page 755

Given,
P0 = ( 5, 5, 5) in.			P1 = ( –5, –5, 5)
d			d		t	0 = 0	t1 = 2
dt----P0	=	( 0, 0, 0)	dt----P1	=	( 0, 0, 0)

Model the path with a function that allows acceleration/deceleration, in this case a third order polynomial will be used. The equation will be parameterized for simplicity (i.e., s = [0,1], where s=0 is the path start, and s=1 is the path end).

P( t) = P0 + ( P1 – P0) s( t)

s( t ) = 0			s( t ) = 1			d	) = 0	d
						----s( t		----s( t ) = 0
0					1	dt	0	dt		1
s( t)	= At	3	+ Bt	2	+ Ct + D	d			2	+ 2Bt + C
						----s( t) = 3At
						dt

Next, numerical values will be entered to find equation values

s( 0) = A( 0) 3 + B( 0) 2 + C( 0) + D = 0

s( 2) = A( 2) 3 + B( 2) 2 + C( 2) + D = 1

d
dt----s( 0) = 3A( 0) + 2B( 0) + C = 0
d		=	3A( 2) + 2B( 2) + C =		0
----s( 2)
dt
8A + 4		–	3		A =	1
			-- A = 1			--
			2			2

This can now be put in the final form,

P( t) = P		+ ( P			t3	3	2
	0		1	– P ) ---		– --t
				0	2	4

D = 0

8A + 4B = 1

C = 0

		–	3	A = B
			--
			2
B =			3
		–	4--

41.4.2 Computer Control of Robot Paths (Incremental Interpolation)

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•Path Planning is a simple process where the path planning methods described before (such as straight line motion) are used before the movement begins, and then a simple real-time lookup table is used.

•The path planner puts all of the values in a trajectory table.

•The on-line path controller will look up values from the trajectory table at predetermined time, and use these as setpoints for the controller.

•The effect of the two tier structure is that the robot is always shooting for the next closest ‘knotpoint’ along the path.

Desired

Set point

configuration

Trajectory

Off-line

Kinematic

table

Current

table

path

Transforms

Configuration

planning

Done before motion begins

Done during motion, and all other times

Time based interrupt

Servo motor routine runs

routine

for each axis

Interrupt

Choose new

Clock

point from

trajectory table

Read θ desired

Set-point

table

θ desired

Compute

error

		Output actuator
Return		signal

• The above scheme leads to errors between the planned, and actual path, and lurches occur when the new setpoints are updated for each servo motor.

page 757

speed

actual position

trajectory table

time

trajectory table time step

position

required

actual

time

•The quantization of the desired position requires a decision of what value to use, and this value is fixed for a finite time.

•The result is that the path will tend to look somewhat bumpy,

The desired path tends to deviate from the points in the trajectory							In this case an
table, and the tool tends to follow a path like
that pictured below. Moreover this scheme							extra step is
will lead to uneven acceleration over					knot 5		added to
the path segments							end compensate for
		knot 4					the position errors
Path segments	knot 3
knot 2
knot 1
start	**Note: this occurs with straight-line motion also