Alexander W. Winkler
Robotics researcher specialized in motion planning for legged systems.
Floating-base Robot Models
Alexander W. Winkler
[ taken from Chapter 1.2 of A.W. Winkler, "Optimization-based motion planning for legged robots", PhD Thesis, ETH Zurich, 2018 ]
\mathbf{\dot{x}} = \mathbf{F}(\mathbf{x}(t), \color{red}{\mathbf{u}(t)})
"Reality" \(\Rightarrow \) Rigid Body Dynamics
\mathbf{M}(\mathbf{q}) \mathbf{\ddot{q}} + \mathbf{h}(\mathbf{q}, \mathbf{\dot{q}}) = \mathbf{S}^T \color{red}{\tau} + \mathbf{J}(\mathbf{q})^T \color{red}{\mathbf{f}}
\mathbf{M}_u(\mathbf{q}) \mathbf{\ddot{q}} + \mathbf{h}_u(\mathbf{q}, \mathbf{\dot{q}}) = \mathbf{J}_u(\mathbf{q})^T \color{red}{\mathbf{f}}
\mathbf{M}_a(\mathbf{q}) \mathbf{\ddot{q}} + \mathbf{h}_a(\mathbf{q}, \mathbf{\dot{q}}) =
\color{red}{\tau} +
\mathbf{J}_a(\mathbf{q})^T \color{red}{\mathbf{f}}
\Leftrightarrow
+ Assumption A1: Bodies do not deform when forces are applied.
Rigid Body Dynamics \(\Rightarrow\) Centroidal Dynamics
\mathbf{M}_u(\mathbf{q}) \mathbf{\ddot{q}} + \mathbf{h}_u(\mathbf{q}, \mathbf{\dot{q}}) = \mathbf{J}_u(\mathbf{q})^T \color{red}{\mathbf{f}}
\mathbf{M}_a(\mathbf{q}) \mathbf{\ddot{q}} + \mathbf{h}_a(\mathbf{q}, \mathbf{\dot{q}}) =
\color{red}{\tau} +
\mathbf{J}_a(\mathbf{q})^T \color{red}{\mathbf{f}}
\Rightarrow \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}
+ zero Assumptions
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})
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}
\Rightarrow
SRBD \( \Rightarrow\) Linear Inverted Pendulum
\Rightarrow
\ddot{\mathbf{r}}(t)
=
\frac{g}{h}(\mathbf{r}(t)- \color{red}{\mathbf{p}_c}(t))
\color{red}{\mathbf{p}_c}
\color{red}{\mathbf{p}_c}
+ Assumption A4: CoM height is constant.
+ Assumption A5: Angular velocity \(\omega\) and acceleration \(\dot{\omega}\) are zero.
+ Assumption A6: Footholds are at constant height \(p_z\)
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})
\Rightarrow \color{red}{p_{c,x}} =
\frac{\sum_{i=1}^{4} \color{red}{f_{i,z}} \color{#1c4587}{p_{i,x}}}
{mg}
\Rightarrow h= r_z - \color{#1c4587}{p_{i,z}}
| (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{r}, \mathbf{\theta},
r_x, r_y
\tau, \mathbf{f}_i
\mathbf{f}_i
\mathbf{f}_i
\mathbf{p}_c
\mathbf{p}_i
\mathbf{x}
\mathbf{\dot{x}} = \mathbf{F}(\mathbf{x}(t), \color{red}{\mathbf{u}(t)})
\mathbf{u}
Centroidal Dynamics Model Comparison
By Alexander W. Winkler
Centroidal Dynamics Model Comparison
PDF: https://www.research-collection.ethz.ch/handle/20.500.11850/272432
