Model Element

Description

Reference_ParamCurve defines a parametric curve element. A parametric curve is defined in terms of one free parameter, u. Referring to the image below, assume a curve C is defined with respect to a coordinate system OXYZ. The coordinates of any arbitrary P point on the curve, as measured in OXYZ, can be represented uniquely in terms of the free parameter u with three functions, f(u), g(u), and h(u), that define the x-, y- and z-coordinates of P. The extent of the curve is governed by the start and end values of u. This is a parametric representation for a curve.

A parametric curve

Parametric curves may be open or closed.  The curve is said to be closed when the closure boundary conditions shown below are satisfied:

f(u_start)       = f(u_end)

f’(u_start)      = f’(u_end)

f’’(u_start)     = f’’(u_end)

g(u_start)       = g(u_end)

g’(u_start)      = g’(u_end)

g’’(u_start)     = g’’(u_end)

h(u_start)       = h(u_end)

h’(u_start)      = h’(u_end)

h’’(u_start)     = h’’(u_end)

There are two types of parametric curves: analytical and spline.  Spline representations require that (x,y,and z) data for the curve be provided in a matrix of data points or directly specified in the XML file.  MotionSolve fits a spline through the data points and automatically parameterizes the curve in terms of a u variable.

Analytical representations require that the functions f(), g(), and h() are defined in a user-written subroutine CURSUB.  The range of u needs to be explicitly specified for analytically defined curves.

When the curves are closed, MotionSolve requires that the first and last points of the spline data are the same.  Similarly, you are required to satisfy the closure boundary conditions mentioned above.

Parametric curves are used to model higher pair constraints in MotionSolve. Reference_ParamCurve is only allowed to be defined on rigid bodies, not on flex bodies.

Format

<Reference_ParamCurve

       id                  = "integer"

       label               = “Name of the Curve”

       is_u_closed         = { "TRUE" | "FALSE" }

       is_curve_points     = { "TRUE" | "FALSE" }

       is_smooth_linear   = { "TRUE" | "FALSE” }

       smoothing_distance = "real"

  [    quintic             = { "TRUE" | "FALSE" }     ]    

  {

       matrix_id           = "integer" >

     | u_start             = "real"

       u_end               = "real"

       usrsub_dll_name     = "valid_path_name"

       usrsub_param_string = "USER( [[par_1][, ...][, par_n]] )"

       usrsub_fnc_name     = "custom_fnc_name" >

     | u_start             = "real"

       u_end               = "real"

       script_name         = valid_path_name

       interpreter         = "PYTHON" | "MATLAB"

       usrsub_param_string = "USER( [[par_1 [, ...][,par_n]] )"

       usrsub_fnc_name     = "custom_fnc_name" >

     | num_xyz_triple      = "integer" >

         x1    y1     z1

         x2    y2     z2

         ...  ...    ...

         xn    yn     zn

  }

/>

Attributes

id

Element identification number (integer>0).  This number is unique to all Reference_ParamCurves.

label

The name of the Reference_ParamCurve elements.

is_u_closed

Select from TRUE or FALSE. If the curve is closed in the U parametric space, select TRUE. If the curve is open in the U parametric space, select FALSE.

is_curve_points

Select from TRUE or FALSE.

TRUE means that the curve passes through the points.

FALSE means that the curve (B-spline) stays close to the points, but does not pass through them in general.

is_smooth_linear

Select from TRUE or FALSE

TRUE means that the curve follows a linear interpolation of the specified points that is smoothed around the corners. Setting is_smooth_linear overwrites the selection of is_curve_points.

FALSE means the interpolation method selected by is_curve_points is active.

See Comment 6.

smoothing_distance

The smoothing distance defines the maximum distance for which the solver may smooth out the curve at the specified curve points.

This option is only active in conjunction with is_smooth_linear, with a default value of 0.1.

See Comment 6.

quintic

Select from TRUE or FALSE.

TRUE indicates that the order of curve interpolation is 5

FALSE indicates that the order of curve interpolation is 4

Default value is FALSE.

matrix_id

Specifies the Reference_Matrix element that contains the curve data.  The matrix should contain only the x-, y-, and z- data for the individual curve points.  For a close curve, the first and last data points must be the same.

u_start

The minimum value of U.  Note that u_start < u_end. Set u_start only when is_curve_points = FALSE. In other words, if a CURSUB is being used to define the curve.

u_end

The maximum value of U.  Note that u_start < u_end. Set u_end only when is_curve_points = FALSE. In other words, if a CURSUB is being used to define the curve.

usrsub_param_string

The list of parameters that are passed from the data file to the user defined CURSUB.  Use this keyword only when is_curve_points = FALSE. In other words, a CURSUB is being used to define the curve.

usrsub_dll_name

Specifies the path and name of the DLL or shared library containing user subroutines.  MotionSolve uses this information to load the user subroutine CURSUB in the DLL at run time.  Use this keyword only if is_curve_points = FALSE. In other words, if a CURSUB is being used to define the curve.

usrsub_fnc_name

Specifies an alternative name for the user subroutine CURSUB.

script_name

Specifies the path and name of the user written script that contains the routine specified by usrsub_fnc_name.

interpreter

Specifies the interpreted language that the user script is written in. Valid choices are MATLAB or PYTHON.

num_xyz_triple

Specifies the number of x-y-z points in the curve.  Use only when the curve points are embedded in the Reference_ParamCurve element itself.  This option cannot be used with MATRIX or usrsub_param_string parameters.

Comments

6.

Specifying is_smooth_linear will construct a curve based on the given points . The curve constructed is clamped at the endpoints and linear between consecutive points and . The curve is smoothed at the interior points based on the value of the parameter smoothing_distance. A larger value for smoothing_distance results in a larger radius being used for smoothing.

Note: If the distance between an interior point and the endpoint is smaller than the smoothing_distance, the linear segments will do a smooth transition in a similar way as done by the TRIM function.

Example curve obtained using 5 points and is_smooth_linear = TRUE

Example

The following example defines how to implement a cycloid that is defined in parametric space in MotionSolve.  The independent parameter for this example is the variable t.  The cycloid is defined as:

x = a * (t – sin(t))

y = a * (1-cos(t))

The cycloid is shown in the image below.  The statement defining the cycloid described above has the following curve attributes:

A parametrically defined cycloid curve

The corresponding MotionSolve XML statement is:

<Reference_ParamCurve

    id                  = "569"

    is_u_closed         = "FALSE"

    is_curve_points     = "TRUE"

    matrix_id           = "23"

/>

The second example shows how to specify exactly the same for implementation in a user defined subroutine CURSUB.

<Reference_ParamCurve

    id                  = "569"

    is_u_closed         = "FALSE"

    is_curve_points     = "FALSE"

    u_start             = "0"

    u_end               = "5.1"

    usrsub_param_string = "USER(22,11)"

    usrsub_dll_name     = "C:\gates\curve.dll"

/>

Model Element

Description

CURVE defines a parametric curve element.  For additional information on the definition of CURVE, please see Reference_ParamCurve.

Declaration

def CURVE(id, LABEL="", TYPE="", FUNCTION="", MINPAR=0.0, MAXPAR=0.0, MATRIX=0, CURVE_POINTS=False, ORDER=4, ROUTINE="", INTERPRETER="", SCRIPT=""):

Attributes

id

Element identification number (integer>0).  This number is unique among all the CURVE elements.

LABEL

The name of the CURVE element.

TYPE

Select from "OPEN" or "CLOSED".

FUNCTION

The list of parameters that are passed from the data file to the user defined CURSUB.  Use this keyword only when CURVE_POINTS = False.  In other words, a CURSUB is being used to define the curve.

MINPAR

The minimum value of U.  Note that MINPAR < MAXPAR.  Set MINPAR only when CURVE_POINTS = False.  In other words, if a CURSUB is being used to define the curve.

MAXPAR

The maximum value of U.  Note that MINPAR < MAXPAR.  Set MINPAR only when CURVE_POINTS = False.  In other words, if a CURSUB is being used to define the curve.

MATRIX

Specifies the ID of MATRIX element that contains the curve data. The matrix should contain only the x-, y-, and z- data for the individual curve points.  For a close curve, the first and last data points must be the same.

CURVE_POINTS

Select from TRUE or FALSE.
TRUE means that the curve passes through the points.
FALSE means that the curve (B-spline) stays close to the points, but does not pass through them in general.

ORDER

Select from 4 or 5.

5 indicates that the order of curve interpolation is 5.

4 indicates that the order of curve interpolation is 4.

Default value is 4.

ROUTINE

Specifies an alternative name for the user subroutine CURSUB.

INTERPRETER

Specifies the interpreted language that the user script is written in.  Valid choices are MATLAB or PYTHON.

SCRIPT

Specifies the path and name of the user written script that contains the routine.

Comments

See Reference_ParamCurve

Example

The first example below demonstrates how a CURVE is defined.

CURVE(569, TYPE="OPEN", MATRIX=23, CURVE_POINTS=True)

The second example below demonstrates how the same CURVE is defined using user defined subroutine CURSUB.

CURVE(569, TYPE="OPEN", FUNCTION=" USER(22,11)", MINPAR=0.0, MAXPAR=5.1, CURVE_POINTS=False, ROUTINE="C:\gates\curve.dll")