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Dynamic Analysis and Impedance Control of a Novel Double-Driven Parallel Mechanism

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Abstract

This paper introduces the design, dynamic model, and impedance control of a novel five degrees of freedom (DOF) double-driven parallel mechanism (DDPM) for the task of grinding aircraft composite skin. Firstly, a novel parallel mechanism consisting of one PUU (P-active prismatic pairs, U-universal joints) limb and two PRRS (R-revolution joints, R-active revolution joints, S-spherical joints) limbs (PUU-2PRRS) is proposed. Secondly, considering the particularity of the sub-closed loop structure of the mechanism, the dynamic model of the proposed PUU-2PRRS parallel mechanism is established based on closed loop vector and Lagrange methods, and then the correctness of the established dynamic model is proved by the comparative analysis of MATLAB and Adams software. Thirdly, an adaptive impedance control law with switching update rate is proposed to achieve constant force tracking in uncertain environments. Considering that the composite skin grinding task requires low maximum overshoot and high convergence speed of force tracking, the selection methods of the update rate are obtained based on the maximum overshoot and convergence speed analysis in frequency domain. Finally, by simulating different working conditions of composite skin grinding, the adaptive impedance control under switching update rate has better force tracking performance compared with the adaptive impedance control under fixed update rate, which proves the effectiveness of the proposed controller.

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Data Availability

The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Funding

This research is financially supported by the Fundamental Research Funds for the Central Universities (No. 3122019188).

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Conceptualization: Mengli Wu and Dezuo Li; Methodology: Mengli Wu, Dezuo Li and Yiran Cao; Formal analysis and investigation: Mengli Wu, Dezuo Li and Xuhao Wang; Writing-original draft preparation: Mengli Wu and Linda Jia; Writing-review and editing: Mengli Wu, Dezuo Li, Yiran Cao and Xuhao Wang; Funding acquisition: Mengli Wu and Linda Jia.

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Correspondence to Yiran Cao or Xuhao Wang.

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Appendices

Appendix 1 Stability proof for position control

Considering the dynamic property of the system, we use the following Lyapunov function candidate,

$$V=\frac{1}{2}{\boldsymbol{r}}^{\textrm{T}}\textbf{M}\left({\textbf{X}}_{\textrm{s}}\right)\boldsymbol{r}$$
(83)

The Lyapunov function is derived as follows

$$\dot{V}={\boldsymbol{r}}^{\textrm{T}}\textbf{M}\left({\textbf{X}}_{\textrm{s}}\right)\dot{\boldsymbol{r}}+\frac{1}{2}{\boldsymbol{r}}^{\textrm{T}}\dot{\textbf{M}}\left({\textbf{X}}_{\textrm{s}}\right)\boldsymbol{r}$$
(84)

According to the characteristics of the dynamic of the mechanism, \(\dot{\textbf{M}}\left({\textbf{X}}_{\textrm{s}}\right)-2\textbf{C}\left({\dot{\textbf{X}}}_{\textrm{s}},{\textbf{X}}_{\textrm{s}}\right)\) is considered to be skew-symmetric yields \(\frac{1}{2}{\boldsymbol{r}}^{\textrm{T}}\dot{\textbf{M}}\left({\textbf{X}}_{\textrm{s}}\right)\boldsymbol{r}={\boldsymbol{r}}^{\textrm{T}}\textbf{C}\left({\dot{\textbf{X}}}_{\textrm{s}},{\textbf{X}}_{\textrm{s}}\right)\boldsymbol{r}\) [48]. Then, substituting \(\frac{1}{2}{\boldsymbol{r}}^{\textrm{T}}\dot{\textbf{M}}\left({\textbf{X}}_{\textrm{s}}\right)\boldsymbol{r}={\boldsymbol{r}}^{\textrm{T}}\textbf{C}\left({\dot{\textbf{X}}}_{\textrm{s}},{\textbf{X}}_{\textrm{s}}\right)\boldsymbol{r}\) into Eq. (84) yields

$${\displaystyle \begin{array}{l}\dot{V}={\boldsymbol{r}}^{\textrm{T}}\left(\textbf{M}\left({\textbf{X}}_{\textrm{s}}\right)\dot{\boldsymbol{r}}+\textbf{C}\left({\dot{\textbf{X}}}_{\textrm{s}},{\textbf{X}}_{\textrm{s}}\right)\boldsymbol{r}\right)\\ {}={\boldsymbol{r}}^{\textrm{T}}\left(-{\textbf{K}}_{\textrm{d}}\boldsymbol{r}- {h}\mathrm{tanh}\left(\frac{\boldsymbol{r}}{\varepsilon}\right)+\Delta \left({\textbf{X}}_{\textrm{s}},{\dot{\textbf{X}}}_{\textrm{s}},{\ddot{\textbf{X}}}_{\textrm{s}}\right)\right)\end{array}}$$
(85)

From Lemmas 1 and 2 (see Appendix 2) [49,50,51], the following relation is given,

$${r}^{\textrm{T}}\left(-h\tanh \left(\frac{r}{\varepsilon}\right)+\varDelta \left({X}_{\textrm{s}},{\dot{X}}_{\textrm{s}},{\ddot{X}}_{\textrm{s}}\right)\right)\le -h\left|r\right|+ h\rho \varepsilon +{r}^{\textrm{T}}\varDelta \left({X}_{\textrm{s}},{\dot{X}}_{\textrm{s}},{\ddot{X}}_{\textrm{s}}\right)\le h\rho \varepsilon, \rho =0.2785$$
(86)

Therefore, the Lyapunov function can be further derived as

$$\begin{aligned}\dot{V} &\le - \boldsymbol{r}^\textrm{T}\boldsymbol{K}_{d}\boldsymbol{r}+ h {\rho}{\varepsilon} \le - {\lambda}_\textrm{min}(\boldsymbol{K}_{d})\boldsymbol{r}^\textrm{T}\boldsymbol{r} + h {\rho}{\varepsilon} \\ &= - \frac{2{\lambda}\;_\textrm{min}(\boldsymbol{K}_\textrm{d})}{{\lambda}_\textrm{max}(\boldsymbol{M}(\boldsymbol{X}_{s}))}\frac{1}{2}\lambda_\textrm{max}(\boldsymbol{M}(\boldsymbol{X}_{s})\boldsymbol{r}^\textrm{T}\boldsymbol{r} + h{\rho}{\varepsilon}\\ & \le - 2{\lambda}V + h{\rho}{\varepsilon}\end{aligned}$$
(87)

where λmin (Kd) and λmax (M(Xs)) are the maximum eigenvalues of Kd and M(Xs), respectively, and \(\lambda =\frac{\lambda {}_{\mathit{\min}}\left({\textbf{K}}_{\textrm{d}}\right)}{\lambda {}_{\mathit{\max}}\left(\textbf{M}\left({\boldsymbol{X}}_s\right)\right)}\).

Considering \(\dot{V}\le -2\lambda V+ h\rho \varepsilon\) and Lemma 3 (see Appendix 2) [52] yield the following relations,

$$V(t)\le {\textrm{e}}^{-2\lambda \left(t-{t}_0\right)}V\left({t}_0\right)+ h\rho \varepsilon {\textrm{e}}^{-2\lambda t}{\int}_{t_0}^t{\textrm{e}}^{-2\lambda \varsigma}\textrm{d}\varsigma ={\textrm{e}}^{-2\lambda \left(t-{t}_0\right)}V\left({t}_0\right)+\frac{h\rho \varepsilon}{2\lambda}\left(1-{\textrm{e}}^{-2\lambda \left(t-{t}_0\right)}\right)$$
(88)

from which the following relation is easily derived, \(\underset{t\to \infty }{\mathit{\lim}}V(t)\le \frac{h\rho \varepsilon}{2\lambda }\). It is obvious that the tracking error converges gradually, and the convergence accuracy depends on ε, λ and h.

Appendix 2 Lemmas for position control stability proof

Lemma 1

For any given real number υ, the following inequality exists [49, 50],

$$\upsilon \mathrm{tanh}\left(\frac{\upsilon }{\varepsilon}\right)=\left|\upsilon \;\mathrm{tanh}\left(\frac{\upsilon }{\varepsilon}\right)\right|=\left|\upsilon \right|\left|\mathit{\tanh}\left(\frac{\upsilon }{\varepsilon}\right)\right|\ge 0$$
(89)

Lemma 2

For any given real number κ, the following inequality exists [51],

$$0\le \left|\kappa \right|-\kappa ;\mathrm{tanh}\left(\frac{\kappa }{\varepsilon}\right)\le \rho \varepsilon, \rho =0.2785$$
(90)

Lemma 3

Let f, V : [0, ∞) ∈ R. If the relation \(\dot{V}\le -\psi V+f,\forall t\ge {t}_0\ge 0\), then the following relation holds [52]:

$$V(t)\le {\textrm{e}}^{-\psi \left(t-{t}_0\right)}V\left({t}_0\right)+{\int}_{t_0}^t{\textrm{e}}^{-\psi \left(t-\tau \right)}f\left(\tau \right)\textrm{d}\tau$$
(91)

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Wu, M., Li, D., Cao, Y. et al. Dynamic Analysis and Impedance Control of a Novel Double-Driven Parallel Mechanism. J Intell Robot Syst 108, 45 (2023). https://doi.org/10.1007/s10846-023-01915-1

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