Rigid Body Trajectories in Different 6D Spaces

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)—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 product—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=⎡⎢⎢⎢⎢⎣𝑣𝑣_𝑦co𝑣s^𝑥(𝑓)_𝑦sin(𝑓)⎤⎥⎥⎥⎥⎦,𝜙⁰=⎡⎢⎢⎢⎢⎣000𝑤00−00𝑤⎤⎥⎥⎥⎥⎦,(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(𝑡)∈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 𝑔⁻¹(𝑡). The tangent field at the origin 𝑋∈𝔤 (where 𝔤 is the Lie algebra) determines the trajectory through the expression 𝑔⁻¹(𝑡)𝑑𝑔𝑑𝑡(𝑡)=𝑋.(2.1) One finds that 𝑔(𝑡)=exp(𝑋𝑡)if𝑋istimeindependent.(2.2) If 𝑋=𝑋(𝑡), 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: 𝑔_𝑛+1=𝑔_𝑛exp𝑋_𝑛𝑠,(2.3) where 𝑠 is the step length, and 𝑛 is the step number so that 𝑋_𝑛=𝑋(𝑡_𝑛) is the tangent field at time 𝑡_𝑛=𝑛𝑠. The trajectory started at 𝑔(0)=𝐼.

Fact. The forward Euler method uses 𝑔_𝑛+1−𝑔_𝑛𝑠=𝑑𝑑𝑔𝑡^𝑛=𝑔^𝑛𝑋_𝑛(2.4) so that 𝑔_𝑛+1=𝑔_𝑛1+𝑠𝑋_𝑛≃𝑔_𝑛exp𝑠𝑋_𝑛.(2.5) Alternatively, assume that 𝑋_𝑛 is constant between times 𝑡_𝑛 and 𝑡_𝑛+1 so that the incremental motion is given by 𝛿𝑔_𝑛=exp(𝑋_𝑛𝑠). This increment is applied to the previous configuration using 𝑔_𝑛+1=𝑔_𝑛𝛿𝑔_𝑛 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 {𝑤_𝑖} for 𝑖∈{1,2,3} about three orthogonal axes {𝑒_𝑖}. The resultant velocity 𝑋 can be written in both coordinate and matrix forms as 𝑋=³_𝑖=1𝑤_𝑖𝑒_𝑖=⎡⎢⎢⎢⎢⎣0−𝑤³𝑤₂𝑤₃0−𝑤₁−𝑤₂𝑤₁0⎤⎥⎥⎥⎥⎦.(2.6) From this notation, the base matrices {𝑒_𝑖} 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 𝜙₀=⎡⎢⎢⎢⎢⎣000𝑤00−00𝑤⎤⎥⎥⎥⎥⎦.(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 𝑜₂=𝑜₃). The angular velocity is unchanging. The axis of rotation does not change (no precession). The attitude of the body Φ(𝑡)∈𝑆𝑂(3) is the solution of Φ⁻¹(𝑡)𝑑Φ(𝑡)𝑑𝑡=𝜙⁰.(2.8) The resulting attitude at time 𝑡 is Φ(𝑡)=exp𝜙₀𝑡=⎡⎢⎢⎢⎢⎣100csoins0((𝑤𝑤𝑡𝑡))−cosisn0(𝑤(𝑤𝑡)𝑡)⎤⎥⎥⎥⎥⎦.(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 𝑣=𝑣₁,𝑣₂,𝑣₃^𝑇=³_𝑖=1𝑣_𝑖𝑒_𝑖,(2.10) where here {𝑒_𝑖} 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 𝑥₁,Φ₁∘𝑥₂,Φ₂=𝑥₁+Φ₁𝑥₂,Φ₁⋅Φ₂(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 ⎡⎢⎢⎣𝑥10₁Φ₁⎤⎥⎥⎦⎡⎢⎢⎣𝑥10₂Φ₂⎤⎥⎥⎦=⎡⎢⎢⎣𝑥10₁+Φ₁𝑥₂Φ₁Φ₂⎤⎥⎥⎦(2.12) which is the format of the Lie group 𝑆𝐸(3) with no structure imposed. The attitude Φ(𝑡) is found using (2.1). The displacement 𝑥 at time 𝑡 is found by integration as 𝑥=^𝑡𝑠=0Φ(𝑠)𝑣(𝑠)𝑑𝑠,(2.13) where Φ(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 4×4 matrix to give the equation 𝑔⁻¹(𝑡)𝑑𝑔𝑑𝑡=⎡⎢⎢⎣𝑣0(𝑡)𝜙0(𝑡)⎤⎥⎥⎦(2.14) which has the solution 𝑆𝐸(3)∋𝑔(𝑡)=^{10𝑥(𝑡)𝜙(𝑡)}. 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=[𝑣_𝑥,𝑣_𝑦cos(𝑓),𝑣_𝑦sin(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 𝑣⋉𝜙=⎡⎢⎢⎢⎢⎣𝑣𝑣_𝑦co𝑣s^𝑥(𝑓)_𝑦sin(𝑓)⎤⎥⎥⎥⎥⎦⋉⎡⎢⎢⎢⎢⎣000𝑤00−00𝑤⎤⎥⎥⎥⎥⎦,(2.15) and the configuration at time 𝑡 is ⎡⎢⎢⎢⎢⎢⎣𝑣_𝑦𝑣_𝑥𝑡𝑣𝑤(sin(𝑤𝑡+𝑓)−sin(𝑓))_𝑦𝑤(cos(𝑓)−cos(𝑤𝑡+𝑓))⎤⎥⎥⎥⎥⎥⎦⋉⎡⎢⎢⎢⎢⎣100csoins0((𝑤𝑤𝑡𝑡))−cosisn0(𝑤(𝑤𝑡)𝑡)⎤⎥⎥⎥⎥⎦.(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 𝑦−𝑧 plane is 𝑣_𝑦 so, at time 𝑡, the origin 𝑂 is seen at an angle 𝑤𝑡 behind the body having traveled a distance 𝑣_𝑦𝑡. The perceived radius of travel 𝑟 is given by 𝑟𝑤𝑡=𝑣_𝑦𝑡(2.17) so the perceived radius is 𝑣_𝑦/𝑤 as seen in Figure 1 and (2.16).

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 ^00𝑣𝜙 to calculate the trajectory, where 𝑣 is the 3×1 column matrix of linear velocities, and 𝜙 is the 3×3 angular velocity matrix given above. In Section 4, the velocity matrix is taken as ⎡⎢⎢⎣0−𝜀𝑣^𝑇𝑣𝜙⎤⎥⎥⎦(2.18) with 𝜖∈{1,0,−1} for 𝔰𝔬(4),𝔰𝔢(3),𝔰𝔬(1,3), respectively. Other variations in the order of columns and rows are possible but produce equivalent results. Alternative values of 𝜀 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 𝐶=∑^6𝑖=1𝑝^2𝑖 where the {𝑝_𝑖} 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 𝐶=∑^3𝑖=1𝑝^2𝑖−∑^6𝑖=4𝑝^2𝑖.(iii)For the Lie group, 𝑆𝐸(3), a Lie-Poisson structure is imposed. The equivalent Casimir is 𝐶=∑^3𝑖=1𝑝^2𝑖. 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 𝐶=∑^3𝑖=1𝑝_𝑖𝑝_𝑖+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=1𝑣_𝑖𝑒_𝑖+𝑤_𝑖𝑒_𝑖+3=⎡⎢⎢⎢⎢⎢⎢⎣0−𝜀𝑣¹−𝜀𝑣₂−𝜀𝑣₃𝑣₁0−𝑤₃𝑤₂𝑣₂𝑤₃0−𝑤₁𝑣₃−𝑤₂𝑤₁0⎤⎥⎥⎥⎥⎥⎥⎦(3.1) with 𝜖∈{1,0,−1} for 𝔰𝔬(4),𝔰𝔢(3),𝔰𝔬(1,3), respectively. The orthogonal basis for the dual Lie algebra 𝔤^∗ is {𝑒^𝑖} given by 𝑒^𝑖=𝕀𝑒_𝑖, where 𝕀=𝐼_4×4, 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: 𝐴𝐵≠𝐵𝐴 in most cases. Structure constants 𝑐^𝑘𝑖𝑗 are used to describe this nonassociative action in a Lie algebra 𝔤. They are defined using the Lie bracket by (see [6, page 56]) 𝑐^𝑘𝑖𝑗𝑒_𝑘=𝑒_𝑖,𝑒_𝑗=𝑒_𝑖𝑒_𝑗−𝑒_𝑗𝑒_𝑖.(3.2) The value of the structure constants is easily shown by matrix multiplication to be, by using 𝜖∈{1,0,−1} for 𝔰𝔬(4), 𝔰𝔢(3), and 𝔰𝔬(1,3), 𝑐𝑐³¹⁵=𝑐¹²⁶=𝑐²³⁴=𝑐³⁴²=𝑐⁶⁴⁵=𝑐¹⁵³=𝑐⁴⁵⁶=𝑐²⁶¹=𝑐⁵⁶⁴=1₂₁₆=𝑐³²⁴=𝑐¹³⁵=𝑐𝑐²⁴³=𝑐⁵⁴⁶=𝑐³⁵¹=𝑐⁶⁵⁴=𝑐¹⁶²=𝑐⁴⁶⁵=−1₆₁₂𝑐=𝑐⁴²³=𝑐⁵³¹=𝜖₅₁₃=𝑐⁶²¹=𝑐⁴³²=−𝜖.(3.3) They are the same for the dual algebra, 𝔤^∗.

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}=𝜕𝐹𝜕𝑞𝜕𝜕𝐸𝑝−𝜕𝜕𝐹𝑝𝜕𝜕𝐸𝑞(3.4) with 𝐹(𝑝,𝑞) and 𝐸(𝑝,𝑞) being functions on the cotangent space with canonical coordinates (𝑞,𝑝), representing position and momentum. To express this in other coordinates, write 𝑞=𝑞(𝑧_𝑖) and 𝑝=𝑝(𝑧_𝑗) and get, using partial differentiation, {𝐹,𝐸}=𝜕𝐹𝜕𝑧_𝑖𝑧_𝑖,𝑧_𝑗𝜕𝐸𝜕𝑧_𝑗.(3.5) If these functions are pulled back to the origin, there is no positional dependence and so {𝐹,𝐸}=𝜕𝐹𝜕𝑝_𝑖𝑝_𝑖,𝑝_𝑗𝜕𝐸𝜕𝑝_𝑗.(3.6) For a Lie algebra with a Poisson structure, it can be shown that {𝑝_𝑖,𝑝_𝑗}=−𝑐^𝑘𝑖𝑗𝑝_𝑘 (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 𝔤^∗ becomes {𝐸,𝐹}=−𝑐^𝑘𝑖𝑗𝑝_𝑘𝜕𝐸𝜕𝑝_𝑖𝜕𝐹𝜕𝑝_𝑗.(3.7)

3.3. Hamiltonian Flow

For a Poisson manifold, Bloch [9, page 121] defines 𝑋_𝐻, the Hamiltonian vector field of 𝐻, as the unique vector field such that 𝑑𝑓𝑑𝑡=𝑋^𝐻(𝑓)={𝑓,𝐻}∀𝑓.(3.8) From (3.7) above, the Hamiltonian vector field is 𝑋_𝐻=−𝑐^𝑘𝑖𝑗𝑝_𝑘𝜕𝐻𝜕𝑝_𝑗𝜕𝜕𝑝_𝑖(3.9) and, for all functions 𝑓, 𝑑𝑓𝑑𝑡=−𝑐^𝑘𝑖𝑗𝑝_𝑘𝜕𝐻𝜕𝑝_𝑗𝜕𝜕𝑝_𝑖𝑓.(3.10) With a Poisson structure naturally or imposed, the coordinate equation for Lie algebras is found by setting 𝑓=𝑝_𝑖 for 𝑖∈{1,2,3,4,5,6}𝑝̇_𝑖=−𝑐^𝑘𝑖𝑗𝑝_𝑘𝜕𝐻𝜕𝑝_𝑗,(3.11) where {𝑝_𝑖} are the components of momentum. The relationship with the velocities is 𝑝_𝑖=𝑚_𝑖𝑣_𝑖 for 𝑖∈{1,2,3} and 𝑝_𝑖+3=𝑜_𝑖𝑤_𝑖, where the linear velocities are {𝑣₁,𝑣₂,𝑣₃}, the angular velocities are {𝑤₁,𝑤₂,𝑤₃}, the moments of inertia are {𝑜₁,𝑜₂,𝑜₃}, and the added masses are {𝑚₁,𝑚₂,𝑚₃} allowing for a different mass in each direction. The Hamiltonian for a natural motion, with no forces, is the kinetic energy so 𝐻=12³_𝑖=1𝑚_𝑖𝑣^2𝑖+𝑜_𝑖𝑤^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 𝐻=12³_𝑖=1𝑚𝑝^2𝑖_𝑖+𝑘^2𝑖𝑜_𝑖,(3.13) where {𝑘_𝑖}={𝑜_𝑖𝑤_𝑖}={𝑝_𝑖+3} are the angular momentum components. The required differential equations for the momentum are, from (3.11) and the structure constants, 𝑑𝑑𝑡⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑝¹𝑝₂𝑝₃𝑘₁𝑘₂𝑘₃⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−𝑝³𝑘₂𝑜₂+𝜖𝑝³𝑘₂𝑚₃+𝑝²𝑘₃𝑜₃−𝜖𝑝²𝑘₃𝑚₂𝑝₃𝑘₁𝑜₁−𝜖𝑝³𝑘₁𝑚₃−𝑝¹𝑘₃𝑚₁+𝜖𝑝¹𝑘₃𝑜₁𝜖𝑝₂𝑘₁𝑚₂−𝑝²𝑘₁𝑜₁+𝑝¹𝑘₂𝑜₂−𝜖𝑝¹𝑘₂𝑚₁𝑘₂𝑘₃𝑜₂−𝑜₃𝑜₂𝑜₃+𝑝²𝑝₃𝑚₂−𝑚₃𝑚₂𝑚₃𝑘₃𝑘₁𝑜₃−𝑜₁𝑜₁𝑜₃+𝑝¹𝑝₃𝑚₃−𝑚₁𝑚₁𝑚₃𝑘₁𝑘₁𝑜₁−𝑜₂𝑜₂𝑜₃+𝑝²𝑝₁𝑚₁−𝑚₂𝑚₂𝑚₁⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(3.14)

Table 1 

Velocity functions for the various groups.

In order to demonstrate the properties of the various groups, the following simplifying assumptions are made: 𝑚_𝑖=𝑚 for all 𝑖. The rotation is assumed to be about the axis of symmetry so 𝑜₃=𝑜₂. The momentum equations become 𝑑𝑑𝑡⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑝¹𝑝₂𝑝₃𝑘₁𝑘₂𝑘₃⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑚−𝜖𝑜²𝑘³𝑝₂−𝑘₂𝑝₃𝑘₁𝑚𝑜₂𝑝₃𝑜₁−𝑘³𝑝₁𝑜₂+𝜖𝑘³𝑝₁−𝑘₁𝑝₃𝑘₂𝑚𝑝₁𝑜₂−𝑘¹𝑝₂𝑜₁+𝜖𝑘¹𝑝₂−𝑘₂𝑝₁𝑘₁0𝑚𝑘₃𝑜₂−𝑜₁𝑜₁𝑜₂−𝑘₁𝑘₂𝑜₂−𝑜₁𝑜₁𝑜₂⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(3.15) Continuing with the simple example from expression (1.1), the initial momentum components are ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑝₁𝑝₂𝑝₃𝑘₁𝑘₂𝑘₃⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑚𝑣_𝑥𝑚𝑚𝑣𝑣_𝑦𝑦cos(𝑓)𝑜sin(𝑓)₁00𝑤⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(3.16) The angular velocities are easily found to be constants ⎛⎜⎜⎜⎜⎝𝑤𝑤₁(𝑡)₂𝑤(𝑡)₃(𝑡)⎞⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎝𝑤00⎞⎟⎟⎟⎟⎠.(3.17) The linear velocity components can be determined from 𝑑𝑑𝑡⎛⎜⎜⎜⎜⎝𝑣¹𝑣₂𝑣₃⎞⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎜⎝1−𝜖𝑜0₁𝑚𝑤𝑣³−1−𝜖𝑜¹𝑚𝑤𝑣²⎞⎟⎟⎟⎟⎟⎠.(3.18) This has solution ⎛⎜⎜⎜⎜⎝𝑣𝑣₁(𝑡)₂𝑣(𝑡)₃(𝑡)⎞⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎜⎝𝑣^𝑥𝑣_𝑦cos1−𝜖𝑜¹𝑣𝑚𝑤𝑡+𝑓_𝑦sin1−𝜖𝑜¹𝑚𝑤𝑡+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 𝑤. 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(𝑡)=𝑣_𝑥𝑒₁+𝑣_𝑦cos𝑤_𝜖𝑡+𝑓𝑒₂+𝑣_𝑦sin𝑤_𝜖𝑡+𝑓𝑒₃+𝑤𝑒₄(4.1) with 𝑤_𝜖=(1−𝜖(𝑜₁/𝑚))𝑤 and 𝜖∈{1,0,−1} for 𝔰𝔬(4),𝔰𝔢(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) 𝑔_𝑛+1=𝑔_𝑛exp𝑋_𝑛𝑠,(4.2) where 𝑠 is the step length, and 𝑛 is the step number so that 𝑋_𝑛=𝑋(𝑡_𝑛) and 𝑡_𝑛=𝑛𝑠.

The figures show the displacements 𝑥, 𝑦 and the rotation 𝜃 for the three Lie groups, with initial velocities {𝑣_𝑥,𝑣_𝑦,𝑤}={0.014,0.01,0.09}, 𝑓=0, 𝑜₁/𝑚=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.

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(𝜃) where 𝜃 is the orientation at time 𝑡.

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: 𝑜₁/𝑚. With 𝑜₁/𝑚=0.6, the difference in angular frequency and amplitude can be seen in Figure 4. The angular frequency is 𝑤_𝑦≃𝑤_𝜖=𝑤(1−𝜖𝑜₁/𝑚), and the amplitude is 𝑦_𝑚≃𝑣_𝑦/𝑤_𝑦.

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.

Table 2 

Approximate displacement functions for small displacements.

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 frame—see Figure 6. The actual and perceived distances traveled are both 𝑣_𝑦𝑡. The perceived angle of rotation is the actual rotation of the trajectory plus the rotation of the body. The actual radius of travel (𝑣_𝑦/𝑤₅) is half the perceived radius since 𝑤₅=2𝑤. 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.

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 (𝜃₁,𝜃₂,𝜃₃). The small angular displacements (𝑥,𝑦,𝑧) from the other axes are the spacial displacements, after scaling down by a factor 𝑅.

Any element 𝑔∈𝑆𝑂(4) can be written as 𝑔_𝑎=expΩ_𝑎=exp⎡⎢⎢⎣0−𝑥^𝑇𝑥𝜃⎤⎥⎥⎦=⎡⎢⎢⎣𝑋𝑎−Φ𝑌_𝑎𝑎⎤⎥⎥⎦,(5.1) where 𝑥=[𝑥,𝑦,𝑧]^𝑇 and 𝜃=^{0−𝜃3𝜃2𝜃}₃^0−𝜃_{1−𝜃2𝜃10}∈𝔰𝔬(3) are the two subelements (and 𝑎=1−‖𝑥‖).

The corresponding group element in 𝑆𝐸(3) is 𝑔_𝑏=expΩ_𝑏=exp⎡⎢⎢⎣𝑥0𝜃0⎤⎥⎥⎦=⎡⎢⎢⎣𝑋10_𝑏Φ_𝑏⎤⎥⎥⎦.(5.2) The difference between the two groups is found from the exponential series expansion to be 𝑔_𝑎−𝑔_𝑏=Ω_𝑎−Ω_𝑏+12Ω^2𝑎=⎡⎢⎢⎣0−𝑥_𝑇−Ω^2𝑏+⋯00⎤⎥⎥⎦+21⎡⎢⎢⎣𝑥𝑥𝑥^𝑇0𝑥_𝑇Φ𝑥⎤⎥⎥⎦+⋯.(5.3) Ignoring the values in the first row, the difference is of order 2 in 𝑥 as follows: 𝑆𝑂(4)∋⎡⎢⎢⎣𝑋𝑎−𝑌_𝑎Φ_𝑎⎤⎥⎥⎦≃⎡⎢⎢⎣𝑋10_𝑏Φ_𝑏⎤⎥⎥⎦∈𝑆𝐸(3)(5.4) with Φ_𝑎=Φ_𝑏=exp𝜃, 𝑋_𝑎=𝑋_𝑏, and 𝑎=1−‖𝑥‖ to second order in 𝑥. The projection 𝜋₃∶𝑆𝑂(4)→𝑆𝐸(3) is 𝜋₃⎛⎜⎜⎝⎡⎢⎢⎣𝑋𝑎−𝑌_𝑎Φ_𝑎⎤⎥⎥⎦⎞⎟⎟⎠=⎡⎢⎢⎣𝑋10_𝑎Φ_𝑎⎤⎥⎥⎦.(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 𝑥, a scaling factor 𝑆∈ℝ (as suggested in [4]) can be chosen so that the plotted displacement 𝑥_𝑝 of the rigid body in ℝ³ is 𝑥_𝑝=𝑆𝑥, and 𝑥 is the calculated displacement which is kept within the required condition |𝑥|<√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 𝑚 and the square of the size of the body 𝑟. So scaling the linear dimensions to be small compared with the unit hypersphere reduces the moment of inertia to order of ‖𝑆²‖.

The angular frequency of the linear velocity terms tends to the angular velocity, as can be seen from Table 1𝑤_𝜖=1−𝜖𝑜¹𝑚𝑤⟶𝑤(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: 𝑔⁻¹(𝑡)𝑑𝑔𝑑𝑡=𝑣^𝑥𝑒₁+𝑣_𝑦cos(𝑤𝑡+𝑓)𝑒₂+𝑣_𝑦sin(𝑤𝑡+𝑓)𝑒₃+𝑤𝑒₄(5.7) has solution ⎡⎢⎢⎢⎢⎢⎢⎣𝑣_𝑦𝑣_𝑥𝑡(sin(2𝑤𝑡+𝑓)−sin(𝑓))𝑣_𝑦𝑤(cos(𝑓)−cos(2𝑤𝑡+𝑓))𝑤⎤⎥⎥⎥⎥⎥⎥⎦⊕⎡⎢⎢⎢⎢⎣100csoins0((𝑤𝑤𝑡𝑡))−cosisn0(𝑤(𝑤𝑡)𝑡)⎤⎥⎥⎥⎥⎦(5.8) for {𝑣_𝑥𝑡,𝑣_𝑦𝑡}≪1. 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 w_𝜖 generates a radius of 𝑣_𝑦/𝑤_𝜖 and maintains a velocity of 𝑣_𝑦 around the screw—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 eliminated—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 𝑥(𝑡)=sin(𝑣_𝑥𝑡) compared with 𝑥(𝑡)=𝑣_𝑥𝑡, which feeds back into the displacements in other directions. As the rate of change of 𝑥(𝑡) decreases, the momentum is transferred to the other directions. The rate of change of 𝑦(𝑡) 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 𝑥(𝑡)=sinh(𝑣_𝑥𝑡) increases exponentially, the displacement in the other directions decreases exponentially. The displacement 𝑦(𝑡) 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+𝑤₁𝑒₄+𝑤₅𝑒₅+𝑤₆𝑒₆.(A.1) By applying a rotation 𝑅 to this, the initial velocities are simplified to 𝑅𝑉𝑅⁻¹=𝑣_𝑥𝑒₁+𝑣_𝑦𝑒₂+𝑣_𝑧𝑒₃+𝑤𝑒₄,(A.2) where 𝑅=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝01𝑤000¹0𝑤0𝑤𝑤𝑤²₃𝑤𝑤³𝑤²−𝑤²¹−𝑤²𝑤²−𝑤²¹0−𝑤²−𝑤²¹𝑤𝑤¹𝑤₂𝑤𝑤²−𝑤²¹𝑤₁𝑤₃𝑤𝑤²−𝑤²¹⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(A.3) The relationships between the general components {𝑣_𝑖,𝑤_𝑖} and the simplified ones {𝑣_𝑥,𝑣_𝑦,𝑣_𝑧,𝑤} can be found by matrix multiplication and justified as follows.(i)Conserving angular momentum 𝑤²=∑^3𝑖=1𝑤^2𝑖: if 𝑤=𝑤₁, then no simplification is required, and 𝑅 is the unit matrix.(ii)From the invariant Casimir ∑^3𝑖=1𝑣_𝑖𝑤_𝑖, the 𝑥 component is fixed at 𝑣_𝑥=1/𝑤∑^3𝑖=1𝑣_𝑖𝑤_𝑖.(iii)Consider 𝑣_𝑦=(𝑣₂𝑤₃−𝑣₃𝑤₂)/𝑤²²+𝑤²³.(iv)Conserving linear momentum 𝑣^2𝑧+𝑣^2𝑦=(∑^3𝑖=1𝑣^2𝑖)−𝑣^2𝑥.It follows that 𝑣^2𝑦+𝑣^2𝑧 is a constant and so the 2 linear components can be written as 𝑣_𝑦=⎷³_𝑖=1𝑣^2𝑖−𝑣^2𝑥𝑣cos(𝑤𝑡+𝑓),_𝑧=⎷³_𝑖=1𝑣^2𝑖−𝑣^2𝑥sin(𝑤𝑡+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 𝑋=⎡⎢⎢⎣1𝑢𝑅0⎤⎥⎥⎦𝑋′⎡⎢⎢⎣1𝑢𝑅0⎤⎥⎥⎦⁻¹,(B.1) where 𝑢 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 𝑋^=⎡⎢⎢⎢⎢⎢⎢⎣𝑣𝑤𝑣sin1(𝑤𝑡)cos0(𝑤𝑡)−sin0(𝑤𝑡)𝑤(1−cos(𝑤𝑡))sin(𝑤𝑡)cos(𝑤𝑡)⎤⎥⎥⎥⎥⎥⎥⎦.(B.2)

If the displacement vector 𝑢 is set to 𝑢=⎡⎢⎢⎢⎢⎣𝑣𝑣𝑤sin(𝑚𝑡𝑤)−sin((𝑛𝑛−si1n)𝑡(𝑤𝑡𝑤/2/2))sin𝑛𝑡𝑤2𝑤1−2cos(𝑚𝑡𝑤)−sin((2𝑛𝑛−sin1)(𝑡𝑡𝑤𝑤//22))sin𝑛𝑡𝑤⎤⎥⎥⎥⎥⎦(B.3) and the rotation 𝑅 is 𝑚𝑡𝑤 for any value of 𝑚, then the trajectory in the alternative frame is given as 𝑋=⎡⎢⎢⎢⎢⎢⎢⎣𝑣𝑤𝑣𝑛sin1(𝑛𝑤𝑡)cos0(𝑤𝑡)−sin0(𝑤𝑡)𝑤𝑛(1−cos(𝑛𝑤𝑡))sin(𝑤𝑡)cos(𝑤𝑡)⎤⎥⎥⎥⎥⎥⎥⎦.(B.4) A trajectory with any value of 𝑛 can be found by changing the displacement vector 𝑢. 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.