A caterpillar graph T(p(1), ..., p(r)) of order n = r + Sigma(r)(i=1) p(i), r >= 2, is a tree such that removing all its pendent vertices gives rise to a path of order r. In this paper we establish a necessary and sufficient condition for a real number to be an eigenvalue of the Randic matrix of T(p(1), ..., p(r)). This result is applied to determine the extremal caterpillars for the Randic energy of T(p(1), ..., p(r)) for cases r = 2 (the double star) and r = 3. We characterize the extremal caterpillars for r = 2. Moreover, we study the family of caterpillars T (p, n - p - q - 3, q) of order n, where q is a function of p, and we characterize the extremal caterpillars for three cases: q = p, q = n - p - b - 3 and q = b, for b is an element of {1, ..., n - 6} fixed. Some illustrative examples are included.