Page 163 - Cam Design Handbook

THB5 8/15/03 1:53 PM Page 151

CAM MOTION SYNTHESIS USING SPLINE FUNCTIONS 151

be continuous everywhere, including the ends (as noted earlier). The general form of the
B-spline surface interpolation can guarantee the continuity only of the internal region of
the interpolated surface but not between the two ends. Closed periodic B-spline functions
produced from uniform knot sequences cited earlier solve this problem.
4. No special consideration of continuity is required at the surface edges in the direc-
tion of translation so the multiple uniform knot sequence is used.
The systematic procedure for implementing the B-spline surface interpolation can be
established as follows:

1. Set up the motion constraints in a n ¥ m rectangular grid. If any data are absent in the
rectangular grid, they can be filled in by using the B-spline interpolation applied along
either the rotating or the translating directions.
2. Select the appropriate order of the B-spline functions. As shown in Eq. (5.37), the
proper value of the order is between two and the number of motion constraints. Recall
that the degrees of the functions are one less than the order.
3. Construct the knot sequence according to the demand of each parametric direction. The
knot sequence, x I, for the closed periodic B-spline (along the rotation coordinate) can
be obtained by

Ï f 2 min i =1
Ô
x = Ì f -f (5.48)
i 2 max 2 min
2
n
Ô Ó x i-1 + n -£ i £+1.
The knot sequence in the translation coordinate can be found as
1 ££ k
j
s Ï 2 min 2
Ô
Ô s - s
1
j
y = Ì y + 2 max 2 min k +£ £ m (5.49)
j j-1 mk +1 2
-
Ô 2
s Ô m +£ £ m k .
+
1
j
Ó 2 max 2
4. Determine the values of B-spline functions corresponding to motion constraints by
using Eqs. (5.38) to (5.41).
5. Collect the values of B-splines and motion constraints to form the matrices
[ N f ()] [ ], and [M s ()].
S
,
2
1
2
6. Obtain the coefficient matrix [p] as follows:
-1
-1
p [] = [ N()] ◊[ S (f , s )]◊[ M s ()] .
f
2 1 2 2 2
7. Evaluate the complete follower motion function by applying Eq. (5.47).
After the follower motion function is determined, kinematic properties for the velocity
v 1 (f 2 , s 2 ), the acceleration A 1 (f 2 , s 2 ), and the jerk J 1 (f 2 , s 2 ) of the synthesized follower
motion can be derived by differentiating Eq. (5.37). When the angular and linear veloci-
ties of the cam are constant, the derivatives can be expressed as