are obtained by forward integrating system dynamics in (performing a rollout) using the latest estimation of the optimal control law with a fixed switching time vector $\bar{\vt}$. For simplicity of notation, in the followings we have dropped the dependencies of the nominal trajectories and consequently the approximated LQ problem with respect to $\bar{\vt}$. The linearized system dynamics of the equivalent system in around these nominal trajectories are defined as $$\begin{aligned}
& \cfrac{d(\delta\vx)}{dz} = (t_{i}-t_{i-1}) \left( \vA_i(z)\delta\vx + \vB_i(z)\delta\vu \right) \quad  i-1 \leq z < i  \notag \\
& \vA_i(z) = \frac{\partial \vf_i(\bar{\vx}(z),\bar{\vu}(z))}{\partial\vx}, \quad \vB_i(z) = \frac{\partial \vf_i(\bar{\vx}(z),\bar{\vu}(z))}{\partial\vu} ,
\label{eq:dynamics_linear_approximation}\end{aligned}$$ where $\delta \vx(z) = \vx(z)-\bar{\vx}(z)$, $\delta\vu(z) = \vu(z)-\bar{\vu}(z)$. The quadratic approximation of the cost function in is $$\begin{aligned}
& \tilde{J} = \tilde{\Phi}(\vx_{t_f})+ \sum_{i=1}^I{\int_{i-1}^{i} { (t_{i}-t_{i-1})\tilde{L}_i(z,\vx,\vu)dz}} \notag \\
&\tilde{\Phi}_f(\vx) = \ q_f + \vq_f^\top \delta\vx + \frac{1}{2} \delta\vx^\top \vQ_f \delta\vx \notag \\
& \tilde{L}_i(z,\vx,\vu) = q_i(z) + \delta\vx^\top \vq_i(z) + \delta\vu^\top \vr_i(z)  + \delta\vx^T \vP_i(z) \delta\vu \notag \\
& \hspace{15mm} + \frac{1}{2} \delta\vx^\top \vQ_i(z) \delta\vx + \frac{1}{2} \delta\vu^\top \vR_i(z) \delta\vu.  \label{eq:cost_quadratic_approximation}\end{aligned}$$ In the above, $q(z)$, $\vq(z)$, $\vr(z)$, $\vQ(z)$, $\vP(z)$, and $\vR(z)$ are the coefficients of the Taylor expansion of the cost function in evaluated at the nominal trajectories. The optimal control law 