Flatness of Maps of Form $R \to R[X_1,...,X_n]/I$

Question

Let $R$ be a regular commutative ring and consider a map of type $\phi: R \to S:=R[X_1,..., X_n]/I$ for finitely generated ideal $I =\langle a_1,...,a_m \rangle \subset R[X_1,..., X_n]$.

Main Question: Are there "standard techniques" in dependence of $I$ & its generators giving criteria when $\phi$ is actually flat?

The question is primarily motivated by this nice construction giving an explicit family over $\Bbb A_k^4 \times \Bbb A_k^N$ (...where $k$ algebraically closed base field and the right $\Bbb A_k^N \cong Mat_{7,2}$ is regarded as standard affine open subscheme of the Grassmanian $Gr(7,10)$).
To recall here Mariano Suárez-Álvarez's construction:

Let $(a_{i,j})$ be an arbitrary $7\times 3$ matrix of scalars, and let $F_1,\dots,,F_7$ be the linear combinations of monomials indicated in the rows of following table: $$\begin{array}{*{12}{c}} X^2 & Y^2 & Z^3 & W^2 & XY & XZ & XW & YZ & YW & ZW \\ \hline 1 & & & & & & & a_{1,1} & a_{1,2} & a_{1,3} \\ & 1 & & & & & & a_{2,1} & a_{2,2} & a_{2,3} \\ & & 1 & & & & & a_{3,1} & a_{3,2} & a_{3,3} \\ & & & 1 & & & & a_{4,1} & a_{4,2} & a_{4,3} \\ & & & & 1 & & & a_{5,1} & a_{5,2} & a_{5,3} \\ & & & & & 1 & & a_{6,1} & a_{6,2} & a_{6,3} \\ & & & & & & 1 & a_{7,1} & a_{7,2} & a_{7,3} \\ \end{array}$$ Now let $a$, $b$, $c$ $d$ be four scalars and consider the ideal generated by the $7$ polynomials $$ F_1(X-a,Y-b,Z-d,W-d), \dots, F_7(X-a,Y-b,Z-d,W-d) $$ and all the polynomials $(X-a)^i(Y-b)^j(Z-c)^k(W-d)^l$ with $i+j+k+l=3$.

This gives you a $25$-dimensional family of ideals of colength $8$, parametrized by a point in $k^4\times M_{7,2}(k)$.
Viewing the entries of the matrix and the coordinates of the point $(a,b,c,d)$ as varibles now, the ideal generated by those seven polynomials in $k[X,Y,Z,W,a,b,c,d,a_{1,1},\dots,a_{7,3}]$ define subscheme $Z$ in $k^4\times M_{7,2}(k)\times k^4$. The restriction of the map $p:k^4\times M_{7,2}(k)\times k^4\to k^4\times M_{7,2}(k)$ projecting on the first two factors to $Z$ is a map $Z\to k^4\times M_{7,2}(k)$ which is a flat family of subschemes of $k^4$, the fiber of $p$.

To phrase it in terms of notations above here $R:=k[a,b,c,d,a_{1,1},\dots,a_{7,3}]$ and $I$ is generated by $ F_1(X-a,Y-b,Z-d,W-d), \dots, F_7(X-a,Y-b,Z-d,W-d) $ and all the polynomials $(X-a)^i(Y-b)^j(Z-c)^k(W-d)^l$ with $i+j+k+l=3$.

So the subquestion is how to see here that $S:=k[X,Y,Z,W,a,b,c,d,a_{1,1},\dots,a_{7,3}]/I=R[X,Y, Z,W]/I$ is flat over $R$. And to which amount this would allow to extract a general technique asked for in Main question to deduce flatness from data of ideal $I$ and its generators?