# Slider: On the Design and Modeling of a 2D Floating Satellite Platform

Avijit Banerjee, Jakub Haluska, Sumeet G. Satpute, Dariusz Kominiak, and
George Nikolakopoulos1 1 Robotics and Artificial Intelligence, Department of
Computer, Electrical and Space Engineering, Luleå University of Technology,
Luleå {aviban, jakhal, sumsat, darkom<EMAIL_ADDRESS>
###### Abstract

In this article, a floating robotic emulation platform for a virtual
demonstration of satellite motion in space is presented. The robotic platform
design is characterized by its friction-less, levitating, yet planar motion
over a hyper-smooth surface. The robotic platform, integrated with sensor and
actuator units, is fully designed and manufactured from the Robotics and
Artificial Intelligence Team at Luleå University of Technology. A detailed
design description along with the mathematical modeling describing the
platform’s dynamic motion is formulated. Finally, the proposed design is
validated in extensive simulation studies, while the overall test bed
experimental setup, as well as the vehicle hardware and software
architectures, are discussed in detail. Furthermore, the entire design,
including 3D printing CAD model and different testbed elements, is provided in
an open-source repository and a test campaign is used to showcase its
capabilities and illustrate its operations.

##  I Introduction

The advancements of space technology in the recent era has revolutionised our
perception about space activities. Space agencies across the globe focus on
the autonomous robotic missions that enable on-orbit servicing, manufacturing
and maintenance of satellites, close monitoring, docking and active debris
removal. Since the past decade with the growing interest of small scale
satellites [1], the present-generation space missions focus on the prospect of
multiple spacecraft formation missions towards autonomous operations in space.
Such ambitious tasks inherit highly complex and challenging objectives that
require efficient and effective autonomous guidance, navigation and control
systems to ensure the overall mission success [2].

To autonomously carry out such highly complex widespread objectives, advanced
navigation, guidance, and control (GNC) techniques are of primary importance
[2] in executing these challenging operations with a high degree of
reliability and accuracy. To ensure higher technology readiness levels before
deploying into space, rigorous validation of the robotic space system through
ground-based high fidelity campaigns in the relevant condition is essential
for cost-effective, low-risk, and potentially high return solutions.

In view of that, a friction-less, micro gravity orbital motion is one of most
the critical aspects of the space environment, that needs to be replicated in
hardware-in-the-loop tests [3]. However, in reality, recreating the space like
micro-gravity condition is indeed challenging to set up in laboratory
conditions on the Earth. A comprehensive study reported in [4] provides a
systematic review on development of such simulating environment. Performing a
parabolic flights test [5] with a free fall maneuver for small period is
presented in [5] as potential micro-gravity research tool. Another possibility
is to conduct a drop-towers test [6], which can provide realistic micro-
gravity condition. A parabolic flight test based experimental validation
related to space robotics and on orbit maintenance were published in [7].
These techniques, however, are constrained by the overall flight time and
limited space that could be provided inside dropped capsule [8]. Moreover,
such approaches are significantly expensive for simulating the micro-gravity
environment.

Researchers have suggested innovative methods for long duration flight tests
for space robotic emulation. One such approach considers an underwater natural
bouncy system to replicate weightlessness [9]. The underwater test facility
provides an significant capacity for an astronaut’s training, the utility of
submerging a satellite is significantly limited. Another approach considers a
weight reducing suspension system [10] to counter balance the gravitational
force. The concept has been validated with a robotic manipulator. However, the
limited allowable motion and disturbances introduced by suspension mechanism
restricts the applicability of such method.

In view of generating an approximate micro-gravity environment, planar air-
bearing based mechanisms provide the most flexible dynamic equivalency with
ideal space representative framework. Robotic vehicles supported by air
bearing, representing spacecrafts/satellites, have some level of manoeuvring
capability to move over a smooth planar nearly friction-less surrounding. Air-
bearings attached with the platform releases pressurized air and creates a
thin film to levitate the platform, and thereby counter balance its weight to
produce a micro-gravity effect (in-plane components of gravity on the test
vehicles are negligible). Thus emulating the drag free and weightless
environment of orbital spaceflight. The only limitation of the design is that
they are restricted to three degrees of freedom, i.e., two translation and one
rotation motion, which is also closely resembles with a space scenario (since
out of plane motion are very limited for actual space mission). Moreover,
these facilities provide various hardware phenomena (e.g. realistic actuation
mechanism, computational constraints, sensor noise, actuator uncertainty,
delay etc.). In this manner, an air bearing based friction less platform
provides GNC testbeds for rigorous validation, both in terms of software
evaluation, as well as hardware-based implementation in high fidelity test
environments that have the capability to emulate realistic conditions in
space. Towards this direction, it should be mentioned that up to best of our
knowledge, there are no such commercially available platforms for these
friction-less micro gravity emulation platform. Various government
organization and university laboratories across the globe has indigenously
constructed their own test bed facilities. Examples can be found in [11],
[12], [13]. Primarily these emulation platform synthesises planner motion of a
robotic vehicle, while in some designs, additional degrees of freedom are
achieved by adding an air-bearing on top of the planar platform [14, 15, 16].
It should be noted that there is usually no thorough characterization of the
test beds in the literature, which restricts a through comparison of the
experimental results obtained by using various test facilities.

In line with the presented background, this article aims in presenting the
design of a novel floating platform, the Slider named from now and on, which
has been fully designed and manufactured developed from the Robotics and
Artificial Intelligence Team [17] at Luleå University of Technology in Sweden.
Figure 1 depicts the hardware-in-loop test-bed facility, which consists of an
epoxy-topped flat table, with a robotic arm connected to a two-dimensional
gantry system and the slider platform on top of the flat table, and an ABB
industrial robot in the vicinity of the flat table. The platform will be a
useful tool to advance the state of the art of GNC evaluation and can be used
to perform end-to-end system-level verification and validation before the
system’s operational deployment.

The main contributions of the article stems from: a) the introduction of a
novel design of a planar floating platform, Slider, with a detailed design
specification of the floating platform based on a limited number of air
bearings, b) a full mathematical model derivation for representing the
transnational and rotational motion of the slider platform over the friction-
less table that is required for the sequential control design step, and c) on
top of that an analytical mathematical model of the framework suitable for
control design is established, while also presenting a low level actuator
design framework, specifically designed for the proposed platform enabling an
accurate actuation. The entire design of our slider platform including 3D CAD
model, data for 3D printing, laser cutting, blueprint diagram and an extensive
list of various components are made available in the GitHub repository [18].
We sincerely believe that our design’s open-sourcing will befit interested
space research communities to rebuild the hardware platform in an individually
customized setup quickly. A visual demonstration of the slider in operational
mode can be found in [19].

The rest of the article is structured as follows. In Section II, a design
description of the physical floating platform equipped with various components
is presented. An initial validation of the overall design is described in
Section III. In Section IV, the mathematical model of the floating platform is
formulated. An open loop simulation with low level actuator selection logic
has been developed and the mathematical model has been validated numerical
simulations in Section V and finally, the article is concluded with its future
direction in Section VI.

##  II Design Description of Slider Platform

The robotic emulator platform is designed to smoothly maneuver over a
friction-less table. The flat top surface of the table (shown in Fig. 1) is
coated with epoxy resins which creates a smooth and flat table surface
required to replicate the friction-less motion of a spacecraft in space
environment. A schematic design of the slider platform is presented in Fig. 2.
The slider is supported with three air-bearings attached at its bottom deck.
The functional surface of each air-bearings is porous in nature. Compressed
air is evenly released through these small holes, which eventually creates an
air cushion. The air-cushion supports the weight of the slider platform and
allow it to levitate of the epoxy-topped table. Indeed, it can not offer a
micro-gravity framework, however, such a mechanism provides a nearly friction-
less environment along the 2-dimensional plane of the flat-table, which
closely resembles a motion in space. Due to this fact, it is preferred as an
emulation platform to demonstrate the state-of-the-art autonomous technologies
for complicated space missions. The slider platform is consisting of the
various subsystems described as follows

Figure 1: Hardware-in-loop testing facility with a $4\times 4~{}\mathrm{m}$
epoxy-topped flat table, floating platform (Slider), and two robotic
manipulators at LuleåUniversity of Technology.

(a) Side View

(b) Bottom View

Figure 2: Physical model of slider platform

Figure 3: Blueprint model of slider platform

###  II-A Structural Design of Slider

The structural design of the platform is constructed in such way that it is
light-weight, supports the necessary payload components (e.g. air bearings,
compressed air tank, thrusters etc.) and provide sufficient rigidity to the
over all assembly. One of the key features of the design is the easiness of
manufacturing. The technologies used for production are primarily 3D printing
and laser cutting. The physical construction of the outline structure is
consist of a circular base along with a ceiling surface, which are connected
by three ‘top-down-frames’ as shown in Fig.2. The circular base of the
structure is made out of $6mm$ poly-carbonate sheet, which is light-weight and
has sufficient stiffness to hold various components mounted over it. The
substantial components, which and relatively bulky (e.g. air tank, battery,
etc.) are placed above the circular base, as close as possible to the center
of the platform. Such compact placement ensures that the centre of gravity of
the slider is placed close to its geometrical center. The base line dimension
of the platform (approximately 350mm in diameter) is largely defined by the
length of the air tank and the size of the air pressure regulators mounted
over it. The components like thruster assembly and regulators are placed over
the periphery of the circular base as shown in Fig.3 that provides maximum
possible torque arm for controlling its motion. The a list of major components
for building the slider platform is given in Table I. A more elaborating
design description is presented in the Table A1, A2 of appendix section.
Various components of the baseline structure used to construct the slider
platform are built in laboratory using 3D printing technology which uses
Polylactic Acid (PLA) [20] as printing material.

TABLE I: List of Components Components | Manufacturer | Product type/Specification
---|---|---
Solenoid valve | Festo | MFH-2-M5
12V coil | Festo | MSFG-12-OD
Regulator | Festo | MS2-LR-QS6-D6-AR-BAR-B
Airsoft regulator | Polarstar | MRS
Tank regulator | Ninja | HP UL Reg 4500psi
Air tank | DYE | UL
Relay module | Seeed | Groove -2-Channel SPDT Relay
Upboard Computer | Arduino | Micro-controller

Figure 4: Schematic representation of Air-management system

###  II-B Air Management System

In order to drive all pneumatic components (primarily consist of air bearings
and thruster assembly), the slider platform carries an air tank of size
$1.2l$, which is filled with pressurized air and placed symmetrically about
the $X_{B}-Z_{B}$ plane of the slider. The air tank structure can stores
compressed air, pressurized up to $300$ bar. The air tank is connected with
air-bearings (attached to bottom deck) and the thrusters through air tubes. A
schematic representation of the air-management system is presented in Fig.4.
Pneumatic flow from air-tank is divided into two separate branches. Each of
the branches is equipped with low-pressure regulator, which are capable to
control the output pressure. The output air pressure, is regulated down to
about $5$ bar for operation of air-bearings and about $7$ bar for thrusting
mechanism. Relatively heavy regulators are mounted on the front side of the
tank, to appropriately counteract the weight of the air tank along
$Y_{B}-Z_{B}$ plane. The compressed air is evenly released through the porous
surface of the air bearing in regulated manner. Three air bearings each of
size $40$ mm are mounted in the bottom deck. While operational the three air-
bearing together makes a plane of air cushion of a thickness of about $15$
microns, resulting in friction less motion of the platform over the friction-
less table.

###  II-C Actuation Mechanism

In order to mobilize the slider platform in a controlled manner, it is
equipped with eight small thrusters that operate in an on-off mode. The
thrusters are synthesized with 3D printed nozzles integrated with $12$ V
solenoid valves. The solenoid valves are controlled by relay modules, which
operates on signals received from on-board computer or RC receiver through
‘Arduino’ board. The energy is stored in $4S$ Lipo $1400$ mAh battery and it
is regulated down to 12V for solenoid valves and down to $5$V for on-board
computer. Engineering resin is used as material for synthesizing nozzles.
Eight thrusters are distributed into four brackets, which are placed wide
apart from the geometric centre, while maintaining a compact footprint of a
square with a side about $39$mm. The placement of the thruster assembly is
presented in Fig.3. Two thrusters sharing a bracket (One aligned with the
$x$-axis and the other with the $y$-axis) maintain a small offset as shown in
the Fig.8. Such wide spread arrangement come up with large torque arms, which
maximizes the magnitude of torque. Each of the thruster can produce a constant
magnitude force of $0.7N$ while activated. The thruster assembly is capable to
provide force and torque that is required to translate, as well to orient the
slider over the friction-less table. An initial set of experiments has been
cried out for construction of force and torque model of small thrusters. The
experimental setup for thruster modeling is presented in Fig.6. A thruster is
mounted over a rod of length about $0.5$ m to provide a sufficiently large
torque-arm. Such arrangement is required to mitigate the the resolution of the
6D Torque/Force sensor. A calibrated response of the thrusters are presented
in Fig.6. It is evident that the delay in the thruster response are fairly
insignificant for engineering practice. However, the undulation in measured
signals are essentially due to presence of unwanted sensor noise. Since the
thrusters operates in a switching mode, a specific combination of thruster
activation results into a particular type of directed motion of the slider.
The specific combination of the thruster required to activate for various
directed motions are presented in Table II.

TABLE II: Thrust activation logic Motion | Thrusters
---|---
| $T_{1}$ | $T_{2}$ | $T_{3}$ | $T_{4}$ | $T_{5}$ | $T_{6}$ | $T_{7}$ | $T_{8}$
Forward |  |  | ✓ |  | ✓ |  |  |
Backward | ✓ |  |  |  |  |  | ✓ |
Left |  | ✓ |  | ✓ |  |  |  |
Right |  |  |  |  |  | ✓ |  | ✓
Clockwise | ✓ |  |  | ✓ | ✓ |  |  | ✓
C-Clockwise |  | ✓ | ✓ |  |  | ✓ | ✓ |

Figure 5: Experimental setup for thruster modeling

Figure 6: Excitation and measurement of output force and torque

##  III Initial Test

An initial trial for the platform has been carried out in the Kiruna space lab
facility. The experiment’s goal is to test and prove the concept and
demonstrate its manoeuvrability over the friction-less table. During the
trial, the platform motion has been controlled manually using a remote control
(RC) transmitter in an open-loop manner. The platform is equipped with an RC
receiver, which is directly connected to the on-board computer. The on-board
computer communicates with the thruster assembly and provides the ‘on/off’
command state to a set of assigned thrusters. The RC transmitter’s sticks are
set to be operated in an ‘on/off’ fashion while Modulation and pulsation of
the thrusters are controlled directly from the operator-end. The control logic
for the RC transmitter has been designed as follows. Three dedicated channels
of the RC transmitter are used to provide the command for the platform’s
directed platform motion. Among these three ones is assigned to control the
forward and backwards movement, while others are used for sidewise (towards
left-right) and rotational (clockwise and counter-clockwise direction) motion.
The actuation logic for selecting the set of the thruster for each directed
motion is described in Table II.

During the initial trial, the Air tank has been filled with compressed air up
to $200$ bar. The operating condition for the platform is set up as follows.
The pressure regulator for the air bearings is set to $5$ bar, and the same
for the thruster assembly is maintained at $7$ bar. With this setup, we
reached about $7$ minutes of flight time. Airflow through the bearings is
allowed continuously during the entire operation period whereas the thrusters
are operated through the on-board computer (as commanded from the RC
transmitter). The on-time for the thrusters is estimated to be $30\%$ of the
entire flight time. Typically, the flight time is primarily dependent on the
consumption rate of the pressurised air and hence on the thrusters usage. The
battery life is over exceeding the lifetime of the air tank heavily; thus, it
will not affect the platform’s flight time. However, refiling of the air tank
is required more often. A proof of demonstration of platform motion during the
initial trial is recorded video-graphically and can be found in [19].

During the initial trial, the platform motion is controlled manually and
operated in open-loop mode. However, the realistic demonstration of various
orbital manoeuvre and synchronised movement of multiple slider platform
requires advanced control law to be operated autonomously in a closed-loop
manner. The formulation of advanced model-based control design requires the
mathematical model of the platform. In view of that, the dynamic model of the
slider platform consisting of the thruster based actuation mechanism is
presented next.

##  IV Equation of Motion

In order to describe the equation of motion of the slider over friction-less
platform, two frame references have been considered. An inertial frame of
reference (denoted as $X-Y-Z$) is assumed to be attached on a corner point of
the friction-less table. The axis of the inertial frames are considered to be
directed along the length, width and height of the table. Another, moving
frame of reference (denoted as $X_{B}-Y_{B}-Z_{B}$) is considered to be
attached to the centre of gravity (CG) of the slider, which moves along the
slider. The slider can translate over the table surface, as well it can rotate
about its $Z_{B}$ axis. The position of the slider (denoted as $x,y$) is
described in the inertial frame of reference. Since the various sensors and
actuators are attached with the slider body, it is preferred to define its
velocity and actuation forces in the body frame. Lets $v_{x}$ and $v_{y}$
denotes the velocity components of the slider expressed along $X_{B}$ and
$Y_{B}$ respectively. Since, the motion of the slider is restricted in $2-$D,
the motion along $z$ component is ignored. Slider transnational kinematic
equations of motion are described as:

Figure 7: Reference frames used to describe the dynamics

Figure 8: Thruster placement in a bracket

$\left[\begin{matrix}{\dot{x}}\\\ {\dot{y}}\\\
\end{matrix}\right]=R_{B}^{I}\left[\begin{matrix}{{v}_{x}}\\\ {{v}_{y}}\\\
\end{matrix}\right]$ (1)

where, $R_{B}^{I}=\left[\begin{matrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\\\ \end{matrix}\right]$ represents the rotation matrix,
that transform a vector from the body frame to inertial frame of reference.
Here, $\theta$ represents the orientation of the slider (heading angle between
$X$ and $X_{B}$ axis). Since the slider can rotate only about its $Z_{B}$
axis, it rotational velocity denoted as $r$ is directed along $Z_{B}$. Based
on the fundamental Newton’s law of motion [21], transnational dynamics of the
slider is formulated as

$\left[\begin{matrix}{{{\dot{v}}}_{x}}\\\
{{{\dot{v}}}_{y}}\end{matrix}\right]=\left[\begin{matrix}r{{v}_{y}}+\frac{{{f}_{x}}}{m}\\\
-r{{v}_{x}}+\frac{{{f}_{y}}}{m}\end{matrix}\right]$ (2)

where $f_{x},f_{y}$ denotes the actuation forces and $m$ represents the mass
of the slider. Note that, the transnational dynamics incorporates the coriolis
effects ($r{{v}_{x}},r{{v}_{y}}$) due to its rotational motion. The rotational
motion of the slider is formulated based on conservation of angular momentum
[22] and presented as follows

$\left[\begin{matrix}{\dot{\theta}}\\\ {\dot{r}}\\\
\end{matrix}\right]=\left[\begin{matrix}r\\\ \frac{\tau}{{{I}_{zz}}}\\\
\end{matrix}\right]$ (3)

where, $\tau$ denotes the applied torque and $I_{zz}$ indicates the principal
moment of inertia along the $Z_{B}$ direction. Note that the slider platform
is designed in a balanced manner such that off diagonal components of moment
of inertia matrix are negligible. combining the Eqs.(1)-(3), the dynamical
equation of motion in compact form is represented as

$\left[\begin{matrix}{\dot{x}}\\\ {\dot{y}}\\\ {\dot{\theta}}\\\
{{{\dot{v}}}_{x}}\\\ {{{\dot{v}}}_{y}}\\\ {\dot{r}}\\\
\end{matrix}\right]=\left[\begin{matrix}{{v}_{x}}\cos\theta-{{v}_{y}}\sin\theta\\\
{{v}_{x}}\sin\theta+{{v}_{y}}\cos\theta\\\ r\\\
r{{v}_{y}}+\frac{{{f}_{x}}}{m}\\\ -r{{v}_{x}}+\frac{{{f}_{y}}}{m}\\\
\frac{\tau}{{{I}_{zz}}}\\\ \end{matrix}\right]$ (4)

The slider’s actuation unit is equipped with a total number of eight small
thrusters attached with the platform. The control action, i.e. forces and
torque components are related with the actuation of thruster units, modeled as

$\left[\begin{matrix}{{f}_{x}}\\\ {{f}_{y}}\\\ \tau\\\
\end{matrix}\right]=\left[\begin{matrix}\sum\limits_{k=1}^{8}{{{T}_{k}}\cos{{\beta}_{k}}}\\\
\sum\limits_{k=1}^{8}{{{T}_{k}}\sin{{\beta}_{k}}}\\\
\left(\sum\limits_{k=1}^{8}{\left({{T}_{k}}r_{{{T}_{k}}}^{y}\cos\beta_{k}-{{T}_{k}}r_{{{T}_{k}}}^{x}\cos{{\beta}_{k}}\right)}\right)\\\
\end{matrix}\right]$ (5)

where, $T_{k}$ denotes the constant thrust magnitude,
$(r_{{{T}_{k}}}^{x},r_{{{T}_{k}}}^{y})$ together indicates the position of the
$k^{th}$ thruster in $X_{B},Y_{B}$ plane and ${{\beta}_{k}}$ represents its
orientation with respect to $X_{B}$ axis.

TABLE III: Numerical values for system and simulation parameters

(a) Position and orientation of individual thruster Thruster |  | Position ($r_{{{T}_{k}}}^{x},r_{{{T}_{k}}}^{y}$)
---
(mm)
| Orientation $\beta_{k}$
---
(deg)
$T_{1}$ | $(195,-140)$ | $0$
$T_{2}$ | $(140,-195)$ | $270$
$T_{3}$ | $(-195,-140)$ | $180$
$T_{4}$ | $(-140,-195)$ | $270$
$T_{5}$ | $(-195,140)$ | $180$
$T_{6}$ | $(-140,195)$ | $90$
$T_{7}$ | $(195,-140)$ | $0$
$T_{8}$ | $(140,195)$ | $90$

(b) System parameters Parameters | Values
---|---
Mass ($m$) |  $4.436$ kg
Moment of inertia ($I_{zz}$) |  $1.092$ $kg-m^{2}$
| Control command
---
time step
$0.5$ s
| System Propagation
---
time step
$0.01$ s
| Minimum
---
on time of thruster
$10$ ms
$T_{lb},T_{ub}$ |  $0,0.7$ N

##  V Model based Open Loop Simulation

In order to validate the mathematical model, an open-loop simulation has been
carried out in this section. Various numerical parameters used for the
simulation study is presented in Table III. Initially the slider platform is
considered to be resting at the corner of the table, i.e.
$[x,y,\theta,v_{x},v_{y},r]=0_{6\times 1}$. It is intended to excite the
platform model with two-step ramp type input command (combining both the
positive and negative actuation). The open-loop control excitation signal is
depicted in the fourth subplot of Fig.11. The identical control input is
considered to be applied along the three channels, i.e. $f_{x},f_{y}$ and
$\tau$. The simulation-based validation has been carried out in the two-step
approach. In the first step, the ideal system response is evaluated by
propagating of the dynamic model Eq.(4) in the presence of continuous-time
force and torque command. Here, the realistic actuator model consisting of
eight thruster assembly, is ignored. Ideal time response of the slider pose
and velocities due to continuous input command are presented in Figs. 11 and
11 respectively. Here the blue dotted lines represent the ideal system
responses. In the next step, the realistic actuator model with its physical
limitations (such as maximum thrust bound, minimum on-time etc.) is accounted
for in the simulation study. The open-loop force and torque command, i.e.
$f_{x},f_{y},\tau$ needs to be realised by actuating the on-off thrusters
assembly, i.e. $T_{1}\cdot T_{8}$. Rewriting the Eq.(5), the relation between
$f_{x},f_{y},\tau$ and $T_{1}\cdot T_{8}$ can be expressed as

$Ax=b$ (6)

where, $b=\left[\begin{array}[]{c}f_{x}\\\ f_{y}\\\
\tau\end{array}\right],x=\left[\begin{array}[]{c}T_{1}\\\ T_{2}\\\ \vdots\\\
T_{8}\end{array}\right],A=\left[\begin{array}[]{ccc}\cos\beta_{1}&&\cos\beta_{8}\\\
\sin\beta_{1}&\vdots&\sin\beta_{8}\\\
{\left(r_{{{T}_{1}}}^{y}\cos\beta_{1}-r_{{{T}_{1}}}^{x}\cos{{\beta}_{1}}\right)}&&{\left(r_{{{T}_{8}}}^{y}\cos\beta_{8}-r_{{{T}_{8}}}^{x}\cos{{\beta}_{8}}\right)}\end{array}\right]$
Note that the matrix $A$ is known and constant, determined by placement of
each thruster’s and its orientation.

Figure 9: Schematic representation of low level actuation logic

### Thruster Selection Logic

The objective of the thruster selection logic is to determine the set of
thrusters required to be activated in order to achieve the desired commanded
force and torque inputs. A pseudo-inverse based solution of Eq. (6) is adopted
in [23]. The method incorporates a null space adjustment depending on the sign
of input command. In this paper, the above problem has been translated into an
optimization framework [24] as follows:

$\displaystyle MinJ=x^{T}x$ (7) $\displaystyle S.T.Ax=b$ (8) $\displaystyle
x_{ub}\geq x\geq x_{lb}$ (9)

Figure 10: Open-loop response of slider pose

Figure 11: Variation of slider velocity profiles and input excitation

The solution of the above optimization problem provides the optimal magnitude
of each thruster ($T_{k}^{*}$) that collectively mitigates the necessary force
and torque demand, while minimizes the actuation effort and ensuring the
physical limitations of each thruster. Note that, the magnitude of each
thruster $T_{k}^{*}$ obtained based on thruster selection logic can be any
value between $x_{b}$ and $u_{b}$ limits. However, in practice these thrusters
operate in an ON-OFF mode, i.e. either it can provide a full actuation with a
maximum thrust or operates in no thrust mode. In order to address this, the
pulse width modulation (PWM) technique is incorporated. Dedicated PWM units
are assigned for each thruster, as shown in Fig.9. The PWM block generates a
sequence of pulses in such a way that the average thrust produced by each
thruster closely follows the required magnitude $T_{k}^{*}$.

The implementation of actuator based simulation has been carried out as
follows. The input commands ($f_{x},f_{y}$ and $\tau$) are processed through
the zero-order-hold mechanism and its magnitude is hold at a constant value
for each control time step, i.e. $0.5$ s. During this period, the input signal
is processed through various intermediate blocks, as presented in Fig. 9. Each
PWM blocks are set to be operated with a frequency of $10$ Hz. The resulting
sequence of output pulses for each thruster $T_{1}-T_{8}$ are presented in
Figs.13 and 13. With the application of pulse input as actuation command for
thrusters, the nonlinear dynamic model is propagated, and the corresponding
time responses are presented in Figs.11 and 11. Here the solid black lines
represent the thrust actuated system responses. In the first subplot of Fig.
11, the slider trajectory is presented where the arrowheads are indicating its
orientation at the given point. Detailed orientation profile is presented in
the fourth subplot of Fig.11. It is evident that the thruster based actuation
mechanism closely resembles the ideal responses. However, the zoomed view in
Fig. 11 shows a non-smooth behaviour, which is due to pulsating mode of
actuation input.

Figure 12: Actuation command for thrusters $T_{1}-T_{4}$

Figure 13: Actuation command for thrusters $T_{5}-T_{8}$

##  VI Conclusion

In this article, a friction-less floating robotic test-bed facility for a
hardware-in-loop experimental study of a planar satellite that has
indigenously developed at Luleå University of Technology is presented. The
details design description of the physical platform, with the each component
specification was presented. Moreover a mathematical model describing its
dynamic motion was formulated. This capability can be used to conduct research
on coordinated control of spacecraft teams. A successful initial test has been
conducted in a open manually operated loop remote control manner. Multiple
simulation studies has been carried out with formulated mathematical model in
various scenarios. An optimization based actuator allocation logic has been
developed and tested in simulation framework. The on-board actuators, composed
of eight thrusters, which is equivalent to the actuators configuration of
typical spacecraft and further establishes close realistic equivalence.
Additionally, multiple such robotic platform can be operated simultaneously
over the friction less table. This state-of-the-art dynamic hardware-in-the-
loop emulation facility will continue to be fruitful for the advancement of
spacecraft proximity operations research.

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##  VII appendix

Additional supplementary details of each component used to construct the
slider platform are presented in this section. Each component is
chronologically identified with a serial number between $1-55$. The location
of each component is indicated with the help of the blueprint model of the
slider platform, as shown in Fig. A1. Various description such as product
specification, materiel used for fabrication etc. are listed for each
components in Tables A1 and A2. A more detail design related supplementary
materials of the slider platform are made available in the GitHub repository
[18], which include 3D CAD model, data for 3D printing, laser cutting,
blueprint diagram and extensive list of various components.

(a) Top view

(b) Side view

Figure A1: Blueprint model of slider platform indicating detailed components TABLE A1: List of Components (serial No. 1- 25) indicated in Figs A1 ITEM | No. of QTY | Product type/Specification | Material | Description
---|---|---|---|---
1 | 1 | Base-6mm | Polycarbonate, Clear | Laser Cut (6mm)
2 | 6 | 130778 QSM-M5-4-100 |  | QSM-push-in fitting
3 | 3 | S104001 |  | Air bearing
4 | 3 | S8013B11 | Stainless Steel | Air bearing bolt
5 | 3 | S8013H04-NuT | Stainless Steel | Air bearing Hex nut
6 | 3 | S8013H04-ScrewNut | Brass, Soft Yellow | Air bearing housing
7 | 1 | Low-top-platform-mount-front-V4-1st part | PLA | 3D print
8 | 1 | Name-plate | PLA | 3D print
9 | 3 | Leg-washer | PLA | 3D print
10 | 2 | Low-top-platform-mount-v2 | PLA | 3D print
11 | 3 | LM2596 DC-DC StepDown Converter v1 |  | Step-down voltage regulator
12 | 8 | 4573 MFH-2-M5 |  | MFH-Solenoid valve
13 | 8 | 320410 MSFG-12-OD—(P) |  | MSFG-p-Solenoid coil
14 | 2 | Valve-holder-v3 | PLA | 3D print
15 | 6 | 153333 QSML-M5-4 |  | QSML-Push-in L-fitting
16 | 8 | Festo-connector-cover | PLA | 3D print
17 | 8 | Nozzle-SLA-base | Resin "Grey pro" | 3D print (SLA)
18 | 8 | 8030314 NPFC-R-G18-M5-FM |  | NPFC-R-Threaded fittings
19 | 8 | Nozzle-bumper-V2 | TPU 95A | 3D print
20 | 4 | 153374 QSMY-6-4 |  | QSMY-Push-in Y-connector
21 | 5 | 153129 QST-6 |  | QST-Push-in T connector
22 | 4 | T-piece-clamp | PLA | 3D print
23 | 4 | Relay-mount | PLA | 3D print
24 | 4 | Grove 2 Channel SPDT Relay |  | Relay
25 | 2 | 153484 QH-QS-6 |  | QH-QS-ball valve
TABLE A2: List of Components (serial No. 26- 55) indicated in Figs A2 ITEM | No. of QTY | Product type/Specification | Material | Description
---|---|---|---|---
26 | 1 | Valve holder-V3 | PLA |
27 | 1 | Zippy compact 1400 |  | Battery
28 | 1 | Battery-case-zippy-1400i-V2 | PLA | 3D print
29 | 1 | Regulator-mount-v2 | PLA | 3D print
30 | 1 | 8086628 MS2-LR-M5-D6-AR-BAR-B |  | Regulator w filter Air bearings
31 | 2 | 5003640 MS2-LR/LFR-B |  | MS2-WR (p)-Mounting bracket
32 | 1 | 8086644 MS2-LFR-QS6-D6-AR-BAR-C-M-B |  | Regulator Thrusters
33 | 1 | Regulator-mount-long-v3 | PLA | PLA 3D print
34 | 1 | T-piece-holder | PLA | 3D print
35 | 2 | 130618 QSW-6HL |  | QSW-HL-Push-in connector
36 | 1 | Splice-sla | Resin "Grey pro" | 3D print (SLA)
37 | 1 | 186096 QS-G1/8-6 |  | QS_G-Push-in fitting
38 | 1 | Bottle mount V3-extended | PLA | 3D print
39 | 1 | Bottle mount V3 | PLA | 3D print
40 | 1 | Arduino shield |  | Arduino shield
41 | 1 | Arduino Nano 33 IoT |  | Arduino
42 | 1 | Arduino-w-shiled-holder | Resin "Black" | 3D print (SLA)
43 | 1 | Polarstar-regulator |  | Airsoft regulator
44 | 1 | paintball air tank-DYE |  | Paintball tank 1,1l
45 | 1 | top ring-6MM-V2 | PLA | 3D print
46 | 1 | UP_BOARD_&_HEAT_SINK_by_JMJV |  | Up-boad computer
47 | 1 | Futaba-reciever | PLA | RC reciever
48 | 1 | Futaba-controller-hodler | PLA | 3D print
49 | 1 | Valve holder-V4 | PLA | 3D print
50 | 1 | holder_ps | PLA | 3D print
51 | 1 | ps-camera |  | Playstation camera
52 | 1 | Gopro-mount | PLA | 3D print
53 | 2 | Valve-holder-v3-mirror | PLA | 3D print
54 | 1 | Spacer | PLA | 3D print
55 | 1 | Low-top-platform-mount-front-V4-2nd part | PLA | 3D print