Efficiently not falling into the hole: walking in the real world

Alexander Werner

German Aerospace Center (DLR)
Institute for Robotics and Mechatronics

Knowledge for Tomorrow


Walking models

Small Model

COM Dynamics


Center of Pressure

Friction Cone

Large Model

$$ \boldsymbol{M}(\boldsymbol{y}) \boldsymbol{\ddot{y}} + \boldsymbol{C}(\boldsymbol{y},\boldsymbol{\dot{y}}) \boldsymbol{\dot{y}} + \boldsymbol{g}(\boldsymbol{y}) = \boldsymbol{S}^T \boldsymbol{\tau} + \sum_{i=0}^{N_c} \boldsymbol{J}^T_{c,i} \boldsymbol{W}_{c,i} $$

Added Constraints

  • Joint Torques
  • Joint velocities

Actuator dynamics (Series-Elastic Robot):

$$ \boldsymbol{M} \boldsymbol{\ddot{q}} = \boldsymbol{\tau} - \boldsymbol{\tau}_{ext} \qquad \boldsymbol{\tau} = \boldsymbol{K} ( \boldsymbol{\theta} - \boldsymbol{q}) \qquad \boldsymbol{B} \boldsymbol{\ddot{\theta}} = \boldsymbol{\tau}_m - \boldsymbol{\tau} $$

Optimal Control: Trajectory generation for walking

Human like gait anyone?

Apply Collocation

1. Parameterize

  • All states: $$ \boldsymbol{y} = f_\text{y}(\boldsymbol{p},t) $$
  • Internal and External Forces $$ \begin{bmatrix} \boldsymbol{\tau} \\ \boldsymbol{W}_\text{C,i} \end{bmatrix} = f_\tau(\boldsymbol{p},t) $$

2. Ensure physics:

$ \boldsymbol{M}(\boldsymbol{y}) \boldsymbol{\ddot{y}} + \boldsymbol{C}(\boldsymbol{y},\boldsymbol{\dot{y}}) \boldsymbol{\dot{y}} + \boldsymbol{g}(\boldsymbol{y}) = \boldsymbol{S}^T \boldsymbol{\tau} + \sum_{i=0}^{N_c} \boldsymbol{J}^T_{c,i} \boldsymbol{W}_{c,i} $
$ d_\mathrm{C,i,N} \cdot F_{C,i,N} + | \dot{d}_\mathrm{C,i,T} | \cdot F_\mathrm{C,i,N} = 0$

IROS 2017: Generation of Locomotion Trajectories for Series Elastic and Viscoelastic Bipedal Robots A Werner, W Turlej and C Ott

Applications: Locomotion using new Collocation Scheme

Dimensionality of walking

Simplified task space for bipedal locomotion
Proposed approach to use off-line computed trajectories in a feedback loop

Generalization of Trajectories

  1. sample task space in all dimensions of $k$
  2. compute $p$ for all $k$ with optimization method
  3. Ensure generate dataset of data is smooth
Result structure for $\Gamma_{\tau}$
Generalize Dataset:
Using Gaussian Processes Regression
Still quite high sample density required

IROS 2015: Generalization of optimal motion trajectories for bipedal walking - A Werner, D Trautmann, D Lee, R Lampariello

CRunner Walking Robustly


Accepted for Humanoid 2017: Optimal and Robust Walking using Intrinsic Properties of a Series-Elastic Robot - A Werner, B Henze, F Loeffl, C Ott

Part 2: Perception and Step Planning

IROS 2016: Multi-contact planning and control for a torque-controlled humanoid robot - Alexander Werner, Bernd Henze, Diego A Rodriguez, Jonathan Gabaret, Oliver Porges, Máximo A Roa

Perception workflow

Step Planning

  • Generated static stepping capabilities via "Posture Generator"
  • Define criterion for foot-surface match
  • Plan steps like A*

Conclusions from Experiments

  • Precision matters, errors accumulate
    Perception: 2-3cm, Control: 1cm
  • Accept partial contact of the foot
  • Low Contact Stiffness makes balancing on one foot hard


Humanoid Robot Toro: Getting up the stairs

Humanoid Robot Toro: Autonomous Applications in Aircraft Manufacturing