Given two Banach spaces \(X\) and \(Y\), a metric space \(P\) and a set-valued map \(F:X\times P\rightrightarrows Y\), define \(G:P\rightrightarrows X\) by \(G(p)=\{x\in X:0\in F(x,p)\}\). One says that \(F\) is locally metrically regular in Robinson's sense around \((\bar{x},\bar{p},0)\in X\times P\times Y\) where \(0\in F(\bar{x},\bar{p})\), if there exist constants \(k,\mu>0\) and neighborhoods \(U\) of \(\bar{x}\), \(V\) of \(\bar{p}\), such that \[ d(x;G(p))\leq kd(0;F(x,p)) \] for all \(x\in U\), \(p\in V\) with \(d(0;F(x,p))<\mu\). This definition covers the well-known notions of metric regularity and metric subregularity of a map \(H:X\rightrightarrows Y\). The main result of this paper gives a sufficient condition for a map \(F\) to be locally metrically regular in Robinson's sense; at the same time, under this condition, it provides an estimation of the neighborhoods \(U\) and \(V\) and the constant \(\mu\). As applications, refinements of recent results on metric regularity and metric subregularity are given.