Human-Inspired Balance Control of a Humanoid on a Rotating Board

Abstract

We present a stability analysis of the upright stance of a model of a humanoid robot balancing on a rotating board and driven by a human-inspired control strategy. The humanoid-board system is modeled as a triple inverted pendulum actuated by torques at the board’s hinge, ankle joint, and hip joint. The ankle and hip torques consider proprioceptive and vestibular angular information and are affected by time delays. The stability regions in different parameter’ spaces are bounded by pitchfork and Hopf’s bifurcations. It is shown that increasing time delays do not affect the pitchfork but they shrink the Hopf bifurcations. Moreover, the human-inspired control strategy is able to control the upright stance of a humanoid robot in the presence of time delays. However, more theoretical and experimental studies are necessary to validate the present results.

Appendix: Mathematical Model

The following are the detailed expressions for the terms used in Eq. 1 of the main text. The components of the inertia matrix ${\mathbf{D}}\left( {\mathbf{q}} \right)$ are given by Eqs. A1–A7. The terms in the Coriolis vector ${\mathbf{C}}\left( {{\mathbf{q}},{\dot{\mathbf{q}}}} \right)$ are given by Eqs. A8–A10. Finally, the terms in the gravity vector ${\mathbf{G}}\left( {\mathbf{q}} \right)$ are given by Eqs. A11–A13.

$$ {\mathbf{D}}\left( {\mathbf{q}} \right) = \left[ {\begin{array}{*{20}c} {D_{11} \left( {\mathbf{q}} \right)} & {D_{12} \left( {\mathbf{q}} \right)} & {D_{13} \left( {\mathbf{q}} \right)} \\ {D_{21} \left( {\mathbf{q}} \right)} & {D_{22} \left( {\mathbf{q}} \right)} & {D_{23} \left( {\mathbf{q}} \right)} \\ {D_{31} \left( {\mathbf{q}} \right)} & {D_{32} \left( {\mathbf{q}} \right)} & {D_{33} \left( {\mathbf{q}} \right)} \\ \end{array} } \right] ; { }{\mathbf{C}}\left( {{\mathbf{q}},{\dot{\mathbf{q}}}} \right) = \left[ {\begin{array}{*{20}c} {C_{1} \left( {{\mathbf{q}},{\dot{\mathbf{q}}}} \right)} \\ {C_{2} \left( {{\mathbf{q}},{\dot{\mathbf{q}}}} \right)} \\ {C_{3} \left( {{\mathbf{q}},{\dot{\mathbf{q}}}} \right)} \\ \end{array} } \right]; \, {\mathbf{G}}\left( {\mathbf{q}} \right) = \left[ {\begin{array}{*{20}c} {G_{1} \left( {\mathbf{q}} \right)} \\ {G_{2} \left( {\mathbf{q}} \right)} \\ {G_{3} \left( {\mathbf{q}} \right)} \\ \end{array} } \right] $$

(A1)

$$ \begin{aligned} D_{11} ({\mathbf{q}}) = & I_{1} + I_{2} + I_{3} + L_{1}^{2} m_{1} + L_{1}^{2} m_{2} + l_{1}^{2} m_{3} - L_{1}^{2} m_{2} + L_{1}^{2} m_{3} - h_{a}^{2} m_{1} + 2l_{1} h_{a} m_{2} \\ & + \,2L_{2} l_{3} m_{3} + 2l_{1} l_{3} m_{3} cos(q_{a} + q_{h} ) + 2l_{1} L_{1} m_{3} cos(q_{a} ) + 2h_{a} l_{3} m_{3} cos(q_{a} + q_{h} ) \\ & + \,2l_{1} l_{2} m_{2} cos(q_{a} ) - L_{2}^{2} m_{3} + 2L_{1} l_{3} m_{3} cos(q_{h} ) + 2h_{a} l_{2} m_{2} cos(q_{a} ) + 2L_{1} l_{2} m_{2} \\ & + \,2L_{1} h_{a} m_{3} cos(q_{a} ) + h_{a}^{2} m_{2} + h_{a}^{2} m_{3} + 2l_{1} h_{a} m_{3} \\ \end{aligned} $$

(A2)

$$ \begin{aligned} D_{12} ({\mathbf{q}}) = & D_{21} ({\mathbf{q}}) = I_{2} + I_{3} - L_{1}^{2} m_{2} + L_{1}^{2} m_{3} - L_{2}^{2} m_{3} + 2L_{1} l_{2} m_{2} + L_{1} h_{a} m_{3} cos(q_{a} ) \\ & + \,2L_{1} l_{3} m_{3} cos(q_{h} ) + 2L_{2} l_{3} m_{3} + l_{1} l_{2} m_{2} cos(q_{a} ) + h_{a} l_{2} m_{2} cos(q_{a} ) \\ & + \,l_{1} L_{1} m_{3} cos(q_{a} ) + h_{a} l_{3} m_{3} cos(q_{a} + q_{h} ) + l_{1} l_{3} m_{3} cos(q_{a} + q_{h} ) \\ \end{aligned} $$

(A3)

$$ \begin{aligned} D_{13} ({\mathbf{q}}) = D_{31} ({\mathbf{q}}) = & - m_{3} L_{2}^{2} + 2l_{3} m_{3} L_{2} + I_{3} { + }l_{1} l_{3} m_{3} cos(q_{a} + q_{h} ) \\ & + \,h_{a} l_{3} m_{3} cos(q_{a} + q_{h} ) + L_{1} l_{3} m_{3} cos(q_{h} ) \\ \end{aligned} $$

(A4)

$$ D_{22} ({\mathbf{q}}) = I_{2} + I_{3} - L_{1}^{2} m_{2} + L_{1}^{2} m_{3} - L_{2}^{2} m_{3} + 2L_{1} l_{2} m_{2} 2L_{2} l_{3} m_{3} + 2L_{1} l_{3} m_{3} cos(q_{h} ) $$

(A5)

$$ D_{23} ({\mathbf{q}}) = D_{32} ({\mathbf{q}}) = - m_{3} L_{2}^{2} + 2l_{3} m_{3} L_{2} + I_{3} + L_{1} l_{3} m_{3} cos(q_{h} ) $$

(A6)

$$ D_{33} ({\mathbf{q}}) = - m_{3} L_{2}^{2} + 2l_{3} m_{3} L_{2} + I_{3} $$

(A7)

$$ \begin{aligned} C_{1} ({\mathbf{q}},{\dot{\mathbf{q}}}) = & - (l_{1} + h_{a} )\left( {l_{2} m_{2} sin(q_{a} ) + l_{3} m_{3} sin(q_{a} + q_{h} )} \right)\dot{q}_{2}^{2} - (l_{1} + h_{a} )\left( {L_{1} m_{3} sin(q_{a} )} \right)\dot{q}_{a}^{2} \\ & - \,2l_{3} m_{3} \left( {l_{1} sin(q_{a} + q_{h} ) + L_{1} sin(q_{h} )} \right)\dot{q}_{a} \dot{q}_{h} - 2l_{3} m_{3} \left( {h_{a} sin(q_{a} + q_{h} )} \right)\dot{q}_{a} \dot{q}_{h} \\ & \left( { - 2sin(q_{a} )(l_{1} + h_{a} )\left( {l_{2} m_{2} + L_{1} m_{3} } \right) - 2(l_{1} + h_{a} )\left( {l_{3} m_{3} sin(q_{a} + q_{h} )} \right)} \right)\dot{q}_{b} \dot{q}_{a} \\ & - \,l_{3} m_{3} \left( {l_{1} sin(q_{a} + q_{h} ) + L_{1} sin(q_{h} )} \right)\dot{q}_{h}^{2} - l_{3} m_{3} \left( {h_{a} sin(q_{a} + q_{h} )} \right)\dot{q}_{h}^{2} \\ & - \,2l_{3} m_{3} \left( {l_{1} sin(q_{a} + q_{h} ) + L_{1} sin(q_{h} )} \right)\dot{q}_{h} \dot{q}_{b} - 2l_{3} m_{3} \left( {h_{a} sin(q_{a} + q_{h} )} \right)\dot{q}_{h} \dot{q}_{b} \\ \end{aligned} $$

(A8)

$$ \begin{aligned} C_{2} ({\mathbf{q}},{\dot{\mathbf{q}}}) = & (l_{1} + h_{a} )\left( {l_{2} m_{2} sin(q_{a} ) + L_{1} m_{3} sin(q_{a} )} \right)\dot{q}_{b}^{2} + (l_{1} + h_{a} )l_{3} m_{3} sin(q_{a} + q_{h} )\dot{q}_{b}^{2} \\ & - \,2L_{1} l_{3} m_{3} sin(q_{h} )\dot{q}_{b} \dot{q}_{h} - L_{1} l_{3} m_{3} sin(q_{h} )\dot{q}_{h}^{2} - 2L_{1} l_{3} m_{3} sin(q_{h} )\dot{q}_{a} \dot{q}_{h} \\ \end{aligned} $$

(A9)

$$ \begin{aligned} C_{3} ({\mathbf{q}},{\dot{\mathbf{q}}}) = & l_{3} m_{3}^{2} \left( {l_{1} sin(q_{a} + q_{h} ) + L_{1} sin(q_{h} )} \right)\dot{q}_{1}^{2} + l_{3} m_{3}^{2} \left( {h_{a} sin(q_{a} + q_{h} )} \right)\dot{q}_{b}^{2} \\ & + \,2L_{1} l_{3} m_{3} sin(q_{h} )\dot{q}_{b} \dot{q}_{a} + L_{1} l_{3} m_{3} sin(q_{h} )\dot{q}_{a}^{2} \\ \end{aligned} $$

(A10)

$$ \begin{aligned} G_{1} \left( \varvec{q} \right) = & - g\left( {h_{a} m_{3} + l_{1} m_{1} + l_{1} m_{2} + l_{1} m_{3} } \right)sin(q_{b} ) - gl_{2} m_{2} sin(q_{b} + q_{a} ) \\ & + \,L_{1} m_{3} sin(q_{b} + q_{a} ) - g\left( {l_{3} m_{3} sin(q_{b} + q_{a} + q_{h} ) + h_{a} m_{2} sin(q_{b} )} \right) \\ \end{aligned} $$

(A11)

$$ G_{2} \left( {\mathbf{q}} \right) = - gsin\left( {q_{b} + q_{a} } \right)\left( {L_{1} m_{3} + l_{2} m_{2} } \right) - gl_{3} m_{3} sin\left( {q_{b} + q_{a} + q_{h} } \right) $$

(A12)

$$ G_{3} \left( {\mathbf{q}} \right) = - gl_{3} m_{3} sin\left( {q_{b} + q_{a} + q_{h} } \right) $$

(A13)