Trajectory Tracking Control of a Class of Underactuated Mechanical Systems with Nontriangular Normal Form Based on Block Backstepping Approach

Abstract

In this paper, the formulation of a block-backstepping control approach is presented to address the trajectory tracking problem for a general class of nonlinear n degrees of freedom (n-DOF) underactuated mechanical systems (UMSs) in nontriangular normal form. First, the Euler-Lagrange model of the general form of UMSs is transformed into block-strict feedback form. Then, control input for the n-DOF UMS will be obtainable by synthesis of the backstepping approach. Additionally, an integral action is incorporated to the proposed controller to enhance the steady state performance of the overall system and also to improve the trajectory tracking precision of the control system. Lyapunov theory is utilizable to prove the stability and convergence of the overall system. To demonstrate the effectiveness of the designed controller, the proposed control algorithm is applied through numerical simulation for the trajectory tracking of a single-link flexible-link flexible-joint manipulator (SFLFJM) as an UMS with the nontriangular normal form.

Appendix: Dynamic Model of the SFLFJM

Mass matrix:

$$M=M_{FI} +M_{FR} $$

where

$$M_{FI} =\left[ {{\begin{array}{*{20}c} {h^{2}I_{r} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \end{array} }} \right], M_{FR} =\left[ {{\begin{array}{*{20}c} 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {\mathrm{M}_{\text{11}} } \hfill & {\mathrm{M}_{12} } \hfill & {\mathrm{M}_{13} } \hfill \\ 0 \hfill & {\mathrm{M}_{21} } \hfill & {\mathrm{M}_{22} } \hfill & {\mathrm{M}_{23} } \hfill \\ 0 \hfill & {\mathrm{M}_{31} } \hfill & {\mathrm{M}_{32} } \hfill & {\mathrm{M}_{33} } \hfill \end{array} }} \right] $$

where

$$\begin{array}{@{}rcl@{}} M_{11} &=&{(\rho L^{3})} / 3+\rho ({\Phi}_{11} {\delta_{1}^{2}} + 2{\Phi}_{12} \delta_{1} \delta_{2} +{\Phi}_{22} {\delta_{2}^{2}} )\\ &&+I_{h} +I_{tip} +m_{tip} L^{2} +m_{tip} (\phi_{_{1} }^{2} (L){\delta_{1}^{2}}\\ &&+ 2\phi_{1} (L)\phi_{2} (L)\delta_{1} \delta_{2} +\phi_{_{2} }^{2} (L){\delta_{2}^{2}} )\\ M_{12} &=&\rho {\Phi}_{1x} +m_{tip} L\phi_{1} (L)+I_{tip} {\phi }^{\prime}_{1} (L) \\ M_{21} &=&M_{12} \end{array} $$

$$\begin{array}{@{}rcl@{}} M_{13} &=&\rho {\Phi}_{2x} +m_{tip} L\phi_{2} (L)+I_{tip} {\phi }^{\prime}_{2} (L) \\ M_{31} &=&M_{13} \\ M_{22} &=&\rho {\Phi}_{11} +m_{tip} {\phi_{1}^{2}}(L)+I_{tip} \phi^{\prime{2}}_{1}(L) \\ M_{23} &=&\rho {\Phi}_{11} +m_{tip} \phi_{1} (L)\phi_{2} (L)+I_{tip} {\phi }^{\prime}_{1} (L){\phi }^{\prime}_{2} (L) \\ M_{32} &=&M_{23} \\ M_{33} &=&\rho {\Phi}_{22} +m_{tip} {\phi_{2}^{2}}(L)+I_{tip} {\phi }^{\prime2}_{2} (L) \end{array} $$

Stiffness matrix:

$$K=K_{FI} +K_{FR} $$

where

$$\begin{array}{@{}rcl@{}} K_{FI} =\left[ {{\begin{array}{*{20}c} {\mathrm{k}_{\text{jo}} } \hfill & {{-k}_{\text{jo}} } \hfill & 0 \hfill & 0 \hfill \\ {{-k}_{\text{jo}} } \hfill & {\mathrm{k}_{\text{jo}} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \end{array} }} \right], K_{FR} =\left[ {{\begin{array}{*{20}c} 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {\text{0}} \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\mathrm{K}_{11} } \hfill & {\mathrm{K}_{12} } \hfill \\ 0 \hfill & 0 \hfill & {\mathrm{K}_{21} } \hfill & {\mathrm{K}_{22} } \hfill \end{array} }} \right] \end{array} $$

where

$$\begin{array}{@{}rcl@{}} K_{11} &=& EI{\Phi}_{11xx} \\ K_{12} &=& EI{\Phi}_{12xx} \\ K_{21} &=& K_{12} \\ K_{22} &=& EI{\Phi}_{22xx} \end{array} $$

Dissipation matrix:

$$\begin{array}{@{}rcl@{}} C=\left[ {{\begin{array}{*{20}c} 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {\mathrm{C}_{11} } \hfill & {\mathrm{C}_{12} } \hfill & {\mathrm{C}_{13} } \hfill \\ 0 \hfill & {\mathrm{C}_{21} } \hfill & {\mathrm{C}_{22} } \hfill & {\mathrm{C}_{23} } \hfill \\ 0 \hfill & {\mathrm{C}_{31} } \hfill & {\mathrm{C}_{32} } \hfill & {\mathrm{C}_{33} } \hfill \end{array} }} \right] \end{array} $$

where

$$\begin{array}{@{}rcl@{}} C_{11} &=&\rho (\dot{{\delta }}_{1} (\delta_{1} {\Phi}_{11} +\delta_{2} {\Phi}_{12} )+\dot{{\delta }}_{2} (\delta_{2} {\Phi}_{22} +\delta_{1} {\Phi}_{12} )) \\ && +m_{tip} ({\phi_{1}^{2}} (L)\dot{{\delta }}_{1} \delta_{1} +\phi_{1} (L)\phi_{2} (L)\dot{{\delta }}_{1} \delta_{2}\\ &&+\phi_{1} (L)\phi_{2} (L)\dot{{\delta }}_{2} \delta_{1} +\delta_{2} \dot{{\delta }}_{2} \phi_{2}^{2} (L)); \\ C_{12} &=&\rho \dot{{\theta }}_{L} (\delta_{1} {\Phi}_{11} +\delta_{2} {\Phi}_{12} )+m_{tip} \dot{{\theta }}_{L} (\delta_{1} {\phi_{1}^{2}} (L)\\ &&+\phi_{1} (L)\phi_{2} (L)\delta_{2} ); \\ C_{21} &=&-C_{12} \\ C_{13} &=&\rho \dot{{\theta }}_{L} (\delta_{2} {\Phi}_{22} +\delta_{1} {\Phi}_{12} )+m_{tip} \dot{{\theta }}_{L} (\delta_{1} \phi_{1} (L)\phi_{2} (L)\\ &&+\delta_{2} {\phi_{2}^{2}} (L)); \\ C_{31} &=&-C_{13} \\ C_{23} &=&0 \\ C_{32} &=&0 \\ C_{22} &=&0 \\ C_{33} &=&0 \end{array} $$

Gravity vector:

$$\begin{array}{@{}rcl@{}} G=\left[ {{\begin{array}{*{20}c} {G_{1} } \hfill \\ {G_{2} } \hfill \\ {G_{3} } \hfill \\ {G_{4} } \hfill \end{array} }} \right] \end{array} $$

where

$$\begin{array}{@{}rcl@{}} G_{1} &=&0; \\ G_{2} &=&\rho g{(L^{2}} / 2)cos(\theta_{L} )-\rho g(\delta_{1} {\Phi}_{10} +\delta_{2} {\Phi}_{20} )sin(\theta_{L} ); \\ G_{3} &=&\rho g{\Phi}_{10} cos(\theta_{L} ); \\ G_{4} &=&\rho g{\Phi}_{20} cos(\theta_{L} ); \end{array} $$

where

$$\begin{array}{@{}rcl@{}} &&{\kern-7.75pt} {\Phi}_{10} = {{\int}_{0}^{L}} {\phi_{1} (x)dx} \\ &&{\kern-7.75pt} {\Phi}_{20} = {{\int}_{0}^{L}} {\phi_{2} (x)dx} \\ &&{\kern-7.75pt} {\Phi}_{11} = {{\int}_{0}^{L}} {{\phi_{1}^{2}} (x)dx} \\ &&{\kern-7.75pt} {\Phi}_{12} = {{\int}_{0}^{L}} {\phi_{1} (x)\phi_{2} (x)dx} \\ &&{\kern-7.75pt} {\Phi}_{22} = {{\int}_{0}^{L}} {{\phi_{2}^{2}} (x)dx} \\ &&{\kern-7.75pt} {\Phi}_{1x} = {{\int}_{0}^{L}} {x\phi_{1} (x)dx} \\ &&{\kern-7.75pt} {\Phi}_{2x} = {{\int}_{0}^{L}} {x\phi_{2} (x)dx} {\Phi}_{11xx} ={{\int}_{0}^{L}} {\left( {\frac{d^{2}\phi_{1} (x)}{dx^{2}}} \right)^{2}dx} \\ &&{\kern-7.75pt} {\Phi}_{12xx} = {{\int}_{0}^{L}} {\left( {\frac{d^{2}\phi_{1} (x)}{dx^{2}}} \right)\left( {\frac{d^{2}\phi_{2} (x)}{dx^{2}}} \right)dx} \\ &&{\kern-7.75pt} {\Phi}_{22xx} = {{\int}_{0}^{L}} {\left( {\frac{d^{2}\phi_{2} (x)}{dx^{2}}} \right)^{2}dx} \end{array} $$

where

$$\begin{array}{@{}rcl@{}} \phi_{\text{1}} (\mathrm{x}) = {\cosh}(\mu_{1} \text{x})-\cos(\mu_{1} \text{x})-\gamma_{\text{1}} ({\sinh}(\mu_{1} \text{x})-\sin(\mu_{1} \text{x})) \\ \phi_{\text{2}} (\text{x})= {\cosh}(\mu_{2} \text{x})-\cos(\mu_{2} \text{x})-\gamma_{\text{2}} ({\sinh}(\mu_{2} \text{x})-\sin(\mu_{2} \text{x})) \end{array} $$

where

$$\begin{array}{@{}rcl@{}} \mu_{1} &=&\frac{\beta_{1} }{L} , \mu_{2} =\frac{\beta_{2} }{L} \\ \beta_{1} &=&1.8751 , \beta_{2} = 4.6941 , \gamma_{1} = 0.7341 , \gamma_{2} = 1.0185 \end{array} $$