Orientation, Position, and Coordinate Convention

The Navigation Toolbox™ enables you to track orientation, position, pose, and trajectory of a platform. A platform refers generally to any object you want to track its state.

Orientation

Orientation is defined by angular displacement. Orientation can be described in terms of point or frame rotation. In point rotation, the coordinate system is static and the point moves. In frame rotation, the point is static and the coordinate system moves. For a given axis and angle of rotation, point rotation and frame rotation define equivalent angular displacement but in opposite directions.

Navigation Toolbox defaults to frame rotation.

In frame representation, orientation is defined as rotation that takes the parent frame to the child frame. The choice of parent and child frames depends on the problem being solved. For example, when you manipulate the mounting of a sensor on a platform, you can select the platform body frame as the parent frame and select the sensor mounting frame as the child frame. The rotation from the platform body frame to the sensor mounting frame defines the orientation of the sensor with respect to the platform.

Navigation Toolbox primarily supports the NED (north-east-down) coordinate frame. You can also use the ENU (east-north-up) coordinate frame in many features.

Frame Rotation

To relate one orientation to another you must rotate a frame. The table summarizes the z-y-x rotation conventions. You can also use other conventions, such as the z-x-z rotation convention. See the rotation sequence (RS) argument of quaternion for more details on these conventions.

Variable	Euler Angle	Symbol	Output Interval (Degrees)
z	Yaw	ψ	−180 ≤ ψ < 180
y	Pitch	θ	−90 ≤ θ ≤ 90
x	Roll	ϕ	−180 ≤ ϕ < 180

A positive rotation angle corresponds to a clockwise rotation about an axis when viewing from the origin along the positive direction of the axis. The right-hand convention is equivalent, where positive rotation is indicated by the direction in which the fingers on your right hand curl when your thumb is pointing in the direction of the axis of rotation.

To define three-dimensional frame rotation, you must rotate sequentially about the axes. Navigation Toolbox uses intrinsic (carried frame) rotation, in which, after each rotation, the axis is updated before the next rotation. For example, to rotate an axis using the z-y-x convention:

1. Rotate the parent frame about the z-axis to yield a new set of axes, (x',y',z), where the x- and y-axes have changed to x'- and y'-axes and the z-axis remains unchanged.

   $$\left[\begin{array}{c}{x}^{\text{'}}\\ y^{\text{'}}\\ z\end{array}\right]={\text{R}}_z\left(\psi \right)\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$$

2. Rotate the new set of axes about the y'-axis, yielding another new set of axes, (x'',y',z').

   $$\left[\begin{array}{c}{x}^{\text{'}\text{'}}\\ y^{\text{'}}\\ z^{\text{'}}\end{array}\right]={\text{R}}_y\left(\theta \right)\left[\begin{array}{c}{x}^{\text{'}}\\ y^{\text{'}}\\ z\end{array}\right]$$

3. Rotate this new set of axes about the x''-axis, arriving at the desired child frame, (x'',y'',z'').

   $$\left[\begin{array}{c}{x}^{\text{'}\text{'}}\\ y^{\text{'}\text{'}}\\ z^{\text{'}\text{'}}\end{array}\right]={\text{R}}_x\left(\varphi \right)\left[\begin{array}{c}{x}^{\text{'}\text{'}}\\ y^{\text{'}}\\ z^{\text{'}}\end{array}\right]$$

This sequence of rotations follows the convention outlined in Rotations, Orientation, and Quaternions. The rotation matrix required to convert a vector in the parent frame to a vector in the child frame for a given yaw, pitch, and roll is computed as:

$$\text{R}\left(\psi ,\theta ,\varphi \right)={\text{R}}_x\left(\varphi \right){\text{R}}_y\left(\theta \right){\text{R}}_z\left(\psi \right)=\left[\begin{array}{ccc}\cos \psi \cos \theta & \sin \psi \cos \theta & -\sin \theta \\ \cos \psi \sin \theta \sin \varphi -\sin \psi \cos \varphi & \sin \psi \sin \theta \sin \varphi +\cos \psi \cos \varphi & \cos \theta \sin \varphi \\ \cos \psi \sin \theta \cos \varphi +\sin \psi \sin \varphi & \sin \psi \sin \theta \cos \varphi -\cos \psi \sin \varphi & \cos \theta \cos \varphi \end{array}\right]$$

For features that support frame-based processing, Navigation Toolbox provides coordinates as an N-by-3 matrix, where N is the number of samples in time and the three columns correspond to the x-, y-, and z-axes. The following calculation rotates a parent frame to a child frame:

$$a_{\text{child}}={\left(\text{R}\left(\psi ,\theta ,\varphi \right)\times {\left(a_{\text{parent}}\right)}^{\text{T}}\right)}^{\text{T}}$$

where a_parent represents a N-by-3 matrix of coordinates expressed in the parent coordinate frame and a_child is the resulting N-by-3 matrix of coordinates expressed in the child frame.

Navigation Toolbox enables efficient orientation computation using the quaternion data type. To create a rotation matrix using quaternions, use the rotmat function.

Express Gravitational Vector in Body Frame

In an NED frame, the gravitational vector can be express as

gNED = [0 0 9.8]; % m/s^2

Consider a body frame obtained by a consecutive rotation of 20 degrees in yaw, 5 degrees in pitch, and 10 degrees in roll from the parent NED frame.

yaw = 20; % degree
pitch = 5; % degree
roll = 10; % degree

To obtain the expression of the gravitational vector in the body frame, you first obtain the quaternion corresponding to the three consecutive Euler angles.

q = quaternion([yaw pitch roll],"eulerd","zyx","frame");

Then, using the rotateframe object function, you can obtain the coordinates of the gravitational vector in the body frame as

gBody = rotateframe(q,gNED)

gBody = 1×3

   -0.8541    1.6953    9.6144

Alternately, you can obtain the coordinates using rotation matrix. First, you use the rotmat object function of quaternion to obtain the corresponding rotation matrix that transforms coordinates from the NED frame to the body frame.

R = rotmat(q,"frame");

Then, obtain the coordinates of the gravitational vector in the body frame as

gBody2 = (R*gNED')'

gBody2 = 1×3

   -0.8541    1.6953    9.6144

Position

Position is defined as the translational distance from a parent frame origin to a child frame origin. For example, take the local NED coordinate system as the parent frame. In the NED coordinate system:

• The origin is arbitrarily fixed to a point on the surface of the Earth. This makes the NED coordinate system local.

• The x-axis points toward the ellipsoid north.

• The y-axis points toward the ellipsoid east.

• The z-axis points downward along the ellipsoid normal (geodetic latitude, ρ).

Azimuth and Elevation

Given a vector in R³:

• Azimuth is defined as the angle from the x-axis to the orthogonal projection of the vector onto the xy-plane. The angle is positive going from the x-axis toward the y-axis. Azimuth is given in degrees in the range [−180, 180).

• Elevation is defined as the angle from the projection onto the xy-plane to the vector. The angle is positive going from the xy-plane to the z-axis. Elevation is given in degrees in the range [−90, 90].

Pose and Trajectory

To specify an object in 3-D space fully, you can combine position and orientation. Pose is defined as the combination of position and orientation. Trajectory defines how pose changes over time. To generate ground-truth trajectories in Navigation Toolbox, use kinematicTrajectory or waypointTrajectory.