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

TranslationalComponents.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 - This connector represents a mechanical flange with position and force as the potential and flow variables, respectively. (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_

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

math w_{crit} = 2\pi f_

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_

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

math w_{crit} = 2\pi f_

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

No test cases defined.

Related

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