Implementation of an Inertial Navigation Systemwith Focus toward GPS IntegrationByMatt GuibordExecutive Summary:Inertial navigation systems (INS) are becoming increasingly popular. Such systems makeuse of an inertial measurement unit (IMU) that is based on accelerometers and gyroscopes. Using the acceleration and rotational velocity data, one can track their speed, position, and attitude. One problem with such a system is the fact that errors in the accelerometers and gyroscopes are double integrated over time. For this reason, IMU’s are very accurate over short distances, but require some external calibration such as from a GPS unit to remain accurate over long periods of time.Keywords: INS, Inertial Navigation, Accelerometers, Gyroscopes, Navigation Equations.I. IntroductionAn inertial navigation system uses inertial measurements to navigate around the globe. As learned in calculus courses, if one takes acceleration and integrates it, the output will be velocity. Similarly, integrating velocity will yield position. The system becomes much more complicated when rotations have to be taken into account. These rotations are in the roll, pitch, and yaw direction of the user or vehicle and will vary how the accelerations and velocities interact. Later, the navigation equations will be discussedthat relate the rotations to directional travel around Earth.II. ObjectiveThe objective of this note is to present a novel approach to creating an inertial navigation system. The required hardware will be discussed as well as the mathematical equations needed to navigate using inertial measurements. Implementation of the navigation equations on a microprocessor will be reviewed.III. Background InformationThe main components of an inertial measurement unit are accelerometers and gyroscopes. Accelerometers measure the acceleration of a mass within the sensor. Whenthe mass is accelerated, it creates an electrical signal that is proportional to the acceleration. Often, a single accelerometer chip can measure accelerations in x, y, and z simultaneously. Similarly, gyroscopes use a vibrating crystal that can sense rotational velocities. The sensor outputs a voltage that is proportional to the rotational velocity experienced by the crystal. Gyroscopes measure rotational velocities in the roll, pitch, and yaw directions. Commercial gyroscopes usually read only one or two of the directions within the same chip, but new three axis gyroscopes are starting to appear. By using these two sensors one can then keep track of their position as they move. The question is: how can one make use of accelerometers and gyroscopes to navigate around the globe?IV. Useful Coordinate FramesFirst, it is important to describe the different coordinate systems that are used in practice as presented by Shin (p. 8-9):A. Body FrameThe first is the body coordinate frame. This frame has its origin at the center of mass of the body of the user. The x-axis or heading, points straight ahead of the user. The z-axis points straight up and the y-axis points to the side tocomplete the right-handed orthogonal frame. Rotations in this frame are called pitch, roll and yaw. Where pitch is a rotation about the y-axis, roll is about the x-axis, and yaw is about the z-axis.B. Navigation FrameThe second coordinate frame is the navigation frame, which in our case will be the North, East, Down (NED) frame. In this frame, the origin is shared with that of the body frame. The positive x-axis points north, the positive z-axis points down, and the positive y-axis points east. This frame is also called the tangential plane frame because the system forms a plane that is tangential to the surface of the Earth as can be seen in figure 1. The East, North, Up (ENU) frame is also used in practice, but the NED frame is more popular.C. Earth-Centered Earth-Fixed FrameThe third frame is the Earth-centered Earth-fixed (ECEF) frame. This frame has its origin at the center of the earth. The x-axis points toward the prime meridian, the z-axis is parallel to the axis of rotation of the Earth, and the y-axis completes the right-hand orthogonal frame. Positions in the frame are measured as latitude (), longitude () and height (h). This is the frame that global position systems (GPS) use. Thus, this frame will be useful when integrating INSand GPS systems together. Figure 1: NED frame and ECEF frame (Shin, 10)D. Inertial FrameFinally, there is the inertial frame which has axes that are independent of the rotation of the earth. The origin is at the center of the Earth as in the ECEF frame. The x-axis points toward the mean vernal equinox, the z-axis is parallel with the rotation of Earth, and the y-axis completes a right-handed orthogonal frame.V. Coordinate Frame TransformationsThe transformation from the body frame to the navigation frame requires the rotation about three axes. A convenient way of describing these rotations is using a direction-cosine matrix (DCM). If one defines three angles, , and as the yaw,pitch, and roll of the body, respectively, with referenced to the navigation frame, one can define a DCM to rotate from the body to the navigation frame. Hence, once the three separate rotation matrices have been multiplied out, the final DCM is given as (Shin, p. 12):coscoscossinsinsinsincoscossinsinsinsincoscossincoscossincossinsincossinsinsincoscoscosnbCTo rotate velocities or accelerations from the body to the navigation frame, one would then use (Stovall, p. 4):bnbnaCa where a x represents the acceleration or velocity vector in the x-frame.It is worth noting that the common method of implementation of the rotational matrix is to use quaternions. A quaternion is a four part complex number which can describe a three-axis rotation as a rotation about a single axis. This method is commonly used because the implementation of quaternions does not require the use of trigonometricfunctions which greatly reduces the computational complexity (Shin, p.18-19).VI. Navigation EquationsThe navigation equations used in our project are a slight variation of those presented by Shin (p.
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