KINEMATICS AND DYNAMICS ANALYSIS FOR A HOLONOMIC

WHEELED MOBILE ROBOT

Ahmed El-Shenawy, Achim Wagner and Essam Badreddin

Computer Engineering, University of Mannheim , B6. 23-27 part B, Mannheim, Germany

Keywords:

Modeling, Robot Kinematics, Robot Dynamics, Analysis.

Abstract:

This paper presents the kinematics and the dynamics analysis of the holonomic mobile robot C3P. The robot

has three caster wheels with modular wheel angular velocities actuation. The forward dynamics model which

is used during the simulation process is discussed along with the robot inverse dynamics solution. The in-

verse dynamics solution is used to overcome the singularity problem, which is observed from the kinematic

modeling. Since both models are different in principle they are analyzed using simulation examples to show

the effect of the actuated and non actuated wheel velocities on the robot response. An experiment is used to

illustrate the performance of the inverse dynamic solution practically.

1 INTRODUCTION

Wheeled mobile robots became an important tool in

our daily life. They are found in a multitude of ap-

plication such as guiding disabled people in muse-

ums (Burgard et al., 2002; Steinbauer and Wotawa,

2004) and hospitals (Kartoun et al., 2006), transport-

ing goods in warehouses, manipulation of army ex-

plosives (Bruemmer et al., 1998), or securing impor-

tant facilities (Chakravarty et al., 2004).

Wheeled mobile robots are categorized into two

main types: holonomic and non-holonomic, which

are the mobility constraints of the mobile robot plat-

form (Yun and Sarkar, 1998). A holonomic conﬁg-

uration implies that the number of robot velocities

DOF (Degrees Of Freedom) is equal to the number

of position coordinates. The main advantage of holo-

nomic mobility is the ability of efﬁcient maneuver-

ing in narrow places. In the last two decades, many

considerable research efforts addressing the mobility

of holonomic wheeled mobile robots were done (Ful-

mer, 2003; Moore and Flann, 2000) and (Yamashita

et al., 2001).

Usually, kinematic modeling is used in the ﬁeld

of WMRs (Wheeled Mobile Robot) to obtain stable

motion control laws for trajectory following or goal

reaching (Khatib et al., 1997; Ram

´

ırez and Zeghloul,

2000). Using the dynamics modeling during the sim-

ulation process results in a better control design. By

comparing its results to the practical implementation,

a better precision and more variables assumption are

achieved in comparison to the kinematics model.

The dynamic modeling is much more complex

than the kinematic modeling. Furthermore, the kine-

matic modeling is required for deriving the dynamic

model. Hence, it is assumed that the velocity and ac-

celeration solutions can be solved without any difﬁ-

culty. Generally, the main property of the dynamic

model is that it involves the forces that act on the

multibody system and its inertial parameters such as

: mass, inertia, with respect to the center of gravity

(Albagul and Wahyudi, 2004; Asensio and Montano,

2002).

The holonomic mobile robot ”C3P” (Caster 3

wheeled Platform) is described geometrically in sec-

tion 2. The singularity problem found in the C3P con-

ﬁguration is illustrated in section 3 through the kine-

matic analysis. The forward dynamic model and the

inverse dynamic solution are presented and analyzed

in section 4 to show their different structure. Few sim-

ulation examples are shown to illustrate the effect of

the actuated and non-actuated variables on the robot

behavior using the inverse dynamic solution. In sec-

tion 5, a lab experiment is presented to show the per-

485

El-Shenawy A., Wagner A. and Badreddin E. (2007).

KINEMATICS AND DYNAMICS ANALYSIS FOR A HOLONOMIC WHEELED MOBILE ROBOT.

In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 485-491

DOI: 10.5220/0001650404850491

Copyright

c

SciTePress

formance of the solution on the practical implemented

module.

2 THE PLATFORM

CONFIGURATION

The C3P WMR is a holonomic mobile robot, which is

previously discussed in (Peng and Badreddin, 2000).

The C3P has three caster wheels attached to a trian-

gular shaped platform with smooth rounded edges as

shown in Fig. (1). Each caster wheel is attached to

each hip of the platform. The platform origin co-

ordinates are located at its geometric center, and the

wheels are located with distance h from the origin and

α

1

= 30

o

, α

2

= 150

o

, and α

3

= 270

o

shifting angles,

Figure 1: C3P platform conﬁguration.

where

X,Y,φ : WMR translation and rotation displace-

ment.

θ

s

i

: the steering angle for wheel i.

r,d : the wheel radius and the caster wheel offset.

The conventional Caster wheel has three DOF due

to the wheel angular velocity

˙

θ

x

i

, the contact point an-

gular velocity

˙

θ

c

i

and the steering angular velocity

˙

θ

s

i

.

The unique thing about the C3P is that it has wheels

angular velocities actuation only (

˙

θ

x

i

) with no steer-

ing angular actuation, which makes the model modu-

lar and more challenging.

3 KINEMATICS MODELING AND

ANALYSIS

For WMRs’ mobility analysis and control, the kine-

matics modeling is needed. Furthermore, calculat-

ing the velocity and acceleration variables are impor-

tant in the dynamics modeling procedures. In order

to analyze and derive the C3P mathematical models,

some variables are assigned, such as the following:

the robot position vector p = [X Y φ ]

T

, the wheel an-

gles vector q

x

= [θ

x

1

θ

x

2

θ

x

3

]

T

, the steering angles

vector q

s

= [θ

s

1

θ

s

2

θ

s

3

]

T

, and the contact angles vec-

tor q

c

= [θ

c

1

θ

c

2

θ

c

3

]

T

. By differentiating the robot

and wheel vectors with respect to time, the robot and

wheel velocities are

˙p =

dp

dt

, ˙q

x

=

dq

x

dt

, ˙q

s

=

dq

s

dt

, ˙q

c

=

dq

c

dt

. (1)

From the generalized inverse kinematic solution

described in (Muir, 1987), the wheel angular velocity

inverse kinematic solution is

˙q

x

= J

in

x

˙p

J

in

x

=

1

r

−S(θ

s

1

) C(θ

s

1

) h C(α

1

− θ

s

1

)

−S(θ

s

2

) C(θ

s

2

) h C(α

2

− θ

s

2

)

−S(θ

s

3

) C(θ

s

3

) h C(α

3

− θ

s

3

)

(2)

while the steering angular actuation is

˙q

s

= J

in

s

˙p

J

in

s

=

−1

d

C(θ

s

1

) S(θ

s

1

) −h S(α

1

− θ

s

1

) + d

C(θ

s

2

) S(θ

s

2

) −h S(α

2

− θ

s

2

) + d

C(θ

s

3

) S(θ

s

3

) −h S(α

3

− θ

s

3

) + d

(3)

and the contact angular velocity inverse solution is

˙q

c

= J

in

c

˙p

J

in

c

=

−1

d

−S(θ

s

1

) C(θ

s

1

) −h C(α

1

− θ

s

1

)

−S(θ

s

2

) C(θ

s

2

) −h C(α

2

− θ

s

2

)

−S(θ

s

3

) C(θ

s

3

) −h C(α

3

− θ

s

3

)

(4)

where ”C” stands for ”cos” and ”S” stands for ”sin”.

The solution (2) shows singularities for some steer-

ing angles conﬁgurations. The singularity appears

only when the steering angles are equal. For example,

when the steering angles are −90

o

, the robot velocity

˙

Y is not actuated (Fig. (2-a)), and when they are 0

o

the

velocity

˙

X is not actuated (Fig. (2-c)). The steering

conﬁguration in Fig. (2-b) gives singular determent

for the matrix J

in

x

with −45

o

steering angles although

all the robot DOFs are actuated.

Obviously, the direction of [−1 1 0]

T

is not ac-

tuated, which concludes the following; if all steering

angles yield the same value, then the robot is not ac-

tuated in the direction parallel to the wheel axes. Fig.

(2-d) represents a non-singular steering wheels con-

ﬁguration condition.

Solutions (3) and (4) can be used to overcome the

singularity practically by adding practical actuation

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

486

Figure 2: Different steering angles conﬁgurations.

to the steering angular velocities, or theoretically by

virtually actuating the steering angular velocities (El-

Shenawy et al., 2006c).

The forward sensed kinematics used in (El-

Shenawy et al., 2006a) shows that the sensed vari-

ables are sufﬁcient for robust sensing and slippage de-

tection through the following equation

˙p = J

f

x

˙q

x

+ J

f

s

˙q

s

, (5)

where J

f

x

and J

f

s

are the sensed forward solutions for

the wheel angular and steering angular velocities re-

spectively. The derivative of equation (5), yields the

robot accelerations,

¨p = J

f

x

¨q

x

+ J

f

s

¨q

s

+ g(q

s

, ˙q

x

, ˙q

s

), (6)

from equations (2), (3) and the inversion method pro-

posed in (Muir, 1987) the inverse actuated kinematic

accelerations is

¨q

x

¨q

s

=

J

in

x

J

in

s

¨p− g

cs

(q

s

, ˙p) (7)

4 DYNAMIC MODELING AND

ANALYSIS

4.1 Euler Lagrange

The dynamic equations of motion are derived using

the Euler-Lagrangian method (Naudet and Lefeber,

2005) based on the Lagrangian function

L = K − P, (8)

where K denotes the kinetic energy and P denotes the

potential energy. Since the C3P is assumed to drive

on a planar surface, P is zero.

The Lagrangian dynamic formulation is described

as

τ =

d

dt

∂L

∂ ˙q

−

∂L

∂q

, (9)

where τ is the vector of actuated torques.

The C3P is considered as a closed chain multi-

body system. To derive the dynamic model of the

C3P, the system is converted into an open chain struc-

ture. First, the dynamic model of each serial chain is

evaluated using the Lagrangian dynamic formulation

(9). Second, the platform constraints incorporate the

open chain dynamics into a closed chain dynamics.

The robot consists of 7 parts; 3 identical wheels, 3

identical links and one platform (Fig.(3)).

Figure 3: The C3P parts structure.

The kinetic energy of the rigid body depends on

its mass, inertia , linear and angular velocities as de-

scribed by the following equation

K =

1

2

m V

T

V +

1

2

Ω

T

I Ω (10)

where,

m, I : mass and inertia of the rigid body.

V, Ω : linear and angular velocity at the center of

gravity of the rigid body.

The sum of the platform kinetic energies equa-

tions results in the following Lagrangian function

L =

3

∑

i=1

K

wi

+

3

∑

i=1

K

li

+ K

pl

, (11)

where K

wi

, K

li

are the wheel and link i kinetic energies

respectively, while K

pl

is the platform kinetic energy.

The wheel co-ordinates q can be considered as the ac-

tuated displacements and ˙q as the actuated velocities,

KINEMATICS AND DYNAMICS ANALYSIS FOR A HOLONOMIC WHEELED MOBILE ROBOT

487

while τ is the external torque/force vector. The overall

dynamics of the robot can be formulated as a system

of ordinary differential equations whose solutions are

required to satisfy the WMR constraints through the

following force/torques vector equation

τ = M(q) ¨q+ G(q, ˙q) (12)

where, M(q) is the inertia matrix, G(q, ˙q) contains the

centripetal and Coriolis terms.

4.2 Control Structure Design

The C3P control structure contains three main mod-

els (Fig.(4)): the forward kinematics for calculating

the C3P velocities, the C3P forward dynamic model

which models the C3P practical prototype on the sim-

ulation level, and the inverse dynamics solution.

The velocity controller (V.C) calculates a control

signal u from the velocity error ˙e = ˙p

r

− ˙p

o

, which

is added to the reference acceleration signal ¨p

r

. The

reference robot velocities ˙p

r

and accelerations ¨p

r

are

used in the inverse dynamic solution to deliver the ac-

tuated wheels torques τ

xa

.

Figure 4: Robot closed-loop structure.

The forward dynamics consists of two main equa-

tions; the Wheels Torques Dynamics (WTD) and the

Dynamic Steering Estimator (DSE), which were pro-

posed in (El-Shenawy et al., 2006b). The wheel angu-

lar velocities are calculated using the wheels torques

dynamic equation of motion

τ

x

= M(q

s

) ¨q

x

+ G

x

(q

s

, ˙q

x

), (13)

with respect to the actuated wheels torques, where

τ

x

= [τ

x

1

τ

x

2

τ

x

3

]

T

. The steering angles and the steer-

ing angular wheel velocities are recursively calculated

by the steering dynamic estimator

¨q

s

= M

−1

ss

M

xx

¨q

x

+ M

−1

ss

G

ssx

(q

s

, ˙q

s

, ˙q

x

) (14)

corresponding to the angular wheels velocities and ac-

celerations generated due to the applied wheel torque

resulting from equation (13).

The C3P dynamic model shown in Fig. (5) has

the actuated wheel torques τ

x

as an input, while the

outputs are the sensed wheel velocities ˙q

x

, the steer-

ing angular velocities ˙q

s

, and the steering angles q

s

.

Since the steering angular velocities are actuated by

the angular wheel velocities, the angular wheel ve-

locities ˙q

x

and accelerations ¨q

x

are the main inputs

of the steering dynamic estimator. The steering an-

gles q

s

and steering angular velocities ˙q

s

are delayed

by unity time interval because the steering dynamic

model is calculated recursively according to (14).

Figure 5: The C3P Dynamic Model.

The inverse dynamic solution proposed is imple-

mented in the velocity control loop to overcome the

singularity with simpler velocity controller and better

performance. The inverse dynamic equation depends

on the platform constraints, which are described in the

forward kinematic solution. They are combined using

Lagrangian formulation and the dynamic torque equa-

tion (9) to obtain the described wheel torques equa-

tion

τ

x

a

=

M

x

a

M

s

a

¨q

x

¨q

s

+ G

sx

a

( ˙q

x

, ˙q

s

,q

s

) (15)

The matrix M

x

a

is the inverse dynamic solution for

actuating the wheels torques τ

x

a

, while the matrix M

s

a

is the inverse dynamic solution for actuating the steer-

ing angular acceleration ¨q

s

using the wheel torques

τ

x

a

. The inverse dynamics solution is a relation be-

tween the desired robot velocities and accelerations

(˙q, ¨q) as an input and the actuated applied torques

of the wheels (τ

x

a

) as an output. However, the dy-

namic torque equation (15) is a function of ¨q

x

, ¨q

s

, ˙q

x

,

and ˙q

s

. By using the velocity and acceleration inverse

kinematic solutions(2, 3, 4 and 7), the desired torque

equation is achieved and the actuation characteristics

of the steering angular velocities and accelerations are

included in the inverse dynamic solution as well. As

a result the actuated torques equation will have the

robot velocities ˙p and accelerations ¨p as input vari-

ables

τ

x

a

= M

x

(q

s

) ¨p+ G

xi

(q

s

, ˙p) (16)

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

488

M

x

=

M

x

a

M

s

a

J

in

x

J

in

s

, (17)

G

xi

= G

sx

a

(q

s

, ˙p) −

M

x

a

M

s

a

g

cs

(q

s

, ˙p) (18)

4.3 Model Analysis

The proposed inverse and forward dynamic solutions

are different in structure and mathematical represen-

tation as well. However, both models should yield

the inversion of each other. Therefore some simula-

tion examples are done and analyzed in this section to

illustrate the performance of the model. The simula-

tions are done using the structure shown in Fig. (4)

with zero values for the velocity (V.C.) and the axes

level control parameters to disable their effects. The

C3P parameters are set to be exactly like the practical

prototype, which are described in Table 1.

Table 1: The C3P parameters.

C3P Parameters Value Units

h 0.343 m

d 0.04 m

r 0.04 m

M

p

(Platform mass) 25 Kg

I

p

(Platform inertia) 3.51 Kg m

2

Fig.(6) shows a comparison between two different

examples. The ﬁrst example is a non singular condi-

tion, where the steering angles are θ

s

1

= θ

s

2

= θ

s

3

=

0

o

as shown in Fig. (2c). The input signal is a ramp

input in Y direction while the X translational and Φ

velocities are zeros. The input V

y

or

˙

Y is a ramp sig-

nal till 3.2 seconds then it is constant. The second

example is a non singular condition but with differ-

ent steering angles, where θ

s

1

= 45

o

,θ

s

2

= −45

o

and

θ

s

3

= 90

o

with the same ramp input signal. Fig. (6-

a) shows the trajectory of the steering angles, which

are indicated as (θ

s−oi

) for example one and (θ

s−i

) for

example two.

The reference input velocities maintain zero steer-

ing angles value. For the ﬁrst example, the steering

angles keep their initial value, while in the second ex-

ample the steering angles were adjusted from their

initial value to the zero value. The steering wheel

adjustment took place due to the the step accelera-

tion input in Y direction. The robot output velocity

and acceleration Out − 1 result from the ﬁrst example

(Fig. (6-b) &(6-c)), which follow the input signal as

well as the steering and wheel angular accelerations

(α

s−oi

,α

x−oi

) as shown in Fig. (6-d) &(6-e).

0 1 2 3 4 5 6

−45

0

45

90

Time (s)

θ

s

(

o

)

a) The steering angles

θ

s−oi

θ

s−1

θ

s−2

θ

s−3

0 2 4 6

0

0.02

0.04

0.06

0.08

0.1

Time (s)

V

y

(m/s)

b) Velocity in Y direction

Reference

Out−1

Out−2

0 2 4 6

−2

0

2

4

6

8

10

12

x 10

−3

Time (s)

a

y

(m/s

2

)

c) Acceleration in Y direction

Reference

Out−1

Out−2

0 2 4 6

−0.1

−0.05

0

0.05

Time (s)

α

s

(r/min

2

)

d) The steering angular acceleration

α

s−oi

α

s−1

α

s−2

α

s−3

0 2 4 6

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time (s)

α

x

(r/min

2

)

e) The wheel angular acceleration

α

x−oi

α

x−1

α

x−2

α

x−3

Figure 6: Simulation comparison for ramp input.

For the second example, the input acceleration and

the initial steering angles produce disturbances and

oscillations in the wheel angular acceleration ((α

x

i

=

0 for i ∈ {1,2,3} (Fig. (6-e)))). Such disturbances

produce oscillations in the steering angular accelera-

tions ((α

s

i

= 0 for i ∈ {1,2,3} (Fig. (6-d)))), which

results negative overshoot in the robot Y acceleration

(Out − 2) (Fig. (6-c))). In addition to the presence of

the dynamic delay in the forward dynamics solution

(Fig. (5)) and some simulation numerical errors the

overshoot appears.

When the robot velocity takes constant value the

desired wheel acceleration suddenly change from

value 0.17 (r/min

2

) to 0 (r/min

2

). Such input does

not cause multiple oscillations or high overshoots in

output signal (Fig. (6-d)).

The next simulation shows the responses of the

robot velocities and accelerations after enabling the

axes and robot level controllers. The robot starts from

the same initial steering angles but the input velocity

is ramp signal in X direction and Zero value in the Y

and the rotational velocities (Fig.(7-d)). Such input

yields the steering angles to reach −90

o

(Fig.(7-a)),

which is adjusted due to the wheel angular accelera-

KINEMATICS AND DYNAMICS ANALYSIS FOR A HOLONOMIC WHEELED MOBILE ROBOT

489

tions oscillations resulted in Fig. (7-b).

0 5 10 15

−90

−45

0

Time (s)

θ

s

(

o

)

a) The steering angles

θ

s

1

θ

s

2

θ

s

3

0 5 10 15

−0.2

−0.1

0

0.1

0.2

0.3

Time (s)

α

x

( r/min

2

)

b) The wheels angular accel.

α

x

1

α

x

2

α

x

3

0 5 10 15

−2

0

2

4

6

8

10

x 10

−3

Time (s)

a

x

(m/s

2

)

c) Accel. in X direction

Reference

Measured

0 5 10 15

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Time (s)

V

x

(m/s)

d) Velocity in X direction

Reference

Measured

Figure 7: simulation responses for ramp input with singu-

larity conﬁguration.

In the ﬁrst seven seconds during the steering an-

gles adjustment, the output acceleration of the robot

X direction oscillates but it reach the desired value

as the steering angles settle with their desired values

(Fig.(7-c)). These oscillations affect the robot veloc-

ity output as well by oscillating around the desired

signal (Fig.(7-d)). In case of having Y as adesired di-

rection, the output signal will be exactly the same like

the input, because the effect of the delay unit in the

forward dynamics solution is negligible.

5 LAB EXPERIMENT

After analyzing the inverse and forward dynamic so-

lutions, the inverse dynamic solution is implemented

on the C3P prototype (Fig. (8)) to test its perfor-

mance practically. Therefore, an experiment was im-

plemented in the lab with the following initial steering

angles θ

s

1

= θ

s

2

= θ

s

3

= 135

o

and input robot veloc-

ities ˙p

r

= [0.5(m/s) 0.5(m/s) 0(r/min)]

T

. Such in-

put velocities yields the steering angles to ﬂip 180

o

to

reach −45

o

or 315

o

values (Fig.(9-a)).

The steering angles θ

s

2

and θ

s

3

ﬂipped after 2 sec-

onds in different directions (Fig.9c), causing oscilla-

tions in the robot velocities (Fig.(9-d), (9-e) & (9-f)).

After the fourth second, the ﬁrst steering angle θ

s

1

ﬂipped producing another overshoots in the robot ve-

locities. The robot velocities (Fig.(9-d), (9-e) & (9-f))

Figure 8: The C3P practical prototype.

0 2 4 6 8

−45

45

135

225

315

Time (s)

θ (

o

)

a) The steering angles

θ

s

1

θ

s

2

θ

s

3

0 2 4 6 8

−100

−50

0

50

100

150

Time (s)

ω

x

(r/min)

b) The wheels angular velocities

ω

x

1

ω

x

2

ω

x

3

0 2 4 6 8

−200

−100

0

100

200

Time (s)

ω

s

(r/min)

c) The steering angular velocities

ω

s

1

ω

s

2

ω

s

3

0 2 4 6 8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (s)

V

x

(m/s)

d) Linear C3P velocity in X direction

Reference

Measured

0 2 4 6 8

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Time (s)

V

y

(m/s)

e) Linear C3P velocity in Y direction

Reference

Measured

0 2 4 6 8

−3

−2

−1

0

1

2

Time (s)

Φ

dot

(r/min)

f) Angular C3P velocity around Z axis

Reference

Measured

Figure 9: Practical Lab results for the C3P prototype.

are measured with respect to the ﬂoor frame of co-

ordinates, which illustrates how the responses follow

the reference value in the steady state.

The main advantage and main problem of the C3P

conﬁguration are the same. This is the non direct ac-

tuation of the steering angular velocities of the wheel.

Such a problem is very challenging from the theoret-

ical and practical point of view. Since the steering

angular velocities are virtually actuated, their behav-

ior can not be exactly predicted. However, the for-

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

490

ward dynamic model is the main factor in deriving

the inverse dynamic solution and designing its con-

trol structure along with tuning its parameters. These

information are very useful during the practical im-

plementation takes place, as in the last experiment.

6 CONCLUSION

The kinematics and the dynamics models of the holo-

nomic mobile robot C3P are presented in this paper.

The singularity problem found in such actuation con-

ﬁguration is described by the inverse kinematic so-

lution. The dynamic analysis showed the effect of

the wheel and steering angular accelerations of the

caster wheels on the inverse dynamic behavior with

and without velocity controllers. Although the inverse

and forward dynamic models are different in struc-

ture, they yield the inversion of each other. The steer-

ing angular acceleration plays a very important rule in

calculating the reference actuated signal to the plat-

form to overcome the robot singularities and to adjust

its steering angles. The simulation examples showed

the model dynamic analysis through the robot accel-

eration variables, and the practical experiment proved

the effectiveness of the control structure.

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