Chapter 9 Strapdown system technology 9.1 Introduction The preceding chapters have described the fundamental principles of strapdown navigation systems and the sensors required to provide the necessary measurements of angular rate and specific force acceleration. In this chapter, aspects of strapdown system technology are discussed. 9.2 The components of a strapdown navigation system As indicated in the earlier discussion, a strapdown inertial navigation system is basically formed from a set of inertial instruments and a computer. However, for reasons which will shortly become clear, such a system may be sub-divided further into the following component parts: instrument cluster; instrument electronics; attitude computer; navigation computer. These components, which form the basic building blocks of a full strapdown navigation system, are shown schematically in Figure 9.1. These units will be mounted in a case, together with the necessary electrical power supplies and interface electronics, which may then be installed in a vehicle requiring an on-board navigation capability. Whilst it is often assumed that the unit will be fixed rigidly in the vehicle, it is usually necessary for it to be installed on anti-vibration (AV) mounts to provide isolation from vehicle motion at frequencies to which the unit is particularly sensitive. Whilst we are primarily concerned here with the implementation of a full inertial navigation system, applications arise in which the full navigation function is not required. For example, in some short range missile applications, inertial
Inertial instrument block Instrument support electronics Attitude computation Navigation computation ^^^^^^^^^^^^^?l I Inertial navigation system (INS) Figure 9.1 Strapdown inertial navigation system building blocks measurements, typically of angular rate and specific force, are required purely for flight control purposes. In such cases, the instrument cluster and instrument electronics blocks alone are used to form what is known as an inertial measurement unit or IMU. For other applications requiring attitude and heading information alone, the IMU is combined with a processor in which the attitude equations are solved. The resulting system is known as an attitude and heading reference system or AHRS, the processor being referred to here as the attitude computer. An AHRS is sometimes used in combination with a Doppler radar to form a navigation system. Finally, the addition of a further computer in which the navigation equations are solved provides a full inertial navigation capability. In the following sections, the components of the strapdown inertial navigation system defined above are described separately in more detail. This includes some discussion of the requirements for internal power supplies and AV mounts. 9.3 The instrument cluster 9.3.1 Orthogonal sensor configurations The instrument cluster usually includes a number of gyroscopes and accelerometers which provide measurements of angular rate and specific force, respectively. The unit may contain either three single-axis gyroscopes or two dual-axis gyroscopes, as well as three single-axis accelerometers, all attached to a rigid block which can be mounted in the body of the host vehicle, either directly or on AV mounts. The sensitive axes of the instruments are most commonly mutually orthogonal in a Cartesian reference frame, as illustrated in Figure 9.2. This arrangement of the instruments allows the components of angular rate and specific force in three mutually orthogonal directions to be measured directly, thus providing the information required to implement the strapdown computing tasks.
Gyroscope 3 Accelerometer Accelerometer 3 Gyroscope Gyroscope Accelerometer 2 Figure 9.2 Orthogonal instrument cluster arrangement As indicated earlier, systems using dual-axis sensors such as the dynamically tuned gyroscope as an alternative to the single-axis rate-integrating gyroscope require one less sensor. A dual-axis sensor configuration also provides an additional rate measurement. Through careful choice of the relative orientation of the two dual-axis sensors, the redundant measurement provided by one gyroscope may be used to monitor the performance of the other as part of a built-in test facility. In practice of course, many other factors will influence the choice between these two gyroscopes and their different characteristics are discussed in Chapter 4. Other instrument arrangements are possible using modern sensing techniques which offer various novel approaches. For instance, a pair of multi-sensors mounted orthogonally or a single laser triad with suitable accelerometers may be used to form an instrument cluster. Such sensors are described in Chapters 5 and 6. 9.3.2 Skewed sensor configurations Theoretically, it is possible to mount the instruments in orientations other than the orthogonal arrangement illustrated in Figure 9.2. This type of configuration will function provided the measurements may be expressed as independent linear combinations of the orthogonal components of angular rate and specific force. The orthogonal components may then be extracted from the measurements as part of the strapdown processing task. Such instrument configurations are referred to as skewed sensor configurations and may be used to advantage in certain applications. A practical implementation is the silicon drive discussed in Section 7.7.1. Skewed sensor arrangements are used primarily in applications requiring on-line failure detection and fail-safe operation as discussed in Section 9.3.4.
However, they may also be used in situations where the turn rate about a singleaxis of a vehicle may exceed the normal operating range of a gyroscope having performance characteristics suitable for the particular application. By mounting the gyroscopes so that their sensitive axes form an angle with the high rate axis of the vehicle, it is possible to ensure that the resolved component of the turn rate falls within the maximum range of the sensor. Given knowledge of the skew angle, it is possible to calculate the turn rate about the vehicle axis using the measurements provided by the skewed sensors. An example of this technique based on a system using two dual-axis gyroscopes is discussed in the following section. 9.3.3 A skewed sensor configuration using dual-axis gyroscopes Two dual-axis gyroscopes may be configured in the symmetrical form illustrated in Figure 9.3. A body axis frame Ox^y^ib is indicated along with gyroscope axes Ox\y\z\ and Ox2y2Z2> The gyroscope spin axes lie in the Ox^y^ plane in the directions Ox \ and Ox2, inclined at angle < to OJq 5. The turn rates about the respective body axes are denoted GO*, Go 3 ; and co z. This particular configuration may be used where the rotation rate (GO*) about the axis x^ exceeds the normal operating range of the gyroscopes. For the arrangement shown in the figure, the turn rates COA, OOB, GOC and COD, sensed by the gyroscopes may be Gyroscope 1 Gyroscope 2 Input axes Spin axis Spin axis Input axes Figure 9.3 Dual-axis gyroscope skewed configuration
expressed in terms of the body rates as follows: Gyroscope 1: (9.1) Gyroscope 2: where O is the absolute value of the angular displacement between the spin axis of each gyroscope and the body axis jq,. Estimates of the body rates may be derived by summing and differencing the measurements of the turn rates provided by the gyroscopes as shown below, where the A notation is used to denote an estimated quantity. (9.2) This equation is in fact a least-squares solution to the measurement eqn. (9.1). The component of oo* sensed by each gyroscope is equal to oo x times the cosine of the direction angle, 0, between the body axis x\, and the gyroscope's input axes. For the instrument configuration considered here, (9.3) If the maximum body rate about the axis Jq 3 is 1200 /s and the maximum rate which can be measured by the gyroscopes is 600 /s, then in the absence of any motion about the other axes, the angle 0 must be greater than 60, that is, the angular displacement of the spin axis (<f>) as shown in the Figure 9.3 should not exceed 45. In general, the value of O will need to be less than this figure to cope with turn rates about the other axes of the body. In order to satisfy a particular set of performance objectives using a skewed sensor configuration of this type, it will be necessary to use higher quality gyroscopes or to compensate the sensors more precisely than would be required for a conventional strapdown arrangement. It can be shown that biases on the measurements of turn rate provided by the sensors and the accuracy of mounting alignment become more critical in inertial systems which use skewed sensor arrangements. From eqn. (9.2), it can be shown that biases in the four rate measurements, denoted $ooa, SOOB, SOOC and SGOD, and an error in the skew angle O will give rise to biases in
the estimates of the rates about the body axes, ScO x, Soo^ and 8oo z, given by (9.4) with the result that the effects of the measurement biases on the estimates of turn rate in the x or y direction can be magnified through the use of the skewed sensor configuration. The system is also particularly susceptible to misalignment of the sensor mounts. Hence, accurate knowledge of the skew angles is required in order to obtain precise estimates of the body rates. This is particularly true in the situation illustrated in the Figure 9.3, where the sensitive axes of the gyroscopes are displaced by large angles with respect to a potentially high rate axis. In general, skewed systems based on conventional angular momentum gyroscopes are expected to be applicable in situations where body rates exceed the sensor maximum angular rate capability only transiently. In many applications where high turn rates are likely to be sustained, it is considered that optical rate sensors, such as the ring laser gyroscope or the fibre gyroscope, now offer the best solution because of the high rotation rate capability and excellent scale-factor linearity offered by these types of sensors. The ring laser gyroscope in particular, offers superior scale factor performance. 9.3.4 Redundant sensor configurations For reasons of safety and reliability, many applications require navigation systems with on-line failure detection and fail-safe operation [1 3]. To satisfy this objective using a strapdown system, additional sensors are required to provide a level of redundancy in the measurements. This may be achieved using an orthogonal sensor arrangement by adding additional sensors to detect the turn rates and accelerations in each vehicle axis. Alternatively, and more commonly, a skewed sensor configuration is often proposed. For instance, a skewed sensor system employing four dual-axis gyroscopes and eight accelerometers may be used to provide quadruplex redundancy for an aircraft's flight control and avionics sensor unit. The gyroscopic input axes are equally distributed on a cone, the axis of which is aligned with the pitch axis of the aircraft for the purposes of this example. The accelerometers may be oriented in a similar manner. The four separate sources of rate information provided by such a system are indicated in Table 9.1 where GO;* and oo^ are the rates measured about the x and y input axes of the ith gyroscope, and K\ and K2 are geometrical constants. For the instrument configuration considered here, K\ = l/>/2 and K^ = 1/2. Similar equations may be written for the acceleration in aircraft body axes. This arrangement
Table 9.1 Sources of rate information provided by a skewed sensor system employing four dual-axis gyroscopes Sources of information Pitch rate Roll rate Yaw rate ^l (wijc + CDiy) K\ (<*>1JC - COi 3 ;) K\ (^3JC - <»3y) K\(CO2JC + 0>2y) (W2jc - C02 y ) - K\ (0)3* - my) (^2x ~ ^2y) ~ %l (^>lx ~ <*ly) K\ (^3JC + CO3 V ) K2(u>2x ~ ^2y) ~ ^2(<*>4jt ~ ^>4y) («>4x ~ <*%) + K\ (^\x ~ <*>\y) K\ (CO4 X + U>4y) (^>4JC ~ ^4y) + K\ (0)3^ - ^y) ^2(^2x ~ U>2y) + ^2(^4JC ~ ^>4y) of sensors, which is discussed in detail in Reference 1, offers high reliability with a 'fail-operational, fail-safe' level of fault tolerance. Fail-operational means that a fault must be detected, localised and the system must be dynamically reconfigured. Fail-safe refers to the capability to detect a fault which must not affect system safety. To achieve the same level of redundancy with an orthogonal sensor arrangement would require up to eight two-axis gyroscopes and twelve accelerometers. The reader interested in redundant strapdown sensor configurations is referred to the many excellent papers on the subject which include References 2 and 3 given at the end of this chapter. Redundant sensor configurations are discussed further in Section 15.4.5 in the context of the Segway machine. 9.4 Instrument electronics The instrument electronics unit contains the dedicated electronics needed to operate the inertial sensors. Typically, this includes instrument power supplies, read-out electronics to provide signals in the form needed by a navigation processor and possibly a computer. The precise requirements vary in accordance with the types of instruments used and the level of performance which is needed. For the vast majority of applications, the electronic signals provided by the inertial sensors are required in a digital format for input directly to a computer. Whilst many sensors naturally provide output in digital form, this is not always the case. Where analogue output is provided, this will need to be converted to digital form. The analogue-to-digital conversion process forms part of the instrument electronics. The output signals from the inertial sensors are often provided in incremental form, that is, as measurements of incremental angle and incremental velocity corresponding to the integral of the measured angular rate and linear acceleration respectively over a short period of time, r. The incremental angle output (W) which may be provided by a gyroscope may be expressed mathematically as: / t+t co dt (9.5)
where GO is the measured turn rate. Similarly, the incremental velocity output (hv) from an accelerometer may be written as: / t+t fdt (9.6) where / is the measured acceleration. This form of sensor output is very convenient since it eases the tasks of updating attitude and velocity. Many contemporary sensors, ring laser gyroscopes for example, naturally provide output signals in this form, whilst for others it is a result of the digitising process carried out within the inertial measurement unit. The way in which the incremental measurements are used is discussed in Chapter 11 together with other aspects of the strapdown processing tasks. Many conventional sensors such as spinning mass gyroscopes and pendulous accelerometers typically operate in a null seeking or re-balance loop mode in order to achieve a linear and accurate response characteristic. In such cases, instrument re-balance electronics will form part of the instrument electronics block along with gyroscope spin motor and pick-off power supplies. The use of a computer within the inertial measurement unit (IMU) [4] enables some form of on-line compensation of the instrument outputs to be performed based on instrument characterisation data obtained during laboratory or production testing, as described in Chapter 8. Since such computing tasks are very specific to the type of instrument used, they may well be implemented here rather than as part of the subsequent attitude and navigation processing. Because instrument characteristics are often temperature dependent, there may also be a need to compensate the instrument outputs for temperature variation in order to achieve satisfactory performance. It follows therefore that instrument temperature monitoring is often required. This is often the subject of some dilemma as to where to monitor the temperatures. In many sensors, the variation in performance with temperature is the result of the temperature sensitivity of magnetic material used in the very core of the sensor. Hence, it may not be adequate to sense temperature outside the instrument cluster or merely close to the case of the instrument. Finally, it may well be advisable for most applications to carry out some form of on-line testing of the inertial sensors and associated electronics. This may involve checks to confirm that the outputs of the sensors remain within certain known limits appropriate to the application and that they continue to vary in the expected manner whilst operational. For instance, a sensor output remaining at a fixed level for an extended period of time may well suggest a failure has occurred and a warning should be given. Such tasks may also be implemented within the IMU processor which would form part of the built-in test equipment (BITE) within the unit. It follows from the above remarks that the instrument electronics block could typically comprise the following: instrument power supplies; re-balance loop electronics; temperature monitoring electronics;
Gyro 1 Gyro 2 Acc'r 1 Acc'r 2 Acc'r 3 Inertial sensor block RO. Monitor electronics I Rotor speed Temp. Re-balance loop electronics Re-balance loop electronics Clock control Gyro output digitisers Acc'r output digitisers Calibration R.O.M. Power supplies Data bus Output interface Address bus Microcomputer Prime power source Compensated body angular rates Compensated body linear acceleration Bite status Clock Figure 9.4 Inertial measurement unit functions instrument compensation processing; analogue-to-digital conversion electronics; output interface conditioning; built-in test facility. These components are illustrated schematically in Figure 9.4 which shows an inertial measurement unit containing two dual-axis gyroscopes and three accelerometers. 9.5 The attitude computer The attitude computer essentially takes the measurements of body rate about three orthogonal axes provided by the inertial measurement unit and uses this information to derive estimates of body attitude by a process of 'integration'. The attitude is usually represented within the computer as a set of direction cosines or quaternion parameters as discussed in Chapter 3, either of which are appropriate for on-line attitude computation. The Euler angle representation described in Chapter 3 is not generally recommended for implementation in strapdown systems. As a result of the preponderance of trigonometric terms in the equations coupled with the presence of singularities for pitch angles of ±90, the Euler equations do not lend themselves to real-time solution in an on-board navigation processor. However, it should be borne in mind that there may well be a requirement to extract the Euler angles from the direction cosines or quaternion parameters for control purposes in some applications. The equations to be solved in the attitude computer are summarised below assuming quaternion parameters are to be used to define the attitude of the vehicle body with respect to the navigation reference frame. The quaternion may be expressed as a four-element vector [a b c d] T, the elements of which are calculated by solving
the following set of differential equations: a = 0.5 (bco^ H- cody -f- d( z ) b = 0.5(000* ^CO 3 ; + coo z ) c = 0.5(^CO x + aa)^ bod z ) d = -0.5(COO x b(x) y aco z ) where (O x, oo^ and oo z are estimates of the components of vehicle turn rate with respect to the navigation reference frame. These quantities are computed by differencing the measurements of body rate output by the IMU and estimates of the turn rate of the navigation frame calculated in the navigation computer. The quaternion parameters may be used to construct the direction cosine matrix which relates the body reference frame to the navigation reference frame (Cg) using: (9.8) It is customary in most strapdown attitude computation schemes to carry out self-consistency checks. In the case of the quaternion, the self-consistency check involves confirming that the sum of the squares of the individual quaternion elements remains equal to unity, that is, a 2 + b 2 + c 2 + d 2 = 1 (9.9) The attitude computation algorithm used for a given application must be able to keep track of vehicle orientation whilst it is turning at its maximum rate and in the presence of all of the motion of the vehicle, including vibration. Algorithms which may be used to implement the attitude computation function in the presence of such motion are described in Chapter 11. 9.6 The navigation computer The solution of the navigation equations is carried out in the navigation computer. To implement the navigation function, it is first necessary to transform, or resolve, the specific force measurements provided by the accelerometers, denoted here by the vector f b, into the navigation reference frame. This can be accomplished using the attitude information provided by the attitude computer. Using the direction cosine representation of attitude for instance, the required transformation is achieved using: fn = cn f b (9.10)
where f n is the specific force expressed in navigation axes and C is the direction cosine matrix described earlier. Both the specific force and the direction cosine matrix are time-varying quantities. Therefore, care must be taken to ensure that all significant movements of the vehicle, including turn rates and vibratory motion, can be accommodated in the computer implementation of this equation. The resolved specific force components form the inputs to the navigation equations which are used to calculate vehicle velocity and position. The navigation equations are described in Chapter 3 but are repeated here for completeness. For a system which is required to navigate in the vicinity of the Earth to provide estimates of north and east velocity, latitude, longitude and height above the Earth, the equations to be solved may be written as follows: (9.11) (9.12) (9.13) (9.14) (9.15) (9.16) where V^,VE,VD, are the north, east and vertical components of vehicle velocity with respect to the Earth, /N, /E, /D, are the components of specific force resolved in the local geographic reference frame, L is the vehicle latitude, t is the vehicle longitude, h is the vehicle height above ground, RQ is the mean radius of the Earth, 2 is the turn rate of the Earth and g is the acceleration due to gravity. Refinements to these equations needed to take account of the shape of the Earth and variation in gravitational attraction over the surface of the Earth are given at the end of Chapter 3. The turn rate of the vehicle with respect to the local geographic navigation frame <o b = [oo x a) y oo z ] T which is required to implement the attitude computation process described above is given by: (9.17) where co^ is the turn rate of the body with respect to inertial frame as measured by the strapdown gyroscopes in the inertial measurement unit and O)^n is the turn rate of
the navigation frame with respect to the inertial frame, which is computed as follows: (9.18) Algorithms which may be used to implement the navigation function are described in Chapter 11. 9.7 Power conditioning The raw power supplies available in the host vehicle, whether it is an aircraft, a ship or a land vehicle, will not usually be sufficiently stable or provide the particular voltage levels required by the inertial navigation system. Therefore, it will be necessary to include power conditioning within the unit to generate the supply voltages required which are smoothed sufficiently and controlled to the desired amplitude to ensure satisfactory operation of the navigation system. 9.8 Anti-vibration mounts A strapdown inertial navigation system will usually be installed on anti-vibration (AV) mounts to provide isolation from vehicle motion at frequencies to which the unit is particularly sensitive. In many applications, the unit may need to be isolated from certain frequencies in the vibration spectrum of the vehicle which may excite resonances within the inertial sensors or give rise to computational errors. The design of suitable AV mounts is frequently a complex task requiring careful matching of the mount design to the characteristics of the inertial sensors within the unit as well as the range and frequency of the perturbing characteristics of the host platform. The effects of vibration are discussed in more detail in Chapter 12 in relation to both instrument errors and overall system performance. 9.9 Concluding remarks A strapdown inertial navigation system providing navigation in three dimensions will have the following components in one form or another: Instrument cluster to sense translational and rotational movements; Instrument electronics to provide the control of the sensors and to produce measurement information; Attitude computer to compute the attitude of the vehicle for resolution of the specific force measurements;
Navigation computer to resolve the specific force data and to solve the navigation equations to generate estimates of position and velocity; Gravitational model to allow compensation for the effects of gravitational attraction on the translational measurements; Power conditioning to provide smoothed and controlled voltage levels needed for satisfactory system operation; Input/output interface to communicate with the host vehicle. These units are mounted in a case which is installed in a vehicle. The case is usually attached to the body of the vehicle via AV mounts. Reduced configurations are possible to produce an attitude and heading reference unit, or a unit for navigation in a single plane. A photograph of a strapdown navigation system incorporating dynamically tuned gyroscopes is shown in Figure 9.5. A modern equivalent to this system based on MEMS sensor technology is shown in Figure 9.6. In a strapdown system, the inertial sensors provide measurements of angular rates and specific force in axes that are usually aligned with the principal body axes of the vehicle. Skewed sensor arrangements may be used in some designs to allow the instruments to cope with very high rates about a single-axis or in systems employing multiple sensors to provide redundancy for fault tolerance purposes. The inertial measurements have to be transformed to the appropriate axis Figure 9.5 Photograph of a strapdown system incorporating dynamically tuned gyroscopes
Figure 9.6 Photograph of a strapdown system incorporating MEMS sensors set for navigation. A variety of reference frames are used depending on the particular application; typically, a local geographic frame is used to provide estimates of latitude, longitude and height, for navigation in the vicinity of the Earth. A variety of methods are available for the transformation procedure, direction cosine matrices or quaternion parameters being most commonly used since both are free from singularities at ±90 pitch angles. Quaternions are generally preferred because they ensure self-consistency. The algorithms required for specific force transformation, the correction for the gravitational attraction and the solution of the navigation equations are implemented in the navigation computer. This processor produces the estimates of vehicle velocity and position in whichever axis set the vehicle is using for its navigation. References 1 KROGMANN, U.: 'Optimal integration of inertial sensor functions for flight control and avionics'. AIAA-DASC, San Jose, October 1988 2 KROGMANN, U.: 'Design considerations for highly reliable hard- and software fault tolerant inertial reference systems'. DGON proceedings, Gyro Technology Symposium, Stuttgart, 1990 3 HARRISON, J.V., and GAI, E.G.: 'Evaluating sensor orientations for navigation performance and failure detection', IEEE Transactions, 1977, AES-13 (6) 4 EDWARDS, CS., and CHAPLIN, RJ.: 'Strapdown dynamically tuned gyroscopes and the use of microprocessors to simplify their application', DGON proceedings, Gyro Technology Symposium, Stuttgart, 1979