Position

Forced movement of a flange according to a reference position

Forces a translational flange to track a reference position input via a second-order Bessel filter.

This component drives the translational position 's' of its 'flange' to follow an external reference signal s_ref. The movement is smoothed by a second-order linear filter, configured as a Bessel filter using the parameters f_crit (critical frequency), af (s-coefficient), and bf (s²-coefficient). The control input s_ref dictates the target position.

The core dynamic relationship, ensuring smooth tracking, is given by the differential equation:

$$\frac{bf}{w_{crit}^2}\frac{d^{2}s}{d{t}^2}+\frac{af}{w_{crit}}\frac{ds}{dt}+s=s_{ref}$$

where the critical angular frequency 'w_crit' is defined as:

$$w_{crit}=2\pi f_{crit}$$

This formulation ensures a smooth transition of the flange position towards the reference, exhibiting the linear phase characteristics of a Bessel filter. The flange's velocity v and acceleration a are also calculated. Initially, the flange position s is set to the reference position s_ref.

Usage

Position(f_crit=0.0, af=1.3617, bf=0.6180, w_crit=2*π*f_crit)

Parameters:

Name	Description	Units	Default value
f_crit	User-defined critical frequency of the Bessel filter in Hertz	Hz	0

Connectors

• s_ref - This connector represents a real signal as an input to a component (RealInput)

• flange - (Flange)

Variables

Name	Description	Units
s	State variable for the actual position of the flange	m
v	State variable for the velocity of the flange	m/s
a	State variable for the acceleration of the flange	m/s2

Behavior

$$\begin{array}{rl}v\left(t\right)& =\frac{\mathrm{d}s\left(t\right)}{\mathrm{d}t}\\ a\left(t\right)& =\frac{\mathrm{d}v\left(t\right)}{\mathrm{d}t}\\ a\left(t\right)& =\frac{(-\mathtt{af}v\left(t\right)+\mathtt{w}\mathtt{\_}\mathtt{crit}(-s\left(t\right)+\mathtt{s}\mathtt{\_}\mathtt{ref}\left(t\right)))\mathtt{w}\mathtt{\_}\mathtt{crit}}{\mathtt{bf}}\\ \mathtt{flange}\mathtt{.}\mathtt{s}\left(t\right)& =s\left(t\right)\end{array}$$

Source

# Forced movement of a flange according to a reference position
#
# Forces a translational flange to track a reference position input via a second-order Bessel filter.
#
# This component drives the translational position 's' of its 'flange' to follow an external reference signal `s_ref`.
# The movement is smoothed by a second-order linear filter, configured as a Bessel filter using the
# parameters `f_crit` (critical frequency), `af` (s-coefficient), and `bf` (s²-coefficient).
# The control input `s_ref` dictates the target position.
#
# The core dynamic relationship, ensuring smooth tracking, is given by the differential equation:
# ```math
# \frac{bf}{w_{crit}^2} \frac{d^2s}{dt^2} + \frac{af}{w_{crit}} \frac{ds}{dt} + s = s_{ref}
# ```
# where the critical angular frequency 'w_crit' is defined as:
# ```math
# w_{crit} = 2\pi f_{crit}
# ```
# This formulation ensures a smooth transition of the flange position towards the reference,
# exhibiting the linear phase characteristics of a Bessel filter. The flange's velocity `v`
# and acceleration `a` are also calculated. Initially, the flange position `s` is set to the
# reference position `s_ref`.
component Position
  # Input connector for the reference position signal
  s_ref = RealInput() [{"Dyad": {"placement": {"icon": {"x1": -50, "y1": 450, "x2": 50, "y2": 550}}}}]
  # Output connector representing the mechanical translational flange
  flange = Flange() [{"Dyad": {"placement": {"icon": {"x1": 950, "y1": 450, "x2": 1050, "y2": 550}}}}]
  # State variable for the actual position of the flange
  variable s::Length
  # User-defined critical frequency of the Bessel filter in Hertz
  parameter f_crit::Frequency = 0.0
  # Pre-defined s-coefficient for the Bessel filter's characteristic polynomial
  final parameter af::Real = 1.3617
  # Pre-defined s^2-coefficient for the Bessel filter's characteristic polynomial
  final parameter bf::Real = 0.6180
  # Calculated critical angular frequency in rad/s (2*pi*f_crit)
  final parameter w_crit::Real = 2*π*f_crit
  # State variable for the velocity of the flange
  variable v::Velocity
  # State variable for the acceleration of the flange
  variable a::Acceleration
relations
  initial s = s_ref
  v = der(s)
  a = der(v)
  a = ((s_ref-s)*w_crit-af*v)*(w_crit/bf)
  flange.s = s
metadata {"Dyad": {"icons": {"default": "dyad://TranslationalComponents/Position.svg"}}}
end

Flattened Source

# Forced movement of a flange according to a reference position
#
# Forces a translational flange to track a reference position input via a second-order Bessel filter.
#
# This component drives the translational position 's' of its 'flange' to follow an external reference signal `s_ref`.
# The movement is smoothed by a second-order linear filter, configured as a Bessel filter using the
# parameters `f_crit` (critical frequency), `af` (s-coefficient), and `bf` (s²-coefficient).
# The control input `s_ref` dictates the target position.
#
# The core dynamic relationship, ensuring smooth tracking, is given by the differential equation:
# ```math
# \frac{bf}{w_{crit}^2} \frac{d^2s}{dt^2} + \frac{af}{w_{crit}} \frac{ds}{dt} + s = s_{ref}
# ```
# where the critical angular frequency 'w_crit' is defined as:
# ```math
# w_{crit} = 2\pi f_{crit}
# ```
# This formulation ensures a smooth transition of the flange position towards the reference,
# exhibiting the linear phase characteristics of a Bessel filter. The flange's velocity `v`
# and acceleration `a` are also calculated. Initially, the flange position `s` is set to the
# reference position `s_ref`.
component Position
  # Input connector for the reference position signal
  s_ref = RealInput() [{"Dyad": {"placement": {"icon": {"x1": -50, "y1": 450, "x2": 50, "y2": 550}}}}]
  # Output connector representing the mechanical translational flange
  flange = Flange() [{"Dyad": {"placement": {"icon": {"x1": 950, "y1": 450, "x2": 1050, "y2": 550}}}}]
  # State variable for the actual position of the flange
  variable s::Length
  # User-defined critical frequency of the Bessel filter in Hertz
  parameter f_crit::Frequency = 0.0
  # Pre-defined s-coefficient for the Bessel filter's characteristic polynomial
  final parameter af::Real = 1.3617
  # Pre-defined s^2-coefficient for the Bessel filter's characteristic polynomial
  final parameter bf::Real = 0.6180
  # Calculated critical angular frequency in rad/s (2*pi*f_crit)
  final parameter w_crit::Real = 2*π*f_crit
  # State variable for the velocity of the flange
  variable v::Velocity
  # State variable for the acceleration of the flange
  variable a::Acceleration
relations
  initial s = s_ref
  v = der(s)
  a = der(v)
  a = ((s_ref-s)*w_crit-af*v)*(w_crit/bf)
  flange.s = s
metadata {"Dyad": {"icons": {"default": "dyad://TranslationalComponents/Position.svg"}}}
end

Test Cases

This is setup code, that must be run before each test case.

using TranslationalComponents
using ModelingToolkit, OrdinaryDiffEqDefault
using Plots
using CSV, DataFrames

snapshotsdir = joinpath(dirname(dirname(pathof(TranslationalComponents))), "test", "snapshots")

"/home/actions-runner-10/.julia/packages/TranslationalComponents/khJb7/test/snapshots"

Related

• Examples

• Experiments

• Analyses