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AAS 22-802
COMPUTATION OF RELATIVE ORBITAL MOTION USING
PRODUCT OF EXPONENTIALS MAPPING
Aryslan Malik*
, David Canales†
, Taylor Yow‡
, Daniel Posada§
, Christopher W.
Hays§
, David Zuehlke§
, and Troy Henderson¶
The Product of Exponentials (PoE) method consists of an exponential mapping
commonly used in robotics. However, it can easily be adapted to describe the
position and orientation of any rigid body in relation to another body or frame. In
prior work, it was demonstrated that the PoE framework is useful as an alternative
method for defining and drawing orbits given a set of orbital elements. This paper
expands the application of the PoE framework to compute relative orbital motion.
It proves to efficiently correlate the relative attitude and position between a chaser
and a target using a compact formulation.
INTRODUCTION
Brockett’s study was the first to adopt the Product of Exponentials (PoE) formulation to express
forward kinematics of a multi-body system.1The PoE formulation’s core concept is to consider a
joint as a screw that moves the rest of the external links (or bodies), which allows for simultaneous
rotation and translation resulting in a concise mathematical formulation.2–7 The PoE has proven to
be useful for robotic control and manipulation.8–22 Recently, the PoE formulation was expanded to
“draw” orbits using each element of the Keplerian orbital elements set [Ω, i, ω, θ, r(θ)], where
each element acts as a separate joint.23 In this work, its application is expanded to the description
of relative orbital motion. The benefit of using PoE formulation is that the underlying matrices
defining rotation and translation are inverted, and therefore, using a succinct notation, the chaser’s
attitude and position is easily related to the target’s, and vice-versa.
The PoE-orbital mechanics framework relates a set of orbital elements ([Ω, i, ω, θ, r(θ)]) to the
full state (position, velocity) of the satellite (peo ∈R3, Veo ∈R3)using exponential mappings.23
Note that Ωis the right ascension of the ascending node, iis the inclination of the orbit, ωis the
argument of periapsis, θis the true anomaly, and r(θ)is the location of the spacecraft defined as:
r(θ) = a(1 −e2)
1 + ecos θ,(1)
*Visiting Assistant Professor, Space Technologies Lab, Embry-Riddle Aeronautical University, 1 Aerospace Blvd., Day-
tona Beach, FL, 32114.
†Assistant Professor, Aerospace Engineering Department, Embry-Riddle Aeronautical University, Daytona Beach, FL,
32114.
‡Masters Student, Space Technologies Lab, Embry-Riddle Aeronautical University, 1 Aerospace Blvd., Daytona Beach,
FL, 32114.
§Ph.D. Candidate, Space Technologies Lab, Embry-Riddle Aeronautical University, 1 Aerospace Blvd., Daytona Beach,
FL, 32114.
¶Associate Professor, Space Technologies Lab, Embry-Riddle Aeronautical University, 1 Aerospace Blvd., Daytona
Beach, FL, 32114.
1
being athe semi-major axis and ethe eccentricity of the orbit. The position of a body frame
expressed in the inertial frame is calculated here leveraging a single PoE formula:
Teo =e[S1]Ωe[S2]ie[S3]ωe[S4]θe[S5]r(θ)Meo ∈SE(3),(2)
where,
Meo =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
=I4×4,(3)
S1=
0
0
1
0
0
0
,S2=
1
0
0
0
0
0
,S3=
0
0
1
0
0
0
,S4=
0
0
1
0
0
0
,S5=
0
0
0
1
0
0
.(4)
The configuration matrices Teo are part of SE (3), which is the Special Euclidean space that
preserves the Euclidean distance between any two points. Such an Euclidean space is described here
by 4×4matrices that store both attitude and position information, as shown in Eq. (6). The subscript
(eo)in Meo corresponds to the “home” configuration of an “object” relative to the equatorial frame
of the major body. The Meo is the 4×4identity matrix given that, initially, it is assumed that
the object’s position and orientation coincides with the equatorial frame’s position and orientation,
which is then modified by exponentials which allow the transformation of our object’s frame. Figure
1 represents the different reference frames used in this investigation: (xe, ye, ze)corresponds to
the geocentric equatorial frame, (xo, yo, zo)to the orbital perifocal frame, and (xb, yb, zb), to the
satellite body frame with xbpointing radially outwards, ybis in the direction of the velocity of the
satellite, and zbbeing normal to the orbital plane.
The inertial velocity of an object in orbit expressed in an inertial frame is given by the following
equation:
Vs=Reo
˙r(θ)
r(θ)˙
θ
0
=ReoVb,(5)
where Reo ∈SO(3) is extracted from the configuration of the body,
Teo =Reo peo
0 1 ∈SE (3).(6)
More generally, the inertial velocity of the body is calculated directly from the spatial twist, Vs∈
R6, as follows:
Vs=S1˙
Ω + Ade[S1]Ω (S2)˙
i+ Ade[S1]Ωe[S2]i(S3) ˙ω(7)
+ Ade[S1]Ωe[S2]ie[S3]ω(S4)˙
θ+ Ade[S1]Ωe[S2]ie[S3]ωe[S4]θ(S5) ˙r(θ),
where,
Vs=ωs
vs∈R6.(8)
2
b
Ω
ω
Orbit Plane Perigee
Equatorial Plane
Ascending Node
i
b
b
Satellite
r(ѳ)
ѳ
Figure 1. Representation of the different reference frames used in this study.
The adjoint operator is also defined as,
AdT=R03×3
[ˆp]R R ,(9)
with [ˆ
·]being the skew-symmetric operator defined in Eq. (10):
[ˆp] =
0−p3p2
p30−p1
−p2p10
.(10)
Therefore, the velocity vector of the satellite expressed in the inertial frame becomes:
Vs=vs+ωs×peo ∈R3.(11)
Alternatively, the inertial velocity may be computed from a body twist:
Vb=ωb
vb= AdT−1
eo
Vs∈R6,(12)
Vs=03×3ReoVb=Reo vb∈R3.(13)
Taking the inertial derivative of the position vector twice yields the inertial acceleration found in
Eq. 14.
As=¨r(θ)
r(θ)peo + ˙ωs×peo +2 ˙r(θ)
r(θ)ωs×peo +ωs×(ωs×peo)(14)
where ˙ωscan be either computed from the derivative of spatial twist in Eq. 7,
As=˙
Vs=˙ωs
˙vs∈R6,(15)
3
or from spatial twist and inertial velocity,
˙ωs=−2Vs·peo
r2(θ)ωs.(16)
To this extent, the PoE formulation is extended in this paper to compute relative orbital motion.
Various relative motion and proximity operations are designed using PoE. The PoE method al-
lows transitioning from orbital elements that would incur particular relative motions. This fact
is accomplished by accommodating the exponentials formulation to the Clohessy-Wiltshire (CW)
equations.24 The resulting methodology could then be exploited for different applications, such as
servicing and proximity operations.
RELATIVE ORBITAL MOTION
The relative orbital motion problem considers two spacecraft operating within close proximity
of one another. The inertial position of the “target” spacecraft A is denoted as pet, and the inertial
position of the “chaser” spacecraft B is denoted as pec. The relative position of spacecraft A with
respect to spacecraft B in inertial frame (e)is denoted as the vector p(e)
ct =pet −pec.
Target A
pet
pec
t
t
t
Inertial Frame
pct
Chaser B
Figure 2. Co-moving reference frame attached to target A, from which body B is observed
Relative motion in orbit usually involves a target spacecraft, which is most of the time passive
and non-maneuvering, and a chaser spacecraft, which is active and performs maneuvers to bring
itself closer to the target in a specific orientation. The configuration of the target spacecraft with
respect to the geocentric equatorial frame is denoted by Tet, such that:
Tet =Ret pet
0 1 ∈SE (3).(17)
where, Ret ∈SO(3) is the orientation and pet ∈R3is the position of the target spacecraft expressed
in geocentric equatorial frame.
4
Similarly, the chaser spacecraft’s configuration is expressed using SE(3) configuration matrix:
Tec =Rec pec
0 1 ∈SE(3).(18)
These configuration matrices can be directly computed from the orbital elements set using the PoE
formula, the following formula demonstrates this for the chaser spacecraft:
Tec =e[S1]Ωce[S2]ice[S3]ωce[S4]θce[S5]r(θ)cMec ∈SE (3) (19)
The orientation of the target spacecraft expressed in chaser spacecraft’s frame is given by the fol-
lowing formula:
Rct = (Rec)−1Ret ∈SO(3) (20)
The result of Eq. 20 is a special orthogonal matrix SO(3) because an inverse of a rotation matrix
R∈SO(3) is also a rotation matrix, and the product of two rotation matrices is also a rotation
matrix. Now, similarly to rotation matrices, the transformation matrices possess the same properties
such that an inverse of a transformation matrix T∈SE(3) is also a transformation matrix:
T−1=R p
0 1−1
=RT−RTp
0 1 ∈SE (3) (21)
Now, the configuration of the the target spacecraft with respect to the chaser can be computed using
the following equation:
Tct = (Tec)−1Tet =RT
ecRet RT
ecpet −RT
ecpec
0 1 ="Rct p(c)
ct
0 1 #∈SE(3) (22)
where, Rct is the orientation and p(c)
ct is the position of the target spacecraft expressed in chaser
spacecraft’s frame - that is how the observer on chaser spacecraft would see the target spacecraft.
Now, the relative velocity can be described using the developed tools. The first step is to compute
inertial velocities of the target Vtand chaser Vcspacecraft using either Eq. 5 or Eq. 11. Next, the
relative velocity of target spacecraft relative to chaser is computed as follows:
V(e)
ct =Vt−Vc−ωc×(pet −pec)(23)
where, ωcis the angular velocity of chaser spacecraft’s frame expressed in inertial frame, and is
computed from spatial twist using Eqs. 7 and 8. This inertial relative velocity as seen from chaser
spacecraft’s frame of reference is computed as:
V(c)
ct =RT
ecV(e)
ct (24)
Secondly, the relative accelerations may be computed as follows:
A(e)
ct =At−Ac−2ωc×(V(e)
ct )−ωc×ωc×(pet −pec)−˙ωc×pct (25)
where Atand Acare the inertial acceleration terms of the target and chaser, respectively. Each of
these terms are calculated using the method presented in our work.25 The relative acceleration term
may also be rotated into the chaser spacecraft’s frame of reference:
A(c)
ct =RT
ecA(e)
ct (26)
5
To validate the developed relative orbit PoE formulation, an example problem from a textbook
was used.26 The chaser and target orbits are described in Table 1. The orbits are visualized in Figure
3. The relative motion of spacecrafts was simulated for 60 orbital periods of the chaser spacecraft
with starting true anomaly of 40◦. The relative position of the target spacecraft expressed in inertial
frame is shown on the left of Figure 4, which is not very intuitive to visualize. The same relative
position expressed in the chaser frame is shown on the right of Figure 4, where the chaser’s LVLH
frame is shown in black, and the target’s position with respect to chaser when t= 0 is shown in
green. Relative velocity V(e)
ct and acceleration A(e)
ct expressed in inertial frame was calculated using
Eqs. 23 and 25, respectively, and plotted by components in Figure 5.
The proposed PoE approach produced relative position, velocity, and acceleration vectors that
were in agreement with those obtained using traditional techniques. Figure 6 demonstrates the error
in the relative position of the target expressed in chaser frame when comparing the output of the
proposed PoE method and an example code in Curtis Matlab library. All of the error components
are cyclic and remain consistent indicating that the culprit of the insignificant difference is purely
numerical (e.g. round-off) - the magnitude of the error does not exceed 0.3 mm.
Table 1. Description of Orbits
Orbital Element Chaser Orbit Target Orbit
a6803 km 6878 km
e0.0257 0.0073
i60◦50◦
ω30◦120◦
Ω 40◦40◦
Figure 3. Chaser and target orbits, with Earth scaled to 80%
6
Figure 4. Relative position of target with respect to chaser spacecraft expressed in
inertial p(e)
ct (left) and chaser p(c)
ct (right) frames
Figure 5. Relative velocity V(e)
ct (left) and acceleration A(e)
ct (right) expressed in inertial frame
Figure 6. Relative errors in relative position expressed in chaser frame p(c)
ct when
comparing the output based on PoE and Curtis’s Matlab library,26 simulated for 60
orbital periods of chaser
7
CONCLUSIONS AND FUTURE WORK
This work approached the relative orbital motion problem using orbital differences within the
PoE framework. The relative position and orientations were described using SE (3) configuration
matrices, and relative velocity and acceleration were developed using spatial twist and spatial ac-
celeration. The presented approach was validated by comparing the output of the PoE method with
the classical approach outlined in Curtis’s work. The results of the comparison demonstrate that the
proposed method is in good agreement with traditional methods.
The main advantage of using the PoE-orbital mechanics framework presented in this investigation
lies in its unified approach of describing both the relative position and orientation using a single
formula. This elegant and concise formulation comes from the existence of an analytical inverse of
the SE (3) configuration matrices (e.g. T−1
eo ). Furthermore, because PoE is inherently a geometric
approach, singularities within the equations of motion or its parameterization do not exist.
Lastly, the presented PoE-orbital mechanics framework allows all orbital elements to vary with
respect to time while still being able to calculate the state of a spacecraft. In future research, it is
planned to investigate the utility of this framework when modeling different sources of orbit decay
and perturbations, such as J2 perturbations.
8
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