This paper presents an algorithm that decomposes an arbitrary set F of multivariate polynomials involving parameters into finitely many sets Γi of (lexicographical) Gröbner bases G ij such that associated with each Γi there is a system Ai of parametric constraints, all the Ai's partition the parameter space, all the Gij's in each Γi remain Gröbner bases under specialization of the parameters satisfying the constraints in Ai, and for each i the Gröbner bases Gij together with their corresponding W-characteristic sets form a normal characteristic decomposition of F. The sets of Gröbner bases computed by the algorithm provide a comprehensive characteristic decomposition of F that is structure-invariant under the associated constraints and possesses many algebraic and geometric properties, including most of the notable properties on comprehensive Gröbner systems and comprehensive triangular decomposition. Some of these properties are highlighted in the paper and advantages of the proposed algorithm are discussed briefly from the aspects of methodological simplicity and computational performance using illustrative examples and preliminary experiments.