# Cutting-Edge Trajectory Optimization through Quantum Annealing
## 3 Transcription Procedure  
The goal of trajectory optimization is to find the optimal control inputs (\(u(t)\)) that steer a system to the desired final state (\(x_{f}\)) while considering the trade-offs between achieving the objective and minimizing the associated cost. Thus, given an initial condition (\(x_{0}\)) and a control input trajectory defined over finite interval \(t\in[t_{0},t_{f}]\), the cost function (\(J\)) in the Bolza form can be computed as follows:  
\[J=\chi(\mathbf{x}(t_{0}),\mathbf{x}(t_{f}))+\int_{t_{0}}^{t_{f}}\mathcal{L}( \mathbf{x}(t),\mathbf{u}(t),t)dt, \tag{7}\]  
where \(\chi(\mathbf{x}(t_{0}),\mathbf{x}(t_{f}))\) is the cost component depending on initial state \(\mathbf{x}(t_{0})\) and final state \(\mathbf{x}(t_{f})\) of the system, and \(\int_{t_{0}}^{t_{f}}\mathcal{L}(\mathbf{x}(t),\mathbf{u}(t),t)dt\) represents the integral of the instantaneous cost function \(\mathcal{L}\) over the entire trajectory, which depends on state \(\mathbf{x}(t)\), control input \(\mathbf{u}(t)\), and time \(t\). Moreover, the trajectory optimization problem must align with the underlying dynamics of the system. In the proposed method, the dynamic systems considered are linear or linearized; thus, the trajectory optimization problem can be expressed in the following comprehensive form:  
Figure 1: Scheme of a simple embedding procedure. \[ \begin{split}\min_{\mathbf{u}(t)}&\chi(\mathbf{x}(t_{0}), \mathbf{x}(t_{f}))+\int_{t_{0}}^{t_{f}}\mathcal{L}(\mathbf{x}(t),\mathbf{u}(t ),t)\,dt\\ \text{subject to}&\dot{\mathbf{x}}(t)=A\mathbf{x}(t)+B \mathbf{u}(t),\quad\forall t\in[t_{0},t_{f}]\\ &\mathbf{x}(t_{0})=\mathbf{x}_{0},\quad\mathbf{x}(t_{f})= \mathbf{x}_{f}.\end{split}  \tag{8}\]  
Equation (8) provides a concise representation of the trajectory optimization problem, encapsulating the goal of reaching a desired final state while considering the cost incurred along the trajectory, subject to the system's dynamic behavior and initial condition. In particular, \(\mathbf{x}=[\mathbf{s}(t),\dot{\mathbf{s}}(t)]^{T}\) represents the state vector, resulting in a total of six components. This vector is composed of both position (\(\mathbf{s}(t)=[x,y,z]^{T}\)) and velocity (\(\dot{\mathbf{s}}(t)=[v_{x},v_{y},v_{z}]^{T}\)) along the three trajectory components. The proposed transcription procedure aims to map the common trajectory optimization problem (Equation (8)) into a binary optimization problem in a QUBO form (Equation (6)), which can be effectively addressed using QA techniques. Central to the transcription procedure is the discretization of the trajectory by representing the state variable and its derivatives through polynomial functions across finite interval \(t\in[t_{0},t_{f}]\). In contrast to the usual practice of determining the polynomial degree (\(n\)) based on the number of boundary conditions, our method increases the polynomial degree by a specific factor. Consequently, the ultimate polynomial degree is obtained as the combination of two components: \(n=l+(m-1)\), where \(m\) represents the number of boundary conditions and \(l\) signifies the count of independent variables. These independent variables are essentially the coefficients of the polynomials that play a crucial role, acting as pivotal crossing points capable of altering the trajectory's shape. This core concept forms the foundation of the proposed optimization approach, where both the trajectory and its derivatives can be expressed as functions of \(l\) independent variables. In our procedure, we chose to minimize the number of variables, thus selecting \(l=1\), leading to polynomial degree \(n\) being simply equal to \(m\). The decision to set \(l=1\) was primarily dictated by a desire to keep the approach simple and practical, making it easier to convert to the QUBO format. Moreover, it is worth noting that in other works, a similar choice was made to set \"L\" to 1 [3; 36]. This approach is commonly adopted in trajectory optimization research for its practical advantages.