1 Introduction. Position Estimation 4: 1.1 History

Transcription

1 1.1History 1 Introduction 1.1 History 1 Inertial Navigation Systems Name comes from use of inertial principles (Newton s Laws). Historical roots in German Peenemunde Group. Modern form credited to Charles Draper et 1.2 Advantages Most accurate dead reckoning available. Useful in wide excursion (outdoor) missions. Work anywhere where gravity is known. Are jamproof - require no external information. Radiate nothing - exhibit perfect stealth. 1.3 Disadvantages Cannot sense accelerations of unpowered space flight. Most errors exhibit Schuler oscillation (advantage?).

2 Most errors are time dependent. Requires input of initial conditions. 1.4 Fundamental Idea Mount three accelerometers along three orthogonal axes. Integrate twice to determine position. 2 Principles of Inertial Navigation 2 1.4Fundamental Idea 2 Principles of Inertial Navigation 2.1 Naive Concept Try strapping three accelerometers to the vehicle and double integrating their outputs: accelerometers This breaks many rules of math and physics: Accelerometers measure wrong quantity. They measure it in wrong reference frame. They represent it in wrong coordinate system.

3 The quest for ever better engineering solutions to these problems is the primary reason for the complexity of the modern INS. 2.2 First Fix: Convert Specific Force to Acceleration Accelerometer is a specific force transducer because the calibrated restraint responds to force in the spring, (which is not the net force on the mass). T W Accelerometer Fundamental equation of inertial navigation is Newton s second law applied to the proof mass: F = T + W = ma i 2 Principles of Inertial Navigation 3 2.2First Fix: Convert Specific Force to Acceleration Specific force is the quantity T m. We can express the inertial acceleration in terms of the specific and gravitational forces as follows: a i T W = = t+ w m m (1) At this point, (if gravity is known), it is possible to convert the specific force into inertial acceleration, so that it can be integrated. Notice that explicit knowledge of the gravitational field strength is required at every position of the vehicle. Inertial navigation is only viable when this field is known (or known to be insignificant). 2.3 Second Fix: Remove Apparent Forces Use the Coriolis Law twice. This is moderately scary math - but its just what you need to put into code to use a modern strapdown IMU. Three frames of reference: i: geocentric (inertial frame).

4 e: earth frame (rotating). v: vehicle frame (fixed to accels). Let the constant rotation of the earth with respect to inertial space be given by Ω. Let the rotation of the vehicle frame with respect to the earth be given by ρ. Let the inertial rotation of the vehicle frame be given by ω. Clearly: ω = Ω+ ρ (2) Let r x, v x, and a x be the position vector, velocity, and acceleration of the vehicle measured in the frame x. Since the position vector is the same in both the inertial and earth frames (the i and e origins are coincident), the inertial velocity of the vehicle can be expressed as: v dr i dr e i = = = i i (3) dr e + Ω r e = v e + Ω r e e 2 Principles of Inertial Navigation 4 2.3Second Fix: Remove Apparent Forces Intuitively, the velocity of the vehicle wrt the inertial frame is that wrt the earth plus the part caused by the rotation of the earth wrt the inertial frame. The result of differentiating the above one more time in the inertial frame is: a i dv i = = i dv e i + Ω dr e The accelerometers inherently provide measurements along the axes of the vehicle frame. Hence, it is most convenient to express derivatives in terms of this frame so that the outputs can be directly integrated. The first term on the right hand side can be referred to the vehicle frame by another application of the Coriolis law: dv e dv e = + ω v (5) e i v By substituting (2) [5] and then (2) [3] into (2) [4]: i (4)

5 a i a i a i Mobile Robot Systems dv i dv e dr e = = + Ω i i dv e dr e = + ω v e + Ω = dv e v v + ω v e + Ω [ v e + Ω r e ] dv e a i = + ( ω+ Ω) v e + Ω ( Ω r e ) v Substituting the specific force equation (equation (1)) and rearranging gives: a i = t + w = dv e v dv e + ( ω+ Ω) v e + Ω ( Ω r e ) v = t ( ω+ Ω) v e + w Ω ( Ω r e ) (7) This is one of the most convenient forms of the equation of motion of the vehicle. We will use it as THE equation of inertial navigation. i i (6) 2 Principles of Inertial Navigation 5 2.3Second Fix: Remove Apparent Forces The last two terms on the right hand side are both functions of only the position and are often grouped together and called gravity. Gravity is part gravitation and part centrifugal force: g w Ω ( Ω r e ) = (8) In a precision inertial guidance system, you do not just use 9.8 m/s/s for g. You have a map of its precise value everywhere on earth. Solving (2) for the quantities of interest (by integrating in the vehicle frame): t v e = [ t ( ω+ Ω) v e + g]dt + r e = 0 t 0 v e dt + r e0 These equations can be solved for the unknown position, and velocity, given the following information: (9) v e0

6 A model of the earth s acceleration due to gravitation w m as a function of position. The earth s sidereal rate of rotation Ω. The specific forces t from the accelerometers. The initial position r e0. The initial velocity. v e0 3 Gravity and Gravitation 6 2.3Second Fix: Remove Apparent Forces 3 Gravity and Gravitation Gravity is defined as the force per unit mass required to keep a test mass in the same position relative to the earth. Gravitation is the force that s proportional to the masses. By Newtons law of gravitation: W = w m --- = GM e R 3 R The practice of measuring and modelling gravity is called geodesy. An object fixed to the surface of the earth experiences a centrifugal force as viewed from a reference frame spinning with the earth. A plumb bob at the surface of the earth does not point toward its center but is rather displaced slightly toward the equator. Gravitation will be distinguished from gravity from now on.

7 4 Generic Mechanization of an INS 7 4.1Vector Formulation Ω Ω Ω R 4 Generic Mechanization of an INS circle of constant latitude Equator R G g For now, consider the problem in its coordinate system independent form. 4.1 Vector Formulation The mechanization equations amount to a need to do four things: Add gravitation w to specific force t. Remove centrifugal force Ω ( Ω r e ) due to the vehicle offset from the center of the earth. Remove coriolis force ω + Ω ( ) v e due to the vehicle s motion on the surface of the earth. Perform two integrations, incorporating initial conditions. 4.2 Temporal Error Propagation Why mess with all these correction terms whose magnitudes are only tiny anyway? Because the computed position is actually extremely sensitive to them.

8 t + - Mobile Robot Systems a i gravitation GM e (_) 3 from the accelerometers Ω Ω _ centrifugal ( ω + Ω) _ For a vehicle at the equator, moving eastward at a velocity of 10 meters per second, and accelerating at 0.1 g, the following table gives the magnitude of each term: Table 1: Term Magnitudes v e0 v e r e0 r e t 0 coriolis dt t 0 dt 4 Generic Mechanization of an INS 8 4.2Temporal Error Propagation Table 1: Term Magnitudes Term Name Expression Nominal Value Centrifugal Ω Ω r 1.5x10-4 e g The process of integration multiplies acceleration by the square of time, and 1 hour is 13 million seconds squared. After 1 hour, neglecting the centrifugal (smallest) term accounts for over 3 kilometers of accumulated error. Point one milli-g equals 3 Km! because t 2 in seconds is 3600*3600 = 10 million. Note that this error occurs even when the system is stationary for that time period - so it is truly time dependent. Term Name Expression Nominal Value Specific Force Gravitational Coriolis t g 2Ω v e 0.1 g 1.0 g 0.03 g

9 5 Implementation 5.1 Third Fix: Mechanize a Particular Coordinate System You can t just add up abstract vectors like t and w. You have to establish some coordinate system to actually do the computing. All inertial navigation systems use: Gyroscopes to measure vehicle rotation in inertial space. Accelerometers to measure inertial specific force. This package is often called the inertial measurement unit (IMU) or inertial reference unit (IRU). We either actively control or passively keeping track of the orientation of the accelerometers as the vehicle moves. The latter is the modern strapdown approach. A trade-off exists between mechanical complexity and computational complexity 5 Implementation 9 5.1Third Fix: Mechanize a Particular Coordinate System 5.2 Stabilization Techniques Three classes of systems exist on this spectrum: Gimballed Systems The first class, known as gimballed or geometric systems, employ a stabilized platform which is actively servoed to the required orientation. This type of system was the first practical class to be developed historically since navigational information was available directly from the gimbal angles. These systems require only minimal computational capacity. In practice, they have been replaced by another class of systems, called semi-analytic Semi-Analytic Systems These are gimballed systems which control the orientation of a platform in order to instrument the navigation frame only. Computations transform the sensor outputs from the inertial frame. Computations are also

10 used to determine the latitude and longitude of the vehicle. Gimballed systems employ gyro torquers which drive the platform to rotate as required. Untorqued gyros naturally implement an inertial reference but this is only useful in space. Instrumenting an earth reference requires gyro torquing to keep the platform level. The total torquer signal may include compensation for: The gyro drift rate. The sidereal rotation of the earth, called earth rate. The angular velocity of the vehicle with respect to the earth, called the vehicle rate Strapdown Systems The third class, known as strapdown or analytic systems, are said to be computationally stabilized. Strapdown systems get their name from the fact that they strap the sensors directly to the 5 Implementation Navigation Coordinate Systems vehicle chassis, eliminating the platform gimbals at the cost of causing the components to suffer the dynamic rotation of the vehicle. Strapdown mechanization places more stringent requirements on the sensory components used, and on the computational throughput, so these systems have appeared relatively recently in the history of development of inertial navigation. Such systems promise to replace semianalytic systems in most applications. 5.3 Navigation Coordinate Systems A second and most important way to classify systems: Space stabilized. Earth stabilized Space Stabilized Systems Employ an inertially fixed coordinate system. Navigation equations are simple (trivial even).

11 Very inconvenient for use in terrestrial applications Earth Stabilized Systems Maintain orientation referenced in various ways to the earth. Earth fixed systems employ a coordinate system fixed with respect to the rotating earth. (aka. base point systems) Local vertical systems are the most common earth stabilized systems and employ coordinates referenced to the local vertical direction with the other two axes horizontal. There are three variants of local vertical systems: North-slaved systems maintain one coordinate axis pointed north at all times. Free azimuth and wander azimuth systems allow the level coordinate axes to rotate with respect to north about the vertical axis as a function of latitude: 5 Implementation Taxonomy Of Inertial Navigation Systems 5.4 Taxonomy Of Inertial Navigation Systems Inertial Navigation Systems space stabilized local vertical (locally level) north slaved earth stabilized earth fixed (base point) free azimuth wander azimuth Note: any system above can be analytic, semi-analytic, or geometric

12 5.5 Common Mechanized Coordinate Systems The three most common terrestrial INS mechanizations differ only in the way that the platform rotates with respect to the earth ( ω ). Let the vehicle latitude be denoted λ and its longitude Λ North Slaved, Locally Level System Ω dλ ω x = Ω + cλ dt x dλ ω y = z y dt R dλ ω z Ω + λ dt vehicle = sλ Λ center at present location x axis horizontal, north y axis horizontal, east z axis vertical, down prime meridian equator Like all locally level systems, avoids gravity computations. Gyro torquers required. Permits easy calculation of lat, long. 5 Implementation Common Mechanized Coordinate Systems Used in long term cruise systems. dλ Singularity at the poles. dt Free Azimuth, Locally Level System ω x ω y = = dλ Ω + dλ cλcα sα dt dt dλ Ω + dλ cλsα d cα dt t ω z = 0 For Wander Azimuth: ω z = Ωsλ prime meridian center at present location x axis horizontal, angle α with north y axis horizontal, angle α with east z axis vertical, down Λ Ω λ equator n x α y z e R vehicle

13 Solves the high polar platform rate problem by providing no inertial platform rate about the vertical axis. Azimuth is allowed to wander with respect to north and computations keep track of the true direction of north. In the wander azimuth variation, only the earth rate is compensated for - vehicle rate is not Tangent Plane, Base Point System ω x = Ωcλ 0 ω y = 0 ω z = Ωcλ 0 one should be sin? center at present location x axis horizontal, initial north y axis horizontal, initial east z axis vertical, initial down prime meridian Ω R Λ λ 0 0 equator z z x y vehicle x y base point 5 Implementation Stable Table While the platform moves with the vehicle, its orientation with respect to the earth remains fixed at its initial value. Takes advantage of the fact that constant platform rates can be generated much more accurately than variable ones. Suitable for limited excursions only. 5.6 Stable Table This is the basis of mechanical stabilization. A feedback loop is used to regulate the table in a such a way as to prevent the precession of the gyroscopes. The loop works as follows: Vehicle yaws, tries to rotate gyro about its input axis. Gyro actually precesses about its output axis. Pickoff senses the output, signal amplified. Servo loop drives motor to rotate the table back to where it started.

14 Although the vehicle yaws, the table does not. 6 Error Dynamics of Inertial Navigation Schuler Tuning 6 Error Dynamics of Inertial Navigation 6.1 Schuler Tuning All INS systems which navigate close to the earth experience horizontal oscillatory errors reflecting the famous Schuler period of 84 minutes. Hence, horizontal error, being oscillatory, is inherently bounded and this is contrary to our intuition. We would expect a quadratic growth with time (because acceleration is integrated twice). Schuler tuning 1 is the tuning of a device to exhibit a natural frequency the same as that of a pendulum whose length is the radius of the earth. This technique, for example, makes the gyrocompass immune to the motions of a ship. A pendulum attached to a vehicle moving on the surface of the earth will be deflected 1. Invented by Max Schuler.

15 6 Error Dynamics of Inertial Navigation Schuler Loop because of the vehicle acceleration as shown below. θ a 2) Spring deflects this way θ 3) Which is apparent motion this way The hypothetical Schuler pendulum, having a length equal to the earth s radius, can be accelerated arbitrarily around the surface of the earth without being disturbed by the motion of the vehicle. In practice, one cannot construct such a pendulum, but any dynamic system with the correct natural frequency will exhibit this same stability property. 6.2 Schuler Loop Consider a stable platform servo, which is called a Schuler loop. It computes the angular velocity of the vehicle with respect to the earth and rotates the table appropriately in order to keep it level. g 1) Platform tilts off-level this way g 4) Which causes the platform to compensate this way to stay level There is an inherent minus sign in the process as illustrated above using a simple spring model of an accelerometer: a + - gθ a gθ Accelerometer g θ dt dt v 1 -- R ω R dt r

16 The feedback path is constructed by recognizing that the accelerometer specific force includes any component of gravity due to an off-level condition. This feedback is inherent in the accelerometer - there is no electrical connection required. If the table is off level, the accelerometer will measure the gravity component: a gsinθ gθ = (for small angles) (10) We can investigate the dynamics by unwinding the loop: g a gθ = a g ωdt = a -- vdt R g a gθ = a -- [ a gθ] R dt (11) 6 Error Dynamics of Inertial Navigation Vertical Instability Differentiate this twice to obtain the characteristic differential equation of the loop: gθ g = -- ( a gθ) θ R (12) g a + -- θ = -- R R Thus, given any forcing function whatever, the table will naturally oscillate at a period of: T = 2π R ḡ - = 84minutes (13) which is the period of a pendulum whose length is the radius of the earth! The same horizontal oscilation happens computationally even if there is no stable table. 6.3 Vertical Instability Vertically, the situation is much worse: Suppose the position estimate is slightly high in altitude. This causes gravity to be underestimated.

17 Which means the actual acceleromeer readings will be interpreted partly as an acceleration upward. Which causes an even higher overestimate of altitude. Yadda yadda! 6.4 Simple Error Analysis of Inertial Navigation Most errors in an INS oscillate with the Schuler period. Lets compute the above behaviors from first principles. Consider a single error source - accelerometer bias Consider the basic navigation equation in vector form - inertial acceleration expressed in terms of the specific force indicated by the accelerometers a, and gravitation g: 2 d a inertial = r = a+ g dt 2 (14) We will use a technique called perturbative analysis. A hypothetical perturbative error is 6 Error Dynamics of Inertial Navigation Simple Error Analysis of Inertial Navigation applied to the sensed specific force and the effect of this on the system output is investigated. Let the indicated specific force include an error denoted δa, and cause errors in the computed position and gravitation denoted by r and g. This is accomplished through the substitutions: a i = a t + δa r i = r t + δr g i = g t + δg (15) Where the subscripts i and t represent indicated and true quantities. Substituting this back into the original equation and cancelling out the original equation yields. 2 d δr = δa + δg (16) dt 2 This is the differential equation which describes the propagation of errors from the accelerometer to the position and gravity computations.

18 The gravitational force is a function only of position. By a Taylor series expansion assuming a spherical homogeneous earth, the gravitation error can be rewritten in terms of the error in the position as follows: g GM = r = r 3 GM r ( r r) 32 / By the product rule of differentiation: δg = g δr = r Substituting this into (2) yields: 2 (17) GM δr + 3 GM ( r δr)r r 3 Further analysis of this equation requires that a coordinate system be adopted. Let the origin be placed at the center of the earth, and the three cartesian axes be oriented arbitrarily. Then the above equation in component form is: r 5 d GM δr δa dt 2 = r 3 δr + 3 GM r 5 ( r δr)r (18) 6 Error Dynamics of Inertial Navigation Simple Error Analysis of Inertial Navigation δx GM r 3 δx 3 GM r 5 ( xδx+ yδy + zδz)x = δa x δy GM r 3 δy 3 GM r 5 ( xδx+ yδy + zδz)y = δa y δz GM r 3 δz 3 GM r 5 ( xδx+ yδy + zδz)z = δa z (19) Any particular solution to these equations requires knowledge of the trajectory followed by the vehicle. Let the start point for the system be along the z axis on the surface of the earth, and let the vehicle trajectory remain close to this point so that x = y = 0 and z = r = R. Under this assumption, the cross coupling terms in the equations cancel and they reduce to: δx + ( GM R 3 )δx = δa x δy + ( GM R 3 )δy = δa y δz ( 2GM R 3 )δz = δa z (20) Let g 0 and R 0 be the gravitational acceleration of and radius to the local region, which for our assumed trajectory are approximately constant. If the accelerometer

19 errors are assumed to be constant biases, the solutions to these equations for zero initial conditions are: 6 Error Dynamics of Inertial Navigation Other Error Dynamics δ h δ v is the horizontal error in any direction and is the vertical error. δx δy = = δa x g 0 R 0 δa y g 0 R 0 cos sin g t R 0 g t R 0 (21) Error, ft X δ v δ h δz = δa z g 0 R 0 cosh 2g t R Time, min Hence, the accelerometer feedback that is inherent when operating in a gravitational field bounds the horizontal error channels (sinusoidal) at the cost of a divergent (exponential) vertical channel. Without such a field, the errors all grow quadratically with time for constant accelerometer bias. The development of position error over time is expected to resemble the following, where 6.5 Other Error Dynamics Similiar analyses can be conducted for many different sources of error. Some interesting results are summarized below. Let ω 0 be the Schuler frequency, g be gravity, R be the earth s radius, be the V c

20 inertial velocity in the crosstrack direction, be the present latitude, and t be time. Then: Table 2: Propagation of INS Errors Error Source Acceleromoeter Bias ε a Position Error ε a ( cosω 2 0 t) ω 0 Initial Velocity sinω Error ε 0 t v ε v ω 0 Physical Source instrument initial condition λ 6 Error Dynamics of Inertial Navigation Other Error Dynamics Table 2: Propagation of INS Errors Error Source Position Error Azimuth Gyro Drift ω z t 2 ω z VI --- cosω 0 t 1 c Physical Source instrument For mobile robots, it would be nice to know the error dynamics formulas for and INS with odometry (aka VMS or vehicle motion sensor ) inputs. ω 0 2 Vertical Gyro sinω Drift ω 0 t v Rω v t ω 0 Initial Vertical Alignment θ 0 Rθ 0 ( 1 cosω 0 t) Initial Azimuthal Alignment ψ 0 ψ 0 VI c t instrument initial condition calibration instrument initial condition calibration instrument

21 7 Real Inertial Systems 7.1 Aided Inertial Systems The use of external measurements to reduce error is referred to as damping, and the systems employing the technique are called aided or augmented inertial systems. In practice, all INS systems are aided in some way. The damping of the vertical channel can be accomplished in many ways. Measurements of position, velocity, and attitude can be used singly or in any combination. The particular solution adopted is application dependent. Some alternatives are: Barometric altitude Radar altimeters Doppler radar velocity GPS Landmarks Map matching 7 Real Inertial Systems Aided Inertial Systems Periodic return to same spot (survey points) Zero velocity update Odometry Magnetic heading 7.2 Alignment When an INS is started, a sequence of operations, taking several minutes, is often performed before the system is operational. Mechanical setup involves spinning up the gyros and waiting for the components to reach operating temperature (often, precision components are environmentally stabilized). After this is completed, the system undergoes a sequence which effects the initial alignment of the accelerometers along the axes of the navigation coordinate system. In self alignment, the local vertical is determined through the gravity vector and the earth s spin provides a reference for north. Levelling is driving the platform until the horizontal accelerometers read zero.

22 North alignment or gyrocompassing 1 is accomplished by rotating the now level platform about the vertical until one gyro senses no component of the earth s rotation. Modern GPS aided systems do moving base alignment where the difference in GPS readings over time can be used to determine vehicle heading. 7.3 Accuracy Commercial cruise systems can achieve accuracies on the order of 0.2 nautical miles of error per hour of operation. Pitch and roll are often accurate to 0.05 and true geographic heading to 0.5. In some cases, position accuracy along the trajectory (alongtrack) and both normal directions (crosstrack and vertical) are distinguished. Land vehicle navigation systems achieve 0.2% to 2% of distance travelled. Pitch and 7 Real Inertial Systems Accuracy roll can be measured to 0.1 and heading to Although the term is used commonly, it does not mean that the gyro operates according to the principle of the gyrocompass. It means that it does the same thing as the gyrocompass, namely, finds north.

23 8 Summary Inertial navigation is based on Newton s laws and works everywhere that gravity is known. It is stealthy and jamproof. The basic process is accelerometer dead reckoning (so-called free inertial operation). Errors of 1 part in 10,000 in measuring acceleration of predicting gravity cause kilometers of position error after 1 hour of operation regardless of any motion. Naive approaches are seriously flawed. Three problems and their solutions are that accelerometers measure: the wrong quantity <-> model gravity wrt the wrong ref frame <-> Coriolis Law in the wrong Coord System <-> gyros track attitude Modern strapdown systems are computationally stabilized. The stable platform is no longer necessary in an INS though it has other uses in pointing systems. 8 Summary Accuracy Wander azimuth systems deliberately do not try to point North. Hence, they have no problems operating near the poles where North can change instantaneously. Some forms of horizontal position error in free inertial mode are oscillatory with the Schuler period of 84 minutes and hence are bounded. Vertical position is unstable in free inertial mode and various other aiding devices are used to damp it.

24 9 Notes - mine perceptor INS unit survey for info. 10 Bibliography [1] Broxmeyer, C., Inertial Navigation Systems, McGraw Hill, 1964 [2] Britting, K., Inertial Navigation System Analysis, Wiley, [3] Draper, C. S., W. Wrigley, and J Hovorka: Inertial Guidance, Pergammon Press, New York 1960 [4] Farrell, J., Integrated Aircraft Navigation, Academic Press, [5] Fernandez M., and Macomber G. R., Inertial Guidance Engineering, Prentice Hall, Englewood Cliffs, NJ, 1962 [6] Ivey, D. G., Physics, University of Toronto Press, 1982 [7] Kuritsky M. M., and Goldstein M. S., Inertial Navigation, Proceedings of the IEEE, vol 71, no 10, [8] McGreevy, J. Fundamentals of Strapdown Inertial Navigation, Litton Systems Inc, Moorpark Ca, Rev C, May 1986 [9] Odonnell, C. F. (ed.), Inertial Navigation Analysis & Design, McGraw Hill, New York, 1964 [10] Parvin, R., Inertial Navigation, Van Nostrand, 1962 [11] Pittman, G. R. Jr (ed.), Inertial Guidance, John Wiley Z& Sons Inc, New York [12] Sutherland, A. A. Jr, and Gelb, A., Application of the Kalman Filter to Aided Inertial Systems, Analytical Sciences Corporation, Winchester, Mass. [13] Savage, P. G., Strapdown Systems Algorithms, AGARD Lecture Series No. 133, Strapdown Associates Inc., Minnetonka, Minnesota. [14] Savant C. J. Jr., R. C. Howard, C. B. Solloway, and C. A. Savant, Principles of Inertial Navigation, McGraw Hill Book Company, Inc, New York, 1961 [15] Unknown, An Introduction to Inertial Navigation, Litton Systems Inc, Woodland Hills, Ca, 9 Notes Accuracy 11 Glossary aided inertial systems - inertial systems which make use of redundant external measurements of navigation quantities. alongtrack - the direction parallel to the direction of motion. analytic - a designation for systems which compute rather than instrument the navigation frame quantities. base point - designation for system which maintain the platform orientation at the initial orientation with respect to the earth. crosstrack - the direction perpendicular to the direction of motion, measured horizontally. damping - the process of filtering redundant external measurements with the INS output in order to reduce the magnitude of system error. direction cosine matrix - the matrix which describes the transformation of the body axes into the navigation frame. earth fixed - see base point. earth-stabilized - designation for systems which maintain a navigation frme which is defined with respect to the earth in some way. easting - eastern coordinate in UTM coordinates. free azimuth - a designation for systems providing no platform rate about the azimuth. geometric - designation for systems which perform geometric stabilization utilising a stable platform.

25 gimballed - designation for systems incorporating a stable platform. gimbal lock - singularity of the platform gimbal mechanism. gravity - vector sum of gravitation and centrifugal force. grid - a regular coordinate grid placed on a map. Unlike latitude and longitude, grid lines are parallel and equally spaced. gyrocompassing - the process of using the earth s spin to find the direction of north. inertial measurement unit - the sensor suite consisting of the gyros and accelerometers. inertial reference unit - see inertial measurement unit. latitude - angle of a point on the earth measured from the equatorial plane. levelling - the process of levelling a stable platform during alignment. locally level - designation for systems which maintain the platform level. local vertical - see locally level. longitude - angle of a point on the earth measured from the prime meridian. mechanization - the process by which the navigation equations are implemented in hardware and software. meridian - a line of constant longitude. northing - northern coordinate in UTM coordinates. north-slaved - designation for systems which physically track the direction of north by rotating the platform. 11 Glossary Accuracy parallel - a line of constant latitude. pickoff - the transducer which measures a gimbal angle. pole problem - the degradation of performance experienced by inertial systems as they approach the earth s poles. rebalance loops - a control loop which actively prohibits deflection of a compliant transducer. reference ellipsoid - an idealized mathematical model of the earth. Schuler loop - the loop which employs gravity feedback in order to level a table. Schuler period - the period of oscillation of the Schuler pendulum, or 84 minutes. Schuler tuning - the process of designing a system to exhibit the Schuler natural frequency. self alignment - the process of aligning an INS without the aid of external measurements. semi-analytic - a designation for systems which instrument only the navigation frame. signal generator - see pickoff. space-stabilized - designation for systems which maintain constant inertial platform orientation. specific force - the net inertial acceleration minus gravitational attractions. stable platform - a platform whose orientation is stable in a particular reference frame. strapdown - see analytic.

26 11 Glossary Accuracy star-tracker - a device which tracks ad reports the bearing of a star/ torquer - device for applying torque to a gyro or gimbal. vehicle rate - the angular velocity of the vehicle with respect to the center of the earth. wander angle - the rotation of the coordinate axes with respect to north. wander azimuth - a designation for a system which torques the azimuth to follow the earth rate only.