3-3 Coordinate Frames

The INS measurements of the AUV’s location need to be described in a certain coordinate frame. In the scenario of an AUV on a search operation, there is more than one coordinate frame that is relevant to the operation. In fact, there are three coordinate frames that are important in this scenario. These three coordinate frames are listed below.

• Global coordinate frame

• Robot coordinate frame

• Sensor coordinate frame

The three coordinate frames are shown in Figure 3-10 [17].

Figure 3-10: Three different coordinate frames.

3-3 Coordinate Frames 25

In Figure 3-10, the axes corresponding to the global coordinate frame are labelled g_x and g_y, the axes corresponding to the robot coordinate frame are labelled r_x and r_y and the axes corresponding to the sensor coordinate frame are labelled s_x and s_y.

The global coordinate frame is a coordinate frame that has its axes fixed on the earth’s surface. The global coordinate frame used by GPS has its origin located somewhere off the coast of Africa, but in the simulation framework the origin is chosen somewhat more conve-nient inside the search area. The location and orientation measurements made by the AUV’s INS are described in this coordinate frame, and this is also the coordinate frame that the locations of the found objects should be described in. Contrary to the global coordinate frame, the robot coordinate frame has its axes fixed on the AUV. This means that when the AUV is moving, a location that is stationary in the robot coordinate frame will be moving in the global coordinate frame. Likewise, the axes of the sensor coordinate frame are fixed on a sonar system. When the sonar system moves relatively to the robot, locations that are stationary in the sensor coordinate frame will be moving in the global and robot coordinate frames. Locations in the sonar images are described in the sensor coordinate frame.

In this final thesis project, the sonar systems are not moving with respect to the AUV.

For this reason, the robot coordinate frame can be discarded when the axes of the sensor coordinate frame are fixed on the AUV, since locations in sonar images obtained from both the sonar systems can be described in this coordinate frame now. This means that from the three coordinate frames shown in Figure 3-10, only the global and sensor coordinate frames are used. Another difference with Figure 3-10 is the way that the sensor coordinate frame is fixed on the AUV. In this final thesis project, the sensor coordinate frame is fixed on the AUV such that the AUV is always facing the positive y-axis of this coordinate frame.

Coordinates described in the sensor coordinate frame can be transformed into coordinates described in the global coordinate frame, and vice versa, provided that the current location and orientation of the sensor coordinate frame axes inside the global coordinate frame are known. Since the INS provides a measurement regarding the location and orientation of the AUV in the global coordinate frame, this measurement can be used to relate the two coordi-nate frames. These two coordicoordi-nate frames are related through homogeneous transformation matrices, such that ^b_aT is the homogeneous transformation matrix that maps homogeneous coordinates from coordinate frame a to coordinate frame b.

The reason that homogeneous transformation matrices are used in this final thesis project, is because these transformation matrices can apply the translation of a vector through a matrix multiplication with this vector. In order to do so, they need to be multiplied with a vector describing homogeneous coordinates instead of Cartesian coordinates. Homogeneous coordi-nates add one dimension to the Cartesian coordicoordi-nates. This extra dimension is a non-zero real number that scales the Cartesian coordinates. In this final thesis project this extra di-mension is set to one, such that the Cartesian coordinates (x, y) in R² are described by the homogeneous coordinates (x, y, 1) in R³.

The sensor coordinate frame can be transformed into the global coordinate frame by ro-tating and translating the sensor coordinate frame such that it becomes equal to the global coordinate frame. The same can be done to transform the global coordinate frame into the

sensor coordinate frame. The required rotation is determined by the difference in orientation of the two coordinate frames. An estimated value for this difference in orientation can be obtained by the INS orientation measurement θ_m. This θ_m is the measured rotation [rad]

of the AUV around the z-axis with respect to the positive x-axis, in the global coordinate frame. The required translation is determined by the difference in location of the origins of the two coordinate frames. An estimated value for this difference can be obtained by the INS location measurements x_m and y_m. These x_m and y_m are the measured x-coordinate and y-coordinate of the AUV’s location, in the global coordinate frame.

Using the θ_m, x_m and y_m INS measurements, the homogeneous rotation and translation transformation matrices to transform coordinates from the sensor coordinate frame to the global coordinate frame are shown in Equation (3-12) [25] and Equation (3-13) [25] respec-tively.

The homogeneous transformation matrix that transforms coordinates from the sensor coor-dinate frame to the global coorcoor-dinate frame can be computed by a matrix multiplication of

gsTtranslation and ^g_sTrotation, as shown in Equation (3-14).

g

The homogeneous transformation matrix that transforms coordinates from the global coor-dinate frame to the sensor coorcoor-dinate frame can be constructed in a similar way. When transforming from the global coordinate frame to the sensor coordinate frame, the homoge-neous rotation and translation transformation matrices are given by Equation (3-15) [25] and Equation (3-16) [25] respectively.

The homogeneous transformation matrix that transforms coordinates from the global coordi-nate frame to the sensor coordicoordi-nate frame can again be computed by a matrix multiplication of the homogeneous translation and rotation transformation matrices, but this time in the reverse order. This is shown in Equation (3-17).

s