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

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

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  • 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.

    Mathematics Subject Classification: Primary: 90C90, 65K05; Secondary: 65Z05.

    Citation:

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  • Figure 1.  The quad-rotor UAV and the coordinate systems

    Figure 2.  Control Parametrization

    Figure 3.  An Illustration of the Local Smoothing Technique

    Figure 4.  Example 1: the planned path of the UAV with Algorithm 1

    Figure 5.  Example 1: the optimal states

    Figure 6.  Example 1: the optimal control inputs

    Figure 7.  Example 2: the planned path of the UAV with Algorithm 1

    Figure 8.  Example 2: the optimal states

    Figure 9.  Example 2: the optimal control inputs

    Figure 10.  Example 3: The planned path of UAV with Algorithm 1

    Figure 11.  Example 3: the optimal states

    Figure 12.  Example 3: the optimal control inputs

    Figure 13.  Example 3: The planned path of UAV with RRT

    Figure 14.  Example 3: The planned path of UAV with A star

    Figure 15.  The structure of PD controller

    Figure 16.  Example 4: the tracking trajectory of the UAV with Algorithm 1

    Figure 17.  Example 4: the tracking trajectory of the UAV with RRT

    Figure 18.  Example 4: the tracking trajectory of the UAV with A star

    Figure 19.  Example 4: the motor speed

    Table 1.  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 $\varepsilon\leq\varepsilon_{min}$.
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    Table 2.  Parameters of the UAV

    $L$ $0.2\ m$ $M$ $1.5\ kg$ $g$ $9.8\ m/ s^{2}$
    ${I}_{x}$ $0.0075\ kg\cdot m^2$ ${I}_{y}$ $0.0075\ kg\cdot m^2$ ${I}_{z}$ $0.013\ kg\cdot m^2$
    ${K}_{1}, {K}_{2}$ $0.06\ N/m/s$ ${K}_{3}$ $0.09\ N/m/s$ ${K}_{4}, {K}_{5}$ $0.002\ N/m/s$
    ${K}_{6}$ $0.1\ N/m/s$ $C$ $10^{-7}$ ${K}_{v}$ $1.5\times10^{-5}\ N/m/s$
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