Elmomc Multi-Axis Motion Controller-Maestro Motion Contro Manual de usuario

Motion Control Library Tutorial January 2007 (Ver. 1.0)

Maestro Motion Library Tutorial MAN-INTUG (Ver. 1.7) 7

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

Figure 1-5: Projection on the XZ plane Example (Motion Mathematic Lib Samples\ Vector_3D \ Helix – www.elmomc.com)

Maestro Motion Library Tutorial MAN-INTUG (Ver. 1.7) 10 Yc = Y - R*sin(Teta) // X coordinate of the helix axis v2.splines()

Inside the polyline operator parenthesis vector_name.starts(trj_name) and vector_name.ends() can be added function calls – addline(), addcircle(), add

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 swit

Maestro Motion Library Tutorial MAN-INTUG (Ver. 1.7) 13

Maestro Motion Library Tutorial MAN-INTUG (Ver. 1.7) 14 Figure 1-8: Recording of

Maestro Motion Library Tutorial MAN-INTUG (Ver. 1.7) 15 Figure 1-9: Three-dimensional polygon drawn in

Maestro Motion Library Tutorial MAN-INTUG (Ver. 1.7) 16 Figure 1-11: Pr

Notice This tutorial is delivered subject to the following conditions and restrictions:  This tutorial contains proprietary information belongi

In fact, the value defined as r ≥ (vse) 2/(vae*vac ) (by default vae = 0.9) must be used in the calculations. 2. Implicitly pre-defined by the us

Input parameters and intersection geometry define the influence of a switch arc on a trajectory. The main cases of shapes intersection are considered

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-1 Chapter 2: Switch Radius Calculation 2.1 Line – line intersection If a traje

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-2 vsr ≤ min(0.5ΔL1, 0.5ΔL2)*tg(γ/2)

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-3 r_max = dmax*tg(γ/2) = 50000* tg(0.5*0.1974) = 4951 This value is limiting a

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-4 vse = [r_switch*vac*vae]1/2 = [4455.9*500000*0.9]1/2 = 44778.9 Example 2.1c

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-5 Line 1 is defined by its init point (300000, 900000) and end point (700000,2

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-6 2.2 Circle – line intersection Note: C – circle arc, L – line, R – circle

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-7 Figure 2-2 Example 2-2 (Motion Mathematic L

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-8 Yp = Yc + K*(Xp – Xc) = 0 +0.7*(-46979 - 0) = -32885 And the perpendicular l

Contents Chapter 1: General Description ...11.1 Introduct

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-9 The length of the perpendicular h should also be calculated. By knowing the

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-10 Figure 2-4 In our calculations was not taken in account add

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-11 r = ρ1ρ2/(ρ1 + ρ2)

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-12 ρ1= 100000 - |C/B| = 100000 - |(-3464101600.0)/(-90000)| = 61509.98222

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-13 Figure 2-7 This condition is not always sufficient. Adequacy depends on a

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-14 Figure 2-8 Example 2-9 (Motion Mathematic Lib Samples\Circle to Line\ Se

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-15 Figure 2-9 Projection of the circle arc init point P1 on the line L does

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-16 Figure 2-10 Example 2-11 (Motion Mat

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-17 2.2.1.3 Line intersects the center of the circle Consider the last case of

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-18 Figure 2-13 Example 2-14 (Motion Mathematic Lib Samples\Circle to Line\ S

Chapter 1: General Description 1.1 Introduction The Motion Library (ML) produces trajectories based on the PVT mechanism. It implements a set of fun

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-19 c) The circle arc sweeps an angle less than 90o and a perpendicular droppe

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-20 By (a1.6) we have Xp = (Yo – Y1 + kX1 – qXo)/(k – q) = (–80000 + 56569 – 5

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-21 ρ[(Xp,Yp),(X1,Y1)] = r

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-22

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-23 that produces r = [R2 – (ρ1)2 – (ρ3)2]/(2R + 2ρ1)

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-24 2.2.2.3 Circle center (Xc,Yc) Є L1 (line L1 intersects the center of the c

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-25 1. Circle init radius intersects with the line L continued in its positive

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-26 or rd = hd – hR – hr

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-27 Figure 2-24 2.2.3.2 Line parallel to the circle arc init radius a) Li

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-28 Figure 2-25b Maximum switch radius is perpendicular to the line L at the

general trajectory time (vtt) switch arc definitions (vsc, vsr, vsd) admissible velocity and position errors definitions (vpe,vve) PVT step low and hi

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-29 3. Know trajectory init point P2(X2,Y2), calculate ρ2 = ρ(p2, p1) = [(X2

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-30 By (a3.6)-(a3.7) from Appendix 3. q1 = ΔX1/ΔY1= (34641-0)/(20000-0) = 1.732

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-31 2.3.1 One of two circle arcs intersects the internal area of the second If

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-32 (Xo – Xc2)2 + (Yo – Yc2)2 = (R2 – r)2

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-33 (C1)2 + (C2)2 – 1 = [(X2 – X1)/d]2 + [(Y2 – Y1)/d]2 – 1 = d2/d2 – 1 = 0

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-34 (rC1 + C3)2 + (rC2 + C4)2 = (R2 – r)2

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-35 (X2 + 65000)2 + (– 35000)2 = 1000002 that produces X2 = -158675. d = |X2 –

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-36 From (2.3.1-27) Figure 2-31 XoR1 – X1R1 = r(Xc1 – X1)

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-37 r2 C12 + (2C1C3)r + C32 + r2C22 + (2C2C4)r + C42 = r2 + (2R2)r + R22

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-38 Substituting into (4.1-32) (X2 + C1r – Xc1)2 + (Y2 + C2r – Yc1)2 =

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)

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-39 2.3.2 Each circle intersects the internal area of the second Figure 2-33 sh

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-40 This system is similar to (2.3.2-2) – (2.3.2-4) and comes to the same solut

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-41 C1 = (X1 – Xc2)/R2 = -0.866025 C2 = (Y1 – Yc2)/R2

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-42 This system is similar to (4.2) – (4.4) and comes to the same solution r =

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-43 Consider the case that the sweep angle of the first circle is β1 < 90 an

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-44 r2C5 + rC6 + C7 = 0

Motion Library Tutorial Switch Radius Calculation MAN-MLT (Ver 2.0) 2-45 So for r, the results are: r = –C7/C6

Appendix A: Projection of a point on a line defined by the end points The line L is defined by its end points P1(X1,Y1) and P2(X2,Y2). Drop a perpendi

Y is from (a1.4). Coordinates (X,Y) of the intersection point line L and perpendicular are coordinates of projection point (Xp,Yp). Having got a proje

Appendix B: Coefficients of the line standard equation for the line defined by the end points If the line L is defined by its end points (X1,Y1) and

Other popular types of splines like Bezier curves, B- splines or NURBS are usually not interpolation but smoothing splines. The spline curve does

Appendix C: Intersection point of two lines defined by the end points Line L1 is defined by its end points P1(X1,Y1) and P2(X2,Y2). Line L2 is defined

or (X3 – X1)/∆X1 = (Y – Y1)/∆Y1 (a3.10) and f

Appendix D: Circle – line intersection points The line is defined by its end points (X1,Y1) and (X2,Y2). The circle is defined by its radius R and c

1.3.3.1 Examples for the two-dimensional spline interpolation Example Example (Motion Mathematic Lib Samples\ Vector_2D \ Spline_Ellipse – www.elmomc.

Maestro Motion Library Tutorial MAN-INTUG (Ver. 1.7) 6 for t = 0:pi/72:2*pi x = R*cos(3*t) y = R*sin(5*t) v1.splinep(x,y) // add spline po