AN155
Equation 3
Common Mistakes
As demonstrated in the preceding section, the linear
velocity profile is not as simple as it would first appear.
The values stored in the linear velocity table must
closely follow the non-linear equation shown in
up and running do not use the proper relation for the
stepper motor table. This is a very common mistake that
is very easy to make.
The most common mistake is to have the step period
decrease linearly with the step number. For example,
one might have an initial step period of 256 timer ticks
and decrease the step period by one each time. This
results in a non-linear velocity that is increasing
hyperbolically as the step period approaches zero. Such
a profile will hardly move at first and then the velocity
will increase much too quickly.
The second most common mistake is to have the step
period decrease with the inverse of the step number.
This results in a velocity that is linear with respect to the
step number. But this ignores the fact that the step
period is constantly changing. The velocity should be
plotted against the cumulative time, not the step
number. If the velocity is plotted against the absolute
time, the resulting curve is a second order function. That
is, the velocity is increasing with the square of time. This
profile also starts out too slow and ends up accelerating
too fast.
2
1
-
n
= --
αt
n
2
Solving Equation 3 for time gives the results shown in
a linear acceleration profile. This would be useful if we
were working in absolute time and scheduling each
commutation point based on a cumulative count from
the beginning. However, we would like to use a relative
count for each step period.
Equation 4
t
n
=
2n
------
-
α
The definition of the step period from Equation 1 is used
with the results in Equation 4 to provide an equation for
the step period listed in Equation 5. The constant
acceleration term has been factored out and is called T
0
as defined in Equation 6. The value of T
0
will determine
the step period of the initial step with n equal to zero.
Thus, we can use a single table for the relationship and
let T
0
be a variable. This means one table can be used
with any stepper motor.
Equation 5
Linear-Velocity vs. Linear-Acceleration
Many engineers hold a preconceived notion that a forth-
order linear-acceleration profile will provide much better
dynamic performance than a linear velocity profile. A
linear-acceleration order profile has an acceleration that
is trapezoidal in nature and velocity shaped in an s-
curve.
A linear-acceleration profiler provides only marginally
better dynamic performance in some systems. Only a
few applications can actually benefit from a forth-order
profile. For example, a printer head driven by an elastic
band might benefit from the improved smoothness. The
linear-acceleration profiler will have a smoother
transition between the acceleration and slewing phases.
The maximum step rate is dictated by motor parameters
and will be the same in either case.
The linear velocity profile has several advantages over
the forth-order profile: The linear velocity profile can be
implemented using a single table. The step table is
fixed. It can use a single multiply function to provide a
variable acceleration. Using the table avoids having to
calculate complex functions like a square root.
Calculating the profile is very simple.
T
n
=
t
n
+
1
–
t
n
=
T
0
(
n
+
1
–
n
)
Equation 6
T
0
=
2
--
-
α
Rev. 1.0
7
SILABS [ SILICON LABORATORIES ]
