Contents

• Part 1: Trajectory generation for walking
• Part 2: Step planning from perception data

Walking models

Small Model

COM Dynamics

Unilaterality

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

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

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

• Capture Image through RGBD Camera
• Cluster flat surfaces (steps)
• Compute convex hulls for each surface

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

Questions

Humanoid Robot Toro: Getting up the stairs

Humanoid Robot Toro: Autonomous Applications in Aircraft Manufacturing