feat(CI): improve readme and code coverage

[Meow404]

changes:

• Add github action to report coverage
• Improve coverage for C3 (atleast 90%)
• Add code formatting check
• Updated Readmes
• Succesfully build and install python wheel

This change is 

[xuanhien070594]

core/c3.h line 203 at r2 (raw file):

   * algorithm for the first trajectory (index 0).
   */
  std::vector<Eigen::VectorXd> GetWarmStartDelta() const {

Do we use those function any where? If not, we can just remove them.

[xuanhien070594]

core/c3.h line 213 at r2 (raw file):

   * algorithm for the first trajectory (index 0).
   */
  std::vector<Eigen::VectorXd> GetWarmStartBinary() const {

Same comment as above.

[xuanhien070594]

examples/c3_example.cc line 187 at r2 (raw file):

    /// print the state
    // std::cout << "state: " << x[i + 1] << std::endl;

We can add a flag verbose for printing out state.

[xuanhien070594]

bindings/pyc3/README.md line 38 at r2 (raw file):

```sh
bazel build //bindings/pyc3:pyc3_wheel

I have a question: will bazel use the python version in the virtual environment or the system python?

[xuanhien070594]

bindings/pyc3/README.md line 65 at r2 (raw file):

Complete examples are available in the `examples/python/` directory:

- **`c3_example.py`**: Basic C3 optimization example

Suggestion:

- **`c3_example.py`**: Basic C3 with toy examples

[xuanhien070594]

README.md line 5 at r2 (raw file):

[**Main Paper: Consensus Complementarity Control**](https://arxiv.org/abs/2304.11259)

<p align="center">

Nit. Can we make the build status and coverage status on the same line?

[xuanhien070594]

README.md line 22 at r2 (raw file):

```shell
cd c3
bazel build ...

Can we add instruction to install bazel or bazelisk

[xuanhien070594]

README.md line 120 at r2 (raw file):

├── multibody/         # algorithms for computation in multibody environments
├── third_party/       # External dependencies
└── WORKSPACE          # Bazel workspace file

We don't have WORKSPACE file.

[xuanhien070594]

README.md line 126 at r2 (raw file):

## Reference

For a detailed explanation of the C3 algorithm, please refer to the [main paper](https://arxiv.org/abs/2304.11259). Additional resources and in-depth examples can be found in the [dairlib repository](https://github.com/DAIRLab/dairlib).

We might consider to add a Citing C3 section where we'll put the BibTeX entry for C3 paper.

[xuanhien070594]

examples/README.md line 3 at r2 (raw file):

# C3 Standalone Example
This example (c3_example.cc) demonstrates how to set up and solve C3 problems for different systems (cartpole, finger gaiting, and pivoting) using only Eigen and the core C3 classes, without Drake dependencies. It is useful for understanding the core algorithm and for running C3 on custom LCS models.

Core C3 is actually using Drake dependencies (e.g MathematicalProgram). We can say in those examples, we derive analytical dynamics equations instead of involving Drake's MultibodyPlant.

[xuanhien070594]

examples/README.md line 40 at r2 (raw file):

# LCS Factory System Example

The cartpole with softwalls problem is modeled as a Linear Complementarity System (LCS) generated by the LCS factory. This LCS forms the basis for running the C3 controller, which stabilizes the cartpole while tracking predefined state trajectories.

We can add more a little bit more details such as: The LCS model provides a piecewise-affine approximation of the nonlinear dynamics, generated using the LCS factory.

[xuanhien070594]

examples/README.md line 64 at r2 (raw file):

#### Cube Pivoting
```sh
# Run the example

We don't need comment here.

[xuanhien070594]

examples/README.md line 53 at r2 (raw file):

### Build the example
```sh
# Build the cartpole LCS factory system example

We also don't need comment here.

[xuanhien070594]

examples/README.md line 75 at r2 (raw file):

C3 Standalone Example (Python)
```sh
# Build and run the Python C3 standalone example

Nit. I think the comments seem to be redundant.

[xuanhien070594]

core/test/core_test.cc line 358 at r2 (raw file):

  ASSERT_TRUE(delta.size() > 0);

  auto binary = optimizer.GetWarmStartBinary();

Are we using these functions GetWarmStartDelta or GetWarmStartBinary any where? If not, I think we can remove them.

[xuanhien070594]

systems/test/systems_test.cc line 328 at r2 (raw file):

}

TEST_F(LCSFactorySystemTest, OutputContactJacobianIsValid) {

Note that after merging #10, this test is no longer valid. We'll need a new test.

[xuanhien070594]

Only have some minor comments. Please note that some of the unit tests for LCSFactorySystem and LCSFactory will be obsolete after merging #10

Reviewed 13 of 22 files at r1, 45 of 45 files at r2, all commit messages.
Reviewable status: all files reviewed, 17 unresolved discussions (waiting on @Meow404)

[Meow404]

Only have some minor comments. Please note that some of the unit tests for LCSFactorySystem and LCSFactory will be obsolete after merging #10

Reviewed 13 of 22 files at r1, 45 of 45 files at r2, all commit messages.
Reviewable status: all files reviewed, 17 unresolved discussions (waiting on @Meow404)

Yeah, I'd want to try and merge this PR first and then add/fix unit tests in #10 after a rebase.

[Meow404]

Reviewable status: all files reviewed, 15 unresolved discussions (waiting on @xuanhien070594)

---

README.md line 22 at r2 (raw file):

Previously, xuanhien070594 (Hien Bui) wrote…

Can we add instruction to install bazel or bazelisk

Done.

---

README.md line 120 at r2 (raw file):

Previously, xuanhien070594 (Hien Bui) wrote…

We don't have WORKSPACE file.

Done.

---

README.md line 126 at r2 (raw file):

Previously, xuanhien070594 (Hien Bui) wrote…

We might consider to add a Citing C3 section where we'll put the BibTeX entry for C3 paper.

Done.

---

bindings/pyc3/README.md line 38 at r2 (raw file):

Previously, xuanhien070594 (Hien Bui) wrote…

I have a question: will bazel use the python version in the virtual environment or the system python?

My understanding is it will use the version specified in the .bazelrc file

---

core/c3.h line 203 at r2 (raw file):

Previously, xuanhien070594 (Hien Bui) wrote…

Do we use those function any where? If not, we can just remove them.

Done.

---

core/c3.h line 213 at r2 (raw file):

Previously, xuanhien070594 (Hien Bui) wrote…

Same comment as above.

Done.

---

examples/c3_example.cc line 187 at r2 (raw file):

Previously, xuanhien070594 (Hien Bui) wrote…

We can add a flag verbose for printing out state.

Done.

---

examples/README.md line 3 at r2 (raw file):

Previously, xuanhien070594 (Hien Bui) wrote…

Core C3 is actually using Drake dependencies (e.g MathematicalProgram). We can say in those examples, we derive analytical dynamics equations instead of involving Drake's MultibodyPlant.

Done.

---

examples/README.md line 40 at r2 (raw file):

Previously, xuanhien070594 (Hien Bui) wrote…

We can add more a little bit more details such as: The LCS model provides a piecewise-affine approximation of the nonlinear dynamics, generated using the LCS factory.

Done.

---

examples/README.md line 53 at r2 (raw file):

Previously, xuanhien070594 (Hien Bui) wrote…

We also don't need comment here.

Done.

---

examples/README.md line 64 at r2 (raw file):

Previously, xuanhien070594 (Hien Bui) wrote…

We don't need comment here.

Done.

---

systems/lcs_factory_system.cc line 3 at r2 (raw file):

Previously, xuanhien070594 (Hien Bui) wrote…

We don't need iostream right?

Done.

---

systems/test/systems_test.cc line 328 at r2 (raw file):

Previously, xuanhien070594 (Hien Bui) wrote…

Note that after merging #10, this test is no longer valid. We'll need a new test.

Hoping to merge this PR first, then resolve the dependencies and coverage in #10

---

core/test/core_test.cc line 358 at r2 (raw file):

Previously, xuanhien070594 (Hien Bui) wrote…

Are we using these functions GetWarmStartDelta or GetWarmStartBinary any where? If not, I think we can remove them.

Done.

---

bindings/pyc3/README.md line 65 at r2 (raw file):

Complete examples are available in the `examples/python/` directory:

- **`c3_example.py`**: Basic C3 optimization example

Done.

[xuanhien070594]

README.md line 154 at r3 (raw file):

      primaryClass={cs.RO},
      url={https://arxiv.org/abs/2304.11259}, 
}

Suggestion:

@article{Aydinoglu2024,
  title = {Consensus Complementarity Control for Multi-Contact MPC},
  author = {Aydinoglu, Alp and Wei, Adam and Huang, Wei-Cheng and Posa, Michael},
  year = {2024},
  month = jul,
  journal = {IEEE Transactions on Robotics (TRO)},
  youtube = {L57Jz3dPwO8},
  arxiv = {2304.11259},
  doi = {10.1109/TRO.2024.3435423},
  url = {https://ieeexplore.ieee.org/document/10614849}
}

[xuanhien070594]

Only need to change the citation BibTeX of C3 paper.

Reviewed 9 of 9 files at r3, all commit messages.
Reviewable status: all files reviewed, 1 unresolved discussion (waiting on @Meow404)

[xuanhien070594]

Reviewed 2 of 2 files at r4, all commit messages.
Reviewable status: all files reviewed, 1 unresolved discussion (waiting on @Meow404)