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doi: 10.3934/jimo.2021201
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 1 School of Electro-Optical Engineering, Changchun University of Science and Technology, Changchun 130012, Jilin, China 2 College of Control Science and Engineering, Zhejiang University, Hangzhou 310027, Zhejiang, China 3 School of Electrical Engineering, Sichuan University, Chengdu 610065, Sichuan, China 4 School of Aeronautics and Astronautics, Sichuan University, Chengdu 610065, Sichuan, China

* Corresponding author: Gaoqi Liu

Received  March 2021 Revised  September 2021 Early access November 2021

Unmanned Aerial Vehicles (UAVs) have been extensively studied to complete the missions in recent years. The UAV trajectory planning is an important area. Different from the commonly used methods based on path search, which are difficult to consider the UAV state and dynamics constraints, so that the planned trajectory cannot be tracked completely. The UAV trajectory planning problem is considered as an optimization problem for research, considering the dynamics constraints of the UAV and the terrain obstacle constraints during flight. An hp-adaptive Radau pseudospectral method based UAV trajectory planning scheme is proposed by taking the UAV dynamics into account. Numerical experiments are carried out to show the effectiveness and superior of the proposed method. Simulation results show that the proposed method outperform the well-known RRT* and A* algorithm in terms of tracking error.

Citation: Yi Cui, Xintong Fang, Gaoqi Liu, Bin Li. Trajectory optimization of UAV based on Hp-adaptive Radau pseudospectral method. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021201
References:
 [1] N. Ahn and S. Kim, Optimal and heuristic algorithms for the multi-objective vehicle routing problem with drones for military surveillance operations, J. Industrial and Management Optimization, 2021. doi: 10.3934/jimo.2021037. [2] A. F. Alkaya and D. Oz, An optimal algorithm for the obstacle neutralization problem, J. Ind. Manag. Optim., 13 (2017), 835-856.  doi: 10.3934/jimo.2016049. [3] R. Austin, Unmanned aircraft systems: UAVs design, development, anddeployment, Journal Publications Chestnet.org, 50 (2010), 31-36. [4] A. Chamseddine, Y. Zhang, C. A. Rabbath, C. Join and D. Theilliol, Flatness-based trajectory planning/replanning for a quadrotor unmanned aerial vehicle, IEEE Transactions on Aerospace and Electronic Systems, 48 (2012), 2832-2848. [5] J. Chen, Y. Cao, N. Du, X. Liu and Y. Han, Gaussian pseudospectral longitudinal trajectory optimization algorithm of a solar powered communication/remote-sensing UAV, 2019 IEEE International Conference on Unmanned Systems and Artificial Intelligence (ICUSAI), (2019), 303–308. [6] W. P. Coutinho, M. Battarra and J. Fliege, The unmanned aerial vehicle routing and trajectory optimisation problem, a taxonomic review, Computers and Industrial Engineering, 120 (2018), 116-128. [7] B. Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge University Press, Cambridge, 1996.  doi: 10.1017/CBO9780511626357. [8] B. T. Gatzke, Trajectory optimization for helicopter unmanned aerial vehicles (UAVs), NPS Thesis, 2012. [9] A. Goli, H. K. Zare, R. Tavakkoli-Moghaddam and A. Sadeghieh, A comprehensive model of demand prediction based on hybrid artificial intelligence and metaheuristic algorithms: A case study in dairy industry, MPRA Paper, 11 (2018), 190-203. [10] A. Goli, H. Khademi-Zare, R. Tavakkoli-Moghaddam, A. Sadeghieh, M. Sasanian and R. M. Kordestanizadeh, An integrated approach based on artificial intelligence and novel meta-heuristic algorithms to predict demand for dairy products: A case study, Network Computation in Neural Systems, 1 (2021), 1-35. [11] M. Y. Hussaini and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics. Springer-Verlag, New York, 1988. doi: 10.1007/978-3-642-84108-8. [12] S. Karaman and E. Frazzoli, Optimal kinodynamic motion planning using incremental sampling-based methods, 49th IEEE Conference on Decision and Control (CDC), (2010), 7681–7687. doi: 10.1109/CDC.2010.5717430. [13] J. Li, G. Deng, C. Luo, Q. Lin, Q. Yan and Z. Ming, A hybrid path planning method in unmanned Air/Ground vehicle (UAV/UGV) cooperative systems, IEEE Transactions on Vehicular Technology, 65 (2016), 9585-9596.  doi: 10.1109/TVT.2016.2623666. [14] B. Li, J. Zhang, L. Dai, K. L. Teo and S. Wang, A hybrid offline optimization method for reconfiguration of multi-UAV formations, IEEE Transactions on Aerospace and Electronic Systems, 57 (2021), 506-520. [15] 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 Anal. Hybrid Syst., 25 (2017), 1-20.  doi: 10.1016/j.nahs.2017.01.006. [16] H. Liu, Q. Chen, N. Pan, Y. Sun and Y. Yang, Three-dimensional mountain complex terrain and heterogeneous multi-UAV cooperative combat mission planning, IEEE Access, 8 (2020), 197407-197419. [17] R. Mahony, V. Kumar and P. Corke, Multirotor aerial vehicles: Modeling, estimation, and control of quadrotor, IEEE Robotics and Automation Magazine, 19 (2012), 20-32. [18] M. A. Patterson, W. W. Hager and A. V. Rao, A ph mesh refinement method for optimal control, Optimal Control Appl. Methods, 36 (2015), 398-421.  doi: 10.1002/oca.2114. [19] M. A. Patterson and A. V. Rao, GPOPS-II: A MATLAB software for solving multiple-phase optimal control problems using hp-adaptive gaussian quadrature collocation methods and sparse nonlinear programming, ACM Trans. Math. Software, 41 (2014), Art. 1, 37 pp. doi: 10.1145/2558904. [20] P. Pharpatara, B. Hérissé and Y. Bestaoui, 3-D trajectory planning of aerial vehicles using RRT*, IEEE Transactions on Control Systems Technology, 25 (2017), 1116-1123.  doi: 10.1109/TCST.2016.2582144. [21] B. Salamat and A. M. Tonello, A modelling approach to generate representative UAV trajectories using PSO, 2019 27th European Signal Processing Conference (EUSIPCO), (2019), 1–5. [22] N. Sariff and N. Buniyamin, An overview of autonomous mobile robot path planning algorithms, 2006 4th Student Conference on Research and Development, (2006), 183–188. doi: 10.1109/SCORED.2006.4339335. [23] Y. Shi, R. Li and H. Xu, Control augmentation design of UAVs based on deviation modification of aerodynamic focus, J. Industrial and Management Optimization, 11 (2015), 231-240. [24] Z. S. Shui, J. Zhou and Z. L. Ge, On-line predictor-corrector reentry guidance law based on Gauss pseudospectral method, J. Astronautics, 6 (2011), 1249-1255. [25] K. P. Valavanis and G. J. Vachtsevanos, Handbook of Unmanned Aerial Vehicles, 1$^st$ edition, Springer Netherlands, 2014. [26] S. Vera, J. A. Cobano, G. Heredia and A. Ollero, An hp-adaptative pseudospectral method for collision avoidance with multiple UAVs in real-time applications, 2014 IEEE International Conference on Robotics and Automation (ICRA), (2014), 4717–4722. [27] N. Wen, X. Su, P. Ma, L. Zhao and Y. Zhang, Online UAV path planning in uncertain and hostile environments, International J. Machine Learning and Cybernetics, 8 (2017), 469-487. [28] F. Yan, Y. Liu and J. Xiao, Path planning in complex 3D environments using a probabilistic roadmap method, International J. Automation and Computing, 6 (2013), 525-533. [29] S. Yang and Z. Wang, Quad-rotor UAV control method based on PID control law, 2017 International Conference on Computer Network, Electronic and Automation (ICCNEA), (2017), 418–421. [30] W. Zeng and R. L. Church, Finding shortest paths on real road networks: The case for A*, International J. Geographical Information Science, 23 (2009), 531-543.  doi: 10.1080/13658810801949850. [31] L. Zhang, F. Deng, J. Chen, Y. Bi, S. K. Pang and X. Chen, Trajectory planning for improving vision-based target geolocation performance using a quad-rotor UAV, IEEE Transactions on Aerospace and Electronic Systems, 55 (2019), 2382-2394. [32] Y. Zhang, C. Yu and Y. Xu, Minimizing almost smooth control variation in nonlinear optimal control problems, J. Ind. Manag. Optim., 16 (2020), 1663-1683.  doi: 10.3934/jimo.2019023. [33] 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. [34] MATLAB, 9.7.0.1190202 (R2019a), Natick, Massachusetts: The MathWorks Inc, 2019.

show all references

References:
 [1] N. Ahn and S. Kim, Optimal and heuristic algorithms for the multi-objective vehicle routing problem with drones for military surveillance operations, J. Industrial and Management Optimization, 2021. doi: 10.3934/jimo.2021037. [2] A. F. Alkaya and D. Oz, An optimal algorithm for the obstacle neutralization problem, J. Ind. Manag. Optim., 13 (2017), 835-856.  doi: 10.3934/jimo.2016049. [3] R. Austin, Unmanned aircraft systems: UAVs design, development, anddeployment, Journal Publications Chestnet.org, 50 (2010), 31-36. [4] A. Chamseddine, Y. Zhang, C. A. Rabbath, C. Join and D. Theilliol, Flatness-based trajectory planning/replanning for a quadrotor unmanned aerial vehicle, IEEE Transactions on Aerospace and Electronic Systems, 48 (2012), 2832-2848. [5] J. Chen, Y. Cao, N. Du, X. Liu and Y. Han, Gaussian pseudospectral longitudinal trajectory optimization algorithm of a solar powered communication/remote-sensing UAV, 2019 IEEE International Conference on Unmanned Systems and Artificial Intelligence (ICUSAI), (2019), 303–308. [6] W. P. Coutinho, M. Battarra and J. Fliege, The unmanned aerial vehicle routing and trajectory optimisation problem, a taxonomic review, Computers and Industrial Engineering, 120 (2018), 116-128. [7] B. Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge University Press, Cambridge, 1996.  doi: 10.1017/CBO9780511626357. [8] B. T. Gatzke, Trajectory optimization for helicopter unmanned aerial vehicles (UAVs), NPS Thesis, 2012. [9] A. Goli, H. K. Zare, R. Tavakkoli-Moghaddam and A. Sadeghieh, A comprehensive model of demand prediction based on hybrid artificial intelligence and metaheuristic algorithms: A case study in dairy industry, MPRA Paper, 11 (2018), 190-203. [10] A. Goli, H. Khademi-Zare, R. Tavakkoli-Moghaddam, A. Sadeghieh, M. Sasanian and R. M. Kordestanizadeh, An integrated approach based on artificial intelligence and novel meta-heuristic algorithms to predict demand for dairy products: A case study, Network Computation in Neural Systems, 1 (2021), 1-35. [11] M. Y. Hussaini and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics. Springer-Verlag, New York, 1988. doi: 10.1007/978-3-642-84108-8. [12] S. Karaman and E. Frazzoli, Optimal kinodynamic motion planning using incremental sampling-based methods, 49th IEEE Conference on Decision and Control (CDC), (2010), 7681–7687. doi: 10.1109/CDC.2010.5717430. [13] J. Li, G. Deng, C. Luo, Q. Lin, Q. Yan and Z. Ming, A hybrid path planning method in unmanned Air/Ground vehicle (UAV/UGV) cooperative systems, IEEE Transactions on Vehicular Technology, 65 (2016), 9585-9596.  doi: 10.1109/TVT.2016.2623666. [14] B. Li, J. Zhang, L. Dai, K. L. Teo and S. Wang, A hybrid offline optimization method for reconfiguration of multi-UAV formations, IEEE Transactions on Aerospace and Electronic Systems, 57 (2021), 506-520. [15] 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 Anal. Hybrid Syst., 25 (2017), 1-20.  doi: 10.1016/j.nahs.2017.01.006. [16] H. Liu, Q. Chen, N. Pan, Y. Sun and Y. Yang, Three-dimensional mountain complex terrain and heterogeneous multi-UAV cooperative combat mission planning, IEEE Access, 8 (2020), 197407-197419. [17] R. Mahony, V. Kumar and P. Corke, Multirotor aerial vehicles: Modeling, estimation, and control of quadrotor, IEEE Robotics and Automation Magazine, 19 (2012), 20-32. [18] M. A. Patterson, W. W. Hager and A. V. Rao, A ph mesh refinement method for optimal control, Optimal Control Appl. Methods, 36 (2015), 398-421.  doi: 10.1002/oca.2114. [19] M. A. Patterson and A. V. Rao, GPOPS-II: A MATLAB software for solving multiple-phase optimal control problems using hp-adaptive gaussian quadrature collocation methods and sparse nonlinear programming, ACM Trans. Math. Software, 41 (2014), Art. 1, 37 pp. doi: 10.1145/2558904. [20] P. Pharpatara, B. Hérissé and Y. Bestaoui, 3-D trajectory planning of aerial vehicles using RRT*, IEEE Transactions on Control Systems Technology, 25 (2017), 1116-1123.  doi: 10.1109/TCST.2016.2582144. [21] B. Salamat and A. M. Tonello, A modelling approach to generate representative UAV trajectories using PSO, 2019 27th European Signal Processing Conference (EUSIPCO), (2019), 1–5. [22] N. Sariff and N. Buniyamin, An overview of autonomous mobile robot path planning algorithms, 2006 4th Student Conference on Research and Development, (2006), 183–188. doi: 10.1109/SCORED.2006.4339335. [23] Y. Shi, R. Li and H. Xu, Control augmentation design of UAVs based on deviation modification of aerodynamic focus, J. Industrial and Management Optimization, 11 (2015), 231-240. [24] Z. S. Shui, J. Zhou and Z. L. Ge, On-line predictor-corrector reentry guidance law based on Gauss pseudospectral method, J. Astronautics, 6 (2011), 1249-1255. [25] K. P. Valavanis and G. J. Vachtsevanos, Handbook of Unmanned Aerial Vehicles, 1$^st$ edition, Springer Netherlands, 2014. [26] S. Vera, J. A. Cobano, G. Heredia and A. Ollero, An hp-adaptative pseudospectral method for collision avoidance with multiple UAVs in real-time applications, 2014 IEEE International Conference on Robotics and Automation (ICRA), (2014), 4717–4722. [27] N. Wen, X. Su, P. Ma, L. Zhao and Y. Zhang, Online UAV path planning in uncertain and hostile environments, International J. Machine Learning and Cybernetics, 8 (2017), 469-487. [28] F. Yan, Y. Liu and J. Xiao, Path planning in complex 3D environments using a probabilistic roadmap method, International J. Automation and Computing, 6 (2013), 525-533. [29] S. Yang and Z. Wang, Quad-rotor UAV control method based on PID control law, 2017 International Conference on Computer Network, Electronic and Automation (ICCNEA), (2017), 418–421. [30] W. Zeng and R. L. Church, Finding shortest paths on real road networks: The case for A*, International J. Geographical Information Science, 23 (2009), 531-543.  doi: 10.1080/13658810801949850. [31] L. Zhang, F. Deng, J. Chen, Y. Bi, S. K. Pang and X. Chen, Trajectory planning for improving vision-based target geolocation performance using a quad-rotor UAV, IEEE Transactions on Aerospace and Electronic Systems, 55 (2019), 2382-2394. [32] Y. Zhang, C. Yu and Y. Xu, Minimizing almost smooth control variation in nonlinear optimal control problems, J. Ind. Manag. Optim., 16 (2020), 1663-1683.  doi: 10.3934/jimo.2019023. [33] 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. [34] MATLAB, 9.7.0.1190202 (R2019a), Natick, Massachusetts: The MathWorks Inc, 2019.
Optimization of columnar obstacles trajectory
Optimal state and control inputs variables
Trajectory optimization by using AStar algorithm
Trajectory optimization by using RRT* algorithm
Optimal state and control inputs variables
Limited time tracking controller structure
Trajectory tracking by using Astar algorithm
Trajectory tracking by using RRT* algorithm
State and control constraints of UAV
 Parameters Value Altitude $z$ (m) [0, 500] Velocity of x direction ${V^x}$ (m/s) [0, 25] Velocity of y direction ${V^y}$ (m/s) [0, 25] Velocity of z direction ${V^z}$ (m/s) [0, 20] pitch angle $\theta (^\circ)$ $\left[ { - \frac{{\pi }}{6}, \frac{\pi }{6}} \right]$ roll angle $\psi (^\circ)$ $\left[ { - \frac{\pi }{6}, \frac{\pi }{6}} \right]$ yaw angle $\varphi (^\circ)$ $\left[ { - \frac{{\pi }}{6}, \frac{{\pi }}{6}} \right]$ Virtual control input 1 $u_1 (N)$ [10,12] Virtual control input 2 $u_2 (N\cdot M)$ [-0.005, 0.005] Virtual control input 3 $u_3 (N\cdot M)$ [-0.005, 0.005] Virtual control input 4 $u_4 (N\cdot M)$ [-0.005, 0.005] Derivative of virtual control input 1 $\dot u_1 (N/s)$ [-1.5, 1.5] Derivative of virtual control input 2 $\dot u_2 (N\cdot M/s)$ [-1, 1] Derivative of virtual control input 3 $\dot u_3 (N\cdot M/s)$ [-1.5, 1.5] Derivative of virtual control input 4 $\dot u_4 (N\cdot M/s)$ [-1.5, 1.5]
 Parameters Value Altitude $z$ (m) [0, 500] Velocity of x direction ${V^x}$ (m/s) [0, 25] Velocity of y direction ${V^y}$ (m/s) [0, 25] Velocity of z direction ${V^z}$ (m/s) [0, 20] pitch angle $\theta (^\circ)$ $\left[ { - \frac{{\pi }}{6}, \frac{\pi }{6}} \right]$ roll angle $\psi (^\circ)$ $\left[ { - \frac{\pi }{6}, \frac{\pi }{6}} \right]$ yaw angle $\varphi (^\circ)$ $\left[ { - \frac{{\pi }}{6}, \frac{{\pi }}{6}} \right]$ Virtual control input 1 $u_1 (N)$ [10,12] Virtual control input 2 $u_2 (N\cdot M)$ [-0.005, 0.005] Virtual control input 3 $u_3 (N\cdot M)$ [-0.005, 0.005] Virtual control input 4 $u_4 (N\cdot M)$ [-0.005, 0.005] Derivative of virtual control input 1 $\dot u_1 (N/s)$ [-1.5, 1.5] Derivative of virtual control input 2 $\dot u_2 (N\cdot M/s)$ [-1, 1] Derivative of virtual control input 3 $\dot u_3 (N\cdot M/s)$ [-1.5, 1.5] Derivative of virtual control input 4 $\dot u_4 (N\cdot M/s)$ [-1.5, 1.5]
Parameters of UAV
 Parameters Value Length from motor to center of mass $L$ (m) 0.2 Total mass of UAV $M$ (kg) 1.5 Acceleration of gravity $g$ $\left( {m/{s^2}} \right)$ 9.8 Moment of inertia about the x axis ${I_x}$ $\left( {kg \cdot {m^2}} \right)$ 0.0075 Moment of inertia about the y axis ${I_y}$ $\left( {kg \cdot {m^2}} \right)$ 0.0075 Moment of inertia about the z axis ${I_z}$ $\left( {kg \cdot {m^2}} \right)$ 0.013 Coefficient of air resistance ${K_1}$ $\left( {N/m/s} \right)$ 0.06 Coefficient of air resistance ${K_2}$ $\left( {N/m/s} \right)$ 0.06 Coefficient of air resistance ${K_3}$ $\left( {N/m/s} \right)$ 0.09 Coefficient of air resistance ${K_4}$ $\left( {N/m/s} \right)$ 0.002 Coefficient of air resistance ${K_5}$ $\left( {N/m/s} \right)$ 0.002 Coefficient of air resistance ${K_6}$ $\left( {N/m/s} \right)$ 0.1
 Parameters Value Length from motor to center of mass $L$ (m) 0.2 Total mass of UAV $M$ (kg) 1.5 Acceleration of gravity $g$ $\left( {m/{s^2}} \right)$ 9.8 Moment of inertia about the x axis ${I_x}$ $\left( {kg \cdot {m^2}} \right)$ 0.0075 Moment of inertia about the y axis ${I_y}$ $\left( {kg \cdot {m^2}} \right)$ 0.0075 Moment of inertia about the z axis ${I_z}$ $\left( {kg \cdot {m^2}} \right)$ 0.013 Coefficient of air resistance ${K_1}$ $\left( {N/m/s} \right)$ 0.06 Coefficient of air resistance ${K_2}$ $\left( {N/m/s} \right)$ 0.06 Coefficient of air resistance ${K_3}$ $\left( {N/m/s} \right)$ 0.09 Coefficient of air resistance ${K_4}$ $\left( {N/m/s} \right)$ 0.002 Coefficient of air resistance ${K_5}$ $\left( {N/m/s} \right)$ 0.002 Coefficient of air resistance ${K_6}$ $\left( {N/m/s} \right)$ 0.1
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