Conference PaperPDF Available

Computation of Relative Orbital Motion using Product of Exponentials Mapping



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.
AAS 22-802
Aryslan Malik*
, David Canales
, Taylor Yow
, Daniel Posada§
, Christopher W.
, 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.
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 is 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,
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.
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)
Meo =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
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
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:
+ 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),
Orbit Plane Perigee
Equatorial Plane
Ascending Node
Figure 1. Representation of the different reference frames used in this study.
The adjoint operator is also defined as,
[ˆp]R R ,(9)
with [ˆ
·]being the skew-symmetric operator defined in Eq. (10):
[ˆp] =
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= AdT1
Vs=03×3ReoVb=Reo vbR3.(13)
Taking the inertial derivative of the position vector twice yields the inertial acceleration found in
Eq. 14.
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,
or from spatial twist and inertial velocity,
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.
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
Inertial Frame
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.
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
RSO(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 TSE(3) is also a transformation matrix:
T1=R p
0 11
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
0 1 ="Rct p(c)
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:
ct =VtVcωc×(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:
ct =RT
ct (24)
Secondly, the relative accelerations may be computed as follows:
ct =AtAc2ωc×(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:
ct =RT
ct (26)
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
Figure 3. Chaser and target orbits, with Earth scaled to 80%
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
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. T1
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.
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... This type of information is essential for estimating the attitude of a target during the monitoring and capture phases, as well as for controlling the object during the post-capture phase. [9]. ...
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This paper presents a trajectory generation algorithm for multibody robotic systems based on the Product of Exponentials (PoE) formulation, also known as screw theory. A PoE formulation is first developed to model the kinematics and dynamics of a multibody robotic manipulator with 7 revolute joints and an end effector. An inverse kinematic algorithm based on the Newton-Raphson iterative method is then applied to generate constrained joint-space trajectories corresponding to straight-line motion of the end effector in Cartesian space with finite jerk. Derivatives of these joint-space trajectories are computed using Bézier curves, which ensures dynamically feasible trajectories. A novel method of Mean Arctangent Absolute Percentage Error (MAAPE) is then used to check accuracy of the derivatives of joint-space trajectories. The Newton-Euler recursive algorithm is then implemented to compute the inverse dynamics, which generates the joint torques required to achieve the reference trajectories. These torques are then incorporated into a closed-loop control algorithm, and simulation studies are performed using a dynamic model of the robotic system. The simulation results demonstrate that the proposed approach is able to successfully generate constrained trajectories, the MAAPE shows that lower order Bézier curves approximate the joint-space derivatives with the same accuracy as higher order polynomials, and the closed-loop controller provides accurate trajectory tracking subject to model uncertainties. This algorithm can be used to generate trajectories for robotic arms performing spacecraft servicing missions, because it also takes into account the variable gravity vector. The significant contribution of this method is integration of all of these techniques applied to a multibody robotic system.
Purpose Developing general closed-form solutions for six-degrees-of-freedom (DOF) serial robots is a significant challenge. This paper thus aims to present a general solution for six-DOF robots based on the product of exponentials model, which adapts to a class of robots satisfying the Pieper criterion with two parallel or intersecting axes among its first three axes. Design/methodology/approach The proposed solution can be represented as uniform expressions by using geometrical properties and a modified Paden–Kahan sub-problem, which mainly adopts the screw theory. Findings A simulation and experiments validated the correctness and effectiveness of the proposed method (general resolution for six-DOF robots based on the product of exponentials model). Originality/value The Rodrigues rotation formula is additionally used to turn the complex problem into a solvable trigonometric function and uniformly express six solutions using two formulas.
Lie-theoretic methods provide an elegant way to formulate many problems in robotics, and the tutorial by Park et al. (2018, “Geometric Algorithms for Robot Dynamics: A Tutorial Review,” ASME Appl. Mech. Rev., 70(1), p. 010803) is simultaneously a complete and concise introduction to these methods as they pertain to robot dynamics. The central reason why Lie groups are a natural mathematical tool for robotics is that rigid-body motions and pose changes can be described as Lie groups, and allow phenomena including robot kinematics and dynamics to be formulated in elegant notation without introducing superfluous coordinates. The emphasis of the tutorial by Park et al. (2018, “Geometric Algorithms for Robot Dynamics: A Tutorial Review,” ASME Appl. Mech. Rev., 70(1), p. 010803) is robot dynamics from a Lie-theoretic point of view. Newton–Euler and Lagrangian formulation of robot dynamics algorithms with O(n) complexity were formulated more than 35 years ago using recurrence relations that use the serial structure of manipulator arms. This was done without using the knowledge of Lie theory. But issues such as why the x terms in rigid-body dynamics appear can be more easily understood in the context of this theory. The authors take great efforts to be understandable by nonexperts and present extensive references to the differential-geometric and Lie-group-centric formulations of manipulator dynamics. In the discussion presented here, connections are made to complementary methods that have been developed in other bodies of literature. This includes the multibody dynamics, geometric mechanics, spacecraft dynamics, and polymer physics literature, as well as robotics works that present non-Lie-theoretic formulations in the context of highly parallelizable algorithms.
We provide a tutorial and review of the state-of-the-art in robot dynamics algorithms that rely on methods from differential geometry, particularly the theory of Lie groups. After reviewing the underlying Lie group structure of the rigid-body motions and the geometric formulation of the equations of motion for a single rigid body, we show how classical screw-theoretic concepts can be expressed in a reference frame-invariant way using Lie-theoretic concepts and derive recursive algorithms for the forward and inverse dynamics and their differentiation. These algorithms are extended to robots subject to closed-loop and other constraints, joints driven by variable stiffness actuators, and also to the modeling of contact between rigid bodies. We conclude with a demonstration of how the geometric formulations and algorithms can be effectively used for robot motion optimization.
A complete, stand-alone text for this core aerospace engineering subject * NEW: updated throughout, with new coverage of perturbations, Lambert's problem, attitude dynamics, and techniques for numerically integrating orbits * NEW: more examples and homework problems, more Matlab algorithms * NEW: improved support material, including instructor solutions manual and lecture PowerPoint slides. * NEW: Reorganized and improved discusions of coordinate systems, new discussion on perturbations and quarternions * NEW: Increased coverage of attitude dynamics, including new Matlab algorithms and examples in chapter 10 * New examples and homework problems * Highly illustrated and fully supported with downloadable MATLAB algorithms for project and practical work; fully worked examples throughout; extensive homework exercises; Instructor's Manual and lecture slides.