As an example, consider a machine starting at feedrate $f_0$ and then performing a very long linear move at a constant feedrate $f_1$. For this example, If $f(t)$ is the machine feedrate over time and $a$ is a constant acceleration, this profile would be defined mathematically as follows:

\begin{equation}

f(t) = \begin{cases}

f_0 & \mbox{if } t < t_0 \\

f_0 + a (t - t_0) \hspace{0.5cm} & \mbox{if } 0 \leq t - t_0 \leq \frac{f_1-f_0}{a} \\

f_1 & \mbox{otherwise}

\end{cases}

\end{equation}

Graphically, here's a plot of the feedrate over time for a move starting at rest at $t=0$ and accelerating up to a feedrate of 2 over one unit of time:

The problem with a constant acceleration profile is that there are sharp kinks in the feedrate plotted over time. These kinks imply instantaneous changes in acceleration, which in turn imply infinite forces for infinitely short periods of time. Of course, there is no mechanical way to produce these forces, so what actually happens is the machine overshoots very slightly and averages the forces out over a short time. For low feedrates with a light machine this actually works okay, but for a heavy machine at high-feedrates, the overshoot can be more than a motor step which causes the machine to lose position. In an open-loop design, once the machine loses position, it never recovers and in all likelihood, the part is ruined.

There are a few ways to address this problem:

- Use lower feedrates
- Use higher torque motors
- Use a closed-loop control scheme, e.g. with encoders on the motors
- Make the acceleration smooth

In order to smoothly transition between accelerations we can simply use a different curve to interpolate the feedrates. The conditions needed are that the feedrates match the desired rates at the beginning and end of the curve and that the slope of the feedrate curves (i.e. the acceleration) is zero at the endpoints. In between the endpoints we want the curve to be smooth.

The one of the simplest classes of functions that meet these requirements are cubic polynomials. These are defined by four coefficients $a$, $b$, $c$ and $d$ using the following equation, where $\tau$ is the fraction of the total time spent accelerating:

\begin{equation}

f(\tau) = a \tau^3 + b \tau^2 + c \tau + d

\end{equation}

We now want to solve for the coefficients needed to reproduce the move. There are four coefficients so we need four equations. Two come from the requirement that we match the feedrates at the curve endpoints:

\begin{eqnarray}

f(\tau=0) = a 0^3 + b 0^2 + c 0 + d &=& f_0 \\

f(\tau=1) = a 1^3 + b 1^2 + c 1 + d &=& f_1

\end{eqnarray}

From these, we see that $d=f_0$ and $a+b+c=f_1-f_0$. The remaining two equations can be found using the requirements that the slope of the feedrate curve is zero at the endpoints. To enforce these constraints we need the derivative of the cubic function:

\begin{equation}

f'(\tau) = 3 a \tau^2 + 2 b \tau + c

\end{equation}

The constraints can now be enforced by requiring that:

\begin{eqnarray}

f'(\tau=0) = 3 a 0^2 + 2 b 0 + c &=& 0 \\

f'(\tau=1) = 3 a 1^2 + 2 b 1 + c &=& 0

\end{eqnarray}

These equations make it clear that $c=0$ and $3 a + 2 b = 0$. Combining these with the previous conditions leaves two equations and two unknowns:

\begin{eqnarray}

a + b &=& f_1 - f_0 \\

3 a + 2b &=& 0

\end{eqnarray}

So $a = -\frac{2 b}{3}$ which means that $b = 3 (f_1-f_0)$ and $a = -2 (f_1 - f_0)$. This gives the following equation for the interpolating curve:

\begin{equation}

f(\tau) = -2(f_1-f_0)\tau^3 + 3(f_1-f_0)\tau^2 + f_0

\end{equation}

The only remaining thing is to define $\tau$ in terms of $t$. This is a simple linear interpolation from the start of the acceleration $t_0$ to the end of the acceleration $t_1=\frac{f_1-f_0}{a}$:

\begin{equation}

\tau = \frac{t-t_0}{t_1-t_0} = \frac{t-t_0}{\frac{f_1-f_0}{a}-t_0}

\end{equation}

Plotting this for the same parameters as before gives a smooth, kink-free curve that considerably reduces the time-rate-of-change of acceleration:

In the post-to-come I will demonstrate applying this to a real stepper motor being driven aggressively. Although seemingly complicated, for a cost of only a few operations per step, it is possible to switch from the linear acceleration profile to the cubic one derived here and get considerably smoother operation.

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