  We show that the `erasing-larger-loops-first' (ELLF) method, which was first
introduced for erasing loops from the simple random walk on the Sierpinski
gasket, does work also for non-Markov random walks, in particular,
self-repelling walks to construct a new family of self-avoiding walks on the
Sierpinski gasket. The one-parameter family constructed in this method
continuously connects the loop-erased random walk and a self-avoiding walk
which has the same asymptotic behavior as the `standard' self-avoiding walk. We
prove the existence of the scaling limit and study some path properties: The
exponent governing the short-time behavior of the scaling limit varies
continuously in the parameter. The limit process is almost surely
self-avoiding, while its path Hausdorff dimension is the reciprocal of the
exponent above, which is strictly greater than 1.
