Dynamic interpolation and application to flight control

Abstract

To simplify the specification of a desired trajectory for some subset of the variables of a dynamic control system, it may be advantageous to designate a set of intercept points that the trajectory is required to pass through. The system controls can then be computed in terms of a spline function to meet these requirements for dynamic interpolation. Optimization of a cost function under continuity constraints can be embedded in the determination of spline coefficients to obtain certain desirable geometric properties of the resulting trajectory.

The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces

“…Then the problem is reduced to determining suitable controls which give rise to such trajectories. We call this procedure the "dynamic interpolation problem," a term that first appeared in a series of papers related to flight paths of aircraft by Crouch and Jackson [11], [12], [13] and Jackson [21]. In our previous paper [14], we started looking at the dynamic interpolation problem in a more abstract setting, in an attempt to understand the geometry of the problem.…”

“…These optimal control problems can be regarded as particular cases of minimum energy problems for systems evolving on general manifolds, by lifting the original system evolving on a manifold M to a system evolving on its tangent bundle TM. There are many papers dealing with minimum energy problems for systems without a drift term ( [2], [5], [18], [21], [31], [33]), The situation in which the system has a drift term is not so well studied ([22] is one case). Our methods also apply to systems which have a drift term and in which the number of controls may be less than the dimension of M. When we consider homogeneous symmetric spaces rather than Lie groups, our methods still apply to systems with full control.…”

“…For systems evolving on the three-dimensional rotation group, we present nonlinear differential equations which the optimal controls must satisfy. This low-dimensional case has strong connections with applications to robotics and the path planning of aircraft [9], [12], [15], [16], [17], [21].…”

Autonomous Path-Following Flight with Uncertainty and Disturbance Compensation

“…For such predetermined trajectories, any approach can be used. [22][23][24][25] This study follows our previous study; that is, a waypoint-based path generation with cubic spline functions 14,17,24,26) is considered. Assume that a UAV with a velocity of UAV V is pursuing a virtual target as shown in Fig.…”

Robust Path-Following for UAV Using Pure Pursuit Guidance

“…It is useful to generate the desired trajectory from several waypoints. For this study, the cubic spline function is used because it gives the minimum curvature (Blajer, 1990;Jackson & Crouch, 1991). Figure 3 portrays an interpolated trajectory by the cubic spline function.…”