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Acceleration (cm/rad/rad)
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Splines Polynomials Constraint 8 0 90 180 Cam rotation angle (deg) 270
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FIGURE 510 Comparison of the accelerations obtained by both the polynomial and the spline synthesis techniques in Example 4
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j =1
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(514)
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In Eq (513), if all the Wj = 1 ( j = 1, , n), Rj,k(x) = Nj,k(x), then, Eqs (511) and (512) become the same as Eqs (51) and (54), respectively Adjustments to these weight values constitute the additional degrees of freedom mentioned earlier and provide additional design exibility as well Note that, in particular, a change in Wj affects the rational B-splines only in the interval of [Tj, Tj+k] As a result, the designer can exercise local control of motion characteristics by adjusting a particular Wj Hence, for a general case, the rational B-spline procedure permits the designer to re ne the synthesized motion by adjusting the order of B-splines and the knot sequence and to exercise local control through the weight sequence, all without violating any of the motion constraints Moreover, motion constraints can also be added or adjusted to tune motion programs When required for the equations above, derivatives of rational B-splines must be evaluated They can be obtained by differentiating Eq (513) For example, the rst and the second derivatives of rational B-splines required for motion constraints of velocity and acceleration become:
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) ) R(j1k ( x ) = Wj N (j1k ( x ) , ,
Wj N j ,k ( x ) - Wj N j ,k ( x ) Wj N (j1,k) ( x )
j =1 n j =1
n Wj N j ,k ( x ) , j =1
(515)
R(j 2k) ( x ) = Wj N (j 2k) ( x ) , ,
W N
j j =1
j ,k
( x) -
n n (1) (1) (2 ) 2Wj N j ,k ( x ) Wj N j ,k ( x ) + Wj N j ,k ( x ) Wj N j ,k ( x ) j =1 j =1
n n Wj N j ,k ( x ) + 2Wj N j ,k ( x ) Wj N (j1,k) ( x ) j =1 j =1 n Wj N j ,k ( x ) j =1
(516)
The derivatives of the B-splines required above can be evaluated using the recurrence relationships described earlier The mth derivative of rational B-splines at a given point x satisfy the following relationship:
j =1
(m ) j ,k
( x ) = 0
(517)
This equation affords a convenient accuracy check when constructing the derivatives of rational B-splines Implementation of the Procedure The systematic procedure for implementing the rational B-spline approach is very similar to the scheme for applying B-splines described earlier The process is as follows, assuming the motion constraints have already been established: 1 2 3 4 Select the appropriate order of the rational B-splines to be used Choose the weight sequence Establish a knot sequence Determine the values of each rational B-spline or rational B-spline derivative at all points where motion constraints are imposed 5 Collect the values of rational B-splines and/or rational B-spline derivatives and form a linear system of equations using the motion constraints in Eqs (511) and (512) 6 Solve the solution of the linear system equations formed in the step above for the coef cients Aj 7 Evaluate rational B-splines and/or their derivatives as needed to determine displacement and/or its derivatives between motion constraints using Eqs (511) and (512) Example Applications The examples that follow illustrate the application of rational Bsplines to the synthesis of the rise portion of a DRD motion program The cases presented illustrate the effects that adjustments to the rational B-spline parameters have on the synthesized motion programs Note that in the cases below, normalized values for both displacement and time are used to provide a convenient basis for comparison of results Accordingly in the examples Sc denotes a normalized value for cam displacement and S the follower output motion dis-
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