In this paper we propose a family of affine scaling interior point algorithms, called galpv4, using a primal approach, based on a large class of differential barrier functions. We show that these algorithms are in fact an extension and generalization of the classical affine scaling algorithm based on the well-known log barrier function. After carrying out a complete convergence analysis, we select some of these algorithms for comparison with the classical affine scaling algorithm, performed with the help of the familiar Netlib test set.