J Intell Robot SystDOI 10.1007/s10846-013-9878-7

Virtual Force Concept in Steering MobileManipulators with Skid-Steering PlatformMoving in Unknown Environment

Alicja Mazur · Mateusz Cholewinski

Received: 15 January 2013 / Accepted: 30 May 2013© The Author(s) 2013. This article is published with open access at SpringerLink.com

Abstract Underactuated control system, namelymobile manipulator mounted on skid-steeringplatform has been considered in the paper. Asa control scheme the concept of virtual force ispresented and discussed. In such concept, full rankof input matrix, describing relationship betweencontrolled variables and number of actuators, hasbeen assumed. The signal of the assumed addi-tional actuator is equal to zero equivalently (be-cause this force is “virtual” and it does not existin practice) but in this way the invertibility ofinput matrix can be achieved. The equation of theimplicit function generates the reference signalsneeded in control algorithm. Mathematical proofof proper action of such control law has been pre-sented using Lyapunov-like function. Efficiencyand robustness of control scheme on parametricuncertainty of terrain, which the mobile manipu-lator is moving on, have been tested in numeroussimulations.

A. Mazur (B) · M. CholewinskiInstitute of Computer Engineering, Controland Robotics, Wrocław University of Technology,ul.Janiszewskiego 11/17, 50-372Wrocław, Polande-mail: alicja.mazur@pwr.wroc.pl

M. Cholewinskie-mail: mateusz.cholewinski@pwr.wroc.pl

Keywords Mobile robot · Nonholonomicconstraints · Parametric uncertainty · Robustcontrol

1 Introduction

Mobile manipulator, which is the subject of con-sideration, consists of a mobile platform and anonboard rigid manipulating arm. Such roboticobject can execute more complicated tasks thanits components. Manipulating arm is fully con-trolled while mobile platform equipped with morethan one axis of fixed wheels is an underactuatedsystem.

Wheeled mobile platforms constitute impor-tant group of robotic objects. They can be treatedas independent robots or as an transportation partof composed robotic assemble, for instance mo-bile manipulators. Depending on wheels’ kind andthe way in which they are fixed to the cart, motionof wheeled mobile platforms can be realized withor without slippage phenomenon. If no slippageeffect between wheels and surface occurs, thenexists some equation describing forbidden direc-tions of realized velocities of the system. Such arelationship is called nonholonomic constraint inplatform’s motion.

Special kind of wheeled mobile platforms areplatforms with more than one axis equipped withfixed wheels, see Fig. 1. These platforms are called

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Fig. 1 Schematic of mobile manipulator with SSMP plat-form – tank

skid-steering mobile platforms (SSMP), due toskidding effect observed in theirs behavior [4] fordetails.

Research activities in the field of modeling anddeveloping of control algorithms for SSMP plat-forms have been continued from the last 90’s ofXX century. First attempt to solve this problemcan be found in the work [2], in which someassumption about instantaneous center of rota-tion (ICR) during the platform’s motion has beendone. Authors made assumption that there ex-ists some non-integrable relationship between an-gular velocity (orientation’s changes) and lateralslipping of wheels. Such equation, although de-rived from slipping effect, could play a role of spe-cial nonholonomic constraint. Similar approach tothis presented by Caracciolo et al. can be found in[4] and in other papers.

Another approach to the problem of modelingand control of SSMP platform has occurred re-cently, e.g. in [8, 9]. The authors have observedthat the assumption introduced by Caraccioloet al. is not always fulfilled in practice and itshould be extended on accelerations in platform’smotion. They have treated SSMP platform asan underactuated system on dynamic level withnon-stationary kinematics (non-stationary veloc-ity constraint). Both concepts, i.e. with station-ary or non-stationary constraint on velocity, needsystem of motion planning, solving nonholonomicconstraints in any time. Such system is sophisti-cated and complicated.

It is possible to describe SSMP platform inanother way. Such idea, known as a singular per-turbation approach, can be found in the paper [3].

The authors have introduced new variable for slip-page and have reformulated mathematical equa-tions including such variable in dynamical model.This method is not very often used in practicalapplications because it is very complex and it triesto strictly evaluate slippage phenomenon whichcan be unpredictable.

In this paper a new concept of control prob-lem of mobile manipulator’s with SSMP platformhas been presented. Instead of motion planner,mobile manipulator with SSMP platform can beconsidered as an underactuated system with rec-tangular input matrix. Problem of inverting ofsuch matrix (standing near control signals) can besolved with the idea of so-called “virtual force”.

The paper is organized in the following way.In Section 2 concept of proposed approach tothe control nonholonomic mobile manipulators ispresented. Section 3 illustrates theoretical designof mathematical model of considered objects. InSection 4 the control problem is formulated. InSection 5 a new control algorithm is designed andits proper action is proved. Section 6 containsthe simulation results. Section 7 presents someconclusions.

2 Concept of Virtual Force

As mentioned earlier in the introduction, mobilemanipulator with SSMP platform is underactu-ated system: number of controlled system vari-ables is bigger than number of control inputs. Itimplies the rectangular form of input matrix Band, consequently, a problem with its inverse andcontrol design. In this paper a new approach tocontrol of SSMP platform will be presented. Inthis method it is being assumed that controlledsystem has full number of input signals and there-fore input matrix is invertible. These control in-puts, which do not exist in reality, will be called“virtual forces”. Since virtual forces are not real(they do not exist in practice), in further consid-erations it will be assumed that they are alwaysequal to zero equivalently, i.e.

ui,virtual ≡ 0, i = 1, m − n. (1)

Equation 1 defining virtual forces are implicitfunctions including state variables, regulation

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parameters and desired trajectories. From Eq. 1it is possible to obtain missing reference signals,needed for different control algorithms.

Similar approach to the shortage in control in-puts in underactuated systems can be found in lit-erature, e.g. for nonholonomic inverted pendulummounted on unicycle [6], in control algorithms us-ing extended Jacobi matrix for redundant manip-ulators [1] or in endogenous configuration spaceapproach for nonholonomic mobile manipulators[10]. However, concept of virtual force for SSMPplatform is introduced and considered in this pa-per the first time.

3 Mathematical Models in Different Coordinatesfor Mobile Manipulator with SSMP Platform

Let’s consider rigid manipulator mounted on theSSMP platform with two axes of fixed wheels,presented in Fig. 1. State of such object can bedescribed by vector of platform generalized vari-ables qm and manipulator variables qr, i.e. q =(qm, qr)

T

qm = (x y θ φ1 φ2

)T, qr = (

q1 q2 . . . qp)T

where (x, y) are cartesian coordinates of masscenter defined relative to global frame X0Y0, θ isan orientation of SSMP platform, φi is an angle ofrotation of wheels located on the right or the leftside, whereas qr is a vector of manipulator jointvariables. For selected generalized coordinates ofSSMP platform there exists some relationship [4]

⎛

⎝xyθ

⎞

⎠ = Rot(z, θ)

⎛

⎝vx

vy

ω

⎞

⎠ .

Symbols vx and vy mean elements of onward ve-locity relative to local axes Xp and Yp while ω isangular velocity of the SSMP platform.

In this paper the only assumption about wheelsmotion without longitudinal slip has been taken.Moreover, we assume that velocity of consideredmobile manipulator is rather small, not bigger that10 [ km

h ].

All wheels of SSMP platform are identical,therefore constraints on lack of longitudinal slipcan be expressed as follows

φ1r = vx + cω right side of the platform, (2)

φ2r = vx − cω left side of the platform, (3)

where r is a radius of wheel and c is a half ofplatform width. Equations 2–3 can be expressedin so-called Pfaffian form in the following way

A(qm)qm =[

cos θ sin θ −c −r 0cos θ sin θ c 0 −r

]

⎛

⎜⎜⎜⎜⎝

xyθ

φ1

φ2

⎞

⎟⎟⎟⎟⎠

= 0.

(4)

3.1 Model in Generalized Coordinates

For nonholonomic, skid-steering vehicles withoutlongitudinal slippage, dynamics could be derivedfrom d’Alembert principle as

M(q)q + C(q, q)q + D(q) + F(q, q) = B(q)u

+AT(qm)λ,

or more in detail

[M11 M12

M21 M22

](qm

qr

)+

[C11 C12

C21 C22

] (qm

qr

)+

(0D

)

+(

F0

)=

[B 00 I

] (um

ur

)+

(ATλ

0

). (5)

Symbol M(q) denotes inertia matrix of mobilemanipulator, matrix C(q, q) describes Coriolis andcentripetal forces, vector D(q) is a vector of grav-ity forces (only for manipulating arm becauseplatform moves on horizontal surface), while vec-tor F(q, q) is a vector of non-potential forces,mostly reaction forces from terrain and frictionforces. Torques B(q)u represent control inputs(actuators) whereas ATλ are forces coming fromnonholonomic constraints. Matrix B(q), so-called

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input matrix, describes relationship between inputsignals u and torques coming from motor shafts

B · um =

⎡

⎢⎢⎢⎢⎣

0 0 − sin θ

0 0 cos θ

0 0 00 1 01 0 0

⎤

⎥⎥⎥⎥⎦

⎛

⎝u1

u2

uv

⎞

⎠ .

Symbols u1 and u2 describe control signal associ-ated with coupled wheels on the right and left sideof the SSMP platform. Third column of matrixB(q) is responsible for hypothetical virtual forceuv , influenced on platform orientation.

Equations 4 and 5 constitute complete modelof special nonholonomic system, namely mobilemanipulator with SSMP platform, expressed ingeneralized coordinates.

3.2 Model in Auxiliary Coordinates

Mobile manipulator with SSMP platform is a con-trol system with nonholonomic constraints. Theseconstraints, called kinematics of nonholonomic sys-tem, can be transformed to auxiliary coordinates

– kinematics – driftless control system

qm = G(q)η =

⎡

⎢⎢⎢⎢⎣

cos θ cos θ − sin θ

sin θ sin θ cos θ1c − 1

c 00 2

r 02r 0 0

⎤

⎥⎥⎥⎥⎦

⎛

⎝η1

η2

η3v

⎞

⎠ .

On the other hand, dynamics expressed in auxil-iary velocities for nonholonomic subsystem havea form

– dynamics

M∗(

η

qr

)+ C∗

(η

qr

)+

(0D

)+

(F∗0

)= B∗

(um

ur

),

(6)

where:

M∗ =[

GT M11G GT M12

M21G M22

],

C∗ =[

GT(C11G + M11G) GTC12

M21G + C21G C22

],

F∗ = GT F,

B∗ =[

GT B 00 I

].

3.3 Model in Linearizing Coordinates

For SSMP platform, it is possible to choose an-other coordinates for description, e.g. so-calledlinearizing outputs

ξ =⎛

⎝ξ1

ξ2

ξ3

⎞

⎠ =⎛

⎝x + l cos θ

y + l sin θ

θ

⎞

⎠ . (7)

From [7] it is known that number of linearizingoutputs is equal to the difference between dimen-sion of state variables and number of nonholo-nomic constraints. It means that it is possible toget three linearizing variables for SSMP platform.

First and second element of ξ are coordinatesof a point lying in distance l to front of the plat-form, whereas last element is an orientation of theplatform.

Relationship between auxiliary and linearizingcoordinates for nonholonomic SSMP platform hasa matrix form [7]

R(qm)ξ = η

with matrix R(qm)

R(qm) =⎡

⎣cos θ

2sin θ

2 − c2

cos θ2

sin θ2

c2− sin θ cos θ −l

⎤

⎦ ,

with determinant equal to − c2 .

In turn, dynamics of considered mobile manip-ulator can be expressed in (ξ, qr) variables in thefollowing way

Jξ

(ξ

qr

)+ Cξ

(ξ

qr

)+

(0D

)+

(Fξ

0

)= Bξ

(um

ur

),

(8)

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where selected elements of model (8) can be cal-culated as follows

Jξ =[

RT GT M11GR RT GT M12

M21GR M22

],

Bξ =[

RT GT B 00 I

],

Cξ =[

RT GT (C11GR + M11GR + M11GR) RT GT C12

M21GR + M21GR + C21GR C22

]

,

Fξ = RT GT F.

Equation 8 is a mathematical model of mobile ma-nipulator with SSMP platform expressed in (ξ, qr)

variables.

4 Control Problem Statement

In this paper mobile manipulator with SSMP plat-form is considered. Such robotic object shouldtrack desired trajectory without longitudinal slipof its wheels. It implies appearance of nonholo-nomic constraints (4) in mobile manipulator’s mo-tion. By designing of control law we take an as-sumption that dynamics are expressed in (ξ, qr)

coordinates, i.e. given by Eq. 8. Our goal is to ad-dress the following control problem to consideredobject:

1. Determine control law u in such a way thatmobile manipulator with SSMP platform, withfully known dynamics, follows the desired tra-jectory on unknown terrain without any longi-tudinal slip of wheels.

2. All virtual forces have to be equal to zero,because they do not exist in reality.

Desired trajectory is defined separately for eachsubsystem, i.e. platform should move along ξd(t)and joints of rigid onboard manipulator have totrack vector of desired trajectories qrd(t). More-over, it was assumed that desired trajectories areadmissible i.e. that it is possible to move alongthem without skidding effect in wheels.

5 Control Algorithms

5.1 Mobile Manipulator Moving in KnownTerrain – Nonadaptive Case

Let’s consider mobile manipulator with fullyknown dynamics, given by Eq. 8. If surface, whichobject is moving on, is known, then reaction forcesF are precisely defined. For such case we proposefollowing control law(

um

ur

)= B−1

ξ

{Jξ

(ξref

qref

)+ Cξ

(ξref

qref

)+

(0D

)

+(

Fξ

0

)− Kds

}(9)

with symbols defined as below

s =(

sm

sr

)=

(ξ

qr

)−

(ξref

qref

),

where

sm =⎛

⎝ξ1 − ξ1ref

ξ2 − ξ2ref

ξ3 − ξ3ref

⎞

⎠ =⎛

⎝e1 + �e1

e2 + �e2

ξ3 − ξ3ref

⎞

⎠ (10)

sr = qr − qref = eq + �eq (11)

e1 = ξ1 − ξ1d,

e2 = ξ2 − ξ2d,

eq = qr − qrd,

� = �T > 0, Kd = KTd > 0.

It is worth emphasizing that dynamics are calcu-lated for whole mobile object i.e. for SSMP plat-form and manipulating arm simultaneously. It is aconsequence of the fact that dynamic interactionsbetween both subsystems occur and can be reallybig.

Proof For closed-loop system (8) with feedbackcontrol (9), described by equation

Jξ s + Cξ s + Kds = 0, (12)

the following Lyapunov-like function can be chosen

V(ξ, qr) = 12

sT Jξ s. (13)

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Time derivative of V(ξ, qr) calculated along tra-jectories of closed-loop system (12) equals to

V = 12

sT Jξ s + sT Jξ s = −sT Kds ≤ 0.

From La Salle invariance principle it can be con-cluded that the biggest invariant set equals tos = 0. Through positive definiteness of matrix �

and definition of sliding variables for platform andonboard manipulator (10)–(11), tracking errors(e1, e2, eq) → 0 asymptotically fast.

It is necessary to mention that the third com-ponent of sm has to be calculated in differentway than other components. Reference trajectoryξ3ref must be obtained from condition u3v = 0 as asolution of implicit function (14). ��

5.2 Expression for Virtual Force –Nonadaptive Case

Virtual force u3v , obtained from proposed controllaw (9), has to be equal to zero

u3v = − sin θ [Jξ11ξ1ref + Jξ12ξ2ref + Jξ13ξ3ref

+Jξ14q1ref + Jξ15q2ref + Cξ11ξ1ref + Cξ12ξ2ref

+Cξ13ξ3ref + Cξ14q1ref + Cξ15q2ref + Fξ1

−Kd(ξ1 − ξ1ref )] + cos θ [Jξ21ξ1ref + Jξ22ξ2ref

+Jξ23ξ3ref + Jξ24q1ref + Jξ25q2ref + Cξ21ξ1ref

+Cξ22ξ2ref +Cξ23ξ3ref +Cξ24q1ref +Cξ25q2ref

+Fξ2 − Kd(ξ2 − ξ2ref )]= 0. (14)

Above expression makes possible to get the thirdreference trajectory ξ3ref after integrating ξ3ref . Itis the consequence of the fact that in reality it isimpossible to control orientation of the platformbecause the third actuator (virtual force) does notexist. The remaining elements of sliding vectors can be calculated from Eqs. 10–11 as a linearcombination of position and velocity tracking er-rors of output functions for platform and joints ofonboard manipulator.

5.3 Mobile Manipulator Moving in UnknownTerrain – Adaptive Case

Vector of reactive forces and torques is equalto [4]

Fξ =⎛

⎝Fs cos θ − Fl sin θ

Fs sin θ + Fl cos θ

Mr

⎞

⎠ (15)

with elements defined below

Fs =4∑

i=1

Fsi =4∑

i=1

μsi Nisign(vx),

Fl =4∑

i=1

Fli =4∑

i=1

μli Nisign(vy) (16)

Mr = b(Fl2 + Fl3) − a(Fl1 + Fl4)

+ c(Fs3 + Fs4 − Fs1 − Fs2).

Symbols μsi and μli denote dry friction coefficientsfor ith wheel in longitudinal and lateral direction;Ni is force acting on ith wheel in vertical direction,depending on weight of the platform; a and b aredistances from platform mass center to front andback axis of wheels (we assume a = b), c is a halfof platform width.

More interesting situation holds if some pa-rameters, e.g. interaction forces between wheeland terrain, are uncertain. We assume that formof reaction forces Fξ defined by Eq. 8 is knownbut friction coefficients μsi and μli are constantbut unknown. Then reaction forces Fξ given byEqs. 15–16 can be linearly parameterized in sucha way

Fξ = Y(ξ )a∗, (17)

where Y is known regression matrix and a∗ is avector of unknown constant real parameters indynamics.

Table 1 Simulation parameters

Parameter Value Unit Parameter Value Unit

mp 60 kg c 0.75 mmk 3 kg Ip 10 kg·m2

Rw 0.15 m Ik 0.034 kg·m2

l1 0.5 m m1 20 kgl2 1 m m2 15 kga 0.6 m b 0.6 m

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For mobile manipulator moving in unknownterrain we propose control algorithm given below

(um

ur

)= B−1

ξ

{Jξ

(ξref

qref

)+ Cξ

(ξref

qref

)+

(0D

)

+(

Fξ (t)0

)− Kds

}

= B−1ξ

{Jξ

(ξref

qref

)+ Cξ

(ξref

qref

)+

(0D

)

+(

Ya(t)0

)− Kds

}. (18)

−2 0 2 4 6 8 10 12−2

0

2

4

6

8

10

12

y [m

]

x [m]

desired trajectoryreal trajectory

a)

b)

Fig. 2 Trajectory of SSMP platform for known parametersof terrain – nonadaptive case: a for a circle, b for a square

In control law (18) estimates a(t) of real parame-ters a∗ are obtained from adaptation law

˙a = −YTs, = T > 0, (19)

where is the matrix of adaptation gains.

Proof Closed-loop system (8) with feedback con-trol (18) can be presented as

Jξ s + Cξ s + Kds − Ya(t) = 0, (20)

where a(t) = a(t) − a∗ is a parameter error.For the system (20) the following Lyapunov-

like function can be chosen

Va(ξ, qr, a) = 12

sT Jξ s + 12

aT−1a. (21)

0 5 10 15 20 25 30 35 40−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

t [s]

Err

orex [m]ey [m]

0 5 10 15 20 25 30 35 40−2

−1.5

−1

−0.5

0

0.5

t [s]

Err

or

ex [m]ey [m]

a)

b)

Fig. 3 Position errors of SSMP platform for known para-meters of terrain – nonadaptive case: a for a circle, b for asquare

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Time derivative of Va(ξ, qr, a) calculated alongtrajectories of the system (19)–(20) equals to

Va = 12

sT Jξ s + sT Jξ s + aT−1 ˙a = −sT Kds ≤ 0.

From La Salle–Yoshizawa theorem it can be con-cluded [5] that the biggest invariant set equals tos = 0. Moreover, through positive definiteness ofmatrix �, tracking errors (e1, e2, eq) go to zeroasymptotically fast. ��

0 5 10 15 20 25 30 35 40−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

t [s]

0 5 10 15 20 25 30 35 40t [s]

Err

or

eq1 [rad]eq2 [rad]

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Err

or

eq1 [rad]eq2 [rad]

a)

b)

Fig. 4 Joint errors of manipulator for known parametersof terrain – nonadaptive case: a for a circle, b for a square

6 Simulation Results

The simulations were run with MATLAB packageand SIMULINK toolbox. As an object of simu-lations we have taken SSMP platform equippedwith two axes of fixed wheels with 2R rigid ma-nipulator. The parameters of the platform were:mass of the platform mp, mass of the wheel mk,platform moment of inertia Ip relative Z p axis,wheel moment of inertia Ik relative Z p axis, halfof platform width c, distances a and b from masscenter to front and back axis of wheels.

−10 −5 0 5 10

−10

−5

0

5

10

y [m

]

x [m]

desired trajectoryreal trajectory

−2 0 2 4 6 8 10 12−2

0

2

4

6

8

10

12

y [m

]

x [m]

desired trajectoryreal trajectory

a)

b)

Fig. 5 Trajectory of SSMP platform for unknown parame-ters of terrain – adaptive case: a for a circle, b for a square

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The first link of manipulator was modeled asa cylinder with radius of Rw, height l1 and massm1 and the second link as a bar with length l2 andmass m2. The parameters of mobile manipulatorare presented in Table 1.

As a desired trajectory for the platform twotrajectories have been chosen: admissible (circle)and inadmissible (square)

– admissible trajectory – circle with radius R =10 [m] and frequency ω = 1 [Hz],

– inadmissible trajectory – square 12 [m]× 12 [m],realized in 40 s.

0 5 10 15 20 25 30−2

−1.5

−1

−0.5

0

0.5

t [s]

Err

or

ex [m]ey [m]

0 5 10 15 20 25 30 35 40−2

−1.5

−1

−0.5

0

0.5

t [s]

Err

or

ex [m]ey [m]

a)

b)

Fig. 6 Position errors of SSMP platform for unknownparameters of terrain – adaptive case: a for a circle, b fora square

Desired trajectories for manipulator joints weredefined as q1d = 0 [rad] and q2d = 0.1 [rad].

During simulations, some features of the con-trolled system has been tested. First, performanceof trajectory tracking has been checked for fullknowledge about terrain, which platform wasmoving on. For such a case friction coefficients μsi

and μli were chosen as equal to 0.5 and they wereknown. Next, the same simulations have beenmade for adaptive case – for unknown groundparameters: μsi = 0.9 and μli = 0.1.

0 5 10 15 20 25 30−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

t [s]

Err

or

eq1 [rad]eq2 [rad]

0 5 10 15 20 25 30 35 40−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

t [s]

Err

or

eq1 [rad]eq2 [rad]

a)

b)

Fig. 7 Joint errors of manipulator for unknown parame-ters of terrain – adaptive case: a for a circle, b for a square

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6.1 Robot Motion in Known Environment

Figs. 2, 3, 4 have been made for a case of knownground parameters – for nonadaptive case. Track-ing of the desired trajectory for the mobile plat-form has been presented in Figs. 2–3 whereas inFig. 4 position errors in manipulator joints havebeen plotted. Regulation parameters were chosenas Kd = 50 and � = 10.

6.2 Robot Motion in Unknown Environment

Figs. 5, 6, 7 have been made for a case of un-known ground parameters, i.e. for adaptive case.Tracking of the desired trajectory for the mobileplatform has been presented in Figs. 5–6 and in

−10 −5 0 5 10

−10

−5

0

5

10

y [m

]

x [m]

desired trajectoryreal trajectory

−2 0 2 4 6 8 10 12−2

0

2

4

6

8

10

12

y [m

]

x [m]

desired trajectoryreal trajectory

a)

b)

Fig. 8 Trajectory of SSMP platform for changing parame-ters of terrain – adaptive case: a for a circle, b for a square

Fig. 7 position errors in manipulator joints havebeen plotted. Regulation parameters were cho-sen as Kd = 50, � = 10 and = 10 for a circleand Kd = 500, � = 10 and = 10 for a squaretrajectory.

6.3 Robot Motion in Changing Environment

Simulation has been made also for changing para-meters of ground. First, friction coefficients havebeen selected in simulated environment as μsi =0.1 and μli = 0.9 for time interval [0,10] s. Next,after 10 s parameters changed rapidly on valuesμsi = 0.4 and μli = 0.6. Nevertheless, trajectorytracking has been realized properly, what can beseen in Fig. 8. Tracking for the mobile platform

0 5 10 15 20 25 30 35 40−2

−1

0

1

2

3

4

5

t [s]

Err

orex [m]ey [m]

0 5 10 15 20 25 30 35 40−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

t [s]

Err

or

eq1 [rad]eq2 [rad]

a)

b)

Fig. 9 Position errors for changing parameters of terrain:a errors for SSMP platform, b joint errors for manipulator

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0 5 10 15 20 25 30 35 40−100

−50

0

50

100

150co

effic

ient

[1]

t [s]

frmir

10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8

0

0.2

0.4

0.6

0.8

1

coef

ficie

nt [1

]

t [s]

frmir

a)

b)

Fig. 10 Estimates of terrain parameters: a in 40 s, b zoomof plot after changing parameters

has been presented in Figs. 8 and 9a whereas inFig. 9b position errors in joints of manipulatorhave been plotted. In Fig. 10 estimates of terrainparameters have been presented.

7 Conclusions

In this paper new approach to control mobilemanipulator with skid-steering mobile platformhas been presented and discussed. Instead of ar-tificial assumption on object velocity, concept ofvirtual force has been proposed. Such descriptionallows to solve control problem for system with

deficit of input signals. Numerous simulationshave confirmed, that such concept can find prac-tical applications for nonholonomic mobile ma-nipulators moving with slippage effect, even if theterrain, which platform is moving on, is unknownand changes during motion.

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