Testing sinusoidal S-curves

-
My main goal here was to keep a simple mechanism to be easy to calculate and able to be included into 8-bit firmware like Marlin. Which by the way already has support for s-curves for quite a while (at least on this mod for Ultimaker).

So the goal is to have a function f(t) that will map the speed at each given moment but in a way that the maximum speed will be reached at the same time as it would using a uniform acceleration, so the rest of the math and motion planning is not affected. In doing so, the peak acceleration is going to be higher than that when using uniform acceleration. However, if that is a problem it is ok to reduce the maximum acceleration so the system can work flawlessly.

I have tested a couple of functions, both based on sinusoidal functions, the former is:
f(t)=V*(1-cos(PI*(t/t1))/2 (where V is max speed and t1 marks the end of the S-curve ramping up). It is the dotted-green curve below. The shape of the acceleration for this speed pattern is the dotted-pink curve.


The second candidate is the solid-red curve, that is smoother on the edges but has a higher slope in the middle. That is the function f(t)=V*(2*PI*t/t1-sin(2*PI*t/t1))/2/PI. The shape of the acceleration for this speed pattern is the solid-blue curve. 

So I did some tests on a sample one-axis machine, running dcservo and moving a bit of liquid that should give us an idea of the vibration each one of the motion patterns will create (I am using the ramp of the trapezoidal motion as control case here).

The following video shows the result, running a random sequence of movements, all with the same maximum speed. Acceleration is modulated in a way that for each case, the same amount of time (t1) is needed for reaching the desired maximum speed.

 

While the code is not thoroughly tested but more alpha code, it is available here.



Comments

Khalid Khattak said…
Wonderful work...
5:20 am
Post a Comment

Popular posts from this blog

VFD control with Arduino using RS485 link

-
Image
Variable Frequency Drives ( VFD ) allow the control of spindles so speed can accurately be controlled and a detailed acceleration profile for the spindle and reverse rotation can all be handled. In essence a VFD is an three phase inverter for three-phase AC motor. I am using a popular (I mean cheap) Chinese VFD and though the reference manual is not great, I could see there is a built-in RS-485 port. I usually control the start, stop and speed selection using the keyboard on the unit but I thought it will be more useful if I could control everything from the same Arduino is doing our CNC table control. Some cheap RS-485 off eBay and some lines of code later, I can start, stop and change the speed from an Arduino. What a cool thing to have! Some configuration of your VFD are needed before you can use it like that.  You need to set PD163=1 (I am using address 1 in the code). PD=164=1 (for setting serial to 9600bps) and PD165=0 (for using ASCII and 8N1 character format)
Read more

How to get sinusoidal s-curve for a stepper motor

-
Image
After some experimentation, and delving a bit into the math of motion, I realized there can be a significant difference on the motion of a machine by just making small changes. A common and popular approach to smooth one axis of motion is to use a trapezoidal speed pattern, where acceleration and deceleration are uniformly accelerated movements (aka constant acceleration motion) separated by a portion happening at cruising speed. The problem comes with the fact that, while that approach is simple to understand and to code, it does contain sudden changes in the acceleration, that show as spikes on the acceleration derivative, called jerk. These sudden changes in the acceleration translate into undesired vibration in our machines. Most of that can be eliminated by selecting a motion pattern that does not contain sudden changes of acceleration. If we chose a mathematical function for our speed whose second derivative is continuous, then we reduce system vibration quite a lot. One ma
Read more

Stepper motor step signal timing calculation

-
Image
While we use stepper motors in many devices, the timing of the step signals is not always well understood. Most of us can see how a fixed speed is a bad idea unless the speed of motion is really slow, but it is fixed speed what is really to generate with simple code. You just can use the Arduino Blink example to send a fixed frequency of pulses to a stepper motor driver. If too slow, just reduce the delay value. But for most real-life applications variable speed controls will perform better. These systems will speed up the motor to a given maximum speed and it will slow it down to stop at the desired step count. But understanding the math behind this process is sometimes not obvious. Let us start with the simple case of the so-called trapezoidal speed motion profile. Here motion takes place in three phases: acceleration, constant speed (cruising) and deceleration. But the point is how can our code create such a behavior. Stepper motor drivers usually accept two signals to control
Read more