import jax
import jax.numpy as jnp
import jaxsim.api as js
import jaxsim.typing as jtp
from . import utils
[docs]
def crba(model: js.model.JaxSimModel, *, joint_positions: jtp.Vector) -> jtp.Matrix:
"""
Compute the free-floating mass matrix using the Composite Rigid-Body Algorithm (CRBA).
Args:
model: The model to consider.
joint_positions: The positions of the joints.
Returns:
The free-floating mass matrix of the model in body-fixed representation.
"""
_, _, s, _, _, _, _, _, _, _ = utils.process_inputs(
model=model, joint_positions=joint_positions
)
# Get the 6D spatial inertia matrices of all links.
Mc = js.model.link_spatial_inertia_matrices(model=model)
# 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 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 = model.kin_dyn_parameters.joint_transforms(
joint_positions=s, base_transform=jnp.eye(4)
)
# Extract the joint motion subspaces.
S = model.kin_dyn_parameters.motion_subspaces
# Allocate the buffer of transforms link -> base.
i_X_0 = jnp.zeros(shape=(model.number_of_links(), 6, 6))
i_X_0 = i_X_0.at[0].set(jnp.eye(6))
# ====================
# Propagate kinematics
# ====================
ForwardPassCarry = tuple[jtp.Matrix]
forward_pass_carry: ForwardPassCarry = (i_X_0,)
def propagate_kinematics(
carry: ForwardPassCarry, i: jtp.Int
) -> tuple[ForwardPassCarry, None]:
(i_X_0,) = carry
i_X_0_i = i_X_λi[i] @ i_X_0[λ[i]]
i_X_0 = i_X_0.at[i].set(i_X_0_i)
return (i_X_0,), None
(i_X_0,), _ = (
jax.lax.scan(
f=propagate_kinematics,
init=forward_pass_carry,
xs=jnp.arange(start=1, stop=model.number_of_links()),
)
if model.number_of_links() > 1
else [(i_X_0,), None]
)
# ===================
# Compute mass matrix
# ===================
M = jnp.zeros(shape=(6 + model.dofs(), 6 + model.dofs()))
BackwardPassCarry = tuple[jtp.Matrix, jtp.Matrix]
backward_pass_carry: BackwardPassCarry = (Mc, M)
def backward_pass(
carry: BackwardPassCarry, i: jtp.Int
) -> tuple[BackwardPassCarry, None]:
ii = i - 1
Mc, M = carry
Mc_λi = Mc[λ[i]] + i_X_λi[i].T @ Mc[i] @ i_X_λi[i]
Mc = Mc.at[λ[i]].set(Mc_λi)
Fi = Mc[i] @ S[i]
M_ii = S[i].T @ Fi
M = M.at[ii + 6, ii + 6].set(M_ii.squeeze())
j = i
FakeWhileCarry = tuple[jtp.Int, jtp.Vector, jtp.Matrix]
fake_while_carry = (j, Fi, M)
# This internal for loop implements the while loop of the CRBA algorithm
# to compute off-diagonal blocks of the mass matrix M.
# In pseudocode it is implemented as a while loop. However, in order to enable
# applying reverse-mode AD, we implement it as a nested for loop with a fixed
# number of iterations and a branching model to skip for loop iterations.
def fake_while_loop(
carry: FakeWhileCarry, i: jtp.Int
) -> tuple[FakeWhileCarry, None]:
def compute(carry: FakeWhileCarry) -> FakeWhileCarry:
j, Fi, M = carry
Fi = i_X_λi[j].T @ Fi
j = λ[j]
M_ij = Fi.T @ S[j]
jj = j - 1
M = M.at[ii + 6, jj + 6].set(M_ij.squeeze())
M = M.at[jj + 6, ii + 6].set(M_ij.squeeze())
return j, Fi, M
j, _, _ = carry
j, Fi, M = jax.lax.cond(
pred=jnp.logical_and(i == λ[j], λ[j] > 0),
true_fun=compute,
false_fun=lambda carry: carry,
operand=carry,
)
return (j, Fi, M), None
(j, Fi, M), _ = (
jax.lax.scan(
f=fake_while_loop,
init=fake_while_carry,
xs=jnp.flip(jnp.arange(start=1, stop=model.number_of_links())),
)
if model.number_of_links() > 1
else [(j, Fi, M), None]
)
Fi = i_X_0[j].T @ Fi
M = M.at[0:6, ii + 6].set(Fi.squeeze())
M = M.at[ii + 6, 0:6].set(Fi.squeeze())
return (Mc, M), None
# This scan performs the backward pass to compute Mbj, Mjb and Mjj, that
# also includes a fake while loop implemented with a scan and two cond.
(Mc, M), _ = (
jax.lax.scan(
f=backward_pass,
init=backward_pass_carry,
xs=jnp.flip(jnp.arange(start=1, stop=model.number_of_links())),
)
if model.number_of_links() > 1
else [(Mc, M), None]
)
# Store the locked 6D rigid-body inertia matrix Mbb ∈ ℝ⁶ˣ⁶.
M = M.at[0:6, 0:6].set(Mc[0])
return M