Step1 (initialize): Sample at least n0 ?‰? m initial locations, where m is
the number of field model basis functions gj . These initial sampling locations
could be randomly chosen close to a starting robot location. Remove these
sampling points from ??–. Denote the remaining sampling space as ??–n0 . After
Step1, the robot will be at location xn0 .
Step2: Pick a starting guess for the parameter set B, and denote that set
as ?†B
n0 .
Step3: For n ?‰? n0 form measurement vector Z = (zj)1?‰¤j?‰¤n after n
samples. Form basis function matrix Mn as shown in equation (4), using the
current parameter set B estimate.
45
Dan O. Popa and Frank L. Lewis
Step4: For all sampling points x in the parameter space ??–n calculate m(x)
using equation (6). Select the next sampling location to be xnext for which
m(x) is minimal.
Step5: Command the robot to move from location xn to location xnext.
While in transit, sample at k discrete locations xn+1, xn+2, ..., xn+k = xnext.
The number k can be chosen depending on the length of travel. Remove these
points from ??–n to form ??–n+k.
Step6: Using a nonlinear iterative root solver (such as fmincon in MATLAB)
solve the constrained nonlinear equation
z1 = a0 + a1g1(x1,Bn+k) + ?· ?· ?· + amgm(x1,Bn+k)
z2 = a0 + a1g1(x2,Bn+k) + ?· ?· ?· + amgm(x2,Bn+k)
...
zn = a0 + a1g1(xn,Bn+k) + ?· ?· ?· + amgm(xn,Bn+k)
,
to find ?† An+k = (a0, a1...am), ?†B
n+k = (b1, ..., bq). Constrain the nonlinear
root-finding problem by having B located inside a cubic element inside its
coarse grid from the q-dimensional space.
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