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Models for DC Motors

The chassis of a robot with three omnidirectional wheels

: Abstract

 This document describes how to model a DC motor, the
acceleration curve for the angular velocity, the acceleration due to PWM
.signals, and how our robot wheels accelerate

: Introduction
It is important to know how DC motors can accelerate a robot. The final velocity
of a DC motor depends on the voltage applied to it, the motor’s characteristics,
and the load on the wheels. Also,sometimes the wheels do not have enough
grip and when the motor rotates faster, the wheels begin slipping. Knowing the
form of the acceleration curve of a wheel rotating without slipping is useful for
.implementing controllers which provide traction control.
We consider DC motors with permanent magnets. The force provided by
the motors is a torque. The stall torque listed in DC motors data sheets is the
maximum torque produced by the motor when the rotator is not held and is
.prevented from rotating.
The rotator in DC motors rotates because the current flowing through a
coil produces a magnetic field repelled by permanent magnets mounted in the
armature. When the coil rotates brushes touching the coil contacts invert the
polarity of the voltage in such a way that the coil is repelled again, and so on.
When the motor is rotating, the voltage V applied to the motor is diminished
by the voltage E produced by the rotator (which behaves like a dynamo when
.rotating). The faster the rotator rotates, the larger the value of E. In general

where k is a constant, and ! is the angular velocity of the motor. The generated
voltage E increases proportionally to the angular velocity. When E comes close
.to V, the motor is rotating fast but the current flowing through the coil is lower.
This means that the force acting on the rotator is also lower. When eventually
V − E = 0, the current flowing through the coil drops to zero and there is no
.torque acting on the motor. The DC motor has reached its final velocity.
The torque on the rotator (the useful torque for a machine) is proportional
: to the current in the coil

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