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ISRN Mathematical Physics
VolumeΒ 2012Β (2012), Article IDΒ 467520, 21 pages
http://dx.doi.org/10.5402/2012/467520
Research Article

Rigid Body Trajectories in Different 6D Spaces

1School of Systems Engineering, University of Reading, Reading RG6 6AY, UK
2Department of Mechanical Engineering, University of Strathclyde, Glasgow G1 1XQ, UK

Received 26 April 2012; Accepted 4 June 2012

Academic Editors: G.Β Goldin, D.Β Ida, and J. A.Β Nieto

Copyright Β© 2012 Carol Linton et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The objective of this paper is to show that the group 𝑆𝐸(3) with an imposed Lie-Poisson structure can be used to determine the trajectory in a spatial frame of a rigid body in Euclidean space. Identical results for the trajectory are obtained in spherical and hyperbolic space by scaling the linear displacements appropriately since the influence of the moments of inertia on the trajectories tends to zero as the scaling factor increases. The semidirect product of the linear and rotational motions gives the trajectory from a body frame perspective. It is shown that this cannot be used to determine the trajectory in the spatial frame. The body frame trajectory is thus independent of the velocity coupling. In addition, it is shown that the analysis can be greatly simplified by aligning the axes of the spatial frame with the axis of symmetry which is unchanging for a natural system with no forces and rotation about an axis of symmetry.

1. Introduction

The motion of a rigid body in Euclidean space 𝐸3 consists of 3 translations and 3 rotations about the centre of mass. The ways in which these are combined determine the calculated trajectory of the body. This paper considers several methods of determining the corresponding velocities and the resultant trajectories and investigates the consequences of each method.

Previous work on Lie groups has used the special Euclidean group 𝑆𝐸(3) with no imposed structure so that it is the semidirect product of the translations in ℝ3 and rotations in the group 𝑆𝑂(3). The mapping between the groups used by Holm [1] and Marsden and Ratiu [2] is expressed in two formats in this paper: as the mapping itself and as the integration using 𝑆𝐸(3). This method gives the trajectory in the body frame, which cannot be used to determine the trajectory in the spatial frame needed for many applications. The body frame trajectory is the independent of the velocity coupling. There is no natural way of weighting the rotations and translations to measure the distance (in 6 dimensions) along a trajectory using the semidirect product.

There is no bi-invariant Riemannian metric on 𝑆𝐸(3). There are natural metrics on 𝑆𝑂(4) and 𝑆𝑂(1,3)β€”the trace form, which can be inherited by 𝑆𝐸(3) with the appropriate scaling. Etzel and McCarthy [3] used a metric on 𝑆𝑂(4) as a model for a metric on SE(3). Larochelle et al. [4] projected 𝑆𝐸(3) onto 𝑆𝑂(4) to obtain a metric. In this paper, the linear displacements are scaled so that they are small compared with a unit hypersphere. This enables 𝑆𝐸(3) to be projected vertically onto 𝑆𝑂(4) and 𝑆𝑂(1,3). This projection is extended by imposing the Lie-Poisson structure on 𝑆𝐸(3) as mentioned in [5].

The 6D Lie groups 𝑆𝑂(4), 𝑆𝑂(1,3), and 𝑆𝐸(3) with imposed Lie-Poisson structure are compared and shown to result in related trajectories, which approximate to the same values for small linear displacements. The trajectories are in a fixed frame which is the requirement for planning and controlling the motions.

Four methods of combining the translations and rotations are investigated:(i)A semidirect productβ€”where the rotation changes the body frame, but not the linear velocity itself: that is, there is no coupling of the angular and translational velocities.(ii)The special Euclidean group 𝑆𝐸(3) with an imposed Lie-Poisson structure, where the rotation induces a change in the linear velocity to conserve angular momentum.(iii)The rotation group 𝑆𝑂(4) which can approximate the previous results if the linear displacements are scaled down.(iv)The Lorentz group 𝑆𝑂(1,3): the latter is included as a continuation from the other groups, and not as a practical alternative.Finding the trajectory of a rigid body has 2 steps:(1)determining how the velocity changes over time as the velocity components couple together and(2)integrating the velocity function down to the base manifold to give the trajectory.Both these depend on the group, for example, is the space that the group represents flat, convex or concave? The formulas for the general case are derived, but the examples are based on the simplest case with no external forces and rotation about an axis of symmetry.

Any combinations of rotations can be represented by rotation about a single axis of rotation. If this axis is also an axis of symmetry, the rotational axis does not change, and there is no precession. By choosing the axes of the fixed frame appropriately, the initial motion of anybody can be defined as an initial rotation about one axis, with some linear motion along and some motion perpendicular to that axis. Details are provided in the Appendix. In a natural system with no forces, the angular momentum remains constant, and the resulting trajectories determined using the various methods can be compared. The linear displacement of the centre of mass is used. The initial velocities used in the examples are 𝑣0=βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘£π‘₯𝑣𝑦𝑣cos(𝑓)π‘¦βŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦sin(𝑓),πœ™0=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦00000βˆ’π‘€0𝑀0,(1.1) where 𝑓 is a phasing constant. Using the rotation identified in the Appendix, this can be generalized to any initial conditions in a system with no forces so long as the rotation is about an axis of symmetry.

Section 2 introduces the basic ideas of geometric control theory and applies them to the rotation group 𝑆𝑂(3). The linear motion is incorporated using the semidirect product. Section 3 discusses the Lie-Poisson structure which is implicit for the rotation group 𝑆𝑂(4) and the Lorentz group 𝑆𝑂(1,3) and can be imposed for 𝑆𝐸(3). With that structure, the differential equations of motion can be derived from the Hamiltonian. The trajectories are found in Section 4. The results for each group are compared. A comparison of the results from 𝑆𝑂(4) and the structured 𝑆𝐸(3) in Section 5 show that they are interchangeable for small displacements. The strength of the coupling between the angular and translational velocities tends to the same value as the accuracy of the scaling is improved. The final Section 6 compares the different methods of combining linear and rotational motions, and when the methods fail.

2. Geometric Control Theory and Notation

This section provides the basic ideas of geometric control theory. A fuller explanation is available in many texts such as [6]. The rotations and translations are considered separately. This is used as an opportunity to introduce the necessary notation for combining them in the following sections.

2.1. Lie Theory

A trajectory is represented by a matrix 𝑔(𝑑)∈𝐺, where 𝐺 is the matrix Lie group which reflects the structure or shape of the space on which the trajectory lies. In Lie theory, the trajectory is pulled back to the origin by the action of π‘”βˆ’1(𝑑). The tangent field at the origin π‘‹βˆˆπ”€ (where 𝔀 is the Lie algebra) determines the trajectory through the expression π‘”βˆ’1(𝑑)𝑑𝑔𝑑𝑑(𝑑)=𝑋.(2.1) One finds that 𝑔(𝑑)=exp(𝑋𝑑)if𝑋istimeindependent.(2.2) If 𝑋=𝑋(𝑑), time dependent, an analytic solution is difficult to find in most cases, so the forward Euler method is used in this paper to demonstrate the general form of the solutions as follows: 𝑔𝑛+1=𝑔𝑛𝑋exp𝑛𝑠,(2.3) where 𝑠 is the step length, and 𝑛 is the step number so that 𝑋𝑛=𝑋(𝑑𝑛) is the tangent field at time 𝑑𝑛=𝑛𝑠. The trajectory started at 𝑔(0)=𝐼.

Fact. The forward Euler method uses 𝑔𝑛+1βˆ’π‘”π‘›π‘ =𝑑𝑔𝑛𝑑𝑑=𝑔𝑛𝑋𝑛(2.4) so that 𝑔𝑛+1=𝑔𝑛1+𝑠𝑋𝑛≃𝑔𝑛exp𝑠𝑋𝑛.(2.5) Alternatively, assume that 𝑋𝑛 is constant between times 𝑑𝑛 and 𝑑𝑛+1 so that the incremental motion is given by 𝛿𝑔𝑛=exp(𝑋𝑛𝑠). This increment is applied to the previous configuration using 𝑔𝑛+1=𝑔𝑛𝛿𝑔𝑛 which gives (2.3).

2.2. Rotations

Rotation in 3 dimensions is used to demonstrate the ideas of the previous section. A body rotating in space about its fixed centre of mass can have angular velocity {𝑀𝑖} for π‘–βˆˆ{1,2,3} about three orthogonal axes {𝑒𝑖}. The resultant velocity 𝑋 can be written in both coordinate and matrix forms as 𝑋=3𝑖=1𝑀𝑖𝑒𝑖=⎑⎒⎒⎒⎒⎣0βˆ’π‘€3𝑀2𝑀30βˆ’π‘€1βˆ’π‘€2𝑀10⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦.(2.6) From this notation, the base matrices {𝑒𝑖} of the Lie algebra 𝔰𝔬(3) can be extracted.

Any rotation in Euclidean space can be represented by a rotation about a single axis, so, by choosing the axes of the body frame appropriately, all initial rotation can be represented about the 𝑒1 axis. The Appendix gives the required rotation of the body frame to achieve this. The initial rotational velocity can then be written as πœ™0=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦00000βˆ’π‘€0𝑀0.(2.7) In a natural motion of a body rotating about an axis of symmetry, there are no forces and no coupling with other motions. (The discussion in Section 3 can be applied to 𝑆𝑂(3) to confirm this, with π‘œ2=π‘œ3). The angular velocity is unchanging. The axis of rotation does not change (no precession). The attitude of the body Ξ¦(𝑑)βˆˆπ‘†π‘‚(3) is the solution of Ξ¦βˆ’1(𝑑)𝑑Φ(𝑑)𝑑𝑑=πœ™0.(2.8) The resulting attitude at time 𝑑 is ξ€·πœ™Ξ¦(𝑑)=exp0𝑑=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦1000cos(𝑀𝑑)βˆ’sin(𝑀𝑑)0sin(𝑀𝑑)cos(𝑀𝑑).(2.9)

2.3. Semidirect Product ℝ3⋉𝑆𝑂(3)

To represent a rigid body rotating about its centre of mass and moving through Euclidean space, the rotational and translational motions need to be combined. The semidirect product is used to represent the motion from the body frame prospective. The rotational motion has already been determined. The linear velocity can be written in matrix and coordinate representations of ℝ3 as 𝑣𝑣=1,𝑣2,𝑣3𝑇=3𝑖=1𝑣𝑖𝑒𝑖,(2.10) where here {𝑒𝑖} is the orthogonal basis in ℝ3.

The semidirect product enables the rotation to influence the linear motion. The two elements act on each other through an action ∘ defined by ξ€·π‘₯1,Ξ¦1ξ€Έβˆ˜ξ€·π‘₯2,Ξ¦2ξ€Έ=ξ€·π‘₯1+Ξ¦1π‘₯2,Ξ¦1β‹…Ξ¦2ξ€Έ(2.11) as used by Marsden [2, page 22] and Holm [1, page 110]. This rotates the body frame. It can also be written in matrix form as ⎑⎒⎒⎣π‘₯101Ξ¦1⎀βŽ₯βŽ₯⎦⎑⎒⎒⎣π‘₯102Ξ¦2⎀βŽ₯βŽ₯⎦=⎑⎒⎒⎣π‘₯101+Ξ¦1π‘₯2Ξ¦1Ξ¦2⎀βŽ₯βŽ₯⎦(2.12) which is the format of the Lie group 𝑆𝐸(3) with no structure imposed. The attitude Ξ¦(𝑑) is found using (2.1). The displacement π‘₯ at time 𝑑 is found by integration as ξ€œπ‘₯=𝑑𝑠=0Ξ¦(𝑠)𝑣(𝑠)𝑑𝑠,(2.13) where Ξ¦(𝑠) is the rotation achieved at time 𝑠 (since integration is the method of adding incremental changes in ℝ3). Alternatively the 2 velocity functions can be combined into one 4Γ—4 matrix to give the equation π‘”βˆ’1(𝑑)𝑑𝑔=⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦π‘‘π‘‘00𝑣(𝑑)πœ™(𝑑)(2.14) which has the solution 𝑆𝐸(3)βˆ‹π‘”(𝑑)=10π‘₯(𝑑)πœ™(𝑑)ξ€». The same result is obtained from the integration in (2.13).

In the simple example defined in expression (1.1), the initial translation velocity is 𝑣0=[𝑣π‘₯,𝑣𝑦cos(𝑓),𝑣𝑦sin(𝑓)]𝑇. In a natural motion of a body moving in Euclidean space, there are no forces and no coupling with other motions. The translational velocity is unchanging. The combined velocity function is βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘£π‘£β‹‰πœ™=π‘₯𝑣𝑦𝑣cos(𝑓)π‘¦βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦β‹‰βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£βŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦sin(𝑓)00000βˆ’π‘€0𝑀0,(2.15) and the configuration at time 𝑑 is βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘£π‘₯𝑑𝑣𝑦𝑀(𝑣sin(𝑀𝑑+𝑓)βˆ’sin(𝑓))π‘¦π‘€βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦β‹‰βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£βŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦(cos(𝑓)βˆ’cos(𝑀𝑑+𝑓))1000cos(𝑀𝑑)βˆ’sin(𝑀𝑑)0sin(𝑀𝑑)cos(𝑀𝑑).(2.16)

Thus, the semidirect product produces a trajectory as perceived in the body frame. Although the translational velocity is fixed at 𝑣0, and the perceived trajectory is curved. In the body frame, the velocity in the π‘¦βˆ’π‘§ plane is 𝑣𝑦 so, at time 𝑑, the origin 𝑂 is seen at an angle 𝑀𝑑 behind the body having traveled a distance 𝑣𝑦𝑑. The perceived radius of travel π‘Ÿ is given by π‘Ÿπ‘€π‘‘=𝑣𝑦𝑑(2.17) so the perceived radius is 𝑣𝑦/𝑀 as seen in Figure 1 and (2.16).

467520.fig.001
Figure 1: Actual motion in determined using semidirect product.

In a spatial frame, a rigid body rotating about its centre of mass and moving through Euclidean space does not move in a straight line but moves by spirally about an axis. If the rotation is about a fixed axis with constant translational speed, then the trajectory is a screw motion with constant pitch as proved by Chasles in his Screw Theory of motion. The angular momentum induces a change in the translational velocity.

In the semidirect product, the two velocity matrices were combined as ξ€Ί00π‘£πœ™ξ€» to calculate the trajectory, where 𝑣 is the 3Γ—1 column matrix of linear velocities, and πœ™ is the 3Γ—3 angular velocity matrix given above. In Section 4, the velocity matrix is taken as ⎑⎒⎒⎣0βˆ’πœ€π‘£π‘‡βŽ€βŽ₯βŽ₯βŽ¦π‘£πœ™(2.18) with πœ–βˆˆ{1,0,βˆ’1} for 𝔰𝔬(4),𝔰𝔒(3),𝔰𝔬(1,3), respectively. Other variations in the order of columns and rows are possible but produce equivalent results. Alternative values of πœ€ create nonclosed groups and are not considered here.

In the next section, a structure is imposed on these groups which creates a relationship between the rotational and linear elements. With the Lie-Poisson structure, the velocities interact, with rotations inducing changes in the linear motion. The motions are described in the spatial frame, rather than the body frame.

3. Structure of the Lie Algebras

In this section, the structure of the Lie groups is identified in terms of structure constants. This structure (or shape of the space) determines how motion in one direction influences motion in another, and how the motions add together to arrive at a configuration in space.(i)The rotation group 𝑆𝑂(4) has an obvious structure in that it represents rotations. The space has positive curvature which is related to the fixed Casimir βˆ‘πΆ=6𝑖=1𝑝2𝑖 where the {𝑝𝑖} are the momentum components. The group has a bilinear map and is a Poisson manifold.(ii)𝑆𝑂(1,3) has some similar characteristics. It is a space with negative curvature in some directions since one of the fixed Casimirs is βˆ‘πΆ=3𝑖=1𝑝2π‘–βˆ’βˆ‘6𝑖=4𝑝2𝑖.(iii)For the Lie group, 𝑆𝐸(3), a Lie-Poisson structure is imposed. The equivalent Casimir is βˆ‘πΆ=3𝑖=1𝑝2𝑖. There is thus no automatic relative weighting between the rotational and linear elements.More comparative data is provided by Jurdjevic [7]. In all cases, there is another fixed Casimir βˆ‘πΆ=3𝑖=1𝑝𝑖𝑝𝑖+3 which pairs the momentum types but adds no information about the relative weighting of the two types.

The base matrices for the 6-dimensional Lie algebras being considered can be seen from the equation 3𝑖=1𝑣𝑖𝑒𝑖+𝑀𝑖𝑒𝑖+3=⎑⎒⎒⎒⎒⎒⎒⎣0βˆ’πœ€π‘£1βˆ’πœ€π‘£2βˆ’πœ€π‘£3𝑣10βˆ’π‘€3𝑀2𝑣2𝑀30βˆ’π‘€1𝑣3βˆ’π‘€2𝑀10⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦(3.1) with πœ–βˆˆ{1,0,βˆ’1} for 𝔰𝔬(4),𝔰𝔒(3),𝔰𝔬(1,3), respectively. The orthogonal basis for the dual Lie algebra π”€βˆ— is {𝑒𝑖} given by 𝑒𝑖=𝕀𝑒𝑖, where 𝕀=𝐼4Γ—4, the unit matrix, is a bilinear form.

From these base matrices, the structure of the group is quantified in terms of the structure constants in Section 3.1. For a group with a Lie Poisson structure, these same constants provide an interaction between the functions on the algebra in Section 3.2. If those functions are the Hamiltonian and the momentum components, the equations of motion can be expressed in terms of the structure constants in Section 3.3. Finally in Section 3.4, the velocity matrix is found for the three Lie algebras.

3.1. Structure Constants

In many situations, the addition of two motions depends on the order in which they occur-rotate then move, or move then rotate. This is reflected in the nonassociative matrix multiplication: 𝐴𝐡≠𝐡𝐴 in most cases. Structure constants π‘π‘˜π‘–π‘— are used to describe this nonassociative action in a Lie algebra 𝔀. They are defined using the Lie bracket by (see [6, page 56]) π‘π‘˜π‘–π‘—π‘’π‘˜=𝑒𝑖,𝑒𝑗=π‘’π‘–π‘’π‘—βˆ’π‘’π‘—π‘’π‘–.(3.2) The value of the structure constants is easily shown by matrix multiplication to be, by using πœ–βˆˆ{1,0,βˆ’1} for 𝔰𝔬(4), 𝔰𝔒(3), and 𝔰𝔬(1,3), 𝑐315=𝑐126=𝑐234=𝑐342=𝑐645=𝑐153=𝑐456=𝑐261=𝑐564𝑐=1216=𝑐324=𝑐135=𝑐243=𝑐546=𝑐351=𝑐654=𝑐162=𝑐465𝑐=βˆ’1612=𝑐423=𝑐531𝑐=πœ–513=𝑐621=𝑐432=βˆ’πœ–.(3.3) They are the same for the dual algebra, π”€βˆ—.

3.2. Poisson Bracket

In order to develop the coordinate equations for these Lie groups, it is necessary to introduce the Poisson bracket and show that it describes the structure in the same way as the Lie bracket.

The canonical form of the Poisson bracket is (see [8, page 20] onwards) {𝐹,𝐸}=πœ•πΉπœ•π‘žπœ•πΈβˆ’πœ•π‘πœ•πΉπœ•π‘πœ•πΈπœ•π‘ž(3.4) with 𝐹(𝑝,π‘ž) and 𝐸(𝑝,π‘ž) being functions on the cotangent space with canonical coordinates (π‘ž,𝑝), representing position and momentum. To express this in other coordinates, write π‘ž=π‘ž(𝑧𝑖) and 𝑝=𝑝(𝑧𝑗) and get, using partial differentiation, {𝐹,𝐸}=πœ•πΉπœ•π‘§π‘–ξ€½π‘§π‘–,π‘§π‘—ξ€Ύπœ•πΈπœ•π‘§π‘—.(3.5) If these functions are pulled back to the origin, there is no positional dependence and so {𝐹,𝐸}=πœ•πΉπœ•π‘π‘–ξ€½π‘π‘–,π‘π‘—ξ€Ύπœ•πΈπœ•π‘π‘—.(3.6) For a Lie algebra with a Poisson structure, it can be shown that {𝑝𝑖,𝑝𝑗}=βˆ’π‘π‘˜π‘–π‘—π‘π‘˜ (see [8, page 50]), and the Poisson bracket describes the structure in a similar way to the Lie Bracket. Hence the Poisson relationship between any functions 𝐸 and 𝐹 on π”€βˆ— becomes {𝐸,𝐹}=βˆ’π‘π‘˜π‘–π‘—π‘π‘˜πœ•πΈπœ•π‘π‘–πœ•πΉπœ•π‘π‘—.(3.7)

3.3. Hamiltonian Flow

For a Poisson manifold, Bloch [9, page 121] defines 𝑋𝐻, the Hamiltonian vector field of 𝐻, as the unique vector field such that 𝑑𝑓𝑑𝑑=𝑋𝐻(𝑓)={𝑓,𝐻}βˆ€π‘“.(3.8) From (3.7) above, the Hamiltonian vector field is 𝑋𝐻=βˆ’π‘π‘˜π‘–π‘—π‘π‘˜πœ•π»πœ•π‘π‘—πœ•πœ•π‘π‘–(3.9) and, for all functions 𝑓, 𝑑𝑓𝑑𝑑=βˆ’π‘π‘˜π‘–π‘—π‘π‘˜πœ•π»πœ•π‘π‘—πœ•πœ•π‘π‘–π‘“.(3.10) With a Poisson structure naturally or imposed, the coordinate equation for Lie algebras is found by setting 𝑓=𝑝𝑖 for π‘–βˆˆ{1,2,3,4,5,6}̇𝑝𝑖=βˆ’π‘π‘˜π‘–π‘—π‘π‘˜πœ•π»πœ•π‘π‘—,(3.11) where {𝑝𝑖} are the components of momentum. The relationship with the velocities is 𝑝𝑖=π‘šπ‘–π‘£π‘– for π‘–βˆˆ{1,2,3} and 𝑝𝑖+3=π‘œπ‘–π‘€π‘–, where the linear velocities are {𝑣1,𝑣2,𝑣3}, the angular velocities are {𝑀1,𝑀2,𝑀3}, the moments of inertia are {π‘œ1,π‘œ2,π‘œ3}, and the added masses are {π‘š1,π‘š2,π‘š3} allowing for a different mass in each direction. The Hamiltonian for a natural motion, with no forces, is the kinetic energy so 1𝐻=23𝑖=1ξ€·π‘šπ‘–π‘£2𝑖+π‘œπ‘–π‘€2𝑖.(3.12)

3.4. Velocity Functions for the Groups

The velocity functions change with time in each direction as determined in this section and are shown in Table 1. The velocities for 𝐸3⋉𝑆𝑂(3) arise from the separate subgroups and were found earlier. For the Lie groups, the differential equations for the momentum components are found using (3.11), the Hamiltonian given above, and the structure constants found by matrix multiplication of the base matrices. First the Hamiltonian is written in terms of momentum components 1𝐻=23𝑖=1𝑝2π‘–π‘šπ‘–+π‘˜2π‘–π‘œπ‘–ξƒͺ,(3.13) where {π‘˜π‘–}={π‘œπ‘–π‘€π‘–}={𝑝𝑖+3} are the angular momentum components. The required differential equations for the momentum are, from (3.11) and the structure constants, π‘‘βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽπ‘π‘‘π‘‘1𝑝2𝑝3π‘˜1π‘˜2π‘˜3⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽβˆ’π‘3π‘˜2π‘œ2+πœ–π‘3π‘˜2π‘š3+𝑝2π‘˜3π‘œ3βˆ’πœ–π‘2π‘˜3π‘š2𝑝3π‘˜1π‘œ1βˆ’πœ–π‘3π‘˜1π‘š3βˆ’π‘1π‘˜3π‘š1+πœ–π‘1π‘˜3π‘œ1πœ–π‘2π‘˜1π‘š2βˆ’π‘2π‘˜1π‘œ1+𝑝1π‘˜2π‘œ2βˆ’πœ–π‘1π‘˜2π‘š1π‘˜2π‘˜3ξ€·π‘œ2βˆ’π‘œ3ξ€Έπ‘œ2π‘œ3+𝑝2𝑝3ξ€·π‘š2βˆ’π‘š3ξ€Έπ‘š2π‘š3π‘˜3π‘˜1ξ€·π‘œ3βˆ’π‘œ1ξ€Έπ‘œ1π‘œ3+𝑝1𝑝3ξ€·π‘š3βˆ’π‘š1ξ€Έπ‘š1π‘š3π‘˜1π‘˜1ξ€·π‘œ1βˆ’π‘œ2ξ€Έπ‘œ2π‘œ3+𝑝2𝑝1ξ€·π‘š1βˆ’π‘š2ξ€Έπ‘š2π‘š1⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(3.14)

tab1
Table 1: Velocity functions for the various groups.

In order to demonstrate the properties of the various groups, the following simplifying assumptions are made: π‘šπ‘–=π‘š for all 𝑖. The rotation is assumed to be about the axis of symmetry so π‘œ3=π‘œ2. The momentum equations become π‘‘βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽπ‘π‘‘π‘‘1𝑝2𝑝3π‘˜1π‘˜2π‘˜3⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽξ€·π‘šβˆ’πœ–π‘œ2ξ€Έξƒ©π‘˜3𝑝2βˆ’π‘˜2𝑝3ξ€·π‘šπ‘œ2ξ€Έξƒͺπ‘˜1𝑝3π‘œ1βˆ’π‘˜3𝑝1π‘œ2+πœ–ξ€·π‘˜3𝑝1βˆ’π‘˜1𝑝3ξ€Έπ‘šπ‘˜2𝑝1π‘œ2βˆ’π‘˜1𝑝2π‘œ1+πœ–ξ€·π‘˜1𝑝2βˆ’π‘˜2𝑝1ξ€Έπ‘š0π‘˜1π‘˜3ξ€·π‘œ2βˆ’π‘œ1ξ€Έπ‘œ1π‘œ2βˆ’π‘˜1π‘˜2ξ€·π‘œ2βˆ’π‘œ1ξ€Έπ‘œ1π‘œ2⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(3.15) Continuing with the simple example from expression (1.1), the initial momentum components are βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽπ‘1𝑝2𝑝3π‘˜1π‘˜2π‘˜3⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽπ‘šπ‘£π‘₯π‘šπ‘£π‘¦cos(𝑓)π‘šπ‘£π‘¦π‘œsin(𝑓)1𝑀00⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(3.16) The angular velocities are easily found to be constants βŽ›βŽœβŽœβŽœβŽœβŽπ‘€1𝑀(𝑑)2(𝑀𝑑)3⎞⎟⎟⎟⎟⎠=βŽ›βŽœβŽœβŽœβŽœβŽπ‘€00⎞⎟⎟⎟⎟⎠(𝑑).(3.17) The linear velocity components can be determined from π‘‘βŽ›βŽœβŽœβŽœβŽœβŽπ‘£π‘‘π‘‘1𝑣2𝑣3⎞⎟⎟⎟⎟⎠=βŽ›βŽœβŽœβŽœβŽœβŽœβŽ0ξ‚€π‘œ1βˆ’πœ–1π‘šξ‚π‘€π‘£3βˆ’ξ‚€π‘œ1βˆ’πœ–1π‘šξ‚π‘€π‘£2⎞⎟⎟⎟⎟⎟⎠.(3.18) This has solution βŽ›βŽœβŽœβŽœβŽœβŽπ‘£1𝑣(𝑑)2𝑣(𝑑)3⎞⎟⎟⎟⎟⎠=βŽ›βŽœβŽœβŽœβŽœβŽœβŽπ‘£(𝑑)π‘₯π‘£π‘¦π‘œcosξ‚€ξ‚€1βˆ’πœ–1π‘šξ‚ξ‚π‘£π‘€π‘‘+π‘“π‘¦π‘œsinξ‚€ξ‚€1βˆ’πœ–1π‘šξ‚ξ‚βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽ π‘€π‘‘+𝑓.(3.19) These velocities are shown in Table 1.

In summary, the initial velocities remain unchanged in the semidirect product. There is no interaction between the components. In all three Lie groups with the Lie-Poisson structure, the rotation induces a change in the linear velocities perpendicular to the axis of rotation. In 𝑆𝐸(3) (with imposed structure), the frequency of the change is equal to the initial rotation 𝑀. In 𝑆𝑂(4) and 𝑆𝑂(1,3), the frequency of the change is influenced by the ratio of the moment of inertia and the mass of the body.

4. Trajectories Determined for the Lie Groups

Having found the velocity functions, they are integrated in this section to determine the trajectory. The resulting trajectories are analyzed in the next section.

From Table 1 using the base matrices shown in (3.1), the velocity matrix 𝑋 is given by 𝑋(𝑑)=𝑣π‘₯𝑒1+𝑣𝑦𝑀cosπœ–ξ€Έπ‘’π‘‘+𝑓2+𝑣𝑦𝑀sinπœ–ξ€Έπ‘’π‘‘+𝑓3+𝑀𝑒4(4.1) with π‘€πœ–=(1βˆ’πœ–(π‘œ1/π‘š))𝑀 and πœ–βˆˆ{1,0,βˆ’1} for 𝔰𝔬(4),𝔰𝔒(3),𝔰𝔬(1,3), respectively. The trajectory is found by (2.1). Because the velocity matrix is time dependent, the trajectory is calculated by the forward Euler method of numerical iteration using the simple equation (2.3) 𝑔𝑛+1=𝑔𝑛𝑋exp𝑛𝑠,(4.2) where 𝑠 is the step length, and 𝑛 is the step number so that 𝑋𝑛=𝑋(𝑑𝑛) and 𝑑𝑛=𝑛𝑠.

The figures show the displacements π‘₯, 𝑦 and the rotation πœƒ for the three Lie groups, with initial velocities {𝑣π‘₯,𝑣𝑦,𝑀}={0.014,0.01,0.09}, 𝑓=0, π‘œ1/π‘š=0.6 and step length 𝑠=0.1.

The displacement along the axis of rotation as shown in Figure 2 is a straight line for 𝑆𝐸(3) (the dot-dash line). It has a sinusoidal function in 𝑆𝑂(4) (the dotted line) with a maximum of one, the unit sphere. In 𝑆𝑂(1,3), it is a sinh curve.

467520.fig.002
Figure 2: Displacement along the axes of rotation.

The rotation of the body is fixed at the original angular velocity, as shown in Figure 3, the plot of sin(πœƒ) where πœƒ is the orientation at time 𝑑.

467520.fig.003
Figure 3: Constant rotation about original axis of rotation.

The trajectories in all 3 spaces approximate a screw motion about the original axis of rotation. The main difference in the trajectories is determined by the ratio of the moment of inertia and the mass: π‘œ1/π‘š. With π‘œ1/π‘š=0.6, the difference in angular frequency and amplitude can be seen in Figure 4. The angular frequency is π‘€π‘¦β‰ƒπ‘€πœ–=𝑀(1βˆ’πœ–π‘œ1/π‘š), and the amplitude is π‘¦π‘šβ‰ƒπ‘£π‘¦/𝑀𝑦.

467520.fig.004
Figure 4: Induced screw motion about the axis of rotation.

For all four combinations of linear and rotations, the linear and rotational displacement functions shown in Table 2 indicate the form of the solutions. There are additional small magnitude terms which add lower-frequency variations and curve the trajectory away from the main axis of the screw. The largest difference is in 𝑆𝑂(4) and is shown in Figure 5.

tab2
Table 2: Approximate displacement functions for small displacements.
467520.fig.005
Figure 5: Comparison of iterated and calculated values of y displacement for 𝑆𝑂(4).

To conclude this section, the trajectory functions shown in Table 2 are summarized. The motion determined in the semidirect product ℝ3⋉𝑆𝑂(3) is that experienced in the body frame. In 𝑆𝐸(3), with the imposed Poisson structure, the motion is in the spatial frameβ€”see Figure 6. The actual and perceived distances traveled are both 𝑣𝑦𝑑. The perceived angle of rotation is the actual rotation of the trajectory plus the rotation of the body. The actual radius of travel (𝑣𝑦/𝑀5) is half the perceived radius since 𝑀5=2𝑀. In order to conserve angular momentum about the origin, the body performs a screw motion, and the linear velocity changes in the spatial frame. In 𝑆𝑂(4) and 𝑆𝑂(1,3), the radius of the screw motion is influenced by the ratio of the moment of inertia and the mass of the body.

467520.fig.006
Figure 6: Actual motion determined using 𝑆𝐸(3).

The perceived trajectory (calculated using the semidirect product) is independent of the actual trajectory (whether calculated using the Lie-Poisson structure of 𝑆𝐸(3) or working with the uncoupled velocities). Further details are given in the Appendix.

Over longer-time frames, the structure of the group causes the trajectories to diverge from this simple model.

5. Comparing the Different Trajectories

In this section, the trajectories identified previously are compared in more detail. The configuration in 𝑆𝑂(4) is shown to approximate the configuration in 𝑆𝐸(3) for small linear displacements. Thus, 𝑆𝐸(3) can be projected onto 𝑆𝑂(4) (and vice versa). The induced Lie-Poisson structure has coupling of the velocities for 𝑆𝐸(3). The variation in the frequency of the spiral motion across the groups is analyzed and found to be eliminated for small displacements.

5.1. Projecting 𝑆𝑂(4) Vertically onto 𝑆𝐸(3)

Intuitively, the 2 linear displacements and one rotation of a planar body can be represented on the surface of a large sphere. Equivalently, the environment can be scaled down to a unit sphere, so that the linear displacement is small compared to 1, using some linear unit (such as kilometer or light-year). Similarly 3D spatial displacements can be approximated on the surface of a unit hypersphere. Three axes are the rotation axes of the body frame, and there is no constraints on the size of those rotations (πœƒ1,πœƒ2,πœƒ3). The small angular displacements (π‘₯,𝑦,𝑧) from the other axes are the spacial displacements, after scaling down by a factor 𝑅.

Any element π‘”βˆˆπ‘†π‘‚(4) can be written as π‘”π‘Ž=expΞ©π‘ŽβŽ‘βŽ’βŽ’βŽ£=exp0βˆ’π‘₯π‘‡βŽ€βŽ₯βŽ₯⎦=βŽ‘βŽ’βŽ’βŽ£π‘‹π‘₯πœƒπ‘Žβˆ’π‘Œπ‘ŽΞ¦π‘ŽβŽ€βŽ₯βŽ₯⎦,(5.1) where π‘₯=[π‘₯,𝑦,𝑧]𝑇 and ξ‚Έπœƒ=0βˆ’πœƒ3πœƒ2πœƒ30βˆ’πœƒ1βˆ’πœƒ2πœƒ10ξ‚Ήβˆˆπ”°π”¬(3) are the two subelements (and π‘Ž=1βˆ’β€–π‘₯β€–).

The corresponding group element in 𝑆𝐸(3) is 𝑔𝑏=expΞ©π‘βŽ‘βŽ’βŽ’βŽ£βŽ€βŽ₯βŽ₯⎦=βŽ‘βŽ’βŽ’βŽ£π‘‹=exp00π‘₯πœƒ10π‘Ξ¦π‘βŽ€βŽ₯βŽ₯⎦.(5.2) The difference between the two groups is found from the exponential series expansion to be π‘”π‘Žβˆ’π‘”π‘=ξ€·Ξ©π‘Žβˆ’Ξ©π‘ξ€Έ+12ξ€·Ξ©2π‘Žβˆ’Ξ©2𝑏=⎑⎒⎒⎣+β‹―0βˆ’π‘₯π‘‡βŽ€βŽ₯βŽ₯⎦+1002⎑⎒⎒⎣π‘₯π‘₯π‘₯𝑇Φ0π‘₯𝑇π‘₯⎀βŽ₯βŽ₯⎦+β‹―.(5.3) Ignoring the values in the first row, the difference is of order 2 in π‘₯ as follows: βŽ‘βŽ’βŽ’βŽ£π‘‹π‘†π‘‚(4)βˆ‹π‘Žβˆ’π‘Œπ‘ŽΞ¦π‘ŽβŽ€βŽ₯βŽ₯βŽ¦β‰ƒβŽ‘βŽ’βŽ’βŽ£π‘‹10π‘Ξ¦π‘βŽ€βŽ₯βŽ₯βŽ¦βˆˆπ‘†πΈ(3)(5.4) with Ξ¦π‘Ž=Φ𝑏=expπœƒ, π‘‹π‘Ž=𝑋𝑏, and π‘Ž=1βˆ’β€–π‘₯β€– to second order in π‘₯. The projection πœ‹3βˆΆπ‘†π‘‚(4)→𝑆𝐸(3) is πœ‹3βŽ›βŽœβŽœβŽβŽ‘βŽ’βŽ’βŽ£π‘‹π‘Žβˆ’π‘Œπ‘ŽΞ¦π‘ŽβŽ€βŽ₯βŽ₯⎦⎞⎟⎟⎠=βŽ‘βŽ’βŽ’βŽ£π‘‹10π‘ŽΞ¦π‘ŽβŽ€βŽ₯βŽ₯⎦.(5.5) A similar projection can be done from 𝑆𝑂(1,3) onto 𝑆𝐸(3).

5.2. Scaling the Linear Displacements

Since the projection from 𝑆𝑂(4) to 𝑆𝐸(3) is accurate to second order in π‘₯, a scaling factor π‘†βˆˆβ„ (as suggested in [4]) can be chosen so that the plotted displacement π‘₯𝑝 of the rigid body in ℝ3 is π‘₯𝑝=𝑆π‘₯, and π‘₯ is the calculated displacement which is kept within the required condition √|π‘₯|<𝛿 where the required accuracy is 𝛿 (a proportion).

This scaling has a consequence on the inertia terms. The moment of inertia is a function of the mass π‘š and the square of the size of the body π‘Ÿ. So scaling the linear dimensions to be small compared with the unit hypersphere reduces the moment of inertia to order of ‖𝑆2β€–.

The angular frequency of the linear velocity terms tends to the angular velocity, as can be seen from Table 1π‘€πœ–=ξ‚€π‘œ1βˆ’πœ–1π‘šξ‚π‘€βŸΆπ‘€(5.6) for all three groups as the accuracy of the projection increases. For small displacements, the three groups produce the same results as follows: π‘”βˆ’1(𝑑)𝑑𝑔𝑑𝑑=𝑣π‘₯𝑒1+𝑣𝑦cos(𝑀𝑑+𝑓)𝑒2+𝑣𝑦sin(𝑀𝑑+𝑓)𝑒3+𝑀𝑒4(5.7) has solution βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘£π‘₯𝑑𝑣𝑦(sin(2𝑀𝑑+𝑓)βˆ’sin(𝑓))𝑀𝑣𝑦(cos(𝑓)βˆ’cos(2𝑀𝑑+𝑓))π‘€βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŠ•βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£βŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦1000cos(𝑀𝑑)βˆ’sin(𝑀𝑑)0sin(𝑀𝑑)cos(𝑀𝑑)(5.8) for {𝑣π‘₯𝑑,𝑣𝑦𝑑}β‰ͺ1. This gives the displacement in the spatial frame.

The scaling factor 𝑆 also has the effect of changing the relative weighting of the rotational and linear displacements. The linear displacements are shrunk and have less weighting in any comparison, such as distance (in 6 dimensions) traveled along the trajectory. This was demonstrated (5.6) where the angular frequency tends to the same value irrespective of the linear size of the body. There is an innate difference between rotations and translations which cannot be overcome by some arbitrary metric. There is no bi-invariant Riemannian metric on 𝑆𝐸(3). The requirements of each system will determine what (if any) a metric can be defined for that situation.

6. Discussion

Four methods of combining linear motion in 3 dimensions with a rotation have been considered. A single rotation is used since any 3 rotations about orthonormal axes can be replaced by a single rotation about one rotated axis, and if that axis is also an axis of symmetry, precession is avoided. The rotation of the axes necessary to achieve this is given on the Appendix. The linear motion results in linear displacements, and these can be scaled by a factor 𝑆 so that they are small compared to a unit hypersphere. Small linear displacements on a sphere are generated by small angular displacements.

The four methods considered are as follows.(1)Semidirect product, where the motion is described in the body frame: there is no coupling of the angular and translational velocities. This is often referred to as 𝑆𝐸(3) because the integration of the velocities can be done using that group.(2)𝑆𝐸(3) with induced Lie-Poisson structure: the angular momentum induces a change in the linear velocity. The resulting trajectory is in a fixed frame and is a spiral motion about an axis parallel to the original axis of rotation. The frequency of the screw motion is twice the original angular rotation. Whatever the frequency of the screw motion, the perceived motion in the body frame is always that produced by the semidirect product. Higher frequencies reduce the radius of the screw maintaining the angular momentum. A frequency of wπœ– generates a radius of 𝑣𝑦/π‘€πœ– and maintains a velocity of 𝑣𝑦 around the screwβ€”see Table 2.(3)𝑆𝑂(4) where the linear motions are interpreted as small angular displacements from three of the six axes: the frequency of the spiral motion is reduced as the motion of inertia about the original axis of rotation increases. When considering small linear dimensions, this reduction in frequency is eliminatedβ€”see (5.6). The trajectories for small linear displacements are the same as those produced using 𝑆𝐸(3) with induced Lie-Poisson structure. The difference arises from the term π‘₯(𝑑)=sin(𝑣π‘₯𝑑) compared with π‘₯(𝑑)=𝑣π‘₯𝑑, which feeds back into the displacements in other directions. As the rate of change of π‘₯(𝑑) decreases, the momentum is transferred to the other directions. The rate of change of 𝑦(𝑑) increases faster. This is not shown in the approximate functions in Table 2.(4)𝑆𝑂(1,3): the same considerations apply to this group with the following differences. The frequency of the spiral motion is increased as the motion of inertia about the original axis of rotation increases. As the displacement π‘₯(𝑑)=sinh(𝑣π‘₯𝑑) increases exponentially, the displacement in the other directions decreases exponentially. The displacement 𝑦(𝑑) decreases by an extraterm with the form of cosh function.

7. Conclusion

Trajectories of a rigid body in Euclidean space are determined by the linear and rotational velocities, which can interact. This paper has separated the step of finding the velocities from integrating the velocities into a trajectory. The semidirect product of the linear and rotational velocities consider the velocities separately without coupling them. The three 6D Lie groups use the Lie-Poisson structure to reflect the rotational influence on the linear velocity. The angular moments of inertia decrease the frequency of the spiral motion in the rotation group 𝑆𝑂(4), have no impact in 𝑆𝐸(3), and increase it in 𝑆𝑂(1,3). For small displacements, the impact on the linear velocity is the same for all three groups.

The velocities of the semidirect product can be integrated using the matrix format of 𝑆𝐸(3) and result in the usual identification between semidirect product and Special Euclidean group. The resulting trajectory is in the body frame, and this cannot be interpreted into the spatial frame since the perceived trajectory is independent of the velocity coupling. The three Lie groups integrate the motion in a spherical, flat, or hyperbolic space to give the trajectory in a spatial frame. For small displacements, these are the same (within any chosen degree of accuracy) and reflect the trajectory in a fixed frame. It differs from the body frame trajectory created by the semidirect product.

Any combinations of rotations can be represented by rotation about a single axis of rotation. If this axis is also an axis of symmetry, the rotational axis is fixed. In a natural system with no forces and a body symmetric about the axis of rotation, the analysis can be simplified by aligning one axis of the spatial frame with the rotation.

Due to the innate differences between rotational and translational motion, there cannot be a metric that applies to a rigid body motion in Euclidean space, without imposing some relationship between the relative motions which will depend on the system being measured.

Appendices

A. Rotation to Generalize the Motion

The simplified example of rotation about one axis, with initial linear motion along and perpendicular to that axis, can be generalized by rotating the coordinate frame. This is valid for 𝑆𝑂(4), 𝑆𝐸(3), and 𝑆𝑂(1,3), and also 𝑆𝑂(3) by dropping the first row and first column of the rotation matrix.

Assume that the generalized initial velocities are given by 𝑉=𝑣1𝑒1+𝑣2𝑒2+𝑣3𝑒3+𝑀1𝑒4+𝑀5𝑒5+𝑀6𝑒6.(A.1) By applying a rotation 𝑅 to this, the initial velocities are simplified to π‘…π‘‰π‘…βˆ’1=𝑣π‘₯𝑒1+𝑣𝑦𝑒2+𝑣𝑧𝑒3+𝑀𝑒4,(A.2) where βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽ0𝑀𝑅=10001𝑀𝑀2𝑀𝑀3𝑀𝑀003𝑀2βˆ’π‘€21βˆ’π‘€2𝑀2βˆ’π‘€210βˆ’π‘€2βˆ’π‘€21𝑀𝑀1𝑀2𝑀𝑀2βˆ’π‘€21𝑀1𝑀3𝑀𝑀2βˆ’π‘€21⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(A.3) The relationships between the general components {𝑣𝑖,𝑀𝑖} and the simplified ones {𝑣π‘₯,𝑣𝑦,𝑣𝑧,𝑀} can be found by matrix multiplication and justified as follows.(i)Conserving angular momentum 𝑀2=βˆ‘3𝑖=1𝑀2𝑖: if 𝑀=𝑀1, then no simplification is required, and 𝑅 is the unit matrix.(ii)From the invariant Casimir βˆ‘3𝑖=1𝑣𝑖𝑀𝑖, the π‘₯ component is fixed at 𝑣π‘₯βˆ‘=1/𝑀3𝑖=1𝑣𝑖𝑀𝑖.(iii)Consider 𝑣𝑦=(𝑣2𝑀3βˆ’π‘£3𝑀2)/𝑀22+𝑀23.(iv)Conserving linear momentum 𝑣2𝑧+𝑣2π‘¦βˆ‘=(3𝑖=1𝑣2𝑖)βˆ’π‘£2π‘₯.It follows that 𝑣2𝑦+𝑣2𝑧 is a constant and so the 2 linear components can be written as 𝑣𝑦=ξ„Άξ„΅ξ„΅βŽ·ξƒ©3𝑖=1𝑣2𝑖ξƒͺβˆ’π‘£2π‘₯𝑣cos(𝑀𝑑+𝑓),𝑧=ξ„Άξ„΅ξ„΅βŽ·ξƒ©3𝑖=1𝑣2𝑖ξƒͺβˆ’π‘£2π‘₯sin(𝑀𝑑+𝑓).(A.4)

B. Conversion from Body Frame to Spatial Frame

Selig [10] states that the body frame 𝑋′ and an alternative frame viewpoints 𝑋 are related through the expression ⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦π‘‹=10𝑒𝑅𝑋′10π‘’π‘…βˆ’1,(B.1) where 𝑒 is the displacement vector between the two spaces, and 𝑅 is the relative rotation between them. The body frame trajectory (in the y-z plane) in the simple example of this paper is π‘‹ξ…ž=βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘£100𝑀𝑣sin(𝑀𝑑)cos(𝑀𝑑)βˆ’sin(𝑀𝑑)π‘€βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦(1βˆ’cos(𝑀𝑑))sin(𝑀𝑑)cos(𝑀𝑑).(B.2)

If the displacement vector 𝑒 is set to βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘£π‘’=𝑀sin(π‘šπ‘‘π‘€)βˆ’sin((π‘›βˆ’1)𝑑𝑀/2)sin𝑛𝑑𝑀𝑣𝑛sin(𝑑𝑀/2)ξ‚΅2𝑀1βˆ’2cos(π‘šπ‘‘π‘€)βˆ’sin((2π‘›βˆ’1)𝑑𝑀/2)sinπ‘›π‘‘π‘€ξ‚ΆβŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘›sin(𝑑𝑀/2)(B.3) and the rotation 𝑅 is π‘šπ‘‘π‘€ for any value of π‘š, then the trajectory in the alternative frame is given as βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£π‘£π‘‹=100𝑣𝑀𝑛sin(𝑛𝑀𝑑)cos(𝑀𝑑)βˆ’sin(𝑀𝑑)⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘€π‘›(1βˆ’cos(𝑛𝑀𝑑))sin(𝑀𝑑)cos(𝑀𝑑).(B.4) A trajectory with any value of 𝑛 can be found by changing the displacement vector 𝑒. The radius of the trajectory in the spatial frame cannot be determined. This confirms the geometric diagrams provided in the paper. The body frame trajectory cannot be interpreted into a spatial frame without additional information.

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