Journal of Robotics, Networking and Artificial Life, Vol. 1, No. 1 (June 2014), 12-16

Variable-poled Tracking Control of a Two-wheeled Mobile
Robot Using Differential Flatness
Liming Chen, Yingmin Jia

The Seventh Research Division, Beihang University (BUAA),
Beijing, 100191, China
E-mails: clmtest@126.com, ymjia@buaa.edu.cn

Abstract
This paper investigates the tracking control of a two-wheeled mobile robot in both kinematic and dynamic models.
Differential flatness and PD-spectral theory are used for controller design. Based on differential flatness, theoriginal system is transformed via a state prolongation and a state transformation into a normal form to apply
feedback linearization. Then using PD-spectral theory, variable poles of tracking error dynamics are assigned torealize the stability of trajectory tracking. Simulation results are presented to demonstrate the effectiveness of theproposed method.
Keywords: mobile robot, nonlinear control, trajectory tracking, differential flatness

1. Introduction
Wheeled mobile robots have been proven to be one of
the most active areas of research since they are much
useful in varieties of applications ranging fromindustrial settings, to military systems, to home robotics,
etc. One of the main lines of research is the trajectory
tracking problem, which is concerned with driving amobile robot as close as possible to a desired explicit
trajectory. And the tracking control approaches include
backstepping,1 sliding mode control,2 linearization,3neural network-based control,4 fuzzy control5 anddifferential flatness-based control6-8 which is also usedin this paper.
Differential flatness has been introduced by Fliess etal.9 It is a very useful tool for nonlinear controller
design. Roughly speaking, a system is differentially flat,if there exists variables of the same dimension as inputs,
called flat output, such that states and inputs can bealgebraically expressed in terms of flat output and its
derivatives. Moreover, this mapping is invertible, andthe system is equivalent to a linear one. If the desired
trajectory of flat output is given, then by performing a
feedback linearization and designing a time invariant
controller for the linearized system around the desired
trajectory, stable tracking error dynamics are achieved.
Kinematic model and dynamic model of a two-wheeledmobile robot have been proven to be differentially flatby choosing the center position of the wheel axle of therobot as the flat output. In Refs. 7, 8, controllers aredesigned in this scheme. However, in these controllers,
the parameters of the tracking error dynamics areconstant, which means that the convergence speed oftracking error is fixed, and this brings limitation for thismethod.
In this paper, the use of PD-spectral theory of linear
time-varying (LTV) systems is proposed for tracking
controller design. The PD-spectral theory has beendeveloped by Zhu,10 which can be seen as a naturalextension of the conventional eigenvalue-eigenvector
theory for linear time-invariant (LTI) systems. After thestate transformation based on differential flatness, thePublished by Atlantis Press
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12L. Chen, Y. Jia

application of PD-spectral theory becomes much easier.
By assigning time-varying “poles”, more generalized
tracking error dynamics can be obtained. And the poles
can be changed at any time as we want.
The rest of this paper is organized as follows: In
section 2, the kinematic model and dynamic model of atwo-wheeled mobile robot are derived and analyzed
with differential flatness. Section 3 presents the designof control law in both models. Simulation results areshown in section 4. Finally, concluding remarks aregiven in section 5.

center position of the wheel axle ( x, y ) as the flat

2. Models of Two-wheeled Mobile Robot withDifferential Flatness(1) need to be extended to 4 dimensions. Apply one
prolongation of v by considering it as an additional
state, then the extended system is given byx  v cos  , v  u1(2)y  v sin  ,   u2.1. Kinematic modeloutputs ( z1 , z2 ) .
With the chosen flat outputs, the states can bez
expressed as x  z1 , y  z2 ,   arctan 2 , and the twoz1inputs can be written as v z12  z 22 ,  z1 z 2  
z1 z 2 .
z12  z 22

It can be noticed that the expression of state  contains
the first order derivatives of both the flat outputs, z1 andz2 . According to differential flatness theory, the system

2

where u1 and u2   are new inputs of the extendedsystem. The new inputs can be calculated asz z  z zz z  z z
u1  1 1 2 2 , u2  1 22 12 2 .
22z1  z2z  z1(3)2The extended system (2) can be transformed via a statetransformation [ x, y, v, ]T  [ z1 , z2 , z1 , z2 ]T into anormal form
Fig.1. The configuration of a two-wheeled mobile robot withno slip.

Fig. 1 shows that the robot's configuration in Cartesian
coordinates is given by q  [ x, y, ]T where ( x, y ) is thecoordinates of the center of the wheel axle and  is theheading angle of the robot. With the assumption of noslip condition at the wheel contact points, the velocity of
the wheel centers are parallel to the heading orientation.
Then the kinematic model can be written as cos  0 v
q  S (q) v   sin  0   (1) 01   
where, v is the heading speed and  turning speed.
To outline how the kinematic model of the twowheeled mobile robot is differentially flat, we need toselect suitable flat outputs and express all state variables
and inputs in terms of the flat outputs and theirderivatives. The dimensions of flat outputs should beequal to that of the inputs. Here, we can choose thezzz1  u1 cos(arctan 2 )  u2 z12  z22 sin(arctan 2 )z1z1zzz2  u1 sin(arctan 2 )  u2 z12  z22 cos(arctan 2 ).
z1z1(4)Since the expressions of new inputs u1 and u2 in (3)are derived from the system equations (2), the statetransformation ensures that (3) satisfies the equations(4). By replacing z1 and z2 in (3) with u x and u y ,respectively:u1 z1u x  z2 u yz12  z22, u2 z1u y  u x z2z12  z22

(5)where u x and u y are parameters to be designed, andsubstituting (5) into the system equations (4), one getsz1  u x , z2  u y .
(6)
Actually, if only the states of the original system (2),[ x, y, v,  ]T , can be estimated, using the relationship ofstate transformation, the inputs (5) can be rewritten as
u1  u x cos   u y sin  , u2  (u y cos   u x sin  ) / v. (7)Published by Atlantis Press
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Wheeled Robot’s Tracking Control

2.2. Dynamic model[ x, y, v, , u1 ,  ]T  [ z1 , z2 , z1 , z2 , z1 , z2 ]T

By ignoring the mass of the wheels, the equations ofmotion can be derived using Euler-Lagrange method as  C (q, q )q  E (q)  C T (q)M (q)q(8)calculate the third order derivatives of the flat outputs
directly using (12) asz1  u (13) C  D 1
 z2 u2 wheremM  0  d m s in 0C  00

0000
mdm cos  d m  c o s   d m  s in   ,0

 d m s in   d m c o s   ,    r 
 l d 2 m  I E  cos 1 s in r  b(9)cos  s in   .
 b and  the constraint force.
  Sv  Sv . Then
By differentiating (1), one gets q
 into (8), pre-multiplying by S T , andby substituting qusing the property S T C T  0 , one can havev  ( S T MS ) 1 S T ( MS  CS ) v  ( S T MS ) 1 S T E . (10)This can finally be calculated as v  A( v)  Bτ , where1


mr.
b


r ( d 2 m  I ) After introducing an input transformation u  [u1 , u2 ]T A( v )  Bτ , the dynamic model can be written asq  S (q) v, v  u.

(11)Here, we can also choose the center position of thewheel axle ( x, y ) as the flat outputs ( z1 , z2 ) and then allstate variables and inputs can be expressed in terms of
the flat outputs and their derivatives. The expression ofstate  contains the second order derivatives of boththe flat outputs, z1 and z2 , which means that the system(11) need to be extended to 6 dimensions so that it canbe transformed into a normal form via a statetransformation. On applying one prolongation of u1 , theextended system is given byx  v cos v  u1y  v sin    u1  u1

  u2Now,wewhere 2u  sin   v 2 cos  cos  v sin  C 1
,D  
 . (14)2
 sin  v cos  
 2u1 cos   v sin  
z1 and 
z2 with u x and u y , respectively,
By replacing 
this yields

Here, m is the robot mass, I the moment of inertia of
the robot about its center of mass, d the distance
between the center of mass and the center of the wheel
axle, r the wheel radius, b half distance between thetwo wheels,  l and  r the motor torques on the wheels,1 d 2mr
, B A( v )   dm v b d 2 m  I  r ( d 2 m  I )

.

(12)

where u1 and u2  u2 are new inputs of the extendedu
 u1 1  x u   D ( u   C ).
 2
 y

(15)

3. Design of Control Law
3.1. Introduction of PD-spectral theory10

Consider SISO LTV systems represented by the n thorder scalar LTV dynamical systems of the form:y ( n )   n (t ) y ( n 1)     2 (t ) y  1 (t ) y  0. (16)
It can be conveniently represented as D { y}  0 using
the scalar polynomial differential operator (SPDO)
D   n   n (t ) n 1     2 (t )  1 (t ) (17)where   d / dt is the derivative operator. The
factorization of SPDO can be represented asD  (  n (t ))  (  2 (t ))(  1 (t )) (18)where a collection {k (t )}nk 1 is called a series Dspectrum(SD-spectrum) for D and an n-parameterfamily { k (t )  1, k (t )}nk 1 is called a parallel Dspectrum(PD-spectrum) for D , where 1, k (t ) are nparticular solutions for 1 (t ) satisfying some nonlinear
independentconstrains.

exp(   k (t )dt )}nk 1Actually,{ yk (t ) constitutes a fundamental set ofsolutions to D { y}  0 .
ThesolutiontoD { y}  0is

uniformly

asymptotically stable if(i) all PD-eigenvalues are of polynomial order orslower, that is, an integer m  0 exists such that
 (t )
lim k m  0 ;t  tsystem. The state transformation can be written as

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L. Chen, Y. Jia(ii) the extended means of real parts of PD-eigenvalues,
1 t0 T
that is em(Rek (t))  lim  Rek (t)dt , are allT  T t0negative.

 1V3  det  1 12  11

2
 22   2
 . and then control32   3 1

3gains are obtained by 1x (t )  3 2 1  3 1  2 1 

3.2. Control law of kinematic model2 1  1 ,  2 x (t )  3 2  3 1  2 1  2  21 ,  3 x (t )If the desired trajectory of flat outputs ( z1 , z2 ) are given 3  2  1 . 1 y (t ) ,  2 y (t ) and  3 y (t ) can beby ( z1d (t ), z2 d (t )) , tracking error can be introduced as

designed in the same way.

e  [e1 , e11 , e2 , e21 ]  [ z1  z1d , z1  z1d , z2  z2 d , z2  z2 d ] .

4. Simulation Result

T

T

Using (6), parameters u x and u y can be designed asu x  z1d  1x (t )e1   2 x (t )e11

The desired trajectory is given by z1d (t )  3.75 (19)u y  z2 d  1 y (t )e2   2 y (t )e21

104 t 3  1.125  102 t 2  0.15t (m) , z2 d (t )  3  104 t 3 
9  103 t 2 (m) , over t  [0, 20](s) .

which yields

e1   2 x (t )e1  1x (t )e1  0e2   2 y (t )e2  1 y (t )e2  0.

(20)

PD-spectral theory is used to design these time-varying
control gains to ensure the error dynamics to beuniformly asymptotically stable.
First appropriate time-varying PD-eigenvalues
which satisfy (i) and (ii) are designed as 1 and  2 .

In the kinematic model, initial states are set as[ x(0), y (0), v(0),  (0)]  [ 0.1, 0.1, 0.05, 0.5] . It should
be pointed out that the additional state v is in thecontroller. PD-eigenvalues are selected as 1  (1 0.25t )  (1  0.25t )i ,  2  (1  0.25t )  (1  0.25t )i . Fig.
2(a) shows that the mobile robot gradually converges toThen corresponding SD-eigenvalues 1 and 2 can becalculated as 1  1 , 2   2 weobtain 2  1. Using (18), 2  11x (t )  2 1  1  1  2 

 2 x (t )  2  1   1   2 

 2  1
 2  1.

1  2   2 1 2  1

1 y (t ),andFig.2. Desired trajectory and tracking trajectory: (a)Kinematic
model. (b)Dynamic model.

 2 y (t ) can be designed in the same way.
3.3. Control law of dynamic model

Similarly, if the desired trajectory of flat outputs are
and tracking error
given by ( z1d (t ), z2 d (t ))e  [e1 , e11 , e12 , e2 , e21 , e22 ]T  [ z1  z1d , z1  z1d , z1  
z1d , z2

 z2 d , z2  z2 d , z2  
z2 d ]T , then parameters u x and u y

can be designed asu x  z1d  1x (t )e1   2 x (t )e11   3 x (t )e12u y  z2 d  1 y (t )e2   2 y (t )e21   3 y (t )e22 .

(21)

By designing appropriate PD-eigenvalues 1 ,  2 and3 , SD-eigenvalues can be calculated as 1  1 ,2   2 V2V2

V3V2,  3   3  V 3  V 2 where V2   2  1 ,Fig.3. Kinematic model: (a)Tracking error e1 . (b)Trackingerror e2 . (c)Heading speed v . (d)Turning speed  .

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Wheeled Robot’s Tracking Control

Acknowledgements

This work was supported by the National Basic Research
ProgramofChina(973Program:
2012CB821200, 2012CB821201) and the NSFC
(61134005, 61221061, 61327807).
References

Fig.4. Dynamic model: (a)Tracking error e1 . (b)Tracking
error e2 . (c)Transformed input u1 . (d)Transformed input u2 .

the desired trajectory and finally moves along it. Fig. 3
demonstrates the tracking errors e1 , e2 which converge
to zero and the control inputs v and  .
In the dynamic model, [ x(0), y (0), v(0),  (0), u1 (0),(0)]  [0.1,0.1,0.05, 0.5,0,0] . Controller is designedwith PD-eigenvalues 1  (1  0.25t )  (10.25t )i ,2  (1  0.25t )  (1 0.25t )i , 3  2  0.25t . Fig. 2(b)shows the tracking trajectory and Fig. 4 depicts thetracking errors and the transformed inputs u1 and u2 .
5. Conclusion

In this paper, we have presented a novel method for
trajectory tracking control of a two-wheeled mobile
robot in its kinematic model and dynamic model. Basedon differential flatness, the systems can be transformed
into normal forms to use feedback linearization. The
application of PD-spectral theory ensures the stability ofthe tracking error dynamics and establishes adjustable
poles to change as we want.

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