Let $P(x, y)$, and since $P$ is the midpoint of $AB$, we can set $A(x-m, y-n), B(x+m, y+n)$, then
$$
\begin{array}{l}
4(x-m)^{2}-(y-n)^{2}=4, \\
4(x+m)^{2}-(y+n)^{2}=4 .
\end{array}
$$

Also, $O A \perp O B, x^{2}+y^{2}=m^{2}+n^{2}$
(1) and (2) subtracted, $4 m x=n y$.
(1) and (2) added, $4 x^{2}-y^{2}+4 m^{2}-n^{2}=4$.

From (3) and (4), we get $m^{2}=\frac{y^{2}\left(x^{2}+y^{2}\right)}{16 x^{2}+y^{2}}, n^{2}=\frac{16 x^{2}\left(x^{2}+y^{2}\right)}{16 x^{2}+y^{2}}$, substituting these into (5), $3\left(4 x^{2}-y^{2}\right)^{2}=4\left(16 x^{2}+y^{2}\right)$, which is the required equation of the locus.
Example 6 Draw two perpendicular rays from the center $O$ of the hyperbola $x^{2}-\frac{y^{2}}{4}=1$, intersecting the hyperbola at points $A$ and $B$. Find the equation of the locus of the midpoint $P$ of the chord $AB$.