Nonlinear adaptive control for an unmanned aerial payload transportation system: theory and experimental validation

Abstract

In this paper, the position control and swing motion control problem are investigated for an aerial payload transportation system which consists of a quadrotor unmanned aerial vehicle (UAV) and a suspended payload. Under the constraints of underactuated properties and unknown system parameters, a nonlinear adaptive control strategy is designed based on the energy methodology, which achieves accurate position control of the UAV as well as the payload’s fast swing suppression during the flight. The stability of the closed-loop system, asymptotic convergence of the UAV’s position error and payload swing suppression are proved via Lyapunov-based stability analysis. Real-time experimental results validate the effectiveness of the developed technique.

Appendix

1.1 Parameter condition derivation

The function $\varLambda (t)$ in (48) can be expressed as follows:

$$\begin{aligned} -\varLambda (t)&=-\,\eta _{1}{\dot{x}}^{2}-\eta _{2}{\dot{y}}^{2}-\eta _{3}{\dot{z}} ^{2}-\eta _{4}{\dot{\gamma }}_{x}^{2}-\eta _{5}{\dot{\gamma }}_{y}^{2}\nonumber \\&\quad -\,\eta _{6} \rho ^{2}\left( \frac{e_{x}}{2}\right) \nonumber \\&\quad -\,\eta _{7}\rho ^{2}\left( \frac{e_{y}}{2}\right) -\eta _{8}\rho ^{2}\left( \frac{e_{z}}{2}\right) +\alpha k_{\text {d}\gamma _{x}}{\dot{\gamma }}_{x}{\dot{x}}\nonumber \\&\quad +\,\alpha k_{\text {d}\gamma _{y}}{\dot{\gamma }}_{y}{\dot{y}}-2m_{p}glC_{x}\left( \sin \frac{\gamma _{y}}{2}\cos \frac{\gamma _{y}}{2}\right) ^{2}\nonumber \\&\quad -\,2m_{p}glC_{y}\left( \sin \frac{\gamma _{x}}{2}\cos \frac{\gamma _{x}}{2}\right) ^{2} \end{aligned}$$

(64)

where $\eta _{1}$, $\eta _{2}$, $\eta _{3}$, $\eta _{4}$, $\eta _{5}$, $\eta _{6}$, $\eta _{7}$, $\eta _{8}$ are defined as

$$\begin{aligned} \left\{ \begin{array} [c]{l} \eta _{1}\overset{\varDelta }{=}\alpha k_{\text {d}x}-\frac{1}{2}k_{\text {d}x}^{2}-\frac{1}{2}\lambda _{M}-2m_{p}^{2}\\ \eta _{2}\overset{\varDelta }{=}\alpha k_{\text {d}y}-\frac{1}{2}k_{\text {d}y}^{2}-\frac{1}{2}\lambda _{M}-\frac{1}{4}m_{p}^{2}\\ \eta _{3}\overset{\varDelta }{=}\alpha k_{\text {d}z}-\frac{k_{\text {d}z}^{2}}{2}-\frac{1}{2}\lambda _{M}-2m_{p}^{2}\\ \eta _{4}\overset{\varDelta }{=}\alpha c_{\gamma _{x}}-\frac{1}{2}\lambda _{M} -2l^{2}-m_{p}l^{2}-\frac{1}{2}k_{\text {d}\gamma _{x}}^{2}\\ \eta _{5}\overset{\varDelta }{=}\alpha c_{\gamma _{y}}-\frac{1}{2}\lambda _{M} -3l^{2}-\frac{1}{2}k_{\text {d}\gamma _{y}}^{2}\\ \eta _{6}\overset{\varDelta }{=}2k_{px}+2\kappa _{x}\varsigma _{x}-1\\ \eta _{7}\overset{\varDelta }{=}2k_{py}+2\kappa _{y}\varsigma _{y}-1\\ \eta _{8}\overset{\varDelta }{=}2k_{pz}+2\kappa _{z}\varsigma _{z}-\frac{1}{2} \end{array} \right. . \end{aligned}$$

(65)

Now, we define S(t) as follows:

$$\begin{aligned} S&=-\,\eta _{1}{\dot{x}}^{2}-\eta _{2}{\dot{y}}^{2}-\eta _{3}{\dot{z}}^{2}-\eta _{4}{\dot{\gamma }}_{x}^{2}\nonumber \\&\quad -\eta _{5}{\dot{\gamma }}_{y}^{2}+\alpha k_{\text {d}\gamma _{x}}{\dot{\gamma }}_{x}\dot{x}+\alpha k_{\text {d}\gamma _{y}}{\dot{\gamma }}_{y}{\dot{y}}\nonumber \\&=sPs^{T} \end{aligned}$$

(66)

where P is a $5\times 5$ matrix, $s=$$\left[ \begin{array} [c]{ccccc} {\dot{x}}&{\dot{y}}&{\dot{z}}&{\dot{\gamma }}_{x}&{\dot{\gamma }}_{y} \end{array} \right] \in {\mathbb {R}}^{1\times 5}$ and denoted via the following equations

$$\begin{aligned} \left\{ \begin{array} [c]{l} P=\left[ \begin{array} [c]{ccccc} \eta _{1} &{} 0 &{} 0 &{} p_{14} &{} 0\\ 0 &{} \eta _{2} &{} 0 &{} 0 &{} p_{25}\\ 0 &{} 0 &{} \eta _{3} &{} 0 &{} 0\\ p_{41} &{} 0 &{} 0 &{} \eta _{4} &{} 0\\ 0 &{} p_{52} &{} 0 &{} 0 &{} \eta _{5} \end{array} \right] \\ p_{14}=p_{41}=-\,\frac{1}{2}\alpha k_{\text {d}\gamma _{x}}\text { }p_{25}=p_{52} =-\,\frac{1}{2}\alpha k_{\text {d}\gamma _{y}} \end{array} \right. , \end{aligned}$$

(67)

Consequently, S(t) in (66) can be described as

$$\begin{aligned} S=-\,{\dot{e}}^{T}(t)P{\dot{e}}(t). \end{aligned}$$

(68)

By substituting (68) into (64), we can obtain

$$\begin{aligned} -\varLambda (t)&=-\,{\dot{e}}^{T}(t)P{\dot{e}}(t)-\eta _{6}\rho ^{2}(\frac{e_{x}}{2})-\eta _{7}\rho ^{2}\left( \frac{e_{y}}{2}\right) \nonumber \\&\quad -\,\eta _{8}\rho ^{2}\left( \frac{e_{z}}{2}\right) -2m_{p}glC_{x}\left( \sin \frac{\gamma _{y}}{2}\cos \frac{\gamma _{y}}{2}\right) ^{2}\nonumber \\&\quad -\,2m_{p}glC_{y}\left( \sin \frac{\gamma _{x}}{2}\cos \frac{\gamma _{x}}{2}\right) ^{2}. \end{aligned}$$

(69)

In order to ensure $-\varLambda (t)$ to be negative definite, it is implied that P need to be positive definite, then the conditions can denote as follows:

$$\begin{aligned} \left\{ \begin{array} [c]{l} \begin{array} [c]{ccc} \eta _{1}>0 &{} \eta _{1}\eta _{2}>0 &{} \eta _{1}\eta _{2}\eta _{3}>0 \end{array} \\ \begin{array} [c]{cc} \eta _{1}\eta _{2}\eta _{3}\eta _{4}+(\frac{1}{2}\alpha k_{\text {d}\gamma _{x}})^{2} \eta _{2}\eta _{3}>0 &{} \eta _{1}\eta _{2}\eta _{3}\eta _{4}\eta _{5}>0 \end{array} \end{array},\right. \nonumber \\ \end{aligned}$$

(70)

and it yields from (70) that

$$\begin{aligned} \begin{array} [c]{ccccc} \eta _{1}>0&\eta _{2}>0&\eta _{3}>0&\eta _{4}>0&\eta _{5}>0 \end{array} . \end{aligned}$$

(71)

Suppose that $k_{\text {d}x}=k_{\text {d}y}=k_{\text {d}z}=\frac{1}{r}\alpha$ where $r\in {\mathbb {R}}^{+}$ is positive constant. Hence, (71) can be denoted as

$$\begin{aligned} \left\{ \begin{array} [c]{l} \eta _{1}=\left( \frac{1}{r}-\frac{1}{2r^{2}}\right) \alpha ^{2}-\frac{1}{2}\lambda _{M}-2m_{p}^{2}>0\\ \eta _{2}=\left( \frac{1}{r}-\frac{1}{2r^{2}}\right) \alpha ^{2}-\frac{1}{2}\lambda _{M} -\frac{1}{4}m_{p}^{2}>0\\ \eta _{3}=\left( \frac{1}{r}-\frac{1}{2r^{2}}\right) \alpha ^{2}-\frac{1}{2}\lambda _{M}-2m_{p}^{2}>0\\ \eta _{4}=-\,\frac{1}{2r^{2}}\alpha ^{2}+c_{\gamma _{x}}\alpha -\frac{1}{2} \lambda _{M}-2l^{2}-m_{p}l^{2}>0\\ \eta _{5}=-\,\frac{1}{2r^{2}}\alpha ^{2}+c_{\gamma _{y}}\alpha -\frac{1}{2} \lambda _{M}-3l^{2}>0 \end{array} \right. . \end{aligned}$$

(72)

By making some mathematical manipulation, it is reduced as

$$\begin{aligned} \left\{ \begin{array} [c]{l} (2r-1)\alpha ^{2}>(\lambda _{M}+4m_{p}^{2})r^{2}\\ \alpha ^{2}-2r^{2}c_{\gamma _{x}}\alpha +r^{2}(4l^{2}+2m_{p}l^{2}+\lambda _{M})>0\\ \alpha ^{2}-2r^{2}c_{\gamma _{y}}\alpha +r^{2}(6l^{2}+\lambda _{M})>0 \end{array} \right. . \nonumber \\ \end{aligned}$$

(73)

Thus, $\alpha$ should be selected to satisfy

$$\begin{aligned} \max \left\{ \beta _{1}\text { }\beta _{3}\text { }r\sqrt{\frac{\lambda _{M} +4m_{p}^{2}}{2r-1}}\right\}<\alpha <\min \left\{ \beta _{2}\text { }\beta _{4}\right\} \nonumber \\ \end{aligned}$$

(74)

with

$$\begin{aligned} \left\{ \begin{array} [c]{l} \beta _{1}=r^{2}c_{\gamma _{x}}-r\sqrt{r^{2}c_{\gamma _{x}}^{2}-(4l^{2}+2\bar{m}_{p}l^{2}+\lambda _{M})}\\ \beta _{2}=r^{2}c_{\gamma _{x}}+r\sqrt{r^{2}c_{\gamma _{x}}^{2}-(4l^{2}+2\bar{m}_{p}l^{2}+\lambda _{M})}\\ \beta _{3}=r^{2}c_{\gamma _{y}}-r\sqrt{r^{2}c_{\gamma _{y}}^{2}-(6l^{2} +\lambda _{M})}\\ \beta _{4}=r^{2}c_{\gamma _{y}}+r\sqrt{r^{2}c_{\gamma _{y}}^{2}-(6l^{2} +\lambda _{M})} \end{array} \right. , \nonumber \\ \end{aligned}$$

(75)

and r should be chosen to satisfy

$$\begin{aligned} r\ge \max \left\{ \frac{1}{2},\frac{\sqrt{4l^{2}+2{\bar{m}}_{p}l^{2}+\lambda _{M}} }{c_{\gamma _{x}}^{2}},\frac{\sqrt{6l^{2}+\lambda _{M}}}{c_{\gamma _{y}}^{2}}\right\} . \nonumber \\ \end{aligned}$$

(76)

On the other hand, considering the negative property of $-\varLambda (t)$, the following inequalities hold

$$\begin{aligned} \left\{ \begin{array} [c]{l} \eta _{6}=2k_{py}+2\kappa _{x}\varsigma _{x}-1>0\\ \eta _{7}=2k_{py}+2\kappa _{y}\varsigma _{y}-1>0\\ \eta _{8}=2k_{pz}+2\kappa _{z}\varsigma _{z}-\frac{1}{2}>0 \end{array} \right. , \end{aligned}$$

(77)

that means

$$\begin{aligned} k_{px}>\frac{1}{2}\text { }k_{py}>\frac{1}{2}\text { }k_{pz}>\frac{1}{4}\text { }\kappa _{x}>0\text { }\kappa _{y}>0\text { }\kappa _{z}>0. \nonumber \\ \end{aligned}$$

(78)

Thus, combined with (32) and (34), we can obtain the conditions for $\alpha$, $k_{\text {d}x}$, $k_{\text {d}y}$, $k_{\text {d}z}$, $k_{px}$, $k_{py}$, $k_{pz}$, $k_{\text {d}\gamma _{x}}$, $k_{\text {d}\gamma _{y}}$, $\kappa _{x}$, $\kappa _{y}$, $\kappa _{z}$ in (49) and (50) based on (71)-(78).