ElmoMC Multi-Axis Motion Controller User Manual

Motion Control

Library

Tutorial

January 2007 (Ver. 1.0)

Notice

This tutorial is delivered subject to the following conditions and restrictions:

 This tutorial contains proprietary information belonging to Elmo Motion Control Ltd.  The text and graphics included in this manual are for the purpose of illustration and

reference only. The specifications on which they are based are subject to change without notice.

 Elmo Motion Control and the Elmo Motion Control logo are trademarks of Elmo

Motion Control Ltd.

 Information in this document is subject to change without notice.

Document No. MAN-MLT

Elmo Motion Control Ltd.

Revision History:

Ver. 2.0 Jan. 2007 (MAN-MLT.pdf) Updates

Ver. 1.0 June 2004 (MAN-MLT.pdf) Initial Release

Elmo Motion Control Inc.

1 Park Drive, Suite 12 Westford, MA 01886 USA Tel: +1 (978) 399-0034 Fax: +1 (978) 399-0035

Elmo Motion Control GmbH

Steinkirchring 1 D-78056, Villingen-Schwenningen Germany Tel: +49 (07720) 8577-60 Fax: +49 (07720) 8577-70

www.elmomc.com

Motion Library Tutorial

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Chapter 1: General Description ..............................................................................................1

1.1 Introduction ............................................................................................................ 1

1.2 Vector properties...................................................................................................... 1

1.3 Trajectory generation ............................................................................................... 3

1.3.1 Line................................................................................................................ 3

1.3.2 Circle ............................................................................................................. 3

1.3.3 Spline............................................................................................................. 3

1.3.3.1 Examples for the two-dimensional spline interpolation .............5

1.3.3.2 Examples of three-dimensional spline interpolation................... 6

1.3.4 Polyline............................................................................................. 10

1.3.4.1 Examples for the two-dimensional polyline...............................12

1.3.4.2 Examples for the three-dimensional polyline.............................14

1.4 Transition to a new trajectory with a non-zero velocity..................................... 16

Chapter 2: Switch Radius Calculation ................................................................................2-1

2.1 Line – line intersection.......................................................................................... 2-1

2.2 Circle – line intersection ....................................................................................... 2-6

2.2.1 Line goes inside the circle ............................................................... 2-6

2.2.1.1 Switch arc center and circle center belong to two different half

planes defined by the line L...................................................................

2.2.1.2Switch arc center and a circle center belong to the same half

plane………............................................................................................

2.2.1.3 Line intersects the center of the circle...................................... 2-17

2.2.2 Switch arc radius calculation by the distance from the intersection point …………………………………………………………………….220

2.2.2.1 Initial circle center and switch arc center belong to the same

half-plane ...............................................................................................

2.2.2.2 Initial circle center and switch arc center belong to two half

planes defined by the line L.................................................................

2.2.2.3 Circle center (Xc,Yc) Є L1 (line L1 intersects the center of the

circle)…...................................................................................................

2.2.3 Line goes outside the circle .................................................................. 2-24

2.2.3.1 Line L and init radius continued in their positive intersecting

directions................................................................................................

2.2.3.2 Line parallel to the circle arc init radius.......................................2-27

2.2.3.3 Line L and init radius continued in their reverse directions intersect2-28

2.3 Circle – circle intersection.................................................................................. 2-30

2.3.1 One of two circle arcs intersects the internal area of the second...... 2-31

2.3.2 Each circle intersects the internal area of the second......................... 2-39

2.3.3 No circle intersects the internal area of the other .............................. 2-42

Appendix A: Projection of a point on a line defined by the end points........................ A-1

Appendix B: Coefficients of the line standard equation for the line defined by the end

points ........................................................................................................................................B-1

Appendix C: Intersection point of two lines defined by the end points ....................... C-1

Appendix D: Circle – line intersection points ...................................................................D-1

2-6

2-12

2-20

2-22

2-24

2-25

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Chapter 1: General Description

1.1 Introduction

The Motion Library (ML) produces trajectories based on the PVT mechanism. It implements a set of functions that calculate trajectories as PVT tables for the vector motion. As a result of a call to the Motion Library functions, the user gets a PVT table for the requested trajectory. It supports two and three-dimensional vector motion.

A PVT table is a two or three-dimensional sequence of PVT points.

Each PVT point is defined by:

• position value

• velocity for this position

• time interval that is necessary to arrive from the current position to the position defined by

the next PVT point

It supports

1. Single shape trajectories:

line (2D,3D)

circle (2D)

2. Trajectory built from an arbitrary set of pointes interpolated by the cubic spline (2D,3D)

3. Polyline trajectory that can include a number of single shapes:

line segments (2D,3D)

circle arc segments (2D)

spline points (2D,3D)

Inside polyline transition from one shape to another can be executed with a non-zero velocity. In this case, an additional element – switch circle arc-- is inserted between two shapes.

In the case of two-dimensional vector motion, switch arcs can be built for the line-line, line-circle, circle-line and circle-circle intersections. In the case of a three-dimensional polyline, switch arcs can be built for the line-line intersection.

1.2 Vector properties

Geometry of trajectory is defined by the set of vector functions such as circle() or line(). The Velocity profile is also influenced by the set of the following parameters (vector properties):

maximum vector acceleration/deceleration (vac/vdc)

maximum vector velocity (vsp)

end velocity (vse)

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general trajectory time (vtt)

switch arc definitions (vsc, vsr, vsd)

admissible velocity and position errors definitions (vpe,vve)

PVT step low and high limits (VNT,VXT )

All of the vector’s properties can be set in a user program or by the Command Interpreter.

Syntax of a property:

Vector_name.property

Examples:

v1.vsp - defines maximum vector velocity

v1.vtt - defines trajectory time

v1.vsc - defines smooth type from one shape to another

(1 – minimal radius switch arc, 2 – fixed radius switch arc, 3 – switch with a fixed distance from

the intersection point).

Single shape trajectories can be executed in one of three modes pre-defined by the value of the input parameter vum: 1 – max velocity, 2 – fixed time, 3 – fixed velocity.

In the maximum velocity mode, velocity defined by the parameter vsp is considered a limiting value that cannot be exceeded. If a trajectory is not long enough to achieve such a value, then a

trajectory with a triangle velocity profile is built and some maximum vector velocity

max

< vsp

is achieved at one point.

The fixed velocity mode (vum=3) is used if the user is interested in building a trajectory with a trapezium velocity profile – the main part of the trajectory (with the exception of possible acceleration/deceleration at the initial and final parts of the trajectory) is executed with a velocity equal to vsp. If a trajectory is not long enough to reach velocity vsp with the given vector acceleration /deceleration (input parameters vac/vdc), the trajectory is not built and the user receives an error message.

In the fixed time mode (vum = 2) the user must define parameter vtt – time in milliseconds for the trajectory execution. The Motion Library chooses a velocity profile that satisfies parameter vtt. If a trajectory with the given length, maximum velocity vsp and vector acceleration/deceleration cannot be executed within time vtt trajectory is not built and the user receives an error message.

The user can set values for the maximum PVT step in milliseconds – parameter vxt and for the minimal PVT step – parameter vnt ≥ 1msec. In this case, the main part of the trajectory will be

executed with the PVT step

ΔT = 0.5(vxt + vnt).

For the switch arc, connecting two shapes can be chosen from one of the three possible modes predefined by the input parameter vsc: 0 – no switch arc to be built, 1 – switch arc with the minimal possible radius, 2 – switch arc with radius pre-defined by the user, 3 – user defined distance from the intersection point (for the line-line or circle-line intersections).

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1.3 Trajectory generation

1.3.1 Line

Target position for a line is defined by the parameters of the function line():

Two-dimensional line

V1.line(x,y) – produces a line trajectory from the current position to the point (x,y), where x and y integer values in counts.

Three-dimensional line

V1.line(x,y,z) – produces line trajectory from the current position to the point (x,y,z), where x,y and z - integer values in counts.

1.3.2 Circle

Radius, initial and sweep angles for a circle must be defined as parameters of a function circle(): V1.circle(radius, init_angle,sweep_angle), where init_angle and sweep_angle must be set in

degrees (float), radius in counts (integer).

Example (Motion Mathematic Lib Samples\ Vector_2D \CircleArc – www.elmomc.com)

v1.vac = 28000000 //max acceleration

v1.vdc = 28000000 //max deceleration

v1.vum = 1 //build trajectory in max. velocity mode

v1.vsp = 250000 //maximum velocity

v1.vse = 0 //end velocity

v1.circle(100000,45,-270) //build circle arc trajectory

v1.bg // start motion

while (a1.ms==2)||(a2.ms==2) //wait until both axes have stopped

wait(10)

end while

1.3.3 Spline

A spline gives the possibility to move a smooth curve through an arbitrary set of points that do not

necessary belong to a particular geometric shape as a circle, ellipse or a line.

The spline that is supported by the Motion Library is an interpolation cubic spline. All the points

Po,P1,...,P

points

given by the user belong to the spline curve. Between each pair of the neighboring

(Pi,P

), cubic spline is defined by a third-order polynomial.

i+1

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Other popular types of splines like Bezier curves, B- splines or NURBS are usually not

interpolation but smoothing splines. The spline curve does not move through the given points but near them.

So cubic splines are piecewise polynomial (built of cubic polynomials for each segment [i,i+1]) and

produce a curve with continuous first and second derivatives at the internal control points

…,

that, in case of motion control, means continuity of the velocity and acceleration. Denote a

n-1

cubic spline polynomial for the segment [i,i+1] as and S

′′(ti) = S′′

), i = 1,2,…,n-1.

i+1(ti

Si(t) than Si(ti) = S

), Si′(ti) = S

i+1(ti

i+1

The spline trajectory is executed in the maximum velocity mode vum = 1. Input parameters that define kinematics of the trajectory are maximum velocity – parameter vsp, vector acceleration – parameter vac and vector deceleration – parameter vdc.

The Motion Library user can define points for the spline interpolation applying the following function calls:

vector_name.splines(trj_name) - starts a sequence of points to be interpolated. The PVT table is saved in a file named trj_name. The function parameter trj_name can be missed. In this case, the trajectory is saved in a temporary file named vector_name.trj (where vector_name – name defined in a resource file).

To interpolate two-dimensional points, use a function call

, P2,

′(ti)

vector_name.splinep(int PosX, int PosY) ) – adds interpolation point with coordinates (PosX, PosY).

In three-dimensional space

vector_name.splinep(int PosX, int PosY, int PosZ) ) – adds 3D interpolation point with

coordinates (PosX, PosY, PosZ).

vector_name.splinee(parameter) - ends the spline trajectory sequence.

– If parameter = 0 the standard PVT table that in each line contains PVT points for the X and Y

axes (for 2D spline) or PVT points for X, Y and Z axes (for 3D spline) is built.

– If parameter ≠ 0 three tables for the axis X, axis Y and for the gear are built. The table for the

gear contains PVT points with the position equal to the distance along the spline from the

initial spline position, vector velocity V = [V

2]1/2

+ V

for this position and a standard time

step (in 3D case four tables for X,Y, Z and gear are built).

Inside the spline operator parenthesis splines(trj_name) and vector_name.splinee(parameter) can be added operators for the position calculation (for instance ellipse points X = X

+ acos(t), Y = Yc + bsin(t)).

The Motion Library generates a trajectory by the cubic spline interpolation

Important note: Current position is not automatically added to the sequence of spline points. Homing must be done to the first spline point.

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1.3.3.1 Examples for the two-dimensional spline interpolation

Example Example (Motion Mathematic Lib Samples\ Vector_2D \ Spline_Ellipse – www.elmomc.com)

Ellipse trajectory (2D spline interpolation)

v1.vum=1 //build trajectory in max velocity mode

v1.vac = 28000000 //max acceleration

v1.vdc = 28000000 //max deceleration

v1.vsp = 50000 //max. velocity

v1.vse = 0 //end velocity

pi = 3.14159265358979

a = 100000

//ellipse axis a

b = 50000 //ellipse axis b

Xc = 0 // ellipse center coordinate by X Yc = 0 // ellipse center coordinate by Y

v1.splines() // start spline trajectory

for teta = 0:pi/18:2*pi

x = Xc + a*cos(teta)

y = Yc + b*sin(teta)

v1.splinep(x,y)

// add point to the spline trajector y

end for

v1.splinee(0) // end spline trajectory

v1.bg //start motion

After homing drives must at the first point of the sequence.

Example Example (Motion Mathematic Lib Samples\ Vector_2D \ Lissajous curves –

www.elmomc.com)

v1.vac = 28000000 //max vector acceleration

v1.vdc = 28000000 //max vector deceleration

v1.vsp = 100000 //set max. velocity

v1.vse = 0 //set end velocity

v1.splines() // start spline sequence

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for t = 0:pi/72:2*pi

x = R*cos(3*t)

y = R*sin(5*t)

v1.splinep(x,y) // add spline point

end for

v1.splinee(0) // end spline sequence

v1.bg //start motion

while (a1.ms==2)||(a2.ms==2) //wait until both axes have stopped

wait(10)

end while

Figure 1-1: Recording of the spline trajectory (Lissajous curves)

1.3.3.2 Examples of three-dimensional spline interpolatio n

For 3D vector motion, there must be a defined resource file with a vector built from three axes. In the Elmo Studio it can be defined as shown in the picture below.

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Figure 1-2: Resources for the 3D vector

Example (Motion Mathematic Lib Samples\ Vector_3D \ Spline_3D – www.elmomc.com)

A spline curve throws a number of arbitrary points

SetAxisStartPos(a1, 0) //set coordinate x to 0

SetAxisStartPos(a2, 0) //set coordinate y to 0

SetAxisStartPos(a3, 0) //set coordinate y to 0

v2.vsp = 50000

v2.vse=0

v2.splines() // start spline sequence

v2.splinep(0, 0, 0) // add spline 3D point

v2.splinep(50000, 100000, 150000) // add spline 3D point

v2.splinep(100000, 50000, 100000) // add spline 3D point

v2.splinep(200000, 150000, 50000) //add spline 3D point

v2.splinee(0) // end spline sequence

Three-dimensional picture can be drawn in Matlab with the use of the following Matlab operators

[n, posX, velX, posY, velY, posZ, velZ, t] = textread('D:\Dir_22_01\trj_file', '%d %d %d %d %d

%d %d %d ', -1) where 'D:\Dir_22_01\trj_file – full path to the PVT table file.

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For this operator to work properly, the first line of the PVT table containing a text header must be removed.

plot3(posX,posY,posZ)

axis square; grid on

Figure 1-3: Three-dimensional picture corresponding to the calculated

PVT table (drawn in Matlab)

Results of the recording

Figure 1-4: Projection on the XY plane

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Figure 1-5: Projection on the XZ plane

Example (Motion Mathematic Lib Samples\ Vector_3D \ Helix – www.elmomc.com)

Helix curve built with the use of the spline interpolation.

SetAxisStartPos(a1, 50000) //set coordinate x to 0

SetAxisStartPos(a2, 0) //set coordinate y to 0

SetAxisStartPos(a3, 0) //set coordinate y to 0

v2.vsp = 50000

v2.vse=0

alpha = 0 //start angle degrees

beta = 3600 // sweep angle degrees

R = 50000

H = 250000 // height

X = 50000 // start pos x

Y = 0 // start pos y

Z = 0 // start pos z

n = 120 // num points

Teta = pi*(alpha/180) //radian

dTeta = (pi*(beta/180))/n // teta increment for one step

dZ = H/n //z increment for one step

Xc = X - R*cos(Teta) // X coordinate of the helix axis

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Yc = Y - R*sin(Teta) // X coordinate of the helix axis

v2.splines() // start spline sequence

for i=0:1:n

v2.splinep(X,Y,Z) // add spline 3D point

Teta = Teta + dTeta // calc teta for the next point

X = Xc + R*cos(Teta) // calc X coordinate for the next point

Y = Yc + R*sin(Teta) // calc Y coordinate for the next point

Z = Z + dZ // calc Z coordinate for the next point

end for

v2.splinee(0) // end spline sequence

v2.bg // begin motion

Figure 1-6: Helix three-dimensional picture for the calculated PVT table drawn in MATLAB

1.3.4 Polyline

To build a polyline, the Maestro user program applies the following calls to the motion library functions:

vector_name.starts(trj_name) – starts the polyline trajectory sequence with saving the PVT table built by the Motion Library in a file named trj_name. Parameter trj_name can be missed – in this case trajectory is saved in a temporary file named vector_name.trj.

vector_name.ends() - ends the polyline trajectory sequence.

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Inside the polyline operator parenthesis vector_name.starts(trj_name) and vector_name.ends() can be added function calls – addline(), addcircle(), addsplinep() and adddwell() to define polyline segments.

For the 2D polyline vector_name.addline (int PosX, int PosY) – adds a line segment

PosX, PosY - destination position of the linear segment (counts)

vector_name.addcircle(int Radius, float StartAngle, float SweepAngle) – adds circle arc

segment

Radius - radius of the circle segment (counts)

start_angle - start angle of the circle segment (degrees)

sweep_angle - sweep angle of the circle segment (degrees)

vector_name.addsplinep(int x, int y) - adds two-dimensional spline point

Important note: The user should take into account that the last point of the previous segment or the first point of the trajectory is automatically added to the spline. The minimal number of the

addsplinep() operators that define the same spline segment inside the polyline must be great equal

2. This requirement is valid for every spline segment inside the polyline – not only for the first one.

For the 3D polyline

vector_name.addline (int PosX, int PosY, int PosZ) – adds a line segment

PosX, PosY, PosZ - destination position of the linear segment (counts).

vector_name.addsplinep(int x, int y, int z) - adds three-dimensional spline point

As in a two-dimensional case, the last point of the previous segment is automatically added to the spline segment and the number of points defining the spline segment cannot be less than 2.

For 2D and 3D polyline

vector_name.adddwell(delay_time) – adding delay (station) between two segments

delay_time - delay value in millisecond

Smooth transition from one curve to another inside polyline.

There are four modes that define transition from one shape to another that are defined by the parameter (vector property) vsc:

1. vsc = 0 - switch arc is not built

2. vsc = 1 – ML builds switch arc with switch radius minimally radius that satisfies kinematics

constraint r > (vse)

vse – parameter that defines end velocity on the segment preceding switch arc

vac – parameter that defines maximum vector acceleration

vae – parameter that defines admissible error for the vector acceleration

/[vac*vae] where

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3. vsc = 2 – ML builds switch arc with the switch radius vsr

(this parameter must be set by the user).

4. vsc = 3 - ML builds a switch arc with the switch radius implicitly pre-defined via parameter

vsd (distance along the line from the intersection point). Parameter vsd must be set by the user.

For vsc = 2 and vsc = 3, the user can check if the values of the parameters vsr and vsd satisfy geometric constraints. Such a check can be done with the use of algorithms described in chapter 2 of this document.

Switch arc building is also influenced by the previous segment end velocity defined by the parameter (vector property) vse.

1.3.4.1 Examples for the two-dimensional polyline

Example (Motion Mathematic Lib Samples\ Vector_2D \ LineCircle – www.elmomc.com)

v1.vac = 28000000 //max vector acceleration

v1.vdc = 28000000 // max vector deceleration

v1.vum = 1 // build trajectory in max velocity mode

v1.starts() // begin polyline trajectory

v1.vsp = 50000 // max velocity for the line segment

v1.vse = 50000 // end velocity

v1.addline(100000, 100000) // request to add line shape

v1.vse = 0 // end velocity for the circle segment

v1.vsc = 2 // smooth intersection with fixed switch radius

v1.vsr = 10000 // switch radius

v1.addcircle(50000,225,180) //request to add circle arc shape

v1.ends() // ends polyline trajectory

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Figure 1-7: Recording of the polyline trajectory

Any trajectory generated by the Motion Library (single shape or polyline) can be rotated relative to init point due to vector property vra – rotation angle in degrees.

Example (Motion Mathematic Lib Samples\ Vector_2D \ LineCircleRotated –

www.elmomc.com)

// The same polyline rotated at –90 degrees relative to init point

v1.vac = 28000000 //max vector acceleration v1.vdc = 28000000 // max vector deceleration v1.vum = 1 // build trajectory in max velocity mode v1.vra = -90 // rotate spline 90 degrees relative to init point

v1.starts() // begin polyline trajectory

v1.vsc = 2 // smooth intersection with fixed switch radius

v1.vsp = 50000 // max velocity for the line segment

v1.vse = 50000 // end velocity

v1.addline(100000, 100000) // request to add line shape

v1.vsp = 50000 // maximum velocity for the circle segment

v1.vse = 0 // end velocity for the circle segment

v1.vsr = 10000 // switch radius

v1.addcircle(50000,225,180) //request to add circle arc shape

v1.ends() // end polyline trajectory

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Figure 1-8: Recording of the rotated polyline

1.3.4.2 Examples for the three-dimensional polyline

Example (Motion Mathematic Lib Samples\ Vector_3D \ Rectangle – www.elmomc.com)

Three-dimensional rectangle

SetAxisStartPos(a1, 0) //set coordinate x to 0

SetAxisStartPos(a2, 0) //set coordinate y to 0

SetAxisStartPos(a3, 0) //set coordinate y to 0

v2.vsc=2

v2.vsr=12000

v2.vsp = 70000 //max. vector velocity

v2.vse = 70000

v2.starts()

v2.addline(50000, 50000, 0) //create line from current point to coordinate

v2.addline(50000, 50000, 50000)

v2.addline(0, 0, 50000)

v2.vse = 0

v2.addline(0, 0, 0)

v2.ends()

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Figure 1-9: Three-dimensional polygon drawn in Matlab

Results of the recording

Figure 1-10: Projection on the XZ plane

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Figure 1-11: Projection on the YZ plane

1.4 Transition to a new trajectory with a non-zero velocity

If a transition from one trajectory to another (for instance from a line to a circle) must be executed with a velocity not equal to zero at the switch point, the Motion Library builds a switch curve to achieve smooth modification of the velocity. Such a curve is implemented as a circle arc.

As a geometrical object switch arc is completely defined by:

• radius

• coordinates of the two limit points (

where (

(

• center coordinates (X

The radius

last

first

, Y

) – last point on the first shape

last

, Y

) – first point on the second shape

first

)

o,Yo

r of the switch arc can be:

last

, Y

) and (X

last

first

, Y

first

)

1. Explicitly pre-defined by the user (vsc = 2, vsr defined).

In this case, it must obey the following restriction:

r ≥ (V

where

) 2/AC

end

- end velocity at the switch point,

end

v -

vector acceleration.

(1-1)

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In fact, the value defined as r ≥ (vse) the calculations.

Implicitly pre-defined by the user via smooth distance d along the line from the intersection

point (vsc = 3, vsd defined). This mode can be implemented to the line – line, circle – line and line – circle trajectory intersections.

3. Calculated as a minimal possible for the given end velocity V

Regardless of the

switch arc is considered to be uniform with the constant tangent velocity angle velocity

(vsc = 1).

r definition (pre-defined by the user or calculated by (1-1)) movement along the

/(vae*vac )

(by default vae = 0.9) must be used in

and vector acceleration

end

V = V

and constant

end

ω = V/r (1-2)

The intersection geometry imposes constraints on the switch arc radius. So the switch radius in use

r must satisfy

) 2/AC

end

< r < r

max

(1-3)

Chapter 2 will consider all cases of the two shapes intersections and methods for the calculation of

the switch

As a base value for the switch velocity is taken the preceding segment end velocity V by the parameter vse. This value is considered as a limiting parameter that cannot be exceeded but

can be decreased. It’s also valid in case that polyline segments (preceding switch arc and the following) are executed in the fixed velocity mode (vum = 3).

In switch mode vsc = 1, the initial value of a switch velocity equal vse and can be decreased to build a switch arc trajectory equal to the integer number of PVT steps or milliseconds.

In switch mode vsc = 1, the radius of the switch arc is calculated as a minimal possible for the intersection geometry and given vector acceleration/deceleration (input parameters vac/vdc) meaning that the calculated value satisfies account the requirement to build switch arc trajectory equal to an integer number of PVT steps. The Motion Library is trying to build switch arc trajectory with velocity as close to the value given by the parameter vse for the preceding segment as possible. The switch arc (or two arcs on both ends of the segment) can take up almost all segment length that sometimes makes trajectory calculation impossible.

As an example consider the following: segment initial velocity equal

radius limit value r

max.

defined

end

(1-3). While the switch arc calculation is also taken into

Vo, end velocity V

and

min(V

(where ΔL – the length of the segment truncated by switch arcs, ΔT – standard PVT step). When

defining switch radius (input parameter vsr) for vsc = 2 mode or end velocity (input parameter vse) for vsc = 1 mode, the user must take into account that it can significantly influences the whole trajectory and in fact replaces the main part of a particular segment by switch arcs.

)ΔT > ΔL (1-4)

o,Ve

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Input parameters and intersection geometry define the influence of a switch arc on a trajectory. The main cases of shapes intersection are considered below. Here as an example to consider two lines intersection. If an angle between two lines is small, even a switch arc with a small radius can significantly change initial trajectory while an arc with the same radius can be insignificant for lines

with intersection angle close to

180

In addition to geometric constraints, the Motion Library imposes limitations on the switch arc length. Each switch arc should not exceed 50% of the segment length. If there are two switch arcs adjacent to some polyline segment, then both arcs should not take more than 80% of its initial length.

Motion Library Tutorial Switch Radius Calculation

MAN-MLT (Ver 2.0)

Chapter 2: Switch Radius Calculation

2.1 Line – line intersection

2-1

If a trajectory contains a switch from line L of the vector velocity cannot be changed at the one intersection point. To implement such a

switch, insert into a trajectory an additional element – circle arc is inserted into a trajectory (see-Figure 2-1).

The switch arc radius must satisfy (1-1).

to line L

with non-zero velocity, the direction

Figure 2-1

In the case of a line-line intersection, parameters

equation

r and d are connected by the simple

r = d*tg(γ/2) (2.1-1

where γ = π – α (α - angle between vectors L

and L2) so the pre-defined parameter

vsd = d must satisfy

vsd ≥ (V

If ΔL1 is the length of the line L1 and ΔL2 is the length of the line L2, then there is an evident geometric constraint for the switch radius r

≤ min(ΔL1, ΔL

max

(2.1-3)

In fact, due to additional limitations (50% of the segment length) used in ML, the following should be used

)2/[ACv tg(γ/2)]

end

)*tg(γ/2)

(2.1-2)

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ElmoMC Multi-Axis Motion Controller User Manual

Specifications and Main Features

Frequently Asked Questions

User Manual

Notice

Contents

Chapter 1: General Description

1.1 Introduction

1.2 Vector properties

1.3 Trajectory generation

1.3.1 Line

1.3.2 Circle

1.3.3 Spline

1.3.3.1 Examples for the two-dimensional spline interpolation

1.3.3.2 Examples of three-dimensional spline interpolatio n

1.3.4 Polyline

1.3.4.1 Examples for the two-dimensional polyline

1.3.4.2 Examples for the three-dimensional polyline

1.4 Transition to a new trajectory with a non-zero velocity

Chapter 2: Switch Radius Calculation

2.1 Line – line intersection