Control in Astrodynamics

The figure shows the Duffing Oscillator system brought to the origin for different amounts of times. The uncontrolled system has the mass oscillating between −1 and 1 indefinitely, displayed in black. The energy-optimal approach has the state of the system rotating in a spiral around the equilibrium point, at a rate that gets slower, the larger the final time. Thus, the required magnitude of control effort decreases as the system is allowed more time to reach stasis.

A second analysis is performed with a fixed required time of  2 seconds, where the initial condition of the state is picked following a grid, and the system is lead to stasis. The figure shows the path to equilibrium from the 12 different initial conditions, and the relative uncontrolled behavior in the background. Therefore, control has been achieved
given different states and time constraints.

The controlled pathway of the chaser spaecraft to meet the target spacecraft can be appreciated in the figure, where it can be noted that the spacecraft slows down and changes its direction to approach the origin with smaller oscillations each time. The velocity profile, the right plot of the figure, shows how the chaser decreases its velocity in a spiral, reaching stasis at the desired final time. The rendezvous is achieved and the control technique has been proven to work successfully.

The figure shows an application that aims to achieve rendezvous when the chaser is 10 km ahead of the target, given the constraint of performing the maneuver in 12 hours. The figure shows the resulting pathways of the chaser subject to different dynamics. The uncontrolled linear propagation results to a constant integration, which is a single point in the figure. Therefore, the uncontrolled linear solution, in green, is just a point because it is static. On the contrary, the uncontrolled nonlinear dynamics, in black, has been evaluated considering the first six potential polynomials. This line shows the actual behavior related to the uncontrolled relative motion of the spacecraft, with the presence of the translational drift. The controlled pathways are reported in the figure with red and blue lines for the linear and the high-order dynamics, respectively. Thus, the correct solution of the controlled chaser can be appreciated by analyzing the blue curves. As the chaser gains velocity by increasing its x components, it is able to rapidly approach the target in the y direction. After passing the half-point, the control starts to slow the chaser down, such that it reaches rendezvous with null velocity. The controlled linear pathway, in red, is reported for comparison purposes. The red curve shows the behavior of the chaser that is propagated under linearized dynamics, which fails to achieve rendezvous when controlled by the accurate nonlinear control law. Thus, the consideration of nonlinear terms is crucial for the high accuracy and precision evaluation of the costates of the system, and the newly developed energy-optimal control technique is able to evaluate such control.

The rendezvous application has been repeated with the chaser’s initial condition at different relative positions, in order to assess the capability of the KO technique to evaluate the optimal control to approach the target from every direction. As such, the figure reports the optimal rendezvous pathways of the chaser starting from a distance of 2 km from the target (dotted circle), with initial condition separated every π∕4. The figure reports solutions for rendezvous in exactly 4 h. The values of the initial relative velocity are evaluated considering the difference in the orbit’s tangential velocity, where the reference orbit radius a is increased by a factor x. This aspect best describes the relative initial condition between chaser and target orbiting in different orbits. Therefore, the figure shows that the KO energy-optimal inverse control technique is able to calculate the exact costates for the chaser approaching from every direction. These solutions have been calculated using a high-order Lagrangian that considers the first six polynomials in the expansion of the potential energy.

Solving Lambert’s Problem using multiple revolutions involves finding transfer orbits that complete one, or more, revolutions around the gravitational body before reaching
the desired final position vector with a given time of flight. A KO polynomial map is created for each number of revolutions without the need to recalculate the KO matrix, a benefit from using the KO theory. The possible solutions to the multiple revolutions Lambert’s problem are shown in the figure for a variety of transfer times and semi-major axes. This figure presents the comprehensive solution space, encapsulating both direct transfer orbits and transfer orbits that complete multiple revolutions. It offers an entire view of the potential transfers a satellite can take between the specified position vectors. There are seven possible orbital trajectories for transfer times greater than about 4800 s: two possible multiple revolutions transfer orbits for 𝑁 = 1, 2 and 3 and one possible solution for a single revolution orbit transfer. Such times are too long for another single revolution orbit transfer solution as this other possible solution approaches the time corresponding to a parabolic transfer orbit for increasing semi-major axis. The 0, 1, 2, and 3 revolutions solutions are plotted in the left figure. These correspond in color to the multiple revolutions solutions plotted in the right figure. Some of these multiple revolution solutions are not feasible due to having small semi-major axes that cause their orbital trajectories to pass through the Earth for the time constraint considered. Using the KO methodology allows for the calculation of multiple revolutions solutions by increasing the number 𝑁 without having to change the time of flight. This entails benefits because the zonal representation uses true anomaly as its independent variable.

By working with true anomaly as the independent variable, the Koopman Operator State Transition Polynomial Map (KOSTPM) finds the curve, expressed as interpolated polynomials, in the velocity space of orbits that are a candidate for Lambert’s problem. The curve in the  velocity domain and the relative family of orbits are reported in blue, respectively. Every solution is propagated using the same Koopman matrix that connects the position vectors in the TPBVP, as the change in the anomaly of the transfer orbit is the same regardless of the transfer time. That is, looking at the family of transfer orbits, the left figure, they all have a different semi-major axis, a different time of flight, and a different velocity vector, but they all cover the same true anomaly span, Δ𝜃. Therefore, for a specific point in the spacecraft orbit, the family of velocities that create orbits connecting the two points is evaluated in the KO framework. This curve, called Ξ(𝜃), is represented in the three-dimensional velocity space in the right figure. The analytical function of the curve suits the proposed methodology for the application of various optimization techniques and analysis. For example, since each orbit starts from the same position vector, i.e., constant range 𝑟, the orbit transfer with the minimum energy is the one with the minimum velocity. The orbit with the minimum energy is connected to the hyperbola’s shortest velocity vector. The figure highlights the point of Ξ(𝜃) closest to the origin, which corresponds to the minimum velocity norm and, therefore, to the minimum energy transfer. The transfer is highlighted in red. At the same time, it identifies the smallest orbit with the lowest semi-major axis since it is the one with the lowest specific energy value.

The family of possible transfer orbits has been parameterized for each point of curve Ξ(𝜃). This representation leads to other kinds of optimizations, such as minimum fuel transfer, with the assumption of impulsive maneuvers. Thus, assume that the spacecraft is moving between two separate orbits rather than two points in space. Knowing the velocity vector at the giving point and the family of transfer orbit Ξ(𝜃), the minimum fuel solution corresponds to the transfer orbit that requires the minimum total impulse to let the spacecraft change from initial to transfer orbit and from transfer to final orbit. The left figure reports in red the minimum fuel solution, while the right one shows the impulse vectors in the velocity space as the difference between the velocity hyperbolas at the initial and final conditions. The minimum condition is obtained with a global search on the sum of two impulses to identify the transfer orbit that requires the smallest overall change in velocity. Knowing the velocity hyperbola at the initial and final position, thanks to the KOSTPM leads to an easy scalar optimization problem among vectors’ magnitudes.

Time optimization can be performed under the assumption that a maximum amount of fuel is available. Assuming a maximum total impulse of Δ𝑉𝑀𝐴𝑋 = 3 km/s, the minimum time of flight optimization is translated to determining the transfer orbit with a total impulse equal to the constraint, as that corresponds to the fastest transfer orbit. That is, all the fuel is used to achieve the maximum change in velocity to set the spacecraft to the fastest route. The figure  shows the maximum impulse constraint on the total impulse curve as a red horizontal line. The constraint crosses the function in two separate points, Δ𝑉1 and Δ𝑉2. One point corresponds to the minimum time-of-flight solution, while the other is its counterpart, i.e., the orbit with the same change in velocity but selected on the opposite side with respect to the minimum impulse orbit. The solution pathway is reported as well, where the light blue orbit is the optimal time orbit, while the red orbit is the minimum impulse one from the previous analysis. The magenta orbit is the counterpart.

It is possible to calculate the minimum Δ𝑣 trajectory given any initial and final position vectors on two generic elliptic orbits. Hence, it is possible to produce “pork-chop” plots that identify the best departure and arrival orbital positions on these orbits. The minimum Δ𝑣 direct transfer orbit connecting each possible initial true anomaly and each possible final true anomaly along the elliptic orbits is calculated. Values are interpolated so that the minimum Δ𝑣 is calculated every 1 deg in true anomaly. The resulting matrix is displayed as a contour map. A generic pattern can be seen in the figure where diagonals across the matrix have similar Δ𝑣 values. The difference between the true anomaly at arrival and the  true anomaly at departure is constant along these lines. The Δ𝑣 required to transfer between these true anomalies is a significant factor in determining the overall Δ𝑣 requirements for the trajectory. The constant true anomaly difference along each diagonal results in similar minimum Δ𝑣 values for each trajectory on that diagonal. A 3-D “pork-chop” plot of the matrix plot is shown as well. The 3-D pork-chop plot and the contour plot can be used to identify the best departure and arrival orbital positions that correspond to the overall minimum Δ𝑣 transfer.