Let $f:\mathbb R^n\times\mathbb R^m\to\mathbb R^n$, $L:\mathbb R^n\times\mathbb R^m\to\mathbb R$, and the endpoint constraint maps be $C^{p+1}$ on a neighbourhood of a feasible trajectory $(x^*,u^*)$, with $x^*\in C^{p+1}([0,T];\mathbb R^n)$ and $u^*\in C^p([0,T];\mathbb R^m)$. Let a direct collocation scheme of order $p$ use a quasi-uniform mesh with maximum step size $h$ and local defect operator $R_{k,h}$ satisfying, for some constants $C_{\mathrm{loc}}>0$ and $h_0>0$ independent of $k$ and $h$, \begin{align*} |R_{k,h}(x^*,u^*)|\le C_{\mathrm{loc}}h_k^{p+1}\qquad\text{for }0<h\le h_0. \end{align*} Then the sampled exact trajectory has local collocation residual bounded by $C_{\mathrm{loc}}h^{p+1}$ in the Euclidean norm on each interval, and its accumulated residual satisfies \begin{align*} \max_{0\le j\le N}\left|\sum_{k=0}^{j-1}R_{k,h}(x^*,u^*)\right|\le C_{\mathrm{glob}}h^p \end{align*} for a constant $C_{\mathrm{glob}}>0$ independent of $h$. If, in addition, $(x^*,u^*)$ is an isolated normal local minimiser, the continuous second variation is coercive on the critical cone, LICQ and strict complementarity hold for the active endpoint and path constraints, and the linearised discrete KKT systems are uniformly invertible for all sufficiently small $h$, then there are locally optimal collocation solutions $(x_h,u_h)$ with \begin{align*} \max_{0\le k\le N}|x_h(t_k)-x^*(t_k)|+\|u_h-u^*\|_{L^2(0,T)}\le C_{\mathrm{sol}}h^p \end{align*} for a constant $C_{\mathrm{sol}}>0$ independent of $h$.