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This work introduces theoretical developments and experimental verification for Guidance, Navigation, and Control of autonomous multiple spacecraft assembly. We here address the in-plane orbital assembly case, where two translational and one rotational degrees of freedom are considered. Each spacecraft involved in the assembly is both chaser and target at the same time. The guidance and control strategies are LQR-based, designed to take into account the evolving shape and mass properties of the assembling spacecraft. Each spacecraft runs symmetric algorithms. The relative navigation is based on augmenting the target's state vector by introducing, as extra state components, the target's control inputs. By using the proposed navigation method, a chaser spacecraft can estimate the relative position, the attitude and the control inputs of a target spacecraft, flying in its proximity. The proposed approaches are successfully validated via hardware-in-the-loop experimentation, using four autonomous three-degree-of-freedom robotic spacecraft simulators, floating on a flat floor.

The technical difficulties presented by the autonomous multiple spacecraft assembly problem relate to the development of robust and reliable guidance, navigation, and control techniques for on-orbit evolving systems. The main open challenges are: (1) propellant-efficient control of an assembling (also known as evolving system), the evolution occurring both in its mass and inertia properties, as well as in its sensors and actuators configuration and (2) accurate relative navigation among the spacecraft, especially in the event of low frequency measurements update and interruptions of measurements due, for example, to relative sensors’ view’s obstruction by other spacecraft. The works of [

The use of Commercial Off-The-Shelve (COTS) relative sensors, such as low-cost cameras, justify the need for robust relative navigation schemes. Many different filters for tracking a maneuvering target have been considered in the literature.

Approaches based on Kalman filter include the work of Singer [

The input estimation filter and the augmented-dimension filter are commonly used in view of their computational efficiency and tracking performance. Among input estimation techniques, the Augmented State estimation approach yields reasonable performance without constant acceleration or small sampling time assumptions. Furthermore, it not only provides fast initial convergence rate, but it can also track a maneuvering target with fairly good accuracy as mentioned by Khaloozadeh and Karsaz [

In space applications, particularly in the spacecraft relative navigation for the autonomous rendezvous and assembly, each vehicle is both the target and the chaser for the other spacecraft. Here, an additional challenge is considered: the frequent loss of communications for the data exchange when the application involves more than one spacecraft. Alternatives means to perform relative navigation may include a vision-based system. These types of sensors require the image processing and may result in low frequency measurement updates, especially for small spacecraft with limited computation capabilities. Such sensors suffer of problems such as limitations on the field of view and/or other spacecraft obstructing the view. Furthermore, each vehicle does not usually know the other vehicles inputs, that is, it does not possess the information about the maneuvers performed by its fellow spacecraft. This missing information needs to be reconstructed in the estimation scheme that would otherwise diverge quickly.

We here focus on the utilization of low frequency update and low-cost sensors, such as COTS devices. In particular, the spacecraft are envisioned to have range and line of sight measurements, and relative attitude measurements. The navigation algorithm here presented build upon our preliminary work of [

In this work, we build upon known techniques in order to develop guidance, navigation, and control approaches to perform three-degree-of-freedom spacecraft assembly maneuvers. Furthermore, the suggested methodologies are validated via hardware-in-the-loop testing, using four robotics spacecraft simulators.

In particular, the guidance and control problems are tackled by continuously linearizing the dynamics about the current relative state vector between two spacecraft, and employing a Linear Quadratic Regulator to suboptimize propellant consumption. The LQR weighting matrices are computed in real time, depending on the relative state vector, acting as a feedback control. The LQR real-time solver developed for this research is an extension of what used during a real on-orbit spacecraft test inside the International Space Station [

As for the relative navigation, we here propose a design based on the augmented state estimation technique. Robustness to frequent signal loss and/or darkening of the sensors is achieved. Furthermore, the suggested approach reconstructs the information of the other vehicles’ maneuvers. A spacecraft is envisioned to run a copy of the augmented state estimation technique algorithm for each other spacecraft in the bunch, every vehicle being chaser and target at the same time.

For the experimental part of this work, two dynamic models for the relative navigation filter are considered: (1) the classical Kalman filtering technique, [

Between the two approaches, the second one proves to be the most successful. It yields satisfactory performances without constant accelerations or small sampling time assumptions. Furthermore, it does not only provide fast initial convergence rate, but it can also track a maneuvering target with a good accuracy under unpredictable loss of the data link and slow data rate, allowing the spacecraft to perform critical maneuvers such as the docking and the multivehicle assembly.

The successful results of the study here presented pave the way for further research and implementation of the new GNC techniques for the full six degrees of freedom spacecraft relative motion.

Main contributions of this work to the state-of-the-art for multiple spacecraft assembly GNC are as follows.

Development of a guidance and control approach flexible to mass and actuators’ configuration changes during the assembly. The methodology is based on a suboptimal LQR for propellant-efficient rendezvous and docking maneuvers.

Implementation of a spacecraft relative navigation scheme based on augmented state estimation, robust to low frequency measurements updates. In particular, the spacecraft are envisioned to have the availability of range and line of sight measurements, and relative attitude measurements. No relative velocities measurements are available. This is the first time, to our knowledge, that augmented state vector estimation is used for spacecraft relative navigation.

The first (to the best of authors’ knowledge) hardware-in-the-loop laboratory experiment involving four spacecraft simulators in a completely autonomous assembly maneuver.

The paper is organized as follows. Section

This section introduces the third generation of spacecraft simulators developed at the Spacecraft Robotics Laboratory of the US Naval Postgraduate School. Figure

Multispacecraft testbed at the Spacecraft Robotics Laboratory of the Naval Postgraduate School.

In order to perform docking experimentations, two separate custom designed docking interfaces have been developed and each is currently undergoing experimental testing (see Figure

(a) Patent pending docking interface design (electromagnet and fluid transfer capability). (b) Concept (male/female) docking interface used for the experiments in this paper.

Type 1

Type 2

The type 1 docking interfaces are designed in order to passively connect the spacecraft through electromagnetic mechanisms, and their design will allow data/power/fluids exchange (see Figure

Main components of the patent pending docking interface.

Other key features of the spacecraft simulators include the following.

Ad-hoc wireless communication. Continuous data exchange amongst each simulator and the external environment over the wireless network provides for in situ communication. This greatly increases the robustness of data collection in the event of communication loss with one of the simulators.

Modularity. The simulators are divided into two modules where the payload can be disconnected from the consumables, thus allowing for a wide range of applications with virtually any kind of different payload (Figure

Small footprint. The .19 m

Light weight.

Rapid Prototyping. The capability to rapidly reproduce further generations of simulators and improve existing designs via computer aided design (CAD) with the in house STRATASYS 3D printing machine.

Detailed collocation of the hardware on the spacecraft simulators.

Most notably, point 1 of the previous list has provided an invaluable contribution to the success of our ongoing experimentation. The ad-hoc wireless communication system, currently employed onboard the simulators, was experimentally verified by a distributed computing test, which demonstrated the wireless communication real-time capability for the SRL (see [

Table

Electronics hardware description.

Part’s name and manufacturer | Details | Description |
---|---|---|

PC104 (plus) Motherboard | Processor | SmartCoreT3-400, 400 Mhz CPU |

Compact Flash | — | 8 Gbyte capacity |

20 Relays Board (IR-104-PBF) | — | High-density optoisolated input + relay output |

8 Serial Ports Board (MSMX104+) | — | — |

Firewire PC104 board (Embedded Designs Plus) | — | IEEE1394 card with 16 Bit PC104 |

Compact Wireless-G USB Adapter (Linksys) | — | 54 Mbps 802.11 b/g wireless USB network interface adapter |

Wireless Pocket Router/AP DWL-G730AP (D-Link) | — | 2.4 Ghz 802.11 g, ethernet to wireless converter |

Fiber Optic Gyro DSP3000 (KVH) | — | Single axis rate, 100 Hz, Asynchronous, RS-232 |

Magnetometer, MicroMag-3Axis (evaluation kit with RS232 board) (PNI) | — | Asynchronous, RS-232 (the evaluation kit is still a development version) |

DC/DC converters: EK-05 Battery Controller and Regulator + DC1U-1VR 24V DC/DC Converter | — | 3.3, 5, 12, 24 Volts outputs. The main board is equipped with a batteries’ status controller. |

Battery (Inspired Energy) | — | Lithium Ion Rechargeable battery (95 Whr) |

Metris iGPS pseudo-GPS indoor system | — | — |

Each robot performs absolute navigation in the laboratory environment employing indoor pseudo-GPS for position, and magnetometer and gyroscope for attitude (Table

The maximum computational power of 400 Mhz listed in Table

Figure

Ad-hoc wireless network at the SRL test-bed.

The Wi-Fi capability of each robot is not only used to communicate with other robots, but it is also necessary for receiving its own absolute position within the laboratory, as sensed by the pseudo-GPS indoor system.

The onboard real-time operating system is RTAI patched Linux (see [

RTAI Linux has been successfully used as an onboard real-time OS. RTAI is a patch to the Linux kernel that allows for the execution of real-time tasks in Linux (see [

The details on the ad-hoc wireless network and hardware-software interfaces developed for the Spacecraft Simulators are available in [

In this section, we provide the dynamics of spacecraft relative motion in the three degrees of freedom case. The dynamics encompasses both the relative translation (two degrees of freedom) and rotation (one degree of freedom). We will refer in the following to a Local Vertical Local Horizontal (LVLH) reference frame (Figure

Local vertical local horizontal and inertial frames.

The dynamics of such motion can be represented in the compact form as

From now on, we will consider the specific application of hardware-in-the-loop testing using the three-degree-of-freedom spacecraft simulators at the Naval Postgraduate School. For the experimental setup, the state vector becomes

Being

It is common use in the literature to linearize the relative motion dynamics and use the Clohessy and Wiltshire linear equation [

For maneuvers confined in the vicinity of the LVLH origin, elapsing a short time in comparison to the orbital period (

Assuming the spacecraft having the same moment of inertia about the

The goal of this work is to develop a GNC approach for driving the state

In this section, the theory for the three-degree-of-freedom augmented state relative navigation is presented. The controls of another vehicle (target) are treated as additional terms in the corresponding state equation, so that the model provides an augmented state vector. The measurements available on each spacecraft are relative positions (from range and line of sight) and relative attitude, and we assume the knowledge of the controls of the chaser, onboard the chaser itself. No relative velocities measurements are available. Observability proof of the vector

In the following developments, the estimated target’s controls are considered constants within every sample time interval. It is worth underlying that the control variables

The same assumption will be used for the observability demonstration. The navigation algorithm is developed using the Kalman filter approach.

The augmented state estimation approach presents numerical efficiency comparable to the standard Kalman Filter applied on the state only. In fact, the augmented state approach introduces a few more variables in the Kalman Filter, without a significant increase on the numerical burden. Additional references with regards to the implementation of Extended Kalman Filters onboard real space missions can be found in [

The assumption is made of independent estimation and control for the attitude and the position, so that we can proceed as follows. For the relative position, the state vector can be written as (see (

The discrete dynamics for the problem is the following:

The expressions of the matrices:

being

The augmented dynamics adds the estimation of

The same algorithm is implemented for target’s attitude and control torque estimation. Assuming the target rotating only about the vertical axis (

The discrete dynamics for the attitude problem is

being

The formulation of the augmentation of the state dynamics adds the estimation of

For sake of simplicity, considering that the controls are constant in each sample time, we provide, for the planar case, the proof of the observability for the continuous models of the relative dynamics. The observability property holds for the discrete models [

The measurements are related to the state as follows:

It is of immediate demonstration that the following observability matrix has full rank:

Similar developments lead to observability for the relative attitude motion. The dynamics can be expressed as

The measurements are related to the state as follows:

The observability matrix is as following:

This section describes guidance and control for the autonomous assembly.

Figure

Relative vectors used in the alignment and assembly logic. All vectors are in the LVLH

The guidance problem is here expressed in terms of desired state vector for each spacecraft, defined dynamically during the maneuver. The state vector error to minimize is

The subscript “des” indicates a desired relative state vector component. The desired state is dynamically changed throughout the assembly maneuver according to the following two-phase guidance logic.

The center of mass trajectory is unconstrained, free to be optimized, unless in the vicinity of the docking phase. As for the attitude, we reproduce a realistic condition in which the spacecraft has to show one particular side (usually the one with the docking port) towards the current target spacecraft. In other words, the docking port side is commanded to be perpendicular to either the

(1)

Body fixed docking port vector.

(2)

If

SUBCASE 1. The distance between the spacecraft is greater than the chosen impingement stand-off range, then

If

The LQR problem (

The cost function in (

The control vector

Locations of controls for the planar assembly.

For the phases described in the previous section the weighting matrices for the LQR are chosen as

Each time step solution of the LQR generates a gain matrix

The values of the constants

Main simulation parameters.

Mass of each simulator | 10,5 | kg |
---|---|---|

Inertia of simulator | 0.063 | kg m^{2} |

Inertia of the two simulators assembled | 0.18 | kg m^{2} |

Single thruster estimated force [ | 0.16 | |

Docking cone semiaperture | 0.75 | degrees |

Force arms (for torque generation) | 5, 10, 21 | cm |

Limit distance for switching off the thrusters | 0.7 | m |

iGPS accuracy | 1 | mm |

Gyroscope accuracy | 0.003 | deg/sec |

Thrusters minimum actuation time | sec |

Figure

LQR Solver Simulink Block [

The LQR solver employed for developing the proposed approach was downloaded from [

In specializing the design of Figure

Thruster coupling on the spacecraft simulators.

Given the choice for the control vector, the control distribution matrix becomes nonlinear, as in (

In order to employ the LQR approach, the control distribution matrix is linearized at each time step, in the vicinity of the desired attitude.

By inserting the matrices defined in (

Once the S/Cs are assembled, the mass and inertia properties along with the thrusters configuration change. Figure

Assembled configurations with reallocated thruster coupling and COM shift.

The input matrices to the LQR solver will be changed once an additional portion of the structure is connected. Also, the new control vector will have maximum and minimum values reduced, due to the increase of mass. For instance, the case represented in Figure

where

The thrusters remained available after docking will be commanded by either spacecraft one or two, thanks to the real-time wireless link (see [

In this section, assembly maneuvers are employed to experimentally test the suggested guidance, navigation, and control schemes. For our experiment, we do not implement any collision avoidance algorithm, which has been, however, previously successfully tested [

Two experimental runs are presented. The first one demonstrates the unsuccessful relative navigation when classical Kalman Filtering is employed, considering the other S/C’s maneuvers as a random process. Only two simulators are involved. The second experiment involves the four vehicles, showing how augmented state estimation can handle low measurement updates and unpredictable interruptions of updates, and still perform correct relative navigation, driving the mission to success. In particular, we are here imposing, via the wireless network, an update of 2 seconds. Once the two couples of robots are docked, each of the assembled structure is considered to be a new vehicle with new mass and geometry. For this reason, the augmented state estimator is reinitialized for the new structure with different mass and inertia as in Table

The time step, or simulation sampling time, was chosen to be: (1) higher than the thrusters minimum actuation time (Tables

Position, augmented state filter parameters.

Process covariance matrix, for the adapted augmented state filter | ||

Measurements covariance matrix, for the adapted augmented state filter | ||

Initial covariance matrix, for the adapted augmented state filter | ||

0.02 sec | Simulation sampling time |

Attitude, augmented state filter parameters.

Process covariance matrix, for the adapted augmented state filter | ||

Measurements covariance matrix, for the adapted augmented state filter | ||

Initial covariance matrix, for the adapted augmented state filter | ||

0.02 sec | Simulation sampling time |

Figure

Experimental Result: bird’s eye view for two spacecraft simulator failed assembly maneuver. The relative navigation is performed via classical Kalman Filtering, no Augmented State Estimation. The bolded lines are employed to help visualize the simulator’s orientation.

Figure

Experimental Result: bird’s eye view for four spacecraft simulator assembly maneuver. The relative navigation is performed via augmented state estimation. The bolded lines are employed to help visualize the simulator’s orientation.

Once the simulators are assembled in couples, they maneuver as a single bigger unit. In particular, the augmented state estimation is reinitialized in order to switch to a new target vehicle in terms of relative navigation. In Figure

In this work, we are suggesting a complete solution for guidance, navigation, and control of planar multiple spacecraft assembly maneuvers. Guidance is performed by dynamically defining a desired state vector, so that the spacecraft can prepare for docking and correctly connect. The control is based on a real time LQR approach. As for the relative navigation, augmented state estimation is proposed, allowing for correct awareness of the other spacecraft configuration, even in the event of low frequency measurements update. The framework adapts itself to the evolving spacecraft, by switching among different values of mass properties and sensors and actuators configuration, when a new unit assembles to the aggregate.

Theoretical developments are presented for the three-degree-of-freedom case, considering a planar motion for the relative position and a single axis of rotation.

The experimental validation of the proposed methodology is presented, via floating spacecraft simulators, using an assembly maneuver as baseline. Experiments show how the augmented state estimation can cope with low frequency measurement updates, correctly performing the relative navigation, driving the mission to success. On the other hand, Classical Kalman Estimation, is not accurate for close distances with low frequency measurement updates as demonstrated in the three-degree-of-freedom experimental section. The dynamic guidance and control demonstrate real-time feasibility and the capability of performing autonomous assembly.

Commercial off the shelve

Degree of freedom

Guidance, navigation, and control

Input estimation

Linear quadratic regulator

Local vertical local horizontal reference frame centered on the chaser spacecraft

Naval postgraduate school

Pulse width modulation

Spacecraft robotics laboratory.

Docking safety cone semiaperture

Commanded orbiting angle around target docking port in docking phase

Control system sample time

Chaser orbital angular velocity

Target attitude angle in chaser S/C body frame

Target angular velocity in chaser S/C body frame

Transition Matrix of the dynamics of the

Augmented transition matrix of the dynamics of the

Attitude state vector of

Augmented attitude state vector of

Scaling factor in

Discrete time index

Single & combined mass of the spacecraft simulator

Torque Arm: Thruster-center of mass arm

Spacecraft-to-spacecraft vector

Docking port-to-corresponding docking spacecraft vector

Center of mass to docking port vector

Spacecraft-to-spacecraft transition distance between far away phase and docking phase

Time

Chaser’s control vector

Target’s control vector

Relative control vector

Optimal control vector

Single engine maximum thrust

Threshold value for required thrust Before using PWM

State matrix

Control distribution matrix

State-output mapping matrix

Control-output mapping matrix

Control matrix referred of

Discretized control matrix referred of

Discretized augmented control matrix referred of

State matrix of

Augmented state matrix of

Input noise matrix

Measurement matrix

Augmented input noise matrix

Augmented measurement matrix

Cost function

Inertia of a single and combined Spacecraft simulator about the vertical axis

LQR resulting gain matrix

Chaser spacecraft torque

Target spacecraft torque

Initial state error covariance matrix

LQR state error weighting matrix

Process noise covariance matrix

LQR control effort weighting matrix

Measurement noise covariance

Maneuver total time

Sampling time

Measurement noise vector, assumed to be gaussian white zero mean with covariance

Scaling factor in

Input noise vector, assumed to be Gaussian white zero mean with covariance

State vector of

Augmented state vector of

Relative state vector between two S/C

Complete

Measurement vector

Augmented measurement vector

Identity matrix,

Zeros matrix

Target Cartesian coordinates in chaser S/C body frame

Target linear velocities in chaser S/C body frame.

This research was performed while Dr. Bevilacqua was holding a National Research Council Research Associateship Award at the Spacecraft Robotics Laboratory of the US Naval Postgraduate School.