Automation and Robots for Disaster Response

Modeling, Guidance, and Robust Cooperative Control of Two Quadrotors Carrying a “Y”-Shaped-Cable-Suspended Payload

By inergency On Mar 19, 2024

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Appendix A. Proof of Theorem 1

The UDE estimation error is defined as

${\tilde{f}}_{i}^{ξ} = f_{i}^{ξ} - {\hat{f}}_{i}^{ξ} .$

(A1)

By combining (48), (51), and (A1), the UDE estimation error can be expressed in the frequency domain as

${\tilde{F}}_{i}^{ξ} (s) = [I - G_{i}^{ξ} (s)] F_{i}^{ξ} (s),$

(A2)

which can be rewritten in the time domain as

$_{\tilde{f}}^{˙} = - T_{i}^{ξ} {\tilde{f}}_{i}^{ξ} + {\dot{f}}_{i}^{ξ},$

(A3)

where $T_{i}^{ξ}$ is given in (54).

From (35), it can be seen that the

ψ

channel is in a cascade connection with the x and y channels. Therefore, we start with the analysis of the

ψ

channel. By plugging the control design (39) into the quadrotor model (28) and then subtracting the model from the global reference systems (29) and (30) for the two quadrotors, we obtain the following tracking error dynamics in the

ψ

channel:

$\underset{︸}{[\begin{matrix} _{\tilde{ψ} ˙ 1} \\ _{\tilde{ψ} ¨ 1} \\ _{\tilde{ψ} ˙ 2} \\ _{\tilde{ψ} ¨ 2} \end{matrix}]} \tilde{ψ} ˙ = \underset{︸}{[\begin{matrix} 0 & 1 & 0 & 0 \\ - k_{1 p}^{ψ} - α_{1}^{ψ} & - k_{1 d}^{ψ} - β_{1}^{ψ} & α_{1}^{ψ} & β_{1}^{ψ} \\ 0 & 0 & 0 & 1 \\ - k_{1 p}^{ψ} + α_{2}^{ψ} & - k_{1 d}^{ψ} + β_{2}^{ψ} & - α_{2}^{ψ} & - β_{2}^{ψ} \end{matrix}]} A_{r}^{ψ} \underset{︸}{[\begin{matrix} {\tilde{ψ}}_{1} \\ _{\tilde{ψ} ˙ 1} \\ {\tilde{ψ}}_{2} \\ _{\tilde{ψ} ˙ 2} \end{matrix}]} \tilde{ψ} + \underset{︸}{[\begin{matrix} 0 & 0 \\ - 1 & 0 \\ 0 & 0 \\ 0 & - 1 \end{matrix}]} B_{r}^{ψ} \underset{︸}{[\begin{matrix} {\tilde{f}}_{1}^{ψ} \\ {\tilde{f}}_{2}^{ψ} \end{matrix}]} {\tilde{f}}^{ψ} .$

(A4)

By substituting the parameter mapping (59) into (A3), we obtain the UDE estimation error dynamics for the two quadrotors in the $ψ$ channel as

$ε \underset{︸}{[\begin{matrix} _{\tilde{f}}^{˙} \\ _{\tilde{f}}^{˙} \end{matrix}]}^{\tilde{f} ˙ ψ} = \underset{︸}{[\begin{matrix} - \frac{1}{T_{1}^{ψ *}} & 0 \\ 0 & - \frac{1}{T_{2}^{ψ *}} \end{matrix}]} A_{b}^{ψ} \underset{︸}{[\begin{matrix} {\tilde{f}}_{1}^{ψ} \\ {\tilde{f}}_{2}^{ψ} \end{matrix}]} {\tilde{f}}^{ψ} + ε \underset{︸}{[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]} B_{b}^{ψ} \underset{︸}{[\begin{matrix} {\dot{f}}_{1}^{ψ} \\ {\dot{f}}_{2}^{ψ} \end{matrix}]} {\dot{f}}^{ψ} .$

(A5)

Up to now, we have derived the tracking error dynamics (A4) and the UDE estimation dynamics (A5), which are exactly in the form of the standard singular perturbation model [18]. Thus, it is natural to exploit singular perturbation theory to analyze the system stability and robustness. By letting $ε = 0$ in (A5), we obtain the quasi-steady state of the ${\tilde{f}}^{ψ}$ dynamics:Then, by plugging the quasi-steady state ${\tilde{f}}^{ψ *}$ into (A4) and letting $ε = 0$ , the following reduced model is obtained:

$\tilde{ψ} ˙ = A_{r}^{ψ} \tilde{ψ} .$

(A7)

It is readily verified that the origin of (A7) is an exponentially stable equilibrium under Stability Condition 1. Meanwhile, the boundary-layer model corresponding to (A5) is

$^{\tilde{f} ˙ ψ} = A_{b}^{ψ} {\tilde{f}}^{ψ},$

(A8)

which is also exponentially stable at its origin by noting (59). Therefore, by using Theorem 11.4 in [18], we conclude that there exists a $ε^{*} > 0$ , such that $\forall 0 < ε < ε^{*}$ , and the tracking error dynamics (A4) and the UDE estimation dynamics (A5) are both stable, provided the boundedness of $f_{i}^{ψ}$ and its derivatives up to the second order. Thus, the boundedness of $ψ_{i}$ and ${\hat{f}}_{i}^{ψ}$ is verified.

Now, we show the robustness of the

ψ

channel. The solution of the reduced model (A7) is

$\tilde{ψ}^(t) = e^{A_{r}^{ψ}} \tilde{ψ} (0) .$

(A9)

Via Theorem 11.2 in [18], we conclude that there exists a positive constant $k_{ψ}$ such that $\forall 0 < ε < ε^{*}$ and $\forall t > 0$ , and the following inequality holds:

${∥\tilde{ψ} (t) - \tilde{ψ}^(t)∥}_{2} \leq k_{ψ} ε .$

(A10)

Hence, we have

${∥\tilde{ψ} (t)∥}_{2} \leq {∥\tilde{ψ} (t) - \tilde{ψ}^(t)∥}_{2} + {∥\tilde{ψ}^(t)∥}_{2} \leq k_{ψ} ε + {∥\tilde{ψ}^(t)∥}_{2} .$

(A11)

Note that (A7) is stable, and we have ${∥\tilde{ψ}^(t)∥}_{2} \to 0$ . Thus, it is clear that there exist $t_{i}^{ψ} > 0$ , $ε^{* *} > 0$ , and ultimate bound $σ_{i}^{ψ} (ε)$ satisfying $σ_{i}^{ψ} (ε) \to 0$ as $ε \to 0$ , such that $\forall 0 < ε < ε^{* *}$ , and the following inequality holds:

$| {\tilde{ψ}}_{i} (t) | \leq {∥\tilde{ψ} (t)∥}_{2} \leq k_{ψ} ε + {∥\tilde{ψ}^(t)∥}_{2} \leq σ_{i}^{ψ} (ε), \forall t > t_{i}^{ψ} .$

(A12)

Since the z channel has an identical form to the

ψ

channel, an analysis of its stability and robustness can be performed in the same way as that presented above, and, thus, it is omitted here. In what follows, we analyze the x and y channels. The tracking error dynamics are

$\underset{︸}{[\begin{matrix} _{\tilde{ξ} ˙ 1} \\ _{\tilde{ξ} ¨ 1} \\ _{\tilde{ξ} ˙ 2} \\ _{\tilde{ξ} ¨ 2} \end{matrix}]} \tilde{ξ} ˙ = \underset{︸}{[\begin{matrix} 0 & 1 & 0 & 0 \\ - k_{1 p}^{ξ} - α_{1}^{ξ} & - k_{1 d}^{ξ} - β_{1}^{ξ} & α_{1}^{ξ} & β_{1}^{ξ} \\ 0 & 0 & 0 & 1 \\ - k_{1 p}^{ξ} + α_{2}^{ξ} & - k_{1 d}^{ξ} + β_{2}^{ξ} & - α_{2}^{ξ} & - β_{2}^{ξ} \end{matrix}]} A_{r}^{ξ} \underset{︸}{[\begin{matrix} {\tilde{ξ}}_{1} \\ _{\tilde{ξ} ˙ 1} \\ {\tilde{ξ}}_{2} \\ _{\tilde{ξ} ˙ 2} \end{matrix}]} \tilde{ξ} + \underset{︸}{[\begin{matrix} 0 & 0 \\ - 1 & 0 \\ 0 & 0 \\ 0 & - 1 \end{matrix}]} B_{r}^{ξ} \underset{︸}{[\begin{matrix} {\tilde{f}}_{1}^{ξ} \\ {\tilde{f}}_{2}^{ξ} \end{matrix}]} \tilde{ξ} + \underset{︸}{[\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \end{matrix}]} E_{r}^{ξ} E^{ξ}, ξ \in {x, y},$

(A13)

where $E^{ξ} = h^{ξ} (ψ_{d}) - h^{ξ} (ψ_{1}) = {\ddot{Δ}}_{d}^{ξ} - {\ddot{Δ}}_{2}^{ξ} + α_{2}^{ξ} (Δ_{d}^{ξ} - Δ_{2}^{ξ}) + β_{2}^{ξ} ({\dot{Δ}}_{d}^{ξ} - {\dot{Δ}}_{2}^{ξ})$ is the formation offset-related term satisfying

$\begin{matrix} E^{x} & = & (L_{d} {\ddot{ψ}}_{d} cos ψ_{d} + {\dot{L}}_{d} {\dot{ψ}}_{d} cos ψ_{d} + {\ddot{L}}_{d} sin ψ_{d} + {\dot{L}}_{d} {\dot{ψ}}_{d} cos ψ_{d} - L_{d} {\dot{ψ}}_{d}^{2} sin ψ_{d}) \\ - (L_{d} {\ddot{ψ}}_{1} cos ψ_{1} + {\dot{L}}_{d} {\dot{ψ}}_{1} cos ψ_{1} + {\ddot{L}}_{d} sin ψ_{1} + {\dot{L}}_{d} {\dot{ψ}}_{1} cos ψ_{1} - L_{d} {\dot{ψ}}_{1}^{2} sin ψ_{1}) \\ - α_{2}^{y} (L_{d} sin ψ_{1} - L_{d} sin ψ_{d}) - β_{2}^{y} (L_{d} {\dot{ψ}}_{1} cos ψ_{1} + {\dot{L}}_{d} sin ψ_{1} - L_{d} {\dot{ψ}}_{d} cos ψ_{d} - {\dot{L}}_{d} sin ψ_{d}), \end{matrix}$

(A14)

$\begin{matrix} E^{y} & = & (L_{d} {\dot{ψ}}_{d}^{2} cos ψ_{d} + L_{d} {\ddot{ψ}}_{d} sin ψ_{d} + {\dot{L}}_{d} {\dot{ψ}}_{d} sin ψ_{d} - {\ddot{L}}_{d} cos ψ_{d} + {\dot{L}}_{d} {\dot{ψ}}_{d} sin ψ_{d}) \\ - (L_{d} {\dot{ψ}}_{1}^{2} cos ψ_{1} + L_{d} {\ddot{ψ}}_{1} sin ψ_{1} + {\dot{L}}_{d} {\dot{ψ}}_{1} sin ψ_{1} - {\ddot{L}}_{d} cos ψ_{1} + {\dot{L}}_{d} {\dot{ψ}}_{1} sin ψ_{1}) \\ - α_{2}^{x} (L_{d} cos ψ_{d} - L_{d} cos ψ_{1}) - β_{2}^{x} ({\dot{L}}_{d} cos ψ_{d} - L_{d} {\dot{ψ}}_{d} sin ψ_{d} - {\dot{L}}_{d} cos ψ_{1} + L_{d} {\dot{ψ}}_{1} sin ψ_{1}) . \end{matrix}$

(A15)

Similar to the parameter mapping (59) employed in the

ψ

channel, by mapping

T_{i}^{ξ} = ε T_{i}^{ξ *}, ξ \in {x, y}

, it is found that the UDE estimation error dynamics are

$ε \underset{︸}{[\begin{matrix} _{\tilde{f}}^{˙} \\ _{\tilde{f}}^{˙} \end{matrix}]}^{\tilde{f} ˙ ξ} = \underset{︸}{[\begin{matrix} - \frac{1}{T_{1}^{ξ *}} & 0 \\ 0 & - \frac{1}{T_{2}^{ξ *}} \end{matrix}]} A_{b}^{ξ} \underset{︸}{[\begin{matrix} {\tilde{f}}_{1}^{ξ} \\ {\tilde{f}}_{2}^{ξ} \end{matrix}]} {\tilde{f}}^{ξ} + ε \underset{︸}{[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]} B_{b}^{ξ} \underset{︸}{[\begin{matrix} {\dot{f}}_{1}^{ξ} \\ {\dot{f}}_{2}^{ξ} \end{matrix}]} {\dot{f}}^{ξ}, ξ \in {x, y} .$

(A16)

Via singular perturbation theory, it is readily concluded that $\forall 0 < ε < ε^{*}$ , and the error dynamics (A13) and (A16) are stable under Stability Condition 1, provided the boundedness of $f_{i}^{ξ}$ and its derivatives up to the second order; thus, the states $ξ_{i}$ and the UDE estimation signals ${\hat{f}}_{i}^{ξ}$ are bounded.

As for the robustness analysis, the reduced model that corresponds to (A13) is

$\tilde{ξ} ˙ = A_{r}^{ξ} \tilde{ξ} + E_{r}^{ξ} E^{ξ}, ξ \in {x, y},$

(A17)

and its solution is denoted by

$\tilde{ξ}^(t) = e^{A_{r}^{ξ}} \tilde{ξ} (0) + \int_{0}^{t} e^{A_{r}^{ξ} (t - τ)} E_{r}^{ξ} E^{ξ} d τ, ξ \in {x, y} .$

(A18)

From the stability of systems (A4) and (A5), we conclude that state $ψ_{1}$ and its derivatives up to the third order are bounded. Furthermore, reference signal $ψ_{d}$ and its derivatives up to the third order can be designed to be bounded. Then, it is clear that the derivative of function $h^{ξ}$ , denoted by ${\dot{h}}^{ξ} (ψ)$ , is bounded by a positive constant ${\dot{h}}_{m a x}^{ξ}$ , and, thus, the following inequality holds by noting (A12):

$| E^{ξ} | = | h^{ξ} (ψ_{d}) - h^{ξ} (ψ_{1}) | \leq {\dot{h}}_{m a x}^{ξ} | \underset{︸}{ψ_{d} - ψ_{1}} {\tilde{ψ}}_{1} | \leq {\dot{h}}_{m a x}^{ξ} σ_{1}^{ψ} (ε) \overset{Δ}{=} σ_{E^{ξ}} (ε), ξ \in {x, y},$

(A19)

where the bound $σ_{E^{ξ}} (T_{1}^{ψ}) \to 0$ as $ε \to 0$ . Then, we have

$\begin{matrix} {∥\tilde{ξ}^(t)∥}_{2} & \leq & {∥e^{A_{r}^{ξ} t} \tilde{ξ} (0)∥}_{2} + {∥e^{A_{r}^{ξ} t} \int_{0}^{t} e^{- A_{r}^{ξ} τ} d τ E_{r}^{ξ} σ_{E^{ξ}} (ε)∥}_{2} \\ \leq & {∥e^{A_{r}^{ξ} t} \tilde{ξ} (0)∥}_{2} + σ_{E^{ξ}} (ε) {∥(e^{A_{r}^{ξ} t} - I) {(A_{r}^{ξ})}^{- 1} E_{r}^{ξ}∥}_{2}, ξ \in {x, y} . \end{matrix}$

(A20)

Based on the boundedness of ${∥(e^{A_{r}^{ξ} t} - I) {(A_{r}^{ξ})}^{- 1} E_{r}^{ξ}∥}_{2}$ provided by the exponential stability of the reduced model (A18), we use singular perturbation theory to conclude that there exist $k_{ξ} > 0$ , a $t_{i}^{ξ} > 0$ , $ε^{* *} > 0$ , and ultimate bound $σ_{i}^{ξ} (ε)$ satisfying $σ_{i}^{ξ} (ε) \to 0$ as $ε \to 0$ , such that $\forall 0 < ε < ε^{* *}$ and $\forall t > t_{i}^{ξ}$ , and the following inequality holds:

$\begin{matrix} | {\tilde{ξ}}_{i} (t) | & \leq & {∥\tilde{ξ} (t)∥}_{2} \leq {∥\tilde{ξ} (t) - \tilde{ξ}^(t)∥}_{2} + {∥\tilde{ξ}^(t)∥}_{2} \\ \leq & k_{ξ} ε + {∥e^{A_{r}^{ξ} t} \tilde{ξ} (0)∥}_{2} + σ_{E^{ξ}} (ε) {∥(e^{A_{r}^{ξ} t} - I) {(A_{r}^{ξ})}^{- 1} E_{r}^{ξ}∥}_{2} \\ \leq & σ_{i}^{ξ} (ε), ξ \in {x, y} . \end{matrix}$

(A21)

Up to now, we have finished the proof of Theorem 1 for the

ξ \in {ψ, x, y, z}

channels, and the next step is to analyze the

η \in {ϕ, θ}

channels. The tracking error dynamics and UDE estimation error dynamics are respectively given by

$\underset{︸}{[\begin{matrix} \tilde{η} ˙ \\ _{\tilde{η} ¨ i} \end{matrix}]}_{\tilde{η} ˙ i} = \underset{︸}{[\begin{matrix} 0 & 1 \\ - k_{i p}^{η} & - k_{i d}^{η} \end{matrix}]} A_{r}^{η} \underset{︸}{[\begin{matrix} {\tilde{η}}_{i} \\ _{\tilde{η} ˙ i} \end{matrix}]} \tilde{η} ¨ + \underset{︸}{[\begin{matrix} 0 \\ - 1 \end{matrix}]} B_{r}^{η} {\tilde{f}}_{i}^{η}, η \in {ϕ, θ},$

(A22)

and

$ε_{\tilde{f}}^{˙} = - \frac{1}{T_{i}^{η *}} {\tilde{f}}_{i}^{η} + ε {\dot{f}}_{i}^{η}, η \in {ϕ, θ} .$

(A23)

Under stability Condition 1, it is clear that the reduced model

$\tilde{η} ˙ = A_{r}^{η} \tilde{η}, η \in {ϕ, θ},$

(A24)

and the boundary-layer model

$_{\tilde{f}}^{˙} = - \frac{1}{T_{i}^{η *}} {\tilde{f}}_{i}^{η}, η \in {ϕ, θ},$

(A25)

are exponentially stable at the origin. The remaining parts can be proven in the same way as the proof of the $ψ$ channel and is omitted. This ends the proof of Theorem 1.

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