Specific Length Trajectories Optimised for Maximum Acceleration using Conic Parameterised Clothoids

One current method of aircraft flightpath generation employs multi-segmented straight trajectories with each consecutive pair of segments filleted with arcs of constant radius. A difficulty of infeasibility immediately arises. In order for an aircraft to make the transition from a straight to a curved path segment (or vice versa) there is a physical requirement for a step acceleration, implying an infinite jerk. Clothoid curves provide a solution and have been investigated to fillet the consecutive segments. However, in this case, the mathematical definition of the clothoid is in terms of an integral that precludes closed-form solutions for critical guidance parameters. The attractiveness of the clothoid is to solve the physical infeasibility problem, however the mathematical intractability inherent in the approach has attracted limited attention despite being recognised as a major concern through early experiments. A method is presented here where double clothoid curves are used to fillet consecutive segments. This work goes beyond present practice to provide closed form solutions by using the hyperbola as a clothoid facsimile. Acceleration constraints have been enforced, both accumulative and maximum, and both the reference bank angle and bank angle rate have been expressed analytically in closed-form in terms of hyperbolic parameters. The solution can be optimised in terms of accumulative acceleration and path length, whilst simultaneously enforcing maximum acceleration constraints. This method of trajectory generation is attractive for implementation with air traffic management separation assurance, tunnel-in-the-sky, and other proposed cockpit display systems.