Page 61 - Electric Drives and Electromechanical Systems

54 Electric Drives and Electromechanical Systems

acceleration, deceleration and at constant velocity are equal, Fig. 2.10B, the peak speed
and acceleration are given by,

3q m
N max ¼ (2.32)
2t m
3N max
a ¼ (2.33)
t m
In order to determine the power requirements of the drive system, the acceleration/
deceleration duty cycle, d, needs to be determined. If the dwell time between each move
is t d , then the duty cycle for a triangular profile is,

t m
d ¼ (2.34)
t m þ t d
and for the trapezoidal profile,
2t m
d ¼ (2.35)
3ðt m þ t d Þ
The general form of the above equations can also be used for linear motions, though
care must be taken to ensure consistency of the units.

nnn
Example 2.3
Compare the triangular and trapezoidal motion profiles for a rotary drive that is required to
move 90 degrees in 10 s.
1
Using a triangular profile, the peak speed required will 0.314 rad s 1 (3 rev min )
2
requiring an acceleration of 0.0628 rad s 2 (36 rev min ).
Using a trapezoidal profile, with one-third of the move at constant speed, the peak
1
speed required is 0.236 rad s 1 (2.5 rev min ), with an acceleration of 0.0708 rad s 2 (40
2
rev min ).
It can be noted that the peak speed is higher for a triangular motion. Even though the
acceleration is higher for the trapezoidal profile, it is applied for a shorter time period, hence
the energy dissipated in the motor will be lower than for triangular motion profile.
nnn

The motion profiles defined above, while satisfactory for many applications, result in
rapid changes of speed. In order to overcome this the motion trajectory can be defined as
a continuous polynomial, the load will be accelerating and decelerating continually to
follow the path specified, giving a smooth speed profile. If a cubic polynomial is used the
trajectory for a rotary application can be expressed as,
2
qðtÞ¼ a 0 þ a 1 t þ a 2 t þ a 3 t 3 (2.36)
The generation of the polynomial’s coefficients can be calculated from defined
parameters, typically the positions and speeds at the start and end of the move. This will
allow the joint’s velocity and accelerations to be determined as a function of time.