New Approach to the Artiﬁcial Force Concept for Skid-steering Mobile

Platform

∗

Alicja Mazur, Wojciech Domski, Mirela Kaczmarek and Mateusz Cholewi´nski

Chair of Cybernetics and Robotics, Electronics Faculty, Wrocław University of Technology,

ul. Janiszewskiego 11/17, 50-372 Wrocław, Poland

Keywords:

Skid-steering Mobile Platform, Nonholonomic Constraints, Artiﬁcial Force.

Abstract:

In the paper control algorithm for skid-steering mobile platform is presented. For mathematical model of such

an object, expressed in auxiliary coordinates, control law based on the idea of artiﬁcial force is introduced.

A skid-steering mobile platform is an underactuated control system with a rectangular input matrix. In the

approach explored in the paper it was assumed that there exists an additional control input, giving an additional

column in input matrix and causing this matrix invertible. Because such an actuator does not exist in reality,

this input was kept equal to zero equivalently. Simulations have proved proper work of this method.

1 INTRODUCTION

Wheeled mobile platforms can be treated as indepen-

dent robots or as a transportation part of complex

robotic systems. Depending on wheels’ type and a

way in which they are ﬁxed to the chassis, motion

of wheeled mobile platforms can be realized with or

without slipping effect. If slippage effect between

wheels and surface does not occur, then exists an

equation describing forbidden directions for realized

velocities of the system. Such an equation is called

nonholonomic constraint in platform’s motion.

A special kind of wheeled mobile platforms are

platforms with tracks. They can be modeled by a

chassis with more than one axis equipped with ﬁxed

wheels. These platforms are called skid-steering mo-

bile platforms (SSMP), due to skidding effect obse-

rved in their behavior.

Designing of a control algorithm for the skid-

steering platform is based on a proper mathematical

model of a considered object. A problem occurs in

this point, namely there is not an adequate model of

objects moving with sleeping effects. There were

some attempts to get strict model of wheeled mo-

bile platforms using slow manifold idea (Campion

and Motte, 2000) or singular perturbation approach

(D’Andrea-Novel et al., 2007). Unfortunately, math-

∗

The works of Alicja Mazur, Wojciech Domski and

Mirela Kaczmarek were supported by the Wrocław Univer-

sity of Technology under the statutory grant S40173 and the

work of Mateusz Cholewi´nski under the grant B40235.

ematical models presented in literature are inadequate

because they don’t cover both cases, i.e. motion with

or without slippage effect. For this reason authors de-

scribing slippage phenomena in skid-steering mobile

platforms use only approximate models with artiﬁ-

cial constraints see e.g. (Caracciolo et al., 1999) or

with artiﬁcial forces (Mazur et al., 2013), (Mazur and

Cholewi´nski, 2013).

In papers treating of skid-steering mobile plat-

forms, another approach to the problem of object

modeling has appeared. In (Pazderski and Kozłowski,

2008) authors observed that an artiﬁcial constraint in-

troduced by Caracciolo et al. is inadequate to real

behavior of SSMP platform and proposed modiﬁca-

tion of such a constraint to modiﬁed form. It implied

nonholonomicconstraint of second order i.e. dynami-

cally constrained model of SSMP platform. The same

idea can be found in (Mohammadpour et al., 2010)

and in (Maalouf et al., 2006).

In this paper a new method of modeling SSMP

platform is used as a base for artiﬁcial force con-

cept. In previous works, e.g. (Mazur et al., 2013) and

(Mazur and Cholewi´nski, 2013), SSMP platform with

four wheels was modeled as a chassis with two axes

of ﬁxed wheels coupled on both sides of platform. In

this case it was impossible to obtain independent mo-

tion of each wheel separately because two wheels on

one side were connected by a transmission belt. In

this paper the chassis with four uncoupled wheels is

considered. Taking into account different manners of

modeling it was possible to verify if the way of mod-

217

Mazur A., Domski W., Kaczmarek M. and Cholewinski M..

New Approach to the Artiﬁcial Force Concept for Skid-steering Mobile Platform.

DOI: 10.5220/0005521002170222

In Proceedings of the 12th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2015), pages 217-222

ISBN: 978-989-758-123-6

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

eling inﬂuences the resulting behavior of the platform.

The paper is organized in the following way. In

Section 2 mathematical model of SSMP platform with

nonholonomic constraints is developed. In Section 3,

the concept of an artiﬁcial force approach to control

such platforms in presented. In Section 4 the control

problem is formulated. Section 5 presents a new con-

trol algorithm based on artiﬁcial force idea. Section

6 contains the simulation results. Section 7 presents

some conclusions.

2 MATHEMATICAL MODEL OF

SKID-STEERING PLATFORM

SSMP platforms are modeled as a cart with more than

one axis of ﬁxed wheels. In the model of such a plat-

form we will treat it as a cart with four independently

driven ﬁxed wheels, see Fig. 1 for details.

Figure 1: Scheme of skid-steering platform.

The state of such an object can be described by a

vector of platform’s generalized variables



x y ϕ θ



where (x, y) are position coordinates of mass center

expressed in relation to global frame X

, ϕ is an

orientation of skid-steering platform and θ

is an angle

of rotation of ith wheel, i = 1, . . . , 4.

In further considerations we will assume that

SSMP platform moves on horizontal surface and no

longitudinal slippage phenomena occur. All wheels

of mobile platform are identical, therefore constraints

related to the absence of longitudinal slippage can be

expressed in so-called Pfafﬁan form

A(q) ˙q = 0, (1)

where Pfaff’s matrix equals to

A(q) =







cosϕ sinϕ −b −r 0 0 0

cosϕ sinϕ −b 0 −r 0 0

cosϕ sinϕ b 0 0 −r 0

cosϕ sinϕ b 0 0 0 −r







. (2)

Symbol r denotes radius of a wheel whereas b is a half

of platform’s width.

It is obvious that equation (1) determines nonholo-

nomic constraints for SSMP platform.

2.1 Model in Generalized Coordinates

Dynamics of SSMP platform can be derived from La-

grange formula

L(q, ˙q) = K(q, ˙q) −V(q).

The platform moves on horizontal surface, therefore

its potential energy is equal zero (V(q) = 0), hence,

the Lagrange formula is composed only of kinetic en-

ergy of the platform and its wheels

L = K(q, ˙q) =

˙q

M(q) ˙q, (3)

where M(q) is positive deﬁnite inertia matrix of the

platform

M =







0 −

α 0 0 0 0

0 m

β 0 0 0 0

−

β I

0 0 0 0

0 0 0 I

0 0 0

0 0 0 0 I

0 0

0 0 0 0 0 I

0 0 0 0 0 0 I







with elements deﬁned below

• I

= I

+ 4I

+ m

∑

i=1

– total inertia moment

of the platform with wheels relative Z

axis,

• m

= m

+ 4m

– total mass of SSMP platform,

• I

– inertia moment of wheel relative to

rotation axis,

• α = asinϕ − bcosϕ,

• β = acosϕ + bsinϕ.

Symbols m

, m

denote mass of a platform and a

wheel respectively. In turn, I

, I

are inertia mo-

ments of a platform and wheel – for wheel expressed

with the respect to horizontal (I

) and vertical (I

)

axis. The distance between the platform‘s center of

mass and the point of contact of ith wheel with the

ground, is marked as d

and a is the distance between

the mass center and the axis of front or back wheels.

Due to nonholonomic nature of constraints,

for obtaining dynamic model of SSMP platform

d’Alembert Principle must be used

M(q) ¨q+C(q, ˙q) ˙q+ F(q, ˙q) = B(q)u + A

(q)λ, (4)

where:

M(q) – inertia matrix of SSMP platform,

C(q, ˙q) – matrix of Coriolis and centrifugal forces,

F(q, ˙q) – ground reaction forces e.g. friction etc.,

A(q) – Pfaff matrix deﬁned by (2),

λ – vector of Lagrange multipliers,

B(q) – input matrix,

u – vector of controls.

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218

2.2 Model in Auxiliary Velocities

According to (1), since the platform velocities ˙q are

always in the null space of A, it is always possible to

ﬁnd a vector of auxiliary velocities η ∈ R

, such that

˙q = G(q)η, (5)

where G(q) is an 7× 3 full rank matrix satisfying re-

lationship A(q)G(q) = 0.

Now we want to express the model of dynamics

using auxiliary velocities (5) instead of generalized

coordinates of the mobile platform. We compute

¨q = G(q)

η+

G(q)η,

and eliminate in the model of dynamics the Lagrange

multiplier using the condition G

≡ 0. Substituting

˙q and ¨q in (4) we get

∗

η+C

∗

η+ F

∗

= B

∗

u (6)

with elements of matrix equation deﬁned in the fol-

lowing way

∗

= G

MG, C

∗

= G

G+ G

CG,

∗

= G

F, B

∗

= G

3 ARTIFICIAL FORCE IDEA

Mobile SSMP platform REX should be considered

as an underactuated system on dynamic level be-

cause it has got a rectangular input matrix (which is

non-invertible). There are possible two different ap-

proaches to solve the problem of inverting the input

matrix B

∗

• introducing additional artiﬁcial nonholonomic

constraints. Then a model in auxiliary velocities

has the same size of state variables as reduced to

the number of control inputs.

• introducing an additional artiﬁcial input to the dy-

namics. Then a model in auxiliary velocities has

the same number of state variables and control in-

puts. However, an artiﬁcial input (so-called “ar-

tiﬁcial force” in further considerations) has to be

equal 0, i.e.

3 add

≡ 0, (7)

because it does not exist in reality.

4 FORMULATION OF CONTROL

PROBLEM

In the paper, our goal is to ﬁnd control law guarantee-

ing the trajectory tracking for SSMP platform. Our

goal is to address the following control problem to

such platforms:

Determine control law u such that SSMP plat-

form with fully known dynamics follows the

desired trajectory.

To design trajectory tracking controller for the consid-

ered mobile platform, it is necessary to observe that a

complete mathematical model of the nonholonomic

system expressed in auxiliary variables is a cascade

consisting of the two groups of equations: kinemat-

ics (5) and dynamics (6), see Fig. 2: For this reason

object

kinematic

controller

dynamic

controller

kinematics

1 stage

dynamics

2 stage

CASCADE

≅

Figure 2: Structure of the proposed control algorithm: cas-

cade with two stages.

the structure of the controller consists of the two parts

working simultaneously:

• kinematic controller η

– a vector of embedded

control inputs, which ensure realization of the task

for the kinematics (nonholonomic constraints) if

the dynamics were not present. Such the con-

troller generates ’velocity proﬁle’ which can be

executed in practice to realize the trajectory track-

ing for nonholonomic SSMP platform.

• dynamic controller – as a consequence of cas-

caded structure of the system model, the system’s

velocities cannot be commanded directly, as it is

assumed in the designing of kinematic control sig-

nals, and instead they must be realized as the out-

put of the dynamics driven by u.

It can be observed that backstepping-like algorithm

(Krsti´c et al., 1995) has been evoked to solve the pre-

sented control problem for SSMP platform. Back-

stepping is well-known and often used approach to

control cascaded systems, e.g. the system (6) with

nonholonomic constraints (5).

5 CONTROL ALGORITHM

As it has been previously mentioned, a control algo-

rithm has to consist of the two parts, i.e. kinematic

controller and dynamic controller.

5.1 Kinematic Controller

Considering nonholonomic constraints (5), for a real

case of the two active controls, they are equivalent to

the unicycle model. On that basis, the kinematic con-

troller is suggested in the form given by (Samson and

NewApproachtotheArtificialForceConceptforSkid-steeringMobilePlatform

219

Ait-Abderrahim, 1991). This algorithm allows trajec-

tory tracking for a simple unicycle vehicle. Unicycle

velocities appropriate for tracking of desired trajec-

tory q

are described by the following equation (ﬁrst

and second column of matrix G(q) in equation (5))

˙q

= G(q

)η





rcosϕ

rsinϕ

0 r









, (8)

where η

and η

are desired linear and

angular velocities of the platform.

The kinematic algorithm of the model described

by (5) and desired velocities (8) requires





− e

− k

sine

, (9)

where

• v

, ω

are reference linear and angular velocities

for a robot vehicle (signals coming from kine-

matic controller),

• v

, ω

are desired linear and angular velocities,

• k

> 0 and k

> 0 are control parameters,

• e

= (e

, e

)

– reference trajectory errors.

The reference trajectory tracking errors are deﬁned as

below









= Rot(z, − ϕ)





x− x

y− y

ϕ− ϕ





The asymptotic convergence of tracking errors e

zero implies asymptotic trajectory tracking. Refer-

ence velocities η

and η

could be obtained from

relationship

, η

The third component η

is responsible for maintain-

ing the apparent force u

3add

at 0. It can be obtained

by solving the equation u

3add

= 0.

5.2 Dynamic Controller

Let’s consider dynamics of SSMP platform (6) ex-

pressed in auxiliary velocities. For such a system,

following passivity-based control law has been pro-

posed

u = (B

∗

)

−1

∗

+ F

∗

− K

), (10)

where

• K

> 0 – matrix of regulation parameters,

• e

= η− η

– velocity tracking error.

Dynamics of the closed-loop system (6)-(10) can

be described as follows

∗

˙e

∗

+ K

= 0. (11)

5.3 Proof of the Convergence

For the system (11) we propose the following

Lyapunov-like function

V(e

) =

∗

(q)e

≥ 0, (12)

which is non-negative deﬁnite.

Time derivativeofV along solutions of the closed-

loop system (11) is equal to

V =

∗

(q)e

+ e

∗

(q) ˙e

= −e

≤ 0. (13)

From La Salle theorem, see (Krsti´c et al., 1995) for

details, it could be concluded that the errors e

con-

verge asymptotically to zero.

From the other side, the convergence of e

to zero

means that the the velocity proﬁle generated by kine-

matic controller is successfully followed, and there-

fore one can conclude that the nonholonomic system,

i.e. skid-steering mobile platform, tracks desired tra-

jectory q

. It ends the proof.

5.4 Artiﬁcial Force – Implicit Function

From deﬁnition of artiﬁcial force i.e.

3 add

≡ 0,

where control inputs are given by the control algo-

rithm (10), the third reference signal η

3ref

can be cal-

culated as an implicit function in the following way

3ref

(4K

− 4K

3ref

+ βm

cosϕ

2ref

+ αm

sinϕ

2ref

+ 4m

1ref

ϕ+ bm

2ref

ϕ). (14)

6 SIMULATIONS

The simulations were run with the MATLAB package

and the SIMULINK toolbox.

Parameters of the platform were equal to: m

= 30

[kg], m

= 3 [kg], I

= 11.3335 [kg·m

], I

= 0.0968

[kg·m

], a = 0.8 [m], b = 0.3 [m], r = 0.127 [m].

Parameters of terrain are set on value µ = f

= 1.

The model of SSMP platform expressed in auxil-

iary velocities is deﬁned by equations (5) and (6).

The elements of the mathematical model of SSMP

platform expressed in auxiliary velocities have de-

tailed form

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220

• kinematics (5)

G =





rcosϕ rsinϕ 0 1 1 1 1

0 0 r −b −b b b

rsinϕ −rcosϕ 0 0 0 0 0





• dynamics (6)

∗





0 M





= 4I

+ m

= −

(αcosϕ− βsinϕ)

= 4b

+ I

= −

(βcosϕ+ αsinϕ)

= m

∗





0 −

ϕ m

ϕ 0

−m

ϕ −

ϕ 0





∗





µ· sgn˙y

bgm

r[sgn( ˙x+ b˙y) − sgn(˙x − b

ϕ)]





∗





1 1 0

−b b 0

0 0 r





6.1 Desired Trajectories

The goal of the simulations was to investigate a be-

havior of a mobile platform with the controller (10)

proposed in the paper. In simulations the two desired

trajectory of the platform has been tested: a circle and

a square trajectory. The ﬁrst one, was a circle with

a radius of R = 10 [m] and frequency ω = 0.1 [

rad

The second desired trajectory was a square with side

equal to 10 meters. The linear velocity v

for this de-

sired trajectory was set to 0.5

The parameters of a kinematic controller given by

(9) were equal to k

= 1, k

= 1 while a dynamic con-

troller (10) has as regulation parameter K

= 50 for

both desired trajectories.

Circle. In Figures from 3 to 6 were presented the

plots from a simulation with the desired trajectory set

to a circle. From the plots presented in Figures 4-6

it can be observed that position tracking errors and

orientation error tend to 0.

Square Trajectory. In Figures 7-10 were presented

the plots for desired trajectory set to a square.

−10 −5 0 5 10

−10

−8

−6

−4

−2

x [m]

y [m]

Real trajectory

Desired trajectory

Figure 3: Real vs desired trajectory.

0 20 40 60 80 100

−2.5

−2

−1.5

−1

−0.5

0.5

time [s]

[m]

Figure 4: Error in cartesian space for x coordinate.

0 20 40 60 80 100

−1

time [s]

[m]

Figure 5: Error in cartesian space for y coordinate.

0 20 40 60 80 100

−1.5

−1

−0.5

0.5

time [s]

[rad]

Figure 6: Error in cartesian space for ϕ coordinate.

NewApproachtotheArtificialForceConceptforSkid-steeringMobilePlatform

221

−2 0 2 4 6 8 10 12

−2

x [m]

y [m]

Real trajectory

Desired trajectory

Figure 7: Real vs desired trajectory.

0 20 40 60 80 100

−2.5

−2

−1.5

−1

−0.5

0.5

time [s]

[m]

Figure 8: Error in cartesian space for x coordinate.

0 20 40 60 80 100

−2

−1.5

−1

−0.5

0.5

time [s]

[m]

Figure 9: Error in cartesian space for y coordinate.

0 20 40 60 80 100

−2

−1.5

−1

−0.5

0.5

time [s]

[rad]

Figure 10: Error in cartesian space for ϕ coordinate.

7 CONCLUDING REMARKS

In the paper the method of an artiﬁcial force was used

for the skid-steering mobile platform. In this method

an additional input to the model of dynamics has been

introduced making input matrix invertible. Differ-

ently than in previous publications, in considered ap-

proach, the platform has been modeled as a cart with

four independent, not coupled wheels.

Used method has shown that it gives good results

for different ways of modeling a skid-steering plat-

form. It implies that in future research, the use of arti-

ﬁcial force concept will lead to proper work of control

algorithm and good enough trajectory tracking.

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