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Space Based Automated Module. Methods of Trajectory and Docking Control.

Authors: M. Pivovarov, A. Zakcharov, G. Veselova, E. Djujeva, L.
Blinova, I. Sidorov, V. Frolov.
Space Research Institute of Russian Academy of Science


Preface

Principal schemes of flight control methods elaborated by specialists from
the Space Research Institute are described in a given address. As an object
of control an automatic Space based module, capable to execute a set of
dynamic operations in the Orbital Station environs is overviewed.
The main technological tasks the module should execute are as following:
1. Flying up to the object, which is moving near the orbital station,
docking with it, towage and assembling an object at a given place at the
orbital station;
2. Carrying out different objects from the station and providing their
trajectory of flight and angular orientation as need be;
3. Re-docking station modules or fragments of construction from one place
to another;
4. Flying around the station and hanging in a given point in order to carry
out the visual inspection of the station's external surface;
5. Performing the maintenance at the external surface of the station.

It is shown that the module's control system and corresponding motions
control algorithms are able to ensure the exact realization of mentioned
above tasks.


Automated Module Main Sub-systems


The free-flying universal automated module (AM) has its own control system
capable to ensure the spatial controlled motion of AM center of masses and
spatial angular turns about center of masses. The system can operates in
two working modes - completely autonomous mode, when the system realize a
preliminary determined set of operations using the corresponding algorithms
of autonomous control; and second one is an operating in a dialog mode by
means of commands received from operator at the Station.
To realize the mentioned above technological activity the AM control system
should consist of the following main blocks (the control system doesn't
include the manipulator itself):

1. Autonomous navigation system (ANS). ANS consists of three
accelerometers; each measures the current AM acceleration along the
corresponding translational coordinate constrains with AM. We assume an
accuracy (error) of acceleration measurements (g ~ 2.10-5 m/s2. ANS also
include three angular velocity detectors for measuring the current
angular velocity about each axis constrains with AM. We assume an
accuracy (error) of angular velocity measurements
([pic] = 20 arc.min/hour. ANS has a specialized on-board computer for
calculating on the base of special program the current values of absolute
translational coordinates, velocities, angular coordinates and
velocities. This information incomes into the AM main on-board computer.
2. Position-sensitive optical system working in complex with marked beacons
(point-like sources) fixed near the setting place of an object to dock
with. Towards each beacon the detector measures two spatial angles (, (
in two planes between the optical axis and line directed towards the
beacon. The geometrical scheme of angle measurements will be represented
below. To provide the satisfied accuracy of angle measures to use the
detector with light-sensitive matrix, having 512 x 512 elements, is
enough. From our point of view to use the focus transform device to
change the field of view angle is desirable. Market beacons should be
located not only near the setting place but also in some predefined
points at the Station and other objects the AM operates with. The
detector (as one of the version) may be supplied with mini-driver for the
optical system angular turns.
3. Low-thrust pulsing jet engines. The engine is a rather small electric-
magnetic valve, capable to realize short pulses. In calculations we
accepted that the minimal duration of pulse the engine is capable to
produce is 0.01 s. The working body of engine is a two components fuel.
To realize the required AM spatial control along translational and
rotational coordinates we examined few versions of engines number and
their relative location at the AM. The possible number of engines may be
12, 16, 20, and 24. But the detail solution of this task and searching
for the optimal scheme should be done during the platform elaboration at
the next stage of the work.
4. On-board computer processes the original data received from ANS and
optical system, executes the corresponding algorithms of control and
sends commands to actuate the particular pulse engines.
5. Radio system for linking AM with operator at the Station and exchanging
the information important for the control. We suppose that radio will be
used when the AM operating in dialog mode.
6. We also include here the automatic system for fixing the platform at the
setting place. The device, we propose as a docking unit, consists of two
magnetic plates having the cross-polarized magnetic field. Depending on
plates relative angular position the force of attraction may be varied
from zero to the nominal value. Here with the magnetic field acts
practically at a distance from the plate surface of about 2 mm. To ensure
the AM fixing the docking place of an object should be supplied with the
small iron plate. The nominal attraction force is 6 kg/sm2 and the plate
with squire of about 50 sm2 installed at the docking place is enough to
provide the attracting force about 300 kg. By the rotating one of the
magnetic plate at an angle ~ 700 an attraction will be reduced up to
zero and the AM may be separated with an object.

From our point of view the AM operating capabilities is convenient to
represent as a set of operating phases the AM should realize in an Orbital
Station environ. These phases are combined into a single technological
cycle, which is represented at fig 1-1




Initial conditions are: at a distance ~ 5000m from the Station flies an
object ( let it be a kind of technological satellite). It is supplies
with marked beacons on the external surface. The AM task is to deliver this
object to the docking place at the Station.
The cycle of operations the AM should perform consists of the following
phases.

Phase 1. The unloaded AM is separated with the Station by means of short
pulse and is installed at the nearby orbit (it may be the circular orbit)
with relative altitude ~ 50m. Using the preliminary information on an
object relative location and relative velocity AM orientates towards it.
Observing the market beacons by means of optical detector AM specifies its
position relative to the object.

Phase 2. By means of selected beforehand control method AM flies to the
object environ.

Phase 3. AM executes the approach and docking with a given object. Let us
call the new system (AM + load) as the loaded AM.

Phase 4. Loaded AM executes the required angular turns and flies back to
the Station environ.

Phase 5. The loaded AM is set at the appropriate trajectory of flight
around the Station to ensure the satisfied dynamical conditions at the
initial stage of approach process. Then on the loaded AM executes approach
and docking at a given setting place at the Station. With this phase the
cycle of operations is finished.

It is easy to see that all others mentioned above operations (except AM
operating in "inspector" mode) may be represented as combinations of tasks
performed within this cycle.

The exact and reliable accomplishment of mentioned above maneuvers is based
on application of corresponding algorithms of AM motion control. The main
contents of these methods are described below.



Long Range Flight Control


1 Position/Orientation Determination


To determine the AM relative spatial coordinates and velocity vector we
propose to use one position-sensitive detector and one marked beacon
located at an object to dock with.

This version will be applied when the distance to the beacon is rather
large (~10000m.) and the detector cannot resolve reliably two beacons
because of the limited angular resolution. In this case the scheme of
consecutive measurements in a given time intervals (ti was elaborated and
rather good results were received.

Overview the calculation schemes, considering that the platform orbit
coincides with the orbit of an object to dock with (orbital station or some
kinds of satellites). Let us call this object - a target object. The
duration of flight is around 5000 s. and typical relative velocity is less
than 10m/s.Under these conditions the position-sensitive detector
continuously observes the marked beacon placed at target object and
measures value of an angle ((t) between the optical axis and line towards
the beacon (optical axis and beacon lie in the orbit plane). Here with we
assume that during the autonomous flight a rather high errors is
accumulated when calculating the platform relative position and velocity.

Calculations are done in coordinates system constrained with the detector.
(Z0 - along optical axis, X0 and Y0 are in the plane of detector's light-
sensitive matrix). The detector location relative to the platform's center
of masses is known a'prior.
To simplify calculations we consider that the platform is orientated
towards an object so that the beacon lies in coordinates plane (Z0,X0) of
the detector, conformably the plane (Z0Y0) is perpendicular to the orbit's
plane. This version of the platform relative location is the most simple
and convenient to calculation. The corresponding scheme is depicted at
fig.2.1-1

The procedure of position and velocity determination in this version is as
following.
At a given initial moment t1 (the platform is far from an object) the
onboard computer has the preliminary information on values dLE and drE
received for example from ANS, where:
dLE - an estimated distance to the object along the detector's optical
axis (Z0);
drE - an estimated distance along X0-axis

Let us consider that at the initial moment t1 the accumulated relative
errors have the following values:
- along Z0-axis (Z and [pic];
- along X0-axis (X and [pic]
The aim of calculations is to estimate errors (Z, [pic], (X and [pic] and
thus to get the accurate data on platform current spatial position and
velocity vector.
At a given time t1 the true values dL and dr are as following:

[pic] (2.1.1)

In a time interval (t1 = t2-t1: ((t1 ~ 100 - 150 s.) at a moment t2 the
second set of
measurements is take place.

[pic]
[pic] (2.1.2)

where : (L1 - an estimated distance along Z0-axis within the interval
(t1 ;
(R1 - an estimated distance along X0-axis within the same
time interval.

Values (L1, (R1 are received from the direct integration of exact motion
equations. The interval between measurements is rather short and we assume
that within this time interval (t1 no errors are accumulated.

In time intervals (t2 =2(t1 and (t3 =3(t1 we have the same relations for
dL and dr at points t3 and t4 respectively:

[pic] (2.1.3)

[pic] (2.1.4)

Detecting the angle (n (n = 1,2,3,4) at the corresponding points t1, t2,
t3, t4 we receive in accordance with the geometrical relations the
following system of linear equations:

[pic]
(2.1.5)

Using relations (2.1.1 - 2.1.4) we may find the solution of the system
(2.1.5) and to determine values (Z, [pic], (X and [pic]. Substituting these
values in equation (2.1.4) we may find the first approximations of relative
co-ordinates and velocities dL, dL', dr, dr' for the current time t4. One
should note that in the scheme we assume that no errors are accumulated
within the time interval (t2 and (t3 as well.
According to accepted scheme of detector spatial position, its Y0 -axis is
perpendicular to the orbit plane. Detecting the corresponding angles ((tn)
in the plane (Z0Y0) we may use relations similar to 2.1.1 - 2.1.5 to
determine errors (Y and [pic] along this direction. We should make at least
two sets of measurements, for instance at point t1 and t2. In this case the
relations for the true values Y(tn) will be as following:

[pic] (2.1.6)
[pic] (2.1.7)

Values YE and (Y1 are the estimates of translational co-ordinates along Y0-
axis received from integration of motion equations as well.
Two additional equations to calculate (Y and [pic] are as following:

[pic]
(2.1.8)

These equations should be added to the system (2.1.5). The final system
consisting of (2.1.5) and (2.1.8) allows to determine the full amount of
errors and to estimate precisely the current platform position.
We may continue the described above sequence of calculations for another
time intervals until the platform is in the environs of an object to dock
with.
The calculations done according this scheme give the following preliminary
results:
When the distance to the docking place is ~ 500 m. we may decrease the
current time intervals (tn up to 20 - 40s. and therefore to upgrade the
accuracy of received results. At the initial step of approximation (for
time interval t1 - t4) the accuracy of relative translational co-ordinates
(L,R) determination is around 5m, the relative velocity accuracy is 3 sm/s.
At the terminal point (the relative distance ( 300 - 500m) the final
accuracy of co-ordinates determination ~ 20 sm., relative velocity ( 0.5
sm/s.

In calculations we assume the acceleration measuring accuracy ~ 2.10-5
m/s2.
On the base of given calculations the preliminary technical requirements to
the detector are formulated:
- system must be capable to record the angular position of beacon within
the distance range 1 and 10000 m.;
- field of view angles range 50 - 150 ;
- the accuracy of beacon angular position measurement 1' - 3 ';
- the rate of data interrogation from the sensor ~ 10 to 20 Hz;
- it is desirable to use the focus transform device to change the field
of view angle.

2 Dynamic Equations Analysis and AM Transition to the Environs of an Object
to Dock with

Within the limits of this paragraph we shall overview the following
procedure of the platform flight control.
The initial conditions are as following:
It is considered that an orbit planes of platform and an object of docking
are coincided. The initial relative distance ( 10000m., the typical
relative velocity ( 10 m/s. The based trajectory of an object flight is a
circular orbit with an altitude 450 km. and period 5610s. (it is
approximately corresponds to the orbit of an International Space Station).
The aim of this phase of flight control is to ensure the platform flight
from initial point having arbitrary relative co-ordinates and velocity to
the terminal point (null point) located near the docking place of target
object.
The optimal method of control at this phase is a three-pulses maneuver
executing during the time interval equal to the period Tp of target
object's revolution round the Earth (in our case Tp=5610s.). Here with the
third pulse as usually is needed to decelerate completely the satellite at
the moment the docking is taking place. The distinction of proposed below
method of control is that to construct the process ensuring in the terminal
point a smooth and without hanging passes to the next phase of flight in
the docking place environ. That is why during the flight only first two
pulses are executed and the final decelerated pulse is partly prolonged to
the next flight phase. Another difference is constrained with selecting the
certain co-ordinate system and dynamic parameters allowing to get rather
simple analytical relations of control pulses. This in turn allows to
simplify the selection of terminal dynamical conditions (this conditions
will be the initial parameters for the second phase) and to pass accurately
to the next phase of control.
Overview the initial system of platform motion equations

[pic] (2.2.1)


Where: v - orbital velocity, r - altitude of flight, ( - pitch angle, L -
distance along the Earth surface, z - deviation towards the perpendicular
to the orbit's plane, Px - engine's power projection at the velocity
vector, Py - engine's power projection at the Earth radius-vector, Pz
engine's power projection at the perpendicular to the orbit's plane, m -
platform's mass, R - Earth radius, g0 - earth gravity at the surface.

[pic] g0 = 9.81m/s2

The system of equations in decrements is received when substituting the
variables in system 2.2.1

[pic]

Where: variables v0, r0, L0, (=0 describe the motion of target object
along the circular orbit.
The corresponding system of equations has the following form:

[pic]
(2.2.2)

Where: c1 =2(/v0 , c2 = (2/v0, (=2(/Tp
The determinant of the system is as following:

D(() = - (2( (2 + c1g1 -v0c2)


The determinant nonzero roots are:
( = ( ( , ( =1.12*10-3 1/s

The value Tp = 2 (/( = 5610s.
Overview the first four equations of system (2.2.2 ) describing the
platform motion in the orbit's plane. The last equation describes the
motion in the perpendicular direction. It doesn't connected with other
equations and will be overviewed later.
It is convenient to make one more substitution of variables in system
(2.2.2 ) using the following relations.

x = ( c1dv +c2dr )/( , u =dv + (dr
(2.2.3)

The new system of dynamical equations is:

[pic]
(2.2.4)

The reverse transformation of variables gives the following relations:
dr = -( x/c2 +c1/c2 u , dv= -u +v0 x


Analyzing system (2.2.4) one should note that the first equation is
separated from others and in case Px = Py =0 the value u=const along the
trajectory of relative motion. Second and third equations describe the
rotating mode of the platform relative motion with period Tp and the last
equation is an integral of motion along dL co-ordinate. These equations for
example allow to make the following simple conclusions. Providing u =0 and
x = const, all over the period Tp we should have the relative motion along
a stable in time elliptic orbit with parameters depended on value x. Here
with the value u (0 determines the ellipse shifting in time along relative
co-ordinates (dL,dr). Besides, this system shows that the power pulse Px is
a two times effectively than Py . Therefore from the point of view of fuel
consumption it is better to apply the control pulses when d( =0 and to
repeat them each half a period (if it is needed).
In general from our point of view the proposed equations (2.2.4) are very
convenient for analyzing the current dynamic process and their application
significantly simplify the control task solution.
Overview in consequent orders the solution of mentioned above control task.
In co-ordinate system constrains with the target object at the initial
moment t0 relative co-ordinates dL0 ,dr0 as well as initial velocity
vector dv=dL' + (dr are calculated (using 2.1.5 and 2.2.2). The task is to
transit the platform during the period Tp from the initial location to the
terminal point, constrains for example with the target object ( dL=0,
dr=0, dv=0). In variables dL, u , x it is equivalent to transition the
platform to a point with co-ordinates dL=0, u=0,x=0.
To determine the velocity pulses values correcting the current trajectory
of flight we shall overview three versions of initial conditions.

1. At the initial moment t0 (let us consider d((t0)=0) values dL(0, u=0,
x=0. It means that the platform and the target object are at the same orbit
at a distance dL and the relative velocity dv=0. To shift the platform at a
distance -dL along the orbit we may get the corrected pulses directly from
the last equation of system 2.2.4

Jm11(t0) =dL/3Tp , Jm13(t0 + Tp) = -dL/3Tp
(2.2.5)

Under the construction of pulses variables (u,x) would have the following
values:
At the initial moment u(t0)= Jm11(t0) , x(t0)= Jm11(t0) c1/(;
At the terminal point u(t0 + Tp)=0, x(t0 + Tp)=0, dL=0. (the required
conditions are achieved).

2. At the initial moment dL=0, u(0, x=0
On the base of equations (2.1.3.4) we shall derive the following values of
velocity pulses:

[pic] (2.2.6)

These sequences of pulses were received with the help of simple
conclusions. After the first pulse variables would have the following
values: u1 = u/4 (remaining of u), x1 = -c13u/4(.

After half a period Tp/2 the velocity u1 and the corresponding distance dL1
should be compensated during the remained time Tp/2. By the second pulse (-
u/2) the remained values will be: u2 = -u/4. Variable x1 after half a
period will change the sign (x1 = c13u/4() and executing the
second pulse the value should be:

x2= (3u/4-u/2)c1/(=c1u/4(.

At the terminal point (t0 +Tp) the distance dL1 will be compensated (the
whole dL=0). By the third pulse (u/4), as it is clear from the previous
relations, variables (u,x) will be set to zero (u=0,x=0). Again the
required conditions are realized.

3. At the initial moment dL=0 , u = 0, x(0.
This version may be excluded by executing an additional first pulse.
From the relation 2.2.3 it is resulting that to set x to zero the current
velocity should be:
dv = -drc2/c1 . Here with value u= u0 =(dr/2
To achieve it one should add at the moment t0 the pulse value

Jm0(t0)=-(dv+drc2/c1),
(2.2.7)

and therefore to provide at the initial stage of control the value x=0. By
this additional pulse we reduce the situation to the previous version with
corresponding value u0.
Finally having the arbitrary initial values dL ,u, x, we may fined the
values of control pulses by summing the corresponding pulses in relations
2.2.5, 2.2.6 and 2.2.7.

[pic]
[pic]
(2.2.8)
[pic]

Using the same system 2.2.4 and applying the same reasoning we may modify
the pulses to ensure the platform transition to the terminal point with
relative co-ordinates (r, (L. It is easy to ague that in this case in
relations 2.2.3 instead of dr we should take drf = dr-(r and in equations
(2.2.8) instead of dL should be dLf = dL - (L - 6((r. Besides, to ensure
at the terminal point the value dv=0, the additional terminal pulse p =
((r should be executed.
Under these conditions the general structure of pulses is as following:

[pic]
[pic]
(2.2.9)
[pic]

Parameter ( may have the following value:
( =1 - relative velocity dv = 0 at the terminal point;
( = - 1 - the platform will be set at an elliptic orbit around the target
object;
( = 0.5 - the platform will be set at a nearby circular orbit with radius
(R+r0)+(r
In the direction (z) perpendicular to the orbit plane we have the following
dynamic equation:

[pic]


At the initial moment t0 the relative co-ordinate and velocity along z are:
[pic]
In this case to coincide orbits two pulses should be executed at the moment
t0 and t0+Tp/4 respectively. The corresponding pulses have the following
values:

[pic]
(2.2.10)
[pic]


The whole amount of pulses determined in 2.2.9 and 2.2.10 ensure the
platform controlled flight from the arbitrary point to an environ of an
object to dock with. Here with the proposed system of equations 2.2.4
provides a rather simple procedure of control pulses determination and
allows to go away of complicated calculations. Manipulation with variables
(u, x) considerably simplify the analyze of platform dynamic, ensuring
therefore a smooth passes to the required initial conditions of the next
stage of control in the close environ of target object.


Approach and Docking Phase Control


1 Position/Orientation Calculating Scheme

To construct the platform control algorithm when flying in the station
environ to have the information on platform current spatial coordinates and
velocities relative to the station is needed. As it was mentioned in
Chapter 1 (task 6) to calculate the AM current position and orientation two
types of measuring instruments are applied.
1. autonomous navigation system consisting at least of three angular
velocity detectors, tree accelerometers and a specialized computer;

2. position-sensitive optical system working in complex with marked
beacons (point like sources) setting near the docking place.
Within the frame of given researches tow versions of calculations
schemes enabling to determine the AM relative spatial position and angular
orientation were examined.

Integrating system of dynamic equations and using data from ANS the on-
board computer has an information on AM current spatial coordinates
(position and orientation) and velocities relative to the Station. Using
the calculation procedure mentioned in paragraph 2 the algorithm of control
at each current time interval has an information on the platform
coordinates and velocities relative to the target object and its docking
place respectively. Here with the rotational coordinates are determined
with high accuracy but the translational values have an insufficient
accuracy to ensure an approach control.
The initial conditions of AM autonomous flight are as following:
- AM autonomous flight time is rather shot (around an hour) and the
original data coming out of angular velocity detector are rather
accurate;
- The initial distance between detector and marked beacons < 200 m.;
- Beacons are in the detector's field of view;
- The platform flies towards the docking place.
In this case to determine the AM location we may use one optical
detector and two marked beacons, fixed near the setting place of the object
AM have to dock with.

At fig.3.1-1 the schematic of AM spatial location relative to the
docking place is depicted.

[pic]
fig.3.1-1


Co-ordinates system (X1Y1Z1) constrained with the platform's centre
of mass. Co-ordinates of detector (D) in the co-ordinates system [pic] are
[pic], i.e. (X0,Y0,Z0) is the shifting of[pic]at value (a) along X1 axis.
The co-ordinates system [pic] is constrained with the center of mass of an
already docked AM. In co-ordinates system (XYZ) co-ordinates of marked
beacons 1, 2 are (a,b,0) and (a,-b,0) respectively.
The detector (D) provides the onboard computer with information on two
pairs of angles ((1, (1) and ((2, (2) between the detector's optical axis
(OZ) and corresponding beacons (1,2) as it is shown at fig.1.
It is convenient to calculate the current AM translational coordinates in
system (XYZ), constrained with the object's docking place.
Solving the geometrical task, we receive in the coordinates system [pic]
coordinates [pic] of the platform's center of mass, its altitude [pic] over
the support place and the angular position [pic]Z about the vertical axis.
[pic] (3.1.1)
[pic] (3.1.2)
[pic] (3.1.3)
[pic] (3.1.4)
where:
[pic].
[pic].

(X and (Y - platform orientation about X and Y axes
respectively.

One should mark that within a given scheme of relative location at
the initial phase of docking control values (X and (Y are rather small
and are reduced to zero near the terminal point. That is why during the
whole process the calculated relative coordinates have a satisfied
accuracy.
Finally along with angular coordinates, received from the angular
velocity detector, we have in real time scale the full amount of data on AM
current spatial location.


2 Algorithm with 'Model' and Terminal Task Methods Application

From our point of view the most critical element of flight is the approach
and docking phase. In this respect to ensure a high stability and accuracy
of dynamic process control an algorithm with "model" and original scheme of
terminal task solution was elaborated. Even this task attracts the main
attention in our researches.

The proposed method of control executes in consecutive order the following
operations. On the base of data, incoming from an autonomous navigation
system or optical system (when observing marked beacons), it determines the
current coordinates of the platform relative to the station, its spatial
angular position, as well as components of the velocity's vector and
disturbing external forces.

Have used a selected beforehand law of the platform moving and solving a
corresponding terminal task, we receive the current values of required
engine's power. Having analyzed the calculated results, the control system
actuates particular engines of the platform and corrects the trajectory of
flight as need be.

The algorithm permanently repeats the above-described procedures with
periodicity ( 0.1-0.2 sec. until the process is completed.
To show the principal scheme of the control algorithm, let us overview a
concrete process of approaching and docking of the platform at a given
place on the surface of the station.

First of all, it is convenient to calculate current translational
coordinates and velocities of the platform in the coordinates system (XYZ),
constrains with the docking place of the target object. Coordinates X, Y
are in the plane of docking surface and plane (X,Z) coincides with the
orbit plane.

Let us overview the processes of control in the plane of the station orbit.
The initial conditions of the task may be as following:
- relative distance along Z-axis is around 150m;
- relative velocity of approach is less than 1.5 m/s;
- relative velocity of docking is less than 0.15 m/s;
- detector's plane is orientated towards the marked beacons.
Within the process the following random disturbances may be: deviation of
pulse power from the nominal value, shifting of the platform's center of
mass and change of inertia moment (due to the fuel expenditure), low-
frequency flexible oscillations of setting place.

In proposed method the platform motion near the setting place may be
described by the simplified system of dynamic equations ("model"), where
the external forces and disturbances are low then the power of a low-thrust
engine. We should mark that depending on particular platform initial
location, along Z-axis might be either relative distance along the orbit
(dL) or relative altitude (dr). Along X-axis would be variables (dr) or
(dL) respectively.

Examine the principal scheme of the control algorithm. Let us assume, for
example, that the docking place is at the "top" side of the station and at
the beginning of approach phase the platform moves towards the docking unit
along the relative altitude dr. Initial versions of approach phase is
depicted at fig 3.2-1 along with the corresponding
controlled approach trajectories
At the initial moment the marked beacons, fixed near the setting place,
come into the detector's field of view. From then on, the algorithm solves
in consecutive order the following tasks.

1) The position-sensitive sensor feed the on-board computer with
information about the angles between the marked beacons (it is enough two
beacons) and
detector's optical axis Oz . Solving the geometric task (described in §
3.1), we receive in the coordinate system [pic] (constrained with the
center of mass of
the already installed platform) coordinates[pic] of the platform's center
of mass, distance L along Z-axis and the angular position (Z about Z-axis.

2) Calculations of components of the velocity's vector and estimation of
disturbing forces values are done with the help of algorithm with "model"
procedure.
In co-ordinates system (X,Z) it is convenient to use the following dynamic
equations of the platform motion ("model"), where external forces are
absent.

[pic][pic][pic][pic]
(3.2.1)
[pic]


where: [pic], Ym, Zm - coordinates of the platform relative to the setting
place;[pic]
[pic], PY, Pz - power of pulse engines;
[pic]- platform's mass.[pic]
Choose the interval of analysis [pic], where:
[pic] - time interval of feeding the on-board computer with data from the
detector
([pic] = 0.1 - 0.2s);
[pic]- number of points at the interval (n = 50- 100).
At the first interval of analysis Px=0
Let us overview the control process along X-axis.
By a point-by-point integration of equations 3.2.1, at each step of
integration we have the estimated values of coordinate Xm(n) and velocity
VXm(n). Then, at the interval of analysis [pic]at each step [pic]
differences are calculated:

[pic]
(3.2.2)

The received values are approximated with parabola with the help of less
squares method and at the end of the interval T increments of coordinate (X
, velocity (Vx (derivative of parabola at the end interval T) and
disturbing external forces FXq (second derivative of parabola) are
calculated:

[pic]
[pic]
(3.2.3)
[pic]

where:
[pic]- weight coefficients (coefficients of approximation) having the
following construction:
[pic]
[pic] (3.2.4)
[pic]

where: N=T/2(t and should be an integer number, n = 1, 2, .2N
a = 2 for even numbers, a = 4 for odd numbers and a = 1 at the end of
interval of analysis.
We take the parabola on the base of assumption that the relative evolutions
of the platform are rather slow. So at each interval T we may consider all
external forces (including random disturbances) not included at equation
(1) are constant. (In some cases, when the relative velocities are rather
high, we may take cubic parabola). From the other hand, as it was mentioned
in § 2.2, variables dr, dL (in our case Zm , Xm) have the sinusoidal law of
motion with the period Tp = 5610s. In our case the time of analysis (T (
10s) consists a small pies of period Tp and therefore curves defined in
3.2.2 may be approximated with parabola with high accuracy.
Finally, the smoothed values of coordinates and velocities for the right
margin of the interval T are as following;

[pic];
(3.2.5)

(It is obvious that for coordinates Z,[pic] the scheme of calculations
according to relations 3.2.2 - 3.2.5 is the same).
We ought to mark, that the proposed procedure with "model" allows to
calculate values (3.2.5) and force Fq with high degree of accuracy. It
becomes possible because at the interval T we manage a relatively rich
statistics, though we use the simple form of dynamic equations (3.2.1) and
calculated on the base of data received from the detector coordinates have
a rather poor accuracy.
3) Calculated values (3.2.5) and force Fq allow to solve the terminal task
and to form commands for actuating particular engines ([pic]) as need be.
Examine the scheme of the terminal task solution along the [pic]-axis.
First of all we should estimate the duration Tk of controlled process.
At the initial moment t0 we have the corresponding boundary conditions:
for [pic]= 0 (right margin of the interval[pic])

coordinate [pic]
(3.2.6)
velocity [pic]

To estimate the value Tk at first approximation we simply integrate the
system 3.2.1 using the initial value 3.2.6. We ought to mark here that the
initial velocity along Z-axis Vz ( 0, that is the platform moves towards
the docking place and as it will be described below the control along Z-
axis is reduced to the task of platform deceleration up to the appropriate
value.
From the first equation 3.2.1 we shall have the following relation:

[pic] (3.2.7)

We propose to solve the terminal task in the following way. Let us choose
the trajectory of flight along X-axis which satisfies the boundary
conditions at the terminal point. These conditions are:
for [pic](the terminal point)
[pic]
[pic]
The corresponding law of motion may have the following form;

[pic]
(3.2.8)

where: C0 and C1 are an unknown constants.

(The problem of selecting the appropriate law of motion will be described
in our next address.)

Using an equation (3.2.8) and its first derivative while t=0 along with
conditions (3.2.6) we obtain the system of algebraic equations relative to
[pic] and[pic]:

[pic] (3.2.9)
[pic]

The second derivative of relation 3.2.8 is:

[pic] (3.2.10)

Substituting C0 and C1 from 3.2.9 into this relation and replacing Tk with
(Tk-t) we shall receive from 3.2.7 the final relation for the required
value of power:

[pic] (3.2.11)

In this relation Xs(t) and VXs(t) are the current coordinate and velocity
along X-axis. The value (Tk - t) is the time remained to the end of
the process. This parameter is permanently recalculated during the control
process. Besides, to avoid the dividing by zero in this relation, when the
platform is near the terminal point, we simply put over the following
arbitrary condition: if (Tk - t) ( 0.05s then (Tk - t) = 0.05s. (but
the calculations show that in fact there are no needs of this condition).
To make sure the marked beacons are in the detector field of view during
the control process one more additional limit is usefully to set up. The
detector field of view is limited by the angle (MAX. If in the orbit's
plane the current angular location (i of one of the beacon is near (MAX
(for instance (i ( 0.9(MAX ) then in relation 3.2.11 instead of (Tk -
t) should be the value:

[pic]

where: Zi - the current altitude over the setting place;
P - the nominal engines power along X-axis;
( - an arbitrary parameter within the range (0.1 ( 1)(t.

Under this condition the calculated power PX is depended on (MAX and the
control system will always ensure beacons location in the detector field of
view.
On the base of calculated value (3.2.11) the control system can ensure a
required trajectory of flight at each time interval [pic] by means of pulse
engines. But at the platform, we have the pulse jet engines with not
adjusting engine's power P, so to execute the correct pulse value the
following rule is stated: during each interval [pic] we ought to produce
the power pulse P(t, where
(t =[pic][pic].
But the value (t is limited by the minimal pulse duration the engine is
capable to realize. For the pulse engines we propose to install at the
platform (tmin = 0.01s. So, if (t ( (tmin the engine is not switched. (The
computer simulation of control process shows that normally the value (t (
0.5 (t )
The engine switching is realized by the command from the computer.

Then on the algorithm repeats the whole above-described procedures through
relations 3.1.1 - 3.2.11 during the second time intervals [pic] (shifting
the current interval of analysis [pic]at step [pic]) and so on until the
process is completed.
One should note that in correlation (3.2.11) the item [pic] in fact plays a
role of dynamic stabilizer of the process, damping different disturbances
occurring during the flight.
Along Y coordinate the relation for power PY is similar to relation 3.2.11

[pic]

Concerning the control along Z-axis we may repeat absolutely the same
procedure through 3.2.1 - 3.2.10 replacing [pic] with [pic] and to get the
same relation for Pz.(but without first term).

[pic] (3.2.12)

The reason of excluding the term connected with the relative coordinate is
the following. According to the accepted dynamical scheme Z-axis is
perpendicular to the docking place and the platform moves practically along
this direction. That is why the process of control is reduced to the
deceleration of the platform, so the term [pic] has no means and should be
excluded in relation 3.2.12. From the other hand during the contact with
the setting place the platform should have a negligible velocity to provide
the docking. To ensure this regime we should receive the following rule: if
VZ (t)( Vmin then VZ=0 (Vmin might be, for example, 0.1m/s.).
The corresponding trajectories of controlled flight for different versions
of initial conditions are represented at fig 3.2-1.

From our point of view the proposed algorithm of control is rather
universal and may be applied in different versions of platform motion in
the object environ, including the platform angular attitude control. In
this case to control for example the angular position and angular velocity
about Z-axis we may use the equation:
[pic]
(3.2.13)
where: [pic] - angular acceleration about Z-axis;
MZ - momentum about Z-axis;
JZ - inertia abour the axis.
Applying the described above procedure of calculations with the terminal
conditions ([pic]) we shall receive the relation for the value MZ

[pic] (3.2.14)


MFz - accidental angular disturbances (also calculated within the algorithm
with "model".
To realize the platform angular stabilization within the limited angle and
limited angular velocity when executing an approaching phase we should
choose the appropriate value (Tk - t). Thus to make the relation (3.2.14)
sensitive towards the angle ( ( ( ( (max the following condition should be
fulfilled:

(Tk -t) ( Tmin = (JZ12(max / pMmax )
(3.2.15)

where: Mmax - the available (nominal) momentum value;
p - parameter which in fact is an equivalent of minimal pulse duration the
engine is capable to realize. In our case p = ( t min / (t ( 0.1. To ensure
the needed control along angular coordinate we should accept the simple
rule: if (Tk - t) ( Tmin then (Tk - t) = Tmin ; if (Tk - t) ( Tmin then
the current value (Tk - t) is taking. The relations similar to 3.2.14 are
used to control angular turns about X and Y-axis.
But in proposed scheme of calculations with one optical detector and two
marked beacons we have no opportunity to get values of angles (X and (Y by
means of an optical system. To measure these angles one more additional
optical detector is required and we may install it at the platform. But as
it was mentioned in paragraph 1 the platform is supplied with the angular
velocity detectors having a rather high accuracy (the error is about 20
arc.min./hour) From our point of view to provide a reliable docking process
this accuracy is quite enough. Thus, during the autonomous flight of about
3 hours the accumulated angular error should be not more than 1 degree. The
results of computer simulations show this level of errors plays practically
no role in the process of angular control and to use in calculations the
initial data on current angles and angular velocity directly from ANS is
possible.

In calculations we took the following values of main parameters:
- mass of unloaded platform 200kg, loaded platform 1000 kg;
- nominal engines power along each translational coordinate is 50 N;
- about each axis ratio (momentum of force) / (inertia) ( 0.3 radian/s2 for
unloaded platform;
- the maximum relative distance along dr - 8000 m., along dL - 10000m.;

Under these values the influence of each pulse in control process is
negligible, but a set of allocated in time pulses realises the exact
trajectory of flight, ensuring a high degree of reliability and accuracy of
terminal task solution. Besides, the control algorithm capable instantly
reacts on accidental disturbances and damps them rapidly. In case of
angular control we also try to operate with rather low angular velocities
(( 0.03 radian/s) and separate in time the rotation about each axis.
As a result the control system realizes a 'flexible' trajectory of flight
and we have a smooth and precise docking of the platform at the setting
place.
We ought to mark, that two circumstances (an accurate calculation of
dynamical characteristics and permanently solving the terminal task at each
time interval [pic]) allow to use a simple form of dynamic equations and
therefore to simplify considerably the whole process of calculations. Thus
the results of the process computer simulations confirm that within the
algorithm of control we may use instead of system (3.2.1) the simplest
dynamic equations:
[pic],[pic][pic][pic]
and the final result will be practically the same.

At present time it is developed a package of platform control algorithms,
taking into account the random shifting of the platform's center of mass, a
rough estimate of an object inertia and its geometry, as well as different
kinds of random external disturbances. The developed method of control has
a high stability towards different kinds of interference and provides a
high accuracy when executing a definite task (for example, errors of
matching of assembly points of the platform and the support is in the
limits of ( 0.5 sm ).
Finally, on the base of described above principles we can realize quite
simple and reliable algorithms of control. These method can provide
accurate maneuvers of the spacecraft, exact flying around the station and
precise docking at a given place.


Loaded AM Dynamic Schemes of Control


In general features the loaded AM flight control is based at the same
principles which were described in paragraphs 2,3. But the control process
of loaded AM has some peculiarities and in some cases is differ with
unloaded AM control. In the system platform + load the center of masses is
shifted aside and the action of pulse engines along at least two
translational co-ordinates excites the corresponding angular disturbances.
That is why two principal schemes of loaded AM angular location when motion
along the trajectory were researched. First one - "rocketry" scheme, when
AM trusters push or tag the load and slightly turn the system (AM+load)
along the current velocity vector direction, ensuring the minimum of
angular disturbances. This scheme of motion is depicted at fig. 4-1



Second one is the direct control along each translational coordinate and
simultaneous angular disturbance damping. The analysis of various dynamic
processes shows that each scheme has its positive and negative aspects.
Thus the execution of three pulses transition of load by means of
"rocketry" scheme raise in practice no hard problems. The time interval
between pulses is about Tp/2 (in our version ~ 2805 s.) and there is
enough time to turn the loaded AM at the appropriate angle. According to
the scheme (fig 1-1) at the beginning of control pulse (d( = 0) X-axis
should be attitude along the current velocity vector, and should stay
constant when pulse execution. Then within the interval (t = 22805 s. the
system is prepared to the next pulse (the turn at 1800 if it is currently
needed) and so on. Besides that, within the interval (t (because of its
large duration) the control system may perform the required angular
evolutions, for example, to set the appropriate angle attitude for marked
beacons detecting and therefore to specify the current relative coordinates
and velocities.
But by means of "rocketry" scheme the system cannot perform simultaneously
the control pulses along all translational coordinates when executing the
terminal task. In this case during the control for instance in the orbital
plane we should provide pulses along at least two axes X, Z (see fig 4-1)
An algorithm of control calculates values of power Px(t) and PZ(t)
according to relations 3.2.11, 3.2.12. To realize the current control
pulses the loaded AM first should be turned at an angle ( about Z-axis
which is determined by the relation:
[pic] , and then to execute the pulse. This scheme is also realized by
means of computer simulation and a rather suitable results of docking
accuracy are received. This scheme is represented at fig. 4-2.
One more version of dynamical scheme of control was examined on the base of
particular example The idea of this scheme is as following. Actually the AM
control along Z-axis needs a relatively poor accuracy. The velocity of
collision with the docking place in practice may lies within the range 0.01
( 0.15 m/s. It means (according to relation 3.2.12) that if at the initial
moment of approach phase the velocity Vz(0) has the definite small value
the control along Z-axis reduced to one or two short pulses. Thus, for
example, calculations show: if at the initial moment the distance to the
docking place is ~ 100 m and approach velocity VZ(0) ~ 0.25m/s then at the
terminal point the value VZ ~ 0.08 m/s and no additional pulses are needed.
Here with the duration of process ~ 700 s. We should mark that at the
initial moment the data received from detector and specified by means of
algorithm with "model" is rather accurate. It allows to set the
corresponding initial velocity VZ(0) with error ( 0.05 m/s. and practically
exclude the control along Z-axis. This fact will significantly simplify the
docking control of loaded AM along X and Y-axis.
In order to reduce the velocity along Z-axis down to the required value the
loaded AM should be orientated as it is shown at fig. 4-3. After the
deceleration pulse the system during the time interval ~ 40 - 60s. is set
to the angular position as it is shown at the same figure. After this
maneuver the described above algorithm begins to realize the approach and
docking control. Within this scheme the control along X and Y-axis may be
realized in a following way. At the end of initial interval of analysis T
values PX and PY are determined. Then on the pulse engines turns the
loaded AM at an angle (, where [pic] , and then the control system
executes pulses as need be. Coordinate Y doesn't constrain with X and Z
coordinates. That is why during the control process the value PY is rapidly
decreased to zero and we shall have in fact the control along the X-axis.


Concerning the scheme of direct control along each axis one should make the
following remarks. Let us assume the same mass and geometry of loaded AM as
it is shown at fig 4-1. When executing the terminal task it is easy to see
that in this case the control along, for instance, X-axis excites the
angular disturbance about Y-axis which should be damp by switching an
additional corresponding pair of engines. Surely, the corresponding
terminal task will be done and the required precision of docking will be
achieved. But in this version of control an additional fuel will be
expended to damp the angular turns. Thus, assuming that the disturbing
momentum (MF) which appears after the control pulse along X-axis is equal
to the nominal momentum about Y-axis (MY) the AM is available to produce.
In this case each correcting pulse (p along X-axis requires the fuel
consumption equal to 2(p value. The same situation with the fuel expense
will be when the control along Y-axis (the perpendicular to the orbit's
plane direction) is take place.
To find the optimal solution in selecting the dynamical scheme of loaded AM
flight control a portion of calculations were done. The preliminary results
allow to make the following selection When applying the three pulses method
of flight control it is convenient to use the "rocketry" scheme of flight
(a rather obvious and commonly used scheme when flying at large relative
distance). But within the approach and docking phase when a proposed above
algorithm with "model" is applied it is much better to use schemes either
the scheme with the permanent angular tuns (fig 4-2) or the scheme with
preliminary deceleration of loaded AM (fig 4-3)

By the end of this paragraph we would like to represent an example of
flight control in the environ of an Orbital Station. The initial conditions
are: the loaded AM is at the distance ~ 300 m from the docking place; the
initial relative velocity ~ 2m/s. The task is to set the load at the
docking place in a time interval ~ 400s.
The flight control is realized by means of algorithm with "model"
(described in § 3) The initial data on relative coordinates and velocity is
derived from integration of motion equation. At the final part of the
process the information on relative position is taken from the optical
system when observing the marked beacons. Two types of trajectories are
calculated and the result is depicted at fig 4-4.



Final Overview of Control Processes Cycle


Then on let us return back to the cycle of operations, which was mentioned
at the beginning of our address. On the base of described methods of flight
control we shall repeat the cycle's phases in a more definite way referring
to the corresponding paragraphs of a given paper.
First of all we should underline that the AM on-board computer actually
uses an exact dynamical equations of relative motion for calculating the
current AM translational coordinates. These equations should take into
account the second and third approximations of Earth gravity potential,
though under the time scale of about 3 - 4 hours an influence of these
terms is negligible. Results of dynamical equations integration is the
based information on AM current location relative to the Orbital Station.


Phase 1. When the AM separates from the Station the on-board computer
starts to integrate dynamical equations and finishes calculations at the
end of operating cycle. The purpose of the given phase is to specify the AM
location and velocity relative to the target object and to determine
coordinates of terminal point AM should fly to. This task is solved by
means of position determination procedure, which is described in paragraph
2.1

Phase 2. Results of computation allow AM to realize the flight towards the
target object on the base of modified "three pulses" flight control method.
The scheme of required pulses determination is described in paragraph 2.2
By the end of this phase the AM is set into appropriate initial position
where from the approach and docking phase is started.

Phase 3. This phase of flight control is realized by applying an original
algorithm of control with "model" and terminal task solution. The method is
described in paragraph 3.2. The initial data on AM position relative to the
docking place is received from position sensitive detector when measuring
the angular position of beacons, located near the setting place. The
corresponding scheme of calculations is represented in 3.1.

Phase 4. Then on in general features the loaded AM repeats in consequent
order phases 2,3. But in case of loaded AM we should select first of all
the dynamic scheme of flight. To realize the flight towards the Station by
means of "three pulses" control method, to use the "rocketry" scheme of
motion is preferable. To perform this version of flight the loaded AM
should execute the angular turn and to ensure the orientation with d( = 0
at the beginning of first control pulse. An angular turn is performed under
the low angular velocity ( ~ 0.05 radian/s).An average duration when turns
at an angle 900 is about 40 s. After the third pulse The loaded AM is
set into the predefined point near the setting place. By an additional
pulse ((dr (see equation 2.2.9) the loaded AM may be set, for example, at
an elliptic orbit around the Station. Here with the initial parameters of
ellipse are determined beforehand and should provide the appropriate
conditions at the beginning of approach phase of flight.

Phase 5. This phase is the most important part of AM trajectory of flight.
The control during this phase is also realized on the base of control
algorithm with "model" But taking into account problems mentioned in
paragraph 4 the dynamical scheme of flight should be selected. A few
versions of loaded AM angular location were examined and as it seams to us
one of the possible scheme may be just an example which was described in
paragraph 3.2. This scheme with prior loaded AM deceleration ensures a
rather simple conditions of control for different types of loads. In fact
it is a "rocketry" scheme of flight with some negligible pulses along Z-
axis (see paragraph 3.2) and simultaneous damping an angular disturbances.

Within this part of our researches the computer simulations of particular
versions of flight control with arbitrary values of random errors (it is
mentioned at the end of paragraph 3.2) were realized. It is easy to see
that the main goal of flight control is to ensure the safety and exact
docking of unloaded and loaded AM. Results of calculations confirm that the
proposed methods of control are capable to perform this task.

At present time, a prototype of the platform, working in complex with a
cable-crane in the test-hall and capable to install masses of about 60 kg,
is developed.
The prototype is depicted at the photograph.




The platform is supplied with four pulse engines. The engine is the
electric-magnetic valve where the power is produced by escaping the
compressed air. The balloon with compressed air is placed aside in the test
hole and connected with engines by pneumatics lines. The power of each
engine ~ 2.5 N. Position-sensitive detector and on-board computer is places
at the platform. Two marked beacons are displaced on the floor near the
setting place. The platform performs the motion control along two
translational axes in horizontal plane and angular attitude about vertical
axis. The on-board computer used the approach and docking control algorithm
similar to that was described in paragraph 3. During the experiments an
accurate of matching the assembly points of the load and the support was in
the range ( 0.6 sm

This technological model is elaborates by specialists from the Space
Research Institute in close cooperation with participators of a given
project.

Results of computer simulations and experimental tests with the platform
prototype confirm the following conclusions:
- The proposed methods of flight control are able to perform the safety and
accurate Automated Module operating in Space.
- Declared properties of the Module are quite reliable;
- There are no technical obstacles in the device creation.


-----------------------
[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Load

Platform

[pic]