# Appendix – Coordinate Systems

The DVL operates in five different coordinate systems. “Magnetic Frame” is the local coordinate system referenced to magnetic north. “World Frame” is the local coordinate frame referenced to true north. “Sensor Frame” is the coordinate system referenced to the DVL sensor array. “Vehicle Frame” is the coordinate system referenced to the vehicle chassis. “Vehicle Frame” is Sensor Frame rotated to compensate for mechanical mounting offsets. These frames are usually given in Euler angles (e.g., pitch, roll, yaw), although quaternions are given for some as described in the message definitions. Euler and quaternions are different representations of equivalent orientations.

## Right-Hand Rule

Right-hand rule is used to describe rotations around an axis in Cartesian coordinate system. In right-hand rule, you conceptually grasp an axis in your right hand, with your thumb laid along the axis pointed in the positive direction. Your fingers curl around the axis, and they point in the direction of increasing (more positive Euler angles) rotation. All rotations in this manual except “Heading” are right-hand rule.

## Left-Hand Rule

Left-hand rule is used to describe rotations around an axis in Cartesian coordinate system. In left-hand rule, you conceptually grasp an axis in your left hand, with your thumb laid along the axis pointed in the positive direction. Your fingers curl around the axis, and they point in the direction of increasing (more positive Euler angles) rotation. In this manual, only “Heading” is given using left-hand rule, since it matches the rule used by magnetic Boy Scout and Mariner compasses for indicating directions.

## Z-Axis Definition

In Magnetic, World, and Body Frames, the Z axis is referenced to local gravity. It passes through the DVL. The negative Z axis points straight down thought the center of the earth. The positive Z axis points straight up. The local horizontal plane is perpendicular to the Z axis.

## Magnetic Frame

In Magnetic, World, and Body Frames, the Z axis is referenced to local gravity. It passes through the DVL. The negative Z axis points straight down thought the center of the earth. The positive Z axis points straight up. The local horizontal plane is perpendicular to the Z axis.

The Y axis lies in the local horizontal plane and is tangent to a great circle that includes the DVL and the magnetic north pole, with the +Y direction toward north. In colloquial terms, the Y axis points at the magnetic north pole, like a Boy Scout compass. The +X axis is rotated -90 degrees (right hand rule) from the Y axis in the local horizontal plane.

The IMU determines the Magnetic Frame using magnetometers and accelerometers.

## World Frame

In Magnetic, World, and Body Frames, the Z axis is referenced to local gravity. It passes through the DVL. The negative Z axis points straight down thought the center of the earth. The positive Z axis points straight up. The local horizontal plane is perpendicular to the Z axis.

The Y axis lies in the local horizontal plane and is tangent to a great circle that includes the DVL and the north pole, with the +Y direction toward north. In colloquial terms, the Y axis points at the north pole. The +X axis is rotated -90 degrees (right hand rule) from the Y axis in the local horizontal plane. World Frame is created from Magnetic Frame by adding the magnetic declination supplied by the user (i.e., you).

## Sensor Frame

Sensor frame is shown in the diagram below. Sensor frame is not directly available via DVL messages. Sensor frame is used to generate Body Frame and Vehicle Frame.

## Vehicle Frame

Vehicle Frame (or “ROV Frame” is Sensor Frame rotated in three axes as specified in the “SET-SENSOR-ORIENTATION” command. This can used to compensate heading, roll, and pitch for side-tracking setups or any mounting offset (e.g., you mount the sensor rotated by 90 degrees compared to the labeling in the diagram above to get a better placement for the cable exiting the sensor head).

## Quaternion Coordinates and Notations

The conversion between Euler and quaternion is shown in the diagram below. Our standard notation for the quaternion is (w, x, y, z), where “w” is the real part of the quaternion. This corresponds to similar notations (a + bi + cj + dk) or (a, b, c, d), where “w” corresponds to “a”, “x” corresponds to “b”, “y” corresponds to “c”, and “z” corresponds to “d”.

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