# The Constraint Space

In Section 10.1.3 we provided a typical list of constraints the design brief of a fixed wing UAV may contain. Some of these may be a little hard to code up into a user-friendly, integrated mathematical model (we gave the example of the storage/transportation constraints), but the most important ones are, and here we focus on those. Namely, we shall discuss a unified treatment of the level, constant-velocity turn constraint, the rate of climb constraint, the ground run constraint, the cruise constraint, and the approach speed constraint (the importance of the latter, which is a good surrogate for the probability of damage on landing, is particularly great when unprepared landing strips are envisaged or the runway is in a confined area, such as on the deck of a vessel).

The challenge here is to find scale factors for the (at the moment) dimensionless aircraft sketch that will strike a suitable balance between the future aircraft meeting these constraints and its weight and cost remaining as low as possible (one way of satisfying all constraints might be, e.g., via an enormous wing area, but the wisdom of this is unlikely to be borne out by subsequent preliminary design calculations!). So why the plural in “scale factors”? Beyond the obvious geometrical scale - usually expressed in terms of the wing area, as this has relevance to all performance constraints - we also have to find the correct ballpark for the powerplant, most commonly in terms of the thrust it must be able to provide.

There is no unique “right” way of plugging these numbers into simple models of the constraints and solving the inequalities simultaneously, but there is a tried and tested convention that makes the solution process easy and facilitates the interpretation of the results. This involves the normalization of both numbers with the weight of the aircraft (let us say the MTOW for now). Specifically, in the interest of being able to plot all the constraints on the same diagram, we seek to express them in terms of weight-normalized thrust, that is, the *thrust to weight ratio ^{[1]} T/W* - and wing area-normalized weight - the

*wing loading W/S.*

Defining the constraint space in terms of these two ratios allows the engineer to encapsulate the three most important numbers of the nascent concept, and it has the advantage that they are both relatively easy to sanity-check due to their intuitive and universal nature. The take-off thrust to weight ratio gives an immediate indication of the performance of the aircraft in the vertical dimension: typical numbers are close to 1 for air-superiority fighters, around 0.3-0.4 for transport aircraft, and under 0.3 for light aircraft and most unmanned aircraft. Typical wing loading values range from under 100kg/m^{2} for light aircraft to 0.75 t/m^{2} for large transports. By comparison, unmanned aircraft tend to have far more lightly loaded wings - our UAVs typically have *W/S* of 16 kg/m^{2}.

- [1] Sometimes the related quantity of the power to weight ratio is used instead, using the relationship P = TV/^p -thrust times airspeed divided by propulsive efficiency - or, to get the result in horsepower PBhp = (1/746) TV/^p.