multiaxis motion - 16.19

speed

actual position

trajectory table

time

trajectory table time step

position

required

actual

time

Figure 16.19 Errors in path following

16.4PATH PLANNING

•When we have simple features, paths are easy to generate. These features include,

-steps

-pockets

-holes

-etc.

•Typically paths for these will repeat as shown below,

multiaxis motion - 16.20

• For complex surfaces we want to contour appropriately. These surfaces will almost always be represented with spline patches.

• Recall that a spline patch can be represented parametrically

multiaxis motion - 16.21

(u=0,v=1)	(u=1,v=1)

xp

p( u, v) = yp

zp

(u=0,v=0) (u=1,v=0)

• A simple algorithm to cut the surface is shown below.

dirn_flag = 1; a direction flag

n=10 ; number of passes to cut the surface step=1.0/n ; step sizes for u and v directions

start=step/2 ; the start offset in the u and v directions [xp,yp,zp] = p(start,start) ; calculate the start position

print(“G00 X”,xp,” Y”,yp,” Z”,zp+0.2) ; move the tool to above the start position for i=0 to (n-1) ; will increment in the u direction

for j=0 to (n-1) ; will increment in the v direction ; calculate next point

if dirn_flag=-1 then [xp,yp,zp]=p(start+i*step,start+j*step)

if dirn_flag=1 then [xp,yp,zp]=p(start+i*step,start+(n-j)*step) print(“G01 X”,xp,” Y”,yp,” Z”,zp) ; instruction to cut to next point

next j ; make next step in v direction until done

dirn_flag = -dirn_flag ; reverse direction to cut in opposite direction next i ; move to next cut line in the u direction

print(“G00 Z”,zp+0.2) ; move the tool to above the end position

16.5CASE STUDIES

-the controller described in the block diagram below uses a model for a DC permanent magnet DC motor to estimate a voltage based upon a predicted velocity and position.

-the desired position and velocity are given in figure xxx based upon the motion

multiaxis motion - 16.22

position control derived in the earlier section

θ

( t)

=

θ start + ( θ end – θ

start) p

t – tstart

-------------------------

t

end

– t

start

p( u) = –2u3 + 3u2

ω

( t)

=

(θ end

·

t – tstart

– θ start) p

-------------------------

t

end

– t

start

·

=

–6u

2

+ 6u

p( u)

Figure 16.20 Motion position control equation

- the controller that uses the desired position and velocity is shown in figure xxxx

real-time control loop

ω

d

JRD + K2

-----------------------

K

+

physical system

θ

d

+

θ a

P

K

-----------------------

-

+

JRD + K2

							1																											Kenc
							----------
											Kenc

Figure 16.21 Feedforward trajectory controller