Add safe norm function and refactor usages

src/jaxsim/api/data.py

@@ -382,9 +382,8 @@ def base_orientation(self, dcm: jtp.BoolLike = False) -> jtp.Vector | jtp.Matrix
         # we introduce a Baumgarte stabilization to let the quaternion converge to
         # a unit quaternion. In this case, it is not guaranteed that the quaternion
         # stored in the state is a unit quaternion.
-        W_Q_B = jnp.where(
-            jnp.allclose(W_Q_B.dot(W_Q_B), 1.0), W_Q_B, W_Q_B / jnp.linalg.norm(W_Q_B)
-        )
+        norm = jaxsim.math.safe_norm(W_Q_B)
+        W_Q_B = W_Q_B / (norm + jnp.finfo(float).eps * (norm == 0))
 
         return (W_Q_B if not dcm else jaxsim.math.Quaternion.to_dcm(W_Q_B)).astype(
             float
@@ -611,11 +610,8 @@ def reset_base_quaternion(self, base_quaternion: jtp.VectorLike) -> Self:
 
         W_Q_B = jnp.array(base_quaternion, dtype=float)
 
-        W_Q_B = jax.lax.select(
-            pred=jnp.allclose(jnp.linalg.norm(W_Q_B), 1.0, atol=1e-6, rtol=0.0),
-            on_true=W_Q_B,
-            on_false=W_Q_B / jnp.linalg.norm(W_Q_B),
-        )
+        norm = jaxsim.math.safe_norm(W_Q_B)
+        W_Q_B = W_Q_B / (norm + jnp.finfo(float).eps * (norm == 0))
 
         return self.replace(
             validate=True,

src/jaxsim/math/__init__.py

@@ -8,5 +8,6 @@
 from .rotation import Rotation
 from .skew import Skew
 from .transform import Transform
+from .utils import safe_norm
 
 from .joint_model import JointModel, supported_joint_motion  # isort:skip

src/jaxsim/math/quaternion.py

@@ -4,6 +4,8 @@
 
 import jaxsim.typing as jtp
 
+from .utils import safe_norm
+
 
 class Quaternion:
     @staticmethod
@@ -111,18 +113,13 @@ def Q_inertial(q: jtp.Vector) -> jtp.Matrix:
             operand=quaternion,
         )
 
-        norm_ω = jax.lax.cond(
-            pred=ω.dot(ω) < (1e-6) ** 2,
-            true_fun=lambda _: 1e-6,
-            false_fun=lambda _: jnp.linalg.norm(ω),
-            operand=None,
-        )
+        norm_ω = safe_norm(ω)
 
         qd = 0.5 * (
             Q
             @ jnp.hstack(
                 [
-                    K * norm_ω * (1 - jnp.linalg.norm(quaternion)),
+                    K * norm_ω * (1 - safe_norm(quaternion)),
                     ω,
                 ]
             )

src/jaxsim/math/rotation.py

@@ -4,6 +4,7 @@
 import jaxsim.typing as jtp
 
 from .skew import Skew
+from .utils import safe_norm
 
 
 class Rotation:
@@ -67,7 +68,7 @@ def from_axis_angle(vector: jtp.Vector) -> jtp.Matrix:
         def theta_is_not_zero(axis: jtp.Vector) -> jtp.Matrix:
 
             v = axis
-            theta = jnp.linalg.norm(v)
+            theta = safe_norm(v)
 
             s = jnp.sin(theta)
             c = jnp.cos(theta)
@@ -81,19 +82,9 @@ def theta_is_not_zero(axis: jtp.Vector) -> jtp.Matrix:
 
             return R.transpose()
 
-        # Use the double-where trick to prevent JAX problems when the
-        # jax.jit and jax.grad transforms are applied.
         return jnp.where(
-            jnp.linalg.norm(vector) > 0,
-            theta_is_not_zero(
-                axis=jnp.where(
-                    jnp.linalg.norm(vector) > 0,
-                    vector,
-                    # The following line is a workaround to prevent division by 0.
-                    # Considering the outer where, this branch is never executed.
-                    jnp.ones(3),
-                )
-            ),
+            jnp.allclose(vector, 0.0),
             # Return an identity rotation matrix when the input vector is zero.
             jnp.eye(3),
+            theta_is_not_zero(axis=vector),
         )

src/jaxsim/math/utils.py

@@ -0,0 +1,22 @@
+import jax.numpy as jnp
+
+import jaxsim.typing as jtp
+
+
+def safe_norm(array: jtp.ArrayLike, axis=None) -> jtp.Array:
+    """
+    Provides a calculation for an array norm so that it is safe
+    to compute the gradient and the NaNs are handled.
+
+    Args:
+        array: The array for which to compute the norm.
+        axis: The axis for which to compute the norm.
+    """
+
+    is_zero = jnp.allclose(array, 0.0)
+
+    array = jnp.where(is_zero, jnp.ones_like(array), array)
+
+    norm = jnp.linalg.norm(array, axis=axis)
+
+    return jnp.where(is_zero, 0.0, norm)

src/jaxsim/rbda/contacts/soft.py

@@ -309,19 +309,16 @@ def hunt_crossley_contact_model(
 
         # Compute the direction of the tangential force.
         # To prevent dividing by zero, we use a switch statement.
-        # The ε, instead, is needed to make AD happy.
-        f_tangential_direction = jnp.where(
-            f_tangential.dot(f_tangential) != 0,
-            f_tangential / jnp.linalg.norm(f_tangential + ε),
-            jnp.zeros(3),
+        norm = jaxsim.math.safe_norm(f_tangential)
+        f_tangential_direction = f_tangential / (
+            norm + jnp.finfo(float).eps * (norm == 0)
         )
 
         # Project the tangential force to the friction cone if slipping.
         f_tangential = jnp.where(
             sticking,
             f_tangential,
-            jnp.minimum(μ * force_normal_mag, jnp.linalg.norm(f_tangential + ε))
-            * f_tangential_direction,
+            jnp.minimum(μ * force_normal_mag, norm) * f_tangential_direction,
         )
 
         # Set the tangential force to zero if there is no contact.

src/jaxsim/terrain/terrain.py

@@ -7,6 +7,7 @@
 import jax_dataclasses
 import numpy as np
 
+import jaxsim.math
 import jaxsim.typing as jtp
 from jaxsim import exceptions
 
@@ -41,7 +42,7 @@ def normal(self, x: jtp.FloatLike, y: jtp.FloatLike) -> jtp.Vector:
             [(h_xm - h_xp) / (2 * self.delta), (h_ym - h_yp) / (2 * self.delta), 1.0]
         )
 
-        return n / jnp.linalg.norm(n)
+        return n / jaxsim.math.safe_norm(n)
 
 
 @jax_dataclasses.pytree_dataclass