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Research Papers

# An Efficient Approach of Time-Optimal Trajectory Generation for the Fully Autonomous Navigation of the Quadrotor

[+] Author and Article Information
Wei Dong

State Key Laboratory of Mechanical
System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: chengquess@sjtu.edu.cn

Ye Ding

State Key Laboratory of Mechanical
System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: y.ding@sjtu.edu.cn

Jie Huang

State Key Laboratory of Mechanical
System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: thk2dth@sjtu.edu.cn

Xiangyang Zhu

State Key Laboratory of Mechanical
System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: mexyzhu@sjtu.edu.cn

Han Ding

State Key Laboratory of Mechanical
System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: hding@sjtu.edu.cn

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received May 12, 2016; final manuscript received December 1, 2016; published online April 13, 2017. Assoc. Editor: Jingang Yi.

J. Dyn. Sys., Meas., Control 139(6), 061012 (Apr 13, 2017) (9 pages) Paper No: DS-16-1245; doi: 10.1115/1.4035453 History: Received May 12, 2016; Revised December 01, 2016

## Abstract

In this work, a time-optimal trajectory generation approach is developed for the multiple way-point navigation of the quadrotor based on the nonuniform rational B-spline (NURBS) curve and linear programming. To facilitate this development, the dynamic model of the quadrotor is formulated first. Then, the geometric trajectory regarding multiple way-point navigation is constructed based on the NURBS curve. With the constructed geometric trajectory, a time-optimal interpolation problem is imposed considering the velocity, acceleration, and jerk constraints. This optimization problem is solved in two steps. In the first step, a preliminary result is obtained by solving a linear programming problem without jerk constraints. Then by introducing properly relaxed jerk constraints, a second linear programming problem is formulated based on the preliminarily obtained result, and the time-optimal problem can be fully solved in this way. Subsequently, a nonlinear trajectory tracking controller is developed to track the generated trajectory. The feasibilities of the proposed trajectory generation approach as well as the tracking controller are verified through both simulations and real-time experiments. With enhanced computational efficiency, the proposed approach can generate trajectory for an indoor environment with the smooth acceleration profile and moderate velocity $V≈1$ m/s in real-time, while guaranteeing velocity, acceleration, and jerk constraints: $Vmax=1 m/s, Amax=2 m/s2$, and $Jmax=5 m/s3$. In such a case, the trajectory tracking controller can closely track the reference trajectory with cross-tracking error less than 0.05 m.

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## Figures

Fig. 1

Free-body diagram

Fig. 6

The jerk evolution of the fourth-order NURBS trajectory: (a) without jerk constraints and (b) with jerk constraints

Fig. 7

The trajectory tracking process when Vmax=1 m/s, Amax=2 m/s2, and Jmax=2 m/s3: (a) trajectory tracking process and (b) dynamic error

Fig. 9

The cross-tracking error when with different jerk constraints: (a) Jmax=1 m/s3, (b) Jmax=2 m/s3, and (c) Jmax=5 m/s3

Fig. 3

The quadrotor test bed

Fig. 4

The fourth NURBS trajectory regarding a cluttered environment with three rings: (a) fourth-order NUBRS trajectory for ring obstacle avoidance and (b) velocities and accelerations

Fig. 5

The velocity (a) and acceleration (b) evolution of the fourth-order NURBS trajectory

Fig. 8

The along-tracking error when with different jerk constraints: (a) Jmax=1 m/s3 and (b) Jmax=5 m/s3

Fig. 2

The trajectory optimization for the trajectory with multiple squares: (a) optimal trajectory and (b) velocity and acceleration evolutions

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