Orbital Rendezvous Research Articles

Output feedback elliptical orbital rendezvous via state‐dependent Riccati differential equations

This article studies spacecraft rendezvous with the target spacecraft in an arbitrary elliptical orbit. Based on the linearised Tschauner-Hempel equations, state-dependent Riccati differential equation (RDE)-based approaches are proposed to solve the problem. An observer-based output feedback controller is established. To implement the proposed controller, only some forward-type RDEs need to be integrated online, indicating that the controllers can be implemented online as long as some positive-definite initial conditions are given. Moreover, it is shown analytically that the closed-loop system under the proposed feedback controller is exponentially asymptotically stable. Numerical examples show the effectiveness of the proposed approach.

Enhanced Linear Fractional Transformation: a Matlab Toolbox for Space System Modeling and Controller Analysis and Synthesis

Through an example of orbital rendezvous between a flexible spacecraft and a rigid target, this paper presents the Enhanced Linear Fractional Transformation Toolbox (ELFTT) developed under ESA contract in order to support advanced design, validation and verification processes of space system control loop. ELFTT will be distributed within the ESA Member States under an Open Source-like license. It is built on (and fully compatible with) Matlab Robust Control Toolbox RCT. It is dedicated to build, verify, manage Linear Fractional Transformations LFT specialized for space applications. On the one hand, it contains a library of generic LFT models relevant for space applications including typical uncertain actuators models (reaction wheels, thrusters.) uncertain dynamics models (flexible spacecraft, rendezvous dynamics.), uncertain sensors models (star trackers, gyrometer. ), etc. On the other hand, it implements new Matlab functions dedicated to manage the LFT models and to support robust controller synthesis and analysis.

Optimization of Time-Open Constrained Lambert Rendezvous Using Interval Analysis

By defining a parking time and a transfer time, a time-open constrained Lambert rendezvous problem is proposed in this paper. The purpose of optimization is to minimize the velocity increment for the orbital rendezvous. The interval branch and bound optimization algorithm is introduced for solving a rendezvous optimization problem with strong nonlinearity and nonconvexity. A numerical example is undertaken to validate the interval optimization algorithm. The results indicate that although the global optimal solution for the time-open constrained Lambert rendezvous problem can’t be found by using only the interval algorithm in the restricted hardware environment because of the inevitable influence of interval dependency and the complex calculation for the cost function, the interval algorithm can narrow the search space and split the feasible space into many small subspaces. Combining the gradient-based optimization method, an improved interval algorithm is used to find the minimal velocity increment for orbital rendezvous, which is better than the optimal solution using the genetic algorithm.

Quadratically Constrained Linear-Quadratic Regulator Approach for Finite-Thrust Orbital Rendezvous

This paper focuses on the design of a quadratically constrained linear-quadratic regulator for finite-thrust orbital rendezvous. The original linear-quadratic optimal control problem is subject to maximum thrust magnitude and quadratic collision avoidance constraints. Thrust arcs are approximated by impulsive velocity increments and the Yamanaka–Ankersen transition matrix propagates the state vector. An explicit closed-loop solution is obtained by performing high-order series expansions of the Hamilton–Jacobi–Bellman equation on subregions of the state space associated with specific sets of active constraints. The algorithm is computationally efficient because the Lagrange multipliers are expressed as polynomial functions of the states and can be computed offline. A rendezvous in an elliptical orbit is considered to demonstrate the application of this method.

Stochastic Optimal Maneuver Strategies for Transfer Trajectories

AbstractAfter being initially launched into the parking orbit, a spacecraft usually implements several maneuvers to insert its nominal orbit as the geostationary orbit for global communication, a Halo orbit to survey the solar wind, and an adjacent orbit to rendezvous with other spacecraft. The transfer problem between two different trajectories is very important in academic research and practical engineering. Trajectory correction maneuvers are necessary to keep actual flight trajectories near the nominal ones because of the errors produced by control maneuvers, measurements, launching, and modeling. However, an inappropriate strategy costs more time and fuel and may even result in mission failure. This study investigates optimal maneuver strategies from the stochastic control viewpoint to track interesting and typical trajectories, including the Halo-transfer and Lambert rendezvous orbits. This study proposes an improved correction algorithm to eliminate modeling errors for reference transfer trajectori...

Use of industrial robots for hardware-in-the-loop simulation of satellite rendezvous and docking

One of the most challenging and risky operations for spacecraft is to perform rendezvous and docking autonomously in space. To ensure a safe and reliable operation, such a mission must be carefully designed and thoroughly verified before a real space mission can be launched. This paper describes the control strategy for achieving high fidelity contact dynamics simulation of a new, robotics-based, hardware-in-the-loop (HIL) rendezvous and docking simulation facility that uses two industrial robots to physically simulate the 6-DOF dynamic maneuvering of two docking satellites. The facility is capable of physically simulating the final approaching within a 25-meter range and the entire docking/capturing process for a satellite on-orbit servicing mission. The key difficulties of using industrial robots for high-fidelity HIL contact dynamics simulation were found and different solution techniques were investigated in the presented project. An admittance control method was proposed to achieve the goal of making the robots in the HIL simulation process match the impedance of the two docking satellites. Simulation study showed the effectiveness and performance of the proposed solution method.

Elliptic Rendezvous in the Chaser Satellite Frame

The analysis of satellite rendezvous in planetary orbits typically derives control laws in a frame rotating with the target satellite. However, because the control law is ultimately required in the chaser satellite’s frame, knowledge of the chaser satellite’s motion with respect to the planet may be required to correctly transform the control laws. The transformation may also result in suboptimal or infeasible control laws if control components have different relative weights. A new approach is described that poses the rendezvous problem in the chaser satellite’s frame directly. A nonlinear transformation between the chaser and target frames, in terms of relative position and velocity variables is derived. This transformation is used to formulate and solve the second-order nonlinear rendezvous problem using optimal power-limited propulsion analytically. Thus, a framework is developed that can be used to solve the orbital transfer problem. The efficacy of the derived control algorithm is demonstrated by means of an example.

Surrogate-based parameter optimization and optimal control for optimal trajectory of Halo orbit rendezvous

This work considers the optimization of rendezvous trajectories for spacecraft starting on different z-amplitude Halo orbits. A surrogate-based parameter optimization strategy is proposed for the optimal trajectory of Halo orbit rendezvous in the Sun–Earth system. The optimal rendezvous problem is transformed into an optimal control problem which fixes the initial flight time and the time of flight for the rendezvous trajectory. Further, the initial flight time and the time of flight are taken as design variables for parameter optimization under the objective of minimum fuel consuming. Since the precise optimization model is typically time consuming and computational expensive, the surrogate model is constructed using data drawn from the precise model, and provides fast approximation of the objective at new design points. Therefore, the surrogate model is feasible and employed for Halo orbit rendezvous in this paper. Numerical simulations show that the surrogate-based parameter optimization takes advantages over the global traversal method and the global optimization method (genetic algorithm). At last, the influences of the different relative z-amplitude Halo orbits for optimal trajectory of Halo orbit rendezvous have been studied by employing the surrogate-based parameter optimization strategy in the numerical simulations.

Tangent Orbital Rendezvous with the Same Direction of Terminal Velocities

Tangent Orbital Rendezvous with the Same Direction of Terminal Velocities

Circular orbital rendezvous with actuator saturation and delay: a parametric Lyapunov equation approach

This study concerns circular orbital rendezvous by using saturated and delayed controls. By identifying the linearised relative motion, known as Clohessy-Wiltshire equations that are asymptotically null controllable by bounded controls, linear feedback laws are designed to semi-globally stabilise the system in the presence of both actuator saturation and delay, namely, the initial condition of the system can be as large as desired as long as it is bounded. To design these controllers, only solutions to a class of parametric Lyapunov equations are required and whose solution can be obtained explicitly. Numerical example is worked out to show the effectiveness of the proposed methodology.

An improved two-manoeuvre method for orbit rendezvous with constant thrust

An improved two-manoeuvre method to the two-point boundary value problem for arbitrary non-coplanar elliptic orbital rendezvous with constant thrust is proposed. The initial required velocity is obtained by solving Lambert's problem. A new method is proposed to determine the optimal thrust direction to nullify the velocity variation. The optimal direction is close to the direction of the velocity-to-be-gained vector. Therefore, the velocity-to-be-gained guidance method is adopted in both the first and the second engine manoeuvres. The total rendezvous process requires a closed-loop feedback control in the first manoeuvre and an open-loop control in the second one. Compared with existing methods based on the linear relative motion equations, this new approach can be used in the cases of far relative ranges and long rendezvous times. Numerical simulations show that the proposed method is feasible.

Spacecraft long-duration phasing maneuver optimization using hybrid approach

Before a manned orbital rendezvous mission, the target spacecraft usually performs several maneuvers to adjust the initial phase angle of the orbital rendezvous and to coordinate the injection of the chaser. This maneuvering process is referred to as the “target phasing mission”. This target phasing presents an orbital long-duration two-point boundary value problem. Further, when the maneuver revolution numbers are used as design variables, the target phasing maneuver's optimization becomes a mixed integer nonlinear programming problem. This paper presents a new optimization method for this phasing maneuver mission, employing a hybrid approach. First, we provide an approximate phasing optimization problem that considers the phase angle influences of node drift and orbital altitude decay. This problem is then optimized using a hybrid approach that integrates branch-and-bound and sequential quadratic programming. Second, a shooting iteration method is adopted to improve the solution to the approximate problem in order to satisfy the terminal constraints of high-precision numerical integration. The proposed method is then applied to an operational target phasing maneuver problem. The results lead to four major conclusions: (1) The proposed approximate phasing optimization model presents a good approximation of the operational mission. (2) The hybrid optimization approach can solve the approximate problem effectively, and the shooting iteration used to arrive at a high-precision solution converges steadily and rapidly. (3) Compared with mixed-code genetic algorithm, the proposed method can obtain a similar result with a lower computation cost and, compared with the approximate model that does not consider node drift and orbital altitude decay, the proposed method has better convergence efficiency. (4) The terminal time of target phasing remains almost constant when the initial semi-major axis increases in a limited interval, and the transition appears only when there is a change in the terminal revolution number.

Angular velocity tracking for satellite rendezvous and docking

Autonomous satellite on-orbit servicing is a very challenging task when the satellite to be serviced is tumbling and has an unknown dynamics model. This paper addresses an adaptive control approach, which can be used to assist the control of a servicing satellite to rendezvous and dock with a tumbling satellite whose dynamics model is unknown. A proximity-rendezvous and docking operation can be assumed to have three steps: (1) prior-dock alignment, (2) soft docking and latching/locking-up, and (3) post-docking stabilization. The paper deals with the first and third steps. Lyapunov-based tracking law and adaptation law are proposed to guarantee the success of the nonlinear control procedures with dynamics uncertainties. Dynamics simulation examples are presented to illustrate the application of the proposed control approach. Simulation results demonstrated that the adaptive control method can successfully track any required angular velocity trajectory even when the dynamics model of the target satellite is unknown.

Necessary and sufficient conditions of the rendezvous between a single spacecraft and three non-coplanar Walker constellation satellites

The probability of the rendezvous between a single spacecraft and three non-coplanar constellation satellites is studied, and the necessary and sufficient conditions of the rendezvous without orbital maneuver are deduced. The rendezvous orbit design can be transformed into the patching of two spacecraft orbits, either of which can achieve the rendezvous with two satellites. Firstly, due to the precious quality of spherical geometry, the unique existence of the rendezvous orbit for two constellation satellites is proved. Then, according to the difference between equispaced and non-equispaced orbital planes of three satellites, the necessary and sufficient conditions are given respectively, and the calculating method of the spacecraft orbit is proposed. At last, the constraint conditions between two different rendezvous orbits is derived, while the relative position of two groups of objects are under specific distribution. The results can be applied to the rendezvous between a single spacecraft and multiple constellation satellites without orbital maneuver.

A solution to two-point boundary value problem for power-limited rendezvous with constant thrust

A new solution to the two-point boundary value problem for power-limited non-coplanar elliptic orbital rendezvous with constant thrust is proposed based on the linear relative motion equations. The process of rendezvous is composed of three time intervals, i.e., the first engine maneuver, a coasting subarc, and the second engine maneuver. An analytical propagation of the relative state is obtained under a constant external acceleration. With the analytical propagation, the required relative velocity in the first maneuver is analytically solved and realized in feedback with a velocity gain guidance method. The thrust in the second engine maneuver is kept in a solved direction. Numerical simulations show that the analytical state propagation is accurate and the proposed rendezvous method is feasible.

Lyapunov Differential Equation Approach to Elliptical Orbital Rendezvous with Constrained Controls

and energy. A parametric Lyapunov differential equation approach is proposed in this paper to solve this constrained regulation problem. After establishing the fact that the Tschauner-Hempel equations are both null controllable with controls of bounded magnitude and energy, this paper proves that the proposed linear periodic controllersemigloballystabilizesthesystem.Equivalently,forany fixedinitialconditions,themagnitudeandenergy of the control can be made as small as desired by tuning some free parameters in the feedback laws. In comparison with the existing quadratic-regulation-based approach, which requires solutions to nonlinear Riccati differential equations,thenewapproachrequiresonlythesolutionoflinearperiodicLyapunovdifferentialequations,whichare investigated inthepaperbyusingtheperiodicgenerator approach.Numericalsimulationsofthe nonlinearmodelof the spacecraft rendezvous instead of a linearized one show that both the magnitude and energy of the control can be reduced to an arbitrarily small level by reducing the values of some parameters in the controller and that the rendezvous mission can be accomplished satisfactorily.

Fuel Optimization for Continuous-Thrust Orbital Rendezvous with Collision Avoidance Constraint

This paper focuses on the issue of minimum-fuel rendezvous between an active chaser satellite with continuous-thrust capability and a passive target satellite. It is first formalized as an optimal control problem subject to a collision-avoidance constraint on the path of the chaser satellite. Then, a new method for dealing with this state constraint is built by adapting to the optimal control framework a recently developed approach for solving inequality-constrained nonlinear programming problems. The resulting method implies solving a sequence of unconstrained optimal control problems, the solutions of which converge toward the solution of the original problem. Convergence of the method is proved, and its efficiency is demonstrated through numerical results obtained in the case of a rendezvous in a highly elliptical orbit.

Optimal four-impulse rendezvous between coplanar elliptical orbits

Rendezvous in circular or near circular orbits has been investigated in great detail, while rendezvous in arbitrary eccentricity elliptical orbits is not sufficiently explored. Among the various optimization methods proposed for fuel optimal orbital rendezvous, Lawden’s primer vector theory is favored by many researchers with its clear physical concept and simplicity in solution. Prussing has applied the primer vector optimization theory to minimum-fuel, multiple-impulse, time-fixed orbital rendezvous in a near circular orbit and achieved great success. Extending Prussing’s work, this paper will employ the primer vector theory to study trajectory optimization problems of arbitrary eccentricity elliptical orbit rendezvous. Based on linearized equations of relative motion on elliptical reference orbit (referred to as T-H equations), the primer vector theory is used to deal with time-fixed multiple-impulse optimal rendezvous between two coplanar, coaxial elliptical orbits with arbitrary large eccentricity. A parameter adjustment method is developed for the prime vector to satisfy the Lawden’s necessary condition for the optimal solution. Finally, the optimal multiple-impulse rendezvous solution including the time, direction and magnitudes of the impulse is obtained by solving the two-point boundary value problem. The rendezvous error of the linearized equation is also analyzed. The simulation results confirmed the analyzed results that the rendezvous error is small for the small eccentricity case and is large for the higher eccentricity. For better rendezvous accuracy of high eccentricity orbits, a combined method of multiplier penalty function with the simplex search method is used for local optimization. The simplex search method is sensitive to the initial values of optimization variables, but the simulation results show that initial values with the primer vector theory, and the local optimization algorithm can improve the rendezvous accuracy effectively with fast convergence, because the optimal results obtained by the primer vector theory are already very close to the actual optimal solution. If the initial values are taken randomly, it is difficult to converge to the optimal solution.

A second-order solution to the two-point boundary value problem for rendezvous in eccentric orbits

A new second-order solution to the two-point boundary value problem for relative motion about orbital rendezvous in one orbit period is proposed. First, nonlinear differential equations to describe the relative motion between a chaser and a target are presented considering the second-order terms in the gravity. Then, by regarding the second-order terms as external accelerations, we establish second-order state transition equations. Moreover, the J2 perturbations effects can also be considered in the state transition equations. Last, the initial relative velocity to fulfill a rendezvous is determined by solving the state transition equations. Numerical simulations show that the new second-order state transition equations are accurate. The second-order solution to the two-point boundary value problem on eccentric orbits is valid even if the relative range is farther than 500 km.

Indirect Optimization for Finite-Thrust Time-Optimal Orbital Maneuver

A N IMPORTANTproblem in astronautics is to transfer a satellite between elliptic orbits, which has been widely studied bymany researchers in both the impulsive case and the continuous low-thrust case [1,2]. Recently, much attention has been focused on the computation of the optimal trajectory for the case of the low-thrust orbital transfer, which can be performed by minimizing the cost of the final time or the fuel-consumption under some additional constraints [2–8]. The dynamics of the satellite are usually described by the positionspeed variables or the modified equinoctial elements. Bonnard et al. [3], Caillau and Noailles [4], and Caillau et al. [5] studied the lowthrust time-optimal and minimum fuel-consumption orbital transfer problem using themodified equinoctial elements. The controllability property of the system, the existence of the optimal control and the singularity observed in the problem were also proposed from the geometrical analysis viewpoint. In the numerical experiment of the minimum-time transfer problem, some researchers found that the minimum time tfmin and the magnitude of the maximal thrust Tmax have the relationship that tfmin Tmax c, where c is a constant [5,7]. However, the problem of whether there exists a positive constant c such that tfmin Tmax tends to c as Tmax tends to zero is still open [8]. The missions of the orbital maneuver include the orbital rendezvous and the orbital intercept, which are different from that of the orbital transfer mainly in the terminal constraint conditions. As early as the 1950s and 1960s, the rendezvous and intercept problems had been widely investigated in the impulsive thrust case [9]. As for the rendezvous problem, the relative dynamics of the satellites (for example, the Hill–Clohessy–Wiltshire equations) have also been concerned [10]. The relative dynamics using the position-speed variables have also been used to deal with the orbital intercept problem [11]. However, the optimization problems for these two maneuver cases are not as widely studied as that for the transfer in the mode of continuous low-thrust using the equations described by the modified equinoctial elements. In this paper, the optimal-time orbital maneuver problems for the cases of the transfer, intercept, and rendezvous under a unified framework using the modified equinoctial elements are considered. The trajectory optimization problem is reduced to a two-point boundary-value (TPBV) problem by using the PontryaginMaximum Principle (PMP) [12], and the corresponding terminal conditions for the cases of the orbital transfer, intercept, and rendezvous are studied, respectively. The main contribution of this paper is twofold. First, the indirect optimization method is applied to the discussion of the finite-thrust orbital intercept and rendezvous problems using the modified equinoctial elements, which is quite different from the previous results on these problems obtained in the impulsive thrustmode and/or by using the relative dynamics of the spacecrafts. To the best knowledge of the authors, this is a novel approach to solve these problems. Second, the numerical simulation results show that the product of the minimum flight time and themaximal thrust is also approximately a constant in the orbital rendezvous and intercept. It is also shown that the orbital transfer and the rendezvous share almost the same optimal trajectory for a fixed maximum magnitude of the thrust. In this paper, the following notations are used: h; i indicates the inner product of two vectors, j j is the finite-dimensional Euclidean norm, and the superscript T means the transpose of a matrix.