent parameters q1, ..., qn;

n degrees of freedomThe holonomic constraints subtract a degree of

freedom for each constraint equationThe non holonomic constraints restrict only velocity:

- they allow to reach any position, so they do notreduce the degrees of freedom. - some paths are not allowed while any position can be reached.imagine a car; whilst it is possible for it to be in any position on the road, it is not possible for it to move sideways.

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WMR holonomic and non holonomic

Holonomic robot =

all its constraints are holonomicA mobile robot on the plane has 3 dofAn holonomic WMR with 3 dof is onlythe omnidirectional one

Many WMR are non holonomicWheels cannot move orthogonally to the direction of travel

Non holonomic constraints reduce the control space (ex: not possible to move laterally)Non-holonomic systems

• differential equations are not integrable to the final position. • the measure of the traveled distance of each wheel is not sufficient

to calculate the final position of the robot. One has also to know how this movement was executed as a function of time.

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Free movement (holonomic robot)

RIV, IspraService robot for nuclear waste store

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Control of WMR• The objective of a kinematic controller is to follow a

trajectory described by its position and/or velocity profiles as function of time.

• Given 2 poses (qi and qf) find a trajectory that followsthe kinematic constraints

there are possible trajectories also for non holonomicrobotsusually the kinematic model is usedusually mobile robots are underactuated – forinstance the differential drive has 2 input Vl and Vrand 3 generalized coordinates x, y, θ

• Often mobile robots use open loop control

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Open Loop Control

• trajectory (path) divided in motion segments of clearly defined shape:

straight lines and segments of a circle• control problem:

pre-compute a smooth trajectory based on line and circle segments

• Disadvantages:It is not an easy task to pre-compute a feasible trajectory limitations and constraints of the robots velocities and accelerationsdoes not adapt or correct the trajectory if dynamical changes of the environment occurThe resulting trajectories are usually not smooth

yI

xI

goal

3.6.1

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Whegs

• Wheels + legs• Simple control

• RHEX, MiniWhegs

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Scout (umn - Darpa)Length: 110 mmDiameter: 40 mmWeight: 200 gTop Speed: 0.31 m/sLocomotion: Primary: 2 Foam

wheelsSecondary: Jumping" mechanismMaximum Jump Height: 30 cmSensors: Camera (Color or B/W)

AccelerometersWheel EncodersMagnetometers

Operational Lifetime:Quiescent Mode: 120 minutesFull Speed: 70 minutesJumping Only: 100 jumps

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LionHell

• All terrain robot• Whegs + passive joint + active neck• Behavior based

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Robots to move in air

• Flying robotsFixed wings vehiclesMobile wings vehiclesquadrotors

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Robots to move in water

Acquatic vehicles: two basic designs are torpedo-like structures, with a single propellerBodys with a collection of thrusters

New design is bioinspired and uses soft materials.In general they operate in a tethered manner, connected to a

support vessel that provides power and communication

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spheroA. Charging Dock The ball rests in a dock that uses an induction system to transfer electricity to the batteries. B. Wheels Two independently controlledrubber-rimmed wheels inside the Sphero steerand drive the ball at up to 1.2 m/sec C. Top Slip Bearing braces the mechanismin order to keep the wheels in firm contactwith the shell.D. Printed Circuit Board A processorcombines data from a three-axisaccelerometer and a gyroscope to produce the precise measurements of the Sphero’s roll, pitch, and yaw. These measurements are required to respond correctly to commandsradioed by a smart phone over a Bluetooth connection E. Bluetooth Radio and Antenna Thissystem, with a maximum range of 50 meters, is used to communicate with mobile devices.F. Multicolor LED The light from a single LED package with red, green, and blueelements is diffused to make the Sphero glow. Different colors signal information.

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Passive robot

• Planets or ocean exploration• Adapt (no or limited actuation)

Example: thumbleweed (NASA)

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conclusion

• Who is investing in new mobile robots?

Media Google (Boston Dynamics)

ElectroMechanical companies

Sony

Honda

Traditional robotics companies

iRobot

Adept Technology

Kuka