Kinematics Model of Kinematics Model of Nonholonomic Wheeled Mobile Nonholonomic Wheeled Mobile Robots for Mobile Manipulation Robots for Mobile Manipulation

TasksTasksDimitar Chakarov

Institute of Mechanics- BAS, 1113 Sofia, “Acad.G.Bonchev” Str., Block 4

Outline

IntroductionIntroduction

Holonomic and nonholonomic WMRHolonomic and nonholonomic WMR Direct and reverse kinematics task of Direct and reverse kinematics task of

nonhoonomic WMRnonhoonomic WMR

ConclusionConclusion

1.Introduction(1/2)The robotized technologies are very quickly spreading for domestic, service and entertainment needs. A big number of scientific investigations and scientific activities form a new scientific field during the recent years devoted to mutual interaction among robots and the human being. This is an interdisciplinary scientific field that covers robotics, computer sciences, and the science of knowledge, physiology and sociology.Robots are going very soon to assist the human being on a wide range of problems, which are not attractive, they are dangerous, not well paid or boring to humans. Robots assistants are going to work in the future as patient sitters, as security guards, as rescuers and fire rescuers, in surgery and rehabilitation, in domestics and in offices, in mining, in building as well as in stores and museums.

1.Introduction(2/2)In order to work together with and to assist and to interact with people the new robot generation must posses a mechanical structure that is suitable for this partnership in the human not organised and unknown surrounding environment. Wheeled mobile robots (WMR) are long ago invented in areas, where they interact with humans as service robots for remote book reading in a library or for serving tea and meals, human following and guiding robots, entertainment and cleaning robots. The compatible with people robots must integrate mobility and manipulation. Mobile manipulator systems hold promise in many industrials and service applications including manipulations, assembly, inspection and work in hazardous environments. The integration of a manipulator and a mobile robot base places special demands on the vehicle's mechanical system. .The objective of the present paper is to evaluate the possible solutions of mobile robot bases and to build a kinematics model of WMR suitable for mobile manipulation tasks.

2.Holonomic and nonholonomic WMR (1/5)

The wheeled mobile robots (WMR) are divided into two basic types - holonomic and nonholonomic. Theoretically, the holonomic mechanical systems comprise links, that impose restrictions on the limb velocities, and after integration these restrictions can be reduced to restrictions only on the limb locations. When WMR do not impose restrictions on the motion velocities in the 2D (planar) solution they are called holonomic.

The holonomic WMR possess maximal number of degrees of freedom in the 2D (planar) solution h=3. In the field of the mobile robots the term holonomic is used as an abstract term for WMR with three degrees of freedom. Thus, every WMR with three degrees of freedom in plane is called a holonomic one.

2.Holonomic WMR (2/5)Various mechanisms are used as universal or omni wheels, orthogonal or ball wheels in order to achieve a holonomic motion. [5]. The holonomic WMR allow easier motion planning in a plane. A holonomic WMR is shown on Fig.1.

Fig.1. Holonomic WMR

2. Nonholonomic WMR (3/5)The nonholonomic mechanical systems comprise links restricting the system velocities; thus these restrictions cannot be integrated. In this way the nonholonomic WMR impose restrictions on the velocities of the motions in plane.Due to this reason the nonholonomic WMR possess less than three degrees of freedom in plane h < 3. They are simpler in construction and thus cheaper, with less controllable axes and ensure the necessary mobility in plane. Due to this reason a kinematics model of nonholonomic WMR with two degrees of freedom h=2 are derived in the present work . A nonholonomic WMR is shown on Fig.2.

Fig.2. Nonholonomic WMR.

2. Nonholonomic WMR (4/5)

X1

X

Y1 Y

Vcr

Vcl

VP

P

b

b

r

r r

ф

l

O

r C

Vc

X1

T

L

Z1

Fig.3. A general model of non-holonomic WMR.

In Fig. 3 a generalised model of a nonholonomic WMR with h=2 is presented. It includes two symmetrically allocated driving wheels with radii r. The nonholonomic WMR include a various number universal wheels for keeping up the balance in plane. This wheel is not driving one and it is not included in the kinematics model.. In the robot centre P is connected a local co-ordinate system PX1Y1 , where X1 is along the axis of symmetry, and Y1 is along the axis of the driving wheels. The angle between the axis X1 and the axis X of the immovable co-ordinate system OXY is denoted with ф. The distance between the driving wheels along the axis Y1 is 2b, and the angular velocities of the left and the right wheel are given , . lθ

2. Nonholonomic WMR (5/5)

X1

X

Y1 Y

Vcr

Vcl

VP

P

b

b

r

r r

ф

l

O

r C

Vc

X1

T

L

Z1

When one wheel is rolling on a straight line without slipping with angular velocity , its

centre is moving with velocity Vc. The velocity of the oscillate point T with the plane L is 0 and thus the equation (1) is fulfilled:

θ

0 rθVV cT (1)

This equation can be integrated and can be presented as a link among the angular and the linear position of the wheel.

When the wheel is rolling along a curved line the linear velocity of its centre Vc, in the base co-ordinate system OXY depends on the wheel orientation in the plane defined by the angle ф. These equations (2) can not be integrated in order to define relations only between the wheel positions. In the plane motion on the wheel velocities are imposed restrictions, thus the mobile devices from the type shown in Fig. 3, are called nonholonomic WMR0sin

0cos

φrθY

φrθX

c

c

(2)

l - θr = 0

3. Direct kinematics task of nonhoonomic WMR.(1/3)

X1

X

Y1 Y

Vcr

Vcl

VP

P

b

b

r

r r

ф

l

O

r C

Vc

X1

T

L

Z1

The nonholonomic mobile devices include two co-axial driving wheels, the velocities of their centres Vcr и Vcl are co-linear with the axis X1 of the local co-ordinate system PX1Y1. The velocity Vp of the centre P of the mobile platform is also co-linear with the axis X1. The plan of velocities of WMR in the plane OXY is presented in Fig.1. The following equations can be derived:

(4)

lc

rc VVb2

)VV(21V lc

rcp (3)

(4) Fig.3. A general model of non-holonomic WMR.

3. Direct kinematics task of nonhoonomic WMR.(2/3)

If we derive the upper equations along the axes of the base co-ordinate system OXY, where the velocity of the centre Vp is presented by the co-ordinates ,

and the velocities Vcr и Vcl of the right and the left wheel is defined with the help of (2)

,

thus equations are derived defining the kinematics of WMR in the base co-ordinate system

Tpp

0p ]Y;X[V

lr

lrp

lrp

b2r

b2r

sin2rsin2

rY

cos2rcos2

rX

(5)

0sin

0cos

φrθY

φrθX

c

c

(2)

3. Direct kinematics task of nonhoonomic WMR.(3/3)

If the velocity Vp of the centre of the mobile platform and

its velocity of rotation we combine in the vector:

T

pp ];Y;X[X

and the velocities of the driving wheels l,

we combine in the vector T

lr ];[

then the direct task of the kinematics of WMR is presented by the vector equation

SX

where

b2r;b2

r

sin2r;sin2

r

cos2r;cos2

r

S

(6)

(7)

(8)

(9)

4. Reverse kinematics task of nonhoonomic WMR.(1/3)

It is necessary to solve the reverse task of the kinematics in order to plan the robot motion and the robot control. In order to find a solution for the reverse task of the kinematics of WMR, the pseudo inverse matrix S+ can be used, because matrix S is not a quadratic one.

It is necessary to use an additional restriction on absolute parameters defined by the nonholonomic links of WMR because in this case the number of the input parameters (6) is bigger than the number of the output parameters (7):

XS

0XA

Tpp ];Y;X[X

(10)

(11)

4. Reverse kinematics task of nonhoonomic WMR.(2/3)

Matrixes S+ and A can be easily defined by using the equalities (3) and (4) presented in the form below:

lcp

rcp

VbV

VbV

(12)

when we define the upper equations along the axes of the local co-ordinate system PX1Y1. The velocity of the centre Vp is defined along the axes X1 and Y1 by means of consideration of their angle of rotation ф. In this way equations (12) along axis X1 present:

lpp

rpp

rbsinYcosX

rbsinYcosX

and along axis Y1 present:

0cosYsinX pp

(13)

(14)

4. Reverse kinematics task of nonhoonomic WMR.(3/3)

Equation (13) express the reverse task in kinematics from where for the matrix S+ we can derive:

Equality (14) expresses the restriction equation between the velocities of the

nonholohomic WMR including the matrix:

XS

b;sin;cos

b;sin;cos

r

1S

0XA

0;cos;sinA

(15)

(16)

5. Conclusion.The developed kinematics model of the nonholonomic WMR can be used for control creation in mobile tasks or in mobile manipulation tasks. Control can be based on the direct kinematics task by means of using equation (5). In this case the equilibrium of the given velocities of the wheels , guarantee motion along a straight line, and the difference between them defines robot rotation. This control mode belongs to a lower level and is more convenient for derivation of on-line tasks of motion. Control can be build up on the inverse kinematics task by using equation (10). In this case control models are derived including tracing a path considering the nonholonomic restrictions (11). These control schemes are more sophisticated, they can consider also the dynamics of WMR, they can include adaptiveness and robustness.

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