In this chapter, we reveal a dual-tensor-based procedure to obtain exact expressions for the six degree of freedom (6-DOF) relative orbital law of motion in the specific case of two Keplerian confocal orbits. The result is achieved by pure analytical methods in the general case of any leader and deputy motion, without singularities or implying any secular terms. Orthogonal dual tensors play a very important role, with the representation of the solution being, to the authors’ knowledge, the shortest approach for describing the complete onboard solution of the 6-DOF orbital motion problem. The solution does not depend on the local-vertical–local-horizontal (LVLH) properties involves that is true in any reference frame of the leader with the origin in its mass center. A representation theorem is provided for the full-body initial value problem. Furthermore, the representation theorems for rotation part and translation part of the relative motion are obtained.
- relative orbital motion
- full body problem
- dual algebra
- Lie group
- Lie algebra
- closed form solution
The relative motion between the leader and the deputy in the relative motion is a six-degrees-of-freedom (6-DOF) motion engendered by the joining of the relative translational motion with the rotational one. Recently, the modeling of the 6-DOF motion of spacecraft gained a special attention [1, 2, 3, 4, 5], similar to the controlling the relative pose of satellite formation that became a very important research subject [6, 7, 8, 9, 10]. The approach implies to consider the relative translational and rotational dynamics in the case of chief-deputy spacecraft formation to be modeled using vector and tensor formalism.
In this chapter we reveal a dual algebra tensor based procedure to obtain exact expressions for the six D.O.F relative orbital law of motion for the case of two Keplerian confocal orbits. Orthogonal dual tensors play a very important role, the representation of the solution being, to the authors’ knowledge, the shortest approach for describing the complete onboard solution of the six D.O.F relative orbital motion problem. Because the solution does not depend on the LVLH properties involves that is true in any reference frame of the Leader with the origin in its mass center. To obtain this solution, one has to know only the inertial motion of the Leader spacecraft and the initial conditions of the deputy satellite in the local-vertical-local-horizontal (LVLH) frame. For the full body initial value problem, a general representation theorem is given. More, the real and imaginary parts are split and representation theorems for the rotation and translation parts of the relative orbital motion are obtained. Regarding translation, we will prove that this problem is super-integrable by reducing it to the classic Kepler problem.
The chapter is structured as following. The second section is dedicated to the rigid body motion parameterization using orthogonal dual tensors, dual quaternions and other different vector parameterization. The Poisson-Darboux problem is extended in dual Lie algebra. In the third section, the state equations for a rigid body motion relative to an arbitrary non-inertial reference frame are determined. Using the obtained result, in the fourth section, the representation theorem and the complete solution for the case of onboard full-body relative orbital motion problem is given. The last section is designated to the conclusions and to the future works.
2. Rigid body motion parameterization using dual Lie algebra
The key notions that will be presented in this section are tensorial, vectorial and non-vectorial parameterizations that can be used to properly describe the rigid-body motion. We discuss the properties of proper orthogonal dual tensorial maps. The proper orthogonal tensorial maps are related with the skew-symmetric tensorial maps via the Darboux–Poisson equation. Orthogonal dual tensorial maps are a powerful instrument in the study of the rigid motion with respect to an inertial and noninertial reference frames. More on dual numbers, dual vectors and dual tensors can be found in [2, 16, 17, 18, 19, 20, 21, 22, 23].
2.1. Isomorphism between Lie group of the rigid displacements and Lie group of the orthogonal dual tensors
Let the orthogonal dual tensor set be denoted by.
where is the set of special orthogonal dual tensors and is the unit orthogonal dual tensor.
Theorem 1. (Structure Theorem). For any a unique decomposition is viable
where and are called structural invariants, , .
Taking into account the Lie group structure of and the result presented in previous theorem, it can be concluded that any orthogonal dual tensor can be used globally parameterize displacements of rigid bodies.
The parameters and are called the natural invariants of . The unit dual vector gives the Plücker representation of the Mozzi-Chalses axis [16, 24] while the dual angle contains the rotation angle and the translated distance .
The Lie algebra of the Lie group is the skew-symmetric dual tensor set denoted by , where the internal mapping is .
The link between the Lie algebra , the Lie group , and the exponential map is given by the following.
Theorem 3. The mapping is well defined and surjective.
Any screw axis that embeds a rigid displacement is parameterized by a unit dual vector, whereas the screw parameters (angle of rotation around the screw and the translation along the screw axis) is structured as a dual angle. The computation of the screw axis is bound to the problem of finding the logarithm of an orthogonal dual tensor , that is a multifunction defined by the following equation:
and is the inverse of (Eq. (4)).
From Theorem 2 and Theorem 3, for any orthogonal dual tensor , a dual vector is computed, represents the screw dual vector or Euler dual vector (that includes the screw axis and screw parameters) and the form of implies that . The types of rigid displacements that is parameterized by the Euler dual vector as below:
roto-translation if ;
pure translation if ;
pure rotation if .
Also, , Theorem 2 and Theorem 3 can be used to uniquely recover the screw dual vector , which is equivalent with computing .
Theorem 4. The natural invariants can be used to directly recover the structural invariants and from (Eq. (2)):
Theorem 5. (Isomorphism Theorem): The special Euclidean group and are connected via the isomorphism of the Lie groups
where , ,
Proof. For any , the map defined in (Eq. (7)) yields
Let . Based on Theorem 1, which ensures a unique decomposition, we can conclude that the only choice for , such that is . This underlines that is a bijection and keeps all the internal operations.
Remark 1: The inverse of is
2.2. Dual tensor-based parameterizations of rigid-body motion
The Lie group admits multiple parameterization and few of them will be discussed in this section.
2.2.1. The exponential parameterization (the Euler dual vector parameterization)
If , then we can construct the Euler dual vector (screw dual vector) which combined with Theorem 2 and Theorem 3 lead to
2.2.2. Dual quaternion parameterization
One of the most important non-vectorial parameterizations for the orthogonal dual tensor is given by the dual quaternions [20, 21]. A dual quaternion can be defined as an associated pair of a dual scalar quantity and a free dual vector:
The set of dual quaternions will be denoted and is organized as a -module of rank 4, if dual quaternion addition and multiplication with dual numbers are considered.
The product of two dual quaternions and is defined by
From the above properties, results that the -module becomes an associative, non-commutative linear dual algebra of rank 4 over the ring of dual numbers. For any dual quaternion defined by (Eq. (12)), the conjugate denoted by and the norm denoted by can be computed. For , any dual quaternion is called unit dual quaternion. Regarded solely as a free -module, contains two remarkable sub-modules: and . The first one composed from pairs , isomorphic with , and the second one, containing the pairs , isomorphic with . Also, any dual quaternion can be written as , where and , or , where are real quaternions. The scalar and the vector parts of a dual unit quaternion are also known as dual Euler parameters .
where and . This representation is the quaternionic counterpart to (Eq. (2)). Also a dual number and a unit dual vector exist so that:
Remark 2: The mapping , is well defined and surjective.
Remark 3: The dual unit quaternions set , by the multiplication of dual quaternions, is a Lie group with being it’s associated Lie algebra (with the cross product between dual vectors as the internal operation).
Using the internal structure of any element from the following theorem is valid:
Theorem 6. The Lie groups and are linked by a surjective homomorphism
Proof. Taking into account that any can be decomposed as in (Eq. (15)), results that . This shows that relation (Eq. (16)) is well defined and surjective. Using direct calculus, we can also acknowledge that .
An important property of the previous homomorphism is that for and we can associate the same orthogonal dual tensor, which shows that (Eq. (16)) is not injective and is a double cover of .
2.2.3. N-order modified fractional Cayley transform for dual vectors
Next, we present a series of results that are the core of our research. These results are obtained after using a set of Cayley transforms that are different than the ones already reported in literature [17, 25, 26, 27].
Theorem 7. The fractional order Cayley map
is well defined and surjective.
Proof. Using direct calculus results that and . The surjectivity is proved by the following theorem.
Theorem 8. The inverse of the previous fractional order Cayley map, is a multifunction with n branches given by
Remark 4: If then is the parameterization of a pure rotation about an axis which does not necessarily pass through the origin of reference system. Meanwhile, if the mapping is the parameterization of a pure translation. Otherwise, is the parameterization of roto-translation.
Taking into account that a dual number and a dual vector exist in order to have
from (Eq. (18)), results that:
The previous equation contains both the principal parameterization , which is the higher order Rodrigues dual vector, while for the dual vectors are the shadow parameterization  that can be used to describe the same pose. Based on and , results that .
If then , which allows the avoidance of any singularity of type .
Theorem 9. If is the parameterization of a displacement obtained from (Eq. (20)), then
In order to evaluate the usefulness of the iterative expressions, we provide the second to third order polynomials and the resulting dual quaternions and dual orthogonal tensors:
2.3. Poisson-Darboux problems in dual Lie algebra and vector parameterization
Consider the functions and to be the parametric equations of any rigid motion. Thus, any rigid motion can be parameterized by a curve in where , where t is time variable. Let embed the Plücker coordinates of a line feature at . At a time stamp the line is transformed into:
Theorem 10. In a general rigid motion, described by an orthogonal dual tensor function , the velocity dual tensor function defined as
is expressed by
Let , then , equivalent with , which shows that .
The dual vector is called dual angular velocity of the rigid body and has the form:
where is the instantaneous angular velocity of the rigid body and represents the linear velocity of the point of the body that coincides instantaneously with the origin of the reference frame. The pair () is usually refereed as the twist of the rigid body.
2.3.1. Poisson-Darboux equation in dual Lie algebra
Theorem 11. For any continuous function a unique dual tensor exists so that
Due to the fact that orthogonal dual tensor completely models the six degree of freedom motion, we can conclude that the Theorem 11 is the dual form of the Poisson-Darboux problem  for the case when the rotation tensor is computed from the instantaneous angular velocity. So, in order to recover , it is necessary to find out how the dual angular velocity vector behaves in time and also the value of at time .
The dual tensor can be derived from , when is positioned in space, or from , which denotes the dual angular velocity vector to be positioned in the rigid body.
Remark 6. The dual angular velocity vector positioned in the rigid body can be recovered from , thus transforming (Eq. (32)) into:
The tensorial (Eq. (32)) and (Eq. (33)) are equivalent with 18 scalar differential equations. The previous parameterizations of the orthogonal dual tensors allow us to determine some solutions of smaller dimension in order to solve the dual Poisson- Darboux problem.
2.3.2. Kinematic equation for Euler dual vector parameterization
where, and is the following dual tensor:
The representation of the Poisson-Darboux problem from (Eq. (33)) is equivalent to
2.3.3. Kinematic equation for high order Rodrigues dual vector parameterization
where , and is the following dual tensor :
3. Rigid body motion in arbitrary non-inertial frame revised
To the author’s knowledge, in the field of astrodynamics there aren’t many reports on how the motion of rigid body can be studied in arbitrary non-inertial frames. Next, we proposed a dual tensors based model for the motion of the rigid body in arbitrary non-inertial frame. The proposed method eludes the calculus of inertia forces that contributes to the rigid body relative state. So, the free of coordinate state equation of the rigid body motion in arbitrary non-inertial frame will be obtained.
Let and be the dual orthogonal tensors which describe the motion of two rigid bodies relative to the inertial frame.
If is the orthogonal dual tensor which embeds the six degree of freedom relative motion of rigid body C relative to rigid body D, then:
Let denote the dual angular velocity of the rigid body C and the dual angular velocity of the rigid body D, both being related to inertial reference frame. In the followings, the inertial motion of the rigid body C is considered to be known. If is the dual angular velocity of the rigid body D relative to the rigid body C, then, conforming with (Eq. (42)):
Considering being the dual angular velocity vector of the rigid body D in the body frame, the dual form of the Euler equation given in  results that:
In (Eq. (44)) , where the force applied in the mass center and is the torque. Also in (Eq. (44)), represents the inertia dual operator, which is given by , where is the inertia tensor of the rigid body D related to its mass center and is the mass of the rigid body D. Combining with (Eq. (44)) results:
Taking into account that , the dual angular velocity vector can be computed from
this through differentiation gives:
If the previous equation is multiplied by , then
which combined with generates:
After a few steps, (Eq. (49)) is transformed into
which combined with (Eq. (45)) gives:
Because , the final equation is:
is a compact form which can be used to model the six D.O.F relative motion problem. In the previous equation the state of the rigid body D in relation with the rigid body C is modeled by the dual tensor and the dual angular velocities field . This initial value problem can be used to study the behavior of the rigid body D in relation with the frame attached to the rigid body C. In (Eq. (53)), all the vectors are represented in the body frame of C, which shows that the proposed solution is onboard and has the property of being coupled in and .
Next, we present a procedure that allows the decoupling of the proposed solution.
In order to describe the solution to (Eq. (53)), we consider the following change of variable:
This change of variable leads to . The result is equivalent with or
Where is the dual torque related to the mass center in the body frame of the rigid body D and . (Eq. (56)) is a dual Euler fixed point classic problem.
For any , the solution of (Eq. (53)) emerges from
Making use of (Eq. (54)), results that . If operator used, the previous calculus is transformed into . After multiplying the last expression by , we obtain the initial value problem:
Let be the unique solution of the following Poisson-Darboux problem:
Considering , a representation theorem of the solution of (Eq. (53)) can be formulated.
where , .
Different representations can be considered for the problem (60).
Using dual quaternion representation , (Eq. (60)) is equivalent with the following one:
For the n-th order of Cayley transform based representation , the (Eq. (60)) becomes:
where the tensor is:
Different particular cases can be analyzed for the (Eq. (62)):
Let be the Rodrigues dual vector for n = 1:
Let be the modified Rodrigues dual vector (Wiener-Milenkovic dual vector) for n = 2:
The initial value problem (62) is a minimum parameterization of the six degrees of freedom motion problem. The singularity cases can be avoided using the shadow parameters of the n-th order Modified Rodrigues Parameter dual vector.
4. A dual tensor formulation of the six degree of freedom relative orbital motion problem
The results from the previous paragraphs will be used to study the six degrees of freedom relative orbital motion problem.
The relative orbital motion problem may now be considered classical one considering the many scientific papers written on this subject in the last decades. Also, the problem is quite important knowing its numerous applications: rendezvous operations, spacecraft formation flying, distributed spacecraft missions [3, 4, 6, 7, 8, 9, 10].
The model of the relative motion consists in two spacecraft flying in Keplerian orbits due to the influence of the same gravitational attraction center. The main problem is to determine the pose of the Deputy satellite relative to a reference frame originated in the Leader satellite center of mass. This non-inertial reference frame, known as “LVLH (Local-Vertical-Local- Horizontal)” is chosen as following: the axis has the same orientation as the position vector of the Leader with respect to an inertial reference frame with the origin in the attraction center; the orientation of the is the same as the Leader orbit angular momentum; the axis completes a right-handed frame. The angular velocity of the LVLH is given by vector , which has the expression:
where vector is
where is the conic parameter, is the angular momentum of the Leader, being the true anomaly and is the eccentricity of the Leader.
We propose dual tensors based model for the motion and the pose for the mass center of the Deputy in relation with LVLH. Both, the Leader satellite and the Deputy satellite can be considered rigid bodies.
Furthermore, the time variation of is:
In order to a more easy to read list of notations, for there will be used the followings:
where is the unity vector of the X-axis from LVLH.
The full-body relative orbital motion is described by the (Eq. (53)) where the dual angular velocity of the Chief satellite is:
and the dual torque related to the mass center of Deputy satellite is:
In (71), we’ve noted and
Theorem 13. (Representation Theorem of the full body relative orbital motion). The solution of (Eq. (53)) results from the application of the tensor from (Eq. (71)) to the solution of the classical dual Euler fixed point problem (60).
4.1. The rotational and translational parts of the relative orbital motion
The complete solution of (Eq. (53)) can be recovered in two steps.
Consider first the real part of (Eq. (53)). This leads to an initial value problem:
which has the solution , the real tensor being the attitude of Deputy in relation with LVLH. In (Eq. (72)), is the angular velocity of the Deputy in relation with LVLH, is the angular velocity of LVLH, is the resulting torque of the forces applied on the Deputy in relation with is mass center, is the inertia tensor of the Deputy in relation with its mass center. The angular velocity of Deputy in respect to LVLH at time is denoted with and is the orientation of Deputy in respect to LVLH at time .
Consider now the dual part of (Eq. (53)). Taking into account the internal structure of , which is given by (Eq. (2)), after some basic algebraic calculus we obtain a second initial value problem that models the translation of the Deputy satellite mass center with respect to the LVLH reference frame:
where is the gravitational parameter of the attraction center and represent the relative position and relative velocity vectors of the mass center of the Deputy spacecraft with respect to LVLH at the initial moment of time .
Based on the representation theorem 12, the following theorem results.
where and are the solutions of the the classical Euler fixed point problem and, respectively, Kepler’s problem:
and is given by (Eq. (65)).
The relative velocity of the translation motion may be computed as:
This result shows a very interesting property of the translational part of the relative orbital motion problem (73). We have proven that this problem is super-integrable by reducing it to the classic Kepler problem [11, 12, 31, 32]. The solution of the translational part of the relative orbital motion problem is expressed thus:
The chapter proposes a new method for the determination of the onboard complete solution to the full-body relative orbital motion problem.
Therefore, the isomorphism between the Lie group of the rigid displacements and the Lie group of the orthogonal dual tensors is used. It is obtained a Poisson-Darboux like problem written in the Lie algebra of the group , an algebra that is isomorphic with the Lie algebra of the dual vectors. Different vectorial and non-vectorial parameterizations (obtained with n-th order Cayley-like transforms) permit the reduction of the Poisson-Darboux problem in dual Lie algebra to the simpler problems in the space of the dual vectors or dual quaternions.
Using the above results, the free of coordinate state equation of the rigid body motion in arbitrary non-inertial frame is obtained.
The results are applied in order to offer a coupled (rotational and translational motion) state equation and a representation theorem for the onboard complete solution of full body relative orbital motion problem. The obtained results interest the domains of the spacecraft formation flying, rendezvous operation, autonomous mission and control theory.
|V3||real vectors set|
|V¯3||dual vectors set|
|V3R||time depending real vectorial functions|
|V¯3R||time depending dual vectorial functions|
|a∼¯||skew-symmetric dual tensor corresponding to the dual vector a¯|
|hc||specific angular momentum of the leader satellite|
|LV¯3V¯3||dual tensor set|
|R||real numbers set|
|R¯||dual numbers set|
|SO3||orthogonal real tensors set|
|SO¯3||orthogonal dual tensor set|
|SO3R||time depending real tensorial functions|
|SO¯3R||time depending dual tensorial functions|
|U||unit quaternions set|
|U¯||unit dual quaternions set|