"Facebook Presentation: Optimization-based motion planning for legged systems"

Optimization-based Motion-Planning for Legged Systems

Alexander Winkler

Jan 22, 2020 $\cdot$ Facebook Reality Labs

$\Rightarrow$ Cubic-Hermite splines

Represent continous motion $\mathbf{u}(t), {\color{red}\mathbf{u}(t)}$ with discrete parameters

\color{black}{\mathbf{w}} = \{ \color{black}{x_0}, \color{black}{\dot{x}_0}, \color{black}{x_1}, \color{black}{\dot{x}_1}, \color{black}{x_2}, \color{black}{\dot{x}_2}, \color{black}{x_T}, \color{black}{\dot{x}_T} \}

\color{black}{\mathbf{w}} = \{ \color{black}{x_0}, \color{black}{\dot{x}_0}, \color{black}{x_1}, \color{black}{\dot{x}_1}, \color{black}{x_2}, \color{black}{\dot{x}_2}, \color{black}{x_T}, \color{black}{\dot{x}_T} \}

Optimization parameters:

\min\limits_{\mathbf{w}} a(\mathbf{w}) \quad \text{subject to} \quad \mathbf{b}(\mathbf{w}) = \mathbf{0}, \quad \mathbf{c}(\mathbf{w})\ge \mathbf{0}

\min\limits_{\mathbf{w}} a(\mathbf{w}) \quad \text{subject to} \quad \mathbf{b}(\mathbf{w}) = \mathbf{0}, \quad \mathbf{c}(\mathbf{w})\ge \mathbf{0}

3rd-order polynomials defined by node values

Dynamic Model

\ddot{\mathbf{r}}(t) = \frac{g}{h}(\mathbf{r}(t)- \color{red}{\mathbf{p}_c}(t))

\ddot{\mathbf{r}}(t) = \frac{g}{h}(\mathbf{r}(t)- \color{red}{\mathbf{p}_c}(t))

Linear Inverted Pendulum

\min\limits_{\mathbf{w}} a(\mathbf{w}) \quad \text{subject to} \quad \mathbf{b}(\mathbf{w}) = \mathbf{0}, \quad \mathbf{c}(\mathbf{w})\ge \mathbf{0}

\min\limits_{\mathbf{w}} a(\mathbf{w}) \quad \text{subject to} \quad \mathbf{b}(\mathbf{w}) = \mathbf{0}, \quad \mathbf{c}(\mathbf{w})\ge \mathbf{0}

\color{red}{\mathbf{p}_c}

\color{red}{\mathbf{p}_c}

Why it's difficult to plan a motion $\{\mathbf{\ddot{q}}_{des}\}_{t=0..T}$ for legged systems

(Base $\in \mathbb{R}^6$ )

\mathbf{M} \mathbf{\ddot{q}}_{des} + \mathbf{h} = \mathbf{J}^T \color{red}{\mathbf{f}}

\mathbf{M} \mathbf{\ddot{q}}_{des} + \mathbf{h} = \mathbf{J}^T \color{red}{\mathbf{f}}

\mathbf{f}

\mathbf{f}

mg

external forces $\mathbf{f}$ drive the system, but strong restrictions:
- Forces can only push
- forces only possible in contact
- position of force fixed during stance
- tangential forces must stay inside friction cone

joint torques $\tau$ can't directly drive the 6 DoF base $\Rightarrow$ underactuated

Unilateral Contact Forces $\Leftrightarrow$ CoP inside Support-Area

\min\limits_{\mathbf{w}} a(\mathbf{w}) \quad \text{subject to} \quad \mathbf{b}(\mathbf{w}) = \mathbf{0}, \quad \mathbf{c}(\mathbf{w})\ge \mathbf{0}

\min\limits_{\mathbf{w}} a(\mathbf{w}) \quad \text{subject to} \quad \mathbf{b}(\mathbf{w}) = \mathbf{0}, \quad \mathbf{c}(\mathbf{w})\ge \mathbf{0}

\color{blue}{\mathbf{p}_4}

\color{blue}{\mathbf{p}_4}

\color{blue}{\mathbf{p}_1}, \color{red}{\mathbf{\lambda}_1}, \color{#7f6000}{c_1}

\color{blue}{\mathbf{p}_1}, \color{red}{\mathbf{\lambda}_1}, \color{#7f6000}{c_1}

\color{blue}{\mathbf{p}_2}

\color{blue}{\mathbf{p}_2}

\color{blue}{\mathbf{p}_3}

\color{blue}{\mathbf{p}_3}

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

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

\mathbf{p}_c

\mathbf{p}_c

1. \quad{\color{red}\mathbf{p}_c} = \sum\limits_{i=1}^{{\color{#7f6000}4}} \color{red}{\lambda_i} \color{blue}{\mathbf{p}_i}

1. \quad{\color{red}\mathbf{p}_c} = \sum\limits_{i=1}^{{\color{#7f6000}4}} \color{red}{\lambda_i} \color{blue}{\mathbf{p}_i}

2. \quad \sum\limits_{i=1}^{{\color{#7f6000}4}} {\color{red}\lambda_i} = 1

2. \quad \sum\limits_{i=1}^{{\color{#7f6000}4}} {\color{red}\lambda_i} = 1

3. \quad 0 \le\color{red}{\lambda_i}

3. \quad 0 \le\color{red}{\lambda_i}

\le c_i

\le c_i

foothold change

Simultaneous Foothold and CoM Optimization

\min\limits_{\mathbf{w}} \mathbf{0} \quad \text{subject to} \quad \mathbf{b}(\mathbf{w}) = \mathbf{0}, \quad \mathbf{c}(\mathbf{w})\ge \mathbf{0}

\min\limits_{\mathbf{w}} \mathbf{0} \quad \text{subject to} \quad \mathbf{b}(\mathbf{w}) = \mathbf{0}, \quad \mathbf{c}(\mathbf{w})\ge \mathbf{0}

\text{find} \quad \mathbf{r}(t) \in \mathbb{R}^3 \quad \text{(CoM)}

\text{find} \quad \mathbf{r}(t) \in \mathbb{R}^3 \quad \text{(CoM)}

\mathbf{\theta}(t) \in \mathbb{R}^3 \quad \text{(Base orientation)}

\mathbf{\theta}(t) \in \mathbb{R}^3 \quad \text{(Base orientation)}

\text{for every foot } i \in \{1,\ldots,n_{ee}\}:

\text{for every foot } i \in \{1,\ldots,n_{ee}\}:

\color{darkblue}{\mathbf{p}_i}(t) \in \mathbb{R}^3 \quad \text{(Foot position)}

\color{darkblue}{\mathbf{p}_i}(t) \in \mathbb{R}^3 \quad \text{(Foot position)}

\color{red}{\mathbf{f}_i}(t) \in \mathbb{R}^3 \quad \text{(Foot force)}

\color{red}{\mathbf{f}_i}(t) \in \mathbb{R}^3 \quad \text{(Foot force)}

\mathbf{p}_1

\mathbf{p}_1

\mathbf{p}_2

\mathbf{p}_2

\mathbf{p}_3

\mathbf{p}_3

\mathbf{p}_4

\mathbf{p}_4

\mathbf{f}_1

\mathbf{f}_1

\mathbf{f}_2

\mathbf{f}_2

\mathbf{r},

\mathbf{r},

\theta

\theta

Gait and Trajectory Optimization for Legged Systems through Phase-based End-Effector Parameterization

IEEE Robotic and Automation Letters (RA-L) $\cdot$ 2018

A. W. Winkler, D. Bellicoso, M. Hutter, J. Buchli

Towards integrated motion-planning

m \, \mathbf{\ddot{r}} \quad \quad \quad \quad \quad = \sum_{i=1}^{4} {\color{red}\mathbf{f}_i} - m \mathbf{g}

m \, \mathbf{\ddot{r}} \quad \quad \quad \quad \quad = \sum_{i=1}^{4} {\color{red}\mathbf{f}_i} - m \mathbf{g}

\mathbf{I}(\theta) \, \dot{\omega} + \omega\!\times\!\mathbf{I}(\theta) \omega = \sum_{i=1}^{4} {\color{red}\mathbf{f}_i}\!\times\!(\mathbf{r}-{\color{#1c4587}\mathbf{p}_i})

\mathbf{I}(\theta) \, \dot{\omega} + \omega\!\times\!\mathbf{I}(\theta) \omega = \sum_{i=1}^{4} {\color{red}\mathbf{f}_i}\!\times\!(\mathbf{r}-{\color{#1c4587}\mathbf{p}_i})

Dynamic Model

Single Rigid Body $\cdot$ Newton-Euler Equations

\mathbf{p}_1

\mathbf{p}_1

\mathbf{p}_2

\mathbf{p}_2

\mathbf{p}_3

\mathbf{p}_3

\mathbf{p}_4

\mathbf{p}_4

\mathbf{f}_1

\mathbf{f}_1

\mathbf{f}_2

\mathbf{f}_2

\mathbf{I}, m

\mathbf{I}, m

\begin{bmatrix} \mathbf{\ddot{r}} \\ \mathbf{\dot{\omega}} \end{bmatrix}

\begin{bmatrix} \mathbf{\ddot{r}} \\ \mathbf{\dot{\omega}} \end{bmatrix}

\min\limits_{\mathbf{w}} \mathbf{0} \quad \text{subject to} \quad \mathbf{b}(\mathbf{w}) = \mathbf{0}, \quad \mathbf{c}(\mathbf{w})\ge \mathbf{0}

\min\limits_{\mathbf{w}} \mathbf{0} \quad \text{subject to} \quad \mathbf{b}(\mathbf{w}) = \mathbf{0}, \quad \mathbf{c}(\mathbf{w})\ge \mathbf{0}

Kinematic Model

{\color{#1c4587}\mathbf{p}_i} \in {\color{#1c4587}\mathcal{R}_i}(\mathbf{r},\theta)

{\color{#1c4587}\mathbf{p}_i} \in {\color{#1c4587}\mathcal{R}_i}(\mathbf{r},\theta)

\mathbf{p}_1

\mathbf{p}_1

\mathbf{p}_2

\mathbf{p}_2

\mathbf{r}, \mathbf{\theta}

\mathbf{r}, \mathbf{\theta}

\mathcal{R}_2

\mathcal{R}_2

\mathcal{R}_1

\mathcal{R}_1

Range-of-Motion Box $\approx$ Joint limits

\min\limits_{\mathbf{w}} \mathbf{0} \quad \text{subject to} \quad \mathbf{b}(\mathbf{w}) = \mathbf{0}, \quad \mathbf{c}(\mathbf{w})\ge \mathbf{0}

\min\limits_{\mathbf{w}} \mathbf{0} \quad \text{subject to} \quad \mathbf{b}(\mathbf{w}) = \mathbf{0}, \quad \mathbf{c}(\mathbf{w})\ge \mathbf{0}

Gait Optimization

R | 0 | R | 2 | R | 2

.... gait defined by continuous phase-durations $\Delta T_i$

without Integer Programming

swing

stance

individual foot always alternates between and

\Delta T_{R,1}

\Delta T_{R,1}

\Delta T_{R,2}

\Delta T_{R,2}

\Delta T_{R,3}

\Delta T_{R,3}

\Delta T_{L,1}

\Delta T_{L,1}

\Delta T_{L,2}

\Delta T_{L,2}

\Delta T_{L,3}

\Delta T_{L,3}

\Delta T_{L,4}

\Delta T_{L,4}

R | 2 | L | R | 2

Sequence:

Terrain constraints

{\color{blue}p_{i,s}^z} = h({\color{blue}p_{i,s}^x}, {\color{blue}p_{i,s}^y})

{\color{blue}p_{i,s}^z} = h({\color{blue}p_{i,s}^x}, {\color{blue}p_{i,s}^y})

{\color{red}\mathbf{f}_i(t)} \cdot \mathbf{n}({\color{blue}p_{i,s}^x}, {\color{blue}p_{i,s}^y}) \ge 0

{\color{red}\mathbf{f}_i(t)} \cdot \mathbf{n}({\color{blue}p_{i,s}^x}, {\color{blue}p_{i,s}^y}) \ge 0

\lvert {\color{red}\mathbf{f}_i(t)}\cdot \mathbf{t}({\color{blue}p_{i,s}^x}, {\color{blue}p_{i,s}^y}) \rvert < \mu {\color{red}\mathbf{f}_i(t)} \cdot \mathbf{n}({\color{blue}p_{i,s}^x}, {\color{blue}p_{i,s}^y})

\lvert {\color{red}\mathbf{f}_i(t)}\cdot \mathbf{t}({\color{blue}p_{i,s}^x}, {\color{blue}p_{i,s}^y}) \rvert < \mu {\color{red}\mathbf{f}_i(t)} \cdot \mathbf{n}({\color{blue}p_{i,s}^x}, {\color{blue}p_{i,s}^y})

Foot can only stand on terrain

Forces can only push

Forces inside friction pyramid

t \in \mathcal{C}

t \in \mathcal{C}

\min\limits_{\mathbf{w}} \mathbf{0} \quad \text{subject to} \quad \mathbf{b}(\mathbf{w}) = \mathbf{0}, \quad \mathbf{c}(\mathbf{w})\ge \mathbf{0}

\min\limits_{\mathbf{w}} \mathbf{0} \quad \text{subject to} \quad \mathbf{b}(\mathbf{w}) = \mathbf{0}, \quad \mathbf{c}(\mathbf{w})\ge \mathbf{0}

Know if polynomial belongs to swing or stance phase

Foot $\mathbf{p}_i(t)$ cannot move while

\color{red}{\mathbf{f}_i} (t\notin\mathcal{C}_i) = \mathbf{0}

\color{red}{\mathbf{f}_i} (t\notin\mathcal{C}_i) = \mathbf{0}

\color{blue}{\dot{\mathbf{p}}_i} (t\in \mathcal{C}) = \mathbf{0}

\color{blue}{\dot{\mathbf{p}}_i} (t\in \mathcal{C}) = \mathbf{0}

Physical Restrictions

Forces $\mathbf{f}_i(t)$ cannot exist while

standing

swinging

Unilateral Contact Forces $\Leftrightarrow$ CoP inside Support-Area

\color{red}{\mathbf{p}_c}

\color{red}{\mathbf{p}_c}

\color{blue}{\mathbf{p}_{1}}

\color{blue}{\mathbf{p}_{1}}

\color{blue}{\mathbf{p}_{8}}, \color{red}{\mathbf{\lambda}_8}, \color{#7f6000}{c_8}

\color{blue}{\mathbf{p}_{8}}, \color{red}{\mathbf{\lambda}_8}, \color{#7f6000}{c_8}

\color{blue}{\mathbf{p}_{2}}

\color{blue}{\mathbf{p}_{2}}

\mathbf{c} = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}

\mathbf{c} = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}

1. \quad{\color{red}\mathbf{p}_c} = \sum\limits_{i=1}^{{\color{#7f6000}|\mathbf{c}|}} \color{red}{\lambda_i} \color{blue}{\mathbf{p}_i}

1. \quad{\color{red}\mathbf{p}_c} = \sum\limits_{i=1}^{{\color{#7f6000}|\mathbf{c}|}} \color{red}{\lambda_i} \color{blue}{\mathbf{p}_i}

2. \quad \sum\limits_{i=1}^{{\color{#7f6000}|\mathbf{c}|}} {\color{red}\lambda_i} = 1

2. \quad \sum\limits_{i=1}^{{\color{#7f6000}|\mathbf{c}|}} {\color{red}\lambda_i} = 1

3. \quad 0 \le\color{red}{\lambda_i}

3. \quad 0 \le\color{red}{\lambda_i}

\le c_i

\le c_i

\mathbf{p}_c

\mathbf{p}_c

\color{blue}{\mathbf{p}_2}

\color{blue}{\mathbf{p}_2}

\color{blue}{\mathbf{p}_1}, \color{red}{\mathbf{\lambda}_1}, \color{#7f6000}{c_1}

\color{blue}{\mathbf{p}_1}, \color{red}{\mathbf{\lambda}_1}, \color{#7f6000}{c_1}

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

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

\color{blue}{\mathbf{p}_4}

\color{blue}{\mathbf{p}_4}

\color{blue}{\mathbf{p}_3}

\color{blue}{\mathbf{p}_3}

\min\limits_{\mathbf{w}} a(\mathbf{w}) \quad \text{subject to} \quad \mathbf{b}(\mathbf{w}) = \mathbf{0}, \quad \mathbf{c}(\mathbf{w})\ge \mathbf{0}

\min\limits_{\mathbf{w}} a(\mathbf{w}) \quad \text{subject to} \quad \mathbf{b}(\mathbf{w}) = \mathbf{0}, \quad \mathbf{c}(\mathbf{w})\ge \mathbf{0}

m \ddot{\mathbf{r}} = \sum_{i=1}^{4} \color{red}{\mathbf{f}_i} - m \mathbf{g}

m \ddot{\mathbf{r}} = \sum_{i=1}^{4} \color{red}{\mathbf{f}_i} - m \mathbf{g}

\mathbf{I}(\theta)\dot{\omega} + \omega\!\times\!\mathbf{I}(\theta) \omega = \sum_{i=1}^{4} \color{red}{\mathbf{f}_i}\!\times\!(\mathbf{r}-\color{#1c4587}{\mathbf{p}_i})

\mathbf{I}(\theta)\dot{\omega} + \omega\!\times\!\mathbf{I}(\theta) \omega = \sum_{i=1}^{4} \color{red}{\mathbf{f}_i}\!\times\!(\mathbf{r}-\color{#1c4587}{\mathbf{p}_i})

Centroidal Dynamics $\Rightarrow$ Single Rigid Body Dynamics

Newton-Euler Equations

+ Assumption A2: Momentum produced by the joint velocities is negligible.

+ Assumption A3: Full-body inertia remains similar to the one in nominal configuration.

\mathbf{A}(\mathbf{q}) \mathbf{\ddot{q}} + \mathbf{\dot{A}}(\mathbf{q},\mathbf{\dot{q}}) \mathbf{\dot{q}} = \begin{bmatrix} \sum_{i=1}^{4} \color{red}{\mathbf{f}_i} - m \mathbf{g} \\ \sum_{i=1}^{n_i} \color{red}{\mathbf{f}_i}\!\times\!(\mathbf{r}(\mathbf{q})-\color{#1c4587}{\mathbf{p}_i}(\mathbf{q})) \end{bmatrix}

\mathbf{A}(\mathbf{q}) \mathbf{\ddot{q}} + \mathbf{\dot{A}}(\mathbf{q},\mathbf{\dot{q}}) \mathbf{\dot{q}} = \begin{bmatrix} \sum_{i=1}^{4} \color{red}{\mathbf{f}_i} - m \mathbf{g} \\ \sum_{i=1}^{n_i} \color{red}{\mathbf{f}_i}\!\times\!(\mathbf{r}(\mathbf{q})-\color{#1c4587}{\mathbf{p}_i}(\mathbf{q})) \end{bmatrix}

	(pos)		Assumptions
Rigid Body Dynamics (RBD)			A1
Centroidal Dynamics (CD)			A1
Single Rigid Body Dynamics (SRBD)			A1, A2, A3
Linear Inverted Pendulum (LIP)			A1, A2, A3, A4, A5, A6

\mathbf{q}_b, \mathbf{q}_j

\mathbf{q}_b, \mathbf{q}_j

\mathbf{q}_b, \mathbf{q}_j

\mathbf{q}_b, \mathbf{q}_j

\mathbf{r}, \mathbf{\theta},

\mathbf{r}, \mathbf{\theta},

r_x, r_y

r_x, r_y

\tau, \mathbf{f}_i

\tau, \mathbf{f}_i

\mathbf{f}_i

\mathbf{f}_i

\mathbf{f}_i

\mathbf{f}_i

\mathbf{p}_c

\mathbf{p}_c

\mathbf{p}_i

\mathbf{p}_i

\mathbf{x}

\mathbf{x}

\mathbf{\dot{x}} = \mathbf{F}(\mathbf{x}(t), \color{red}{\mathbf{u}(t)})

\mathbf{\dot{x}} = \mathbf{F}(\mathbf{x}(t), \color{red}{\mathbf{u}(t)})

\mathbf{u}

\mathbf{u}

x(t) = a_0 + a_1t + a_2t^2 + a_3t^3

x(t) = a_0 + a_1t + a_2t^2 + a_3t^3

\{

\color{black}{x_0}

\color{black}{x_0}

\color{black}{\dot{x}_0}

\color{black}{\dot{x}_0}

\{

\{

\{

-\color{#7f6000}{\Delta T_1}^{-2} [ 3(\color{black}{x_0} - \color{black}{x_1}) + \color{#7f6000}{\Delta T_1}(2\color{black}{\dot{x}_0} + \color{black}{\dot{x}_1}) ]

-\color{#7f6000}{\Delta T_1}^{-2} [ 3(\color{black}{x_0} - \color{black}{x_1}) + \color{#7f6000}{\Delta T_1}(2\color{black}{\dot{x}_0} + \color{black}{\dot{x}_1}) ]

\color{#7f6000}{\Delta T_1}^{-3} [ 2(\color{black}{x_0} - \color{black}{x_1}) + \color{#7f6000}{\Delta T_1}( \color{black}{\dot{x}_0} + \color{black}{\dot{x}_1}) ]

\color{#7f6000}{\Delta T_1}^{-3} [ 2(\color{black}{x_0} - \color{black}{x_1}) + \color{#7f6000}{\Delta T_1}( \color{black}{\dot{x}_0} + \color{black}{\dot{x}_1}) ]

\color{black}{\mathbf{w}_j} = \{ \color{black}{x_0}, \color{black}{\dot{x}_0}, \color{#7f6000}{\Delta T_1}, \color{black}{x_1}, \color{black}{\dot{x}_1}, \color{#7f6000}{\Delta T_2}, \color{black}{x_2}, \color{black}{\dot{x}_2}, \color{#7f6000}{\Delta T_3}, \color{black}{x_T}, \color{black}{\dot{x}_T} \}

\color{black}{\mathbf{w}_j} = \{ \color{black}{x_0}, \color{black}{\dot{x}_0}, \color{#7f6000}{\Delta T_1}, \color{black}{x_1}, \color{black}{\dot{x}_1}, \color{#7f6000}{\Delta T_2}, \color{black}{x_2}, \color{black}{\dot{x}_2}, \color{#7f6000}{\Delta T_3}, \color{black}{x_T}, \color{black}{\dot{x}_T} \}

Cubic-Hermite Spline for $\color{red}{f_{\{x,y,z\}}(t)}, \color{blue}{p_{\{x,y,z\}}(t)}$

\mathbf{p}_c

\mathbf{p}_c

\mathbf{p}_c

\mathbf{p}_c

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

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

Constraint: Unilateral Contact Forces $\Leftrightarrow$ CoP inside Support-Area

Difficult for single point-contacts or lines

\color{blue}{\mathbf{p}_1}

\color{blue}{\mathbf{p}_1}

\color{blue}{\mathbf{p}_2}

\color{blue}{\mathbf{p}_2}

\color{blue}{\mathbf{p}_3}

\color{blue}{\mathbf{p}_3}

Ordering of contact points

1. \quad{\color{red}\mathbf{p}_c} = \sum\limits_{i=1}^4 \color{red}{\lambda_i} \color{blue}{\mathbf{p}_i}

1. \quad{\color{red}\mathbf{p}_c} = \sum\limits_{i=1}^4 \color{red}{\lambda_i} \color{blue}{\mathbf{p}_i}

2. \quad \sum\limits_{i=1}^{4} {\color{red}\lambda_i} = 1

2. \quad \sum\limits_{i=1}^{4} {\color{red}\lambda_i} = 1

3. \quad 0 \le\color{red}{\lambda_i}

3. \quad 0 \le\color{red}{\lambda_i}

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

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

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

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

\le c_i

\le c_i

\mathbf{p}_c

\mathbf{p}_c

\mathbf{p}_c

\mathbf{p}_c

Vertex-Based Zmp-Constraint Formulation

\color{blue}{\mathbf{p}_1}, \color{red}{\mathbf{\lambda}_1}, \color{#7f6000}{c_1}

\color{blue}{\mathbf{p}_1}, \color{red}{\mathbf{\lambda}_1}, \color{#7f6000}{c_1}

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

IEEE Robotic and Automation Letters (RA-L) $\cdot$ 2017

A. W. Winkler, F. Farshidian, D. Pardo, M. Neunert, J. Buchli

Optimization-based Motion-Planning for Legged Systems Alexander Winkler Jan 22, 2020 ⋅ Facebook Reality Labs

Optimization-based Motion-Planning for Legged Systems

Advantages of legged systems

?

Represent continous motion $\mathbf{u}(t), {\color{red}\mathbf{u}(t)}$ with discrete parameters

Dynamic Model

Why it's difficult to plan a motion $\{\mathbf{\ddot{q}}_{des}\}_{t=0..T}$ for legged systems

Unilateral Contact Forces $\Leftrightarrow$ CoP inside Support-Area

Motion-Plan Search Space

Towards integrated motion-planning

Dynamic Model

Kinematic Model

Gait Optimization

Terrain constraints

Character Simulation in Game Engines

Summary

Recreating Dust 2

Unilateral Contact Forces $\Leftrightarrow$ CoP inside Support-Area

Centroidal Dynamics $\Rightarrow$ Single Rigid Body Dynamics

Constraint: Unilateral Contact Forces $\Leftrightarrow$ CoP inside Support-Area

Vertex-Based Zmp-Constraint Formulation