intensity, signal attenuation, and position of each signal source within the seeker FOV. The signal intensity of each individual source is established by inputs and may be con- trolled by simulation logic depending on factors such as range to the target, time, or relative aspect.
Some IR missiles bias the commanded seeker tracking rate to cause the seeker to track ahead of the IR source, which typically is the exhaust plume of the target, as shown in Fig. 8-5. This is called rate bias, and it is implemented by adding a commanded tracking rate component in the direc- tion that causes the seeker boresight axis to rotate toward alignment with the missile axis. That is, rate bias reduces the seeker gimbal angle. (The seeker gimbal angle is defined in subpar. 8-2.1.1.)
As discussed in subpar. 8-2.1.2, a time lag exists between the commanded tracking rate and the achieved tracking rate that is caused by seeker rotational dynamics, and an addi- tional time lag results from signal processing in the guid- ance and autopilot systems. As with the simple guidance models, these lags can be simulated in a digital model with intermediate fidelity by passing the commanded tracking rate wC through a pair of digital, first-order lag (low-pass) filters in series (Eqs. 8-4) with appropriate time constants to represent mechanical and signal processing lags. The achieved seeker tracking rate vector wach is assumed to act during the current computation time step and is employed to find the new boresight axis vector at the end of the time step. The final processed seeker rate signal vector mf from
Figure 8-5. Effect of Rate Bias
8-8
Eqs, 8-4 is employed in Eq. 8-10 or 8-13 to calculate the missile lateral acceleration command AC.
acceleration of gravity normal to the flight path is calculated in a simulation by using
In the general case, when target and missile velocities are not coplanar, the line-of-sight angular rate vector is not perpendicular to the missile flight path. Since aerodynamic lift is perpendicular to the flight path, the missile maneuver can respond only to that component of the line-of-sight-rate vector that is perpendicular to the missile velocity vector.
The vector product (ωσ x VM) gives the correct component of ωσ multiplied by VM and has the proper direction for the commanded acceleration. putting Ac (Eq. 8-6) in vector form and adding a component of acceleration opposite the perpendicular component of gravity give the general equa- tion for the commanded maneuver acceleration for “per- fect” guidance in three-dimensional space:
Since the ideal maneuver-acceleration-command vector is in the lift direction, AC is substituted into Eq. 7-15 to deter- mine the required lift force for three-degree-of-freedom simulations.
8-2.2.2 Practical Proportional Navigation
In a practical application of proportional navigation, the actual angular line-of-sight rate wÓ is replaced by the pro- cessed-seeker-head-angular-rate signal wf In addition, the missile velocity vector VM usually is not known onboard the actual missile, and thus makes it impossible to imple- ment Eq. 8-8 directly. Various methods have been employed for approximating proportional navigation in a practical seeker when VM is not available. Two such methods are described-one for missiles having RF sensors that can measure closing speed and one for IR seekers that cannot.
8-2.2.2.1 Missiles With RF Seekers
Radio frequency systems have the potential to measure the magnitude of the closing velocity, i.e., magnitude of the velocity of the missile relative to the target, and missiles with RF seekers sometimes implement Eq. 8-8 approxi- mately by substituting the closing speed, i.e., the magnitude of the closing velocity VC for that of the missile velocity VM
The closing speed is calculated by using
This takes care of the magnitude of the velocity vector to be used in Eq. 8-8, but the direction of the velocity vector must also be approximated. The usual approximation is to substitute the missile axis for the direction of the missile velocity vector in the guidance equation. The missile axis and the velocity vector coincide when the angle of attack at is zero, and the error in the approximation is small for usual angles of attack.
A practical implementation of proportional navigation for RF seekers is obtained, then, by employing the processed seeker angular rate as a measure of the line-of-sight angular rate, substituting closing speed for missile speed, and using the missile centerline axis to approximate the direction of the missile velocity. The equation for simulating this imple- mentation is
The gravity compensation term has not been included here;
many surface-to-air missiles have no onboard instruments to measure the direction of gravity. For applications that do compensate for gravity, the gravity term can be added as in Eq. 8-8, with gn. calculated by using in place of uvM in Eq. 8-7.
8-9
The effect of substituting the direction of the missile axis vector for the direction of the missile velocity vector is that the acceleration command vector, determined by Eq. 8-10, becomes perpendicular to the missile axis rather than per- pendicular to the missile flight path as required by pure pro- portional navigation. Thus the component of aerodynamic force required to produce the commanded acceleration is a normal force rather than lift. Therefore, the commanded-lat- eral-acceleration vector AC calculated by Eq. 8-10 should be substituted into Eq. 7-8 to calculate the required normal force for three-degree-of-freedom simulations. For five- and six-degree-of-freedom simulations, the components of AC are substituted into Eqs. 8-15 and 8-14, respectively.
If the practical implementation of proportional navigation is compared with ideal proportional navigation, the major difference (aside from the dynamic and processing lags) is that the navigation ratio NR has been replaced by an effec- tive navigation ratio NReff To show this difference, again consider the coplanar case, and assume that . Then Eq. 8-10 can be written as the scalar equation
By comparing Eq. 8-12 with Eq. 8-6, it is shown that when VC is substituted for VM, the effective navigation ratio NReff
is equal to NR(Vc/VM) (Ref. 2).
8-2.2.2.2 Missiles With IR Seekers
Missiles with IR seekers typically do not contain instru- mentation to measure missile velocity, and they have no means by which to measure closing velocity. Conceptually, an a priori estimate of missile speed as a function of time could be used in place of the actual missile speed VM to implement proportional navigation in a missile; however, such a priori information is not generally programmed into a guidance processor. Instead this information is taken into account implicitly in the design of the guidance and control components to give an approximation of proportional navi- gation. That is, the relationships among the missile compo- nents are designed—without the expense and complexity of an explicit guidance processor—to cause the missile flight path angular rate to be approximately proportional to the
seeker angular rate. The approximation is made as good as is reasonably feasible within the design constraints of sim- plicity and cost. Thus the essence of proportional navigation is implemented in a very simple way, with no guidance computer and only minimal signal processing.
The result is that a guidance law very close to ideal pro- portional navigation is mechanized in the actual missile, except that the navigation ratio varies with Mach number and altitude, depending on the specific design of the control system and on the aerodynamics of the missile. The designs of the missile subsystems are planned to give navigation ratios as close as possible to the desired ratio within the Mach number regions that are most critical. Atypical curve of navigation ratio versus Mach number at a given altitude for a small IR missile is shown in Fig. 8-6 (Ref. 3).
The navigation ratio multiplied by the velocity is called the system gain. The system gain is a steady state lateral acceleration that the actual missile achieves per unit of seeker angular rate. Fig. 8-7 shows typical system gain
Figure 8-6. Naviagation Ratio Achieved by Typi- cal IR Missile Design
Figure 8-7. System Gain for Typical IR Missile 8-10
curves as functions of Mach number and altitude. These curves are constructed by actual measurements of accelera- tions and of seeker angular rates in flight tests or by very detailed analyses (or simulations) of the missile guidance, control, and aerodynamics. Multiplying the system gain by the seeker angular rate yields the steady state lateral acceler- ation of the missile. System gain curves are useful in less detailed missile flight simulations because they contain a great deal of a priori information about how the missile per- forms that need not be recalculated every time the simula- tion is run. System gain curves are input to the simulation as lookup tables. Actually the system gain also varies depend- ing on the weight and center of mass, and if a very accurate simulation of system gain during the motor bum period is needed, additional system gain curves that cover the burn period must be included in the system gain tabular data for interpolation.
For three-degree-of-freedom simulations using tabular system gain input tables, the commanded-lateral-accelera- tion vector AC is calculated by using
Published system gain data for existing missiles are sometimes presented in units of acceleration g per degree per second g/(deg/s) in which case the units must be con- verted before substitution into Eq. 8-130 The acceleration vector AC calculated by Eq. 8-13 is perpendicular to the mis- sile velocity vector and should therefore be substituted into Eq. 7-15 to calculate the aerodynamic lift.
To simulate the guidance process when the system gain curves are not known requires detailed simulation of the entire guidance and control sequence of events-horn the seeker output through the control servos and fin deflections to the aerodynamic response. An often-selected alternative is to substitute as much of the actual system hardware as is feasible in place of mathematical modeling.