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THB5 8/15/03 1:52 PM Page 137

CAM MOTION SYNTHESIS USING SPLINE FUNCTIONS 137

higher order implicit methods for stiff equations have been made in an effort to avoid step
size limitations due to stability considerations. Both collocation and the implicit Runge-
Kutta (Butcher, 1964) methods have been investigated for such cases (Wright, 1970).
It has been shown that the collocation method for the general first-order system yields
results that are equivalent to those obtained by implicit Runge-Kutta methods (Weiss,
1974; Wright, 1970). However, the ease and generality of application of the collocation
method make it very suitable for practical, difficult (stiff) ordinary differential equations
(Shampine and Gear, 1979; Hulme and Daniel, 1974). Further, the solution requires only
algebraic operations to find the unknown coefficients. In addition, for higher order differ-
ential equations the solution can be obtained without reducing the equation to a system of
first-order equations (Cerutti, 1974).
In applying the method the domain of interest is divided into small elements (sub-
th
domains). Also it has been suggested that the roots (Gaussian points) of the P Legendre
orthogonal polynomial be chosen as p collocation points within each of these elements (de
Boor and Swartz 1973; Stroud and Secrest, 1986). When this is done, the resulting global
p+q
th
truncation error in the solution of a q order differential equation will be of order 0(d ),
where d is the maximum length of an element. At the ends of each element, the approxi-
2P
mation solution and its first q - 1 derivatives have errors of order 0(d ).
As is the case in the synthesis of output motions, B-splines are used as the known func-
tions. Evaluation of B-splines and their derivatives, using recurrence relations was dis-
cussed earlier. Because both the spline interpolation used in the output motion synthesis
and spline collocation use B-splines as basis functions and because both require the solu-
tion of linear systems, much of the same software can be used for both procedures. In the
collocation procedure, the solution (U(t)) of a general differential equation is represented
as a linear combination of B-splines, written as follows:
n
U t () = Â A N (). (5.25)
t
j
j k
,
j=1
It is obvious that these expressions for the solution of a differential equation are similar
to those of the spline interpolation procedure described earlier. The procedure for apply-
ing the spline collocation method to the solution of differential equations is very simple
and is described in the following outline:
1. Discretize the domain of interest of the differential equation into e subdomains
(elements). In each element, p Gaussian points are chosen as collocation points. The p
th
Gaussian points in each element are the roots of the p orthogonal Legendre polynomial.
Not including initial or boundary conditions, there are p·e collocation points on the range
of the differential equation.
2. Establish the knot sequence at both ends (t = 0 and t = 1) of the domain and at
the ends of the e elements above. Following de Boor (1978), the knots at t = 0 and at
t = 1 are repeated k times for a B-spline of order k. Also, at the ends, mesh points
(not including points at t = 0 and t = 1) of each element have the knots repeated p times
for p Gaussian points. Hence, there are a total of 2k + p(e - 1) knots. For q conditions
(initial or boundary) and p·e collocation points to be satisfied, n B-splines are required
and n = q + p·e. The order of B-splines equals q + p. In addition, the number of knots,
2k + p(e - 1), equals k + n.
3. Determine the values of each B-spline and its derivatives at the points where
conditions are given and at collocation points.
4. Formulate a linear system of equations that can be solved for the coefficients A j
in Eq. (5.25). For example, the differential equation ax≤(t) + bx¢(t) + cx(t) = F(t) with