## 7 Receding Horizon Trajectory Planning
### Disturbance Estimation  
In the context of optimal trajectory planning, simultaneously computing optimal control policy, which is required to respond to unknown, sudden disturbances, and handling kinematics (i.e., obstacle avoidance) as well as dynamics (i.e., satisfying velocity and acceleration constraints) yields a challenging problem, especially for quadrotors. While geometry-based path planning techniques [95; 96] ensure the asymptotical optimality of a path, they however do not consider quadrotor dynamics. However, it is essential that the generation of an optimal control policy ensures dynamic feasibility. Thus, in [97; 98], LQR was incorporated into path planning, by which both dynamic feasibility and local optimality were guaranteed. However, local optimality does not necessarily yield global optimality [99]. In [26; 32], a set of motion primitives was used to find feasible trajectories ensuring both global and local optimality. When dealing with unknown disturbances, MPC is a more robust technique than LQR. In [32], MPC-based trajectory planning approach was proposed, ensuring both the local and global optimality. However, none of the aforementioned approaches formally guarantees stability and safety. [PERSON]'s analysis can be applied to confirm the local stability. Moreover, the terminal constraints set [100] can be incorporated. However, those measures are time consuming, which directly affects the real-time performance [101]. A set of CBFs was proposed for improving real-time performance without affecting the system stability in [102; 103; 104]. Recently, reference governors-based techniques were proposed in [105; 106], enforcing safety constraints. It is natural that designing a path planer is followed by the actual controller to maneuver MAV. In such approaches, a reference governor can be used to handle the stability and constraint satisfaction separately to ensure system stability [107]. Figure 8: The basic idea of MPC-based receding horizon planning. MPC-based receding horizon planning predicts the optimal control policy \(\mathbf{u}^{*}\) at each iteration to minimize the given cost function \(J\). The above approaches are employed to estimate optimal control policy for safe navigation while imposing stability either using Lyapunov functions or reference governors. On the other hand, [PERSON] et al. [108] proposed to obtain an optimal control policy using a state-dependent distance metric (SDDM). They have modeled the system dynamics as a linear, time-invariant as follows:  
\[\dot{\mathbf{x}}=A\mathbf{x}+B\mathbf{u}, \tag{61}\]  
where \(\mathbf{u}\) indicates the control input. The system state, i.e., \(\mathbf{x}:=(\mathbf{p}(t),\mathbf{y}(t))\), consists of two parts: \(\mathbf{p}\) and \(\mathbf{y}\), where \(\mathbf{p}(t)\) denotes the quadrotor position at a given time \(t\) and \(\mathbf{y}(t)\) describes the higher-order terms, e.g., velocity, acceleration, etc. In the latter work, the quadratic norm was utilized to represent the error between robot position and close-in obstacles positions. The quadratic norm is defined as \(\left\|\mathbf{p}\right\|_{R}:=\sqrt{\mathbf{p}^{T}R\mathbf{p}}\), where \(R\) is a symmetric positive matrix. \(R[\psi_{z}]\) is fully determined by the MAV heading direction \(\psi_{z}\) at a given time instance as follows:  
\[R[\psi_{z}]= \begin{cases}c_{2}I+(c_{1}-c_{2})\frac{\psi_{z}\psi_{z}^{T}}{\| \psi_{z}\|^{2}},&\text{if }\psi_{z}\
eq 0\\ c_{1}I,&\text{otherwise}\end{cases} , \tag{62}\]  
where both \(c_{1}\) and \(c_{2}\) are predefined scales such that \(c_{2}>c_{1}>0\); this process is called the SDDM, trajectory will be bounded incorporating SDDM information. Since quadrotor dynamics are linear, a reference governor [106] is introduced to maintain safety and stability. Other than LQR and MPC, there exist several receding horizon-based techniques for optimal trajectory planning, as given in Table 2.