1. Introduction
The great majority of the robotic arms used nowadays
are rigid arms with articulated joints which introduce
certain limitations: Spatial motion limitation due to its
joint type, the number of joints, and the lengths
between these joints. The space/weight occupied by
the arm is non-negligible: presence of actuators on the
arm itself, global rigidity etc… Our idea [1] is to
introduce a new kind of totally flexible “arm” with no
actuators on the beam itself. A shape is given to this
flexible beam as it is deployed, enabling a wide range
of shapes and lengths. The actuators are located on the
spacecraft itself, where they push the beam out and
give it his shape. We can divide the mechanism on the
spacecraft in two distinct sub-mechanisms: the shaper
part, which bends the beam to give it the actual shape,
and the deployment part which pushes the stored beam
from its storage compartment, through the shaper and
to the outside of the spacecraft (Fig. 1). Contrarily to
our device, an arm is by definition an articulated
mechanism with joints on it. We thus decided to avoid
the use of this terminology and named it “Morphable
Beam Device” or MBD. The beam is free of any
actuators: only the actuators located on the satellite
will bend the beam while releasing it more and more.
Fig. 1 Morphable Beam Device and Application
To demonstrate this new concept, a first
experimental device was developed, basic experiments
had been done, and we found that the system has a
retro-bending behavior[1]. In this paper, finding a
behavior model, setting up a control framework for
this device, including a control algorithm and an
efficient control system, and the measurement of the
curvature of the beam shape are conducted, and the
relationship between bending displacement and the
resulted beam curvature is obtained. A beam shape
modeling made by the MBD taking into the
retro-bending and desired beam shape generation
algorithms are described.
2. Definition and Mathematical Model
To understand and model the relationship between
deployment and bending, the better way is to work
with regard to the surface of deployment. Figure 2
shows the effective output surface and the resulting
cones in which the deployment direction vector is
contained.
Fig. 2 Ouput Surface and Zones
Fig. 3 shows the measurement of the curvature of the
beam shape after lateral bending and deployment of
the beam, which shows that at a fixed bending position,
when deploying the beam, the curvature obtained is
constant along the deployed part. The relationship
between bending displacement and the resulted beam
curvature is obtained as in Fig. 4, where the curvature
drastically changes with the bending position: we call
it the retro-bending. This behavior must be considered
for generating beam shape algorithms. The relationship
between the lateral displacement and the radius can be
approximated well by combination of polynomial curves
of order four and three as shown in Fig. 4(b).
Fig. 3 Beam Shape Measurement
(a) Lateral Bending Displacements
and Generated Circular Shapes
(b) Relationship Between Lateral Displacement and
Radius, and Approximated Curve Model
Fig. 4 Generated Beam Shapes
3. Beam Shape Generating Algorithms
In this section, a beam shape modeling made by the
MBD taking into the retro-bending and desired beam
shape generation algorithms are described. The
objective is to build a satisfying 3D curve passing by
the points the user inputs under the condition that the
curvature must be continuous along the beam to model
reality. We focus on polynomial interpolation and
chose the method of relaxed cubic spline curves: cubic
bezier curves glued together so that the continuity of
the first and second derivatives is verified. We then
compute the characteristics (curvature, etc.) along the
discrete beam model.
3.1 Constant Curvature Algorithm
3.1.1 Position Specified (CCA-I)
The user inputs are the followings as shown in Fig. 5:
1) Specify the direction of bending α, namely normal
plane P including the beam.
2) Specify the position (one point) of the beam end.
When the output tangent is vertical, we can define a
circle from two points and the tangent to one of the point,
and we compute the corresponding curve, curvature and
bending position[4].
Figure 6 shows an example of the algorithm CCA-I, and
Fig. 7 shows an example of experimental verification, and
the error is about 2-3 mm. Figure 8 shows a slice
representation of the beam end positions accessible
with the algorithm CCA-I and maximum beam length
of 300mm.
Fig. 5 Constant Curvature Generation Method
(Position Specified)
Fig. 6 Bending Displacement and Deployment for Beam
Shape Generation Algorithm CCA-I
Fig. 7 Experimental Result
(The elliptic denotes the linear part of the beam)
Fig. 8 Accessible Space of the Beam End
3.1.2 Position and Direction Specified (CCA-II)
The user inputs are the followings as shown in Fig. 9:
1) Specify the direction of bending α, namely normal
plane P including the beam.
2) Specify the position and direction (one point) of the
beam end.
Figure 10 shows an example of the algorithm CCA-II.
Fig. 9 Constant Curvature Generation Method
(Position and Direction Specified)
Fig. 10 Bending Displacement and Deployment for
Beam Shape Generation Algorithm CCA-II
For a given beam end direction, a given maximum
beam length, let’s see what are the different targets
accessible in a given vertical plane in a given direction
by the MBD. We must note that there is also a
restriction on the tangent angle at the exit point. To
visualize the problem, Fig. 11 shows the maximum
output cone for the tangents at point O, the exit of the
deployer (this cone is shown in pink).
Fig. 11 Arcs satisfying the specified direction and
arbitrary positions
Fig. 12 Evolution with the direction and Superposition
We can see that these conditions can be really
restrictive. In fact, in the case, only one of the curves is
acceptable regarding all these criterions (except the
maximum beam length which isn’t specified.)
Fig. 12 shows an evolution with the direction at fixed
maximum beam length and superposition for all directions
in a given deployment plane P.
3.2 Variable Curvature Algorithm, VCA
The algorithm VCA implemented into the control
station works the following way.
1) First, the user inputs: a series of points (n points)
that define the curve, and the number of points (Nbp)
to discrete the curve model into.
2) For each set of two points, a cubic Bezier curve is
defined using the process earlier and such that the first
and second derivatives of the curve will be continuous
at the junctions.
3) A discrete representation of the curve is obtained,
then the vectors t, b, n of the Frenet reference frame
along the curve for each points, and the position of the
center and the corresponding Radius and curvature of
the osculation circle at each point are computed.
4) In order to compute the direction in which bend the
beam at point p as to produce the final expected shape,
we compute for each point to different angle values,
Theta and Beta. By processing all our points
recursively starting from the deployer-end of the entire
curve we want to have, we can for each step j define:
a) The angle Theta_j corresponding to the rotation of
the vertical plane around the vertical axis such that the
new plane contains tj
b) The angle Beta_j corresponding to the angle of the
next tangent with the vertical axis.
Using this angle Beta_j we then rotate the whole
spline such that the next tangent becomes the vertical
axis. Our beginning condition is the following: we start
from the endpoint of the beam at the deployer end,
with an initial vertical tangent. (the 3d cubic spline
computed from the user points is rotated such that this
is the case). The computation of these angle Theta_j
and Beta_j can be summarized in Fig. 13.
Figure 14 shows an example of the algorithm VCA,
and Fig. 15 is a variation of the curvature along the
modeled beam.
Representation of the angles along the discrete curve
Fig. 13 Recursive computing of the angles along the curve
Fig. 14 Variable Curvature Shape Generation Example
Fig. 15 Curvature Distribution Along the Beam
4. An Application Example: Remote Inspecting
In this section, the case is considered where the
satellite would approach a spacecraft’s orbit
sufficiently near to have a zero relative speed seen
from the spacecraft. Let’s take for example the
space-shuttle. Our satellite’s mission would be to
inspect the surface of the spacecraft and particular
targets. In a usual case, with no Morphable Beam
Device (MBD), we would have to move the satellite
around in order to inspect the surface all around the
shuttle: front, back, top, etc. This would be very costly
in propellant, which is very limited on small
spacecrafts like satellites. Furthermore these
maneuvers can be dangerous, with risks of collision,
loss of control, etc. Such a mission would be very
limited and would be very difficult to adapt and update
the objectives during the mission itself.
Suppose now that our satellite is equipped with an
MBD system and that we managed to position it at a
relatively close, but not too much, distance from the
spacecraft, as shown in Fig. 16. By deploying our
beam, which if we found a good material could be tens
of meters long, we can now go and inspect all around
the spacecraft without moving the satellite and wasting
our precious propellant. Since the spacecraft’s position,
once stabilized, doesn’t move relatively to the shuttle,
we can enter a rough three dimensional model of the
shuttle’s position as seen from the MBD in our control
software.
Fig. 16 Application Example: Remote Inspection
Using MBD
5. Conclusions
In this paper, a beam shape modeling, beam shape
generation algorithms by specifying the beam end position
and/or direction, and also an application example of
Morhable Beam Device (MBD) were discussed. In the
future, we will develop a flight model of the MBD
applicable to install on a satellite and will demonstrate it
on orbit.
References
[1] S. Matunaga, S. Sugita and T. Iljic, "Morphable Beam:
Concept and Initial Hardware for Spacecraft Inspection,"
2006 ISAS 16th Workshop on Astrodynamics and Flight
Mechanics, Sagamihara, August 1-2, C-2, 2006.
[2] T. Ilijic and S. Matunaga, “Visual and Positioning
System Study for a Morphable Beam Device,” 2006
ISAS 16th Workshop on Astrodynamics and Flight
Mechanics, Sagamihara, August 1-2, A-16, 2006.
[3] 松永三郎, 杉田沙織，トーマ・イルジック, "衛星
搭載用外観検査用アームの試作," 日本機械学会2006
年度年次大会講演論文集, 熊本, No.06-1, Vol.5, 1021,
Sep., 2006, pp.335-336.
[4] Iljic Thomas, “Morphable Beam Device and its
Vision and Positioning Sysytem,” Master Thesis,
Department of Mechanical and Aerospace Engineering,
Tokyo Institute of Technology, 2007.