As represented by Hu and Huo, many nonholonomic mechanical systems can be described by the following dynamic equations [
And the nonholonomic kinematic constraints are as:
where, 
and 
are respectively the generalized configuration and the control input vectors, respectively; 
is the constraint force vector; 
is a positive definite matrix; 
is the term which may include centripetal and Coriolis forces; 
is a 
full rank transformation input matrix; 
is a 
full rank matrix associated with the constraints. Let 
i = 
is a set of smooth and linearly independent functions such that:
Considering 
as the distribution spanned by the vectors
, then from (2) it follows that
, that is, there exists an 
dimensional pseudo-velocity vector, 
such that:
where,
.
Differentiating (4) results in:
Substituting (5) into (1) and then pre-multiplying by 
gives:
Through pre-multiplying (6) by
, the nonholonomic mechanical system (1) and (2) reduces to:
where, 
is a 
matrix; 
is a 
matrix and
. Now, the proposed system (7) is used to dynamic modeling of the WMR.
For the purpose of dynamical modeling, 
and 
should be determined. The driving wheels of the considered WMR rotate by independent actuator motors. According to the schematic model in 
is the distance between the centroid, C of the robot and the connection center of the driving wheels. The WMR’s centroid in the global coordinate system with axes X-Y is indicated as
, and x-y stand for a local coordinate system which its origin coincides to C and the axes are fixed to the robot body as
The symbol, 
is the angle between x-axis and Xaxis representing the heading angle. 
denotes the velocity of the robot along x-axis and 
denotes the angular velocity. 
and 
are the angular velocities of the right and left driving wheels, respectively. 
and 
denote the length of driving axel and the radius of every driving wheel, respectively.
The position vector of the WMR, 
is defined as:
According to the recent research works [
The following rotation matrix transforming the veloc-
ity components between the global and local coordinate systems, play an important role in developing the kinematics of WMR:
Obtaining, 
in terms of 
from (12) and using (9) and (10), the nonholonomic constraint of the WMR is represented as:
According to (8) and (13), 
and
; and therefore, z is a two dimensional vector. Considering 
and 
as internal state variables gives:
Therefore, (4) can be constructed in the following form which is known the kinematic model of the WMR.
Now the following Lagrangian is considered for the WMR dynamic modeling [5,8].
where, 
is the total mass of the robot. 
is the moment of inertia around the axis crossing C and perpendicular to X-Y plane. The moment of inertia of driving wheels is shown by
. By applying the Lagrangian approach, an equation like (1) is achieved, therefore, by comparing it with (1), 
and 
can be determined. Then, using (6) and (7), 
and 
can be easily computed. Finally, the dynamical model of the WMR is obtained as:
is the input torque vector in which 
and 
are the produced torques by the right and left driving wheels, respectively.
which make the WMR to track a feasible trajectory with bounded errors. The three dimensional posture variables of the reference trajectory are considered as,
; the reference velocity and the acceleration vectors are derived by 
as, 
and, 
respectively. The real world robotic systems have inherent system perturbations such as parameter uncertainties and external disturbances. Therefore, dynamical equations of the WMR are represented as:
considers the uncertainty and disturbance effects in the dynamical model. It is assumed that 
is energy bounded and satisfies the uncertainty matching condition as:
and 
are the upper bounds of the perturbations.
and 
are positive fixed parameters. If 
is asymptotically stable, then 
and 
will converge to zero asymptotically. Because if 
then
. Therefore, if 
then 
and if 
then
. Therefore, the equilibrium state of 
is asymptotically stable. Similarly, if 
is asymptotically stable, 
and 
asymptotically converge to zero. Thus, if 
and 
become stabilized, the convergence of the WMR to the predetermined reference trajectory is guaranteed.
is the control law. Applying the control input (21) into the dynamic equation of WMR (18), the feedback-linearized dynamic equation is given as:
and 
which stabilize the sliding surfaces 
and 
are proposed as:
are greater than
, respectively; and
, 
are real positive constant values. Lyapunov’s direct method is used to prove the stability of 
and 
by imposing 
and
. Substituting 
and 
in (22) and (23), respectively, yields 
and 
as:
and 
from (25) and (26) in (28) results in the following negative definite,
.
and 
are asymptotically stable and therefore, the WMR will converge to the desired reference trajectories. It should be noted that due to using the sign term in the designed controller, the chattering phenomenon may occur when the posture state errors are negligible values. To weaken the unwanted chattering phenomenon, some continuous functions, for example a saturation term could be used to approximate the sign term.
. Furthermore, according to Figures 8 and 15, the bounded input torques of the DSMC even under measurement noises shows its significance for real implementations.