Add new RBDA to compute the doubly-left full Jacobian derivative

src/jaxsim/rbda/__init__.py

@@ -2,6 +2,10 @@
 from .collidable_points import collidable_points_pos_vel
 from .crba import crba
 from .forward_kinematics import forward_kinematics, forward_kinematics_model
-from .jacobian import jacobian, jacobian_full_doubly_left
+from .jacobian import (
+    jacobian,
+    jacobian_derivative_full_doubly_left,
+    jacobian_full_doubly_left,
+)
 from .rnea import rnea
 from .soft_contacts import SoftContacts, SoftContactsParams

src/jaxsim/rbda/jacobian.py

@@ -4,7 +4,7 @@
 
 import jaxsim.api as js
 import jaxsim.typing as jtp
-from jaxsim.math import Adjoint
+from jaxsim.math import Adjoint, Cross
 
 from . import utils
 
@@ -199,3 +199,120 @@ def compute_full_jacobian(
     B_J_full_WL_B = J.squeeze().astype(float)
 
     return B_J_full_WL_B, B_H_L
+
+
+def jacobian_derivative_full_doubly_left(
+    model: js.model.JaxSimModel,
+    *,
+    joint_positions: jtp.VectorLike,
+    joint_velocities: jtp.VectorLike,
+) -> tuple[jtp.Matrix, jtp.Array]:
+    r"""
+    Compute the derivative of the doubly-left full free-floating Jacobian of a model.
+
+    The derivative of the full Jacobian is a 6x(6+n) matrix with all the columns filled.
+    It is useful to run the algorithm once, and then extract the link Jacobian
+    derivative by filtering the columns of the full Jacobian using the support
+    parent array :math:`\kappa(i)` of the link.
+
+    Args:
+        model: The model to consider.
+        joint_positions: The positions of the joints.
+        joint_velocities: The velocities of the joints.
+
+    Returns:
+        The derivative of the doubly-left full free-floating Jacobian of a model.
+    """
+
+    _, _, s, _, ṡ, _, _, _, _, _ = utils.process_inputs(
+        model=model, joint_positions=joint_positions, joint_velocities=joint_velocities
+    )
+
+    # Get the parent array λ(i).
+    # Note: λ(0) must not be used, it's initialized to -1.
+    λ = model.kin_dyn_parameters.parent_array
+
+    # Compute the parent-to-child adjoints and the motion subspaces of the joints.
+    # These transforms define the relative kinematics of the entire model, including
+    # the base transform for both floating-base and fixed-base models.
+    i_X_λi, S = model.kin_dyn_parameters.joint_transforms_and_motion_subspaces(
+        joint_positions=s, base_transform=jnp.eye(4)
+    )
+
+    # Allocate the buffer of 6D transform base -> link.
+    B_X_i = jnp.zeros(shape=(model.number_of_links(), 6, 6))
+    B_X_i = B_X_i.at[0].set(jnp.eye(6))
+
+    # Allocate the buffer of 6D transform derivatives base -> link.
+    B_Ẋ_i = jnp.zeros(shape=(model.number_of_links(), 6, 6))
+
+    # Allocate the buffer of the 6D link velocity in body-fixed representation.
+    B_v_Bi = jnp.zeros(shape=(model.number_of_links(), 6))
+
+    # Helper to compute the time derivative of the adjoint matrix.
+    def A_Ẋ_B(A_X_B: jtp.Matrix, B_v_AB: jtp.Vector) -> jtp.Matrix:
+        return A_X_B @ Cross.vx(B_v_AB).squeeze()
+
+    # ============================================
+    # Compute doubly-left full Jacobian derivative
+    # ============================================
+
+    # Allocate the Jacobian matrix.
+    J̇ = jnp.zeros(shape=(6, 6 + model.dofs()))
+
+    ComputeFullJacobianDerivativeCarry = tuple[
+        jtp.MatrixJax, jtp.MatrixJax, jtp.MatrixJax, jtp.MatrixJax
+    ]
+
+    compute_full_jacobian_derivative_carry: ComputeFullJacobianDerivativeCarry = (
+        B_v_Bi,
+        B_X_i,
+        B_Ẋ_i,
+        J̇,
+    )
+
+    def compute_full_jacobian_derivative(
+        carry: ComputeFullJacobianDerivativeCarry, i: jtp.Int
+    ) -> tuple[ComputeFullJacobianDerivativeCarry, None]:
+
+        ii = i - 1
+        B_v_Bi, B_X_i, B_Ẋ_i, J̇ = carry
+
+        # Compute the base (0) to link (i) adjoint matrix.
+        B_Xi_i = B_X_i[λ[i]] @ Adjoint.inverse(i_X_λi[i])
+        B_X_i = B_X_i.at[i].set(B_Xi_i)
+
+        # Compute the body-fixed velocity of the link.
+        B_vi_Bi = B_v_Bi[λ[i]] + B_X_i[i] @ S[i].squeeze() * ṡ[ii]
+        B_v_Bi = B_v_Bi.at[i].set(B_vi_Bi)
+
+        # Compute the base (0) to link (i) adjoint matrix derivative.
+        i_Xi_B = Adjoint.inverse(B_Xi_i)
+        B_Ẋi_i = A_Ẋ_B(A_X_B=B_Xi_i, B_v_AB=i_Xi_B @ B_vi_Bi)
+        B_Ẋ_i = B_Ẋ_i.at[i].set(B_Ẋi_i)
+
+        # Compute the ii-th column of the B_Ṡ_BL(s) matrix.
+        B_Ṡii_BL = B_Ẋ_i[i] @ S[i]
+        J̇ = J̇.at[0:6, 6 + ii].set(B_Ṡii_BL.squeeze())
+
+        return (B_v_Bi, B_X_i, B_Ẋ_i, J̇), None
+
+    (_, B_X_i, B_Ẋ_i, J̇), _ = (
+        jax.lax.scan(
+            f=compute_full_jacobian_derivative,
+            init=compute_full_jacobian_derivative_carry,
+            xs=model.kin_dyn_parameters.link_indices[1:],
+        )
+        if model.number_of_links() > 1
+        else [(_, B_X_i, B_Ẋ_i, J̇), None]
+    )
+
+    # Convert adjoints to SE(3) transforms.
+    # Returning them here prevents calling FK in case the output representation
+    # of the Jacobian needs to be changed.
+    B_H_L = jax.vmap(lambda B_X_L: Adjoint.to_transform(B_X_L))(B_X_i)
+
+    # Adjust shape of doubly-left free-floating full Jacobian derivative.
+    B_J̇_full_WL_B = J̇.squeeze().astype(float)
+
+    return B_J̇_full_WL_B, B_H_L