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Design and Verification of Parallel Hip Exoskeleton Considering Output Torque Anisotropy

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Abstract

In rehabilitation training, it is crucial to consider the compatibility between exoskeletons and human legs in motion. However, most exoskeletons today adopt an anthropomorphic serial structure, which results in rotational centers that are not precisely aligned with the center of the hip joint. To address this issue, we introduce a novel exoskeleton called the Parallel Hip Exoskeleton (PH-Exo) in this paper. PH-Exo is meticulously designed based on the anisotropic law of output torque. Considering the friction of the drive components, a dynamic model of the human–machine complex is established. Simulation analysis demonstrates that PH-Exo not only exhibits outstanding torque performance but also achieves high controllability in both flexion/extension and adduction/abduction directions. Additionally, a robust controller is designed to address model uncertainty, friction, and external interference. Wearing experiments indicate that under the control of the robust controller, each motor achieves excellent tracking performance.

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Data Availability

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

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Funding

This study was funded by the Natural Science Foundation of Hebei Province (Grant no. F2022203043) and the Provincial Key Laboratory Performance Subsidy Project (Grant no. 22567612 H).

Author information

Authors and Affiliations

  1. Engineering Research Center of the Ministry of Education for Intelligent Control System and Intelligent Equipment, Yanshan University, Qinhuangdao, 066004, China

    Jilong Xu & Fucai Liu

  2. Key Laboratory of Industrial Computer Control Engineering of Heibei Province, Yanshan University, Qinhuangdao, 066004, China

    Jilong Xu & Fucai Liu

  3. College of Mechanical Engineering, Yanshan University, Qinhuangdao, 066004, China

    Yunzhan Niu

Authors
  1. Jilong Xu
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  2. Yunzhan Niu
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  3. Fucai Liu
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Correspondence to Fucai Liu.

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Appendices

Appendix A

\({{{{\varvec{G}}}}_{M0}} = {\left[ {\begin{array}{*{20}{c}} 1&{}0&{}0&{}{ - 2{{\textrm{s}}_{21}}}&{}1&{}0\\ 0&{}{{{\textrm{s}}_{11}}}&{}{ - 2}&{}0&{}0&{}{{{\textrm{s}}_{31}}}\\ 0&{}{{{\textrm{c}}_{11}}}&{}0&{}{ - 2{{\textrm{c}}_{21}}}&{}0&{}{{{\textrm{c}}_{31}}}\\ {{u_1}}&{}{{v_1}}&{}0&{}0&{}0&{}0\\ 0&{}0&{}{{u_2}}&{}{{v_2}}&{}0&{}0\\ 0&{}0&{}0&{}0&{}{{u_3}}&{}{{v_3}} \end{array}} \right] ^{ - 1}}\), \({{{{\varvec{G}}}}_{M1}} = \left[ {\begin{array}{*{20}{c}} 0&{}{2{L_{2{\textrm{c}}}}{{\textrm{c}}_{21}}}&{}0\\ { - {L_{1{\textrm{c}}}}{{\textrm{c}}_{11}}}&{}0&{}{ - {L_{3{\textrm{c}}}}{{\textrm{c}}_{31}}}\\ {{L_{1{\textrm{c}}}}{{\textrm{s}}_{11}}}&{}{ - 2{L_{2{\textrm{c}}}}{{\textrm{s}}_{21}}}&{}{{L_{3{\textrm{c}}}}{{\textrm{s}}_{31}}}\\ { - {L_{1{\textrm{c}}}}{p_{11}}}&{}0&{}0\\ 0&{}{{L_{2{\textrm{c}}}}{p_{21}}}&{}0\\ 0&{}0&{}{ - {L_{3{\textrm{c}}}}{p_{31}}} \end{array}} \right]\).

In \({{{{\varvec{G}}}}_{M0}}\) and \({{{{\varvec{G}}}}_{M1}}\), \({u_1} = {l_{{\mathrm{12\,s}}}} - {x_o} + a\); \({u_2} = {l_{{\mathrm{22\,s}}}} + {y_o} - b\); \({u_3} = {l_{{\mathrm{32\,s}}}} - {x_o} - a\); \({p_{k1}} = {y_o}{{\textrm{c}}_{k1}} + {z_o}{{\textrm{s}}_{k1}}\); \({p_{21}} = {x_o}{{\textrm{c}}_{21}} - {z_o}{{\textrm{s}}_{21}}\); \({v_i} = {L_{i{\textrm{c}}}} - {({-1})^i}{p_i}\). In \({v_i}\), \({p_k} = {y_o}{{\textrm{s}}_{k1}} - {z_o}{{\textrm{c}}_{k1}}\); \({p_2} = {x_o}{{\textrm{s}}_{21}} + {z_o}{{\textrm{c}}_{21}}\).

Appendix B

\({{{\varvec{G}}}}_q^{i1} = \left[ {\begin{array}{*{20}{c}} {{{{{\varvec{s}}}}_{i1}}}\\ {{{{{\varvec{m}}}}_{i11}}} \end{array}} \right] {{{\varvec{g}}}}_\theta ^i\), \({{{\varvec{G}}}}_q^{i2} = \left[ {\begin{array}{*{20}{c}} {{{{{\varvec{s}}}}_{i1}}}&{}{{{{{\varvec{s}}}}_{i2}}}\\ {{{{{\varvec{m}}}}_{i21}}}&{}{{{{{\varvec{m}}}}_{i22}}} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} {{{{\varvec{g}}}}_\theta ^i}\\ {{{{{\varvec{G}}}}_{Bi}}\left( {{\textrm{2}},:} \right) } \end{array}} \right]\), \({{{\varvec{G}}}}_q^{i3} = \left[ {\begin{array}{*{20}{c}} {{{{{\varvec{s}}}}_{i1}}}&{}{{{{{\varvec{s}}}}_{i2}}}&{}{{{\varvec{0}}}}\\ {{{{{\varvec{m}}}}_{i31}}}&{}{{{{{\varvec{m}}}}_{i32}}}&{}{{{{{\varvec{s}}}}_{i3}}} \end{array}} \right] {\left[ {\begin{array}{*{20}{c}} {{{{\varvec{g}}}}{{_\theta ^i}^{\textrm{T}}}}&{{{{\varvec{G}}}}_{Bi}^{\textrm{T}}\left( {{\textrm{2}},:} \right) }&{{{{\varvec{G}}}}_{Bi}^{\textrm{T}}\left( {{\textrm{1}},:} \right) } \end{array}} \right] ^{\textrm{T}}}\).

In \({{{\varvec{G}}}}_q^{ic}\), \({{{\varvec{g}}}}_\theta ^1 = \left[ {\begin{array}{*{20}{c}} 1&0&0 \end{array}} \right]\); \({{{\varvec{g}}}}_\theta ^2 = \left[ {\begin{array}{*{20}{c}} 0&1&0 \end{array}} \right]\); \({{{\varvec{g}}}}_\theta ^3 = \left[ {\begin{array}{*{20}{c}} 0&0&1 \end{array}} \right]\); \({{{{\varvec{s}}}}_{i2}} = {{{\varvec{R}}}_{i1}}{{{\varvec{y}}}}\); \({{{{\varvec{s}}}}_{i3}} = {{{\varvec{R}}}_{i2}}{{{\varvec{z}}}}\); \({{{{\varvec{m}}}}_{icj}} = {{{{\varvec{s}}}}_{ij}} \times \left( {{{{{\varvec{P}}}}_{ic}} - {}^A{{{\varvec{r}}}_{A,ij}}} \right)\); \({}^A{{{\varvec{r}}}_{A,i2}} = {}^A{{{\varvec{r}}}_{A,i1}} + {}^A{{{\varvec{r}}}_{i1,i2}}\).

Appendix C

\({{{\varvec{J}}}}{\mathrm{= }}{{{{\varvec{J}}}}_\omega }{{{{\varvec{J}}}}_\chi }\). In \({{\varvec{J}}}\), \({{{{\varvec{J}}}}_\omega }{\mathrm{= }}\left[ {\begin{array}{*{20}{c}} {{\textrm{c}}\beta {\textrm{c}}\gamma }&{}{ - {\textrm{s}}\gamma }&{}0\\ {{\textrm{c}}\beta {\textrm{s}}\gamma }&{}{{\textrm{c}}\gamma }&{}0\\ { - {\textrm{s}}\beta }&{}0&{}1 \end{array}} \right]\); \({{{{\varvec{J}}}}_\chi }{\mathrm{= }}{{{{\varvec{J}}}}_0}\left( {{{{{\varvec{J}}}}_1} + {{{{\varvec{J}}}}_2}{{{{\varvec{G}}}}_{MC}}} \right)\).

In \({{{{\varvec{J}}}}_\chi }\), \({{{{{\varvec{J}}}}_0}} = {\left[ {\begin{array}{*{20}{c}} 0&{}{r{\textrm{c}}\beta }&{}0\\ 0&{}{ - r{\textrm{s}}\beta {\textrm{s}}\gamma }&{}{r{\textrm{c}}\beta {\textrm{c}}\gamma }\\ {\left( {{b_1}{\textrm{c}}\alpha - {c_1}{\textrm{s}}\alpha } \right) {\textrm{c}}\beta }&{}{{x_o}{\textrm{c}}\beta - {b_1}{\textrm{s}}\alpha {\textrm{s}}\beta - {c_1}{\textrm{c}}\alpha {\textrm{s}}\beta }&{}0 \end{array}} \right] ^{ - 1}}\); \({{{{\varvec{J}}}}_1} = \left[ {\begin{array}{*{20}{c}} {{L_{1{\textrm{c}}}}{{\textrm{s}}_{11}}}&{}0&{}{ - {L_{3{\textrm{c}}}}{{\textrm{s}}_{31}}}\\ { - {L_{1{\textrm{c}}}}{{\textrm{c}}_{11}}}&{}0&{}{{L_{3{\textrm{c}}}}{{\textrm{c}}_{31}}}\\ 0&{}{ - {L_{2{\textrm{c}}}}{{\textrm{s}}_{21}}}&{}0 \end{array}} \right]\); \({{{{\varvec{J}}}}_2}{\mathrm{= }}\left[ {\begin{array}{*{20}{c}} { - {{\textrm{c}}_{11}}}&{}0&{}{{{\textrm{c}}_{31}}}\\ { - {{\textrm{s}}_{11}}}&{}0&{}{{{\textrm{s}}_{31}}}\\ 0&{}{{{\textrm{c}}_{21}}}&{}0 \end{array}} \right]\); \({{{{\varvec{G}}}}_{MC}} = \left[ {\begin{array}{*{20}{c}} {{{{{\varvec{G}}}}_M}\left( {2,:} \right) }\\ {{{{{\varvec{G}}}}_M}\left( {4,:} \right) }\\ {{{{{\varvec{G}}}}_M}\left( {6,:} \right) } \end{array}} \right]\).

Appendix D

\({{{{\varvec{E}}}}_0} = {\left[ {\begin{array}{*{20}{c}} 0&{}{n_1^ * }&{}{{n_1}}&{}{{e_1}}&{}{e_{21}^ * }&{}{e_{31}^ * }\\ { - 2n_2^ * }&{}0&{}{ - 2{n_2}}&{}{e_{12}^ * }&{}{{e_2}}&{}{e_{32}^ * }\\ 0&{}{n_3^ * }&{}{{n_3}}&{}{e_{13}^ * }&{}{e_{23}^ * }&{}{{e_3}} \end{array}} \right] ^{\textrm{T}}}\). In \({{{{\varvec{E}}}}_0}\), \({n_i} = {L_{i{\textrm{c}}}}{{\textrm{c}}_{i1}}{\dot{{\theta }}_{i1}} + 2{{\textrm{s}}_{i1}}{\dot{L}_{i{\textrm{c}}}}\); \(n_i^ * = {L_{i{\textrm{c}}}}{{\textrm{s}}_{i1}}{\dot{{\theta }}_{i1}} - 2{{\textrm{c}}_{i1}}{\dot{L}_{i{\textrm{c}}}}\); \({e_i} = {\left( { - 1} \right) ^{i + 1}}\left( {{L_{i{\textrm{c}}}}{p_i}{{\dot{{\theta }}}_{i1}} - 2{{\dot{L}}_{i{\textrm{c}}}}{p_{i1}}} \right) - {\dot{l}_{{\textrm{i2s}}}}G_M^{2i - 1,i} - {\dot{L}_{i{\textrm{c}}}}G_M^{2i,i}\); \(e_{ij}^ * = - {\dot{L}_{i{\textrm{c}}}}G_M^{2i,j} - {\dot{l}_{i{\mathrm{2\,s}}}}G_M^{2i - 1,j}\); \(G_M^{i,j}\) is the element in the \(\textit{i}\)-th row and \(\textit{j}\)-th column in matrix \({{{{\varvec{G}}}}_M}\).

Appendix E

\({{{\varvec{E}}}}_q^{i1} = {\dot{{\theta }}_{i1}}\left[ {\begin{array}{*{20}{c}} {{{\varvec{0}}}}\\ {{{{{\varvec{s}}}}_{i1}} \times {{{{\varvec{m}}}}_{i11}}} \end{array}} \right] {{{\varvec{g}}}}_\theta ^i\), \({{{\varvec{E}}}}_q^{i2} = {{{\varvec{G}}}}_\theta ^{i2}\left[ {\begin{array}{*{20}{c}} {{{\varvec{0}}}}\\ {{{{{\varvec{E}}}}_{Bi}}\left( {2,:} \right) } \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {{{\dot{{\theta }}}_{i1}}\left( {{{{{\varvec{s}}}}_{i1}} \times {{{{\varvec{s}}}}_{i2}}} \right) {{{{\varvec{G}}}}_{Bi}}\left( {{\textrm{2}},:} \right) }\\ {{{\dot{{\theta }}}_{i1}}\left( {{{{{\varvec{s}}}}_{i1}} \times {{{{\varvec{m}}}}_{i21}}} \right) {{{\varvec{g}}}}_\theta ^i + {{{\varvec{E}}}}_1^ * } \end{array}} \right]\), \({{{\varvec{E}}}}_q^{i3} = {{{\varvec{G}}}}_\theta ^{i3}\left[ {\begin{array}{*{20}{c}} {{{\varvec{0}}}}\\ {{{{{\varvec{E}}}}_{Bi}}\left( {2,:} \right) }\\ {{{{{\varvec{E}}}}_{Bi}}\left( {1,:} \right) } \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {{{\dot{\theta }}_{i1}}\left( {{{{{\varvec{s}}}}_{i1}} \times {{{{\varvec{s}}}}_{i2}}} \right) {{{{\varvec{G}}}}_{Bi}}\left( {{\textrm{2}},:} \right) }\\ {{{\dot{\theta }}_{i1}}\left( {{{{{\varvec{s}}}}_{i1}} \times {{{{\varvec{m}}}}_{i31}}} \right) {{{\varvec{g}}}}_\theta ^i + {{{\varvec{E}}}}_2^ * + {{{\varvec{E}}}}_3^ * } \end{array}} \right]\).

In \({{{\varvec{E}}}}_q^{i2}\) and \({{{\varvec{E}}}}_q^{i3}\), \({{{\varvec{E}}}}_1^ * = \left[ {2{{\dot{\theta }}_{i1}}\left( {{{{{\varvec{s}}}}_{i1}} \times {{{{\varvec{m}}}}_{i22}}} \right) + \left( {{{{{\varvec{s}}}}_{i2}} \times {{{{\varvec{m}}}}_{i22}}} \right) {{{\dot{\varvec{\theta }}}}^{\textrm{T}}}{{{\varvec{G}}}}_{Bi}^{\textrm{T}}\left( {{\textrm{2}},:} \right) } \right] {{{{\varvec{G}}}}_{Bi}}\left( {{\textrm{2}},:} \right)\); \({{{\varvec{E}}}}_2^ * = \left[ {2{{\dot{\theta }}_{i1}}\left( {{{{{\varvec{s}}}}_{i1}} \times {{{{\varvec{m}}}}_{i32}}} \right) + \left( {{{{{\varvec{s}}}}_{i2}} \times {{{{\varvec{m}}}}_{i32}}} \right) {{{\dot{\varvec{\theta }}}}^{\textrm{T}}}{{{\varvec{G}}}}_{Bi}^{\textrm{T}}\left( {{\textrm{2}},:} \right) } \right] {{{{\varvec{G}}}}_{Bi}}\left( {{\textrm{2}},:} \right)\); \({{{\varvec{E}}}}_3^ * = 2\left[ {\left( {{{{{\varvec{s}}}}_{i2}} \times {{{{\varvec{s}}}}_{i3}}} \right) {{{\dot{\varvec{\theta }}}}^{\textrm{T}}}{{{\varvec{G}}}}_{Bi}^{\textrm{T}}\left( {{\textrm{2}},:} \right) + {{\dot{\theta }}_{i1}}\left( {{{{{\varvec{s}}}}_{i1}} \times {{{{\varvec{s}}}}_{i3}}} \right) } \right] {{{{\varvec{G}}}}_{Bi}}\left( {1,:} \right)\).

Appendix F

\({{{{\varvec{M}}}}_q} = \left[ {\begin{array}{*{20}{c}} {{{\varvec{N}}}}\\ {\left[ {\left( {{}^A{{{\varvec{r}}}_{A,O}} - {{{{\varvec{P}}}}_0}} \right) \times } \right] {{{\varvec{N}}}} + \left[ {\left( {\left[ {\left( {{}^A{{{\varvec{r}}}_{A,O}} - {{{{\varvec{P}}}}_0}} \right) \times } \right] {{{\varvec{J}}}{\dot{\varvec{\theta }}}}} \right) \times } \right] {{{\varvec{J}}}}} \end{array}} \right]\).

In \({{{{\varvec{M}}}}_q}\), \({{{\varvec{N}}}} = {{{{\varvec{J}}}}_\omega }{{{{\varvec{N}}}}_\chi } + {{{{\varvec{N}}}}_\omega }{{{{\varvec{J}}}}_\chi }\). In \({{{\varvec{N}}}}\), \({{{{\varvec{N}}}}_\chi } = {{{{\varvec{J}}}}_0}\left( {{{{{\varvec{N}}}}_1} - {{{{\varvec{N}}}}_2} + {{{{\varvec{J}}}}_2}{{{{\varvec{E}}}}_{MC}}} \right)\); \({{{{\varvec{N}}}}_\omega } = \left[ {\begin{array}{*{20}{c}} { - {\textrm{c}}\gamma {\textrm{s}}\beta \dot{\beta }- {\textrm{c}}\beta {\textrm{s}}\gamma \dot{\gamma }}&{}{ - {\textrm{c}}\gamma \dot{\gamma }}&{}0\\ { - {\textrm{s}}\gamma {\textrm{s}}\beta \dot{\beta }+ {\textrm{c}}\beta {\textrm{c}}\gamma \dot{\gamma }}&{}{ - {\textrm{s}}\gamma \dot{\gamma }}&{}0\\ { - {\textrm{c}}\beta \dot{\beta }}&{}0&{}0 \end{array}} \right]\). In \({{{{\varvec{N}}}}_\chi }\), \({{{{\varvec{N}}}}_1} = \left[ {\begin{array}{*{20}{c}} 0&{}{r{\textrm{s}}\beta \dot{\beta }}&{}0\\ 0&{}{r{\textrm{s}}\gamma {\textrm{c}}\beta \dot{\beta }}&{}{r\left( {2{\textrm{c}}\gamma {\textrm{s}}\beta \dot{\beta }+ {\textrm{s}}\gamma {\textrm{c}}\beta \dot{\gamma }} \right) }\\ {{a_4}}&{}{{a_5}}&{}{{c_1}{\textrm{c}}\alpha {\textrm{c}}\beta \dot{\beta }} \end{array}} \right] {{{\varvec{J}}}}\); \({{{{\varvec{N}}}}_2} = \left[ {\begin{array}{*{20}{c}} { - {n_1}}&{}0&{}{{n_3}}\\ { - n_1^ * }&{}0&{}{n_3^ * }\\ 0&{}{{n_2}}&{}0 \end{array}} \right]\); \({{{{\varvec{E}}}}_{MC}} = \left[ {\begin{array}{*{20}{c}} {{{{{\varvec{E}}}}_M}\left( {2,:} \right) }\\ {{{{{\varvec{E}}}}_M}\left( {4,:} \right) }\\ {{{{{\varvec{E}}}}_M}\left( {6,:} \right) } \end{array}} \right]\). In \({{{{\varvec{N}}}}_1}\), \({a_4} = \left( {{b_1}{\textrm{s}}\alpha + {c_1}{\textrm{c}}\alpha } \right) {\textrm{c}}\beta \dot{\alpha }\); \({a_5} = 2\left( {{b_1}{\textrm{c}}\alpha - {c_1}{\textrm{s}}\alpha } \right) {\textrm{s}}\beta \dot{\alpha }+ \left( {{x_0}{\textrm{s}}\beta + {b_1}{\textrm{s}}\alpha {\textrm{c}}\beta + {c_1}{\textrm{c}}\alpha {\textrm{c}}\beta } \right) \dot{\beta }\).

Appendix G

\({{{\varvec{F}}}} = \sum \limits _{i = 1}^3 {\sum \limits _{c = 1}^3 {\left( {{m_{ic}}{{{\varvec{v}}}}_{ic,\dot{\theta }}^{\textrm{T}}{{{\varvec{g}}}}} \right) } } + \sum \limits _{i = 1}^3 {{{\varvec{\omega }}}_{i1,\dot{\theta }}^{\textrm{T}}{\tau _{i1}}{{{{\varvec{s}}}}_{i1}}} + {m_0}{{{\varvec{v}}}}_{0,\dot{\theta }}^{\textrm{T}}{{{\varvec{g}}}} + {{{\varvec{v}}}}_{0,\dot{\theta }}^{\textrm{T}}{{{{\varvec{f}}}}_0} + {{{{\varvec{J}}}}^{\textrm{T}}}{{{{\varvec{n}}}}_0}\),

\({{{{\varvec{F}}}}^*} = \sum \limits _{i = 1}^3 {\sum \limits _{c = 1}^3 {\left( { - {m_{ic}}{{{\varvec{v}}}}_{ic,\dot{\theta }}^{\textrm{T}}{{{{\varvec{a}}}}_{ic}} - {{\varvec{\omega }}}_{ic,\dot{\theta }}^{\textrm{T}}\left( {{{{{\varvec{I}}}}_{ic}}{{{\varvec{\varepsilon }}}_{ic}} + {{{\varvec{\omega }}}_{ic}} \times {{{{\varvec{I}}}}_{ic}}{{{\varvec{\omega }}}_{ic}}} \right) } \right) - {m_0}} } {{{\varvec{v}}}}_{0,\dot{\theta }}^{\textrm{T}}{{{{\varvec{a}}}}_0} - {{{{\varvec{J}}}}^{\textrm{T}}}\left( {{{{{\varvec{I}}}}_0}{{{\varvec{\varepsilon }}}_0} + {{{\varvec{\omega }}}_0} \times {{{{\varvec{I}}}}_0}{{{\varvec{\omega }}}_0}} \right)\).

where \(\textit{m}_{ic}\) and \({{\varvec{I}}}_{ic}\) are the mass and inertia matrix of the link \(\textit{ic}\), and \({{{{\varvec{I}}}}_{ic}} = {{{\varvec{R}}}_{ic}}{}^{ic}{{{\varvec{IR}}}}_{ic}^{\textrm{T}}\); \({}^{ic}{{{\varvec{I}}}}\) is the inertia matrix of the link \(\textit{ic}\) in \({\textit{P}_{ic}}-{{{x}_{ic}}{{y}_{ic}}{{z}_{ic}}}\); \(\textit{m}_{0}\) and \({{{\varvec{I}}}}_0\) are the mass and inertia matrix of the motion complex, and \({{{{\varvec{I}}}}_{0}} = {{{\varvec{R}}}_{0}}{}^{0}{{{\varvec{IR}}}}_{0}^{\textrm{T}}\); \({}^{0}{{{\varvec{I}}}}\) is the inertia matrix of the motion complex in \({\textit{C}_{23}}-{xyz}\); \({\tau _{i1}}\) is the driving torque of limb \(\textit{i}\); \({{\varvec{g}}}\) is the acceleration of gravity; \({{{\varvec{f}}}}_0\) and \({{{\varvec{n}}}}_0\) are the external force and torque equivalent to the barycenter of the motion complex; \({{{\varvec{\omega }}}_{ic,\dot{\theta }}}\) and \({{{{\varvec{v}}}}_{ic,\dot{\theta }}}\) are the sub-matrices consisting of the first three rows and the last three rows of \({{{\varvec{G}}}}_q^{ic}\); \({{{{\varvec{v}}}}_{0,\dot{\theta }}}\) is the sub-matrix consisting of the last three rows of \({{{{\varvec{G}}}}_q}\).

Appendix H

\({{{\varvec{M}}}} = \sum \limits _{i = 1}^3 {\sum \limits _{c = 1}^3 {\left( {{m_{ic}}{{{\varvec{v}}}}_{ic,\dot{\theta }}^{\textrm{T}}{{{{\varvec{v}}}}_{ic,\dot{\theta }}} + {{\varvec{\omega }}}_{ic,\dot{\theta }}^{\textrm{T}}{{{{\varvec{I}}}}_{ic}}{{{\varvec{\omega }}}_{i1,\dot{\theta }}}} \right) + {{{{\varvec{J}}}}^{\textrm{T}}}{{{{\varvec{I}}}}_0}{{{\varvec{J}}}} + {m_0}{{{\varvec{v}}}}_{0,\dot{\theta }}^{\textrm{T}}{{{{\varvec{v}}}}_{0,\dot{\theta }}}} }\),

\({{{\varvec{C}}}} = \sum \limits _{i = 1}^3 {\sum \limits _{c = 1}^3 {\left( {{m_{ic}}{{{\varvec{v}}}}_{ic,\dot{\theta }}^{\textrm{T}}{{{\varvec{E}}}}_{qv}^{ic} + {{\varvec{\omega }}}_{i1,\dot{\theta }}^{\textrm{T}}{{{{\varvec{H}}}}_1}} \right) } } + {{{{\varvec{J}}}}^{\textrm{T}}}{{{{\varvec{H}}}}_2} + {m_0}{{{\varvec{v}}}}_{0,\dot{\theta }}^{\textrm{T}}{{{{\varvec{M}}}}_{qv}}\),

\({{{\varvec{G}}}} = - \sum \limits _{i = 1}^3 {\sum \limits _{c = 1}^3 {\left( {{m_{ic}}{{{\varvec{v}}}}_{ic,\dot{\theta }}^{\textrm{T}}{{{\varvec{g}}}}} \right) } } - {{{\varvec{v}}}}_{0,\dot{\theta }}^{\textrm{T}}\left( {{m_0}{{{\varvec{g}}}} + {{{{\varvec{f}}}}_0}} \right) - {{{{\varvec{J}}}}^{\textrm{T}}}{{{{\varvec{n}}}}_0}\).

In \({{\varvec{C}}}\), \({{{{\varvec{H}}}}_1} = {{{{\varvec{I}}}}_{ic}}{{{\varvec{E}}}}_{q\omega }^{ic} + \left[ {{{{\varvec{\omega }}}_{i1,\dot{\theta }}} \times } \right] {{{{\varvec{I}}}}_{ic}}{{{\varvec{\omega }}}_{i1,\dot{\theta }}}\); \({{{{\varvec{H}}}}_2} = {{{{\varvec{I}}}}_0}{{{\varvec{N}}}} + \left[ {{{{\varvec{\omega }}}_0} \times } \right] {{{{\varvec{I}}}}_0}{{{\varvec{J}}}}\); \({{{\varvec{E}}}}_{q\omega }^{ic}\) and \({{{\varvec{E}}}}_{qv}^{ic}\) are the sub-matrices consisting of the first three rows and the last three rows of \({{{\varvec{E}}}}_q^{ic}\); \({{\varvec{M}}}_{qv}\) is the sub-matrix consisting of the last three rows of \({{\varvec{M}}}_q\).

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Xu, J., Niu, Y. & Liu, F. Design and Verification of Parallel Hip Exoskeleton Considering Output Torque Anisotropy. J Bionic Eng (2024). https://doi.org/10.1007/s42235-024-00500-y

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