[SSC23-VII-02]

Development of On-Ground Hardware In Loop Simulation Facility for Space
Robotics

Roshan Sah, Raunak Srivastava, Kaushik Das
Space Systems, Tata Consultancy Services(TCS)-Research
Bangalore, India
<EMAIL_ADDRESS>
ABSTRACT

Over a couple of decades, space junk has increased rapidly, which has caused
significant threats to the LEO operation satellites. A mitigating measure
should be taken to protect the LEO space environment. An Active Debris Removal
(ADR) concept continuously evolves for space junk removal. One of the ADR
methods is Space Robotics, whose function is to chase, capture and de-orbit
the space junk. This paper presents the development of an on-ground space
robotics facility in the TCS Research for on-orbit refueling and debris
capture experiments. A Hardware-in-Loop Simulation (HILS) system will be used
for integrated system development, testing, and demonstration. HILS is the
most effective and vital system to test the on-orbit docking mechanism’s
reliability, usability, and safety. The HiLS test facility of TCS Research Lab
will use two Universal Robot(UR)5e and UR10 manipulators in which one
manipulator is attached to the robotic-RG2 gripper, and the other is attached
to a force-torque sensor named Hexa-E Onrobot and with a scaled mock-up
satellite model. The first UR5 manipulator will be mounted on a 7-axis linear
rail and contain the docking probe. First UR5 manipulator with the suitable
gripper has to interface its control boxes. The grasping algorithm was run
through the ROS interface line to demonstrate and validate the On-orbit and
Debris removal operation. The manipulator will be mounted with LIDAR and a
Real sense camera to visualize the mock-up model, find the target model’s pose
and rotational velocity estimation, and a gripper that will move relative to
the target model. The other manipulator has the UR10 control, providing
rotational and random motion to the mock-up satellite, enabling a dynamic
simulator fed by force-torque data. The dynamic simulator is fed up with the
orbit propagator model SGP4, which will provide the orbiting environment to
the target model. For the simulation of the docking and grasping of the target
model, a 7-axis linear rail of a 6-meter setup is still in the procurement
process. Once reaching proximity, the grasping algorithm will be launched to
capture the target model after reading the random motion of the mock-up
satellite model. The HILS system proposed in this paper helps develop on-orbit
servicing (OOS) like repairing, upgrading, transporting, rescuing
technologies, on-orbit refueling, and berthing and debris removals.

## INTRODUCTION

The growth of spacecraft launches has increased substantially over the past
couple of decades, and the statistics results indicated an average of more
than one hundred satellites were launched annually. Most spacecraft launches
have accomplished their mission goals, but some experienced failures and
irregularities [1]. In the past, a launcher failure is the most typical
failure of the spacecraft. However, an on-orbit failure has recently overtaken
launch failures, resulting in billions of dollars lost in space organizations
like NASA, ESA, ISRO, etc. Eventually, all launched satellites run out of fuel
after their end-of-life (EOL) span, which causes the formation of space
junks/debris at different orbit regions.

In recent decades, the spaces junks (residual of the spent satellite, rocket
staging, body and booster, junk particles from the collision of debris
objects) have increased drastically in Low Earth Orbit (LEO) region, which can
cause a severe threat to the operation satellite at LEO orbit. Several works
of literature have shown that servicing, like repairing, refueling, upgrading,
etc., of damaged spacecraft in flight is cost-effective and extends the
existing satellite’s life span. In addition, the post-mission removal of the
spent satellite, rockets staging, etc., will keep the operational satellite in
a favorable orbital environment for upcoming space missions. An On-Orbit
Servicing (OOS) has become instrumental in increasing the life span of many
satellites, thereby helping in the sustainable utilization of space orbits.

To do the servicing of damaged spacecraft, NASA realized, as early as the
1980s, the significance of the robotics on-orbit servicing activities to
protect assets in space [2]. It was termed Space Robotics, which is the
combination of spacecraft and robotics manipulators. Recently, Roshan et al.
have developed a spacecraft with single/dual manipulators, which are termed a
“Debris Chaser Satellite” [3][4] [5] whose primary function is to chase,
capture, and de-orbit[6] the space junks to a shallow earth orbit. However,
the debris chaser satellite can also be used for servicing applications.
Autonomous control of space robots for precise in-orbit operations (satellite
servicing, active debris removal, etc.) has become critical for long-term
sustainable use of the Earth’s orbits due to a multi-fold increase in space
debris. However, the spacecraft-manipulator couple system produces non-
linearity to space systems’ complex control problem. Manipulator motion
induces reaction forces and moments on the satellite, which disturbs its
position and attitude, affecting the end-effector’s pose. Therefore, the
dynamic interaction between the manipulator and the base satellite must be
considered while designing a control law for such space manipulators.

The space robot consists of three different modes of approach[7];

1\. Free Flying space robot:

It refers to the class of satellite-manipulator systems wherein the cold gas
thrusters/jets facilitate[8] [9] a stabilized and controlled base for robotic
manipulator motion. The fixed base is favorable for trajectory formation of
manipulators, however, at the expense of fuel consumption and restricted
workspace.

2\. Free Floating space robot:

It refers to the class of satellite-manipulator systems wherein the combined
system momentum remains conserved due to the absence of an external force or
torque in the terminal phase when the arm is deployed to reach the target.
Only the manipulator arm motion is actively controlled in a Free Floating
configuration. At the same time, the satellite base is left uncontrolled to
freely translate and rotate as per the forces and moment acted upon by the
arms motion. In contrast, there is no fuel consumption at all.

3\. Rotation Floating Space Robot:

Rotation floating space robots[10] are a comparatively newer approach wherein
the base satellite attitude is controlled using actuators like reaction wheels
(RW) which preserve the system’s momentum and control the manipulator’s arms.
The satellite base is still left free to translate. This approach causes fuel
consumption in a limited amount.

These three approaches are primarily used for the manipulation in orbit for
OOS applications. Ground-based experiments will be required to verify and
explore the capability, feasibility, and limitations of the critical
trajectory planning and control algorithm before servicing the satellite i.e.,
the space robot is launched for OOS missions. A ground-based experiment helps
us to ensure that the space robot can successfully implement the mission tasks
like orbit capturing and docking operations. Hence, setting up a suitable
ground-based experiment system for a space robot performing fine manipulation
becomes essential.

The ground-based experiment for the evaluation of planning and control
algorithms for space robots is challenging in two ways;

1\. the coupled space robot system needs to be permitted to move in the 6DOF
system, and

2\. the effect of gravity cannot be canceled as space considers the micro-
gravity environment.

Over the last decade, there have been five methods to emulate the micro-
gravity environment on the ground [11];

1\. Free fall motion or airplane flying at the stratosphere.

2\. Neutral buoyancy

3\. suspension system

4\. air-bearing table.

5\. Hybrid method.

The first four methods to emulate the micro-gravity environment are expensive
and time-consuming. Therefore most of the space research organizing focuses on
the hybrid method, which is the mathematical model combined with the
mechanical model. The trajectory of the space robot in a micro-gravity
environment is calculated using the gravity-based mathematical model. Then the
mechanical model is forced to move according to the calculation. A precise
dynamic model can emulate the behaviors of a space robot coupled system. The
hybrid method is relatively cheap and can be set up using ground-based robots
like UR5, UR10, and KUKA.

Organizations like NASA, ESA, DLR (German Aerospace Center), Surrey Space
Center, and JAXA have set up several research facilities to execute the
ground-based experiment to verify the space robotics operation for OOS
applications. The DLR Robotics Laboratory has set up a European Proximity
Operations Simulator (EPOS), which consist of two KUKA robot in which; one is
used for simulating target and the other for simulation chaser, linear rail,
sun simulator, and sensors. The EPOS facility can be used to test the
autonomous GNC for rendezvous and mission inspection[12] [13]. Similarly, the
STAR lab of the University of Surrey has developed the Orbital robotics
testbed, which is used to simulate the 6DOF manipulation by using multiple
robotics arms or air-bearing tables. Their setup has also considered orbital
disturbance and tries to grasp the 3D printed mock-up model using a range of
control modes from tele-operation to full autonomy [14].

In this paper, we have established on ground Hardware In Loop Simulation
(HiLS) facility to explore and verify the capability of the space robot to
execute fine manipulation, non-cooperative target capturing, and the end phase
of docking. This HiLS facility is the first space robotics laboratory in India
set up by TCS Research India and will be used for future ISRO OOS missions.
Using ground-based robots, a hybrid simulation approach has been used to
verify our trajectory and control algorithms. These verified algorithms can be
used for upcoming missions.

The rest of the paper has been organized into the following sections; Section:
Hybrid Simulation Method describes the simulation and hardware concept of
dynamics emulation of chaser-target motion using the Pybullet and URs
manipulator. The section: Space Robotics Laboratory(SRL) explains about
objective, lab layout, and the different SRL components required for on-ground
experiment systems. In the section: Results and Discussion, the initial
results of the Hardware-in-loop System (HiLS)of URs replicating the chaser’s
and target’s motion were explained. The last section deals with the conclusion
and future works that will be executed in the coming days by considering the
multiple scenarios of space rendezvous and docking in a more real-time
consideration.

## HYBRID SIMULATION METHOD

Figure 1 shows a snapshot of the PyBullet environment created to simulate the
chaser-target space robot. A red target is being tracked by a Rotation
Floating space robot[15] with a UR5 manipulator on-board it. Three orthogonal
reaction wheels are mounted on the space robot to control the attitude of the
base satellite. The objective is to use a two-layered MPC-based control [16]
architecture that controls the motion of the satellite base and UR5
manipulator to enable it to reach the moving target. After that, the target
will be grabbed using an impedance controller, followed by bringing the arm
(with the target) back to the initial configuration. PyBullet simulations have
been performed, and the results are shown later. An orbital environment [17]
[18] needs to be incorporated with the Pybullet environment by using STK tools
which will be part of the future work.

Figure 1: Space Robot in Pybullet Environment.

The dynamic equation of the space robot is generally expressed in the
following form;

$\begin{bmatrix}H_{b}&H_{bm}\\\
H_{bm}^{T}&H_{m}\end{bmatrix}\begin{bmatrix}\ddot{x_{b}}\\\
\ddot{\Theta_{s}}\end{bmatrix}+\begin{bmatrix}c_{b}\\\
c_{m}\end{bmatrix}=\begin{bmatrix}F_{b}\\\
\tau_{m}\end{bmatrix}+\begin{bmatrix}J_{b}^{T}\\\ J_{m}^{T}\end{bmatrix}$ (1)

whereas

$H_{b}\in R^{6x6}$ and $H_{m}\in R^{nxn}$ are the inertial matrices of the
base and the manipulator’s arm;

$H_{b}m\in R^{6xn}$ is the couple inertial matrix;

$c_{b}\in R^{6}$ and $c_{m}\in R^{6}$ are the velocity-dependent non-linear
terms for base and arm;

$F_{b},F_{e},\tau_{m}\in R^{6}$ are force and torques on the centroid of the
base and exert on the manipulator’s hand, torque on the manipulator’s joints.

$\centering\dot{x_{e}}=J_{b}\dot{x_{b}}+J_{m}\dot{\Theta_{s}}\centering\@add@centering\@add@centering$
(2)

Whereas, $J_{b}$ and $J_{m}$ are the Jacobian matrices dependent on the motion
of the base and the manipulator; $\dot{x_{b}}$ and$\dot{x_{e}}$ are the
absolute velocity of the space base and end-effectors. $\Theta_{s}$ is the
joint angle of the space manipulator.

Since no external forces and torques are acting on the free-floating system of
the space robot, the linear and angular momentum is considered zero initially.
Therefore, the system momentum keeps zero according to the conservation law.
The equation of such a free-floating space robot is given as

$\centering
H_{b}\dot{x_{b}}+H_{b}m\dot{\Theta_{s}}=0\centering\@add@centering\@add@centering$
(3)

Hence, the absolute velocity of the space robot can be solved from equation 3
as

$\centering\dot{x_{b}}=-H_{b}^{-1}H_{b}m\dot{\Theta_{s}}=H\dot{\Theta_{s}}\centering\@add@centering\@add@centering$
(4)

Substituting the value of the absolute velocity of space base for equation 4
to 3, the following relationship can be formed,

$\centering\dot{x_{e}}=(J_{m}-JbH_{b}^{-1}H_{b}m)\dot{\Theta_{s}}=J^{*}(\Phi_{b},\Theta_{s})\dot{\Theta_{s}}\centering\@add@centering\@add@centering$
(5)

Whereas, $J^{*}(\Phi_{b},\Theta_{s})$ is the Generalized Jacobian Matrix (GJM)

And by GJM, the manipulator’s hand can be controlled by resolved motion rate
or acceleration control in the inertial space.

The simulations are followed by hardware experiments to validate the results
obtained from PyBullet. Towards this end, a Hardware-in-the-loop-based Hybrid
approach is followed, the most commonly used approach for space robots. Figure
2 shows the philosophy behind the hardware experiments. The motion of the
chaser end-effector and the target is calculated concerning a frame attached
to the base of the chaser satellite. The transformed 6 DOF motion is
replicated on two Universal Robot arms: Chaser on a UR5 arm and target motion
on the UR10 arm. The actual motion of the hardware arms is then fed back to
the PyBullet environment as real-time feedback. The 6 DOF motion is
transformed back into the inertial frame, which is then acted upon by the
proposed controller. The cycle repeats until and unless the chaser arm (UR5)
reaches the target arm (UR10). In the preliminary results, independent motions
of the UR5 (chaser motion) and UR10 (target motion) were executed in sync with
the PyBullet environment in real time. Both arms produced perfect results and
reproduced the PyBullet results. In the future, they will be run together to
complete the pipeline executed in PyBullet.

Figure 2: Hybrid Approach of Hardware in the loop.

## SPACE ROBOTICS LAB(SRL) SETUP

The space robotics lab is developed under a cross-section area of 12.2m x 5.8m
in the TCS Research Building. The concept of a space robotics lab was taken
from the German Aerospace Center(DLR) [19] robotics test facility where they
developed European Proximity Operations Simulator [12]. The main aim of the
EPOS is to execute the cooperative rendezvous and docking system for Automated
Transfer Vehicle (ATV) to the ISS. The EPOS test facilities consist of the
test bed of two KUKA robots, 25m linear rail, sun simulation, realistic
background, and HiLS setup for the execution of the end-to-end testing of
rendezvous and docking.

A similar concept has been used to develop our robotic space lab in TCS
Research. The SRL is the first space robotics test facility in India that will
be diversely working in the field of space robots. The test setup will test
and validate cooperative and non-cooperative rendezvous and docking operations
used in On-Orbit Servicing applications like Refueling, debris removal,
maintenance, berthing, and assembly. The schematic diagram of the SRL lab is
shown in figure 3.

Figure 3:  Space Robotics Lab Layout.

### Objective

The main objective of the SRL system is to test and verify the On-Orbit
Servicing (OOS) tasks like refueling, assembly, debris removal, and
maintenance by using the space robots. The experiment system typically
considers the cooperative target, like a communication satellite that needs
refueled or battery replacement or maintenance, and the non-cooperative
target, like the spent satellite at LEO. The primary operations are near-field
rendezvous, capture, fine manipulation, and docking to the target spacecraft
for OOS.

The main experiment which will be carried out in the robotic space lab is
listed below:

1\. Testing and Verification of trajectory plan and control algorithm in
service-target space robots.

2\. Calibration, Testing, and Validation of force torque sensor in servicer
space robot.

3\. Testing and Verification of guidance algorithm while executing the
proximity rendezvous between space robots

4\. Testing and Validation of visual servoing algorithm to track non-
cooperative targets.

5\. Near field rendezvous and robotic capturing of non-cooperative targets
autonomously.

6\. Maintenance and replacement operation of faulty spacecraft in orbit.

### Components of SRL.

The space robotics lab in TCS research consists of the following components
for the experiment:

#### Universal Robot Manipulator

The universal robot manipulators set up the robotics lab’s capture and fine
manipulation operation. We will be using UR5e and UR10 manipulators for our
experiment; the UR5e will act as the chaser space robot and will be
replicating the chaser motion, and the UR10 will act as the target space
robot, which will have the 3U mockup satellite model and will be replicating
target motion. The lab UR5e and UR10 manipulators are shown in Figures 4 and
5.

Figure 4: UR5 manipulator replicating chaser motion.

Figure 5: UR10 manipulator replicating target motion.

#### Linear Rail System

The 7th-axis linear rail system of 6 m in length is used to simulate the near-
end rendezvous and proximity operation. A Rack and Pinion mechanism uses the
servo-based control to move the URs robot in a linear direction. This linear
rail system can withstand a payload of 20-30 kg with a speed of 1m/s and $\pm$
0.1mm of positional accuracy.

Figure 6: Linear Rail System.

The schematic diagram of the linear rail system is shown in figure 6. The
linear system is made of the mild-steel attached to the floor using the
grouting method. In the linear rail system, the UR5e and UR10 supporting
plates are there, which will help to provide linear movement to the robot. The
relative velocity between the chaser and target object can be simulated using
the linear rail system on a ground-based platform.

####  Robotics Arm Gripper

The robotics gripper aims to grasp the target object at its grasping points.
An Onrobot RG2 gripper is used in chaser space robots which can withstand a
payload of 2kg with a stroke of 110mm. The figure of the Onrobot RG2 gripper
is shown in figure 7 on the left side. This gripper has inbuilt firmware,
which will be directly used by the URs manipulator end-effector and provides
intelligence, fast deployment, easy customization, and programming.

Figure 7: Robotic Grippers for UR5e and UR10.

Similarly, an Onrobot RG2 v2 gripper is used in the UR10 robot, where the
mock-up satellite will be hung. This gripper, through UR10 control, primarily
provides the tumbling motion to the 3U mock-up satellite body. The image of
the Onboard RG2 v2 gripper is shown on the right side of figure 7.

#### Force Torque Sensor

A Hex-E Onrobot 6-axis force/torque sensor is used for our UR5e manipulator.
It provides accurate force and torque measurement along all 6DOF and gives us
precise control for assembly, debris capturing, and refueling application.
This sensor includes force control, path recording, high accuracy, and special
features for insertion tasks.

The Hex-E’s primary function in our experiment is determining the impact force
or torque acting on the chaser space robot end-effector while grasping the
target object. It will help us to find the optimal impact force or torque
acting on the chaser robot while docking and helps us to model the contact
dynamics between the two space robot. This sensor can withstand the force of
220N and torques of 10Nm during the experimental phase. Figure 8 shows the
Hex-E force-torque sensor.

Figure 8: Hex-E Onrobot 6-axis force/torque sensor.

#### 3U Mock-up satellite

A 3U standard CubeSat platform is used to make the mock-up satellite model. An
acrylic sheet is used to make the 3U satellite model, and a solar panel is
attached to the topmost surface of the model. Some grasping points have been
made to the mock-up satellite model so that the gripper will grasp at given
grasping points. The 3U satellite model is shown in figure 9. The mock-up
model will be placed on the UR10 manipulator, which provides the tumbling
motion.

Figure 9: 3U mock-up satellite model.

#### Vision Based Camera

An Intel depth camera real sense D415 is used for the vision-based guidance
and pose estimation of the target body from the chaser manipulator motion.
Figure 10 shows the depth camera which will be used for our HiLS experiment.
This camera will be placed on the end-effector of the manipulator of the
chaser. Its maximum range is approximately 10m, providing 1280 x 720 active
stereo depth resolution with RGB image.

Figure 10: Real Sense Depth Camera D415.

Figure 11: Zed Camera.

An additional ZED camera is used for the motion sensing of the mock-up
satellite model, which gives us 6DOF IMU for accurate motion tracking. The Zed
camera used for our experiment is shown in figure 11, and it provides ultra-
depth sensing mode with high-speed data transfer and improved reliability. It
has a small form factor of 6.5 cm eye separation, and provides an accurate
6DOF of a tumbling mock-up model.

#### Control Facility Room

The space robotics lab consists of a PC-based real-time facility control room
where 2 PCs will be used to inspect the UR5e and UR10 robots and control their
motion in a linear rail platform. Tables and chairs will organize the control
facility room with LED light movement, which will help us detect the target
from different angles.

#### Miscellaneous accessories

The miscellaneous accessories consist of a black curtain, LED light,
preparation area, window arrangement in the control facility room, wiring of
230V ac power, and GPU setup for running the hybrid experiment approach. A
black curtain encloses the experiment areas to mimic the space environment. An
LED light setup is made at the experiment area, where the light will
illuminate the target mock-up satellite to replica the Sunlight. The
preparation area is mainly used for making plans and preparing the
experimental procedures before performing any specified experiment.

## RESULTS $\&$ DISCUSSION

The results section presents the complete pipeline of the workflow from the
moment of the rendezvous phase of the rotation floating space robot ends. Once
the target is in the workspace of the stowed manipulator arm, the Model
Predictive Controller extends the arm with the gripper towards the target in
the desired orientation. When this proximity operation ends, an on-off
controller closes the gripper, followed by a PD controller, which brings the
arm and the target to their initial orientation.

### Preliminary Hardware-in-the-loop Results

The hardware in the loop experiments is needed to validate the simulation
results obtained from the bullet pipeline. Following the hybrid method
philosophy described above, hardware experiments are independently conducted
to replicate the chaser arm motion on a UR5 arm and target motion on a UR 10
arm. As explained, these motions concern a frame attached to the base
satellite in the PyBullet environment. The feedback is obtained from the arms
and given to the PyBullet to get the MPC command and update the system’s
dynamics. Figure 12 shows the motion of the UR5 arm while replicating the
chaser motion leading to the plot 13 obtained. In figure 12 a., the initial
hardware motion of the UR5 manipulator is shown, which shows the deviation
from the Pybullet simulation command. As the chaser motion proceeds, the UR5
starts following the trajectory similar to the Pybullet simulation, shown in
figure 12b. and figure 12c. Although stage II and stage III show the
difference in x,y, and z position values with the simulation results, the
trajectory path remains the same.

a. Stage I

b. Stage II

c. Stage III

Figure 12 d. shows the final position of the UR5, similar to the Pybullet
simulation results. It was found that the UR5 is reaching the target point
similar to the Pybullet trajectory path with slight deviation caused by the
time latency, hardware friction, etc.

d. Stage IV Figure 12: UR5 motion while replicating chaser motion.

Figure 13 shows the motion of the UR5 arm replicating the chaser arm motion.
Solid lines are the actual trajectories of the UR5 arm, while dotted lines are
the trajectories computed by the MPC, which are given as commands to the UR5
robot. It can be seen that the chaser can follow the commanded trajectory with
good precision.

Figure 13: Commanded vs Followed Position of UR5 arm representing Chaser.

Figure 14 shows the motion of the UR10 arm while replicating the chaser motion
leading to the plot 15 obtained. In figure 14 a., the initial hardware motion
of the UR5 manipulator is shown, which shows the deviation from the Pybullet
simulation command. As the target motion proceeds, the UR10 follows the
trajectory similar to the Pybullet simulation, shown in figure 14 b. and
figure 14 c. Although stage II and stage III show the difference in x,y, and z
position values with the simulation results, the trajectory path remains the
same.

a. Stage I

b. Stage II

c. Stage III

Figure 14 d. shows the final position of the UR5, similar to the Pybullet
simulation results. It was found that the UR10 is reaching the target point
similar to the Pybullet trajectory path with slight deviation caused by the
time latency, hardware friction, etc.

d. Stage IV Figure 14: UR10 motion while replicating target motion.

Similar results are obtained for the target motion using the UR10 arm. Figure
15 shows the motion of the UR10 arm (solid lines) against the motion of the
target obtained from PyBullet. These motions concern an intermediate frame
attached to the base satellite in PyBullet.

Figure 15: Commanded vs Followed Position of UR10 arm representing Target.

We intend to complete the PyBullet pipeline on the hardware-in-the-loop setup
for the final results. For the same, we are developing a mock satellite model
to act as the target. Force and torque sensors will be applied on the walls of
the UR5 (chaser gripper), and the values will be recorded. These will help us
design an impedance controller for the contact dynamics.

## CONCLUSION $\&$ FUTURE WORK

The paper presents the development process of the HiLS facility setup for the
space robotics application. TCS Research, India’s first space robotics
laboratory, has developed the space robotics lab to execute the proximity
rendezvous and docking process. Numerous objective has been set up that will
be implemented in the space robotics lab. A hybrid simulation approach has
been utilized to emulate the micro-gravity environment on the ground with
minimum cost. This approach has been promising because free base motion will
be captured in a simulation environment like Pybullet, and the manipulators
will mimic the chaser’s and target’s motion in hardware. UR5e and UR10
manipulators are used to replicate the motion of the chaser and target object.
A 3U CubeSat mock-up model is made, which will be mounted on UR10 and provide
the tumbling motion to it. The MPC controller extends the arm with the gripper
as an end effector toward the target in the desired orientation. Once reaching
the proximity, an on-off controller closes the gripper, followed by a PD
controller, which brings the arm and the target to their initial orientation.
The paper also presents the initial results of replicating the chaser and
target motion by UR5 and UR10 manipulators. The results show that the motion
of the UR5 and UR10 arms against the motion of the chaser and target obtained
from the Pybullets are in the same trajectory line.

In future work, we intend to develop the mock-up model and mount it at UR10,
which will act as a target. A plan to add the force torque sensor will be
applied on the end-effector of UR5, i.e., the chaser manipulator, and the
values will be recorded while capturing the target objects. The multiple
scenarios will be run using an F/T sensor to find efficient values to design
the impedance controller for the contact dynamics. In the later phase of the
experiment, a confirmed case of the thrust characteristics [20] [21] [22] [23]
obtained from the STK simulation [24] [25] will be incorporated into the
chaser manipulator to execute the rendezvous trajectory close to the target
object by using the linear rail.

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# Geometric aspects of Young Integral:
decomposition of flows

Lourival Lima111E-mail<EMAIL_ADDRESS>Research supported by CAPES
88882.329064/2019-01. Paulo Ruffino222E-mail<EMAIL_ADDRESS>Research
partially supported by CNPq 305212/2019-2, FAPESP 2020/04426-6 and
2015/50122-0. Pedro Catuogno333E-mail<EMAIL_ADDRESS>Research partially
supported by CNPq 302.704/2008-6, 480.271/2009-7 and FAPESP 07/06896-5

(April 2022)

###### Abstract

In this paper we study geometric aspects of dynamics generated by Young
differential equations (YDE) driven by $\alpha$-Hölder trajectories with
$\alpha\in(1/2,1)$. We present a number of properties and geometrical
constructions on this low regularity context: Young Itô geometrical formula,
horizontal lift in principal fibre bundles, parallel transport, covariant
derivative, development and anti-development, among others. Our main
application here is a geometrical decomposition of flows generated by YDEs
according to diffeomorphisms generated by complementary distributions
(integrable or not). The proof of existence of this decomposition is based on
an Young Itô-Kunita formula for $\alpha$-Hölder paths proved by Castrequini
and Catuogno (Chaos Solitons Fractals, 2022).

##  1 Introduction

In this paper we study geometric aspects of dynamics generated by Young
differential equations (YDE) driven by $\alpha$-Hölder trajectories with
$\alpha\in(1/2,1]$. More precisely, given a smooth manifold $M$, we focus on
geometrical properties of equations of the type:

$dx_{t}=X(x_{t})\,dZ_{t},$ (1)

with initial condition $x_{0}\in M$ at $t=0$, where $x\rightarrow
X(x)\in\mathcal{L}(\mathbb{R}^{d},T_{x}M)$ is a smooth assignment of $d$
vector fields on $M$ and $Z\in C^{\alpha}([0,T],\mathbb{R}^{d})$ is an
$\alpha$-Hölder continuous trajectory in $\mathbb{R}^{d}$. We say that a path
$x:[0,T]\rightarrow M$ is a solution of equation (1) if for all test function
$f\in C^{\infty}(M;\mathbb{R})$ we have that

$\displaystyle f(x_{t})=f(x_{0})+\int_{0}^{t}Xf(x_{s})\,dZ_{s},$ (2)

where $Xf$ is a short term for $\sum Df(x)X(x)e_{i}$, with $e_{i}$’s the
elements of the canonical basis of $\mathbb{R}^{d}$. The last term of equation
(2) is an integral in the Young sense, see e.g. the classical [30], or more
recent Hairer and Friz [13], Gubinelli et al. [14], Lyons [20], Castrequini
and Russo [7], Castrequini and Catuogno [8], Cong [10], Ruzmaikina [28], among
many others.

In this scenario of low regularity of trajectories, the geometric Young Itô
Formula, Theorem 2.3 opens possibility to many basic geometric constructions
on this dynamics. These topics are explored in the next section, where we
prove the existence of horizontal lifts on principal fiber bundles with an
affine connection. In particular, considering a Riemannian manifold and its
orthonormal bundle, parallel transport and covariant derivatives can be
established along $\alpha$-Hölder trajectories. Development and anti-
development can also be constructed.

Motivated by the fact that in many kinds of dynamical systems, in order to
obtain local or asymptotic parameters of the dynamics, one performs a
befitting decomposition of the associated flow, our main example of
application of this low regularity techniques in manifolds concerns a
decomposition of the associated Young flow. In fact, many examples of this
kind of decomposition appear in the literature related to distinct geometrical
or analytical contexts. We mention few of them: given a system in a semi-
simple Lie group, we get a lot of information if we decompose the system into
each component of the Iwasawa decomposition (see, e.g. in the stochastic
context Malliavin and Malliavin [23]); given a stochastic flow in a Riemannian
manifold, one can write this flow (up to some conditions) as a Markovian
process in the group of isometries of the manifold composed with a process in
the Lie group of diffeomorphisms which fix the initial condition and has
derivatives at this point given by an upper triangular matrix, see Ming Liao
[21], [22]. Also, given a flow in an $m$-dimensional manifold with a pair of
complementary foliation (i.e. locally the manifold and foliations are
diffeomorphic to $\mathbb{R}^{k}\times\mathbb{R}^{m-k}$) then, locally in time
and space, a stochastic flow can be written as a composition of
diffeomorphisms which preserve each of these foliations, see [24], [25]. We
are going to make this last example more precise and explore its potential in
the Young integral context.

The decomposition of Young flows is allowed thanks to an Itô-Ventzel-Kunita
type formula in this low regularity context, Theorem 3.4 due to Castrequini
and Catuogno [8]. The framework where we apply this formula is a pair of
geometrical distributions (involutive, i.e. which generate a foliation, or
not). The main result in Section 3 establishes the local decomposition of an
Young flow of diffeomorphisms as one component given by a diffeomorphism
generated by vector fields in one distribution and another component given by
diffeomorphism generated by the other distribution. Precise definition are
given in Section 3.

The article is organized as follows: in the next section we recall basic
properties of the Young integral and prove the relevant geometric results we
use latter on. In Section 3 we prove the decomposition of flows given
complementary distributions. In Section 4 we present examples. Initially
linear systems are treated with a pair of foliations given by affine parallel
hyperplanes. We present conditions to the existence of global decomposition
for any time in this context. The last example provides explicit calculations
in the case of fiber bundles over homogeneous space $G\rightarrow M=G/H$ where
$G$ is a Lie group and $H<G$ is a closed subgroup.

##  2 Geometric set up

###  2.1 Young differential equation on manifolds

We recall that for a general metric space $(M,d)$, a curve
$\sigma:[0,T]\rightarrow M$ is $\alpha$-Hölder continuous, with $\alpha>0$ if
there exists a constant $C>0$, such that

$\displaystyle d(\sigma(t),\sigma(s))\leq C|t-s|^{\alpha},$ (3)

for all $s,t\in[0,T]$. This concept extends naturally to a Riemaniann
manifold, since it carries the well known induced metric $d(x,y)$ given by

$d(x,y)=\inf\Big{\\{}\int_{0}^{1}||\gamma^{\prime}(t)||dt;\
\gamma:[0,1]\rightarrow M\mbox{ differentiable such that }\gamma(0)=x,\
\gamma(1)=y\Big{\\}}.$

See e.g. [11] among many other classical books. Hence, naturally,
$\alpha$-Hölder paths are also well defined in Riemannian manifolds. Most of
the classical analytic results on this regularity theory also holds for
$\alpha$-Hölder paths in a Riemannian manifold. For instance, composition of a
differentiable function with an $\alpha$-Hölder trajectory is also an
$\alpha$-Hölder path. Particularly, in a geometrical context, for readers
convenience we prove the following

###### Proposition 2.1

Let $M$ and $N$ be Riemannian manifolds, $\dim N\geq 1$. A path
$\sigma:[0,T]\rightarrow M$ is $\alpha$-Hölder continuous on $M$ if and only
if, for all differentiable map $f:M\rightarrow N$, the path $f(\sigma(t))$ is
$\alpha$-Hölder continuous on $N$.

Proof: There are many interesting ways to prove this result. Here, we use an
embedding argument. Initially consider that $N$ is an Euclidean space
$\mathbb{R}^{n}$ and take $\sigma(t)$ an $\alpha$-Hölder trajectory on $M$.
There exists an isometric embedding $i:M\rightarrow\mathbb{R}^{d}$ for a
sufficiently large integer $d$ (Nash theorem). For sake of notation we write
$\sigma_{t}:=\sigma(t)$.

Since $\|i(x)-i(y)\|_{\mathbb{R}^{d}}\leq d_{M}(x,y)$ for all $x,y\in M$, we
have the following inequalities:

$\|i(\sigma_{t})-i(\sigma_{s})\|_{\mathbb{R}^{d}}\ \leq\
d_{M}(\sigma_{t},\sigma_{s})\ \leq\ C|t-s|^{\alpha},$

which implies that $i(\sigma_{t})$ is $\alpha$-Hölder in $\mathbb{R}^{d}$.
Now, for any differentiable function $f:M\rightarrow\mathbb{R}^{n}$, use the
fact that it can be extended to a differentiable function
$\bar{f}:U\rightarrow\mathbb{R}^{n}$ defined in a tubular neighbourhood $U$ of
$i\circ\sigma([0,T])$ in $\mathbb{R}^{N}$. Hence,
$f(\sigma_{t})=\bar{f}(i(\sigma_{t}))$. Since Hölder regularity is preserved
by differentiable functions on Euclidean spaces, $f(\sigma_{t})$ is
$\alpha$-Hölder continuous in $\mathbb{R}^{n}$. Mind that, in fact, in the
compact set $i(\sigma_{t})$ the metrics $d_{M}$, and $\ell_{2}$ in
$\mathbb{R}^{d}$ are uniformly equivalents, see Lemma 2.2 [18]. Hence,
$\ell_{2}$-norm Hölder regularity in $\mathbb{R}^{d}$ is equivalent to Hölder
regularity on $(M,d_{M})$.

For a general Riemannian manifold $N$ and a differentiable map $f:M\rightarrow
N$, consider another isometric embedding
$i^{\prime}:N\rightarrow\mathbb{R}^{d^{\prime}}$ for an integer $d^{\prime}$
sufficiently large. Then, the last paragraph shows that $i^{\prime}\circ
f(\sigma)$ is $\alpha$-Hölder in $\mathbb{R}^{d^{\prime}}$. From Lemma 2.2
[18] we have that there exists a positive constant $C_{1}$ such that

$d_{N}(f(\sigma_{t}),f(\sigma_{s}))\leq
C_{1}\|i(f(\sigma_{t}))-i(f(\sigma_{t}))\|_{\mathbb{R}^{d^{\prime}}}\leq
C_{2}|t-s|^{\alpha},$

for a positive constant $C_{2}$, which shows that $f(\sigma_{t})$ is
$\alpha$-Hölder continuous in $N$.

Conversely, suppose that $f(\sigma_{t})\in N$ is $\alpha$-Hölder for all
differentiable function $f:M\rightarrow N$. Denote the projections of
$i(\sigma_{t})\in\mathbb{R}^{d}$ by $\sigma_{t}^{j}:=p_{j}\circ i(\sigma_{t})$
for each $1\leq j\leq d$. Let $\varphi:V\rightarrow W\subset N$ be a local
parametrization for $N$, with $V$ an open set in an Euclidean space. There
exist another local parametrization obtained from the previous one, just
enlarging the domain by homothety, if necessary, which we call again by
$\varphi:\tilde{V}\rightarrow W\subset N$ such that the set
$\\{(x,0,\ldots,0);x=\sigma^{j}(t)\mbox{ for some
}t\in[0,T]\\}\subset\tilde{V}$ for all $1\leq j\leq d$. Consider the
differentiable functions $f_{j}:M\rightarrow N$ given by
$f_{j}(x):=\varphi(p_{j}(i(x)),0,\ldots,0))$. Then
$f_{j}(\sigma_{t}):=\varphi(\sigma^{j}_{t},0,\ldots,0))$ is $\alpha$-Hölder by
hypothesis. By metric equivalence in compact sets in the domain of the local
parametrization, we have that $\sigma^{j}_{t}$ is $\alpha$-Hölder for all
$1\leq j\leq d$. We conclude that $\sigma(t)\in M$ is $\alpha$-Hölder
continuous on $M$.

$\Box$

Before we show conditions for existence and uniqueness of solutions for
equation (1), we state the main geometric theorem that is a version of Itô’s
formula for $\alpha$-Hölder continuous paths. We start with the definition of
the Young integral of a real 1-form:

######  Definition 2.2 (Integration of real 1-forms)

_Let $N$ be an $n$-dimensional differentiable manifold with $\bigwedge^{1}(N)$
the space of real $1$-forms. Consider $\beta\in\bigwedge^{1}(N)$ and a chart
$(U,(y_{1},\ldots,y_{n}))$ in $N$ such that_

$\displaystyle\beta=\sum_{i=1}^{n}\beta_{i}\,dy^{i}.$ (4)

_The integral of $\beta$ along an $\alpha$-Hölder path $x:[0,T]\rightarrow N$
is defined by_

$\displaystyle\int_{0}^{T}\
\beta(x_{t})\,dx_{t}=\sum_{i=1}^{n}\int_{0}^{T}\beta_{i}dx^{i}_{t},$ (5)

_where the above integrals are Riemann-Stieltjes integral of $\beta_{i}$ with
respect to the $i$-th coordinate of the path $x_{t}$. Among others properties,
this integration is independent of the local chart, see e.g, Abraham, Marsden
and Ratiu [1] and Ikeda and Manabe [15]_.

The integration of real 1-forms above allows one to integrate many tensor
fields in a manifold. In particular, if $F:M\rightarrow\mathbb{R}^{d}$ is a
smooth function, the integration

$\int_{0}^{t}DF(x_{s})\ dx_{s}$

makes sense, looking at each coordinate of $\mathbb{R}^{d}$. Next Theorem is
the basic property of the the $\alpha$-Hölder calculus we are treating in this
paper.

######  Theorem 2.3 (Young Itô Formula)

Let $M$ and $N$ be Riemannian manifolds. Consider
$x\in\mathcal{C}^{\alpha}([0,T],M)$ and a smooth function $F:M\rightarrow N$.
Then

$dF(x_{t})=DF(x_{s})\ dx_{s}.$ (6)

Remark: We highlight that formula (6) above means that if $\beta$ is a 1-form
in $N$ then

$\int_{0}^{t}\beta\ dF(x_{s})=\int_{0}^{t}(dF(x_{s}))^{*}\beta\ dx_{s}.$ (7)

In particular, if $N$ is an Euclidean space:

$F(x_{t})=F(x_{0})+\int_{0}^{t}DF(x_{s})dx_{s}.$ (8)

Proof: Initially we prove the result for an Euclidean space
$N=\mathbb{R}^{d}$. We use again the embedding argument from Nash’s theorem:
there exists a sufficiently large $p\in\mathbb{N}$ such that $M$ can be
isometrically embedded into $\mathbb{R}^{m+p}$. Abusing notation, we have
$x\in\mathcal{C}^{\alpha}([0,T],\mathbb{R}^{m+p})$, $F$ is defined in a
tubular neighbourhood of the image of $M$ and $DF(x)\in
L(\mathbb{R}^{m+p},\mathbb{R}^{d})$. By Taylor’s formula in Euclidean space,

$F(x_{t})-F(x_{s})=DF(x_{s})\cdot(x_{t}-x_{s})+R(x_{s},x_{t}),$

with

$R(x_{s},x_{t})=\int_{0}^{1}(1-u)\,\mathrm{Hess}\,(F)(x_{s}+u(x_{t}-x_{s}))(x_{t}-x_{s},x_{t}-x_{s})\
du.$

Since $F$ is smooth,

$\|R(x_{s},x_{t})\|\leq C\|x_{t}-x_{s}\|^{2}\leq C^{\prime}|t-s|^{2\alpha}.$

Let $\pi=\\{s_{i}\\}$ be a partition of $[0,T]$. Then

$F(x_{t})-F(x_{0})=\sum_{i}F(x_{s_{i+1}})-F(x_{s_{i}})=\sum_{i}DF(x_{s_{i}})\cdot(x_{s_{i+1}}-x_{s_{i}})+\sum_{i}R(x_{s_{i}},x_{s_{i+1}}).$
(9)

We have that

$\sum_{i}\|R(x_{s_{i}},x_{s_{i+1}})\|\leq
C^{\prime}\sum_{i}|s_{i+1}-s_{i}|^{2\alpha}\leq
C^{\prime}T\sup_{i}|s_{i+1}-s_{i}|^{2\alpha-1}.$

Thus, since $\alpha>1/2$ we have that

$\lim_{|\pi|\rightarrow 0}\sum_{i}\|R(x_{s_{i}},x_{s_{i+1}})\|=0.$

Take the limit $|\pi|\rightarrow 0$ in equation (9) and the definition of
Stieltjes (Young) integral to finish the proof in this context.

In general, when $N$ is a Riemannian manifold, consider a local chart $\phi$.
The previous calculations hold with $\phi\circ F$ whose integration is
independent of the coordinate system.

$\Box$

Note that multidimensional forms of Itô formula above, integration by parts
etc can be obtained from formula (6) considering the manifold $M$ above as
appropriate product spaces. We proceed to prove a theorem of existence and
uniqueness of solution for equation (1).

###### Theorem 2.4

Given an initial condition $x_{0}\in M$, there exists a unique maximal
solution of the Young differential equation _( 1)_ such that $x(0)=x_{0}$.
Moreover, there exists a flow of (local) diffeomorphisms associated to the
solutions.

Proof: A simple way to proof the result for local solutions is based on the
existence and uniqueness results in the Euclidean space. In fact, given the
initial condition $x_{0}$, let $(U,\Psi)$ be a chart on $M$ with $x_{0}\in U$.
Let $\tilde{X}:=D\Psi(X(\Psi^{-1}(p)))$ be the induced vector field in the
image of $\Psi$. The Young differential equation
$dy_{t}=\tilde{X}(y_{t})dZ_{t}$ has a unique solution local solution $y_{t}$
with $y_{0}=\Psi(x_{0})$. See e.g. Lejay [19], Caruana, Lyons and Thierry [5],
Li and Lyons [20], Friz and Hairer [13] and references therein. Take
$x_{t}=\Psi^{-1}(y_{t})\subset U$. We claim that $x_{t}$ is a solution of
equation (1). In fact, consider a test function $f\in C^{\infty}(M)$. By
Theorem 2.3, it follows that

$\displaystyle f(\Psi^{-1}(y_{s}))$ $\displaystyle=$ $\displaystyle
f(\Psi^{-1}(y_{0}))+\int_{0}^{t}D(f\circ\Psi^{-1})(y_{s})dy_{s}$
$\displaystyle=$ $\displaystyle
f(\Psi^{-1}(y_{0}))+\int_{0}^{t}Df\Psi_{*}^{-1}\Psi_{*}X(y_{s})dZ_{s}$
$\displaystyle=$ $\displaystyle f(x_{0})+\int_{0}^{t}Xf(x_{s})dZ_{s}.$

Moreover, the solution $x_{t}$ does not depend on the choice of local
coordinate. In fact, let $(V,\Phi)$ be another chart on $M$, with $x_{t}\in
U\cap V$ and let $z_{t}$ be the solution of the Young differential equation
$\displaystyle{dz_{t}=\Phi_{*}X(z_{t})dZ_{t}}$. Then

$\displaystyle dy_{t}$ $\displaystyle=$ $\displaystyle\Psi_{*}X(y_{t})dZ_{t}$
$\displaystyle=$ $\displaystyle\Psi_{*}\Phi_{*}^{-1}\Phi_{*}X(z_{t})dZ_{t}$
$\displaystyle=$ $\displaystyle\Psi_{*}\Phi_{*}^{-1}dz_{t}.$

Hence $\displaystyle{y_{t}=\Psi\Phi^{-1}(z_{t})}$ and therefore
$\displaystyle{\Phi^{-1}(z_{t})=\Psi^{-1}(y_{t})=x_{t}}$. A maximal solution
is obtained in the classical way by extending a local solution up to its
explosion time. The existence of local flow of (local) diffeomorphisms is also
concluded from the Euclidean case using the same local chart argument.

$\Box$

###  2.2 Horizontal lifts

Let $\displaystyle{\\{P,M,G,\pi\\}}$ be a principal fibre bundle with base
$M$, structure group $G$ and total space $P$. In this case $M$ is a smooth,
connected and paracompact manifold. The projection $\pi$ is taken as
$\pi:P\rightarrow M$. The group $G$ acts freely on $P$ on the right by the
action $\displaystyle{R_{g}:P\rightarrow P}$ defined by
$\displaystyle{R_{g}(u)=ug}$, for $u\in P$ and $g\in G$. Let $\mathfrak{g}$ be
the Lie algebra of $G$, then an element $A\in\mathfrak{g}$ generates the
exponential $\displaystyle{\\{\exp tA,\ t\in\mathbb{R}\\}}$, which induces a
vector field on $P$ by

$A^{*}u=\frac{d}{dt}R_{\exp(tA)}u\ \big{|}_{t=0},$

If $\Gamma^{\infty}(TP)$ is the section of all smooth vector fields on $P$,
then the map $A\rightarrow A^{*}$, from $\mathfrak{g}$ into
$\Gamma^{\infty}(TP)$ is a Lie algebra homomorphism. For more details, see
e.g. Shigekawa [29], the classical Kobayashi and Nomizu [17] among many
others. The tangent space $TP$ has a naturally defined subspace called the
vertical tangent bundle $VTP$ given by $VT_{u}P:=\ker d\pi_{u}$ for all $u\in
P$. Note that $A^{*}u\in VT_{u}P$ for all $A\in\mathfrak{g}$.

A connection in the principal fibre bundle is an assignment of a horizontal
subspace $HT_{u}P$ of $T_{u}P$ which is the kernel of a $\mathfrak{g}$-valued
$1$-form $\omega$ in $P$ with the following properties:

  * (i)

(well-behaved vertically) $\omega dR_{g}=\mathrm{Ad}(g^{-1})\omega$, for all
$g\in G$. Here the linear map
$\displaystyle{\mathrm{Ad}(g^{-1}):\mathfrak{g}\rightarrow\mathfrak{g}}$ is
the derivative at the identity of the adjoint
$\mathrm{Ad}(g^{-1}):G\rightarrow G$ defined by
$\displaystyle{\mathrm{Ad}(g^{-1})a=g^{-1}ag}$.

  * (ii)

(vertical calibration) $\omega(A^{*})=A$, where $A^{*}$ is a vector field on
$VTP$.

Such $1$-form $\omega$ is called a connection form in the principal fibre
bundle $\\{P,M,G,\pi\\}$. Moreover, $\omega$ defines the horizontal tangent
bundle $HTP$ given by $HT_{u}P=\ker\omega_{u}$. Hence, for all $u\in P$, the
tangent space $T_{u}P$ splits into $HT_{u}P\oplus VT_{u}P$ and
$dR_{g}(HT_{u}P)=HT_{ug}P$.

Now we have the geometric set up to define the horizontal lift of
$\alpha$-Hölder continuous paths.

###### Definition 2.5

Let $x:[0,T]\rightarrow M$ be an $\alpha$-Hölder continuous path. Consider
$u\in P$, with $\pi(u)=x_{0}$. The horizontal lift of $x_{t}$ starting at $u$
is a path $\widetilde{x}:[0,T]\longrightarrow P$ such that:

  * (i)

$\widetilde{x}_{0}=u$.

  * (ii)

$\pi(\widetilde{x}_{t})=x_{t}$ for all $t\in[0,T]$.

  * (iii)

$\displaystyle{\int_{0}^{t}\omega\ d\widetilde{x}_{s}=0}$ for all $t\in[0,T]$.

Next result shows the existence and uniqueness of the horizontal lift for an
$\alpha$-Hölder continous paths in a manifold. In the proof we apply the same
technique used in Kobayashi and Nomizu [17] and in Shigekawa [29] where the
existence and uniqueness of horizontal lift were proved in the context of
$C^{1}$ paths and semimartingales respectively.

###### Theorem 2.6

Given an $\alpha$-Hölder continuous path $x:[0,T]\rightarrow M$ and an element
$u$ in the fibre $\pi^{-1}(x_{0})$, there exists (up to a explosion time) a
unique horizontal lift $\widetilde{x}:[0,T]\rightarrow P$ with
$\tilde{x}_{0}=u$.

Proof:

Consider a local trivialization $\phi:\pi^{-1}(U)\rightarrow U\times G$ with
$x_{0}\in U$ and take the $\alpha$-Hölder path $\nu_{t}=\phi^{-1}(x_{t},e)$.
If the horizontal lift of $x_{t}$ exists at all, it has to be of the form
$\widetilde{x}_{t}=\nu_{t}a_{t}$, where $a_{t}\in G$ is an appropriate path
which makes $\widetilde{x}_{t}$ horizontal and $\nu_{0}\,a_{0}=u$.

Let $\Psi:P\times G\rightarrow P$ be the right free action of $G$ on $P$.
Then, by Theorem 2.3 we have that

$d\tilde{x}_{t}=\partial_{1}\Psi(\nu_{t},a_{t})\
d\nu_{t}+\partial_{2}\Psi(\nu_{t},a_{t})\ da_{t}.$

Hence:

$\displaystyle\int_{0}^{t}\omega\ d\,\widetilde{x}_{t}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}(\partial_{1}\Psi(\nu_{t},a_{t}))^{*}\omega\
d\nu_{t}+\int_{0}^{t}(\partial_{2}\Psi(\nu_{t},a_{t}))^{*}\omega\ da_{t}.$
(10) $\displaystyle=$ $\displaystyle\int_{0}^{t}R^{*}_{a_{t}}\omega\
d\nu_{t}+\int_{0}^{t}\theta\ da_{t},$

by the vertical calibration of the connection $\omega$, where $\theta$ is the
canonical Cartan 1-form given by $\theta_{g}(dR_{g}A)=A$ for all $g\in G$ and
$A\in\mathfrak{g}$. The lift $\tilde{x}_{t}$ is horizontal if and only if
equation (10) vanishes for all $t\in[0,T]$, i.e. if and only if
$\mathrm{Ad}(a_{t}^{-1})\omega\ d\nu_{t}=-\theta\ da_{t}$. Let
$F_{1},\ldots,F_{n}$ be a basis of the right invariant Lie algebra
$\mathfrak{g}$. For all $t\in[0,T]$, there exist $\alpha$-Hólder continuous
real functions $\alpha^{1}_{t},\ldots,\alpha^{n}_{t}$, such that:

$\displaystyle\int_{0}^{t}\omega\ d\nu_{s}=\sum_{i=1}^{n}F_{i}\alpha^{i}_{t}.$
(11)

Using this notation we have that a necessary and sufficient condition such
that equation (10) vanishes is that

$\displaystyle\int_{0}^{t}\omega\ d\,\widetilde{x}_{t}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\int_{0}^{t}\mathrm{Ad}(a^{-1}_{t})F_{i}\
d\alpha^{i}+\int_{0}^{t}dR_{a_{t}^{-1}}\ da_{t}=0,$

for all $t\in[0,T]$, i.e., trajectory $a_{t}$ has to satisfy

$da_{t}=-\sum_{i=1}^{n}dR_{a_{t}}\ \mathrm{Ad}(a_{t}^{-1})F_{i}\
d\alpha^{i}_{t},$

with initial condition $a_{0}$. There exists a unique solution by Theorem 2.4,
hence there exists a unique horizontal lift $\tilde{x}_{t}$ up to a explosion.
Mind that at the border of the local trivialization, one can extend further
the solution applying again the same construction above. The maximal solution
covers the whole interval $[0,T]$ (by compactness) if there is no explosion in
the fibre.

Note that for initial element in the fibre $a_{0}g$, the horizontal lift is
given by $a_{t}g$.

$\Box$

Besides the dynamics and the principal fiber bundle approach presented so far
(which are basic to the next Sections), this low regularity Itô-Young calculus
of Theorem 2.3 allows one to develop further geometrical properties. We
mention the following three classical geometric aspects:

A. Parallel Transport and covariant derivative: Given a smooth manifold $M$,
consider the frame bundle $BM\rightarrow M$ of basis
$u:\mathbb{R}^{n}\rightarrow T_{p}M$, with $p\in M$, with the structure group
$G=Gl(n,\mathbb{R})$. Last Theorem applied in this context establishes a
parallel transport along $\alpha$-Hölder path $x_{t}\in M$. In fact, given a
horizontal lift $u_{t}$, the parallel transport of a vector $v\in T_{x_{0}}M$
is obtained by

$\Big{/}\Big{/}_{t}\,v=u_{t}\circ u_{0}^{-1}(v)\in T_{x(t)}M.$

It does not depend on the choice of the horizontal lift. Moreover, if we take
the orthonormal frame bundle $OM\rightarrow M$ of basis orthonormal basis
given by linear isometries $u:\mathbb{R}^{n}\rightarrow T_{x}M$, with $x\in
M$, with the structure group $G=O(n,\mathbb{R})$, the parallel transport is
also an isometry.

Covariant derivative can now be defined along an $\alpha$-Hölder path
$x_{t}\in M$. Given a differentiable vector field $Y$, we have that its
covariant derivative along $x(t)$ is given by:

$DY(x_{t})=\Big{/}\Big{/}_{t}\ \ d\ \Big{/}\Big{/}_{t}^{-1}\ \ Y(x_{t}).$

where the differentials are interpreted in the sense of Young (Definition
2.2).

B. Development and anti-development: Let $M$ be an $m$-dimensional Riemannian
manifold, and consider an $\alpha$-Hölder continuous path
$x:[0,T]\rightarrow\mathbb{R}^{m}$. Take the horizontal operator
$H:OM\times\mathbb{R}^{m}\rightarrow HTOM$ where $H(u,v)$ is the horizontal
lift of $u(v)\in T_{\pi(u)}M$ up to $HT_{u}OM$. The development of $x_{t}$ on
$M$ with initial orthonormal frame $u_{0}$ is obtained from $u_{t}$, the
solution of the YDE:

$d\,u_{t}=H(u_{t},dx_{t}),$

i.e. $\pi(u_{t})$ is the development of $x(t)$ on $M$ (rolling without
slipping, with initial “contact plane” given by $u_{0}$). On the other hand,
the anti-development of an $\alpha$-Hölder continuous path $x:[0,T]\rightarrow
M$ is described using its horizontal lift $\tilde{x}_{t}$ (Theorem 2.6) with
initial condition $\tilde{x}_{0}$:

$y_{t}=\int_{0}^{t}\tilde{x}_{s}^{-1}\ d\,x_{s}.$

Note that, as expected, $y_{t}$ depends on the choice of $\tilde{x}_{0}$.
Compare this approach with the classical Brownian motion approach by Eells and
Elworthy [12], and the isotropic Lévy processes approach in Applebaum and
Estrade [2], among many others.

C. Continuous $\alpha$-Hölder paths in $M$ are solutions of Young differential
equations: As established before, solutions of Young equations driven by
$\alpha$-Hölder paths on a manifold are also $\alpha$-Hölder continuous paths.
Reciprocally, every $\alpha$-Hölder continuous paths on $M$ is a solution of a
Young differential equation (YDE) driven by an $\alpha$-Hölder functions. In
fact, take an embedding $i:M\rightarrow\mathbb{R}^{m+p}$ of $M$ into a
sufficiently large dimensional Euclidean space. Let $U$ be a tubular
neighbourhood with $\pi:U\rightarrow i(M)$ a projection of $U$ into $i(M)$.
Given an $\alpha$-Hölder path $y_{t}$ on $M$, let $z_{t}=i(y_{t})$. Then
$z_{t}$ is an $\alpha$-Hölder trajectory in $\mathbb{R}^{m+p}$. Consider the
YDE in $i(M)$:

$dx_{t}=D\pi(x_{t})\,dz_{t}.$

Then $z_{t}$ is the solution of this YDE with initial condition $x_{0}=z_{0}$:
just check that the YDE is the differential version of the identity
$z_{t}=\pi(z_{t})$, according to Young Itô formula of Theorem 2.3. If the
projection $\pi$ is orthogonal, as in Elworthy [12] then the vector fields are
gradients of the embedding. In general, the dynamics of other trajectories
starting at $x_{0}\neq y_{0}$ depends on the embedding and on the projection.
This is an interesting topic to be studied further.

##  3 Decomposition of flow generated by Young differential equation

Let $\mbox{Diff}(M)$ be the infinite dimensional Lie group of smooth
diffeomorphisms of a compact connected manifold $M$. The Lie algebra
associated to $\mbox{Diff}(M)$ is the infinite dimensional space of smooth
vector fields on $M$, see e.g. Neeb [26], Omori [27], among others. The
exponential map $\exp\\{tY\\}\in\mbox{Diff}(M)$ is the associated flow of
diffeomorphisms generated by the smooth vector field $Y$. In this context,
given an element $\varphi\in\mbox{Diff}(M)$ the derivative of the right
translation is given by $R_{\varphi*}Y=Y(\varphi)$ for any smooth vector $Y$.
The derivative of left translation $L_{\varphi*}Y=D\varphi(Y)$, and
$\mathrm{Ad}(\varphi)Y=\varphi_{*}(Y(\varphi^{-1}))$.

In this Lie group notation, a solution flow $\varphi_{t}$ of an YDE is written
as the solution of a right invariant Young differential equation in the Lie
group of diffeomorphisms $\mbox{Diff}(M)$:

$d\varphi_{t}=R_{\varphi_{t*}}X\,dZ_{t}.$ (12)

Here we abuse notation in the sense that (using the same notation as in
equation 1) one can write

$X\,dZ_{t}=\sum_{i=1}^{d}\ X_{i}\,dZ_{t}^{i},$

where $X_{j}=X(e_{j})$ with $e_{j}$ the elements of the canonical basis.
Hence, equation (12) have to be interpret as

$\displaystyle d\varphi_{t}=\sum_{i=1}^{d}R_{\varphi_{t*}}X_{i}\,dZ_{t}^{i}.$

Interesting problems arise when one decomposes a (flow of) diffeomorphism
$\varphi\in\mbox{Diff}(M)$, into composition of convenient prescribed
components. This kind of decomposition appears in the literature, for example,
in Bismut [4], Kunita [16] and many others. In particular, it is also relevant
when each component of the decomposition belongs to prescribed subgroups of
$\mbox{Diff}(M)$, see e.g Melo et al [25], Catuogno et al [8], Iwasawa and
non-linear Iwasawa decomposition [9], Ming Liao [21] among many others.

In this section, we explore the Young calculus to proof the existence of a
geometrical decomposition of flows generated by $\alpha$-Hölder systems
$\varphi_{t}$ given by equation (1). Suppose that locally $M$ is endowed with
a pair of regular differentiable distributions: i.e., every point $x\in M$ has
a neighbourhood $U$ and differentiable mappings $\Delta^{1}:U\rightarrow
Gr_{k}(M)$ and $\Delta^{2}:U\rightarrow Gr_{{m-k}}(M)$ respectively, where

$Gr_{p}(M)=\bigcup_{x\in M}Gr_{p}(T_{x}M)$

is the Grasmannian bundle of $p$-dimensional subspaces over $M$, with $1\leq
p\leq m$. We assume that $\Delta^{1}$ and $\Delta^{2}$ are complementary in
the sense that $\displaystyle{\Delta^{1}(x)\oplus\Delta^{2}(x)=T_{x}M}$, for
all $x\in U$. With this notation we define the subgroup of $\mbox{Diff}(M)$
which is generated by a certain distribution $\Delta$ by:

$\mbox{Diff}(\Delta,M)=\mbox{cl}\bigl{\\{}\exp(t_{1}X_{1})\ldots\exp(t_{n}X_{n}),\mbox{
with }X_{i}\in\Delta,t_{i}\in\mathbb{R},\forall n\in\mathbb{N}\bigr{\\}}.$

Note that if a distribution $\Delta$ is involutive, then each element of the
group $\mbox{Diff}(\Delta,M)$ preserves the leaves of the corresponding
foliation.

In particular, in this Section we focus on the subgroups
$\mbox{Diff}(\Delta^{1},M)$ and $\mbox{Diff}(\Delta^{2},M)$. The main result
of this paper (Theorem 3.4) establishes a local decomposition of the solution
flow $\varphi_{t}$ into two components: a curve (solution of an autonomous
YDE) in $\mbox{Diff}(\Delta^{1},M)$ composed with a non-autonomous path in
$\mbox{Diff}(\Delta^{2},M)$.

###### Definition 3.1

We say that an element $\eta\in\mbox{Diff}(M)$ preserves transversality of
$\Delta^{1}$ and $\Delta^{2}$ in a neighbourhood $U\subset M$ if
$\eta_{*}\Delta^{2}\left(\eta^{-1}(p)\right)\cap\Delta^{1}(p)=\\{0\\}$, for
all $p\in U$.

By continuity, for any pair of complementary distributions, there always
exists a neighbourhood of the identity $1d\in\mbox{Diff}(M)$ where all
elements in this neighbourhood preserve transversality. Moreover, if the
distribution $\Delta^{1}$ is involutive then all elements in
$\mbox{Diff}(\Delta^{1},M)$ preserves tranversality of $\Delta^{1}$ and
$\Delta^{2}$: in fact, the derivative $\eta_{*}$ above is a linear isomorphism
which sends tangent spaces of the associated foliation to tangent spaces in
the same leaf. In the sequence, we state an extended scope of the Itô-Kunita
formula (see [16] ) in the geometrical Young calculus.

######  Theorem 3.2 (Young Itô-Kunita formula)

Let $X$, $Y$ $\in C^{2}(M,\mathcal{L}(\mathbb{R}^{d},TM))$ and $Z\in
C^{\alpha}([0,T],\mathbb{R}^{d}])$ and suppose that $\eta_{t}$ and $\psi_{t}$
are solutions maps associated to the Young differential equations
$d\eta_{t}=X(\eta_{t})dZ_{t}$ and $d\psi_{t}=Y(\psi_{t})dZ_{t}$ respectively.
Then, $\varphi_{t}=\eta_{t}\circ\psi_{t}$ is the solution map associated with
the Young differential equation

$d\varphi_{t}=X(\varphi_{t})dZ_{t}+\mathrm{Ad}(\eta_{t})Y(\varphi_{t})dZ_{t}.$
(13)

For a proof in this low regularity context, see Castrequini and Catuogno [6,
Thm. 4.1]. Next Corollary shows that the inverse of the solution flow of an
YDE is also $\alpha$-Hölder continuous.

###### Corollary 3.3

If $\eta_{t}$ is the solution flow the Young differential equation on $M$

$\displaystyle dx_{t}=X(x_{t})\,dZ_{t},$ (14)

then, the inverse map $\eta^{-1}_{t}$ is the solution of the Young
differential equation on $M$

$\displaystyle dz_{t}=-D\eta^{-1}_{t}(z_{t})X(\eta_{t}(z_{t}))dZ_{t}.$ (15)

Proof: In fact, just apply expressions (14) and (15) into equation (13).

$\Box$

For a constructive proof of last Corollary see [6, Thm. 4.2]. Next Theorem
states the main result of this section:

######  Theorem 3.4 (Decomposition of flows of YDE)

Up to a life time $\tau\in[0,T]$, the solution flow $\varphi_{t}$ can be
locally decomposed as

$\varphi_{t}=\eta_{t}\circ\psi_{t},$

where $\eta_{t}$ is solution of an (autonomous) Young differential equation in
$\emph{Diff}(\Delta^{1},M)$ and $\psi_{t}$ is a path in
$\emph{Diff}(\Delta^{2},M)$.

Proof:

Given $p\in M$, take $\eta\in\mbox{Diff}(\Delta^{1},M)$ sufficiently close to
the identity such that it preserves tranversality, i.e.
$\mathrm{Ad}(\eta_{t})\Delta^{2}$ and $\Delta^{1}$ are complementary. The
tangent vector(s) $X(p)$ can be decomposed uniquely as

$\displaystyle X(p)=h(p)+V(\eta_{t},p),$ (16)

where $h(p)\in\Delta^{1}(p)$ and
$V(\eta_{t},p)\in\mathrm{Ad}(\eta_{t})\Delta^{2}(p)$, for all $p\in M$. We
take the first component $\eta_{t}$ as the solution map of the following Young
differential equation in $\mathrm{Diff}(\Delta^{1},M)$:

$\displaystyle d\eta_{t}=R_{\eta_{t*}}\,h\,dZ_{t},$ (17)

with initial condition $\eta_{0}=1d$, the identity. Even though the equation
above is described in terms of a right translation, it is not a right
invariant equation since $h$ in general depends on $\eta_{t}$. We obtain the
second component of decomposition of $\varphi_{t}$ using that
$\psi_{t}=\eta^{-1}_{t}\circ\varphi_{t}$. Applying Corollary 3.3, it follows
that:

$\displaystyle d\eta^{-1}_{t}=-L_{\eta^{-1}_{t*}}\,h\,dZ_{t},$

where $L_{\eta^{-1}_{t*}}$ is the derivative of the left translation at the
identity by $\eta^{-1}_{t}$. Finally, we find a equation for $\psi_{t}$ by
applying Theorem 3.2:

$\displaystyle d\psi_{t}$ $\displaystyle=$
$\displaystyle(\eta^{-1}_{t}h\,\eta_{t}\,\psi_{t}-\eta^{-1}_{t}X\,\eta_{t}\,\psi_{t})\,dZ_{t}$
(18) $\displaystyle=$ $\displaystyle\mathrm{Ad}(\eta^{-1}_{t})V(\eta_{t})\
dZ_{t}.$

Note that $V(\eta,p)$ does not necessarily belong to $\Delta^{2}$. Still,
$d\psi_{t}\in\Delta^{2}$ since
$d\psi_{t}\in\mathrm{Ad}(\eta^{-1})\mathrm{Ad}(\eta)\Delta^{2}=\Delta^{2}$.
Then $\psi_{t}$ is the $\Delta^{2}$-component of $\varphi_{t}$.

$\Box$

###### Corollary 3.5

If the distributions $\Delta^{1}$ and $\Delta^{2}$ are integrable, then the
decomposition of Theorem 3.4 is unique.

Proof: In fact, in this case
$\mathrm{Diff}(\Delta^{1},M)\cap\mathrm{Diff}(\Delta^{2},M)=\\{1d\\}$.

$\Box$

##  4 Examples of decomposition of flows of YDE

In this section we consider the same geometric structure used in section 2,
i.e, a principal fibre bundle $\\{P,M,G,\pi\\}$, with base $M$, structure
group $G$ and total space $P$. Our main goal is to apply the decomposition
which was proposed in theorem 3.4 in general fibre bundles. Important notions
such as $1$-forms connection on fibre bundles, horizontal and vertical tangent
bundles and others were discussed briefly on section $2$, more details can be
found for example in Kobayashi and Nomizu [17].

###  4.1 Linear systems

Consider an Euclidean space $\mathbb{R}^{n}$, with a pair of complementary
foliations given by the trivial Cartesian product
$\mathbb{R}^{k}\times\mathbb{R}^{\ell}$, with $k+\ell=n$. More precisely, the
horizontal foliation $\mathcal{F}_{H}$ is given by parallel leaves generated
by affine translations $x+(\mathbb{R}^{k}\times\\{0\\}),$ with
$x\in\mathbb{R}^{n}$. Analogously, the vertical foliation $\mathcal{F}_{V}$ is
given by parallel vertical leaves $x+(\\{0\\}\times\mathbb{R}^{\ell})$, for
all $x\in\mathbb{R}^{n}$. We consider the linear Young differential equation:

$dx_{t}=A\,x_{t}\ dZ_{t},$ (19)

with $x_{0}\in\mathbb{R}^{n}$ and $Z_{t}$ an $\alpha$-Hölder continuous
trajectory in the real line. The Young calculus presented in the previous
section shows that the fundamental linear solution flow of (19) is the
exponential

$F_{t}=\exp{\\{A(Z_{t}-Z_{0})\\}}.$ (20)

$A=\left(\begin{array}[]{ll}\Big{(}A_{1}\Big{)}_{k\times
k}&\Big{(}A_{2}\Big{)}\\\
\Big{(}A_{3}\Big{)}&\Big{(}A_{4}\Big{)}_{\ell\times\ell}\end{array}\right)$

The decomposition we are interested here is

$F_{t}=\eta_{t}\circ\psi_{t}$

such that $\eta_{t}\in Dif^{H}$ and $\psi_{t}\in Dif^{V}$. In general
$\eta_{t}$ and $\psi_{t}$ does not have to be linear, even in quite symmetric
situations. For example, if the pair of foliations in
$\mathbb{R}^{n}\setminus\\{0\\}$ are given by radial and spherical
coordinates, the components of the decomposition are not necessarily linear:
in fact, the linear radial diffeomorphisms is reduced to a one dimensional
group of uniform contractions and expansions $\lambda 1d$, with $\lambda>0$,
which, obviously, is not big enough to perform the decomposition. For the
Cartesian pair of foliation $\mathbb{R}^{k}\times\mathbb{R}^{\ell}$ considered
in this section, we do have that $\eta_{t}$ and $\psi_{t}$ are linear. In
fact, in coordinates, write

$F_{t}=\left(\begin{array}[]{ll}\Big{(}F_{1}(t)\Big{)}_{k\times
k}&\Big{(}F_{2}(t)\Big{)}_{k\times\ell}\\\ &\\\
\Big{(}F_{3}(t)\Big{)}_{\ell\times
k}&\Big{(}F_{4}(t)\Big{)}_{\ell\times\ell}\end{array}\right).$

Since $\eta_{t}$ does not change the last $\ell$ coordinates the
diffeomorphisms $\psi$ must satisfies

$\psi_{t}=\left(\begin{array}[]{ll}\Big{(}1d\Big{)}_{k\times k}&0\\\ &\\\
F_{3}(t)&F_{4}(t)\end{array}\right).$

Hence diffeomorphisms $\psi_{t}$ and $\eta_{t}$, when exist, are global and
linear.

A simple example: A system which illustrates not only these formulae, but also
the lifetime of the decomposition is the pure rotation in $R^{2}$ given by

$dx_{t}=\left(\begin{array}[]{cc}0&-1\\\ 1&0\end{array}\right)x_{t}\,dZ_{t},$

whose decomposition of flow can be easily calculated as:

$\displaystyle\left(\begin{array}[]{cc}\cos Z_{t}&-\sin Z_{t}\\\ \sin
Z_{t}&\cos Z_{t}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\sec Z_{t}&-\tan Z_{t}\\\
0&1\end{array}\right)\left(\begin{array}[]{cc}1&0\\\ \sin Z_{t}&\cos
Z_{t}\end{array}\right).$ (27)

Note that if $Z_{t}\in\\{\frac{\pi}{2}+k\pi,k\in\mathbb{Z}\\}$, then the
decomposition (27) no longer exists at the corresponding time t, i.e. we have
explosion of the solutions of equations (17) or (18).

$\Box$

Back to the general linear case, the components of the decomposition in fact
lie in the Lie group:

$\psi_{t}\in G_{V}=\left\\{g\in
Gl(n,\mathbb{R});g=\left(\begin{array}[]{ll}\Big{(}1d\Big{)}_{k\times k}&0\\\
g_{3}&\Big{(}g_{4}\Big{)}_{\ell\times\ell}\end{array}\right)\right\\}$

whose Lie algebra is given by the vector space generated by

$\left(\begin{array}[]{cc}\Big{(}0\Big{)}_{k\times k}&0\\\ &\\\
\Big{(}*\Big{)}&\Big{(}*\Big{)}_{\ell\times\ell}\end{array}\right),$

where $(*)$ means nonzero matrices of the appropriate dimension. Analogously
for the horizontal component:

$\eta_{t}\in G_{H}=\left\\{g\in
Gl(n,\mathbb{R});g=\left(\begin{array}[]{ll}\Big{(}g_{1}\Big{)}_{k\times
k}&g_{2}\\\ 0&\Big{(}1d\Big{)}_{\ell\times\ell}\end{array}\right)\right\\}$

whose Lie algebra is given by the vector space generated by

$\left(\begin{array}[]{cc}\Big{(}*\Big{)}_{k\times
k}&\Big{(}*\Big{)}_{k\times\ell}\\\ &\\\
0&\Big{(}0\Big{)}_{\ell\times\ell}\end{array}\right),$

Using the properties of the Young integral, we find the differential equations
for the constituents submatrices $g_{1},g_{2}$ and $g_{3},g_{4}$ of $\eta_{t}$
and $\psi_{t}$ respectively. Let
$\pi_{2}:R^{k}\times\mathbb{R}^{\ell}\rightarrow\mathbb{R}^{\ell}$ be the
projection on the second subspace. From formula (16) we have that

$V(\eta,\cdot)=\eta\circ\pi_{2}\circ A(\cdot).$

In fact, it is enough to check that $V(\eta,\cdot)$ is in the image of the
vertical component by $\eta$ and that $\pi_{2}V(\eta,\cdot)=\pi_{2}A(\cdot)$.
From this formula, equations (17) and (18) we find the autonomous equation:

$d\eta_{t}=(1d-\eta_{t}\circ\pi_{2})A\,\eta_{t}\ dZ_{t},$

and the well expected nonautonomous vertical diffeomorphisms:

$d\psi_{t}=\pi_{2}\,A\,\eta_{t}\circ\psi_{t}\ dZ_{t}.$

Rewriting each constituent submatrices we find:

$\displaystyle\begin{array}[]{ll}dg_{1}(t)=\big{[}A_{1}\ g_{1}(t)-g_{2}(t)\
A_{3}\
g_{1}(t)\big{]}\,dZ_{t},&dg_{2}(t)=\big{[}A_{1}g_{2}(t)+A_{2}-g_{2}(t)\,A_{4}-g_{2}\,A_{3}\,g_{2}(t)\big{]}\,dZ_{t},\\\
&\\\
dg_{3}(t)=\big{[}A_{3}\,g_{1}+A_{3}\,g_{2}\,g_{3}+A_{4}g_{3}\big{]}\,dZ_{t},&dg_{4}(t)=\big{[}A_{3}\,g_{2}\,g_{4}+A_{4}\,g_{4}\big{]}\,dZ_{t}.\end{array}$
(31)

Explosion in the solutions of the equations of $g_{1}$ and $g_{2}$ can appear
if $A_{3}$ is not zero (see example of equation (27), where $A_{3}=[1]$).
Otherwise, if $A_{3}=0$ then there exists the decomposition for all time
$t\geq 0$. Using this feature, and the Jordan canonical form we can extend the
scope of the decomposition in the next Proposition. Before that let us fix a
notation. Given two complementary subspaces $E_{1}\oplus
E_{2}=\mathbb{R}^{n}$, let us denote by $\mathcal{F}(E_{1})$ and
$\mathcal{F}(E_{2})$ the corresponding pair of complementary parallel
foliations in $\mathbb{R}^{n}$.

###### Proposition 4.1

Consider a Young linear system in $\mathbb{R}^{n}$

$dx_{t}=A\,x_{t}\ dZ_{t}.$ (32)

If dimension $n>2$, then there exist a pair of parallel foliations
$\mathcal{F}(E_{1})$, $\mathcal{F}(E_{2})$ generated by complementary
subspaces $E_{1}$ and $E_{2}$ such that the decomposition of the flow of
equation (32) exists for all time $t\in[0,T]$, i.e. there is no explosion time
of the decomposition. Dimension of $E_{1}$ can be chosen as a number of the
form $(a+2b)$ where $a=0,1,\ldots,r=\\#\\{\mbox{real eigenvalues with
multiplicities}\\}$, and $b=0,1,\ldots,(n-r)/2$.

Proof: Let $A=P\,JP^{-1}$ be the canonical real Jordan form of $A$, with the
choice of bases $P$ such that the nilpotent component has, if necessary, $1$’s
and identities $I_{2}$’s above the diagonal. The change of coordinates
$y=P\,x$ establishes the conjugate Young system:

$dy_{t}=J\,y_{t}\ dZ_{t}.$

If $n>2$, it is possible to write

$J=\left(\begin{array}[]{ll}\Big{(}J_{1}\Big{)}_{k\times
k}&\Big{(}J_{2}\Big{)}\\\
\Big{(}J_{3}\Big{)}&\Big{(}J_{4}\Big{)}_{\ell\times\ell}\end{array}\right)$

with $k=a+2b$ and its complementary $\ell=n-k$, such that the submatrix
$(J_{3})_{\ell\times k}=0$. The number $a$ represents the number of real
eingenvalues in the block $J_{3}$ and $b$ represents the number of pairs of
conjugate nonreal eigenvalues in this block. Hence, equations (31) guarantee
the there is no explosion in the decomposition of $y_{t}$. By conjugacy, there
is also no explosion in the decomposition of the linear fundamental solution
$F_{t}$ of (32) along the foliations generated by
$E_{1}=P\,(\mathbb{R}^{k}\times\\{0\\})$ and
$E_{2}=P\,(\\{0\\}\times\mathbb{R}^{l})$. This proves the Proposition.

$\Box$

Using the notation in the proof of last Proposition, the decomposition of
$F_{t}=\eta_{t}\circ\psi_{t}$ above are such that $\eta_{t}$ lies in the group
$P\,G_{H}\,P^{-1}$ and $\psi_{t}$ lies in $P\,G_{V}\,P^{-1}$.

###  4.2 Principal fibre bundles over homogeneous spaces

Let $G$ be a connected Lie group with a closed subgroup $H$ and denote by
$\mathfrak{g}$ and $\mathfrak{h}$ their Lie algebras of right invariant vector
fields, respectively. The group $G$ acts on $H$ by left translation $gH$, for
all $g\in G$ and the orbits generate the homogeneous space $M:=G/H$, see e.g.
[17]. We have a principal fibre bundle given by the canonical projection
$\pi:G\rightarrow M$. Given an element $A\in\mathfrak{g}$ consider the right
invariant YDE:

$\displaystyle d\,g_{t}=Ag_{t}\ dZ_{t}.$ (33)

As it has done in Section 2.2, here, we consider a connection $\omega$ in the
principal fibre bundle $\pi:G\rightarrow M$. In this example we construct our
decomposition of flow according to the vertical subspaces (involutive) and the
horizontal subspace established by this connection. The solution flow (global
in $G$ up to lifetime of $Z_{t}$) is given by left action:

$\varphi_{t}(x)=g_{t}x,$

where $g_{t}=\exp\\{A\,Z_{t}\\}$. In this example the distributions
$\Delta^{1}$ and $\Delta^{2}$ in the tangent space $TG$ are given by the
horizontal subspaces with respect to the connection $\omega$ and the tangent
to the fibres $gH$ (involutive). In order to decompose the flow $\varphi_{t}$
as in Theorem 3.4, one has to identify the vector fields $V$ and $h$ as in
equation (16) in the proof of the theorem, i.e.:

$Ax:=h+V(\eta,x)$

Elements $\eta\in\mathrm{Diff}(\Delta^{1},G)$ can be written pointwise (with
respect to $x\in G$) as a left action of elements of $G$ at $x$. This action
preserves the vertical component, i.e. $g_{*}\Delta^{2}=\Delta^{2}$ for all
$g\in G$. Hence, vector field $V$ above is independent of $\eta$ and one can
easily calculate:

$V(x)=\omega(Ax)^{*}\ \ \ \ \mbox{ and }\ \ \ \ h=Ag-\omega(Ax)^{*}.$

By equations (17) and (18) we have that each component of the decomposition
$\varphi_{t}(\cdot)=\eta_{t}\circ\psi_{t}(\cdot)$ are given by:

$d\,\eta_{t}=R_{\eta_{t}*}(A\eta_{t}(\cdot)-\omega(A\eta_{t}(\cdot))^{*})$
(34)

and

$d\,\psi_{t}=\mathrm{Ad}(\eta_{t})\ \omega(A\eta_{t}(\cdot))^{*}.$ (35)

Let denote by $g^{H,x}_{t}\in G$ the $\alpha$-Hölder curve in $G$ such that
$g^{H,x}_{t}x$ is the horizontal lift of $\pi(g_{t}\,x)$ starting at $x$, i.e.
$g^{H,x}_{t}x$ is horizontal and $g^{H,x}_{t}x=g_{t}\,x\,v_{t}$ for some
$v_{t}\in H$. With this notation, fixing the action at a point $x\in G$, the
equations above reduce to well known finite dimensional equations (in G). This
is the content of the following

###### Proposition 4.2

Consider the decomposition $\varphi_{t}(\cdot)=\eta_{t}\circ\psi_{t}(\cdot)$
of the solution flow of equation (33) according to horizontal and vertical
distribution of the fibre bundle in the sense of Theorem 3.4. Then, at each
point $x\in G$, the first component can be written as the left action:

$\eta_{t}(x)=g^{H,x}_{t}x,$

and the second component can be written as the right action:

$\psi_{t}(x)=x\,h_{t}$

where $h_{t}=x^{-1}\,(g^{H,x}_{t})^{-1}\,g_{t}\,x.$

Proof:

The proof of the first equation follows straightforward when one applies
equations (34) at a fixed inicial condition $x\in G$: it is the horizontal
lift of $\pi(g_{t}x)$, cf. Theorem 2.6, using Itô formula 2.3. Regarding the
second equation of the statement, one sees that
$(x^{-1}\,(g^{H,x}_{t})^{-1}\,g_{t}\,x)\in H$ by definition of the horizontal
lift: $g^{H,x}_{t}x=g_{t}\,x\,v_{t}$ for some $v_{t}\in H$. One checks that it
solves (35) at a fixed point $x$.

$\Box$

Trivial fibre bundles: As a particular case, consider a trivial principal
fibre bundle $\pi:G\times H\rightarrow H$ with strutural group $H$, where $G$
and $H$ are connected Lie groups. The trivial connection is given by
$\omega_{(x,y)}(g_{t}^{\prime},h_{t}^{\prime})=y^{-1}h_{t}^{\prime}\in\mathfrak{h}$.
Consider a right invariant YDE in $G\times H$:

$d(x_{t},y_{t})=(A\times B)\ (x_{t},y_{t})\,dZ_{t}$

where $A\in\mathfrak{g}$ and $B\in\mathfrak{h}$, the Lie algebras of $G$ and
$H$ respectively, with an initial condition $(x_{0},y_{0})$. Since the
connection in this case is invariant by left action of $G\times\\{1d\\}$, the
factor $g^{H,x}_{t}\in G\times H$ of Proposition 4.2 does not depend on
$(x,y)$. One recovers the trivial components of the decomposition. In fact we
get a global decomposition where the first component is given by the left
action:

$\eta_{t}(\cdot,\cdot)=(\exp(AZ_{t}),1d)(\cdot,\cdot).$

And the second (vertical) component is given in terms of the right action:

$\psi_{t}(\cdot,\cdot)=(\cdot,\cdot)(1d,h_{t})$

where $h_{t}=y^{-1}\,\exp(BZ_{t})\,y,$ according to Proposition 4.2.

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