10 a review of models and structures for wheeled mobile robots four case studies

Next, the model for omnimoble robots with Swedish wheels is presented to illustrate holonomic omnidirectional motion.. These five classes are: • Type 3,0 robots or omnidirectional robots

A Review of Models and Structures for Wheeled Mobile Robots:

Four Case Studies

Ramiro Vel´azquez and Aim´e Lay-Ekuakille

Abstract— This paper reviews the mathematical models of

four commonly encountered designs for wheeled mobile robots

(WMR) These designs belong to two generic classes of wheeled

robot structures: differential-drive and omnimobile First, the

two wheel differential-drive model is presented to show how

zero turning radius is achieved with only bidirectional

move-ment Three particular designs are addressed: the popular

two-active-fixed wheels and one-passive-caster wheel, a simple

belt-drive, and sprocket-chain system Next, the model for

omnimoble robots with Swedish wheels is presented to illustrate

holonomic omnidirectional motion All four models are based

on physical parameters easily measured and are useful to

understand the internal dynamics of these WMR and to

accurately visualize their motion in 2D environments They

can be therefore used as a practical reference to predict the

accessibility of physical prototypes to selected places and to

test different algorithms for control, path planning, guidance,

and obstacle avoidance

I INTRODUCTION Understanding how wheeled mobile robots (WMR) move

in response to input commands is essential for feedback

control design and many navigation tasks such as path

planning, guidance, and obstacle avoidance

Campion and Chung classified in [1] the mobility of WMR

into five generic structures corresponding to a pair of indices

(m, s): mobility degree m and steerability degree s The first

one refers to the number of degrees of freedom the WMR

could have instantaneously from its current position without

steering any of its wheels while the second refers to the

number of steering wheels that can be oriented independently

in order to steer the WMR These five classes are:

• Type (3,0) robots or omnidirectional robots have no

steering wheels (s=0) and are equipped only with

Swedish or active caster wheels They have full mobility

in the plane (m=3), which means that they are able

to move in any direction without any reorientation

Representative examples of such robots are [2] and [3]

• Type (2,0) robots have no steering wheels (s=0) but

either one or several fixed wheels with a common

axle The common axle restricts mobility to a

two-dimensional plane (m=2) Examples of type (2,0) robots

are [4] and [5]

• Type (2,1) robots have no fixed wheels and at least one

steering wheel If there is more than one steering wheel,

R Vel´azquez is with the Mechatronics and Control Systems Lab

(MCS), Universidad Panamericana, 20290, Aguascalientes, Mexico

Con-tact:rvelazquez@ags.up.mx

A Lay-Ekuakille is with the Department of Innovation

Engineering, Universit`a del Salento, 73100, Lecce, Italy Contact:

aime.lay.ekuakille@unisalento.it

their orientations must point to the same direction (s=1) Therefore, mobility is restricted to a two-dimensional plane (m=2) An example is the synchronous drive WMR in [6]

• Type (1,1) robots have one or several fixed wheels

on a common axle and also one or several steering wheels, with two conditions for the steering wheels: their centers must not be located on the common axle of the fixed wheels and their orientations must

be coordinated (s=1) Mobility is restricted to a one-dimensional plane determined by the orientation angle

of the steering wheel (m=1) Examples of this type are the tricycle, the bicycle, and the car-like WMR

• Type (1,2) robots have no fixed wheels, but at least two steering wheels If there are more than two steering wheels, then their orientation must be coordinated in two groups (s=2) Mobility is restricted to a one-dimensional plane (m=1) determined by the orientation angles of the two steering wheels

This paper particularly address type (3,0) and (2,0) robots Taking as example our own prototypes (and some practical lessons learned from their implementation), we derive the mathematical models of four commonly encountered designs for these two types of WMR

The rest of the paper is organized as follows: in Sec-tion 2, the popular two wheel differential-drive model is obtained using the general two-active-fixed wheels and one-passive-caster wheel structure Next, two other differential-drive designs are presented to illustrate some other efficient locomotion systems: a simple belt-drive system which shows how frictional forces transfer torque to generate motion and

a sprocket-chain system which offers another method for transferring motion when frictional forces are insufficient

to transfer power In Section 3, the omnimobile robot with Swedish wheels is analyzed The resulting model shows how holonomic omnidirectional motion is achieved Finally, the conclusion summarizes the paper main concepts

II DESIGNS ANDPROTOTYPES Let us start addressing type (2,0) robots There are many design alternatives; however, the two-wheel differential-drive robot is by far the most popular design

Let us consider our prototype IVWAN (Fig 1(a)) Its mechanical structure is based on a differential-drive con-figuration consisting of two independently controlled front-active wheels and one-rear-caster wheel (Fig 1(b)) Active wheels are driven by two high-power DC motors which allow IVWAN to achieve a maximum speed of 20 km/hr

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(a) (b) (c) Fig 1 Type (2,0) WMR IVWAN (Intelligent Vehicle With Autonomous Navigation): (a) prototype and (b) its differential-drive structure Two front wheels each driven by its own motor A third wheel is placed in the rear to passively roll along while preventing the robot from falling over The wheels

exhibit three speeds: u, ¯u, and ω (c) Free-body diagram The first subscript stands for front f and caster c wheel while the second subscript stands for

right r and left l wheel.

IVWAN exhibits both manual and autonomous operation:

it can be tele-operated or self-guided by a color camera

and an array of ultrasonic sensors that allow the machine

to detect and follow visual patterns and negotiate obstacles,

respectively [7]

Fig 1(c) shows a schematic representation of the

differential-drive structure Here, B represents the center of

the axis connecting both traction wheels; G represents the

vehicle’s center of mass and for simplicity, it is considered

as the point to control in position (x, y) and orientation (ϕ).

Resultant forces and momentum in the structure can be

expressed by eq (1):



F x = m( ˙u − ¯uω) = F f rx + F f lx + F cx + F Gx



F y = m( ˙¯u + uω) = F f ry + F f ly + F cy + F Gy



M z = I ˙ω = d2(F f rx − F f lx ) − b(F f ry + F f ly) +

where m is the vehicle’s total mass, I is the moment

of inertia around point G, and u, ¯u and ω are the robot’s

linear, transverse sliding, and angular speeds, respectively

(Fig 1(b)) Speed ¯u can be reasonable neglected assuming

that the wheels do not slip during motion Concerning u and

ω, they can further be defined by eq (2):

2[r(ω r + ω l ) + (u r + u l)]

d [r(ω r − ω l ) + (u r − u l)] (2) where r is the traction wheel radius, d is the distance

between the traction wheels (see Fig 1(c)),ω r, and ω l are

the angular speeds of the right and left wheels respectively,

and ur and ul are the linear speeds of the right and left

wheels respectively

Kinematics of point G is related to u andω by eq (3):

˙x = ucosϕ − bωsinϕ

˙y = usinϕ + bωcosϕ

As aforementioned, traction wheels are powered by DC motors These can be modeled by eq (4):

τ r = k a

R a (E r − k b ω r)

τ l = k a

R a (E l − k b ω l) (4) whereτ r andτ r are the torques developed by the motors

on the right and left wheels upon input DC voltages Er and El respectively, ka and kb are the motor’s torque and electromotive force constants, and Ra is the motor’s electric resistance Inductive voltages have been neglected

Equations describing the wheel-motor system can be sim-ply written as shown in eq (5):

I e ω˙r + D e ω r = τ r − F f rx ˆr

I e ˙ω l + D e ω l = τ l − F f lx ˆr (5) where Ie and De are the moment of inertia and the coefficient of viscous friction of the wheel-motor system, respectively andˆr is the nominal radius of the traction wheel

tires Using and combining eqs (1) to (5), the differential-drive model can be summarized by eq (6):

⎡

⎢

⎢

⎣

˙x

˙y

˙ϕ

˙u

˙ω

⎤

⎥

⎥

⎡

⎢

⎢

⎣

ucosϕ − bωsinϕ usinϕ + bωcosϕ ω

a3

a1ˆrrω2− 2 a4

a1u

−2 a3

a2ˆrruω − a4

a2d2ω

⎤

⎥

⎥

⎡

⎢

⎢

⎣

2r

a1 0

a2

⎤

⎥

⎥

⎦

E u

E ω (6) with inputs:

E u = E r + E l

2

E ω = E r − E l

2

(a) (b) Fig 2 (a) Block diagram reference for differential-drive robots (b) Summary of motion upon voltages Er and El.

and constants:

a1 = R a

a2 = R a

k a [I e d2+ 2ˆrr(I + mb2)] [V · m2· s2]

a3 = R a

a4 = R a

k a(k a k b

Note that eq (6) relates the robot’s motion to the motors’

input voltages The block diagram model for

differential-drive robots is shown in fig 2(a) This diagram identifies the

electronics, DC motors, and the vehicle’s kinematics

Fig 2(b) summarizes how differential-drive robots are

controlled by the input voltages Er and El When both

voltages are equal, the two driving wheels turn at the same

angular speed and in the same direction, which causes a

translation movement If one voltage is set to zero, one of

the wheels turns while the other remains motionless, then the

robot describes a circle centered on the motionless wheel If

both voltages are equal in magnitude but opposite sign, the

wheels turn at the same speed but in opposite direction which

causes a rotation around the center of the axis connecting

both wheels (point B) Note a zero turning radius in this

case

Numerical values of the parameters involved in eq (6)

can be easily measured from an existent prototype and

the specifications of the DC motors can be obtained from

the manufacturer As illustrative example, consider all gain

blocks of fig 2(a) as unity gains Fig 3(a) shows a computer

simulation of a certain trajectory in the XY plane Fig 3(b)

shows the driving signals supplied to the DC motors Note

the correspondence with fig 2(b)

There are other design alternatives for differential-drive

robots, subsections A and B present two different approaches.

A Belt-drive

Belts have long been used for the transfer of mechanical

power Today’s flat belts are relatively light, inexpensive,

tolerant of alignment errors, and ensure a long operating

(a)

(b) Fig 3 (a) A simulated trajectory of the differential-drive robot and (b) the corresponding driving signals.

life They transmit power through frictional contacts They function best at moderate speeds (20 to 30 m/s) under static loads Their efficiencies drop slightly at low speeds and centrifugal effects limit their capacities at high speeds [8] Let us consider our prototype Enyo (Fig 4(a)) Its mecha-nical structure is based on a four wheel differential-drive configuration driven by a belt system Two active-front-wheels transfer rotating motion to the two passive-rear-wheels through belts (Fig 4(b)) This motor-belt system allows Enyo to achieve a maximum speed of 30 km/hr Even though Enyo seems a car-like type (1,1) WMR, it is a type (2,0) WMR because none of its wheels are steerable Fig 4(c) shows a schematic representation of the belt-drive system Here, the belt is modeled as a spring with constant

526

(a) (b) (c) Fig 4 Type (2,0) WMR Enyo: (a) prototype and (b) its belt-drive system (c) Schematic of the locomotion system.

k The radii of the pulleys are r1 and r2 while their inertias

are J1and J2, respectively As in the previous case, traction

pulleys are power by DC motors The angular speed of the

motor and hence, of the active pulley isθ m and the angular

speed of the passive pulley is θ p

Eqs (7) to (9) describe this system:

J1θ¨m = k t i − b ˙ θ m − r1(F1− F2) (8)

J2θ¨p = −b ˙θ p + r2(F1− F2) (9)

where E is the input voltage to the motor, R and L are

the motor’s electric resistance and inductance, kt and ke

are the motor’s torque and electromotive force constants,

respectively The motor-pulley friction is denoted by b while

F1 and F2 are the forces exerted by the belt on the pulleys

These forces can be further expressed as eq (10):

F1 = k(x1− x2)

Knowing that x1 = r1θ m and x2 = r2θ p, and further

considering that pulleys are identical (J1 = J2 = J and

r1= r2= r), eqs (8) and (9) become eqs (11) and (12):

J ¨ θ m = k t i − b ˙ θ m − 2kr2(θ m − θ p) (11)

J ¨ θ p = −b ˙θ p + 2kr2(θ m − θ p) (12)

The motor-pulley-belt system can be summarized by state

eq (13):

⎡

⎢

⎢

⎣

i 

¨

θ m

˙θ m

¨

θ p

˙θ p

⎤

⎥

⎥

⎡

⎢

⎢

⎣

− R

L − k e

k t

J − 2kr2

J

J − 2kr2

J

⎤

⎥

⎥

⎦

⎡

⎢

⎢

⎣

i

˙

θ m

θ m

˙θ p

θ p

⎤

⎥

⎥

⎦+

⎡

⎢

⎢

⎣

1

L

0 0 0 0

⎤

⎥

⎥

⎦E (13) Essentially, the goal of belt-drive systems is to transfer

efficiently mechanical power between pulleys so that their

angular speeds are the same (θ m = θ p)

Using Enyo’s parameters given in table 1, fig 5 examines

the effect of varying the spring constant k on the pulleys’

angular speeds Note that, the higher the values of k, the best

match betweenθ mandθ p In practice, this means that, even

TABLE I

S UMMARY OF THE PARAMETERS INVOLVED IN THE BELT - DRIVE MODEL

Parameter Value Unit

k e 0.99 V-s/rad

k t 15.55 N-m/A

b 0.25 N-m-s/rad

Fig 5 Effect of varying the spring constant k on the belt-drive system Plots for k=10, 50, 150, 1000 N/m.

under constant rotation, belts tend to creep Thus, these drives must be kept under substantial tension to function properly One possible physical implementation to increase k is the one shown in fig 4(b) A third pulley or V-belt pulley forces the belt to increase it spring constant k

B Sprocket and chain drive

Sprockets and chains offer another option for transfer-ring rotating motion between shafts when the friction of

a drive-belt is insufficient to transfer power Contrary to

(a) (b) (c) Fig 6 Type (2,0) WMR Connor: (a) prototype and (b) its sprocket and chain system (c) Schematic of the locomotion system.

belts, sprockets and chains transmit power through bearing

forces while maintaining a fixed phase relationship between

the input and output shafts The main drawback is the

contact between the sprocket and chain: the contact can slip

significantly as the chain rollers and sprocket teeth move in

and out [9]

Let us consider our prototype Connor (Fig 6(a)) Its

mechanical structure is based on a sprocket and chain

differential-drive configuration Using chains, one

active-front shaft transfers rotating motion to two passive-rear

shafts (Fig 6(b)) This drive allows Connor to achieve a

maximum speed of 20 km/hr Its caterpillar type structure

makes Connor a type (2,0) WMR as none of its sprockets

are steerable

Fig 6(c) shows a schematic representation of Connor’s

sprocket and chain system As in a belt-drive, the chain can

be modeled as a spring with constant k The radii of the

sprockets are r1, r2, and r3 while their inertias are J1, J2,

and J3 Angular speeds are θ m, θ p, and θ q, respectively

Considering again a DC motor with constants kt and ke

as actuator, then the motor-sprocket-chain system can be

summarized by eqs (14) to (17):

J1θ¨m = k t i − b ˙ θ m − r1k(2r1θ m − r2θ p − r3θ q)(15)

J2θ¨p = −b ˙θ p + r2k(2r2θ p − r3θ q − r1θ m) (16)

J3θ¨q = r3k(2r3θ q − r1θ m − r2θ p) (17)

In principle, all contact drives experience frictional losses

The friction between all-metal drives without any lubricant

is high enough to damage the drive On the other hand,

belt-drives have little frictional losses and the sprocket and chain

drive has none In eqs (15) and (16), b represents the friction

between the sprocket-chain drive and the rolling surface

As in a belt-drive, simulation of eqs (14) to (17) would

illustrate the speed relationships between the three sprockets

of different radii as well as the importance of having the

chain under substantial tension Note that the third sprocket

(J3) is meant for this purpose

III OMNIMOBILEWMR Omnimobile WMR correspond to type (3,0) robots The main advantage of these WMR is that they exhibit holo-nomicity, i.e the ability to move in any direction without an orientation change The holonomicity and omnidirectional properties come from the use of Swedish wheels or active caster wheels Conventional wheels limit WMR motion as they exhibit 2 DOF and a no side-slip condition (see Fig 1(b)) However, special wheels such as the Swedish actually ensure 3 DOF

Let us consider our prototype NG (Fig 7(a)) Its mecha-nical design is based on an equilateral triangle-like structure Three omniwheels are disposed at each one of its vertices and are directly coupled to DC motors (Fig 7(b)) NG can be tele-operated via an RF point-to-point connection or autonomously guided by an array of ultrasonic sensors It was conceived to participate in the 2009 Robocup middle-size soccer league

Fig 7(c) shows a schematic representation of its omni-mobile structure Here, five coordinate systems are defined:

SI the inertial reference system with coordinates (X,Y), Sc the robot’s center of mass with coordinates (xc, yc), and Si

at each wheel with coordinates (xi, yi ), i=1,2,3 L is the

distance between the coordinate systems SI and Si, ϕ i is

the wheel-i angle in the S c system while θ is the robot’s

orientation angle in the SI system The robot’s position in space is then defined by posture vectorβ I=[X Y θ] T The relationship between the robot’s velocity on the SI and Sc coordinate systems is defined by eq (18):

˙

with:

R(θ) =

⎡

⎣cos(θ) −sin(θ) 0 sin(θ) cos(θ) 0

⎤

⎦

where R(θ) is an orthogonal rotation matrix from S cto SI Similarly, velocity components on Scare function of velocity components on the Si coordinate system This relationship

is described by eq (19):

528

(a) (b) (c) Fig 7 Type (3,0) WMR NG: (a) prototype, (b) equilateral triangle-like structure with Swedish wheels, and (c) schematic of the robot’s kinematics.

⎡

⎣˙x ˙y c c

˙θ

⎤

⎦ =

⎡

⎣cos(ϕ sin(ϕ i i ) −sin(ϕ) cos(ϕ i)i) −Lcos(ϕ Lsin(ϕ i)i)

⎤

⎦

⎡

⎣˙x ˙y i i

˙θ

⎤

⎦ (19)

Considering thatϕ1= 90◦,ϕ2= 180◦, andϕ3= 330◦in

NG and applying the coordinate transformation described in

eq (18), the robot’s inertial velocities can be expressed as a

function of the wheels angular velocitiesω i (eq (20)):

⎡

⎣

˙

X

˙Y

˙θ

⎤

⎦ = r

⎡

⎣−

2

3c 13c − √1

3s 13c + √1

3s

−2

3s 13s + √1

3c 13s − √1

3c

1

⎤

⎦

⎡

⎣ω ω12

ω3

⎤

where c=cos( θ), s=sin(θ), and r is the omniwheels’ radius.

Simulation of eq (20) allows motion visualization of

omni-WMR like NG As example, consider L=0.3 m, r=0.05 m.

Fig 8 shows a set of trajectories in the XY plane As it can

be inferred, resulting motion is a combination of the angular

velocities of the three wheels To move the omnidirectional

robot in a straight line, the angular speeds of two wheels

have to be set equal in magnitude but opposite in sign while

the angular speed of the third wheel has to be set as zero

To rotate the robot around any point, only one wheel has to

be actuated To rotate it around its center of mass, all three

angular speeds have to be equal in magnitude and sign

Note that omni-WMR offer a wide range of motion

possibilities which make them superior to differential-drive

WMR in terms of dexterity and driving capabilities [10]

IV CONCLUSION This paper aimed to present simple and reliable

mathe-matical models for different designs of type (3,0) and (2,0)

WMR It intends to be a practical reference for fast and easy

understanding of the main equations governing these WMR

In particular, this paper addressed differential-drive and

omnimobile WMR For the first, the kinematics and

dynam-ics of popular configurations such as the general

two-active-fixed wheels and one-passive-caster wheel, the belt-drive, and

sprocket-chain were obtained For the second, the kinematics

of WMR with Swedish wheels was presented All models

were illustrated using physical prototypes

Fig 8 Simulated trajectories for omnidirectional robot NG.

Computer simulation of these models allows motion vi-sualization and can be therefore exploited for the design of control algorithms, path planning, obstacle avoidance, etc

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