TRAJECTORY TRACKING CONTROL FOR 4 WHEEL SKID-STEERING MOBILE ROBOT

By applying a nonholonomic constraints and Lagrange equation for nonholonomic system, a method is given to model and control the 4-wheel skid-steering mobile robot which tracks a given trajectory. First at all, a fundamental of nonholonomic system is introduced. Next, the skid steering robot’s kinematic model and dynamic model are considered. To control the robot tracking a trajectory, a new algorithm is given by applying feedback linearization and PD control. In addition, simulation results show the good performance in tracking trajectories.


INTRODUCTION
The skid steering robot is considered as allterrain vehicle, and has many advantages than other off-road robots, for example, a high maneuverability, high-power, an ability of working in hard environmental conditions but the mechanism is quite simple. The following figure and table show major steering types and a steering system evaluation [1].  Table 1

. A steering system evaluation
The skid steering robot is navigated by the angular velocity difference between left wheels and right wheels [2]. Because of lateral skidding, velocity constraints occurring in skid steering robot are quite different from the ones met in other mobile platforms wheels are not supposed to skid. An example for this steering type is ATRV-J robot designed by Irobot company.
Kozlowski extended new time differentiable and time-varying control scheme based on the strategy of forcing some transformed states to track an exogenous exponentially decaying signal produced by a tunable oscillator [6], [7].
In this paper, a new control algorithm based on feedback linearization and PD control is presented. It allows us to control a reference point fixing in the 4 wheel skid steering mobile robot tracks a given trajectory.

Trang 85
Where q is the n-dimensional generalized coordinates A(q) is an m x n dimensional matrix Because the constraints are assumed to be nonholonomic, (1) is not integrable. It will be assumed that these constraints are independent.
In another words, A(q) has rank m.
Using the vector λ of Lagrange multiplier, the equations of motion of nonholonomically constrained systems are governed by: Where: M(q) is the n x n dimensional positive definite inertia matrix. It has been established that nonholonomic system described by the constraint equation (1) and the motion equation (2). [8]   The notation is shown in fig. 2, 3.

Kinematic model
Select the inertial frame (COM l where COM is center of mass. Let (X, Y, Z) to be robot's barycentric coordinates in the world frame,  We have: The i-th wheel rotates with an angular velocity ( ) i t ω ,where i=1;2;3;4.
The longitudinal velocity can be obtained: In contrast to most wheeled mobile robot, the lateral velocity of the skid steering robot iy v is generally nonzero.
The radius vector to the local frame from the instantaneous center of rotation (IRC). Thus: Coordinates of ICR in the local frames: Otherwise, from the figure 4 we have: Assuming that 1 So, let L ω , R ω be respectively angular velocities of lefts and right wheels. We can write: 1 .
Combining (10) and (11), a control input at kinematic level is defined as: To complete the kinematic model, nonholonomic constraint is considered.
From (6), the velocity constraint characterized by: The kinematic equation of the robot is obtained: Where S is the following matrix ir ir os sin Wheel forces are depicted in Fig.6 The active force is obtained .

Dynamic model
Where m denotes the robot mass and g is the gravity acceleration. Using the symmetry along the longitudinal midline, we obtain The friction acting one wheel is obtained:  It is assumed that the potential energy of the robot 0 ∏ = because of the planar motion.
Neglecting the energy of rotating wheels, the kinetic energy of this robot can be rewritten: Considering the forces causing the dissipation of energy: Consequently, the active force generated by actuators can be calculated in the inertial frame as follow: The active torque around the center of mass is obtained: The vector of active forces has the following form: The term τ is defined by: We have: ( ). .

Operational Constraint
Let o x be an arbitrary constant which The constraint equation (13) Combining (15)  . .
This state equation can be further simplified as: where os sin sin os sin os os sin From (58) we get: By using equations (54), (55), (61), (62), a new algorithm has been presented. It is easy to control the angular velocities of wheels in other that a skid steering robot tracks a given trajectory.

SIMULATION RESULTS
To  Similarly, the reference point's trajectory quickly converges to the given trajectory.

CONCLUSION
In this paper, a new algorithm of trajectory