Compute the quaternion derivative by using the Hamilton product

[flferretti]

This pull request includes several changes to the quaternion derivative calculations. The primary focus is on computing it by using the Hamilton product of quaternions, which allowed removing the omega_in_body_fixed parameter

Refactoring and simplification:

• src/jaxsim/math/quaternion.py: Removed the omega_in_body_fixed parameter and refactored the Quaternion.derivative method to simplify the quaternion multiplication process. This includes the removal of the conditional logic for frame representation and the introduction of a more efficient computation using Einstein summation. [1] [2]

Associated method calls updated:

• src/jaxsim/api/integrators.py: Removed the omega_in_body_fixed parameter from the call to Quaternion.derivative in the semi_implicit_euler_integration function.
• src/jaxsim/api/ode.py: Removed the omega_in_body_fixed parameter from the call to Quaternion.derivative in the system_position_dynamics function.
• tests/test_api_frame.py: Removed the omega_in_body_fixed parameter from the call to Quaternion.derivative in the compute_q̇ function.
• tests/test_api_link.py: Removed the omega_in_body_fixed parameter from the call to Quaternion.derivative in the compute_q̇ function.
• tests/test_api_model.py: Removed the omega_in_body_fixed parameter from the call to Quaternion.derivative in the compute_q̇ function.

[github-actions]

Benchmark

Benchmark suite	Current: ee47611	Previous: 8d5c896	Ratio
tests/test_benchmark.py::test_forward_dynamics_aba[1]	34.13918586184124 iter/sec (stddev: 0.00025844325456496147)	33.99270710150131 iter/sec (stddev: 0.00009160345295732999)	1.00
tests/test_benchmark.py::test_forward_dynamics_aba[128]	15.42665795795003 iter/sec (stddev: 0.00026084749533837517)	15.450388973393041 iter/sec (stddev: 0.00039276707628969594)	1.00
tests/test_benchmark.py::test_free_floating_bias_forces[1]	28.12431323331991 iter/sec (stddev: 0.0002876628795958804)	28.072185819254948 iter/sec (stddev: 0.0001405275314719085)	1.00
tests/test_benchmark.py::test_free_floating_bias_forces[128]	10.63214596055781 iter/sec (stddev: 0.0009786770011172435)	10.661461909768686 iter/sec (stddev: 0.0003774590436268307)	1.00
tests/test_benchmark.py::test_forward_kinematics[1]	81.58479573356809 iter/sec (stddev: 0.00011051067747936885)	80.47867539657375 iter/sec (stddev: 0.0008406595365800215)	0.99
tests/test_benchmark.py::test_forward_kinematics[128]	23.158854049680517 iter/sec (stddev: 0.00023411841080499604)	22.963981536068786 iter/sec (stddev: 0.00018004443135162003)	0.99
tests/test_benchmark.py::test_free_floating_mass_matrix[1]	41.813242189874465 iter/sec (stddev: 0.00014013725831552208)	41.68963883859595 iter/sec (stddev: 0.00016661815474607527)	1.00
tests/test_benchmark.py::test_free_floating_mass_matrix[128]	40.856369708906506 iter/sec (stddev: 0.00038055938203840995)	41.30544074968093 iter/sec (stddev: 0.00006869051943032785)	1.01
tests/test_benchmark.py::test_free_floating_jacobian[1]	54.47142828422767 iter/sec (stddev: 0.000210504736766508)	54.710482035025365 iter/sec (stddev: 0.0000771513049637593)	1.00
tests/test_benchmark.py::test_free_floating_jacobian[128]	54.17190886799261 iter/sec (stddev: 0.000691060852744495)	54.13600959838642 iter/sec (stddev: 0.0002066391631624143)	1.00
tests/test_benchmark.py::test_free_floating_jacobian_derivative[1]	32.47999356675131 iter/sec (stddev: 0.00031821527188359165)	32.31611675089207 iter/sec (stddev: 0.0004465427329110755)	0.99
tests/test_benchmark.py::test_free_floating_jacobian_derivative[128]	32.70290558462924 iter/sec (stddev: 0.00018015314776348158)	32.8020875749547 iter/sec (stddev: 0.00027199927237599305)	1.00
tests/test_benchmark.py::test_soft_contact_model[1]	30.537867571311576 iter/sec (stddev: 0.0002900156302158126)	30.416550779490947 iter/sec (stddev: 0.00009594224392172117)	1.00
tests/test_benchmark.py::test_soft_contact_model[128]	14.221300968596557 iter/sec (stddev: 0.0006225228895490654)	14.145201786398033 iter/sec (stddev: 0.0005232496065842119)	0.99
tests/test_benchmark.py::test_rigid_contact_model[1]	6.325169073509058 iter/sec (stddev: 0.0017415660424438904)	6.438860629793804 iter/sec (stddev: 0.00028284443651919447)	1.02
tests/test_benchmark.py::test_rigid_contact_model[128]	0.8389991707911736 iter/sec (stddev: 0.003516428975757313)	0.8450260471582455 iter/sec (stddev: 0.0007584783160614406)	1.01
tests/test_benchmark.py::test_relaxed_rigid_contact_model[1]	5.878850818731636 iter/sec (stddev: 0.002103425649990964)	5.901695040090421 iter/sec (stddev: 0.0008691354003239028)	1.00
tests/test_benchmark.py::test_relaxed_rigid_contact_model[128]	3.34933903431996 iter/sec (stddev: 0.0005094982934624128)	3.3393481643233303 iter/sec (stddev: 0.0005133033064496388)	1.00
tests/test_benchmark.py::test_simulation_step[1]	4.819335835168546 iter/sec (stddev: 0.0007924232957427523)	4.792670117574774 iter/sec (stddev: 0.00041546735946859745)	0.99
tests/test_benchmark.py::test_simulation_step[128]	2.6202375003404446 iter/sec (stddev: 0.0013637367458051838)	2.6082535675050296 iter/sec (stddev: 0.0013605955461495007)	1.00

This comment was automatically generated by workflow using github-action-benchmark.

[CarlottaSartore]

This PR changes the API which should be used only internally by jaxsim, indeed from

 jaxsim.math.Quaternion.derivative(
            quaternion=data.base_orientation,
            omega=W_ω_WB,
            omega_in_body_fixed=False,
        )

We pass to

 jaxsim.math.Quaternion.derivative(
            quaternion=data.base_orientation,
            omega=W_ω_WB,
        )
        

Let me allert @ami-iit/darwin

[flferretti]

Friendly ping @ami-iit/darwin

[xela-95]

Thanks @flferretti ! Just a question: I didn't understood if with this PR we made the choice of computing the quaternion derivative using only angular velocities only in body-fixed representation, since you dropped the conditional to handle the omega in inertial case (that is also the case for mixed representation of 6D velocities).
Did this change has the goal of better support for differentiability of performance increase?