Qguidance is a method of missile guidance used in some U.S. ballistic missiles and some civilian space flights. It was developed in the 1950's by J. Halcombe Laning and Richard Battin at the MIT Instrumentation Lab.
Qguidance is used for missiles whose trajectory consists of a relatively short boost phase (or powered phase) during which the missile's propulsion system operates, followed by a ballistic phase during which the missile coasts to its target under the influence of gravity. (Cruise missiles use different guidance methods). The objective of Qguidance is to hit a specified target at a specified time (if there is some flexibility as to the time the target should be hit then other types of guidance can be used).
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At the time Qguidance was developed the main competitive method was called Deltaguidance. According to Mackenzie^{[1]}, Titan, some versions of Atlas, Minuteman I and II used Deltaguidance, while Qguidance was used for Thor IRBM and Polaris, and presumably Poseidon. It appears, from monitoring of test launches, that early Soviet ICBMs used a variant of Deltaguidance.
This is not the place for a detailed discussion of Deltaguidance; suffice it to say that it is based on adherence to a preplanned trajectory. Using groundbased computers a reference trajectory is developed before the flight and stored in the missile's guidance system. In flight, the actual trajectory is modeled mathematically as a Taylor series expansion around the reference trajectory. The guidance system attempt to zero the linear terms of this expression, i.e. to bring the missile back to the preplanned trajectory. For this reason Deltaguidance is sometimes referred to as "fly [along] the wire", where the (imaginary) wire refers to the reference trajectory.^{[2]}
In contrast Qguidance is a dynamic method, reminiscent of the theories dynamic programming or state based feedback. In essence it says "never mind where we were supposed to be, given where we are what should we do to make progress towards the goal of reaching the required target at the required time". To do this is relies on the concept of "velocity to be gained".
At a given time t and for a given vehicle position r, the correlated velocity vector V_{c} is defined as follows: if the vehicle had the velocity V_{c} and the propulsion system was turned off, then the missile would reach the desired target at the desired time under the influence of gravity. In some sense, V_{c} is the desired velocity.
The actual velocity of the missile is denoted by V_{m} and the missile is subject to both the acceleration due to gravity g and that due to the engines a_{T}. the velocity to be gained is defined as the difference between V_{c} and V_{m}:
A simple guidance strategy is to apply acceleration (i.e. engine thrust) in the direction of V_{TBG}. This will have the effect of making the actual velocity come closer to V_{c}. When they become equal (i.e. when V_{TBG} becomes identically zero) it is time to shut off the engines, since the missile is by definition able to reach the desired target at the desired time on its own.
The only remaining issue is how to compute V_{TBG} easily from information available onboard the vehicle.
A remarkably simple differential equation can be used to compute the velocity to be gained:
where the Q matrix is defined by
where Q is a symmetric 3 by 3 timevarying matrix. (The vertical bar refers to the fact that the derivative must be evaluated for a given target position r_{T} and time of free flight t_{f}).^{[3]} The calculation of this matrix is nontrivial, but can be performed offline before the flight; experience shows that the matrix is only slowly time varying, so only a few values of Q corresponding to different times during the flight need to be stored onboard the vehicle.
In early applications the integration of the differential equation was performed using analog hardware, rather than a digital computer. Information about vehicle acceleration, velocity and position is supplied by the onboard Inertial measurement unit.
Derivation of the equation
Notation:
t the current time
r the current vehicle position vector
V_{m} the current vehicle velocity vector
T the time the vehicle will reach the target
t_{f} the time of free flight for the correlated vehicle, i.e. tT
[...]
A reasonable strategy to gradually align the thrust vector to the V_{TBG} vector is to steer at a rate proportional to the cross product between them. A simple control strategy that does this is to steer at the rate
where κ is a constant. This implicitly assumes that V_{TBG} remains roughly constant during the maneuver. A somewhat more clever strategy can be designed that takes into account the rate of time change of V_{TBG} as well, since this is available from the differential equation above.
This second control strategy id based on Battin's insight^{[4]} that "If you want to drive a vector to zero, it is [expedient] to align the time rate of change of the vector with the vector itself". This suggests setting the autopilot steering rate to
Either of these methods are referred to as crossproduct steering, and they are easy to implement in analog hardware.
Finally, when all components of V_{TBG} are small, the order to cutoff engine power can be given.
