(ZMP)

(Capture Point)

Trotting

Push recovery

Walking

\mathbf{v}_0

\mathbf{v}_0

Pacing

Bounding

(Capture Point)

Single Formulation/Planner

Motivation

Alexander W. Winkler et al.

M. Kalakrishnan et al, “Learning, planning, and control for quadruped locomotion over challenging terrain,” IJRR, 2010

J. Pratt et al, “Capture point: A step toward humanoid push recovery,” in Humanoids, 2006.

Trajectory Optimization

\text{find} \quad \color{black}{x(t)}, \color{black}{u(t)} , \quad \text{for } t \in [0,T]

\text{find} \quad \color{black}{x(t)}, \color{black}{u(t)} , \quad \text{for } t \in [0,T]

\text{subject to} \quad \mathbf{x}(0) - \mathbf{x}_0 = 0

\text{subject to} \quad \mathbf{x}(0) - \mathbf{x}_0 = 0

\dot{x} - f(x(t), u(t)) = 0

\dot{x} - f(x(t), u(t)) = 0

h(x(t), u(t)) \leq 0

h(x(t), u(t)) \leq 0

x(T) - x_T = 0

x(T) - x_T = 0

x(t), u(t) = \text{arg min } J(x,u)

x(t), u(t) = \text{arg min } J(x,u)

given initial state

dynamic Model

Path Constraints

Desired Final State

Objective

\text{find} \quad \color{red}{x(t)}, \color{red}{u(t)} , \quad \text{for } t \in [0,T]

\text{find} \quad \color{red}{x(t)}, \color{red}{u(t)} , \quad \text{for } t \in [0,T]

Alexander W. Winkler et al.

Dynamic Consistency

\text{find} \quad x(t), u(t) , \quad \text{for } t \in [0,T]

\text{find} \quad x(t), u(t) , \quad \text{for } t \in [0,T]

\text{subject to} \quad \mathbf{x}(0) - \mathbf{x}_0 = 0

\text{subject to} \quad \mathbf{x}(0) - \mathbf{x}_0 = 0

\dot{x} - f(x(t), u(t)) = 0

\dot{x} - f(x(t), u(t)) = 0

h(x(t), u(t)) \leq 0

h(x(t), u(t)) \leq 0

x(T) - x_T = 0

x(T) - x_T = 0

x(t), u(t) = \text{arg min } J(x,u)

x(t), u(t) = \text{arg min } J(x,u)

Linear Inverted Pendulum Model

S. Kajita et al, “Biped walking pattern generation by using preview control of zero-moment point,” IEEE International Conference on Robotics and Automation, 2003.

\ddot{\mathbf{c}}(t) = \frac{g}{h}(\mathbf{c}(t)- \color{red}{\mathbf{u}}(t))

\ddot{\mathbf{c}}(t) = \frac{g}{h}(\mathbf{c}(t)- \color{red}{\mathbf{u}}(t))

Alexander W. Winkler et al.

Unilateral Contact Forces

\text{find} \quad x(t), u(t) , \quad \text{for } t \in [0,T]

\text{find} \quad x(t), u(t) , \quad \text{for } t \in [0,T]

\text{subject to} \quad \mathbf{x}(0) - \mathbf{x}_0 = 0

\text{subject to} \quad \mathbf{x}(0) - \mathbf{x}_0 = 0

\dot{x} - f(x(t), u(t)) = 0

\dot{x} - f(x(t), u(t)) = 0

h(x(t), u(t)) \leq 0

h(x(t), u(t)) \leq 0

x(T) - x_T = 0

x(T) - x_T = 0

x(t), u(t) = \text{arg min } J(x,u)

x(t), u(t) = \text{arg min } J(x,u)

\color{green}{\mathbf{c}} = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \end{bmatrix}

\color{green}{\mathbf{c}} = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \end{bmatrix}

\color{green}{\mathbf{c}} = \begin{bmatrix} 0 \\ 1 \\ 1 \\ 0 \end{bmatrix}

\color{green}{\mathbf{c}} = \begin{bmatrix} 0 \\ 1 \\ 1 \\ 0 \end{bmatrix}

u

u

u

u

\color{red}{\mathbf{u}}^T \mathbf{n}_i(\mathbf{p}) + \text{offset}(\mathbf{p}) > 0

\color{red}{\mathbf{u}}^T \mathbf{n}_i(\mathbf{p}) + \text{offset}(\mathbf{p}) > 0

Unilateral Contact Forces CoP inside Support-Area

\Leftrightarrow

\Leftrightarrow

M. Kalakrishnan et al, “Learning, planning, and control for quadruped locomotion over challenging terrain,” IJRR, 2010

Alexander W. Winkler et al.

Cannot represent single point-contacts or lines
- Heuristic expansion of points or lines into areas
Ordering of contact points necessary.

\color{red}{\mathbf{u}} = \sum\limits_{i=1}^4 \lambda_i \mathbf{p}_i

\color{red}{\mathbf{u}} = \sum\limits_{i=1}^4 \lambda_i \mathbf{p}_i

\sum\limits_{i=1}^{4} \lambda_i = 1

\sum\limits_{i=1}^{4} \lambda_i = 1

0 < \lambda_i

0 < \lambda_i

\color{green}{\mathbf{c}} = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \end{bmatrix}

\color{green}{\mathbf{c}} = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \end{bmatrix}

\color{green}{\mathbf{c}} = \begin{bmatrix} 0 \\ 1 \\ 1 \\ 0 \end{bmatrix}

\color{green}{\mathbf{c}} = \begin{bmatrix} 0 \\ 1 \\ 1 \\ 0 \end{bmatrix}

< c_i

< c_i

u

u

u

u

Vertex-Based Zmp-Constraint Formulation

Alexander W. Winkler et al.

Kinematic Range

\text{find} \quad x(t), u(t) , \quad \text{for } t \in [0,T]

\text{find} \quad x(t), u(t) , \quad \text{for } t \in [0,T]

\text{subject to} \quad \mathbf{x}(0) - \mathbf{x}_0 = 0

\text{subject to} \quad \mathbf{x}(0) - \mathbf{x}_0 = 0

\dot{x} - f(x(t), u(t)) = 0

\dot{x} - f(x(t), u(t)) = 0

h(x(t), u(t)) \leq 0

h(x(t), u(t)) \leq 0

x(T) - x_T = 0

x(T) - x_T = 0

x(t), u(t) = \text{arg min } J(x,u)

x(t), u(t) = \text{arg min } J(x,u)

kinematic Range:

\mathbf{p}_{i=1..4} \in \mathcal{R}(\mathbf{c})

\mathbf{p}_{i=1..4} \in \mathcal{R}(\mathbf{c})

\mathbf{p}_{LF}

\mathbf{p}_{LF}

\mathcal{R}_{LF}

\mathcal{R}_{LF}

\mathcal{R}_{RF}

\mathcal{R}_{RF}

\mathcal{R}_{RH}

\mathcal{R}_{RH}

\mathcal{R}_{LH}

\mathcal{R}_{LH}

\mathbf{c}

\mathbf{c}

Alexander W. Winkler et al.

Robust Motions

\text{find} \quad x(t), u(t) , \quad \text{for } t \in [0,T]

\text{find} \quad x(t), u(t) , \quad \text{for } t \in [0,T]

\text{subject to} \quad \mathbf{x}(0) - \mathbf{x}_0 = 0

\text{subject to} \quad \mathbf{x}(0) - \mathbf{x}_0 = 0

\dot{x} - f(x(t), u(t)) = 0

\dot{x} - f(x(t), u(t)) = 0

h(x(t), u(t)) \leq 0

h(x(t), u(t)) \leq 0

x(T) - x_T = 0

x(T) - x_T = 0

x(t), u(t) = \text{arg min } J(x,u)

x(t), u(t) = \text{arg min } J(x,u)

\mathbf{u}

\mathbf{u}

\mathbf{\lambda} = \begin{bmatrix} 0 \\ 0.5 \\ 0.5 \\ 0 \end{bmatrix}

\mathbf{\lambda} = \begin{bmatrix} 0 \\ 0.5 \\ 0.5 \\ 0 \end{bmatrix}

\mathbf{\lambda}^* = \begin{bmatrix} 0.33 \\ 0.33 \\ 0.33 \\ 0 \end{bmatrix}

\mathbf{\lambda}^* = \begin{bmatrix} 0.33 \\ 0.33 \\ 0.33 \\ 0 \end{bmatrix}

J = \sum\limits_{k=1}^{n} |\!| \mathbf{\lambda}_k - \mathbf{\lambda}_k^* |\!|_2^2

J = \sum\limits_{k=1}^{n} |\!| \mathbf{\lambda}_k - \mathbf{\lambda}_k^* |\!|_2^2

\mathbf{u}^*

\mathbf{u}^*

"If possible, Avoid The Extremes"

Alexander W. Winkler et al.

Fast Trajectory Optimization for Legged Robots using Vertex-Based ZMP Constraints

Trotting

Push recovery

Walking

Pacing

Bounding

Single Formulation/Planner

Motivation

Single Formulation/Planner

Trajectory Optimization

given initial state

dynamic Model

Path Constraints

Desired Final State

Objective

OPtimization Variables

Dynamic Consistency

Linear Inverted Pendulum Model

Simultaneous Base and Feet Optimization

Unilateral Contact Forces

Unilateral Contact Forces CoP inside Support-Area

Vertex-Based Zmp-Constraint Formulation

Future Work

Take-Away MSGs

Walk

Trot

Bound

Pace

Kinematic Range

kinematic Range:

Robust Motions

"If possible, Avoid The Extremes"

Biped with non-point-feet:

Generating Torques to Track the Optimized Motion