Pseudospectral Optimal Control Theory Makes Debut Flight, Saves NASA $1M in Under Three Hours

If you think today’s gas prices are high, consider the fuel used by spacecraft, at about $10,000/pound. The obvious need to use spacecraft fuel sparingly puts a recent achievement into perspective: On March 3, 2007, by tracking an attitude trajectory developed with pseudospectral optimal control theory, the International Space Station completed a 180-degree maneuver without using any propellant! At nearly 500,000 pounds and approximately 150 feet by 200 feet, the ISS is massive. To maintain its orbit and perform necessary attitude maneuvers, it is equipped with thrusters and gyroscopes, i.e., spinning wheel momentum-storage devices. Small attitude adjustments can be accomplished with gyroscopes, which are driven by renewable solar energy. Large-angle rotations carried out with the flight software, however, require more momentum than gyroscopes can provide; unless thrusters are used, attitude control is lost. But thrusters have significant downsides: consumption of precious propellant and the risk of solar-array contamination. How, then, was the ISS maneuvered without propellant? The key was to realize that the cost of performing a rotation depends directly on the specific trajectory followed and that, by using optimal control theory, the trajectory can be shaped to minimize fuel use. The ISS flight software uses an eigenaxis trajectory, which is the shortest-distance kinematic path. Most spacecraft use this approach, as it is simple to implement in flight software. According to Euler’s rotation theorem, any two orientations are related by a common axis, the eigenaxis, about which rotation by a specific angle, the eigen-angle, is possible. To follow the eigenaxis trajectory, the attitude control system must overcome environmental dynamics, such as torques due to gravity or aerodynamics. As a result, gyroscopes reach momentum cap-acity even if maneuvering at a slower rate. An alternative to an eigenaxis rotation is a “longer” path that takes advantage of these dynamics to maintain the gyroscopes within their limits, thereby avoiding propellant use. What makes this possible is the attitude-dependence of the environmental torques acting on the ISS. If we think of the ISS as a large orbiting dumbbell, the end nearest the Earth experiences a stronger gravitational pull. When the dumbbell is rotated relative to the orbit radius vector, the differential gravity force creates a moment about its center of mass. The aerodynamic drag force, which depends on the angle between the spacecraft and the airstream, also creates a torque. Control of environmentally induced torques can thus be achieved by modifying the orientation of the ISS. The key problem is how to coordinate and modulate these torques while reaching the desired target without overloading the gyroscopes. This longer attitude trajectory can be generated by formulating the ISS maneuver as an optimal control problem. Despite considerable progress in modern optimal control theory, solving this nonlinear problem is quite difficult. For instance, a feedback control law requires a solution to the famously difficult Hamilton–Jacobi–Bellman equation. Analytic solutions can rarely be found for systems with nonlinear dynamics, and numerical approximations suffer from the well-known curse of dimensionality. A practical alternative—and the one used for the ISS maneuver described here—is to combine an inner-loop stabilizing controller designed by conventional methods with an outer loop that generates optimal trajectories. A critical challenge in this two-layer control architecture is a reliable outer loop that generates the optimal reference trajectory in a sufficiently short time. Both reliability and computational speed can be mapped to convergence properties of the computational method. Over the last decade, pseudospectral (PS) techniques have emerged as one of the leading computational methods for solving optimal control problems. To generate and implement the optimal ISS maneuver, the Legendre PS method was used. With a PS method, the original optimal control problem is discretized to formulate a nonlinear programming problem, which is then solved numerically for approximate optimal solutions. PS methods differ from other computational methods in their special discretization at carefully selected nodes. For ex-ample, the so-called Legendre–Gauss–Lobatto (LGL) nodes consist of the points t0 = –1, tN = 1, and tk, being the roots of d/dt LN(t) in the interval [–1, 1] for k = 1,2 . , N – 1, where LN(t) is the Nth-order Legendre polynomial. The state and control functions are approximated by Nth-order Lagrange polynomials based on interpolation at these nodes. Existing literature in approximation theory shows that PS methods are easy and accurate methods for the approximations of smooth functions, integrations, and differentiations that are critical to optimal control problems. For differentiation, the derivatives of the state functions at the LGL nodes are easily computed with the differentiation matrix. Thus, the differential equation of the control system is approximated by algebraic equations at the LGL nodes. The integration in the cost functional of an optimal control problem is approximated by the Gauss–Lobatto integration rule. In addition, it is straightforward to discretize the state-control constraints. The collective application of these approximation methods to an optimal control problem results in a nonlinear programming problem with algebraic constraints. PS methods are able to approximate continuous functions with relatively smaller numbers of nodes, thus resulting in lower-dimensional nonlinear programming problems. Given any real-valued function f(t): [a,b]→R, it is well known in approximation theory that sophisticated node selection can lead to very accurate interpolation. For instance, the polynomial interpolation IN( f ) at the LGL nodes has an error