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Specifying pose in SDFormat

A fundamental tool for robot modeling is the ability to concisely and intuitively express the relative position and orientation of model components in 3-D.

Throughout the documentation, the suffix _A is used to indicate a kinematic quantity that is expressed w.r.t (with respect to) frame A. In addition, the suffix _BA is used to indicate a kinematic quantity that changes from being expressed w.r.t frame A to being expressed w.r.t frame B. For example, the pose of frame A expressed w.r.t frame B is given by the transform X_BA ∈ SE(3). An alternative description of X_BA alludes to X being a linear operator that transforms a frame that is coincident to frame B to being coincident to frameA. In this sense,X_BA is said to be "the transform from B to A". Note that both descriptions are valid interpretations of the same mathematical object.

The SDFormat specification has the <pose> element which accepts 6 numbers in total to represent a coordinate transform X_PC from parent frame P to child frame C:

<pose>x y z roll pitch yaw</pose>

The elements x y z specify the position vector (in meters), and the elements roll pitch yaw are Euler angles (in radians) that specify the orientation, which can be computed by an extrinsic X-Y-Z rotation as shown by the following equation for the rotation matrix R_PC and figure:

$$ R_{PC} = \begin{bmatrix} \cos(yaw) & -\sin(yaw) & 0 \\ \sin(yaw) & \cos(yaw) & 0 \\ 0 & 0 & 1 \end{bmatrix} * \begin{bmatrix} \cos(pitch) & 0 & \sin(pitch) \\ 0 & 1 & 0 \\ -\sin(pitch) & 0 & \cos(pitch) \end{bmatrix} * \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos(roll) & -\sin(roll) \\ 0 & \sin(roll) & \cos(roll) \end{bmatrix} $$

The SDFormat <pose/> element is similar to the <origin/> element from URDF, which both share the same definition of roll, pitch, and yaw angles:

<!-- URDF -->
<origin xyz="x y z" rpy="roll pitch yaw" />

See Also

For a command-line utility to convert between roll-pitch-yaw angles, quaternions, and rotation matrices, please see the quaternion_from_euler and quaternion_to_euler example programs in Gazebo math.

Software implementations for converting between this Euler angle convention and quaternions can be found in gz::math::Quaternion C++ class, and the urdf::Rotation C++ class.

Some other implementations: