American Institute of Mathematical Sciences

March  2022, 18(2): 1079-1100. 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  March 2022 Early access  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 and Management Optimization, 2022, 18 (2) : 1079-1100. doi: 10.3934/jimo.2021009
References:
 [1] X. Cheng, H. Li and R. Zhang, Autonomous trajectory planning for space vehicles with a Newton Kantorovich/convex programming approach, Nonlinear Dynamics, 89 (2017), 2795-2814.  doi: 10.1007/s11071-017-3626-7. [2] K. Daniel, A. Nash, S. Koenig and A. Felner, Theta*: Any-angle path planning on grids, Journal of Artificial Intelligence Research, 39 (2010), 533-579.  doi: 10.1613/jair.2994. [3] A. Dehghan and M. Shah, Binary quadratic programing for online tracking of hundreds of people in extremely crowded scenes, IEEE Transactions on Pattern Analysis and Machine Intelligence, 40 (2018), 568-581. [4] H. Ergezer and and K. Leblebiciolu, 3D path planning for multiple UAVs for maximum information collection, Journal of Intelligent and Robotic Systems, 73 (2014), 737-762. [5] L. D. Filippis, G. Guglieri and F. Quagliotti, Path planning strategies for UAVS in 3D environments, Journal of Intelligent and Robotic Systems, 65 (2012), 247-264. [6] J. L. Foo, J. Knutzon, V. Kalivarapu, J. Oliver and E. Winer, Path planning of unmanned aerial vehicles using b-splines and particle swarm optimization, Journal of Aerospace Computing Information and Communication, 6 (2009), 271-290.  doi: 10.2514/1.36917. [7] B. T. Gatzke, W. Kang and H. Zhou, Trajectory optimization for helicopter unmanned aerial vehicles (UAVs), NPS Thesis, (2010). [8] L. Jennings, K. L. Teo, M. Fisher and C. J. Goh, MISER3 version 2, optimal control software, theory and user manual, Department of Mathematics. The University of Western Australia, Australia, (1997). [9] T. Ju, S. Liu, J. Yang and D. Sun, Rapidly exploring random tree algorithm-based path planning for robot-aided optical manipulation of biological cells, IEEE Transactions on Automation Science and Engineering, 11 (2014), 649-657. [10] V. Kroumov and J. Yu, 3D path planning for mobile robots using annealing neural network, International journal of innovative computing information and control, 6 (2009). [11] Y. Liang, J. T. Qi, J. Z. Xiao and Y. Xia, A literature review of UAV 3D path planning, Proceeding of the 11th World Congress on Intelligent Control and Automation, (2014), 2376–2381. [12] C. Y. Liu, Z. H. Gong, K. L. Teo, J. Sun and L. Caccetta, Robust multi-objective optimal switching control arising in 1, 3-propanediol microbial fed-batch process, Nonlinear Analysis Hybrid Systems, 25 (2017), 1-20.  doi: 10.1016/j.nahs.2017.01.006. [13] J. Omer and J. Farges, Hybridization of nonlinear and mixed-integer linear programming for aircraft separation with trajectory recovery, IEEE Transactions on Intelligent Transportation Systems, 14 (2013), 1218-1230. [14] B. Oommen, S. Iyengar, N. Rao and R. Kashyap, Robot navigation in unknown terrains using learned visibility graphs. Part Ⅰ: The disjoint convex obstacle case, IEEE Journal on Robotics and Automation, 3 (1987), 672-681. [15] P. Pharpatara, B. Hriss and Y. Bestaoui, 3-D trajectory planning of aerial vehicles using RRT, IEEE Journal on Robotics and Automation, 25 (2017), 1116-1123.  doi: 10.1109/TCST.2016.2582144. [16] K. L. Teo, C. J. Goh and K. H. Wong, A unified computational approach for optimal control problems, Pitman Monographs and Surveys in Pure and Applied Mathematics, John Wiley and Sons, Inc., New York, 1991. [17] J. Votion and Y. Cao, Diversity-based cooperative multivehicle path planning for risk management in costmap environments, IEEE Transactions on Industrial Electronics, 66 (2019), 6117-6127. [18] F. Yang, K. L. Teo, R. Loxton, V. Rehbock, B. Li, C. Yu and L. Jennings, Visual MISER: An efficient user-friendly visual program for solving optimal control problems, Journal of Industrial and Management Optimization (JIMO), 12 (2016), 781-810.  doi: 10.3934/jimo.2016.12.781. [19] C. Yu, K. L. Teo, L. Zhang and and Y. Bai, On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem, Journal of Industrial and Management Optimization (JIMO), 8 (2012), 485-491.  doi: 10.3934/jimo.2012.8.485. [20] J. Yuan, C. Liu, X. Zhang, J. Xie, E. Feng, H. Yin and Z. Xiu, Optimal control of a batch fermentation process with nonlinear time-delay and free terminal time and cost sensitivity constraint, Journal of Process Control, 44 (2016), 41-52.  doi: 10.1016/j.jprocont.2016.05.001. [21] B. Zhao, B. Xian, Y. Zhang and X. Zhang, Nonlinear robust adaptive tracking control of a quadrotor UAV via immersion and invariance methodology, IEEE Transactions on Industrial Electronics, 62 (2015), 2891-2902.

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References:
 [1] X. Cheng, H. Li and R. Zhang, Autonomous trajectory planning for space vehicles with a Newton Kantorovich/convex programming approach, Nonlinear Dynamics, 89 (2017), 2795-2814.  doi: 10.1007/s11071-017-3626-7. [2] K. Daniel, A. Nash, S. Koenig and A. Felner, Theta*: Any-angle path planning on grids, Journal of Artificial Intelligence Research, 39 (2010), 533-579.  doi: 10.1613/jair.2994. [3] A. Dehghan and M. Shah, Binary quadratic programing for online tracking of hundreds of people in extremely crowded scenes, IEEE Transactions on Pattern Analysis and Machine Intelligence, 40 (2018), 568-581. [4] H. Ergezer and and K. Leblebiciolu, 3D path planning for multiple UAVs for maximum information collection, Journal of Intelligent and Robotic Systems, 73 (2014), 737-762. [5] L. D. Filippis, G. Guglieri and F. Quagliotti, Path planning strategies for UAVS in 3D environments, Journal of Intelligent and Robotic Systems, 65 (2012), 247-264. [6] J. L. Foo, J. Knutzon, V. Kalivarapu, J. Oliver and E. Winer, Path planning of unmanned aerial vehicles using b-splines and particle swarm optimization, Journal of Aerospace Computing Information and Communication, 6 (2009), 271-290.  doi: 10.2514/1.36917. [7] B. T. Gatzke, W. Kang and H. Zhou, Trajectory optimization for helicopter unmanned aerial vehicles (UAVs), NPS Thesis, (2010). [8] L. Jennings, K. L. Teo, M. Fisher and C. J. Goh, MISER3 version 2, optimal control software, theory and user manual, Department of Mathematics. The University of Western Australia, Australia, (1997). [9] T. Ju, S. Liu, J. Yang and D. Sun, Rapidly exploring random tree algorithm-based path planning for robot-aided optical manipulation of biological cells, IEEE Transactions on Automation Science and Engineering, 11 (2014), 649-657. [10] V. Kroumov and J. Yu, 3D path planning for mobile robots using annealing neural network, International journal of innovative computing information and control, 6 (2009). [11] Y. Liang, J. T. Qi, J. Z. Xiao and Y. Xia, A literature review of UAV 3D path planning, Proceeding of the 11th World Congress on Intelligent Control and Automation, (2014), 2376–2381. [12] C. Y. Liu, Z. H. Gong, K. L. Teo, J. Sun and L. Caccetta, Robust multi-objective optimal switching control arising in 1, 3-propanediol microbial fed-batch process, Nonlinear Analysis Hybrid Systems, 25 (2017), 1-20.  doi: 10.1016/j.nahs.2017.01.006. [13] J. Omer and J. Farges, Hybridization of nonlinear and mixed-integer linear programming for aircraft separation with trajectory recovery, IEEE Transactions on Intelligent Transportation Systems, 14 (2013), 1218-1230. [14] B. Oommen, S. Iyengar, N. Rao and R. Kashyap, Robot navigation in unknown terrains using learned visibility graphs. Part Ⅰ: The disjoint convex obstacle case, IEEE Journal on Robotics and Automation, 3 (1987), 672-681. [15] P. Pharpatara, B. Hriss and Y. Bestaoui, 3-D trajectory planning of aerial vehicles using RRT, IEEE Journal on Robotics and Automation, 25 (2017), 1116-1123.  doi: 10.1109/TCST.2016.2582144. [16] K. L. Teo, C. J. Goh and K. H. Wong, A unified computational approach for optimal control problems, Pitman Monographs and Surveys in Pure and Applied Mathematics, John Wiley and Sons, Inc., New York, 1991. [17] J. Votion and Y. Cao, Diversity-based cooperative multivehicle path planning for risk management in costmap environments, IEEE Transactions on Industrial Electronics, 66 (2019), 6117-6127. [18] F. Yang, K. L. Teo, R. Loxton, V. Rehbock, B. Li, C. Yu and L. Jennings, Visual MISER: An efficient user-friendly visual program for solving optimal control problems, Journal of Industrial and Management Optimization (JIMO), 12 (2016), 781-810.  doi: 10.3934/jimo.2016.12.781. [19] C. Yu, K. L. Teo, L. Zhang and and Y. Bai, On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem, Journal of Industrial and Management Optimization (JIMO), 8 (2012), 485-491.  doi: 10.3934/jimo.2012.8.485. [20] J. Yuan, C. Liu, X. Zhang, J. Xie, E. Feng, H. Yin and Z. Xiu, Optimal control of a batch fermentation process with nonlinear time-delay and free terminal time and cost sensitivity constraint, Journal of Process Control, 44 (2016), 41-52.  doi: 10.1016/j.jprocont.2016.05.001. [21] B. Zhao, B. Xian, Y. Zhang and X. Zhang, Nonlinear robust adaptive tracking control of a quadrotor UAV via immersion and invariance methodology, IEEE Transactions on Industrial Electronics, 62 (2015), 2891-2902.
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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