Let $x=x_0{x}_1\cdots$ be a fixed point of a substitution on the alphabet $\left\{a,b\right\},$ and let $U_a=\left(\begin{array}{cc}\hfill -1& \hfill -1\\ \hfill 0& \hfill 1\end{array}\right)$ and $U_b=\left(\begin{array}{cc}\hfill 1& \hfill 1\\ \hfill 0& \hfill 1\end{array}\right)$. We give a complete classification of the substitutions $\sigma :{\left\{a,b\right\}}^{☆}$ according to whether the sequence of matrices ${\left(U_{x_0}{U}_{x_1}\cdots U_{x_n}\right)}_{n=0}^{\infty }$ is bounded or unbounded. This corresponds to the boundedness or unboundedness of the oriented walks generated by the substitutions.