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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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