The kinematic model of a mobile robot describes its motion without considering forces and torques. For a differential drive robotic vehicle, which consists of two independently controlled wheels and a free caster wheel, the motion is defined by the velocities of the two wheels and the robot’s orientation relative to a global reference frame. Kinematic modeling is essential for path planning, localization, and trajectory tracking in mobile robotics [31].
Coordinate System and Motion Representation
The position of the robot is represented in a global coordinate frame(X, Y, θ), where:
• X, Y define the position of the robot’s center in Cartesian space.
• θis the robot’s orientation (heading angle) relative to the x-axis.
The movement of the robot is controlled by the linear and angular velocities of the wheels. The forward velocityvand angular velocityωdefine the robot’s motion equations as:
X˙ =vcosθ Y˙ =vsinθ θ˙=ω
where:
• X,˙ Y˙ represent the rate of change of the robot’s position.
• θ˙represents the rate of change of the heading angle.
Wheel Kinematics and Velocity Equations
Each wheel of the differential drive robot contributes to its movement. Ifωl andωr denote the angular velocities of theleftandrightwheels, respectively, andris the wheel radius, then the linear velocities of each wheel are:
vl=rωl, vr=rωr
The overall forward velocityvand angular velocityωof the robot are given by:
v= r
2(ωr+ωl)
ω= r
L(ωr−ωl) whereLis the distance between the two wheels.
These equations describe the instantaneous motion of the robot based on wheel speeds. They are funda- mental in designing control algorithms that ensure smooth trajectory tracking and precise movement [32].
Special Cases of Motion
Depending on the velocities of the wheels, the robot exhibits different movement behaviors:
• Straight-line motion:Ifωr=ωl, the robot moves forward or backward without rotation (ω= 0).
• Pure rotation: Ifωr=−ωl, the robot rotates around its center without translation.
• Curved trajectory:Ifωr̸=ωl, the robot follows a curved path. The radius of curvatureRis given by:
R= L 2
ωr+ωl
ωr−ωl
By controlling the wheel speeds, the robot can follow predefined paths, making this model crucial for trajectory planning and motion control systems in autonomous navigation [36].
Inverse Kinematics for Motion Planning
In real-world applications, rather than controlling the wheel velocities directly, we often specify the desired trajectory of the robot in terms ofvandω. Using the inverse kinematic equations, we can determine the required wheel velocities:
ωr=2v+ωL
2r , ωl= 2v−ωL 2r
These equations allow the robot to track a given trajectory by computing appropriate wheel commands. This inverse kinematic approach is widely used in autonomous mobile robots, ensuring smooth transitions between motion states [37].
3.2.2 Dynamic Model of DC Motor
While the kinematic model provides a high-level description of motion, the dynamic model considers forces and torques, enabling the design of precise motor control strategies.
Electrical Model of DC Motor
The motion of each wheel is driven by abrushed DC motor, whose behavior is described using an equivalent electrical circuit. Applying Kirchhoff’s Voltage Law, the electrical equation of the motor is:
V =Ldi
dt+Ri+Eb where:
• V is the applied voltage.
• LandRare the inductance and resistance of the motor windings.
• Ebis theback electromotive force (back EMF), which is proportional to the motor’s rotational speed:
Eb=Keω
whereKeis the motor’s back EMF constant andωis the angular velocity of the motor shaft [33].
Mechanical Model of DC Motor
The torqueτproduced by the motor is proportional to the currenti:
τ=Kti
whereKtis the motor torque constant. The equation of motion for the rotor is given by:
Jdω
dt +Bω=τ where:
• J is the moment of inertia of the rotor.
• Bis the damping coefficient accounting for friction losses.
These equations describe how electrical inputs (voltage, current) are converted into mechanical motion (speed, torque), forming the basis for motor control design in robotic applications [38].
Motor Control Using PWM and H-Brigde
Since direct voltage control is inefficient,Pulse Width Modulation (PWM)is used to adjust motor speed.
The average voltageVavgapplied to the motor is:
Vavg=DãVsupply
whereDis theduty cycle(percentage of the time the signal is high).
To enable bidirectional motion, an H-Bridge circuit is employed. By toggling the transistors in the H-Bridge, the direction of the motor current can be reversed, allowing the robot to move forward and backward seamlessly.
This technique is widely used in embedded motor controllers like L298N and L293D, ensuring efficient and precise speed control [39].
Conclusion
Thekinematic modelprovides a high-level representation of robot movement, while thedynamic model enables precise control of motor behavior. These mathematical models form the foundation for designingmo- tion planning algorithms, trajectory tracking controllers, and real-time motor actuation systems. The next section exploressensor fusion techniquesused to enhance motion accuracy and obstacle avoidance ca- pabilities.