Math

Math#

class jaxsim.math.adjoint.Adjoint[source]#
static from_quaternion_and_translation(quaternion=Array([1., 0., 0., 0.], dtype=float64), translation=Array([0., 0., 0.], dtype=float64), inverse=False, normalize_quaternion=False)[source]#

Create an adjoint matrix from a quaternion and a translation.

Parameters:

  • quaternion (jtp.Vector) – A quaternion vector (4D) representing orientation.

  • translation (jtp.Vector) – A translation vector (3D).

  • inverse (bool) – Whether to compute the inverse adjoint. Default is False.

  • normalize_quaternion (bool) – Whether to normalize the quaternion before creating the adjoint. Default is False.

Returns:

The adjoint matrix.

Return type:

jtp.Matrix

static from_rotation_and_translation(rotation=Array([[1., 0., 0.], [0., 1., 0.], [0., 0., 1.]], dtype=float64), translation=Array([0., 0., 0.], dtype=float64), inverse=False)[source]#

Create an adjoint matrix from a rotation matrix and a translation vector.

Parameters:

  • rotation (jtp.Matrix) – A 3x3 rotation matrix.

  • translation (jtp.Vector) – A translation vector (3D).

  • inverse (bool) – Whether to compute the inverse adjoint. Default is False.

Returns:

The adjoint matrix.

Return type:

jtp.Matrix

static from_transform(transform, inverse=False)[source]#

Create an adjoint matrix from a transformation matrix.

Parameters:

  • transform (Union[Array, ndarray, bool, number, bool, int, float, complex, tuple]) – A 4x4 transformation matrix.

  • inverse (bool) – Whether to compute the inverse adjoint.

Return type:

Array

Returns:

The 6x6 adjoint matrix.

static inverse(adjoint)[source]#

Compute the inverse of an adjoint matrix.

Parameters:

adjoint (jtp.Matrix) – The adjoint matrix.

Returns:

The inverse adjoint matrix.

Return type:

jtp.Matrix

static to_transform(adjoint)[source]#

Convert an adjoint matrix to a transformation matrix.

Parameters:

adjoint (jtp.Matrix) – The adjoint matrix (6x6).

Returns:

The transformation matrix (4x4).

Return type:

jtp.Matrix

class jaxsim.math.cross.Cross[source]#
static vx(velocity_sixd)[source]#

Compute the cross product matrix for 6D velocities.

Parameters:

velocity_sixd (jtp.Vector) – A 6D velocity vector [v, ω].

Returns:

The cross product matrix (6x6).

Return type:

jtp.Matrix

Raises:

ValueError – If the input vector does not have a size of 6.

static vx_star(velocity_sixd)[source]#

Compute the negative transpose of the cross product matrix for 6D velocities.

Parameters:

velocity_sixd (jtp.Vector) – A 6D velocity vector [v, ω].

Returns:

The negative transpose of the cross product matrix (6x6).

Return type:

jtp.Matrix

Raises:

ValueError – If the input vector does not have a size of 6.

class jaxsim.math.inertia.Inertia[source]#
static to_params(M)[source]#

Convert a 6x6 inertia matrix to mass, center of mass, and inertia matrix.

Parameters:

M (jtp.Matrix) – The 6x6 inertia matrix.

Returns:

A tuple containing mass, center of mass (3D), and inertia matrix (3x3).

Return type:

tuple[jtp.Float, jtp.Vector, jtp.Matrix]

Raises:

ValueError – If the input matrix M has an unexpected shape.

static to_sixd(mass, com, I)[source]#

Convert mass, center of mass, and inertia matrix to a 6x6 inertia matrix.

Parameters:

  • mass (jtp.Float) – The mass of the body.

  • com (jtp.Vector) – The center of mass position (3D).

  • I (jtp.Matrix) – The 3x3 inertia matrix.

Returns:

The 6x6 inertia matrix.

Return type:

jtp.Matrix

Raises:

ValueError – If the shape of the inertia matrix I is not (3, 3).

class jaxsim.math.quaternion.Quaternion[source]#
static derivative(quaternion, omega, omega_in_body_fixed=False, K=0.1)[source]#

Compute the derivative of a quaternion given angular velocity.

Parameters:

  • quaternion (jtp.Vector) – Quaternion in XYZW representation.

  • omega (jtp.Vector) – Angular velocity vector.

  • omega_in_body_fixed (bool) – Whether the angular velocity is in the body-fixed frame.

  • K (float) – A scaling factor.

Returns:

The derivative of the quaternion.

Return type:

jtp.Vector

static from_dcm(dcm)[source]#

Convert a direction cosine matrix (DCM) to a quaternion.

Parameters:

dcm (jtp.Matrix) – Direction cosine matrix (DCM).

Returns:

Quaternion in XYZW representation.

Return type:

jtp.Vector

static integration(quaternion, dt, omega, omega_in_body_fixed=False)[source]#

Integrate a quaternion in SO(3) given an angular velocity.

Parameters:

  • quaternion (Union[Array, ndarray, bool, number, bool, int, float, complex, tuple]) – The quaternion to integrate.

  • dt (Union[float, Array, ndarray, bool, number, bool, int, complex]) – The time step.

  • omega (Union[Array, ndarray, bool, number, bool, int, float, complex, tuple]) – The angular velocity vector.

  • omega_in_body_fixed (Union[bool, Array, ndarray, bool, number, int, float, complex]) – Whether the angular velocity is in body-fixed representation as opposed to the default inertial-fixed representation.

Return type:

Array

Returns:

The integrated quaternion.

static to_dcm(quaternion)[source]#

Convert a quaternion to a direction cosine matrix (DCM).

Parameters:

quaternion (jtp.Vector) – Quaternion in XYZW representation.

Returns:

Direction cosine matrix (DCM).

Return type:

jtp.Matrix

static to_wxyz(xyzw)[source]#

Convert a quaternion from XYZW to WXYZ representation.

Parameters:

xyzw (jtp.Vector) – Quaternion in XYZW representation.

Returns:

Quaternion in WXYZ representation.

Return type:

jtp.Vector

static to_xyzw(wxyz)[source]#

Convert a quaternion from WXYZ to XYZW representation.

Parameters:

wxyz (jtp.Vector) – Quaternion in WXYZ representation.

Returns:

Quaternion in XYZW representation.

Return type:

jtp.Vector

class jaxsim.math.rotation.Rotation[source]#
static from_axis_angle(vector)[source]#

Generate a 3D rotation matrix from an axis-angle representation.

Parameters:

vector (jtp.Vector) – Axis-angle representation as a 3D vector.

Returns:

3D rotation matrix.

Return type:

jtp.Matrix

static x(theta)[source]#

Generate a 3D rotation matrix around the X-axis.

Parameters:

theta (jtp.Float) – Rotation angle in radians.

Returns:

3D rotation matrix.

Return type:

jtp.Matrix

static y(theta)[source]#

Generate a 3D rotation matrix around the Y-axis.

Parameters:

theta (jtp.Float) – Rotation angle in radians.

Returns:

3D rotation matrix.

Return type:

jtp.Matrix

static z(theta)[source]#

Generate a 3D rotation matrix around the Z-axis.

Parameters:

theta (jtp.Float) – Rotation angle in radians.

Returns:

3D rotation matrix.

Return type:

jtp.Matrix

class jaxsim.math.skew.Skew[source]#

A utility class for skew-symmetric matrix operations.

static vee(matrix)[source]#

Extract the 3D vector from a skew-symmetric matrix (vee operator).

Parameters:

matrix (jtp.Matrix) – A 3x3 skew-symmetric matrix.

Returns:

The 3D vector extracted from the input matrix.

Return type:

jtp.Vector

static wedge(vector)[source]#

Compute the skew-symmetric matrix (wedge operator) of a 3D vector.

Parameters:

vector (jtp.Vector) – A 3D vector.

Returns:

The skew-symmetric matrix corresponding to the input vector.

Return type:

jtp.Matrix

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