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Kinematic optimization for bipedal robots: a framework for real-time collision avoidance

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Abstract

Bipedal locomotion is more than dynamically stable walking. The redundant kinematic design of humanoid robots allows for complex motions in complex scenarios. One challenge of current robotic research is the exploitation of the capacities of redundant robots in real-time applications. In this paper, we present and evaluate methods for real-time motion generation of redundant robots. The proposed methods are based on a model-predictive approach. We propose and compare methods for optimization of robot motions defined by parameterized task-space trajectories and for redundancy resolution. The approaches are successfully combined in a novel algorithm. The methods are introduced with the help of a minimal model. It shows their applicability for a wide range of complex robotic systems. We apply and validate their effectiveness and their real-time character in several experiments with different environments with the humanoid robot Lola.

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Notes

  1. ASUS Xtion PRO LIVE, see http://www.asus.com/Multimedia/Xtion_PRO_LIVE/.

  2. Available under https://github.com/am-lola/lepp3.

  3. Compare the videos of the experiments in https://youtu.be/6diLLVv41Vw or in https://youtu.be/rKsx8HKvBkg.

  4. The whole planning process has to be done in less than \(T_{Step}\).

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Acknowledgements

We would like to acknowledge the DAAD and the Deutsche Forschungsgemeinschaft (DFG—Project BU 2736/1-1) for their support of this project.

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Authors and Affiliations

  1. Technical University of Munich, 85747, Garching, Germany

    Arne-Christoph Hildebrandt, Simon Schwerd, Robert Wittmann, Daniel Wahrmann, Felix Sygulla, Philipp Seiwald, Daniel Rixen & Thomas Buschmann

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  1. Arne-Christoph Hildebrandt
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  2. Simon Schwerd
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  3. Robert Wittmann
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Appendix

Appendix

1.1 Gradients for optimization of redundancy

Considering the function \(\varvec{f}= \varvec{f}(\varvec{q},\varvec{u},p) = \dot{\varvec{q}}\) as shown in (10) and the Moore–Penrose pseudoinverse \(\varvec{J}^{\#} = \varvec{J}^T(\varvec{J}\varvec{J}^T)^{-1} \), the analytical gradients necessary for redundancy optimization can be formulated as follows:

$$\begin{aligned} \frac{\partial \varvec{f}}{\partial \varvec{q}}= & {} \frac{\partial \dot{\varvec{q}}}{\partial \varvec{q}} = \frac{\partial \varvec{J}^{\#}}{\partial \varvec{q}}\dot{\varvec{w}} - \left( \frac{\partial \varvec{J}^{\#}}{\partial \varvec{q}} \varvec{J}- \varvec{J}^{\#}\frac{\partial \varvec{J}}{\partial \varvec{q}} \right) \varvec{u}\end{aligned}$$
(31)
$$\begin{aligned} \frac{\partial \varvec{f}}{\partial \varvec{u}}= & {} \varvec{I}- \varvec{J}^{\#}\varvec{J}\end{aligned}$$
(32)
$$\begin{aligned} \frac{\partial \varvec{J}^{\#}}{\partial \varvec{q}}= & {} \frac{\partial \varvec{J}^{T}}{\partial \varvec{q}} \left( \varvec{J}\varvec{J}^T\right) ^{-1} + \varvec{J}^T \left[ - \left( \varvec{J}\varvec{J}^T\right) ^{-1} \dots \right. \nonumber \\&\left. \left( \frac{\partial \varvec{J}}{\partial \varvec{q}} \varvec{J}^{T} + \varvec{J}\frac{\partial \varvec{J}^{T}}{\partial \varvec{q}} \right) \left( \varvec{J}\varvec{J}^T \right) ^{-1} \right] \end{aligned}$$
(33)
$$\begin{aligned} \left( \frac{\partial L_{cmf}}{\partial \varvec{q}}\right) ^{T}= & {} 2\left( \varvec{q}- \varvec{q}_{cmf}\right) \end{aligned}$$
(34)
$$\begin{aligned} \left( \frac{\partial L_{vel}}{\partial \varvec{q}}\right) ^{T}= & {} 2 \left( \frac{\partial \varvec{f}}{\partial \varvec{q}}\right) ^T\dot{\varvec{q}} \end{aligned}$$
(35)
$$\begin{aligned} \left( \frac{\partial L_{vel}}{\partial \varvec{u}}\right) ^{T}= & {} 2 \left( \frac{\partial \varvec{f}}{\partial \varvec{u}}\right) ^T\dot{\varvec{q}} \end{aligned}$$
(36)

with \(\varvec{q}_{cmf}\) a user defined comfort pose. For cost gradients regarding collision avoidance and joint limits, we refer to Buschmann et al. (2009) and Schuetz et al. (2014) respectively.

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Hildebrandt, AC., Schwerd, S., Wittmann, R. et al. Kinematic optimization for bipedal robots: a framework for real-time collision avoidance. Auton Robot 43, 1187–1205 (2019). https://doi.org/10.1007/s10514-018-9789-3

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