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A momentum-based balance controller for humanoid robots on non-level and non-stationary ground

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Abstract

Recent research suggests the importance of controlling rotational dynamics of a humanoid robot in balance maintenance and gait. In this paper, we present a novel balance strategy that controls both linear and angular momentum of the robot. The controller’s objective is defined in terms of the desired momenta, allowing intuitive control of the balancing behavior of the robot. By directly determining the ground reaction force (GRF) and the center of pressure (CoP) at each support foot to realize the desired momenta, this strategy can deal with non-level and non-stationary grounds, as well as different frictional properties at each foot-ground contact. When the robot cannot realize the desired values of linear and angular momenta simultaneously, the controller attributes higher priority to linear momentum at the cost of compromising angular momentum. This creates a large rotation of the upper body, reminiscent of the balancing behavior of humans. We develop a computationally efficient method to optimize GRFs and CoPs at individual foot by sequentially solving two small-scale constrained linear least-squares problems. The balance strategy is demonstrated on a simulated humanoid robot under experiments such as recovery from unknown external pushes and balancing on non-level and moving supports.

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Notes

  1. The normal torque τ n also affects \(\boldsymbol {\dot {k}}\) in the transverse plane. Actually f, p, and τ n together constitute the 6 variables that correspond to the 6 variables of the spatial momentum. Usually τ n is omitted in the discussion for simplicity because its magnitude is small.

  2. During single support, the support base is identical to the foot contact area, whereas during double support on level ground, the support base is equivalent to the convex hull of the support areas of the two feet.

  3. \(\boldsymbol {\dot {q}}\) is a slight abuse of notation because we do not define nor use a vector q. However, since \(\mathsf {se(3)}\), the Lie algebra of \(\mathsf {SE(3)}\), is isomorphic to ℝ6, we will use a single vector form of \(\boldsymbol {\dot {q}}\in \mathbb {R}^{6+n}\) for convenience. [ω 0]× represents a skew-symmetric matrix of a vector ω 0.

  4. The vector δ i expresses angular momentum rate change (16) in terms of ρ i as follows:

  5. Specifically, \(\boldsymbol {\varPsi }_{k} = [\boldsymbol {\varPsi }_{k}^{0} \ldots \boldsymbol {\varPsi }_{k}^{5}]\) where

    and \(\boldsymbol {\kappa }_{k} = \boldsymbol {\dot {k}}_{\tau,d} + h ( {\boldsymbol{R}}_{r}^{1} {\boldsymbol{f}}^{b}_{r,X} - {\boldsymbol{R}}_{r}^{0} {\boldsymbol{f}}^{b}_{r,Y} + {\boldsymbol{R}}_{l}^{1} {\boldsymbol{f}}^{b}_{l,X} - {\boldsymbol{R}}_{l}^{0} {\boldsymbol{f}}^{b}_{l,Y} )\). \({\boldsymbol{R}}_{i}^{j}\) is j-th column vector of R i (i=r,l), \({\boldsymbol{f}}^{b}_{i}= {\boldsymbol{R}}_{i}^{T} {\boldsymbol{f}}_{i}\), and h is the height of foot frame from the foot sole.

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Acknowledgements

This work was mainly done while S.H.L. was with HRI. S.H.L. was also supported in part by the Global Frontier R&D Program on “Human-Centered Interaction for Coexistence” funded by the National Research Foundation of Korea (NRF-M1AXA003-2011-0028374).

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Authors and Affiliations

  1. School of Information and Communications, Gwangju Institute of Science and Technology, Gwangju, South Korea

    Sung-Hee Lee

  2. Honda Research Institute, Mountain View, CA, USA

    Ambarish Goswami

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  1. Sung-Hee Lee
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  2. Ambarish Goswami
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Correspondence to Ambarish Goswami.

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Lee, SH., Goswami, A. A momentum-based balance controller for humanoid robots on non-level and non-stationary ground. Auton Robot 33, 399–414 (2012). https://doi.org/10.1007/s10514-012-9294-z

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