# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021009

## A control parametrization based path planning method for the quad-rotor uavs

 1 College of Electrical Engineering, Sichuan University, Sichuan 610065, China 2 School of Aeronautics and Astronautics, Sichuan University, Sichuan 610065, China

* Corresponding author: jiyuandong@aliyun.com

Received  July 2020 Revised  October 2020 Published  January 2021

Fund Project: The first author is supported in part by the National Natural Science Foundation of China under grant 62071317, in part by the Science and Technology on Space Intelligent Control Laboratory under grant KGJZDSYS-2018-03, in part by the Sichuan Science and Technology Program under grant 2019YJ0105, and in part by the Strategic Rocket Innovation Fund

A time optimal path planning problem for the Quad-rotor unmanned aerial vehicles (UAVs) is investigated in this paper. A 3D environment with obstacles is considered, which makes the problem more challenging. To tackle this challenge, the problem is formulated as a nonlinear optimal control problem with continuous state inequality constraints and terminal equality constraints. A control parametrization based method is proposed. Particularly, the constraint transcription method together with a local smoothing technique is utilized to handle the continuous inequality constraints. The original problem is then transformed into a nonlinear program. The corresponding gradient formulas for both of the cost function and the constraints are derived, respectively. Simulation results show that the proposed path planning method has less tracking error than that of the rapid-exploring random tree (RRT) algorithm and that of the A star algorithm. In addition, the motor speed has less changes for the proposed algorithm than that of the other two algorithms.

Citation: Jinlong Guo, Bin Li, Yuandong Ji. A control parametrization based path planning method for the quad-rotor uavs. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021009
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##### References:
The quad-rotor UAV and the coordinate systems
Control Parametrization
An Illustration of the Local Smoothing Technique
Example 1: the planned path of the UAV with Algorithm 1
Example 1: the optimal states
Example 1: the optimal control inputs
Example 2: the planned path of the UAV with Algorithm 1
Example 2: the optimal states
Example 2: the optimal control inputs
Example 3: The planned path of UAV with Algorithm 1
Example 3: the optimal states
Example 3: the optimal control inputs
Example 3: The planned path of UAV with RRT
Example 3: The planned path of UAV with A star
The structure of PD controller
Example 4: the tracking trajectory of the UAV with Algorithm 1
Example 4: the tracking trajectory of the UAV with RRT
Example 4: the tracking trajectory of the UAV with A star
Example 4: the motor speed
Algorithm 1: an iteration algorithm for solving Problem $P(p)$
 Initialization: Set $\varepsilon=\varepsilon_0$, $\gamma=\varepsilon/3$ and $\varepsilon_{min}=10^{-3}\varepsilon_0$. Step 1. Solve the Problem $P_{\varepsilon, \gamma}(p)$ for the optimal solution $K_{\varepsilon, \gamma}^{*}$}. Step 2. For each $i$, check the feasibility of $g_{i}({\bf x}(t))\ge 0$ with $K_{\varepsilon, \gamma}^{*}$. Step 3. If all the constraints in Step 2 are satisfied, then go to the Step 5. Otherwise, go to the Step 4. Step 4. Set $\gamma=\gamma/2$ and go to Step 1. Step 5. Set $\varepsilon=\varepsilon/10$, $\gamma=\gamma/10$, and go to Step 1. Stopping criterion: Algorithm 1 stops when $\varepsilon\leq\varepsilon_{min}$.
 Initialization: Set $\varepsilon=\varepsilon_0$, $\gamma=\varepsilon/3$ and $\varepsilon_{min}=10^{-3}\varepsilon_0$. Step 1. Solve the Problem $P_{\varepsilon, \gamma}(p)$ for the optimal solution $K_{\varepsilon, \gamma}^{*}$}. Step 2. For each $i$, check the feasibility of $g_{i}({\bf x}(t))\ge 0$ with $K_{\varepsilon, \gamma}^{*}$. Step 3. If all the constraints in Step 2 are satisfied, then go to the Step 5. Otherwise, go to the Step 4. Step 4. Set $\gamma=\gamma/2$ and go to Step 1. Step 5. Set $\varepsilon=\varepsilon/10$, $\gamma=\gamma/10$, and go to Step 1. Stopping criterion: Algorithm 1 stops when $\varepsilon\leq\varepsilon_{min}$.
Parameters of the UAV
 $L$ $0.2\ m$ $M$ $1.5\ kg$ $g$ $9.8\ m/ s^{2}$ ${I}_{x}$ $0.0075\ kg\cdot m^2$ ${I}_{y}$ $0.0075\ kg\cdot m^2$ ${I}_{z}$ $0.013\ kg\cdot m^2$ ${K}_{1}, {K}_{2}$ $0.06\ N/m/s$ ${K}_{3}$ $0.09\ N/m/s$ ${K}_{4}, {K}_{5}$ $0.002\ N/m/s$ ${K}_{6}$ $0.1\ N/m/s$ $C$ $10^{-7}$ ${K}_{v}$ $1.5\times10^{-5}\ N/m/s$
 $L$ $0.2\ m$ $M$ $1.5\ kg$ $g$ $9.8\ m/ s^{2}$ ${I}_{x}$ $0.0075\ kg\cdot m^2$ ${I}_{y}$ $0.0075\ kg\cdot m^2$ ${I}_{z}$ $0.013\ kg\cdot m^2$ ${K}_{1}, {K}_{2}$ $0.06\ N/m/s$ ${K}_{3}$ $0.09\ N/m/s$ ${K}_{4}, {K}_{5}$ $0.002\ N/m/s$ ${K}_{6}$ $0.1\ N/m/s$ $C$ $10^{-7}$ ${K}_{v}$ $1.5\times10^{-5}\ N/m/s$
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