diff --git a/.gitmodules b/.gitmodules deleted file mode 100644 index 1350b67..0000000 --- a/.gitmodules +++ /dev/null @@ -1,3 +0,0 @@ -[submodule "deep-contact-estimator"] - path = morpho_symm/datasets/contact_dataset/deep-contact-estimator - url = https://github.com/UMich-CURLY/deep-contact-estimator/ diff --git a/README.md b/README.md index e683888..6acaa01 100644 --- a/README.md +++ b/README.md @@ -1,19 +1,113 @@ -# Morphological Symmetries (MorphoSymm) in Locomoting Dynamical Systems +# Morphological Symmetries - _MorphoSymm_ +The Morphological Symmetries (MorphoSymm) repository offers a comprehensive toolkit for the detection, analysis, and exploitation of morphological symmetries in the modeling and control of robotic systems. These symmetries are a common characteristic of both biological and artificial locomotion systems, including legged, swimming, and flying animals, robots, and virtual characters. Essentially, a morphological symmetry is a transformation that connects two equivalent states under the governing dynamics, meaning two states that will evolve identically when influenced by equivalent moving forces. For example, consider these two motion trajectories from the mini-cheetah quadruped robot, which are related by a symmetry transformation. -Welcome to the Morphological Symmetries (MorphoSymm) repository! Here, you will find a comprehensive set of tools for the identification, study, and exploitation of morphological symmetries in locomoting dynamical systems. These symmetries are commonly found in a wide range of systems, including legged, swimming, and flying animals, robotic systems, and animated characters. As such, this repository aims to provide valuable resources for researchers and practitioners in the fields of Robotics, Computer Graphics, and Computational Biology. +

+ +

-This repository holds the code for the paper: [On discrete symmetries of robotic systems: A data-driven and group-theoretic analysis](https://scholar.google.it/scholar?q=on+discrete+symmetries+of+robotic+systems:+a+data-driven+and+group-theoretic+analysis&hl=en&as_sdt=0&as_vis=1&oi=scholart). -Accepted to *Robotics Science and Systems 2023 (RSS 2023)*. For reproducing the experiments of the paper, please see the master branch. +These symmetries carry significant implications. Notably, they offer a valuable geometric bias, since by modeling and controlling the dynamics of one state, we can effectively identify and control the dynamics of all its symmetric counterparts (see [our paper](https://danfoa.github.io/MorphoSymm/) for details). -#### Contents: + +

+ +

+ +## Group and representation theory + +To exploit these symmetries, we employ the language of group and representation theory, enabling us to represent these transformations via linear algebra. The set of morphological symmetries of a robotic system forms the system's [discrete (or finite) symmetry group](https://en.wikipedia.org/wiki/Discrete_group). For example, the mini-cheetah robot's symmetry group $\mathbb{G}$ contains $8$ distinct transformations generated by combinations of the transformations $\{e, g_s, g_t, g_f\}$: + + + + + + +
+ Morphological Symmetries Mini-Cheetah quadruped robot, Caley Diagram, Discrete Symmetry Group + + Morphological Symmetries Mini-Cheetah quadruped robot, State symmetry, Discrete Symmetry Group +
+ +As depicted above, each symmetry transformation, represented by a group element $g \in \mathbb{G}$, influences the system's configuration and any proprioceptive and exteroceptive observations associated with the system's dynamic evolution. These observations include aspects such as joint torques, center of mass momentum, ground reaction forces, and terrain elevation/orientation. To numerically transform these observations, the appropriate group representation is required. This representation maps each symmetry transformation to an invertible matrix in the space where the observations resides. For example, to transform the robot's joint space generalized velocity coordinates $\mathbf{v}_ {js} \in \mathcal{T}_ {\mathbf{q}}\mathcal{Q}_ {js} \subset \mathbb{R}^{12}$, the group representation $\rho_ {\mathcal{T}_ {\mathbf{q}}\mathcal{Q}_ {js}}: \mathbb{G} \rightarrow \mathbb{GL}(\mathbb{R}^{12})$ is needed. This defines the symmetry transformations in the $12$ dimensional space in which $\mathbf{v}_ {js}$ evolves. In MorphoSymm, when you load a robotic system, you automatically receive its symmetry group and all the relevant group representations required to transform observations used in robotics. For our example, this appears as follows: + +```python +from morpho_symm.utils.robot_utils import load_symmetric_system + +robot, G = load_symmetric_system(robot_name='mini_cheetah') +# Get joint space position and velocity coordinates (q_js, v_js) | q_js ∈ Qjs, dq_js ∈ TqQjs +_, v_js = robot.get_joint_space_state() + +# Get the group representation on the space of joint space generalized velocity coordinates +rep_TqQ_js = G.representations['TqQ_js'] +for g in G.elements: + # Transform the observations + g_v_js = rep_TqQ_js(g) @ v_js # rep_TqQ_js(g) ∈ R^12x12 +``` + +For more details follow the [getting started guide](#getting-started) + +## Uses of morphological symmetries in robotics +There are three use cases of morphological symmetries supported by MorphoSymm. For details and tutorials see the getting started guide + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Data Augmentation
+Any data recording of proprioceptive and exteroceptive observations can be augmented using the appropriate group representations. This means that if you collect data from a symmetric robotic system, you can use the system's symmetries to analitically augment/multiply the recorded data. + + Data augmentation robotics atlas bipedal morphological symmetries +
Equivariant/Invariant Function Approximation
+Given the geometric nature of the data processed in robotics, most applications of machine learning to supervised, unsupervised, and reinforcement learning in robotics rely on the approximation of equivariant/invariant functions. To approximate these functions using deep learning, we should use equivariant/invariant neural networks that respect the symmetries of the learning problem. These networks feature enhanced sample efficiency and generalization, as demonstrated in our paper results. This repository relies on pytorch and the escnn library to offer equivariant and invariant neural network architectures. + + Equivariant function robotics, equivariant neural networks, contact detection +
Dynamics Harmonics Analysis
+Through the application of abstract harmonic analysis, the symmetries of a robotic system enable us to decompose any motion into a superposition of simpler symmetric synergistic motions. Each of these motions evolves in a lower-dimensional space, known as an isotypic subspace. By projecting entire motion trajectories into these spaces, we can characterize each motion as a superposition of lower-dimensional and synergistic motions. For further details on the use of abstract harmonic analysis in robotics see our paper Dynamics Harmonic Analysis of Robotic Systems: Application in Data-Driven Koopman Modeling + + Dynamics Harmonic Analysis of the Mini-cheetah quadruped robot locomotion +
+ Dynamics Harmonic Analysis of the Mini-cheetah quadruped robot locomotion +
+ +## Contents: - [Installation](#installation) - [Library of symmetric dynamical systems](#library-of-symmetric-dynamical-systems) -- [Tutorial](#tutorial) +- [Getting Started](#getting-started) - [Loading symmetric dynamical systems](#loading-symmetric-dynamical-systems) - - [Exploiting Morphological Symmetries](#exploiting-morphological-symmetries) - - [Equivariant Neural Networks](#equivariant-neural-networks) -- [Citation](#citation) + - [Available group representations](#available-group-representations) + - [Data-Augmentation](#data-augmentation) + - [Equivariant/Invariant Neural Networks](#equivariantinvariant-neural-networks) +- [How to cite us?](#how-to-cite-us) - [Contributing](#contributing) ## Installation: @@ -23,163 +117,311 @@ git clone https://github.com/Danfoa/MorphoSymm.git cd MorphoSymm pip install -e . ``` + ## Library of symmetric dynamical systems The following is a non-exhaustive and expanding list of dynamical systems with Discrete Morphological Symmetries. Each example can be reproduced in a 3D interactive environment running: ```python python morpho_symm.robot_symmetry_visualization.py robot= gui=True ``` -This script functions as an introductory tutorial showing how we define the representations of Discrete Morphological Symmetries in order to perform symmetry transformations on the robot state, and proprioceptive and exteroceptive measurements. -### $\mathcal{G}=\mathcal{C}_2$: Reflection Symmetry -| Cassie | Atlas | Bolt | Baxter | -|:----------------------------------------------------------------------------:|:----------------------------------------------------------------------------------------------------------------:|:------------------------------------------------------------:|:----------------------------------------------------------------------------:| -| ![cassie](docs/static/animations/cassie-C2-symmetries_anim_static.gif) | ![atlas](https://user-images.githubusercontent.com/8356912/200183197-94242c57-bd9d-41cb-8a0b-509dceef5cb9.gif) | ![bolt](docs/static/animations/bolt-C2-symmetries_anim_static.gif) | ![baxter](docs/static/animations/baxter-C2-symmetries_anim_static.gif) | -| [Solo](https://open-dynamic-robot-initiative.github.io/) | **A1** | **HyQ** | Mini-Cheetah | -| ![solo](docs/static/animations/solo-C2-symmetries_anim_static.gif) | ![a1](docs/static/animations/a1-C2-symmetries_anim_static.gif) | ![hyq](docs/static/animations/hyq-C2-symmetries_anim_static.gif) | ![mini-cheetah](docs/static/animations/mini_cheetah-C2-symmetries_anim_static.gif) | -| **Anymal-C** | **Anymal-B** | **B1** | Go1 | -| ![anymal_c](docs/static/animations/anymal_c-C2-symmetries_anim_static.gif) | ![anymal_b](docs/static/animations/anymal_b-C2-symmetries_anim_static.gif) | ![b1](docs/static/animations/b1-C2-symmetries_anim_static.gif) | ![go1](docs/static/animations/go1-C2-symmetries_anim_static.gif) | -| **UR-3** | **UR5** | **UR10** | KUKA-iiwa | -| ![ur3](docs/static/animations/ur3-C2-symmetries_anim_static.gif) | ![ur5](docs/static/animations/ur5-C2-symmetries_anim_static.gif) | ![ur10](docs/static/animations/ur10-C2-symmetries_anim_static.gif) | ![iiwa](docs/static/animations/iiwa-C2-symmetries_anim_static.gif) | - -### $\mathcal{G}=\mathcal{C}_n$: Symmetric Systems with Cyclic Group Symmetries -| [Trifinger](https://sites.google.com/view/trifinger/home-page)-C3 | -|:-------------------------------------------------------------------------------:| -| ![trifinger-edu](docs/static/animations/trifinger_edu-C3-symmetries_anim_static.gif) | - -### $\mathcal{G}=\mathcal{K}_4$: Klein-Four Symmetry -| [Solo](https://open-dynamic-robot-initiative.github.io/)| HyQ | Mini-Cheetah | Anymal-C | Anymal-B | -|:---------------------------------------------------------------------------------------------------------------:|:---------------------------------------------------------------------------------------:|:------------------------------------------------------------------------------------------:|:------------------------------------------------------------------------:|:------------------------------------------------------------------------:| -| ![Solo-K4](docs/static/animations/solo-Klein4-symmetries_anim_static.gif) | ![hyq](docs/static/animations/hyq-Klein4-symmetries_anim_static.gif) | ![Mini-Cheetah-K4](docs/static/animations/mini_cheetah-Klein4-symmetries_anim_static.gif) | ![anymal_c](docs/static/animations/anymal_c-Klein4-symmetries_anim_static.gif) | ![anymal_c](docs/static/animations/anymal_b-Klein4-symmetries_anim_static.gif) | - -### $\mathcal{G}=\mathcal{C}_2\times\mathcal{C}_2\times\mathcal{C}_2$: Regular cube symmetry -| [Solo](https://open-dynamic-robot-initiative.github.io/) | Mini-Cheetah | -|:--------------------------------------------------------------------------------------------------------------------:|:---------------------------------------------------------------------------------------------:| -| ![solo-c2xc2xc2](docs/static/animations/solo-C2xC2xC2-symmetries_anim_static.gif) | ![bolt](docs/static/animations/mini_cheetah-C2xC2xC2-symmetries_anim_static.gif) | +This script functions as an introductory tutorial showing how we define the representations of Discrete Morphological Symmetries in order to perform symmetry transformations on the robot state, and proprioceptive and exteroceptive observations. +### $\mathbb{G}=\mathbb{C}_2$: Reflection Symmetry + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Cassie Atlas KUKA-iiwa Mini-Cheetah Bolt Baxter Solo A1 HyQ Anymal-C Anymal-B Go1 B1 UR-3 UR-5 UR-10
+ Morphological symmetries cassie bipedal robot, State symmetry, Discrete Symmetry Group + + Morphological symmetries atlas bipedal robot, State symmetry, Discrete Symmetry Group + + Morphological symmetries iiwa robot, State symmetry, Discrete Symmetry Group + + Morphological symmetries mini-cheetah quadruped robot, State symmetry, Discrete Symmetry Group + + Morphological symmetries bolt bipedal robot, State symmetry, Discrete Symmetry Group + + Morphological symmetries baxter bipedal robot, State symmetry, Discrete Symmetry Group + + Morphological symmetries solo bipedal robot, State symmetry, Discrete Symmetry Group + + Morphological symmetries a1 bipedal robot, State symmetry, Discrete Symmetry Group + + Morphological symmetries hyq quadruped robot, State symmetry, Discrete Symmetry Group + + Morphological symmetries anymal-c quadruped robot, State symmetry, Discrete Symmetry Group + + Morphological symmetries anymal-b quadruped robot, State symmetry, Discrete Symmetry Group + + Morphological symmetries go1 quadruped robot, State symmetry, Discrete Symmetry Group + + Morphological symmetries b1 quadruped robot, State symmetry, Discrete Symmetry Group + + Morphological symmetries ur3 robot, State symmetry, Discrete Symmetry Group + + Morphological symmetries ur5 robot, State symmetry, Discrete Symmetry Group + + Morphological symmetries ur10 robot, State symmetry, Discrete Symmetry Group +
+ +### $\mathbb{G}=\mathbb{C}_n$: Cyclic group symmetries + + + + + + + + +
Trifinger
+ Morphological symmetries trifinger manipulator robot, State symmetry, Discrete Symmetry Group +
+ +### $\mathbb{G}=\mathbb{K}_4$: Klein-Four Symmetry + + + + + + + + + + + + + + + + +
SoloHyQMini-CheetahAnymal-CAnymal-B
+ Morphological symmetries solo bipedal robot, State symmetry, Discrete Symmetry Group + + Morphological symmetries hyq quadruped robot, State symmetry, Discrete Symmetry Group + + Morphological symmetries mini-cheetah quadruped robot, State symmetry, Discrete Symmetry Group + + Morphological symmetries anymal-c quadruped robot, State symmetry, Discrete Symmetry Group + + Morphological symmetries anymal-b quadruped robot, State symmetry, Discrete Symmetry Group +
+ +### $\mathbb{G}=\mathbb{C}_2\times\mathbb{C}_2\times\mathbb{C}_2$: Regular cube symmetry + + + + + + + + + + +
SoloMini-Cheetah
+ Morphological symmetries solo bipedal robot, State symmetry, Discrete Symmetry Group + + Morphological symmetries mini-cheetah quadruped robot, State symmetry, Discrete Symmetry Group +
### Addition of new dynamical systems to the library. -This repository aims at becoming a central tool in the exploitation of Morphological Symmetries in Robotics, Computer Graphics and Computational Biology. -Therefore, here we summarize the present and future efforts to enlarge the library of dynamical systems used in each of these fields. +If you want a new robot to be introduced in the library create an issue request indicating where to find the URDF of the system. +___________________________________________ +## Getting Started -## Tutorial ### Loading symmetric dynamical systems Each symmetric dynamical system has a configuration file in the folder `morpho_symm/cfg/supervised/robot`. To load one of these systems, simply use the function `load_symmetric_system` as follows: + ```python from morpho_symm.utils.robot_utils import load_symmetric_system -from hydra import compose, initialize - robot_name = 'solo' # or any of the robots in the library (see `/morpho_symm/cfg/robot`) -initialize(config_path="morpho_symm/cfg/supervised/robot", version_base='1.3') -robot_cfg = compose(config_name=f"{robot_name}.yaml") -# Load robot instance and its symmetry group -robot, G = load_symmetric_system(robot_cfg=robot_cfg) + +robot, G = load_symmetric_system(robot_name=robot_name) ``` The function returns: - `robot` an instance of the class [`PinBulletWrapper`](https://github.com/Danfoa/MorphoSymm/blob/devel/morpho_symm/robots/PinBulletWrapper.py) (utility class bridging [`pybullet`](https://pybullet.org/wordpress/) and [`pinocchio`](https://github.com/stack-of-tasks/pinocchio)). - `G`: the symmetry group of the system of instance [`escnn.group.Group`](https://quva-lab.github.io/escnn/api/escnn.group.html#group) -#### Getting and resetting the state of the system - -The system state is defined as $`(\mathbf{q}, \mathbf{v}) \;|\; \mathbf{q} \in \mathrm{Q}, \; \mathbf{v} \in T_{q}\mathrm{Q}`$, being $\mathrm{Q}$ the space of generalized position coordinates, and $T_{q}\mathrm{Q}$ the space of generalized velocity coordinates. Recall from the [paper convention](https://arxiv.org/abs/2302.10433) that the state configuration can be separated into base configuration and joint-space configuration $`\mathrm{Q} := \mathrm{E}_d \times \mathrm{Q}_J`$, such that $` -\mathbf{q} := -\begin{bsmallmatrix} -\mathbf{X}_B \\ \mathbf{q}_{js} -\end{bsmallmatrix} -\begin{smallmatrix} -\in \; \mathbb{E}_d \\ \in \; \mathrm{Q}_J -\end{smallmatrix} -`$. Where, $\mathrm{E}\_d$ is the Euclidean space in which the system evolves, and $`\mathrm{Q}_J`$ is the joint-space position coordinates. To access these quantities in code we do: + +### Getting and resetting the state of the system + +The system state is defined as $(\mathbf{q}, \mathbf{v}) | \mathbf{q} \in \mathcal{Q}, \mathbf{v} \in T_{q}\mathcal{Q}$, being $\mathcal{Q}$ the space of generalized position coordinates, and $\mathcal{T}_ {q}\mathcal{Q}$ the space of generalized velocity coordinates. Recall from the [paper convention](https://arxiv.org/abs/2302.10433) that the state configuration can be separated into base configuration and joint space configuration $\mathcal{Q} := \mathbb{E}_ d \times \mathcal{Q}_ {js}$. Where, $\mathbb{E}_ d$ is the Euclidean space in which the system evolves, and $\mathcal{Q}_ {js}$ is the joint space position coordinates. This enables to express every system state as $\mathbf{q} := \[\mathbf{X}_ B, \mathbf{q}_ {js}\]^ T$, where $\mathbf{X}_ B \in \mathbb{E}_ d$ and $\mathbf{q}_ {js} \in \mathcal{Q}_ {js}$. To access these quantities in code we do: ```python # Get the state of the system q, v = robot.get_state() # q ∈ Q, v ∈ TqQ # Get the robot's base configuration XB ∈ Ed as a homogenous transformation matrix. XB = robot.get_base_configuration() -# Get joint space position and velocity coordinates (q_js, v_js) | q_js ∈ QJ, dq_js ∈ TqQJ +# Get joint space position and velocity coordinates (q_js, v_js) | q_js ∈ Qjs, dq_js ∈ TqQjs q_js, v_js = robot.get_joint_space_state() ``` -### Exploiting Morphological Symmetries - - > _This section shows how to get the group representations required for doing *data augmentation* and for the construction of *equivariant neural networks*_. +### Available group representations -The system's symmetry group instance `G` contains the group representations required to transform most proprioceptive and exteroceptive measurements (e.g., joint positions/velocities/accelerations, joint forces/torques, contact locations & forces, linear and angular velocities, terrain heightmaps, depthmaps, etc). These are: - - - - $`\rho_{\mathbb{E}_d}: \mathcal{G} \rightarrow \mathbb{E}(d)`$: Representation mapping symmetry actions to elements of the Euclidean group $\mathbb{E}(d)$. Essentially homogenous transformation matrices describing a rotation/reflection and translation of space (Euclidean isometry) in $d$ dimensions. - - $`\rho_{\mathrm{Q}_J}: \mathcal{G} \rightarrow \mathcal{GL}(\mathrm{Q}_J)`$ and $`\rho_{T_q\mathrm{Q}_J}: \mathcal{G} \rightarrow \mathcal{GL}(T_q\mathrm{Q}_J)`$: Representations mapping symmetry actions to transformation matrices of joint-space position $`\mathrm{Q}_J`$ and velocity $`T_{q}\mathrm{Q}_J`$ coordinates. - - $`\rho_{\mathrm{O}_d}: \mathcal{G} \rightarrow \mathcal{\mathrm{O}_d}`$: Representation mapping symmetry actions to elements of the Orthogonal group $\mathrm{O}(d)$. Essentially rotation and reflection matrices in $d$ dimensions. - - $`\rho_{reg}: \mathcal{G} \rightarrow \mathbb{R}^{|\mathcal{G}|}`$: The group regular representation. - - $`\hat{\rho}_{i}: \mathcal{G} \rightarrow \mathcal{GL}(|\hat{\rho}_{i}|)`$: Each of the group irreducible representations. - -In practice, products and additions of these representations are enough to obtain the representations of any proprioceptive and exteroceptive measurement. For instance, we can use these representations to transform elements of: - -- $\mathrm{E}\_d$: The Euclidean space (of $d$ dimensions) in which the system evolves. +The system's symmetry group instance `G` contains the group representations required to transform most proprioceptive and exteroceptive observations (e.g., joint positions/velocities/accelerations, joint forces/torques, contact locations & forces, linear and angular velocities, terrain heightmaps, depthmaps, etc). These are: + + + - $\rho_ {\mathbb{E}_d}: \mathbb{G} \rightarrow \mathbb{E}_d$: Representation mapping symmetry actions to elements of the Euclidean group $\mathbb{E}_d$. Essentially, **homogenous transformation matrices** describing a rotation/reflection and translation of space (Euclidean isometry) in $d$ dimensions. + + - $\rho_ {\mathcal{Q}_ {js}}: \mathbb{G} \rightarrow \mathcal{GL}(\mathcal{Q}_ {js})$ and $\rho_ {\mathcal{T}_ {q}\mathcal{Q}_ {js}}: \mathbb{G} \rightarrow \mathcal{GL}(\mathcal{T}_ {q}\mathcal{Q}_ {js})$: Representations mapping symmetry actions to transformation matrices of joint space position $\mathcal{Q}_ {js}$ and velocity $T_{q}\mathcal{Q}_ {js}$ coordinates. + + - $\rho_ {\mathbb{R}^d}: \mathbb{G} \rightarrow \mathcal{GL}(\mathcal{\mathbb{R}^d})$: Representation mapping symmetry actions to elements to invertible transformations in $d$ dimensions. In practise, **rotation** and **reflection** matrices . + + - $\rho_ {reg}: \mathbb{G} \rightarrow \mathbb{R}^{|\mathbb{G}|}$: The group regular representation. + + - $\hat{\rho}_ {i}: \mathbb{G} \rightarrow \mathcal{GL}(|\hat{\rho}_ {i}|)$: Each of the group irreducible representations. + +In practice, direct products and direct sums (block-diagonal stacking) of these representations are enough to obtain the representations of any proprioceptive and exteroceptive measurements/observations. That is, these representations are sufficient to perform [data-augmentation](#data-augmentation) and to construct [equivariant/invariant neural networks](#equivariantinvariant-neural-networks). + +### Data-Augmentation + +Here, we provide some basic examples of augmenting observations frequently used in robotics, such as the configuration of a rigid-body, the joint space position, velocity, and accelerations, contact forces, joint torques, etc. To gain intiution, focus on the simplest symmetry group, the relfection group $\mathbb{C}_ 2 = \\{e, g_s\\}$, characteristic of the symmetries of a bipedal system: + + + + + +
+ Morphological Symmetries Atlas bipedal robot, State symmetry, Discrete Symmetry Group + + Morphological Symmetries Atlas bipedal robot, State symmetry, Discrete Symmetry Group +
+ +Any observation from this robot has a symmetric equivalent observation. In this tutorial we will show how to obtain these symmetric observations. You can also check the script [robot_symmetry_visualization.py](https://github.com/Danfoa/MorphoSymm/blob/devel/morpho_symm/robot_symmetry_visualization.py), where we use data-augmentation to generate the animation displayed above. This script works for any robot in the library. + +#### Observations evolving in the Euclidean group of $d$ dimensions $\mathbb{E}_d$. - The representation $`\rho_{\mathbb{E}_d}`$ can be used to transform: - - The system base configuration $\mathbf{X}\_B$. If you want to obtain the set of symmetric base configurations $` - \{{g \cdot \mathbf{X}_B:= \rho_{\mathbb{E}_d}(g) \; \mathbf{X}_B \; \rho_{\mathbb{E}_d}(g)^{-1} \;|\; \forall\; g \in \mathcal{G}}\}`$ [(1)](https://arxiv.org/abs/2302.1043), you can do it with: +The homogenous matrix describing the configuration of any rigid body, including the system's base configuration $\mathbf{X}_ B$ is an observation evolving in $\mathbb{E}_ d$. To obtain the set of symmetric base configurations, i.e. the observation group orbit: $\mathbb{G} \cdot \mathbf{X}_ B = \\{g \cdot \mathbf{X}_ B := \rho_ {\mathbb{E}_ d}(g) \\; \mathbf{X}_ B \\; \rho_ {\mathbb{E}_ d}(g)^{-1} | \forall \\; g \in \mathbb{G}\\}$ [(1)](https://danfoa.github.io/MorphoSymm/), you can do the following: ```python - rep_Ed = G.representations['Ed'] # rep_Ed(g) is a homogenous transformation matrix ∈ R^(d+1)x(d+1) - # The orbit of the base configuration XB is a map from group elements g ∈ G to base configurations g·XB ∈ Ed - orbit_X_B = {g: rep_Ed(g) @ XB @ rep_Ed(g).T for g in G.elements()} +rep_Ed = G.representations['Ed'] # rep_Ed(g) ∈ R^(d+1)x(d+1) is a homogenous transformation matrix +# The orbit of the base configuration XB is a map from group elements g ∈ G to base configurations g·XB ∈ Ed +orbit_X_B = {g: rep_Ed(g) @ XB @ rep_Ed(g).T for g in G.elements()} ``` -- - - Points in $\mathbb{R}^d$. These can represent contact locations, object/body positions, etc. To obtain the set of symmetric points you can do it with: + +Another example of an observation transfromed by $\rho_ {\mathbb{E}_ d}$ are **points** in $\mathbb{R}^d$. These can represent contact locations, object/body positions, etc. To obtain the point orbit, $\mathbb{G} \cdot \mathbf{r} = \\{g \cdot \mathbf{r} := \rho_ {\mathbb{E}_ d}(g) \\; \mathbf{r} | \forall \\; g \in \mathbb{G} \\}$, you can do: ```python - r = np.random.rand(3) # Example point in Ed, assuming d=3. - r_hom = np.concatenate((r, np.ones(1))) # Use homogenous coordinates to represent a point in Ed - # The orbit of the point is a map from group elements g ∈ G to the set of symmetric points g·r ∈ R^d - orbit_r = {g: (rep_Ed(g) @ r_hom)[:3] for g in G.elements} +r = np.random.rand(3) # Example point in Ed, assuming d=3. +r_hom = np.concatenate((r, np.ones(1))) # Use homogenous coordinates to represent a point in Ed +# The orbit of the point is a map from group elements g ∈ G to the set of symmetric points g·r ∈ R^d +orbit_r = {g: (rep_Ed(g) @ r_hom)[:3] for g in G.elements} ``` -- $`\mathrm{Q}_J`$ and $`T_{q}\mathrm{Q}_J`$: The spaces of joint-space position $`\mathrm{Q}_J`$ and velocity $`T_{q}\mathrm{Q}_J`$ generalized coordinates. +#### Observations evolving in Joint Space (e.g. joint positions, velocity, torques). - To transform joint-space states $` - (\mathbf{q}_{js}, \mathbf{v}_{js}) \;|\; \mathbf{q}_{js} \in \mathrm{Q}_J, \;\mathbf{v}_{js} \in T_{q}\mathrm{Q}_J - `$ we use the representations $`\rho_{\mathrm{Q}_J}`$ and $`\rho_{T_q\mathrm{Q}_J}`$. For instance, for a given joint-space configuration $` (\mathbf{q}_{js}, \mathbf{v}_{js})`$, the set of symmetric joint-space configurations (orbit) is given by $` - \{ - (\rho_{\mathrm{Q}_J}(g) \; \mathbf{q}_{js}, \;\rho_{T_q\mathrm{Q}_J}(g) \;\mathbf{v}_{js}) \; | \; \forall \; g \in \mathcal{G} - \} - `$. Equivalently, in code we can do: +Joint space observations usually evolve in the space of generalized position $\mathcal{Q}_ {js}$ and velocity $T_{q}\mathcal{Q}_ {js}$ generalized coordinates. These include joints position, velocity, acceleration, and generalized forces/torques. In this example we will transfrom the joint space state, composed of the joint position and velocity coordinates $(\mathbf{q}_ {js}, \mathbf{v}_ {js}) \\; | \\; \mathbf{q}_ {js} \in \mathcal{Q}_ {js}, \mathbf{v}_ {js} \in \mathcal{T}_ {q}\mathcal{Q}_ {js} +$, using the representations $\rho_ {\mathcal{Q}_ {js}}$ and $\rho_ {\mathcal{T}_ {q}\mathcal{Q}_ {js}}$. To obtain the orbit of the joint space state in code you can do: ```python - rep_QJ = G.representations['Q_js'] - rep_TqJ = G.representations['TqQ_js'] - # Get joint space position and velocity coordinates (q_js, v_js) | q_js ∈ QJ, dq_js ∈ TqQJ - q_js, v_js = robot.get_joint_space_state() - # The joint-space state orbit is a map from group elements g ∈ G to joint-space states (g·q_js, g·v_js) - orbit_js_state = {g: (rep_QJ(g) @ q_js, rep_TqJ(g) @ v_js) for g in G.elements} +rep_Qjs = G.representations['Q_js'] +rep_TqQjs = G.representations['TqQ_js'] +# Get joint space position and velocity coordinates (q_js, v_js) | q_js ∈ Qjs, dq_js ∈ TqQjs +q_js, v_js = robot.get_joint_space_state() +# The joint space state orbit is a map from group elements g ∈ G to joint space states (g·q_js, g·v_js) +orbit_js_state = {g: (rep_Qjs(g) @ q_js, rep_TqQjs(g) @ v_js) for g in G.elements} ``` -- Vectors, Pseudo-vectors in $\mathrm{E}\_d$. - Vector measurements can represent linear velocities, forces, linear accelerations, etc. While [pseudo-vectors](https://en.wikipedia.org/wiki/Pseudovector#:~:text=In%20physics%20and%20mathematics%2C%20a,of%20the%20space%20is%20changed) (or axial-vectors) can represent angular velocities, angular accelerations, etc. To obtain symmetric measurements we transform vectors with $` - \rho_{\mathrm{O}_d}`$. Likewise, to obtain symmetric pseudo-vectors we use $`\rho_{\mathrm{O}_{d,pseudo}}(g) := |\rho_{\mathrm{O}_d}(g)| \rho_{\mathrm{O}_d}(g) \; | \; g \in \mathcal{G}`$. Equivalently, in code we can do: +#### Obervations evolving $\mathbb{R}_ d$ (e.g. vectors, pseudo-vectors). + +Observations evolving in $\mathbb{R}_ d$ include contact and ground reaction forces, linear and angular velocity of rigid bodies, distance vectors to target positons/locations, etc. To tranform vectors we use the representation $\rho_ {\mathbb{R}_ {d}}$. While to transform [pseudo-vectors](https://en.wikipedia.org/wiki/Pseudovector#:~:text=In%20physics%20and%20mathematics%2C%20a,of%20the%20space%20is%20changed) (or axial-vectors) we use the representation $\rho_ {\mathbb{R}_ {d,pseudo}}$ (these can represent angular velocities, angular accelerations, etc.). To obtain the orbit of these observations you can do: ```python - rep_Rd = G.representations['Rd'] # rep_Rd(g) is an orthogonal matrix ∈ R^dxd - rep_Rd_pseudo = G.representations['Od_pseudo'] - - v = np.random.rand(3) # Example vector in Ed, assuming d=3. E.g. linear velocity of the base frame. - w = np.random.rand(3) # Example pseudo-vector in Ed, assuming d=3. E.g. angular velocity of the base frame. - # The orbit of the vector is a map from group elements g ∈ G to the set of symmetric vectors g·v ∈ R^d - orbit_v = {g: rep_Rd(g) @ v for g in G.elements} - # The orbit of the pseudo-vector is a map from group elements g ∈ G to the set of symmetric pseudo-vectors g·w ∈ R^d - orbit_w = {g: rep_Rd_pseudo(g) @ w for g in G.elements} +rep_Rd = G.representations['Rd'] # rep_Rd(g) is an orthogonal matrix ∈ R^dxd +rep_Rd_pseudo = G.representations['Rd_pseudo'] + +v = np.random.rand(3) # Example vector in R3, E.g. linear velocity of the base frame. +w = np.random.rand(3) # Example pseudo-vector in Ed, assuming d=3. E.g. angular velocity of the base frame. +# The orbit of the vector is a map from group elements g ∈ G to the set of symmetric vectors g·v ∈ R^d +orbit_v = {g: rep_Rd(g) @ v for g in G.elements} +# The orbit of the pseudo-vector is a map from group elements g ∈ G to the set of symmetric pseudo-vectors g·w ∈ R^d +orbit_w = {g: rep_Rd_pseudo(g) @ w for g in G.elements} ``` -> As an example you can check the script [robot_symmetry_visualization.py](https://github.com/Danfoa/MorphoSymm/blob/devel/morpho_symm/robot_symmetry_visualization.py), where we use the symmetry representations to generate the animations of all robot in the library (with the same script). +### Equivariant/Invariant Neural Networks + +In robotics, any data-driven application of supervised, unsupervised, and reinforcement learning involves approximating a function. This function maps a set of proprioceptive/exteroceptive observations (e.g., system state, contact state, contact points/forces) to an output set of similar observations (e.g., control position/torque setpoints, linear/angular velocity vector, energy). If the robotic system possesses a symmetry group $\mathbb{G}$, these observations will also feature a symmetry group, making the target function either [$\mathbb{G}$-equivariant or $\mathbb{G}$-invariant](https://youtu.be/03MbWVlbefM?t=1393). In this brief tutorial we will show you how to construct a $\mathbb{G}$-equivariant and a $\mathbb{G}$-invariant neural network to approximate these functions from data. -### Equivariant Neural Networks - > _In this section we briefly show how to construct G-equivariant multi-layer perceptron E-MLP architectures. Future tutorials will cover G-equivariant CNNs and GNNs._ +#### $\mathbb{G}$-equivariant$ neural network +Let's consider the example from [(1)](https://arxiv.org/abs/2302.1043) of approximating the Center of Mass (CoM) momentum from the joint space state observations. That is we want to use a neural network to approximate the function $\mathbf{y} = f(\mathbf{x}) = f(\mathbf{q}_ {js}, \mathbf{v}_ {js})$ for a robot evolving in 3 dimensions, say the robot `solo`. Defining $\mathbf{y} := [\mathbf{l}, \mathbf{k}]^T \subseteq \mathbb{R}^6$ as the CoM momentum linear $\mathbf{l} \in \mathbb{R}^3$ and angular $\mathbf{k} \in \mathbb{R}^3$ momentum, and $\mathbf{x} = (\mathbf{q}_ {js}, \mathbf{v}_ {js}) \\; | \\; \mathbf{q}_ {js} \in \mathcal{Q}_ {js}, \mathbf{v}_ {js} \in \mathcal{T}_ {q}\mathcal{Q}_ {js}$ as the joint space position and velocity generalized coordinates. -Let's consider the example from [(1)](https://arxiv.org/abs/2302.1043) of approximating the Center of Mass (CoM) momentum from the joint-space state measurements. That is we want to use a neural network to approximate the function $` -\mathbf{y} = f(\mathbf{x}) = f(\mathbf{q}_{js}, \mathbf{v}_{js}) -`$ for a robot evolving in 3 dimensions, say the robot `solo`. Defining $`\mathbf{y} := [\mathbf{l}, \mathbf{k}]^T \subseteq \mathbb{R}^6`$ as the CoM momentum linear $`\mathbf{l} \in \mathbb{R}^3`$ and angular $`\mathbf{k} \in \mathbb{R}^3`$ momentum, and $` -\mathbf{x} = (\mathbf{q}_{js}, \mathbf{v}_{js}) \;|\; \mathbf{q}_{js} \in \mathrm{Q}_J, \;\mathbf{v}_{js} \in T_{q}\mathrm{Q}_J -`$ as the joint-space position and velocity generalized coordinates. + + + + + +
+ Morphological symmetries solo bipedal robot, State symmetry, Discrete Symmetry Group + + Morphological symmetries mini-cheetah quadruped robot, State symmetry, Discrete Symmetry Group +
-For this example, you can build an equivariant MLP as follows: +To construct a G-equivariant architecture you need to: +1. Identify the representations of the observations in your input and output spaces. +2. Define the input and output [`FieldType`](https://quva-lab.github.io/escnn/api/escnn.nn.html#field-type) instances using the representations of each geometric object. +3. Instanciate a [EMLP](https://github.com/Danfoa/MorphoSymm/blob/fc42a19654a9385b1037d1a2678aa95829a47a06/morpho_symm/nn/EMLP.py#L14) architecture with the input and output `FieldType` instances. This class handles the appropiate parametrization of the hidden layer group representations and activation functions. ```python import numpy as np @@ -191,54 +433,41 @@ from morpho_symm.nn.EMLP import EMLP from morpho_symm.utils.robot_utils import load_symmetric_system # Load robot instance and its symmetry group -initialize(config_path="morpho_symm/cfg/supervised/robot", version_base='1.3') -robot_name = 'solo' # or any of the robots in the library (see `/morpho_symm/cfg/robot`) -robot, G = load_symmetric_system(robot_name=robot_name) +robot, G = load_symmetric_system(robot_name='solo-k4') # We use ESCNN to handle the group/representation-theoretic concepts and for the construction of equivariant neural networks. gspace = escnn.gspaces.no_base_space(G) # Get the relevant group representations. -rep_QJ = G.representations["Q_js"] # Used to transform joint-space position coordinates q_js ∈ Q_js -rep_TqQJ = G.representations["TqQ_js"] # Used to transform joint-space velocity coordinates v_js ∈ TqQ_js -rep_R3 = G.representations["Od"] # Used to transform the linear momentum l ∈ R3 -rep_R3_pseudo = G.representations["Od_pseudo"] # Used to transform the angular momentum k ∈ R3 -rep_E3 = G.representations["Ed"] # Homogenous transformation matrix - -x = np.random.rand(3) # -g = G.sample() -g_x = rep_EG(g) @ np.concatenate([x, np.ones(1)]) # Transform x with the representation of the Euclidean group - -e, gs, gr, gt = G.elements -A = rep_R3(gs) # 3x3 matrix. +rep_Qjs = G.representations["Q_js"] # Used to transform joint space position coordinates q_js ∈ Q_js +rep_TqQjs = G.representations["TqQ_js"] # Used to transform joint space velocity coordinates v_js ∈ TqQ_js +rep_R3 = G.representations["Rd"] # Used to transform the linear momentum l ∈ R3 +rep_R3_pseudo = G.representations["Rd_pseudo"] # Used to transform the angular momentum k ∈ R3 + # Define the input and output FieldTypes using the representations of each geometric object. -# Representation of x := [q, v, base_vel, base_ang_vel] ∈ Q_js x TqQ_js x R3 x R3 => ρ_X_js(g) := ρ_Q_js(g) ⊕ ρ_TqQ_js(g) | g ∈ G -in_field_type = FieldType(gspace, [rep_QJ, rep_TqQJ, rep_R3, rep_R3_pseudo]) -# Representation of y := [l, k] ∈ R3 x R3 => ρ_Y_js(g) := ρ_O3(g) ⊕ ρ_O3pseudo(g) | g ∈ G +# Representation of x := [q_js, v_js] ∈ Q_js x TqQ_js => ρ_X_js(g) := ρ_Q_js(g) ⊕ ρ_TqQ_js(g) | g ∈ G +in_field_type = FieldType(gspace, [rep_Qjs, rep_TqQjs]) +# Representation of y := [l, k] ∈ R3 x R3 => ρ_Y_js(g) := ρ_R3(g) ⊕ ρ_R3pseudo(g) | g ∈ G out_field_type = FieldType(gspace, [rep_R3, rep_R3_pseudo]) -out_field_type = FieldType(gspace, [G.trivial_representation]) # Construct the equivariant MLP model = EMLP(in_type=in_field_type, out_type=out_field_type, - num_layers=5, # Input layer + 3 hidden layers + output/head layer - num_hidden_units=128, # Number of hidden units per layer + num_layers=5, # Input layer + 3 hidden layers + output/head layer + num_hidden_units=128, # Number of hidden units per layer activation=escnn.nn.ReLU, # Activarions must be `EquivariantModules` instances - with_bias=True # Use bias in the linear layers + with_bias=True # Use bias in the linear layers ) print(f"Here is your equivariant MLP \n {model}") ``` -____________________________________ -In summary, to construct a G-equivariant architecture you need to: -1. Identify the types of geometric objects in your input and output spaces. -2. Identify the representations of each geometric object. -3. Define the input and output `FieldType` instances using the representations of each geometric object. ## How to cite us? -If you find this repository or the [paper](https://scholar.google.it/scholar?q=on+discrete+symmetries+of+robotic+systems:+a+data-driven+and+group-theoretic+analysis&hl=en&as_sdt=0&as_vis=1&oi=scholart) useful, please cite us as: +If you find this repository or any our our papers relevant please cite us as: + +### [On discrete symmetries of robotics systems: A group-theoretic and data-driven analysis](https://danfoa.github.io/MorphoSymm/) ``` @INPROCEEDINGS{Ordonez-Apraez-RSS-23, - AUTHOR = {Daniel F Ordonez-Apraez AND Martin, Mario AND Antonio Agudo AND Francesc Moreno}, + AUTHOR = {Daniel F Ordo{\~n}ez-Apraez AND Martin, Mario AND Antonio Agudo AND Francesc Moreno}, TITLE = {{On discrete symmetries of robotics systems: A group-theoretic and data-driven analysis}}, BOOKTITLE = {Proceedings of Robotics: Science and Systems}, YEAR = {2023}, @@ -247,6 +476,16 @@ If you find this repository or the [paper](https://scholar.google.it/scholar?q=o DOI = {10.15607/RSS.2023.XIX.053} } ``` +### [Dynamics Harmonic Analysis of Robotic Systems: Application in Data-Driven Koopman Modeling](https://danfoa.github.io/DynamicsHarmonicsAnalysis/) + +``` +@article{ordonez2023dynamics, + title={Dynamics Harmonic Analysis of Robotic Systems: Application in Data-Driven Koopman Modelling}, + author={Ordo{\~n}ez-Apraez, Daniel and Kostic, Vladimir and Turrisi, Giulio and Novelli, Pietro and Mastalli, Carlos and Semini, Claudio and Pontil, Massimiliano}, + journal={arXiv preprint arXiv:2312.07457}, + year={2023} +} +``` ## Contributing @@ -260,26 +499,13 @@ It utilizes the [robot_descriptions.py](https://github.com/robot-descriptions/ro third-party robotics packages. This package provides a convenient interface for loading URDF files into GUI visualization tools, robot dynamics packages (such as Pinocchio), and physics simulators. To add a new robotic system to our library 1. The system URDF must be contributed to robot_descriptions.py. -2. The corresponding robot configuration file should be added to `cfg/supervised/robot/` describing the system' symmetry group and joint-space representation generators, should also be added. +2. The corresponding robot configuration file should be added to `cfg/supervised/robot/` describing the system' symmetry group and joint space representation generators, should also be added. In summary, we support: - [x] Loading of URDF files in `pybullet` and `pinocchio` through `robot_descriptions.py` - [x] Visualization of robot Discrete Morphological Symmetries in `pybullet`. Other physics simulators and visualization tools will come soon. -- [x] Utility functions to define symmetry representations of proprioceptive and exteroceptive measurements. -- [x] Construction of equivariant neural networks processing proprioceptive and exteroceptive measurements, using the `escnn` library. - -#### Computer Graphics - -The field of computer graphics does not widely employs URDF descriptions for the definition of dynamical systems. Although covering different description standards is within the goal of this repository, -for now, our main objective is: - -- [ ] Integration of [STAR](https://star.is.tue.mpg.de/) model in the library, to automatically process sequence of data and obtain symmetric sequences. - By defining the sagittal symmetry of all the model parameters. This will enable the use of DMSs in all applications of human motion prediction, shape reconstruction, etc. - If you are interested in contributing to this effort, please contact us. -- [ ] Integration of Motion Capture (MoCap) data formats. Including `.fbx`, `.bvh`, and `.c3d`. - -#### Computational Biology +- [x] Utility functions to define symmetry representations of proprioceptive and exteroceptive observations. +- [x] Construction of equivariant neural networks processing proprioceptive and exteroceptive observations, using the `escnn` library. +- [x] Use abstract harmonic analysis to decompose motion trajectories and recorded proprioceptive and exteroceptive observations into isotypic components. -For computational biology and bio-mechanics, we believe the most relevant format to provide support for is: -- [ ] Coordinate 3D files `.c3d` format. diff --git a/docs/index.html b/docs/index.html index e526b1c..6bfbc76 100644 --- a/docs/index.html +++ b/docs/index.html @@ -19,10 +19,10 @@ + content="On discrete symmetries of robotic systems: A data-driven and group-theoretic analysis, Robotics Science and Systems RSS-2023. This project studies Morphological Symmetries in Robotic Systems, these are Discrete (or finite) symmetry groups of the state space of a robotic and dynamical systems These symmetries are usefull for data-augmentation and equivariant function approximation. Morphological Symmetries (MorphoSymm, Morpho Symm, Morphosymm)"> + content="On discrete symmetries of robotic systems: A data-driven and group-theoretic analysis, Robotics Science and Systems RSS-2023. This project studies Morphological Symmetries in Robotic Systems, these are Discrete (or finite) symmetry groups of the state space of a robotic and dynamical systems These symmetries are usefull for data-augmentation and equivariant function approximation. Morphological Symmetries (MorphoSymm, Morpho Symm, Morphosymm)"/> @@ -32,7 +32,7 @@ + content="On discrete symmetries of robotic systems: A data-driven and group-theoretic analysis, Robotics Science and Systems RSS-2023. This project studies Morphological Symmetries in Robotic Systems, these are Discrete (or finite) symmetry groups of the state space of a robotic and dynamical systems These symmetries are usefull for data-augmentation and equivariant function approximation. Morphological Symmetries (MorphoSymm, Morpho Symm, Morphosymm)"> @@ -40,7 +40,7 @@ + content="discrete morphological symmetries, robotics, computer graphics, symmetries, equivariance, morphosymm, MorphoSymm, morpho symm, dynamics harmonics analysis, harmonic analysis, locomotion, control, equivariant neural networks, invariant neural networks, group theory"> @@ -200,8 +200,9 @@

Symmetries of robotic systems

- - Morphological Symmetries Mini-Cheetah quadruped robot, Caley Diagram, Discrete Symmetry Group + Morphological Symmetries Mini-Cheetah quadruped robot, State symmetry, Discrete Symmetry Group
@@ -213,7 +214,7 @@

Symmetries of robotic systems

@@ -257,8 +258,10 @@

Caley Diagrams of Morphological Symmetries