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A MATHEMATICAL MODEL OF A LEGO DIFFERENTIAL DRIVE

ROBOT

Ph.D. Stud. Eng. Gheorghe GÎLCĂ, Faculty of Automation, Computers and Electronics,

University of Craiova, gigi@robotics.ucv.ro

Prof. Ph.D. Eng. BÎZDOACĂ Nicu George, Faculty of Automation, Computers and

Electronics, University of Craiova, nicu@robotics.ucv.ro

Abstract: This paper details the development of a model for a mobile robot constructed from Lego®

Mindstorms®. The equations representing the dynamics and kinematics of the robot are derived. In

addition, the motors and wheels are represented in the model. The mobile robot is programmed in

graphical programming language NXT-G and can follow a black line without problems, even if the route

to achieve is difficult.

Keywords: Mobile robots, Nonlinear model, Mechanical equation, Driving force.

1. Introduction

Mobile robots are complex electromechanical devices that can be very difficult to

construct and control efficiently. The control problem of Wheeled Mobile Robots (WMRs) is

a topic of great research interest and has been studied extensively during the past few years. A

WMR is a typical nonholonomic system characterized by kinematic constraints that are not

integrable, i.e., the constraints cannot be written as time derivatives of some functions of the

generalized coordinates. In the literature, the research for control of WMRs has been centered

on three basic problems: trajectory tracking, path following and point stabilization. The goal

of trajectory tracking is to control the mobile robot in order to follow a reference trajectory.

Yu Hao et al. propose in their paper a fuzzy method for controlling the tracking of a

wheeled mobile robot. Also using Lagrange equations to express kinematic and dynamic

robot model [1]. Edison Orlando Cobos et al. develop in their work a model of traction based

on the simple friction Coulumb. The linear velocities of the wheel are used in calculating

these traction forces [2].

In the paper [3], the authors project a robot model equipped with a camera. The robot can

detect and track the object through real-time processing of images from the camera. A

Kalman filter is used for object tracking accuracy. Eka Maulana et al. presents in their work

the inverse kinematic model for a mobile robot with differential driving on two wheels. A

mobile robot orientation correction is made with closed loop control using a sensor matrix

applied to the follower line in front of the robot chassis [4].

Abhishek Jha and Manoj Kumar have proposed in their work a method based on the

odometry of two wheels in differential mode. Their method can estimate the estimated

relative position of the mobile robot wheels in relation to the start position [5]. For the

kinematic model, the authors use the Taylor series of second order. In the work [6] they

present two algorithms based on non iterative linearization with application in the trajectory

tracking of mobile robots. These two algorithms, extended Rauch–Tung–Striebel (ERTS) and

unscented Rauch–Tung–Striebel (URTS), compare the nonlinear model predictive control

with the iterative linear quadratic regulator controller and then approximates the inference

approaches.

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Sheelu Trees Mathewl et al. develop a control system for an inverted pendulum robot with

two wheels using the LEGO Mindstorm kit [7]. This robot has a rotating encoder and a

gyroscope sensor, the angular velocity of the body and the wheel rotation angle measurement

is available as an output.

In [8] it is presented a model of robot which has the control proportional with a variable

model reference to the second order derivatives, in which the movement of the robot is

adapted online. A trajectory learning algorithm based on sonar is experienced on the drive

differential LEGO NXT robot, which is equipped with position sensors [9].

2. Model of the Wheeled Mobile Robot

The robot used is constructed from Lego® and can be seen in Fig. 1. Lego® has been chosen

as it enables quick construction of the mobile robot. The robot is a two wheeled differential

drive robot, where each wheel is driven independently. Forward motion is produced by both

wheels being driven at the same rate, turning right is achieved by driving the left wheel at a

higher rate than the right wheel and viceversa for turning left.

Fig. 1: Lego differential drive robot.

2.1. Kinematics and dynamics

The WMR shown in Fig.2 is a typical example of a nonholonomic mechanical system. It

consists of two rear driving wheels mounted on the same axis and a passive front wheel. The

motion and orientation are achieved by the torques provided by the independent actuators,

e.g., DC motors of the rear wheels. OXY is the reference coordinate system; PX 'Y ' is the

coordinate system fixed to the mobile robot; P is the middle of the rear axis; Pc is the center of

mass of the robot body; d is the distance between P and Pc; 2b is the distance between the two

driving wheels, e.g., length of the rear axis; r is the radius of the wheel. The configuration of

the mobile robot can be described by five generalized coordinates: ,

where (x, y) are the coordinates of P in OXY . θ is the heading angle of the mobile robot.

and denote the angles of the right and left driving wheels, respectively. Under the

assumption that the wheels do not slip, there exist three constraints [10]:

(1)

(2)

, (3)

These constraints can be written in matrix form:

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, where

(4)

Fig. 2: Wheeled Mobile Robot and coordinate systems

The assembly kinematic energy of the various components of the mobile robot is given by:

(5)

, where mc is the mass of robot platforms without the driving wheels and the rotors of motors;

mw is the mass of each driving wheel plus the rotor of its motor; Ic, Im, Iw are the moment of

inertia of the body about the vertical axis through Pc, the driving wheel with motor rotor about

the wheel diameter, and the driving wheel with motor rotor about the wheel axis, respectively.

(xc , yc ) are the coordinates of Pc in OXY ; (xrw , yrw ) and (xlw , ylw ) are coordinates of right

and left driving wheels in OXY , respectively. We apply the Lagrange’s equations to derive

the dynamic equations of the mobile robot. The constraint forces added as input terms are

responsible for not allowing the wheels to slip sideways. The constrained dynamics can be

written as:

(6)

, where is the Lagrange multiplier vector corresponds to the constraint forces;

represents the externally applied forces including the torques provided by the independent

actuators and the viscous friction. Expressing (5) in terms of the generalized coordinates and

substituting the result into (6), we obtain the system equations:

(7)

, where is the inertia matrix; is the matrix of velocity-dependent forces.

is the input transformation matrix; represents the torque input of the

right and the left motor; is the viscous friction torque vector.

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2.2. Motor and Wheel Model

The robot model has two inputs: the force generated by each wheel. To calculate these forces

the actuators, in this case two Lego 71427 DC motors, need to be modelled along with the

tires that are being used. The motors are standard DC motors and as such the standard

equations of DC motors can be used to represent them. Equation (8) represents the

electromechanics of the motor [11]:

, (8)

, where R = 22Ω, L = 0.01H, Ke= 0.2367, I is the current (A), ω is the wheel angular velocity

(rad s-1

) and Va = input voltage (V).

The mechanical equation for the motor contains the friction term for the robot. The friction

equation is Equation (9):

, (9)

Where Ff is the frictional force generated, m is the mass of robot, represents the Friction

Coefficient, g is gravity, wheel_r is the radius of the wheel (m) and ω is the wheel angular

velocity (rad s-1

). The final term, ( ), has the effect of scaling the frictional term to

suit the current wheel velocity. The mechanical equation for the motors is given in Equation

(10):

(10)

Where dω/dt is the angular acceleration (rad s-2

), Kt=0.2367, bS= 2.1975e-5

, I is the current

(A), ω is the wheel angular velocity (rad s-1

), Ff is the frictional force, wheel_r is the radius of

the wheel (m) and J =2.8302e-4

kg m-2

.

The equation used to calculate the driving force from a rotating wheel is Equation (11):

, (11)

Where F is the force generated (N), T is the torque of the wheel (Nm) and wheel_r the radius

of the wheel (m). Since the wheel radius is known the torque needs to be calculated. Equation

(12) is used to calculate the torque:

, (12)

Where T is the torque of the wheel (Nm), Tmax is the current maximum torque that can be

generated, Equation (13), Tstall is the stall torque of the motor, ωabs max is the absolute

maximum angular velocity the motor can run at and ω is the current angular velocity.

, (13)

Where Vin = input voltage (V), Vmax is the maximum voltage that can be applied to the motor

and Tstall is the stall torque of the motor.

3. Programming and Implementation

We have implemented a differential mobile robot using Lego Mindstorm kit. Its structure was

presented in Fig. 1 and comprises: an intelligent brick, two engines, a light sensor, connection

cables and components for construction. Brick intelligence is the operational heart of the

system. It executes user programs and controls communication with sensors, actuators, with

PC or other NXT units. The two engines are intended to make the robot walk, DC motors are

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powered from 9V or 12V (short periods). Light sensor is to see variations of light for the

mobile robot to be able to move on the desired trajectory. The role of interconnection cables is

to connect actuators and sensors to central processing unit, these can support analog and

digital interface. For programming we used the minstorm lego nxt 2.0 software. In Fig. 3

shows the scheme for programming of a differential mobile robot so it can follow a route of

black line. The functioning of the scheme is as follows: Light sensor is active. It works as a

breaker in such a way that: if the light variation is less than 40 it goes one way and the robot

is driven by one engine plugged in port B while the second engine is stopped; if the variation

is greater than 40 it will go the second way and the robot is driven by the second engine is

plugged in port C, while the first motor is stopped. The process is repeated indefinitely due to

the repetition loop.

Fig. 3: Diagram for programming of the differential mobile robot

In Fig. 4 shows the results of mobile robot programmed by us:

Fig. 4: Followed trajectory by mobile robot

4. CONCLUSIONS

The differential mobile robot proposed by us can smoothly track any given route. The

Kinematic and dynamic mathematical model can be applied to any robot in this category. The

modeling engine and wheels are important in driving differential. The robot programming is

made in the language of graphical programming NXT-G, which is simple and effective. In

future we want to implement this robot a video camera and a system for the recognition of

human emotions.

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