6 Inertial Navigation 6.1 BACKGROUND A more basic introduction to the fundamental concepts of inertial navigation can be found in Chapter 2, Section 2.2. Readers who are not already familiar with inertial navigation should review that section before starting this chapter. 6.1.1 History of Inertial Navigation Inertial navigation has had a relatively short but intense history of development, much of it during the half-century of the Cold War, with contributions from thousands of engineers and scientists. The following is only an outline of develop- ments in the United States. More details can be found, for example, in [22, 43, 75, 83, 88, 107, 135]. 6.1.1.1 Gyroscopes The word ``gyroscope'' was ®rst used by Jean Bernard Le  on Foucault (1819±1868), who coined the term from the Greek words for turn (g  uroB) and view (sk  opB). Foucault used one to demonstrate the rotation of the earth in 1852. Elmer Sperry (1860±1930) was one of the early pioneers in the develop- ment of gyroscope technology. Gyroscopes were applied to dead reckoning naviga- tion for iron ships (which could not rely on a magnetic compass) around 1911, to automatic steering of ships in the 1920s, for steering torpedos in the 1920s, and for heading and arti®cial horizon displays for aircraft in the 1920s and 1930s. Rockets designed by Robert H. Goddard in the 1930s also used gyroscopes for steering, as 131 Global Positioning Systems, Inertial Navigation, and Integration, Mohinder S. Grewal, Lawrence R. Weill, Angus P. Andrews Copyright # 2001 John Wiley & Sons, Inc. Print ISBN 0-471-35032-X Electronic ISBN 0-471-20071-9 did the autopilots for the German V-1 cruise missiles and V-2 ballistic missiles of World War II. 6.1.1.2 Relation to Guidance and Control Navigation is concerned with determining where you are relative to where you want to be, guidance with getting yourself to your destination, and control with staying on track. There has been quite a bit of synergism among these disciplines, especially in the development of missile technologies where all three could use a common set of sensors, computing resources, and engineering talent. As a consequence, the history of development of inertial navigation technology has a lot of overlap with that of guidance and control. 6.1.1.3 Gimbaled INS Gimbals have been used for isolating gyroscopes from rotations of their mounting bases since the time of Foucault. They have been used for isolating an inertial sensor cluster in a gimbaled inertial measurement unit (IMU) since about 1950. Charles Stark Draper at the Instrumentation Laboratory at MIT (later the Charles Stark Draper Laboratory) played a major role in the development of gyroscope and INS technology for use on aircraft and ships. Much of the early INS development was for use on military vehicles. An early impetus for INS tech- nology development for missiles was the Navaho Project, started soon after World War II by the U.S. Air Force for a supersonic cruise missile to carry a 15,000-lb payload (the atomic bomb of that period), cruising at Mach 3.25 at 90,000 ft for 5500 miles, and arriving with a navigation accuracy of about 1 nautical mile. The project was canceled in 1957 when nuclear devices had been shrunk to a size that could be carried by the rockets of the day, but by then the prime contractor, North American Aviation, had developed an operational INS for it. This technology was soon put to use in the intercontinental ballistic missiles that replaced Navaho, as well as in many military aircraft and ships. The navigation of the submarine Nautilus under the polar ice cap in 1958 would not have been possible without its INS. It was a gimbaled INS, as were nearly all such systems until the 1970s. 6.1.1.4 Early Strapdown Systems A gimbaled INS was carried on each of nine Apollo command modules from the earth to the moon and back between December 1968 and December 1972, but a strapdown INS was carried on each of the six 1 Lunar Excursion Modules (LEMs) that shuttled two astronauts from lunar orbit to the lunar surface and back. 6.1.1.5 Navigation Computers Strapdown INSs generally require more powerful navigation computers than their gimbaled counterparts. It was the devel- opment of silicon integrated circuit technology in the 1960s and 1970s that enabled strapdown systems to compete with gimbaled systems in all applications but those demanding extreme precision, such as ballistic missiles or submarines. 1 Two additional LEMs were carried to the moon but did not land there. T he Apollo 13 LEM did not make its intended lunar landing but played a far more vital role in crew survival. 132 INERTIAL NAVIGATION 6.1.2 Performance Integration of acceleration sensing errors causes INS velocity errors to grow linearly with time, and Schuler oscillations (Section 2.2.2.3) tend to keep position errors proportional to velocity errors. As a consequence, INS position errors tend to grow linearly with time. These errors are generally not known, except in terms of their statistical properties. INS performance is also characterized in statistical terms. 6.1.2.1 CEP Rate A circle of equal probability (CEP) is a circle centered at the estimated location of an INS on the surface of the earth, with radius such that it is equally likely that the true position is either inside or outside that circle. The CEP radius is a measure of position uncertainty. CEP rate is a measure of how fast position uncertainty is growing. 6.1.2.2 INS Performance Ranges CEP rate has been used by the U.S. Air Force to de®ne the three ranges of INS performance shown in Table 6.1, along with corresponding ranges of inertial sensor performance. These rough order-of-magni- tude sensor performance requirements are for ``cruise'' applications, with accelera- tion levels on the order of 1 g. 6.1.3 Relation to GPS 6.1.3.1 Advantages=Disadvantages of INS The main advantages of iner- tial navigation over other forms of navigation are as follows: 1. It is autonomous and does not rely on any external aids or on visibility conditions. It can operate in tunnels or underwater as well as anywhere else. 2. It is inherently well suited for integrated navigation, guidance, and control of the host vehicle. Its IMU measures the derivatives of the variables to be controlled (e.g., position, velocity, and attitude). 3. It is immune to jamming and inherently stealthy. It neither receives nor emits detectable radiation and requires no external antenna that might be detectable by radar. TABLE 6.1 INS and Inertial Sensor Performance Ranges System or Sensor Performance Units Performance Ranges High Medium Low INS CEP Rate (NMI=h) 10 À1 % 1 !10 Gyros deg=h 10 À3 % 10 À2 !10 À1 Accelerometers g a 10 À7 % 10 À6 !10 À5 a 1 g % 9:8m=s=s. 6.1 BACKGROUND 133 The disadvantages include the following: 1. Mean-squared navigation errors increase with time. 2. Cost, including: (a) Acquisition cost, which can be an order of magnitude (or more) higher than GPS receivers. (b) Operations cost, including the crew actions and time required for initializ- ing position and attitude. Time required for initializing INS attitude by gyrocompass alignment is measured in minutes. Time-to-®rst-®x for GPS receivers is measured in seconds. (c) Maintenance cost. Electromechanical avionics systems (e.g., INS) tend to have higher failure rates and repair costs than purely electronic avionics systems (e.g., GPS). 3. Size and weight, which have been shrinking: (a) Earlier INS systems weighed tens to hundreds of kilograms. (b) Later ``mesoscale'' INSs for integration with GPS weighed a few kilo- grams. (c) Developing micro-electromechanical sensors are targeted for gram-size systems. INS weight has a multiplying effect on vehicle system design, because it requires increased structure and propulsion weight as well. 4. Power requirements, which have been shrinking along with size and weight but are still higher than those for GPS receivers. 5. Heat dissipation, which is proportional to and shrinking with power require- ments. 6.1.3.2 Competition from GPS In the 1970s, U.S. commercial air carriers were required by FAA regulations to carry two INS systems on all ¯ights over water. The cost of these two systems was on the order of 10 5 U.S. dollars at that time. The relatively high cost of INS was one of the factors leading to the development of GPS. After deployment of GPS in the 1980s, the few remaining applications for ``stand- alone'' (i.e., unaided) INS include submarines, which cannot receive GPS signals while submerged, and intercontinental ballistic missiles, which cannot rely on GPS availability in time of war. 6.1.3.3 Synergism with GPS GPS integration has not only made inertial navigation perform better, it has made it cost less. Sensor errors that were unacceptable for stand-alone INS operation became acceptable for integrated operation, and the manufacturing and calibration costs for removing these errors could be eliminated. Also, new low-cost manufacturing methods using micro- electromechanical systems (MEMSs) technologies could be applied to meet the less stringent sensor requirements for integrated operation. 134 INERTIAL NAVIGATION The use of integrated GPS=INS for mapping the gravitational ®eld near the earth's surface has also enhanced INS performance by providing more detailed and accurate gravitational models. Inertial navigation also bene®ts GPS performance by carrying the navigation solution during loss of GPS signals and allowing rapid reacquisition when signals become available. Integrated GPS=INS have found applications that neither GPS nor INS could perform alone. These include low-cost systems for precise automatic control of vehicles operating at the surface of the earth, including automatic landing systems for aircraft and autonomous control of surface mining equipment, surface grading equipment, and farm equipment. 6.2 INERTIAL SENSORS The design of inertial sensors is limited only by human imagination and the laws of physics, and there are literally thousands of designs for gyroscopes and acceler- ometers. Not all of them are used for inertial navigation. Gyroscopes, for example, are used for steering and stabilizing ships, torpedoes, missiles, gunsights, cameras, and binoculars, and acceleration sensors are used for measuring gravity, sensing seismic signals, leveling, and measuring vibrations. 6.2.1 Sensor Technologies A sampling of inertial sensor technologies used in inertial navigation is presented in Table 6.2. There are many more, but these will serve to illustrate the great diversity of technologies applied to inertial navigation. How these and other example devices function will be explained brie¯y. A more thorough treatment of inertial sensor designs is given in [118]. TABLE 6.2 Some BasicInertial Sensor Technologies Sensor Gyroscope Accelerometer Physical Effect Used a Conservation of angular momentum Coriolis effect Sagnac effect Gyroscopic precession Electro- magnetic force Strain under load Sensor Implementation Methods Angular displacement Vibration Ring laser Angular displacement Drag cup Piezo- electric Torque rebalance Rotation Fiber optic Torque rebalance Electro- magnetic Piezo- resistive a All accelerometers use a proof mass. The physical effect is the manner in which acceleration of the proof mass is sensed. 6.2 INERTIAL SENSORS 135 6.2.2 Common Error Models 6.2.2.1 Sensor-Level Models Some of the more common types of sensor errors are illustrated in Fig. 6.1. These are (a) bias, which is any nonzero sensor output when the input is zero; (b) scale factor error, often resulting from aging or manufacturing tolerances; (c) nonlinearity, which is present in most sensors to some degree; (c) scale factor sign asymmetry, often from mismatched push±pull ampli®ers; (e) a dead zone, usually due to mechanical stiction or lock-in [for a ring laser gyroscope (RLG)]; and (f) quantization error, inherent in all digitized systems. Theoretically, one should be able to recover the input from the sensor output so long as the input=output relationship is known and invertible. Dead-zone errors and quantization errors are the only ones shown with this problem. The cumulative effects of both types (dead zone and quantization) often bene®t from zero-mean input noise or dithering. Also, not all digitization methods have equal cumulative effects. Cumulative quantization errors for sensors with frequency outputs are bounded by Æ 1 2 LSB, but the variance of cumulative errors from independent sample-to-sample A=D conversion errors can grow linearly with time. 6.2.2.2 Cluster-Level Models For a cluster of three gyroscopes or acceler- ometers with nominally orthogonal input axes, the effects of individual scale factor Fig. 6.1 Common input=output error types. 136 INERTIAL NAVIGATION deviations and input axis misalignments from their nominal values can be modeled by the equation z output  S nominal I  Mfgz input  b z ; 6:1 where the components of the vector b z are the three sensor output biases, the components of the z input and z output vectors are the sensed values (accelerations or angular rates) and output values from the sensors, respectively, S nominal is the nominal sensor scale factor, and the elements m ij of the ``scale factor and misalign- ment matrix'' M represent the individual scale factor deviations and input axis misalignments as illustrated in Fig. 6.2. The larger arrows in the ®gure represent the nominal input axis directions (labeled #1, #2, and #3) and the smaller arrows (labeled m ij ) represent the directions of scale factor deviations (i  j) and misalign- ments (i T j). Equation 6.1 is in ``error form.'' That is, it represents the outputs as functions of the inputs. The corresponding ``compensation form'' is z input  1 S nominal I  Mfg À1 fz output À b z g6:2  1 S nominal fI À M  M 2 À M 3  ÁÁÁgfz output À b z g6:3 % 1 S nominal I À Mfgfz output À b z g6:4 if the sensor errors are suf®ciently small (e.g., <10 À3 rad misalignments and <10 À3 parts=part scale factor deviations). Fig. 6.2 Directions of modeled sensor cluster errors. 6.2 INERTIAL SENSORS 137 The compensation form is the one used in system implementation for compensat- ing sensor outputs using a single constant matrix M in the form z input  Mfz output À b z g6:5 M  def 1 S nominal I  Mfg À1 : 6:6 6.2.3 Attitude Sensors 6.2.3.1 Nongyroscopic Attitude Sensors Gyroscopes are the attitude sensors used in nearly all INSs. There are other types of attitude sensors, but they are primarily used as aids to INSs with gyroscopes. These include the following: 1. Magnetic sensors, used primarily for coarse heading initialization. 2. Star trackers, used primarily for space-based or near-space applications. The U-2 spy plane, for example, used an inertial-platform-mounted star tracker to maintain INS alignment on long ¯ights. 3. Optical ground alignment systems used on some space launch systems. Some of these systems used Porro prisms mounted on the inertial platform to maintain optical line-of-sight reference through ground-based theodolites to reference directions at the launch complex. 4. GPS receiver systems using antenna arrays and carrier phase interferometry. These have been developed for initializing artillery ®re control systems, for example, but the same technology could be used for INS aiding. The systems generally have baselines in the order of several meters, which could limit their applicability to some vehicles. 6.2.3.2 Gyroscope Performance Grades Gyroscopes used in inertial navi- gation are called ``inertial grade,'' which generally refers to a range of sensor performance, depending on INS performance requirements. Table 6.3 lists some TABLE 6.3 Performance Grades for Gyroscopes Performance Units Performance Grades Parameter Inertial Intermediate Moderate Maximum deg=h10 2 ±10 6 10 2 ±10 6 10 2 ±10 6 Input deg=s10 À2 ±10 2 10 À2 ±10 2 10 À2 ±10 2 Scale Factor part=part 10 À6 ±10 À4 10 À4 ±10 À3 10 À3 ±10 À2 Bias deg=h10 À4 ±10 À2 10 À2 ±1010±10 2 Stability deg=s10 À8 ±10 À6 10 À6 ±10 À3 10 À3 ±10 À2 Bias deg=  h p 10 À4 ±10 À3 10 À2 ±10 À1 1±10 Drift deg=  s p 10 À6 ±10 À5 10 À5 ±10 À4 10 À4 ±10 À3 138 INERTIAL NAVIGATION generally accepted performance grades used for gyroscopes, based on their intended applications but not necessarily including integrated GPS=INS applications. These are only rough order-of-magnitude ranges for the different error character- istics. Sensor requirements are largely determined by the application. For example, gyroscopes for gimbaled systems generally require smaller input ranges than those for strapdown applications. 6.2.3.3 Sensor Types Gyroscope designers have used many different approaches to a common sensing problem, as evidenced by the following samples. There are many more, and probably more yet to be discovered. Momentum Wheels Momentum wheel gyroscopes use a spinning mass patterned after the familiar child's toy gyroscope. If the spinning momentum wheel is mounted inside gimbals to isolate it from rotations of the body on which it is mounted, then its spin axis tends to remain in an inertially ®xed direction and the gimbal angles provide a readout of the total angular displacement of that direction from body-®xed axis directions. If, instead, its spin axis is torqued to follow the body axes, then the required torque components provide a measure of the body angular rates normal to the wheel spin axis. In either case, this type of gyroscope can potentially measure two components (orthogonal to the momentum wheel axle) of angular displacement or rate, in which case it is called a two-axis gyroscope. Because the drift characteristics of momentum wheel gyroscopes are so strongly affected by bearing torques, these gyroscopes are often designed with innovative bearing technologies (e.g., gas, magnetic, or electrostatic bearings). If the mechanical coupling between the momentum wheel and its axle is ¯exible with just the right mechanical spring rateÐdepending on the rotation rate and angular momentum of the wheelÐthe effective torsional spring rate on the momentum wheel can be canceled. This type of dynamical ``tuning'' isolates the gyroscope from bearing torques and generally improves gyroscope performance. Coriolis Effect The Coriolis effect is named after Gustave Gaspard de Coriolis (1792±1843), who described the apparent acceleration acting on a body moving with constant velocity in a rotating coordinate frame [26]. It can be modeled in terms of the vector cross-product (de®ned in Section B.2.10) as a Coriolis ÀO v 6:7 À O 1 O 2 O 3 P T T R Q U U S  v 1 v 2 v 3 P T T R Q U U S 6:8  ÀO 2 v 3  O 3 v 2 ÀO 3 v 1  O 1 v 3 ÀO 1 v 2  O 2 v 1 P T T R Q U U S ; 6:9 6.2 INERTIAL SENSORS 139 where v is the vector velocity of the body in the rotating coordinate frame, O is the inertial rotation rate vector of the coordinate frame (i.e., with direction parallel to the rotation axis and magnitude equal to the rotation rate), and a Coriolis is the apparent acceleration acting on the body in the rotating coordinate frame. Rotating Coriolis Effect Gyroscopes The gyroscopic effect in momentum wheel gyroscopes can be explained in terms of the Coriolis effect, but there are also gyroscopes that measure the Coriolis acceleration on the rotating wheel. An example of such a two-axis gyroscope is illustrated in Fig. 6.3. For sensing rotation, it uses an accelerometer mounted off-axis on the rotating member, with its acceleration input axis parallel to the rotation axis of the base. When the entire assembly is rotated about any axis normal to its own rotation axis, the accelerometer mounted on the rotating base senses a sinusoidal Coriolis acceleration. The position and velocity of the rotated accelerometer with respect to inertial coordinates will be xtr cosO drive t sinO drive t 0 P T T R Q U U S ; 6:10 vt d dt xt6:11  rO drive À sinO drive t cosO drive t 0 P T T R Q U U S ; 6:12 where O drive is the drive rotation rate and r is the offset distance of the accelerometer from the base rotation axis. Fig. 6.3 Rotating Coriolis effect gyroscope. 140 INERTIAL NAVIGATION [...]... commonly used for in-air INS alignment for missiles launched from aircraft and for on-deck INS alignment for aircraft launched from carriers Alignment of carrier-launched aircraft may also use the direction of the velocity impulse imparted by the steam catapult 4 GPS- aided alignment, using position matching with GPS to estimate the alignment variables It is an integral part of integrated GPS= INS implementations... ooutput ˆ oinput ‡ dobias …6:17† dobias ˆ doconstant ‡ doturn-on ‡ dorandomwalk ; …6:18† where doconstant is a known constant, doturn-on is an unknown constant, and dorandomwalk is modeled as a random-walk process: d do ˆ w…t†; dt randomwalk …6:19† where w…t† is a zero-mean white-noise process with known variance Bias variability from turn-on is called bias stability, and bias variability after turnon... bias stability from turn-on to turn-on, which may result from thermal cycling of the gyroscope and its electronics, among other causes; and 3 bias drift after turn-on, which is usually modeled as a random walk (de®ned in p Section 7.5.1.2) and speci®ed in such units as deg=h= h or other equivalent units suitable for characterizing random walks After each turn-on, the general-purpose gyroscope bias... coordinate systems used in GPS= INS integration and navigation are given in Appendix C These include coordinate systems used for representing the trajectories of GPS satellites and user vehicles in the near-earth environment and for representing the attitudes of host vehicles relative to locally level coordinates, including the following: 1 Inertial coordinates: (a) Earth-centered inertial (ECI), with... directions of the vernal equinox (de®ned in Section C.2.1) and the rotation axis of the earth (b) Satellite orbital coordinates, as illustrated in Fig C.4 and used in GPS ephemerides 2 Earth-®xed coordinates: (a) Earth-centered, earth-®xed (ECEF), with origin at the center of mass of the earth and principal axes in the directions of the prime meridian (de®ned in Section C.3.5) at the equator and the... ‡ c2 a2 output ‡ Á Á Á : |{z} |‚‚‚‚{z‚‚‚‚} |‚‚‚‚{z‚‚‚‚} bias scalefactor g-squared …6:29† Some accelerometers also exhibit second-order output errors called cross-axis coupling errors, which are proportional to the product of the input acceleration component and an acceleration component orthogonal to the input axis: dai;cross-axis G ai aj ; …6:30† where ai is the input acceleration along the input... model used by GPS is 7,292,115,167Â10À14 rad=sec, or about 15.04109 deg=h This is its sidereal rotation rate with respect to distant stars Its mean rotation rate with respect to the nearest star (our sun), as viewed from the rotating earth, is 15 deg=h 6.4.3.2 GPS Gravity Models Accurate gravity modeling is important for maintaining ephemerides for GPS satellites, and models developed for GPS have been... However, spatial resolution of the earth gravitational ®eld required for GPS operation may be a bit coarse compared to that for precision inertial navigation, because the GPS satellites are not near the surface and the mass concentration anomalies that create surface gravity anomalies GPS orbits have very little sensitivity to surface-level undulations of the gravitational ®eld 6.4 SYSTEM IMPLEMENTATIONS... acceleration For pulse-integrating accelerometers, the feedback current is supplied in discrete pulses with very repeatable shapes, so that each pulse is proportional to a ®xed change in velocity An up=down counter keeps track of the net pulse count between samples of the digitized accelerometer output Fig 6.8 Single-axis accelerometers 6.2 INERTIAL SENSORS 149 Integrating Accelerometers The pulse-feedback electromagnetic... will slew the platform unless a fourth gimbal axis is provided for this contingency Floated-Ball Systems The function of gimbals is to isolate the stable platform from the rotations of the host vehicle Floated-ball systems achieve the same effect by ¯oating the platform (now shaped like a ball) inside a ¯uid-®lled sphere using ¯uid thrusters attached to the stable ball to control its attitude and keep . & Sons, Inc. Print ISBN 0-4 7 1-3 5032-X Electronic ISBN 0-4 7 1-2 007 1-9 did the autopilots for the German V-1 cruise missiles and V-2 ballistic missiles of. initializ- ing position and attitude. Time required for initializing INS attitude by gyrocompass alignment is measured in minutes. Time-to-®rst-®x for GPS receivers