Non-singular terminal sliding mode control of an omnidirectional mobile manipulator based on extended state observer

Abstract

This paper presents a non-singular terminal sliding mode control (NTSMC) design based on an improved extended state observer (IESO) with application to an omnidirectional mobile manipulator (OMM) for trajectory tracking control. Firstly, a unified dynamic model is derived based on Lagrange method for an OMM prototype. An IESO that can reduce the initial peaking phenomenon is applied to estimate the model uncertainties and external disturbances. Then a non-singular terminal sliding mode controller is applied for trajectory tracking control. Stability of the closed-loop system is analyzed using Lyapunov theory. Finally, both simulations and experimental tests verify the effectiveness of the proposed control scheme.

7. Appendix

This appendix shows the matrix in Eqs. (13) and (14).

1.1 7.1 For Eq. (13), i.e., ${{\varvec{M}}_{0}}\ddot{{\varvec{q}}}+{{{\varvec{C}}}_{0}\dot{{\varvec{q}}}}+\varvec{G}=\varvec{\tau }$

Matrix ${\varvec{M}}_0$:

$$\begin{aligned} \varvec{M}_{0} = \left[ \begin{array}{ l l l } {\varvec{M}}_{011} &{} {\varvec{M}}_{012}\\ {\varvec{M}}_{021} &{} {\varvec{M}}_{022} \end{array} \right] , \end{aligned}$$

(44)

where:

$$\begin{aligned} \varvec{M}_{011}& = \left[ \begin{array}{ l l l } { m _ { 11 } } &{} { m _ { 12 } } &{} { m _ { 13 } } \\ { m _ { 21 } } &{} { m _ { 22 } } &{} { m _ { 23 } } \\ { m _ { 31 } } &{} { m _ { 32 } } &{} { m _ { 33 } } \end{array} \right] ,\\ \varvec{M}_{012}& = \left[ \begin{array}{ l l } { m _ { 14 } } &{} { m _ { 15 } } \\ { m _ { 24 } } &{} { m _ { 25 } } \\ { m _ { 34 } } &{} { m _ { 35 } } \end{array} \right] ,\\ \varvec{ M } _ { 021 }& = \left[ \begin{array}{ l l l } { m _ { 41 } } &{} { m _ { 42 } } &{} { m _ { 43 } } \\ { m _ { 51 } } &{} { m _ { 52 } } &{} { m _ { 53 } } \end{array} \right] , \varvec{ M } _ { 022 } = \left[ \begin{array}{ l l } { m _ { 44 } } &{} { m _ { 45 } } \\ { m _ { 54 } } &{} { m _ { 55 } } \end{array} \right] , \\ m_{11}& = m_{0}+m_{1}+m_{2}+m_{3}+m_{4};\\ m_{12}& = 0;\\ m_{13}& = \left( -\frac{1}{2}l_{1}m_{1}-l_{1}m_{3}+\frac{1}{2}l_{3}m_{3}-l_{1}m_{4}\right) \cos \theta _{1}\sin \varphi \\&+\left( \frac{1}{2}l_{2}m_{2}+l_{4}m_{3} +\frac{1}{2}l_{4}m_{4}\right) \sin \theta _{2}\sin \varphi ;\\ m_{14}& = \left( -\frac{1}{2}l_{1}m_{1}-l_{1}m_{3}+\frac{1}{2}l_{3}m_{3}-l_{1}m_{4}\right) \cos \varphi \sin \theta _{1}\\ m_{15}& = -\left( \frac{1}{2}l_{2}m_{2}+l_{4}m_{3}+\frac{1}{2}l_{4}m_{4}\right) \cos \theta _{2}\cos \varphi ;\\ m_{21}& = 0;\\ m_{22}& = m_{0}+m_{1}+m_{2}+m_{3}+m_{4};\\ m_{23}& = \left( \frac{1}{2}l_{1}m_{1}+l_{1}m_{3}-\frac{1}{2}l_{3}m_{3}+l_{1}m_{4}\right) \cos \theta _{1}\cos \varphi \\&-\left( \frac{1}{2}l_{2}m_{2}+l_{4}m_{3}+\frac{1}{2}l_{4}m_{4}\right) \cos \varphi \sin \theta _{2};\\ m_{24}& = \left( -\frac{1}{2}l_{1}m_{1}-l_{1}m_{3}+\frac{1}{2}l_{3}m_{3}-l_{1}m_{4}\right) \sin \theta _{1}\sin \varphi ;\\ m_{25}& = -\left( \frac{1}{2}l_{2}m_{2}+l_{4}m_{3}+\frac{1}{2}l_{4}m_{4}\right) \cos \theta _{2}\sin \varphi ;\\ m_{31}& = m_{13};m_{32}=m_{23};\\ m_{33}& = I+\frac{1}{6}\sum _{i=1}^{4}l_{i}^{2}m_{i}+\frac{1}{2}(l_{1}^{2}m_{3}-l_{1}l_{3}m_{3}+l_{4}^{2}m_{3}+l_{1}^{ 2}m_{4})\\&+\frac{1}{6}\cos ^{2}\theta _{1}(l_{1}^{2}m_{1}+3l_{1}^{2}m_{3}-3l_{1}l_{3}m_{3}+l_{3}^{2}m_{3} \\&+3l_{1}^{2}m_{4})-\frac{ 1 }{ 6 } \cos ^ { 2 } \theta _ { 2 } ( l _ { 2 } ^ { 2 } m _ { 2 } + 3 l _ { 4 } ^ { 2 } m _ { 3 } + l _ { 4 } ^ { 2 } m _ { 4 } ) \\&+\frac{1}{6}\sin ^{2}\theta _{1}(-l_{1}^{2}m_{1} -3l_{1}^{2}m_{3}+3l_{1}l _ { 3 } m _ { 3 } - l _ { 3 } ^ { 2 } m _ { 3 } - 3 l _ { 1 } ^ { 2 } m _ { 4 } ) \\&- \cos \theta _ { 1 } \sin \theta _ { 2 }( 2 l _ { 1 } l _ { 4 } m _ { 3 } -l _ { 3 } l _ { 4 } m _ { 3 } + l _ { 1 } l _ { 4 } m _ { 4 } ) \\&+\frac{ 1 }{ 6 } \sin ^ { 2 } \theta _ { 2 }( l _ { 2 } ^ { 2 } m _ { 2 } +3l_{4}^{2}m_{3}+l_{4}^{2}m_{4});\\ m_{34}& = m_{35}=0;\\ \\ m_{41}& = m_{14};m_{42}=m_{24};m_{43}=0;\\ m_{44}& = \frac{1}{3}l_{1}^{2}m_{1}+l_{1}^{2}m_{3}-l_{1}l_{3}m_{3}+\frac{1}{3}l_{3}^{2}m_{3}+l_{1}^{2}m_{4};\\ m_{45}& = \left( l_{1}l_{4}m_{3}-\frac{1}{2}l_{ 3 } l _ { 4 } m _ { 3 } + \frac{ 1 }{ 2 } l _ { 1 } l _ { 4 } m _ { 4 }\right) \sin (\theta _1-\theta _2);\\ m_{51}& = m_{15};m_{52}=m_{25};m_{53}=0;m_{54}=m_{45};\\ m_{55}& = \frac{1}{3}l_{2}^{2}m_{2}+l_{4}^{2}m_{3}+\frac{1}{3}l_{4}^{2}m _{4}; \end{aligned}$$

Matrix ${\varvec{C}}_0$:

$$\begin{aligned} \varvec{C}_{0}=\left[ \begin{array}{ c c } {\varvec{C}}_{011} &{} {\varvec{C}}_{012}\\ {\varvec{C}}_{021} &{} {\varvec{C}}_{022} \end{array}\right] , \end{aligned}$$

(45)

where:

$$\begin{aligned} \varvec{ C } _ { 011 }& = \left[ \begin{array}{ l l l } { c _ { 11 } } &{} { c _ { 12 } } &{} { c _ { 13 } } \\ { c _ { 21 } } &{} { c _ { 22 } } &{} { c _ { 23 } } \\ { c _ { 31 } } &{} { c _ { 32 } } &{} { c _ { 33 } } \end{array} \right] ,\\ {{\varvec{C}}}_{012}& = \left[ \begin{array}{cc}{c_{14}}&{}{c_{15}}\\ {c_{24}}&{}{c_{25}}\\ {c_{34}}&{}{c_{35}}\end{array}\right] ,\\ {{\varvec{C}}}_{021}& = \left[ \begin{array}{lll} {c_{41}}&{}{c_{42}}&{}{c_{43}}\\ {c_{51}}&{}{c_{52}}&{}{c_{53}}\end{array}\right] ,\\ {{\varvec{C}}} _ { 022 }& = \left[ \begin{array}{ l l } { c _ { 44 } } &{} { c _ { 45 } } \\ { c _ { 54 } } &{} { c _ { 55 } } \end{array} \right] ,\\ c_{11}& = c_{12}=0;\\ c_{13}& = \left( - \frac{ 1 }{ 2 } l _ { 1 } m _ { 1 } - l _ { 1 } m _ { 3 } + \frac{ 1 }{ 2 } l _ { 3 }m_{3}-l_{ 1}m_{4}\right) \cos \theta _{1}\cos (\varphi ){\dot{\varphi }}\\&+\left( \frac{1}{2}l_{2}m_{2}+l_{4 }m_{3}+\frac{1}{2}l_{4}m_{4}\right) \cos \varphi \sin (\theta _{2}){\dot{\varphi }};\\ c_{14}& = \left( -\frac{1}{2}l_{1}m_{1}-l_{1}m_{3}+\frac{1}{2}l_{3}m_{3}-l_{1}m_{4}\right) \cos \theta _{1}\cos (\varphi ){\dot{\theta }}_{1}\\&+(l_{1}m_{1}+2l_{1}m_{3}-l_{3}m_{3}+2l_{1}m_{4})\sin \theta _{1}\sin (\varphi ){\dot{\varphi }};\\ c_{15}& = \left( \frac{ 1 }{ 2 } l _ { 2 } m _ { 2 } + l _ { 4 } m _ { 3 } + \frac{ 1 }{ 2 } l _ { 4 } m _ { 4 } \right) \cos \varphi \sin (\theta _{2}){\dot{\theta }} _ { 2 } + ( l _ { 2 } m _ { 2 } \\&+2 l _ { 4 } m _ { 3 } + l _ { 4 } m _ { 4 } ) \cos \theta _ { 2 } \sin (\varphi ){\dot{\varphi }};\\ c_{21}& = c_{22}=0;\\ c_{23}& = \left( - \frac{ 1 }{ 2 } l _ { 1 } m _ { 1 } - l _ { 1 } m _ { 3 } + \frac{ 1 }{ 2 } l _ { 3 } m _ { 3 } - l _ { 1 } m _ { 4 } \right) \cos \theta _ { 1 } \sin (\varphi ){\dot{\varphi }} \\&+\left( \frac{ 1 }{ 2 } l _ { 2 } m _ { 2 } + l _ { 4 } m _ { 3 } + \frac{ 1 }{ 2 } l _ { 4 } m _ { 4 } \right) \sin \theta _ { 2 } \sin (\varphi ){\dot{\varphi }};\\ c_{24}& = \left( - \frac{ 1 }{ 2 } l _ { 1 } m _ { 1 } - l _ { 1 } m _ { 3 } + \frac{ 1 }{ 2 } l _ { 3 } m _ { 3 } - l _ { 1 } m _ { 4 }\right) \cos \theta _ { 1 } \sin (\varphi ){\dot{\theta }} _ { 1 }\\&-( l _ { 1 } m _ { 1 } + 2 l _ { 1 } m _ { 3 } - l _ { 3 } m _ { 3 } + 2 l _ { 1 } m _ { 4 } ) \cos \varphi \sin (\theta _{1}){\dot{\varphi }};\\ c_{25}& = \left( \frac{ 1 }{ 2 } l _ { 2 } m _ { 2 } + l _ { 4 } m _ { 3 } + \frac{ 1 }{ 2 } l _ { 4 } m _ { 4 }\right) \sin \theta _{2}\sin (\varphi ){\dot{\theta }} _ { 2 } -( l _ { 2 } m _ { 2 }\\&+2 l _ { 4 } m _ { 3 } + l _ { 4 } m _ { 4 }) \cos \theta _ { 2 } \cos (\varphi ){\dot{\varphi }}; \\ c_{31}& = c_{32}=c_{33}=0;\\ c_{34}& = \left( - \frac{ 2 }{ 3 } l _ { 1 } ^ { 2 } m _ { 1 } - 2 l _ { 1 } ^ { 2 } m _ { 3 } + 2 l _ { 1 } l _ { 3 } m _ { 3 } - \frac{ 2 }{ 3 } l _ { 3 } ^ { 2 } m _ { 3 } - 2 l _ { 1 } ^ { 2 } m _ { 4 }\right) \\&\quad \cos \theta _ { 1 }\sin ( \theta _ { 1 } ) {\dot{\varphi }}+(2 l_1 l_4 m_3-l_3 l_4 m_3+l_1 l_4 m_4)\sin \theta _{ 1}\\&\quad \sin (\theta _{2}){\dot{\varphi }};\\ c_{35}& = -( 2 l _ { 1 } l _ { 4 } m _ { 3 } - l _ { 3 } l _ { 4 } m _ { 3 } + l _ { 1 } l _ { 4 } m _ { 4 }) \cos \theta _ { 1 } \cos ( \theta _ { 2 } ) {\dot{\varphi }} \\&+\left( \frac{ 2 }{ 3 } l _ { 2 } ^ { 2 } m _ { 2 } + 2 l _ { 4 } ^ { 2 } m _ { 3 } + \frac{ 2 }{ 3 } l _ { 4 } ^ {2} m _ { 4 }\right) \cos \theta _ { 2 } \sin ( \theta _ { 2 }){\dot{\varphi }};\\ c_{41}& = c_{42}=0;\\ c_{43}& = \left( \frac{ 1 }{ 3 } l _ { 1 } ^ { 2 } m _ { 1 } + l _ { 1 } ^ { 2 } m _ { 3 } - l _ { 1 } l _ { 3 } m _ { 3 } + \frac{ 1 }{ 3 } l _ { 3 } ^ { 2 } m _ { 3 } + l _ { 1 } ^ { 2 } m _ { 4 }\right) \cos \theta _ { 1 } \\&\quad \sin ( \theta _ { 1 } ) {\dot{\varphi }}+( - l _ { 1 } l _ { 4 } m _ { 3 } + \frac{ 1 }{ 2 } l _ { 3 } l _ { 4 } m _ { 3 } - \frac{ 1 }{ 2 } l _ { 1 } l _ { 4 } m _ { 4 }) \sin \theta _ { 1 }\\&\quad \sin ( \theta _ { 2 }) {\dot{\varphi }};\\ c_{44}& = 0;\\ c_{45}& = \left( -l_{1}l_{ 4 } m _ { 3 } + \frac{ 1 }{ 2 } l _ { 3 } l _ { 4 } m _ { 3 } - \frac{ 1 }{ 2 } l _ { 1 } l _ { 4 } m _ { 4 }\right) \cos ( \theta _ { 1 } - \theta _ { 2 }) {\dot{\theta }} _ { 2 };\\ c_{51}& = c_{52}=0;\\ c_{53}& = \left( l _ { 1 } l _ { 4 } m _ { 3 } - \frac{ 1 }{ 2 } l _ { 3 } l _ { 4 } m _ { 3 } + \frac{ 1 }{ 2 } l _ { 1 } l _ { 4 } m _ { 4 } \right) \cos \theta _ { 1 } \cos ( \theta _ { 2 } ) {\dot{\varphi }} \\&- \left( \frac{ 1 }{ 3 } l _ { 2 } ^ { 2 } m _ { 2 } + l _ { 4 } ^ { 2 } m _ { 3 } + \frac{ 1 }{ 3 } l _ { 4 } ^ { 2 } m _ { 4 } \right) \cos \theta _ { 2 } \sin ( \theta _ { 2 } ) {\dot{\varphi }};\\ c_{54}& = \left( l _ { 1 } l _ { 4 } m _ { 3 } - \frac{ 1 }{ 2 } l _ { 3 } l _ { 4 } m _ { 3 } + \frac{ 1 }{ 2 } l _ { 1 } l _ { 4 } m _ { 4 } \right) \cos ( \theta _ { 1 } - \theta _ { 2 } ) {\dot{\theta }} _ { 1 };\\ c_{55}& = 0; \end{aligned}$$

Matrix ${\varvec{G}}$:

$$\begin{aligned} {{\varvec{G}}} = \left[ \begin{array}{ c } { 0 } \\ { 0 } \\ { 0 } \\ {\left( \dfrac{ 1 }{ 2 } l _ { 1 } m _ { 1 } + l _ { 1 } m _ { 3 } - \dfrac{ 1 }{ 2 } l _ { 3 } m _ { 3 } + l _ { 1 } m _ { 4 }\right) g \cos \theta _ { 1 } } \\ { - \left( \dfrac{ 1 }{ 2 } l _ { 2 } m _ { 2 } + l _ { 4 } m _ { 3 } + \dfrac{ 1 }{ 2 } l _ { 4 } m _ { 4 }\right) g \sin \theta _ { 2 } } \end{array} \right] , \end{aligned}$$

(46)

1.2 7.2 For equation (14), i.e., ${{\varvec{M}}\ddot{{\varvec{q}}}+{{\varvec{C}}}\dot{{\varvec{q}}}+{{\varvec{G}}}}+{\varvec{d}}(t)=\varvec{Bu}$

Matrix ${\varvec{B}}$:

$$\begin{aligned} \varvec{ B } = \left[ \begin{array}{ c c } { \dfrac{ n k _ { t } }{ r R _ { a } } {{\varvec{B}}} _ { 0 } } &{} { {\mathbf {0}} _ { 3 \times 2 } } \\ { {\varvec{0}} _ { 2 \times 3 } } &{} { \dfrac{ nk _ {t} }{ R _ { a } } {{\varvec{I}}} _ { 2 \times 2 } } \end{array} \right] \end{aligned}$$

(47)

where:

$$\begin{aligned} \begin{aligned} {{\varvec{B}} } _ { 0 } = \left[ \begin{array}{ c c c } { - \sin \varphi } &{} { - \dfrac{ \sqrt{ 3 } }{ 2 } \cos \varphi + \dfrac{ 1 }{ 2 } \sin \varphi } &{} { \dfrac{ \sqrt{ 3 } }{ 2 } \cos \varphi + \dfrac{ 1 }{ 2 } \sin \varphi } \\ { \cos \varphi } &{} { - \dfrac{ \sqrt{ 3 } }{ 2 } \sin \varphi - \dfrac{ 1 }{ 2 } \cos \varphi } &{} { \dfrac{ \sqrt{ 3 } }{ 2 } \sin \varphi - \dfrac{ 1 }{ 2 } \cos \varphi } \\ { L _ { 0 } } &{} { L _ { 0 } } &{} { L _ { 0 } }\end{array} \right] . \end{aligned} \end{aligned}$$

The relationship between the input voltage $u_i$ and the motor output torque $T_i$ is shown as follows:

$$\begin{aligned} \frac{k_t}{R_a}u_i=\left\{ \begin{array}{ll} I_{md}\cdot {w}_i+\left( b_{0d}+\dfrac{k_t k_b}{R_a}\right) w_i+\dfrac{r}{n}T_i,&{} i=1,2,3\\ I_{mu}\cdot {w}_i+\left( b_{0u}+\dfrac{k_t k_b}{R_a}\right) w_i+\dfrac{T_i}{n},&{} i=4,5 \end{array} \right. \end{aligned}$$

where $w_i$ represents the output angular velocity of the motor.

The relationship between the generalized torque ${\varvec{\tau }}$ and the motor output torque ${\varvec{T}}=[T_1,T_2,T_3,T_4,T_5]^T$ is:

$$\begin{aligned} {\varvec{\tau }}= \left[ \begin{array}{cc} {\varvec{B}}_0 &{} {\varvec{0}}_{3\times 2}\\ {\varvec{0}}_{2\times 3} &{} {\varvec{I}}_{2\times 2} \end{array} \right] {\varvec{T}}. \end{aligned}$$

Matrix ${\varvec{M}}$:

Define ${\varvec{B}}_{1}={{\varvec{B}}_0}^T.$ Then:

$$\begin{aligned} \varvec{M}=\left[ \begin{array}{cc}{{{\varvec{M}} } _ { 011 } + \dfrac{ n ^ { 2 } I _ { m d} }{ r ^ { 2 } } {{\varvec{B}} } _ { 0 } {{\varvec{B}}}_{1} } &{} { {{\varvec{M}} } _ { 012 } } \\ { {{\varvec{M}} } _ { 021 } } &{} { {{\varvec{M}} } _ { 022 } + n ^ { 2 } I _ { m u} {{\varvec{I}} } _ { 2\times 2 } } \end{array} \right] \end{aligned}$$

(48)

Matrix ${\varvec{C}}$ and ${\varvec{G}}$:

$$\begin{aligned} \varvec{C}=\left[ \begin{array}{cc} {{\varvec{C}}}_{11}&{} {{\varvec{C}}}_{12}\\ {{\varvec{C}}}_{21}&{}{{\varvec{C}}}_{22} \end{array}\right] , \end{aligned}$$

(49)

$$\begin{aligned} \begin{aligned} {{\varvec{C}}}_{11}=&{{\varvec{C}}}_{011}+\dfrac{n^{2}I_{md}}{r^2}{{\varvec{B}}_0}{\dot{{\varvec{B}}}_{1}}+\dfrac{n^2}{r^2}\left( b_{0d}+\dfrac{k_t k_b}{R_a}\right) {\varvec{B}}_{0} {\varvec{B}}_{1},\\ {\varvec{C}}_{12}=&{\varvec{C}}_{012},\\ {\varvec{C}}_{21}=&{\varvec{C}}_{021},\\ {{\varvec{C}}}_{22}=&{{\varvec{C}}}_{022}+n^2(b_{0u}+\frac{k_t k_b}{R_a}){\varvec{I}}_{2\times 2}.\\ \end{aligned} \end{aligned}$$

${\varvec{G}}$ is shown in Eq. (46).