An experimental maneuver control for rendezvous and autonomous docking of a small spacecraft

This paper deals with maneuver control for the autonomous docking of two small spacecraft in a rendezvous flight. Due to hardware constraints, maneuver control suppressing fuel consumption and computational cost is a significant issue for small spacecraft. Here, the maneuver control for the approaching motion used a model predictive control system. The spacecraft's maneuvers in two-dimensional plane motion were performed in a frictionless environment with air bearings to verify the control performance.


Introduction
In recent years, in addition to deep space exploration by large spacecraft, small spacecraft missions are rapidly increasing in space development.Therefore, the establishment of motion control technology using multiple small spacecraft is becoming a trend in space engineering [1][2][3][4][5][6][7][8].Autonomous docking technology is one key technology in multiple small spacecraft cooperative missions.This paper deals with the maneuver control for autonomous docking of small spacecraft, which realizes the approach from the rendezvous flight to just before the docking execution between spacecraft.However, autonomous docking of small spacecraft has more severe hardware constraints than that of large spacecraft.In other words, it is difficult for small spacecraft to carry many propulsion thrusters, sufficient power supplies, and high-precision relative position sensors.In addition, the processing power of the control computer cannot be as mighty as that of a large spacecraft.Under these constraints, the autonomous docking maneuver control system must be designed in the optimal control framework [9][10][11][12][13][14][15][16].In this study, a maneuver for the autonomous approaching of two small spacecraft from orbital rendezvous flight to just before docking is realized using model predictive control (MPC).That is, we designed the motion controller of the spacecraft while predicting the control inputs that minimize the energy consumption of the system's attitude control actuators and fuel consumption for propulsion.The small-sized spacecraft treated in this study is assumed to be a microcomputer with low computational power.Furthermore, the small-sized spacecraft is assumed to have thrusters for propulsion and an ultra-compact CMG (Control Moment Gyro) for attitude control torque generated by the gimbal mechanism that supports the flywheel's rotating axis [17].
Recent studies on the control design of mechanical systems formulated the model predictive control in the optimal control framework and confirmed its effectiveness in actual equipment.A cost function is first set in the optimal control framework and the control input that minimizes it is obtained.The optimal control problem is to numerically solve a two-point boundary value problem under these conditions, given the starting and ending points of the state variables.In general, the offline solution involves a vast amount of iterative computations.In addition, the obtained control input is open-loop control and sensitive to model errors and disturbances.In contrast, model predictive control is a robust algorithm that sets a specific prediction time, solves an optimal problem within this prediction time, and finds the control input in real-time, allowing correction against model errors and disturbances.
On the other hand, according to space development guidelines, ground tests using hardware models as well as numerical simulations are required to evaluate the performance and verify the mission feasibility of a spacecraft in the development phase or in basic research to accomplish a given space mission [18].This study is significant because it verifies the feasibility of maneuver control in the framework of optimal control of a small spacecraft with limited hardware resources, using an actual hardware model for experimental ground tests and numerical simulations.
This study aims to obtain primary data for evaluating the effectiveness and performance of the maneuver control system obtained in the framework of optimal control of a small spacecraft through numerical simulations and ground tests using mock-up hardware similar to the actual machine.
The constraints to be satisfied in the control of spacecraft motion in this study can be summarized as follows: (1) Small spacecraft do not have abundant fuel and power sources for docking maneuvers, so consumption must be suppressed.(2) Since the processing power of a control computer is limited to about that of a microcomputer, and it is challenging to construct a control system that requires advanced numerical calculations, the control law must be feasible for a control cycle with a processing unit as small as a microcomputer.
(3) Spacecraft thrusters can only control the on/off of the ejection jet, and the propulsion dynamics are complex.(4) The initial relative position for docking with the target is assumed to be in a rendezvous state.(5) In maneuver control, the final position with the target should be approaching a distance close enough to the docking ports to execute the docking procedure.
The critical point of this paper is to verify the performance of spacecraft maneuver control using model predictive control under these conditions through hardware experiments on the ground.
In this study, the energy consumption of attitude control actuators and fuel consumption of propulsion thrusters were used as evaluation functions for model predictive control.The state variables were not used in the cost function because the path of the state variables from the control start point to the endpoint does not need to be restricted in the orbital maneuver of the spacecraft.
Using the designed model predictive control system, we implemented hardware experiments of maneuver control of a small spacecraft.Since the hardware experiment was conducted under the earth's gravity, the motion control experiment was conducted under a micro-friction environment in a two-dimensional plane using air bearings [19,20].The small mock-up spacecraft has four small thrusters and one axis CMG.A small onboard camera was used to identify the target spacecraft's docking port and relative position.We used the model predictive control to reduce the thrusters' propellant consumption and the CMG's drive energy consumption and to ensure the autonomous approach to the target docking port with sufficient accuracy.
This article is an extended version presented as a regular paper at the 61st SICE Annual conference 2022.In the conference, an offline optimal control system for maneuver control for the docking of small spacecraft was designed [21].Its effectiveness was confirmed by simulations and hardware experiments using control inputs obtained by iterative calculations.In this article, the control system is the model predictive control that can be executed in real-time.Hardware experiments are conducted under conditions closer to the actual machine, such as using an actuator with only two thruster inputs (ON/OFF) to verify the effectiveness of the maneuver control.In addition, the performance evaluation of computing cost, energy consumption, Etc., was also verified through hardware experiments.

Model
Figure 1 shows the geometric relationship between the chaser and the target spacecraft.First, let a 01 , a 02 , a 03 be the orthogonal unit column vectors in the world coordinate system 0 fixed in inertial space.Then, these vectors have the following relationship.
Using the unit vectors a 0i , i = 1, 2, 3, we defined the following coordinate system matrix [a 0 ] ∈ R 3×3 . [ In the same way, we define the following Cartesian coordinate system matrices for the coordinate system 1 fixed to the centre of gravity of the spacecraft body and 2 fixed to the flywheel of the CMG.
• [a 1 ]: The coordinate system matrix for the coordinate 1 which is fixed on the spacecraft at the centre of mass • [a 2 ]: The coordinate system matrix for the coordinate 2 , which is fixed on the CMG's flywheel at the centre of mass The docking port of the target spacecraft is the target, which is the origin of the world coordinate system.Initially, the target and chaser are in a rendezvous flight and are assumed to be apart at a certain distance.Then, we consider the problem of controlling the position and attitude of the chaser spacecraft to approach the vicinity of the target port position.
Figure 2 expresses the definition of the variables and physical parameters.
The following vectors are defined.
• r 0 = [a 0 ] T r 0 = [a 0 ] T [x, y, z] T : the position of the main body's centre of mass from the world coordinate origin.
wheel's position from the main body's centre of mass.
tion of the docking port from the target port.• θ 0 = [θ 01 , θ 02 , θ 03 ] T : Euler angle expressing the attitude of the spacecraft to the world coordinate.
The state variable for equations of motion is defined as follows: where ω 10 is the angular velocity vector of the main body.
We defined the input force/torque vector u corresponding to the state variable Z as follows: The equation of motion of the spacecraft is described as follows: where ω 21 is the angular velocity vector of CMG's flywheel.m 1 and m 2 are the mass of the main body and that of CMG's wheel, respectively.I 3 , O 3 ∈ R 3×3 are the unit and zero matrices.A 21 , A 10 ∈ R 3×3 are rotation matrices from the main body to the CMG wheel, and that from the world coordinate to the main body, respectively.For a given vector x = [x 1 , x 2 , x 3 ] T we define a matrix x as follows.
T are the thruster's input force vector and the input torque of the CMG Gimbal mechanism.The input torque of the CMG generates the gyro moment of the flywheel, and it moves the attitude motion of the main body.
The angular velocity vector and the time derivative of the Euler angle have the following relation.
It should be noted that the degrees of freedom of the spacecraft coincide with those of the actuator's inputs.In addition, the system's equations of motion are linear concerning the input.

Realtime Euler-Lagrange formulation
This study assumed that the spacecraft has CMG for attitude control and chemical thrusters for position control.However, a small spacecraft has little energy and fuel resources.Therefore, we designed the optimal controller to reduce the actuator power and thruster's fuel consumption.
State variable X ∈ R 12 for the controller is defined using the position and Euler angle of the main body as follows: We designed a model predictive control (MPC) system here.First, we introduce an auxiliary variable, s, related to time.The model prediction time (the horizon) is given as T. At the moment of t, s = 0, and the end time of s is set to s = T. Using this auxiliary variable s, let the cost function be the evaluation function in the interval [0, T] concerning s as follows: We formulated the cost L(u) as the thruster's fuel consumption and CMG's electrical energy consumption.It should be noted here that the cost function L(u) is a function of the control input u only and not the state variable X.This is because, in this study, the spacecraft is in a rendezvous state in Earth's orbit.Therefore, there are no restrictions on the path to docking, so there is no need to include the state variable in the cost function.X(t + T) is the state variable at the end time of the model predictive horizon T. The controller only has to approach the spacecraft close enough.After the approach maneuver, the succeeding docking procedure will be implemented.The objective of the approach maneuver in this study is not to reach the docking port strictly.Therefore, this study introduced the constraints on the state variables at the end time of the horizon using a quadratic form in the cost function.
ψ(X(t + T)) is formulated as the quadratic form with weight parameter P. On the other hand, the thruster's fuel consumption and the CMG's energy consumption are evaluated by the cost function L(u).
L(u) and ψ(X(T)) are defined as follows: where Q ∈ R 6×6 and P ∈ R 12×12 are positive definite matrices.By using the Lagrangian multiplier λ ∈ R 12 , Hamiltonian is defined as follows: where f (X, u, s) is denoted from Equation (5) as follows: The constraint condition is derived as follows: Let time t be the initial time and time t + T be the final time.This time interval is divided into N. Then t is the sampling time and T + k t, (k = 0, 1, . . ., N − 1) is the optimal control input ûk (t) at discrete time t + kδt, (k = 0, 1, . . ., N − 1).The state variable and Lagrangian multiplier when the system receives the input Xk (t) and λk (t), respectively.
The vector U(t) summarizing the discrete control inputs is defined as follows.
For the vector U(t), Equation ( 16) becomes the following algebraic equation.

∂H ∂u
We can find an unknown vector U(t) that satisfies Equation ( 18), but we need to reduce the amount of computation for this purpose.Therefore, we consider a differential equation for F and a mechanism to converge to zero with a time constant of 1/ζ (> 0).
By viewing Equation ( 19) as an algebraic equation for U(t) and obtaining U(t) numerically, and integrating U(t) we can obtain the discrete-time control input U(t).In this study, we utilized the Generalized Minimal Residual method (GMRES) to solve Equation (19) algebraically.The GMRES method is an approximate solution algorithm that can solve simultaneous linear equations quickly and is commonly used in MPC.Therefore, we also used this method in this study.

Numerical simulations
Numerical simulations verified the effectiveness of the rendezvous approach until just before docking.In order to keep the correspondence with the two-dimensional hardware experiment, we performed numerical simulations under the condition of two-dimensional motion.The state variable is denoted as a two-dimensional variable as follows:

Lyapunov control for comparison
In order to compare the performance with the model predictive control system, a Lyapunov control system was designed as an example of a commonly used feedback control system with low computational cost.We defined the candidate of the Lyapunov function V as follows: where K P is a feedback gain matrix and is a positive definite symmetric matrix.By differentiating the function V with time along the equation of motion, we obtain V The control input u L was designed so V as to be zero at the equilibrium point(Y = Ẏ = 0) and negative definite otherwise.
where, K D is a feedback gain matrix and is a positive definite symmetric matrix.

Simulation conditions
This study proposed a method for the rendezvous docking of a small spacecraft using the model predictive control law.The optimal input û(t) was numerically calculated using Equations ( 18) and ( 19) for given predictive time duration T.
Here, we treat the two-dimensional motion.The weight parameters have reduced the dimension as follows: In this study, the final state variables must only reach a feasible range for the passive docking stage by the electromagnet.Therefore, we evaluated the numerical solution of the model predictive control input to determine whether the final position and angle were close enough to the target port for docking.
The numerical simulations were implemented under the following conditions.
For comparison, the performance of a Lyapunov controller was also verified.The feedback gains for the Lyapunov control system were determined.With the determined feedback gains, the convergence time constant is almost the same as for the model predictive control system.In other words, the time for the spacecraft position to reach halfway from the initial position to the target position is almost equal in both control systems.

Results
Figure 3 shows examples of spacecraft movement trajectories in numerical simulations using model predictive control and Lyapunov control, respectively.In the figures, the box-shaped drawings express the shape of the spacecraft.The spacecraft got an approach to close enough to the target using the model predictive control input to the target port.The predictive time duration T for the predictive model control is given as 1.0 s.

Hardware experiments
To verify the performance and validity of the controller, we implemented hardware experiments in a two-dimensional environment.The specification of the spacecraft and experimental environment are summarized in Table 1.The target port is fixed, and the   centre of it is the world coordinate's origin.With air bearings, the friction between the mockup spacecraft and the experimental field is so tiny as to be negligible.
Figure 4 shows the structure of a mockup model of the small spacecraft for a hardware experiment.The mockup has a microcomputer as a control unit and an onboard camera to recognize the docking target.In addition, the spacecraft is equipped with four thrusters as actuators for spacecraft motion and a small CMG with one axis for attitude control.
The output of the thruster in the hardware experiment is a binary On/Off control.Propulsion was realized from the control inputs obtained by the model predictive control system calculations by opening and closing the fuel valves of the actual hardware thruster.
In the hardware experiments, = 0.01 N was used.Although this value is arbitrary, it considers that in preliminary experiments, the spacecraft in the experimental setup hardly moved at thrusts below this level due to the very slight friction of the air bearings.F ON is the thruster output when the valve for the thruster gas is turned on.The gas ejection dynamics is complex and not constant, but the maximum measured value was about 1.2 N. Figure 5 shows a snapshot of the experimental hardware setup.The mockup spacecraft hovers on the experimental field with air bearings by 6 μ m gap.The mockup spacecraft has air tanks to supply the air to the air bearings and thrusters.The capacity of the air tank is 2.7 × 10 −3 m 3 .The system has its battery and control devices and autonomously controls its motion onboard.The host computer monitors the system and gets realtime images captured by the onboard camera.In this study, the target and docking ports have no controlled device to dock or latch mechanism.Therefore, the final approach point for the spacecraft is the target port.
Figure 6 shows the architecture of the hardware system.The central controller onboard is a microcomputer with Quad-core Cortex-A72 1.5GHz.The microcomputer controls the thruster valves and CMG.A CMOS camera is equipped in the center of the docking to detect the target port.The relative position and the spacecraft's attitude to the target port are computed using real-time image data.The centre of the target port is equipped with an ArUco marker [22].The position and attitude of the spacecraft are measured using the ArUco marker, which is available in the OpenCV software library.
Figure 7 shows sequential photographs of the hardware experiment.The system first measured its relative position and attitude from the target port using image processing for the ArUco marker.Then the system calculated the model predictive control input to generate the target's approach motion.Finally, the spacecraft got close enough to the target port.
Figure 8 shows the time history of the main body's position and attitude angle in the hardware experiment.In the figure, the numerical simulation result is also plotted for comparison.The system first measured its relative position and attitude from the target port using image processing for the ArUco marker.Then the system calculated the model predictive control input to generate the target's approach motion, reducing the   thruster's fuel and CMG's energy consumption.Finally, the spacecraft got close enough to the target port.The final position and attitude had a convergence error because of the model error of the hardware environment.However, the result showed good performance of the system.
Figure 9 is the time history of the control inputs for the system.In Figure 9(a), the thruster's air valve operates as ON/OFF switching like bang-bang control during the given time.In Figure 9(b), it can be seen that the model predictive control system requires inputs with larger absolute values than those of the Lyapunov control system.This result is because of the tendency caused as the spacecraft gets closer to the end time to satisfy the state variable's target value at the end time.As can be seen from the figure, the maneuvers are most successful when the spacecraft's initial position  is more than about 0.5 m from the target port.Conversely, when the initial position of the spacecraft is within about 0.5 m, the maneuver almost always fails.These results are because the initial position is too close to the target.Therefore, a highly accurate model predictive control is necessary to approach the target while controlling the attitude and position.However, the thrusters and CMGs used in the experiment had significant model errors and were unable to compensate for them during the approach during the maneuver.
Figures 11 and 12 show the position and angular error of the spacecraft's docking port relative to the target at the end time in the hardware experiment.In the experiment, the position and angle errors at the end time are compared between the case with the model predictive control system and the case with the Lyapunov control system.In both cases, it is clear that MPC performs better with limited hardware resources.
Figure 13 compares the thrusters' fuel consumption and the CMG's electrical energy consumption for each control law.Both results are summarized from 50 hardware experiments.Figure 13(a) shows that the thruster fuel consumption is more suppressed with the model predictive control than with the simple Lyapunov control.On the other hand, Figure 13(b) shows that the energy consumption of the CMG is suppressed less by the Lyapunov control law.In this hardware experiment, the model predictive control law is weighted to the thruster fuel consumption due to the small capacity of  the thruster fuel tanks.Therefore, in Figure 13(b), the energy consumption of the CMG for attitude control is used more.
We also estimated the computational cost of the computer in the hardware experiment in this study.The results of the comparison of CPU time for each control system are shown in Figure 14.The data in Figure 14 estimates the time the controller CPU takes to calculate the control input during the 10 ms control sampling cycle using the compiler's timer function.Although the accuracy of the timer function may not be sufficient, the comparison was made based on the judgment that it is sufficient for comparative verification.Unfortunately, the figure shows that model predictive control is more computationally expensive than simple Lyapunov feedback control.In the present study, the Gaussian Elimination calculated the inverse matrix in the Lyapunov feedback control law (Equation ( 23)).Although there is no significant difference in computational complexity between the two control laws, the results require further improvement as the next research step.

Conclusions
This study verified the performance and feasibility of an autonomous approach to docking a small spacecraft using a model predictive control law in numerical simulations and hardware experiments with an actual mockup of a spacecraft.We used the control input of the model predictive controller in a general way with the real-time Euler-Lagrange equation.The cost function consisted of the norm of the control input and the position error.Hardware experiments verified the effectiveness of the rendezvous approach until just before docking in the actual situation within the admissible maneuver errors.The formulated model predictive control established approach maneuvers with enough precision in a broad approach area, suppressing the fuel consumption of the thrusters.
In the hardware experiment, the model predictive control law is weighted to the thruster fuel consumption due to the small capacity of the thruster fuel tanks.Therefore, the energy consumption of the CMG for attitude control is used more.This performance strongly depends on the hardware spec and resources.In this study, CMG has enough energy capacity, while the fuel capacity is limited.This spec depends on the actual mission of the spacecraft.
However, the computational cost for the model predictive controller requires more than or is comparable with the simple Lyapunov feedback controller.Therefore, the next step of this study is to reduce the computational cost of the controller.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Notes on contributors
Asumi Nishimura, a master's course student at the Osaka Institute of Technology, specializes in space engineering and robotics.
Katsuyoshi Tsujita, a professor, specializes in robotics and space engineering.In recent years, in addition to his research on space robots, he has been involved in research on bioinspired robots and autonomous robot systems.

Figure 1 .
Figure 1.Schematic model of the chaser spacecraft and docking target.

Figure 2 .
Figure 2. Definition of the variables and parameters.

Figure 3 .
Figure 3. Two types of examples of simulation results.(a) and (b) show Model Predictive Control (MPC) and Lyapunov control, respectively.

Figure 4 .
Figure 4.The design of the mockup spacecraft.

Figure 5 .
Figure 5. Experimental setup.The mockup spacecraft is floating on the surface plate with air bearings.The target port is fixed on the frame structure of the experimental field.

Figure 6 .
Figure 6.Mockup spacecraft system architecture for hardware experiment.

Figure 7 .
Figure 7. Sequential photographs of the onboard camera view on the docking port in the hardware experiment.The upper left is the initial state, and the lower right is the final state.

Figure 8 .
Figure 8.A result of the hardware experiment.The time history of the main body's position (a)(b) and attitude angle (c).The solid and dashed lines express the hardware experiment and the numerical simulation cases, respectively.

Figure 9 .
Figure 9.An example of thrust valve operation and CMG's input torque in a hardware experiment.(a) Thruster's air valve operation (ON/OFF) (b) CMG's input torque.

Figure 10 .
Figure 10.The results of hardware experiments with maneuver control from various initial positions to the target (N = 50 trials).

Figure 10
Figure10shows the results of hardware experiments with maneuver control from various initial positions to the target.The circles and triangles in the figure indicate successful and unsuccessful initial positions, respectively.Success was judged based on whether the distance between the target port and the spacecraft docking port was closer than 0.05 m during the specified control time (10.0 sec).The reason for setting the judgment distance to 0.05 m is that the distance can be considered a distance at which soft docking by electromagnets is possible in the second stage of the docking mission.As can be seen from the figure, the maneuvers are most successful when the spacecraft's initial position

Figure 11 .
Figure 11.Position error of the docking port against the target (N = 50 trials).

Figure 12 .
Figure 12.Orientation error of the docking port against the target (N = 50 trials).

Figure 13 .
Figure 13.Comparison of thrusters' fuel consumption CMG's energy consumption in each control method.(a) Thrusters' fuel consumption (b) CMG's energy consumption.

Figure 14 .
Figure 14.Comparison of the CPU time for calculation of control input in each sampling interval (N = 50 trials).

Table 1 .
Specification of the experimental setup.