scurve speed eval

Show how s-curve speed type can be obtained.

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• Description
• Performance considerations
• Example
• Prerequisites
• Quickstart
• How this project was built
• References

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Description

For symbolic calculus AngouriMath library was used.

For preliminary analysis sympy with this test was used.

• lets name

  • t (time)
  • x (distance)
  • d (total duration accel/decel)
  • s (final speed)

• suppose [0,d/2] the time for acceleration while [d/2,d] the time for deceleration to produce a target final speed s starting from speed=0 and pos=0

• from base acceleration function 1-cos(t)

• expanding the period domain from [0,2pi] to [0,d/2] through this subst

• stretching the accel function so that its integral, the speed, achieve targetspeed

• acceleration results as this

👉 hit F5 if math not rendered

$$ \large accel(t)=\frac{2\cdot s}{d}\cdot\left(1-\cos\left(\frac{4\cdot\pi\cdot t}{d}\right)\right) $$

• positive max accel value is at midpoint ( t=1/4d because accel positive in [0,d/2] ) thus above can be simplified into

$$ \large maxAccel=\frac{4\cdot s}{d} $$

• integration of accel gives speed

$$ \large speed(t) = \frac{2\cdot s}{d} \cdot \left( t - \frac{d\cdot \sin\left(\frac{4\cdot\pi\cdot t}{d}\right)}{4\cdot\pi} \right) $$

• integration of speed gives pos ( normalized removing integration constant computed for t=0 )

$$ \large pos(t) = \frac{s\cdot d \cdot \left( \cos\left(\frac{4\cdot\pi\cdot t}{d}\right)-1 \right)}{8\cdot\pi^2} + \frac{s\cdot t^2}{d} $$

• targetspeed required from known final position p to reach in required time duration d

$$ \large s = \frac{2\cdot p}{d} $$

• from base deceleration function cos(t)-1

• sets th as

$$ \large th=t-\frac{d}{2} $$

• deceleration results as this

$$ \large deAccel\left(th\right) = \frac{2\cdot s}{d}\cdot\left(\cos\left(\frac{4\cdot\pi\cdot th}{d}\right)-1\right) $$

• despeed results as this ( given from deAccel integral subtracting integration constant and adding speed achieved by accel at d/2 )

$$ \large deSpeed\left(th\right) = \frac{2\cdot s\cdot\sin\left(\frac{4\cdot\pi\cdot th}{d} \right)}{4\cdot\pi}-\frac{2\cdot s\cdot th}{d}+s $$

• depos results as this ( given from deSpeed integral subtracting integration constant and adding pos achieved by accel at d/2 )

$$ \large dePos\left(th\right) = \frac{s\cdot d\cdot\left( 1-\cos\left(\frac{4\cdot\pi\cdot th}{d}\right) \right)}{8\cdot\pi^2}-\frac{s\cdot th^2}{d}+s\cdot th+\frac{s\cdot d}{4} $$

Performance considerations

For a realtime purpose a bezier approach should be used like the one implemented in the Marlin.

Example

This example can be executed through

dotnet run --project examples/scurve-xlsx

and it will produce follow output.xlsx by applying above formulas and doing some test calc about max allowable torque based on motion and supposed load.

Prerequisites

• vscode
• dotnet 8.0

Quickstart

git clone https://github.com/devel0/netcore-util
cd netcore-util
git checkout bd8626104f4b5a9cb9de5db624cebbcd77a8b384
cd ..

git clone https://github.com/devel0/netcore-sci
cd netcore-sci
git checkout 151451d5cd40e8ccebf93718379e8d72dbadce86
cd ..

git clone https://github.com/devel0/scurve-speed-eval
cd scurve-speed-eval
dotnet build
dotnet run --project gui

How this project was built

dotnet new console -n scurve-speed-eval
cd scurve-speed-eval
dotnet sln add scurve-speed-eval.csproj
dotnet sln add .
dotnet add reference ../netcore-util/netcore-util
dotnet add reference ../netcore-sci/netcore-sci
dotnet add package AngouriMath --version 1.3.0
dotnet add package Avalonia.Desktop --version 0.10.0-preview6
dotnet add package OxyPlot.Avalonia --version 2.1.0-20200725.1
dotnet add package UnitsNet --version 4.76.0
dotnet run

References

• How to get sinusoidal s-curve for a stepper motor