Mixed Energy Model for a Differential Guide Mobile Robot Evaluated

with Straight and Curvature Paths

Mauricio F. Jaramillo Morales an d Juan B. G´omez Mendoza

Department of Electrical, Electronic and Computer Engineering, National University, Manizales, Colombia

Keywords:

Energy Model, Dynamic Model, Gaussian Path Planning, Mobile Robot.

Abstract:

Energy consumption is an important issue for mobile robots that carry a limited energy sources, like batteries,

for a long period of time. An energy model can relate t he kinematic movements of the robot with energy

values, giving an estimation of the energy needed for the robot to fulﬁll a speciﬁc task. In this study an energy

model is proposed, based on the dynamic parameters of the mobile robot, as well as the motors, given an

energy value close to real energy consumption. Mixed energy model is tested with a well-known motor energy

model, using the velocities related to straight and curvature paths as i nput. In the r esults, a higher energy

consumption value is identiﬁed by the mi xed energy model, especially when the acceleration of the mobile

robot increases. Energy models are conﬁgured with P3-DX robot mobile parameters.

1 INTRODUCTION

For mobile robots, it is critical to know th e amount

of energy that it mu st carry in order to accomplish a

long-ter m task. Many researches in the literature un-

derline that major ene rgy consumption is generated

for the DC moto r, which governs wheel movements.

However, the inﬂuence that the d ynamic p a rameters

of the mobile robot can exert in total energy con-

sumption, has been ignored (P. Tokekar and Isler,

2014), especially for the differential guide mobile ro -

bot conﬁgur ation (Kim and Kim, 2008). In this paper,

a goo estimation of power and energy consumptio n,

using a mixed energy model that takes into ac c ount

the DC moto r and the mobile robot dynamic para-

meters, incorporating in a path planning, is presen-

ted. An energy model can be calculated using the me-

chanical and kinetic energy f ormulation, based on the

mass and velocity of the mobile robot (Liu and Sun,

2014; G. Kladis and Guerra, 2011), or friction esti-

mation (Dogru and Marques, 2016; Dogru and Mar-

ques, 2018). H owever, th e energy con sumption is not

related to the dynamic parameters of the robot as mo-

ments of inertia. Instead, Chuy and Bensekane’s in-

vestigation presents a power consumption modeling,

using a 2- dimensional, second order differential equa-

tion, that describes a four wheel steering robot for-

ces (O. Chuy Jr. and Ord onez, 2009; I. Bensekrane

and Merzouk, 2017). But, the dynamic p arameters of

the DC motor model are no taken into account in the

energy values, related for example, with motor ch a-

racteristics such as voltage and torque constants.

In Kim and Tokekar’s work, velocity proﬁles that

minimize mobile robot energy consumption for a gi-

ven p ath, is calculate d (Kim and Kim, 2008; P. Toke-

kar and Isler, 2014). The researchers use the energy

motor model as a cost function to op timize, but in the

energy saving values presented, the contribution of

mobile ro bot dynamic param eters, such as the weight

of the mobile robo t or load weight, is not reﬂected.

In this paper, the d ynamic mobile robot model b a-

sed on the Lagrange formalism, and the dynamic mo-

tor model based on electrica l and torque characteris-

tics, are calculated. Th e n, the models torque variables

are mixed. Finally a space state realization (Yun and

Yama moto, 1993) is proposed, in ord er to expand the

state variables and simplify the Lagrange multipliers.

This tran sformation permits d escription of the mixed

energy model with ordinary differential equations, so

that energy consumption values can be calculated. In

the re sults sectio n, the mixed energy model is com-

pared to the well-known ene rgy motor model. Both

models a re tested, using the typical trapezoidal velo-

city proﬁle for the straight path, an adap te d Gaussian

function for the curvature path, and a different load

weights.

The rest of this paper is organiz ed as follows: in

Section 2 the mobile robot dynamic mo del, motor dy-

namic model, and mixed energy model formulation

is pre sented. Section III presents the calculation of

Morales, M. and Mendoza, J.

Mixed Energy Model for a Differential Guide Mobile Robot Evaluated with Straight and Cur vature Paths.

DOI: 10.5220/0006912004730479

In Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2018) - Volume 2, pages 473-479

ISBN: 978-989-758-321-6

473

Figure 1: Schematic of the differential guide mobile robot.

the velocities, b a sed on the desired straight a nd curva-

ture paths. In Section IV, the simulations o f the motor

energy model and the mixed energy model are pre-

sented. Finally, conclusions are presented in Section

2 MIXED DYNAMIC AND

ENERGY MODELS

In this section , the mixed dynamic and energy models

are presented. The dynamic mobile robot mo del and

the dy namic DC motor model are calculated . Then,

both dynamic mo dels are mixed, matching the torque

values. Finally a space state realization (Yun and Ya -

mamoto, 1993) is applied to obtain the ene rgy model

formu lation.

2.1 Dynamic Mobile Robot Model

based on the Lagrange Formulation

The system of nonlinear differential equations that re-

presents the dynamic mobile robot model ( Yun and

Yama moto, 1993; Yamamoto and Yun, 1994), can be

determined by the Lagrange formulation to establish

equations of motion for the mobile robot:



∂T

∂ ˙q



−



∂T

∂q



= τ

− a

(1)

with i= 1, ...4

Where:

T is the equa tion for mobile kinematic ene rgy.

q is the vector of the state variables.

λ is the vector of the Lagrange’s multip liers.

τ is the vector of the torques ap plied to the wheels.

i j

are the matrix movement restriction com ponents.

In Figure 1, the differential wheeled mob ile robot

is shown. The mobile robot conﬁguration has two

movement restrictions:

• The mobile robot can not move in lateral di-

rection.

˙xsin φ− ˙ycosφ = 0 (2)

• The two d riving wheels of the mobile robot, roll

and do not slip.

˙xcosφ+ ˙ysinφ+ l

φ = r

(3)

˙xcosφ+ ˙ysinφ− l

φ = r

(4)

Where (x, y) are the coordin a te s of point Po in the

ﬁxed reference coordinated frame X −Y , φ is the he-

ading angle o f the mobile robot measured from the

X-axis, and θ

,θ

are the angular position s of the left

and right driving wheels.

The equa tions 3 and 4 are added and the equation 5 is

obtained.

˙xcosφ+ ˙ysinφ =

(

) (5)

The equations 2 and 5 can be written in the matr ix

form:











˙x

˙y







= 0 (6)

Where the components of the restriction mo-

vement matrix are:





−sinφ cosφ 0 0

−cos φ −sin φ

(7)

Matrix e quation 6, can be expressed as:

A(q) ˙q = 0 (8)

Where

q =

























(9)

Now, the total kinetic energy equation may be calcu -

lated with the equation:

T =

m( ˙x

+ ˙y

) + m

cd(

−

)( ˙y cosφ− ˙x sin φ)

(

) +

(

−

)

(10)

Equation 10 can b e rewritten as:

T = T

+ T

(11)

Where:

= is the kinetic energy of translation.

= is the kinetic energy of rotation.

= is the mobile inertia moment without wheels.

= is the Wh eel inertia moment.

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

474

The notation of the equation is:

b = is the displacement f rom each of the dri-

ving wheels to the axis of symmetry.

d = is the displacement from po int P

to the ma ss

center of the mo bile robo t, which is assumed to be on

the ax is of symmetry.

r = is the radius of the driving wheels.

c = is a constant equal to

= is the mass of the mobile robot witho ut the

driving wheels an d the r otors of the motors.

= is the mass o f each driving wheel plus the rotor

of its motor.

= is the moment of inertia of the mobile robot

without the driving wheels and the rotors of the

motors abou t a vertical axis thro ugh the intersection

of the axis of sy mmetry with the dr iving wheel axis.

= is the moment of inertia of each driving wheel

and the motor roto r about the wheel axis.

= is the moment of inertia of each driving wheel

and the motor roto r about a wheel diameter.

Then, the derivatives of th e Lagra nge movement

equation a re made for i, from one to four. Finally, the

nonlinear differential system equatio n, which r epre-

sents the dynamic mobile robo t model is:

sinφ+ λ

cosφ = m ¨x − m

φsin φ+

cosφ)

−λ

cosφ+ λ

sin φ = m ¨y + m

φcos φ −

sin φ)

− cbλ

= m

cd( ¨y cos φ − ¨x sin φ)

+ (I

+ I

)

− I

− cbλ

= − m

cd( ¨y cos φ− ¨x sin φ)

− I

+ (I

+ I

)

(12)

2.2 The Dynamic Motor Model and

Dynamic Model Mixture

The dynamic model o f a DC motor can be expressed

by the differential equations (Electro-Craft, 1977)







+ Ri + K

θ = V

i − I

θ− ν

θ = τ

(13)

where:

V and i = armature voltage and current.

R and L = armature resistance and inductance.

ν = is the viscous f riction coefﬁcient.

τ = is the dynamic lo a d applied to the motor.

= is the motor torque constant.

= is the motor voltage c onstant.

= is the motor shaf t inertia.

θ = [θ

] = are the angular positions of the wheels.

The ﬁrst expression in 13 is voltage equation for a

DC mo tor, and the second expression reﬂects torque

forces a pplied to the DC motor. In several studies as

in kim’s r esearch (K im and Kim, 2008), the to rque va-

riable is neglected, which seem s to be problematic for

a real objec t. In the proposed model, the torque va-

lue of a DC dynamic m otor model is ca lc ulated using

mobile ro bot dynamic mode l.

A reduced order model ca n b e achieved for dyn-

amic behavior, as the electric time constant L/R may

be neglected, if compared to the me chanical time con-

stant i/ν. He nce, one may consider L = 0, and the ﬁrst

equation yields.

i =

V − K

(14)

The e quation (14) can be replaced in the second

DC motor model equation, and by isolating τ

τ =

V − I

θ−



+ ν



θ (15)

In order to achieve the mixtur e of dynamic mo-

dels, the equation for the DC m otor torque τ (15), may

be rep la ced in the dynamic robot mobile model sy-

stem equation (12). Also the equation

φ = c(

−

calculated from the subtraction of the equations 3 and

4, may be used for ordering and writing the r esulting

mixed dynamic model in the following matrix equa-

tion:

M(q) ¨q + F ˙q + C(q, ˙q) = TV − A(q)λ (16)

Where:

M(q) =







m 0 −α

o m α

−α

+ I

−I

−α

−I

+ I







= m

cd sinφ

= m

cd cosφ

F =







0 0 0 0

0 0

+ ν

0 0 0

+ ν







C(q, ˙q) =







−m

cosφ

−m

sinφ







T =







0 0







A(q) =







−sinφ −cosφ

cosφ −sinφ

0 cb







(17)

Mixed Energy Model for a Differential Guide Mobile Robot Evaluated with Straight and Curvature Paths

475

2.3 The Mixed Energy Model

In this section, a space state realization is proposed

(Yun and Yamamoto, 1993), in order to transform the

nonlinear differential equation system that represents

the mixed dynamic model, into an ordinary differen-

tial equation system, and can be tested numerically.

In the process, the state space variable is increased,

and the Lagrange multipliers are simpliﬁed, using the

null space S(q) of the restric tion matrix A(q). If η

is the vector of the new variables, it can be said th a t

A(q)S(q )η = 0, and using the e quation A(q) ˙q = 0 (8),

it may also be said that ˙q = S(q)η.

The vector η was chosen as:

η =

θ where,

θ = [

]. (18)

Being S(q)



S(q)





(q) s

(q)









cb cos(φ) cb cos(φ)

cb sin(φ) cb sin(φ)

1 0

0 1







Now, multiplyin g b oth sides of equation (16) by

(q) and using the result S

(q)A(q) = 0, it can be

said that:

(q)M(q) ¨q + S

(q)F ˙q + S

(q)C(q, ˙q) = S

(q)TV

− S

(q)A(q)λ

(19)

Derived from equation ˙q = S(q)η again, te rm ¨q is

obtained.

¨q = S(q)

η+

S(q)η (20)

Replacing ˙q and ¨q.

M(S

η+

Sη) + S

F(Sη) + S

C = S

TV (21)

Isolating

η from (21), the following is obtained.

η = S

TV − S

C − S

FSη − S

Sη

η = (S

MS)

−1

TV − S

C − S

FSη− S

Sη)

(22)

Therefore the dynamic mode l can be re presented

with these new states variables.

x =





























(23)

The mo tion equation (22) and the equa tion ˙q =

S(q)η may be rep resented in the state space form

˙x = f (x) + g(x)V (24)

Where:

f (x) =



S(q)η

−(S

MS)

−1

C + S

FSη+ S

Sη)



g(x) =



MS)

−1



(25)

The voltage variable can b e obtained, isolating V

from equation (21).

V = (S

T )

−1

η+ S

Sη+ S

FSη + S

(26)

Finally, the mixed ene rgy model can be calcula-

ted with the power integral, using the current equ ation

(14) and the voltage e quation ( 26).

E(t) =

V (t) i(t) dt (27)

3 VELOCITY INPUTS FOR THE

ENERGY MODELS

In this section, the calculation of the angular veloci-

ties of the mobile robot wheels, re la ted with straight

and curvature paths, are shown. For the straig ht path,

a typical trap e zoidal velocity proﬁle is used, so the ro-

bot m obile moves a speciﬁc linear distance, taking it

into account that the maximum linear velocity of the

robot P3- D X is 1.21 m/s, and th e m aximum velocities

of the wheels (ﬁgure 3 (a)).

For the curvature path, a path planning from pre-

vious work, based on the G aussian function , is propo-

sed. In this case a cube is selected as an obstacle in

the environment where the mobile robot has to travel.

The base of the cube can be easily delimited by a ci-

rcumference. When the mobile robo t m oves from an

initial point to a goal point an d ha s to avoid the obsta-

cle, a Gaussian func tion is adapted to pass through the

circumference, as shown in the Fig ure 2. The equa-

tion th at represents the Gaussian function is:

= a exp

−(x

− f )

(28)

Where x

and y

represents the desired pa th, f is the

position of the center of the peak, and a is the h eight

of the curve’s peak. In order to accomplish the adapta-

tion o f the Gaussian functio n with the circumference

that delimited the obstacle, f is the same center of the

circumference, and a is the same r adius of the circum-

ference. The variable h is the standard deviation that

controls the wid th of the bell. The correct estimation

of this variable prevent the desired path from passing

through the circumference’s area.

Once the curvature path is obtained, the angula r

velocities of the wheels can be calculated using the

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

476

0.5

1.5

2.5

3.5

0.5

1.5

Obstacle

Circumference

Adapted Gaussian path

Initial and final position

Figure 2: Adapted Gaussian path.

Table 1: Mobile parameters of the WMR, P3-DX.

Parameter Value Parameter Value

r [m] 0.095 m

[kg] 6.04

b [m] 0.165 m

[kg] 1.48

d [m] 0

Table 2: Motor parameters of the WMR, P3-DX.

Parameter Value Parameter Value

R [ω] 0.7 K

[(rad/s)/V] 0.88

ν [Nm/(rad/s)] 0. 035 K

[Nm/A] 0.88

[kgm

] 0.0713

kinematic inverse of the mobile robot, which is repre-

sented by the following matrix:











cosφ

sinφ

cosφ

sinφ

−











˙x

˙y





(29)

4 SIMULATION AND FINAL

RESULTS

The en ergy consumption of the mixed energy mod el

represented by Equation (27) and the motor energy

model (Kim and Kim, 2008; P. Tokekar and Isler,

2014) is compared, when the mobile robot travels on

straight an d curvature paths, a path planning usually

has these two kinds of paths. In order to set up both

energy models, the mob ile robot and motor parame-

ters of the P3-DX mo bile platform present in Kim’s

work, were used. These par a meters are shown in ta-

bles 1, 2.

In the mixed energy model, when traveling a

straight path, the angular velocity in both wheels has

to be the same, in this case

= v/r, with v as the

linear velocity of the mobile robot.

Figure 3 shows the energy consumption of both

energy models , in the critical case when the mo-

Table 3: Simulation results of the energy models with a dif-

ferent kind of maximum linear velocities of the mobile ro-

bot.

Trapezoidal velocity Energy value

Maximum velocity Motor Mixe d

0.66 m/s 53.65 J 54.51 J

0.88 m/s 71.89 J 73.68 J

1.21 m/s 98.85 J 102.9 J

bile robot travels on a straight path of 10 meters, at

the maximum trapezoidal velocity proﬁle (1.21 m/s),

with the ma ximum load weight (7.6 kg) allowed by

the P3-DX mobile rob ot, and a ﬁxed mobile weight

of 6.04 kg.

Figure 3 also shows that the energy consump tion

values given by the mixed energy model increase fas-

ter during the acceleration of the mobile robot linear

velocity, but are the same when the velocity remains

at the maximum value. During deceleration, during

the time c orresponding to the negative phase of power

consumption, a certain amount of energy is regenera-

ted and stored in the batteries. The energy regenera-

ted is equal to 1.05 joules for the motor ene rgy model,

and 9.07 joules for the mixed energy model.

The rec overed energy allows for reduction of the

total energy consumption d ifference between energy

models. Total energy consumption in the motor

energy model is 98.84 joules, and in the mixed energy

model 104.3 jo ules. It is important to un derline that

the highest difference of energy consumption ha ppens

during acceleration. In this phase, e nergy consump-

tion in the motor energy model is equal to 22.03 jou-

les, a nd in the mixed energy model, 36.74 joules.

Table 3 shows the energy consumption values gi-

ven by the energy models when the mo bile robot tra-

vels on a straight path of 10 meters, with a mobile

robot weight of 6.04 kg, with a ﬁxed load weight of

3.76 kg and with different maximum set of velocities

of the trapezoidal veloc ity proﬁle of th e mobile robot.

As the energy models dep ends of the kinematic

mobile robot model, energy consumption raises in

both models, when the maximum veloc ity proﬁle in-

creases as well. However, the ene rgy values are hig-

her in the mixed energy m odel because only this mo-

del depends on the dynamic parameters of the robot

as moments of inertia and weight.

It is for this reason that in Table 4, only the mixed

energy model is considerably affected in its energy

consumption value, when the trapezoidal velocity is

ﬁxed at a velocity of 0.88 m/s, and th e lo ad weight

increases.

For the study of energy consumption when a m o-

bile robot travels on a curvature path, a n initial and

Mixed Energy Model for a Differential Guide Mobile Robot Evaluated with Straight and Curvature Paths

477

−2 0 2 4 6 8 10 12 14

Time in secs

Velocity in rad/s

Trapezoidal velocity of the wheels

0 2 4 6 8 10 12

−10

−5

Power consumption

Time in secs

Power in watts

Mixed model

Motor model

0 2 4 6 8 10 12

100

120

Energy consumption

Energy in joules

Time in secs

Mixed energy model

Motor energy model

Figure 3: Simulation with a maximum trapezoidal angular velocity of the wheels (12.73 rad/s), maximum load weight carried

by the mobile robot (7.6 kg), and a mobile weight of 6,04 kg, for a linear distance of 10 meters. (a) Trapezoidal angular

velocity of the wheels (b) Power consumption given by the energy models. (c) Energy consumption given by the energy

models.

0 2 4 6 8 10 12

acceler

Time in secs

ation in m

0 2 4 6 8 10 12

Power consumption

Time in secs

Power in watts

Mixed model

Motor model

0 2 4 6 8 10 12

Energy consumption

Time in secs

Energy in joules

Mixed energy model

Motor energy model

Figure 4: Simulation wi th a G aussian f unction adapted to a circumference that delimited the obstacle, maximum load weight

carried by the mobile robot (7.6 kg), and a mobile wei ght of 6,04 kg. (a) Linear acceleration of the wheels for the circumfe-

rence radius of 0.5 m, 0.75 m, and 1 m. (b) Power consumption given by the energy models, for a radius circumference of 0.5

m. (c) Energy consumption given by the energy models, for a radius circumference of 0.5 m.

Table 4: Simulation results of the energy models with a dif-

ferent load weights.

Total weight energy consumption

Load and robot Motor Mixed

6.04 k g 71.9 J 73.15 J

9.8 kg 71.89 J 73.68 J

13.64 kg 71.88 J 74.25 J

Table 5: Energy consumption of the models when the radius

circumference of t he adapted Gaussian function changing.

Parameters Energy consumption

Radius Distance Motor Mixed

0.5 m 7.06 m 51.26 J 56.96 J

0.75 m 7,17 m 50.13 J 52.04 J

1 m 7.28 m 50.74 J 51.66 J

goal position for the desired path is (x

= 6, y

= 4),

the position of the circ umference center is (x

= 3, y

= 2), the edge length of the o bstacle cube is 0.7 me-

ters. The only variable that changes for this study’s

proposes is the circumference radius tha t delimited

the obstacle. In the Figure 4(a) is shows that the li-

near acc eleration of the mobile robot is inversely re-

lated to the circumference radius, because a smaller

radius represents a narrow bell of the adapted Gaus-

sian function, forcing the mobile robot to accelerate

to reach the pe ak. Is for that reason that in th e ta-

ble 5 with a sma ller circumference radius, power and

energy consu mption is h igher, despite the facts that, in

the other cases, the travel distance is greater. Finally,

in the Figure 4 the behavior of the energy models

when travels on a curva ture path re mains the same

as on the straight path, the mixed energy mo del incre-

ases its energy values compared to the mo tor energy

model, when th e mobile robo t accelerate, because the

dynamic parameters of the robot, are taken into con-

sideration in the energy model proposed.

5 CONCLUSIONS

In this paper, an energy model that takes robot and

motor dynamic parameters into account, is proposed.

The en ergy consumption of the mixed energy mod el

was compared to a typical motor energy model, using

the angular velocities of the wheels as they relate re-

lated to the travel on straight and curvature paths. The

simulation results show that the highest p e rcentage of

energy consumption came from the motors. However,

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

478

during mobile robot acce le ration, the dyn a mic para -

meters of the robot, such as inertia moments, r obot

weight, or load weight, also inﬂuenced total energy

consumption. It is f or that r eason th a t the energy va-

lues in ﬁgure 3 (b), (c), 4 ( b), (c), and in tables 3 ,

4, 5 are higher in the mixed energy model than those

of the DC motor energy model, because the proposed

model con sid er the DC motor and the mobile robot

dynamic models. However, in the de c eleration phase,

it was proven that a certain a mount of energy was re-

generated and stored in the batteries, allowing for the

reduction of the total difference in ene rgy consump-

tion betwe e n energy models. A good estimation of

power and ene rgy c onsumption as pre sented, can be

more easily related to the real-time a utonomy of dif-

ferential guide mobile r obots, which c arry their own

energy source.

For the fu rther study, the problem of testing the

mixed energy model, on straight and curvature paths

that minimize the energy consumption, and a experi-

ment validation using the Nomad Super Scout mobile

robot, remains.

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