Hindawi Publishing Corporation The Scientific World Journal Volume 2013, Article ID 292787, 9 pages http://dx.doi.org/10.1155/2013/292787

Research Article Riemannian Means on Special Euclidean Group and Unipotent Matrices Group

Xiaomin Duan,1 Huafei Sun,1 and Linyu Peng2

1 School of Mathematics, Beijing Institute of Technology, Beijing 100081, China 2 Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK

Correspondence should be addressed to Huafei Sun; [email protected]

Received 1 August 2013; Accepted 16 September 2013

Academic Editors: R. Abu-Saris and P. Bracken

Copyright © 2013 Xiaomin Duan 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.

Among the noncompact matrix Lie groups, the special Euclidean group and the unipotent matrix group play important roles in both theoretic and applied studies. The Riemannian means of a finite set of the given points on the two matrix groups are investigated, respectively. Based on the left invariant metric on the matrix Lie groups, the geodesic between any two points is gotten. And the sum of the geodesic distances is taken as the cost function, whose minimizer is the Riemannian mean. Moreover, a Riemannian gradient algorithm for computing the Riemannian mean on the special Euclidean group and an iterative formula for that on the unipotent matrix group are proposed, respectively. Finally, several numerical simulations in the 3-dimensional case are given to illustrate our results.

1. Introduction exponential mapping from the arithmetic mean of points on the Lie algebra se(3) to the Lie group SE(3) was constructed A matrix Lie group, which is also a differentiable manifold to give the Riemannian mean in order to get a mean filter. simultaneously, attracts more and more researchers’ atten- In this paper, the Riemannian means on SE(𝑛) and those tion from both theoretic interest and its applications [1–5]. on UP(𝑛), which are both important noncompact matrix The Riemannian mean on the matrix Lie groups is widely Lie groups [13, 14], are considered, respectively. Especially, studied for its varied applications in biomedicine, signal SE(3) is the spacial rigid body motion, and UP(3) is the processing, and robotics control [6–9]. Fiori and Tanaka 3-dimensional Heisenberg group 𝐻(3).Basedontheleft [10] suggested a general-purpose algorithm to compute the invariant metric on the matrix Lie groups, we get the geodesic average element of a finite set of matrices belonging to distance between any two points and take their sum as a any matrix Lie group. In [11], the author investigated the cost function. And the Riemannian mean will minimize Riemannian mean on the compact Lie groups and proposed a it. Furthermore, the Riemannian mean on SE(𝑛) is gotten globally convergent Riemannian gradient descent algorithm. using the Riemannian gradient algorithm, rather than the Different invariant notions of mean and average rotations on approximate mean. An iterative formula for computing the SO(3) (it is compact) are given in [9]. Recently, Fiori [12]dealt Riemannian mean on UP(𝑛) is proposed according to the with computing averages over the group of real symplectic Jacobi field. Finally, we give some numerical simulations on matrices, which found applications in diverse areas such as SE(3) and those on 𝐻(3) to illustrate our results. optics and particle physics. However, the Riemannian mean on the special Euclidean group SE(𝑛) and the unipotent matrix group UP(𝑛),whichare 2. Overview of Matrix Lie Groups the noncompact matrix Lie groups, has not been well studied. Fletcher et al. [6]proposedaniterativealgorithmtoobtain In this section, we briefly introduce the Riemannian frame- the approximate solution of the Riemannian mean on SE(3) work of the matrix Lie groups [15, 16], which forms the by use of the Baker-Cambell-Hausdorff formula. In [7], the foundation of our study of the Riemannian mean on them. 2 The Scientific World Journal

2.1. The Riemannian Structures of Matrix Lie Groups. Agroup where 𝐿 denotes the left translation, 𝑅 denotes the right 𝐺 is called a Lie group if it has differentiable structure: the translation, and (𝐿𝐴)∗ and (𝑅𝐴−1 )∗ are the tangent mappings group operators, that is, 𝐺 × 𝐺 → 𝐺, (𝑥,𝑦) 󳨃→𝑥⋅𝑦and associated with 𝐿𝐴 and 𝑅𝐴−1 , respectively. The adjoint action −1 𝐺→𝐺,𝑥󳨃→𝑥 , are differentiable, 𝑥, 𝑦.AmatrixLie ∈𝐺 Ad𝐴 : s → s is group is a Lie group with all elements matrices. The tangent −1 space of 𝐺 at identity is the Lie algebra g,wheretheLiebracket Ad𝐴𝑋=𝐴𝑋𝐴 . (5) is defined. The exponential map, denoted by exp, is a map from It is also easy to see the formula that the Lie algebra g to the group 𝐺. Generally, the exponential =𝐿 𝑅 . map is neither surjective nor injective. Nevertheless, it is Ad𝐴 𝐴 𝐴 (6) a diffeomorphism between a neighborhood of the identity 𝐼 on 𝐺 and a neighborhood of the identity 0 on g.The Then, the left invariant metric on 𝑆 is given by (local) inverse of the exponential map is the logarithmic map, ⟨𝑋, 𝑌⟩ =⟨(𝐿 −1 ) 𝑋, (𝐿 −1 ) 𝑌⟩ denoted by log. 𝐴 𝐴 ∗ 𝐴 ∗ 𝐼 The most general matrix Lie group is the general linear 𝑇 (7) (𝑛, R) 𝑛×𝑛 =⟨𝐴−1𝑋, 𝐴−1𝑌⟩ := ((𝐴−1𝑋) 𝐴−1𝑌) group GL consisting of the invertible matrices 𝐼 tr with real entries. As the inverse image of R −{0}under the 𝐴 󳨃→ (𝐴) (𝑛, R) continuous map det ,GL is an open subset with 𝑋, 𝑌𝐴 ∈𝑇 𝑆 and tr denoting the trace of the matrix. of the set of 𝑛×𝑛real matrices, denoted by 𝑀𝑛×𝑛,whichis 𝑆 𝑛×𝑛 Similarly, we can define the right invariant metric on as well. isomorphic to R , it has a differentiable manifold structure It has been shown that there exist the left invariant metrics on (submanifold). The group multiplication of GL(𝑛, R) is the all matrix Lie groups. usual matrix multiplication, the inverse map takes a matrix 𝐴 (𝑛, R) 𝐴−1 on GL to its inverse , and the identity element is the 2.2. Compact Matrix Lie Group. A Lie group is compact if identity matrix 𝐼.TheLiealgebragl(𝑛, R) of GL(𝑛, R) turns 𝑈(𝑛) 𝑀 its differential structure is compact. The unitary group , out to be 𝑛×𝑛 with the Lie bracket defined by the matrix the special unitary group SU(𝑛), the orthogonal group 𝑂(𝑛), commutator the special orthogonal group SO(𝑛),andthesymplecticgroup Sp(𝑛) are the examples of the compact matrix Lie groups [17]. [𝑋, 𝑌] =𝑋𝑌−𝑌𝑋, gl∀𝑋,𝑌∈ (𝑛, R) . (1) Denote a compact Lie group by 𝑆1 anditsLiealgebrabys1. ⟨⋅, ⋅⟩ 𝑆 All other real matrix Lie groups are subgroups of There exists an adjoint invariant metric on 1 such that GL(𝑛, R), and their group operators are subgroup restrictions ⟨Ad𝐴𝑋, Ad𝐴𝑌⟩ = ⟨𝑋, 𝑌⟩ (8) of the ones on GL(𝑛, R). The Lie bracket on their Lie algebras is still the matrix commutator. with 𝑋, 𝑌 ∈ s1. Notice the fact that the left invariant metric Let 𝑆 denote a matrix Lie group and s its Lie algebra. of any adjoint invariant metric is also right invariant; namely, The exponential map for 𝑆 turns out to be just the matrix it is a bi-invariant metric; so all compact Lie groups have exponential; that is, given an element 𝑋∈s, the exponential bi-invariant metrics. Furthermore, if the left invariant and map is the adjoint invariant metrics on 𝑆1 deduce a Riemannian ∇ ∞ 𝑋𝑚 connection , then the following properties are valid: exp (𝑋) = ∑ . (2) 𝑚! 1 𝑚=0 ∇ 𝑌= [𝑋, 𝑌] , 𝑋 2 (9) Theinversemap,thatis,thelogarithmicmap,isdefinedas 1 follows: ⟨R (𝑋, 𝑌) 𝑋, 𝑌⟩ =− ⟨[𝑋, 𝑌] , [𝑋, 𝑌]⟩ , 4 ∞ 𝑚 𝑚+1 (𝐴−𝐼) log (𝐴) = ∑ (−1) , (3) where R(𝑋, 𝑌) is a curvature operator about the smooth 𝑚=1 𝑚 tangent vector field on the Riemannian manifold (𝑆1,∇). Therefore, the section curvature K is given by for 𝐴 in a neighborhood of the identity 𝐼 of 𝑆. The exponential ofamatrixplaysacrucialroleinthetheoryoftheLiegroups, ⟨[𝑋, 𝑌] , [𝑋, 𝑌]⟩ which can be used to obtain the Lie algebra of a matrix Lie K (𝑋, 𝑌) = ≥0, 4(⟨𝑋, 𝑋⟩⟨𝑌, 𝑌⟩ − ⟨𝑋, 𝑌⟩2) (10) group, and it transfers information from the Lie algebra to the Lie group. K The matrix Lie group also has the structure of a Rieman- which means that is nonnegative on the compact Lie nian manifold. For any 𝐴, 𝐵 ∈𝑆 and 𝑋∈𝑇𝐴𝑆,thetangent group. space of 𝑆 at 𝐴,wehavethemapsthat In addition, according to the Hopf-Rinow theorem, a compact connected Lie group is geodesically complete. It 𝐿𝐴𝐵 = 𝐴𝐵, (𝐿𝐴)∗𝑋=𝐴𝑋, means that, for any given two points, there exists a geodesic curve connecting them and the geodesic curve can extend −1 (4) 𝑅𝐴𝐵=𝐵𝐴 ,(𝑅𝐴−1 )∗𝑋=𝑋𝐴, infinitely. The Scientific World Journal 3

3 2.3. The Riemannian Mean on Matrix Lie Group. Let 𝛾: with 𝜔=(𝜔𝑥,𝜔𝑦,𝜔𝑧)∈R . ‖𝜔‖𝐹 gives the amount of [0, 1] → 𝑆 be a sufficiently smooth curve on 𝑆. We define rotation with respect to the unit vector along 𝜔,where‖⋅‖𝐹 the length of 𝛾(𝑡) by denotes the Frobenius norm. Physically, 𝜔 represents the 1 angularvelocityoftherigidbody,whereasV corresponds to ̇ ̇ ℓ(𝛾):=∫ √⟨𝛾 (𝑡) , 𝛾 (𝑡)⟩𝛾(𝑡)𝑑𝑡 the linear velocity [19]. In [18], the author presents a closed- 0 form expression of the exponential map se(3) → SE(3) by (11) 1 𝑇 1− 𝜃 𝜃− 𝜃 = ∫ √ {(𝛾(𝑡)−1𝛾̇ (𝑡)) 𝛾(𝑡)−1𝛾̇ (𝑡)}𝑑𝑡, cos ( ) 2 sin ( ) 3 tr exp (𝑉) =𝐼4 +𝑉+ 𝑉 + 𝑉 (18) 0 𝜃2 𝜃3 where 𝑇 denotes the transpose of the matrix. The geodesic 2 2 2 2 with 𝑉∈se(3) and 𝜃 =𝜔𝑥 +𝜔𝑦 +𝜔𝑧.Notethatitcan distance between two matrices 𝐴 and 𝐵 on 𝑆 considered as be regarded as an extension of the well-known Rodrigues a differentiable manifold is the infimum of the lengths ofthe formula on SO(3).ThelogarithmicmapSE(3) → se(3) is curves connecting them; that is, yielded as 𝑑 (𝐴,) 𝐵 := inf {ℓ(𝛾)|𝛾:[0, 1] 𝑄 =𝑞 (𝑞 𝐼 −𝑞 𝑄+𝑞 𝑄2 −𝑞 𝑄3), (12) log ( ) 1 2 4 3 4 5 (19) 󳨀→ 𝑆 with 𝛾 (0) =𝐴,𝛾(1) =𝐵}. where According to the Euclidean analogue (mean on Euclidean 1 3 𝜃 𝜃 space), a definition of the mean of 𝑁 matrices 𝑅1,...,𝑅𝑁 is 𝑞 = ( ) ( ), 1 8csc 2 sec 2 the minimizer of the sum of the squared distances from any matrix to the given matrices 𝑅1,...,𝑅𝑁 on 𝑆. Now, we define 𝑞2 =𝜃cos (2𝜃) − sin (𝜃) , theRiemannianmeanbasedonthegeodesicdistance(12). (20) 𝑞3 =𝜃cos (𝜃) +2𝜃cos (2𝜃) − sin (𝜃) − sin (2𝜃) , Definition 1. The mean of 𝑁 given matrices 𝑅1,...,𝑅𝑁 on 𝑆 in the Riemannian sense corresponding to the metric (7)is 𝑞4 =2𝜃cos (𝜃) +𝜃cos (2𝜃) − sin (𝜃) − sin (2𝜃) , defined as 𝑞 =𝜃 (𝜃) − (𝜃) , 𝑁 5 cos sin 1 2 𝑅=arg min ∑𝑑(𝑅𝑘,𝑅) . (13) (𝑄) = 2 + 2 (𝜃) −𝜋<𝜃<𝜋 𝑅∈𝑆 2𝑁 tr cos ,for . 𝑘=1 (𝑛) (𝑛) 3.2. Algorithm for the Riemannian Mean on SE . Denote 3. The Riemannian Mean on SE 𝑃, 𝑄 ∈ SE(𝑛) by In this section, we discuss the Riemannian mean on the 𝐴 𝑏 𝐴 𝑏 𝑃=( 1 1), 𝑄=( 2 2). special Euclidean group SE(𝑛),whichisasubgroupofGL(𝑛+ 01 01 (21) 1, R).Moreover,thespecialrigidbodymotiongroupSE(3) is taken as an illustrating example. Taking the corresponding exponential mappings on man- 𝑛 ifolds SO(𝑛) and R into consideration, the geodesic 𝛾𝑃,𝑄 𝑛 𝑃 𝑄 (𝑛) 3.1. About SE(𝑛). The special Euclidean group SE(𝑛) in R is between and on the Lie group SE is given by the semidirect product of the special orthogonal group SO(𝑛) 𝑛 𝛼𝐴 ,𝐴 (𝑡) 𝛽𝑏 ,𝑏 (𝑡) R 𝛾 (𝑡) =( 1 2 1 2 ) with itself [18]; that is, 𝑃,𝑄 01 (𝑛) = (𝑛) ⋉ R𝑛. SE SO (14) 𝑡 (22) 𝐴 (𝐴𝑇𝐴 ) 𝑏 +(𝑏 −𝑏)𝑡 (𝑛) =( 1 1 2 1 2 1 ), The matrix representation of elements of SE is 01 𝐴𝑏 𝑛 (𝑛) ={( )|𝐴∈ (𝑛) ,𝑏∈R }. 𝑛 SE 01 SO (15) where 𝛼:[0,1]→ SO(𝑛) and 𝛽:[0,1]→ R are the geodesics expressed, respectively, by An element of SE(𝑛) physically represents a displacement, 𝐴 𝑇 𝑇 𝑡 where corresponds to the orientation, or attitude, of the 𝛼𝐴,𝐵 (𝑡) = exp𝐴 (𝑡 log (𝐴 𝐵)) = 𝐴(𝐴 𝐵) ,𝐴,𝐵∈SO (𝑛) , rigid body and 𝑏 encodes the translation. The Lie algebra se(𝑛) of SE(𝑛) canbedenotedby 𝛽 (𝑡) = (𝑡 (𝑏 −𝑏)) 𝑏1,𝑏2 exp𝑏1 2 1 Ω V (𝑛) ={( )|Ω𝑇 =−Ω,V ∈ R𝑛}. =𝑏 +(𝑏 −𝑏)𝑡, 𝑏 ,𝑏 ∈ R𝑛. se 00 (16) 1 2 1 1 2 (23) Specially, when 𝑛=3, the skew-symmetric matrix Ω can 𝑃 𝑄 be uniquely expressed as Then, the midpoint of and is defined by 𝑇 1/2 1 0−𝜔𝑧 𝜔𝑦 𝐴 (𝐴 𝐴 ) (𝑏 +𝑏) 1 1 2 2 1 1 Ω=(𝜔𝑧 0−𝜔𝑥) (17) 𝑃∘𝑄=( ). (24) −𝜔𝑦 𝜔𝑥 0 01 4 The Scientific World Journal

Before the geodesic distance on SE(𝑛) is given, we first In addition, it is valuable to mention that the distance 𝑇 introduce a lemma which is a known conclusion in linear ‖log(𝐴1𝐴2)‖𝐹, induced by the standard bi-invariant metric algebra [20]. on SO(𝑛), stands for the rotation motion from the point 𝑃 to 𝑄 and the distance ‖𝑏2 −𝑏1‖𝐹 stands for the translation 𝑛×𝑛 𝑚×𝑚 𝑛 Lemma 2. If 𝐸∈R and 𝐻∈R are invertible motion on R . Therefore, considering an object undergoing matrices, then the block matrix a rigid body Euclidean motion, then, this motion can be decomposed into a rotation with respect to the center of mass 𝐸𝐹 ( ) of the object and a translation of the center of mass of the 0𝐻 (25) object. 𝑛×𝑚 is invertible, where 𝐹∈R .Furthermore, Theorem 4. For 𝑁 given points on SE(𝑛)

−1 −1 −1 −1 𝐸𝐹 𝐸 −𝐸 𝐹𝐻 𝐴𝑘 𝑏𝑘 ( ) =( ). 𝑃𝑘 =( ), (32) 0𝐻 0𝐻−1 (26) 01 𝑛 where 𝐴𝑘 ∈ SO(𝑛) and 𝑏𝑘 ∈ R , 𝑘 = 1,2,...,𝑁,ifthe Now, we give the geodesic distance on SE(𝑛) as follows. Riemannian mean of 𝐴1,𝐴2,...,𝐴𝑁 and the Riemannian 𝑏 ,𝑏 ,...,𝑏 Lemma 3. The geodesic distance between two points 𝑃 and 𝑄 mean of 1 2 𝑁 (i.e., arithmetic mean) are denoted by 𝐴 𝑏 𝑃 on SE(𝑛) induced by the scale-dependent left invariant metric and , respectively, then, one has the Riemannian mean of 𝑃 ,𝑃,...,𝑃 ∈ (𝑛) (7) is given by 1 2 𝑁 SE by 𝐴 𝑏 󵄩 󵄩2 󵄩 󵄩2 1/2 𝑃=( ). 𝑑 (𝑃, 𝑄) =(󵄩 (𝐴𝑇𝐴 )󵄩 + 󵄩𝑏 −𝑏󵄩 ) . (27) (33) 󵄩log 1 2 󵄩𝐹 󵄩 2 1󵄩𝐹 01

Proof. As mentioned above, the geodesic distance between Proof. In the Riemannian sense, by (13), the mean 𝑃 is defined two matrices 𝑃 and 𝑄 on SE(𝑛) is achieved by the length of as geodesics connecting them; thus, we will compute it through 𝑁 1 2 substituting (22)into(11). 𝑃=arg min ∑𝑑(𝑃𝑘,𝑃) 𝑃∈ (𝑛) 2𝑁 From Lemma 2,weget SE 𝑘=1

−1 𝑁 1 󵄩 󵄩2 󵄩 󵄩2 𝛾𝑃,𝑄 (𝑡) 󵄩 𝑇 󵄩 󵄩 󵄩 = arg min ∑ (󵄩log (𝐴𝑘𝐴)󵄩 + 󵄩𝑏𝑘 −𝑏󵄩 ) 𝑃∈ (𝑛) 2𝑁 󵄩 󵄩𝐹 𝐹 SE 𝑘=1 𝑇 −𝑡 𝑇 𝑇 −𝑡 𝑇 (𝐴1𝐴2) 𝐴1 −(𝐴1𝐴2) 𝐴1 (𝑏1 +(𝑏2 −𝑏1)𝑡) (34) 𝑁 =( ). 1 󵄩 󵄩2 󵄩 𝑇 󵄩 01 = arg min ∑󵄩log (𝐴𝑘𝐴)󵄩 𝐴∈ (𝑛) 2𝑁 󵄩 󵄩𝐹 SO 𝑘=1 (28) 𝑁 1 󵄩 󵄩2 Then, according to the principle about the derivatives ofthe + arg min ∑󵄩𝑏𝑘 −𝑏󵄩 . 𝑏∈R𝑛 2𝑁 𝐹 matrix-valued functions, the following formula is valid: 𝑘=1

𝑡 From [9], the geodesic distance between 𝐴𝑘 and 𝐴 on SO(𝑛) 𝐴 (𝐴𝑇𝐴 ) (𝐴𝑇𝐴 )𝑏−𝑏 𝛾̇ (𝑡) =( 1 1 2 log 1 2 2 1). is given by 𝑃,𝑄 00(29) 󵄩 󵄩2 𝑑(𝐴 ,𝐴)2 = 󵄩 (𝐴𝑇𝐴)󵄩 , 𝑘 󵄩log 𝑘 󵄩𝐹 (35) Moreover, we have that so we have that −1 𝑇 −1 tr ((𝛾 (𝑡) 𝛾𝑃,𝑄̇ (𝑡)) 𝛾 (𝑡) 𝛾𝑃,𝑄̇ (𝑡)) 𝑁 𝑃,𝑄 𝑃,𝑄 1 󵄩 󵄩2 󵄩 𝑇 󵄩 2 (30) arg min ∑󵄩log (𝐴𝑘𝐴)󵄩 = arg min 𝑑(𝐴𝑘,𝐴) = 𝐴. 2 𝑇 𝑇 𝐴∈ (𝑛) 2𝑁 󵄩 󵄩𝐹 𝐴∈ (𝑛) SO 𝑘=1 SO =−log (𝐴1𝐴2)+(𝑏2 −𝑏1) (𝑏2 −𝑏1). (36) Therefore, the geodesic distance on SE(𝑛) between 𝑃 and 𝑛 On the other hand, for 𝑏𝑘 ∈ R , 𝑘 = 1,2,...,𝑁,itiseasyto 𝑄 is given by see that 1 1/2 𝑁 𝑁 𝑑 (𝑃, 𝑄) = ∫ (− 2 (𝐴𝑇𝐴 )+(𝑏 −𝑏)2) 𝑑𝑡 1 󵄩 󵄩2 1 log 1 2 2 1 arg min ∑󵄩𝑏𝑘 −𝑏󵄩 = ∑𝑏𝑘 = 𝑏. (37) 0 𝑏∈R𝑛 2𝑁 𝐹 𝑁 (31) 𝑘=1 𝑘=1 󵄩 󵄩2 󵄩 󵄩2 1/2 =(󵄩 (𝐴𝑇𝐴 )󵄩 + 󵄩𝑏 −𝑏󵄩 ) . 󵄩log 1 2 󵄩𝐹 󵄩 2 1󵄩𝐹 Therefore, the fact is shown that the Riemannian mean 𝑏 of {𝑏𝑘} is equivalent to the arithmetic mean. This completes the proof of Lemma 2. Consequently, we prove that equality (33)isvalid. The Scientific World Journal 5

𝐿 In addition, let denote the cost function of the mini- 3 (𝑛) mization problem (34)onSE ;thatis, Q 𝐿 (𝑃) =𝐿 (𝐴) +𝐿 (𝑏) 2.5 rota trans 𝑁 𝑁 2 1 󵄩 𝑇 󵄩2 1 󵄩 󵄩2 (38) = ∑󵄩 (𝐴 𝐴)󵄩 + ∑󵄩𝑏 −𝑏󵄩 , 2𝑁 󵄩log 𝑘 󵄩𝐹 2𝑁 󵄩 𝑘 󵄩𝐹 𝑘=1 𝑘=1 z 1.5 𝐿 𝐿 where rota and trans standfortherotationandthetransla- 1 tion components of the cost function 𝐿,respectively.Wehave 𝐿 (𝐴) 𝐴∈ (𝑛) the gradient of rota for SO as follows [21]: 0.5

𝑁 𝑇 0 P grad (𝐿 )=−𝐴∑ log (𝐴 𝐴𝑘). (39) 4 rota −1 𝑘=1 2 0 0 2 1 −2 4 3 x y Consequently, the Riemannian gradient descent algorithm is applied to calculate 𝐴, taking the geodesic on SO(𝑛) as Figure 1: The rigid motion 𝐷(𝑡) from 𝑃 to 𝑄. the trajectory and the negative gradient (39) as the descent direction. Finally, we achieve the following algorithm for computing 3 the Riemannian mean 𝑃 on SE(𝑛). Q 2.5 Algorithm 5. Given 𝑁 matrices 𝑃𝑘, 𝑘 = 1,2,...,𝑁,on SE(𝑛), their Riemannian mean 𝑃 is computed by the following 2 iterative method. 𝑁 z 1.5 (1) Store (1/𝑁) ∑𝑘=1 𝑏𝑘 to 𝑏. P ∘ Q

(2) Set 𝐴=𝐴1 as an initial input, and choose a desired 1 tolerance 𝜀>0. 𝑁 𝑇 0.5 (3) If ‖∑𝑘=1 log(𝐴 𝐴𝑘)‖𝐹 <𝜀,thenstop. P 𝑇 𝐴=𝐴 {−𝜀 ∑𝑁 (𝐴 𝐴 )} 0 (4) Otherwise, update exp 𝑘=1 log 𝑘 , 4 andgotostep(3). 2 0 −1 0 2 1 x −2 4 3 y 3.3. Simulations on SE(3). Let us consider a rigid object 𝑊 in the Euclidean space undergoing a rigid body Euclidean Figure 2: The Riemannian mean 𝑃∘𝑄. motion SE(3). Suppose that the coordinate of the center of 3 gravity in 𝑊 is 𝑑𝑊 ∈ R ;then,theoptimaltrajectoryfrom the configuration 𝑃 to 𝑄 is the curve 𝐷(𝑡) such that 4.1. About UP(𝑛). The set of all of the uppertriangular 𝑛× 𝐷 (𝑡) 𝑑 𝑛 matrices with diagonal elements that are all one is called ( )=𝛾 (𝑡) ( 𝑊), 1 𝑃,𝑄 1 (40) unipotent matrices group, denoted by UP(𝑛). In fact, given an invertible matrix 𝐶∈UP(𝑛),thereis 𝑡 ∈ [0, 1] 𝛾 (𝑡) a neighborhood 𝑈 of 𝐶 such that every matrix in 𝑈 is also where and 𝑃,𝑄 denotes the geodesic connecting 𝑛×𝑛 𝑃 and 𝑄 on SE(3) (see Figure 1). For the configuration of two in UP(𝑛),soUP(𝑛) is an open subset of R .Furthermore, points 𝑃 and 𝑄,asshowninFigure 2,givenbytheangular the matrix product 𝑃⋅𝑄is clearly a smooth function of 𝑃 𝑄 𝑃−1 velocity 𝜔𝑃, 𝜔𝑄 of the rigid body and the linear velocity the entries of and ,and is a smooth function of 𝑃 (𝑛) V𝑃, V𝑄,wechoose𝜔𝑃 = (𝜋/2)(0, 1, 1), V𝑃 = (0, 0, 0), 𝜔𝑄 = the entries of .Thus,UP is a Lie group. On the other (𝑛) 𝑛(𝑛 − 1)/2 𝜋(1/4, 0, −1/2),andV𝑄 = (4.380, −1.348, 3.690);then,we hand, it can be verified that UP is of dimension obtain their Riemannian mean according to Algorithm 5, andisnilpotent.Sincewecanusethenonzeroelements which is just the middle point 𝑃∘𝑄from (24). 𝐶𝑖𝑗 ,𝑖<𝑗, directly as global coordinate functions for UP(𝑛), 𝑛(𝑛−1)/2 the manifold underlying UP(𝑛) is diffeomorphic to R . 4. The Riemannian Mean on UP(𝑛) Therefore, UP(𝑛) is not compact, but simply connected. The Lie algebra up(𝑛) of UP(𝑛) consists of uppertriangu- In this section, the Riemannian mean of 𝑁 given points on lar matrices 𝑇 with diagonal elements 𝑇𝑖𝑖 = 0, 𝑖 = 1,...,𝑛. the unipotent matrix group UP(𝑛) is considered. UP(𝑛) is It is an indispensable tool which gives a realization of the anoncompactmatrixLiegroupaswell.Moreover,inthe Heisenberg commutation relations of quantum mechanics in special case 𝑛=3,itistheHeisenberggroup𝐻(3). the 3-dimensional case [17]. 6 The Scientific World Journal

𝑘 Moreover, it is the fact that both 𝐶−𝐼and 𝑇 are all so the Riemannian mean 𝐴 of the 𝑁 matrices {𝐵 } should nilpotent matrices, for any 𝐶∈UP(𝑛) and 𝑇∈up(𝑛).Thus, satisfy from (2)and(3), the infinite series representations of the (𝑛) exponential mapping in up and the logarithm mapping in 𝑁 (𝑛) −1 𝑘 UP canbegiven,respectively,by ∑ log (𝐴 𝐵 )=0. (49) 𝑘=1 𝑛 (𝐶−𝐼)𝑚 (𝐶) = ∑ (−1)𝑚+1 , log 𝑚 (41) 𝑚=1 From the logarithm of the matrices on UP(𝑛) given by (41), we can rewrite (49)as where 𝐶∈UP(𝑛), ‖𝐶𝐹 −𝐼‖ <1,and

𝑛 𝑚 𝑇 −1 𝑘 𝑚 (𝑇) = ∑ 𝑁 𝑛 (𝐴 𝐵 −𝐼) exp 𝑚! (42) ∑ ∑ (−1)𝑚+1 =0. (50) 𝑚=0 𝑚 𝑘=1𝑚=1 with 𝑇∈up(𝑛). (𝑛) Notice that UP is connected, which means that, for any Therefore, the Riemannian mean 𝐴 of the 𝑁 given matrices 𝐴, 𝐵 𝛾(𝑡) 𝑘 given pair , we can find a geodesic curve such that {𝐵 } can be given explicitly by solving (50). 𝛾(0) = 𝐴 and 𝛾(1) = 𝐵, namely, by taking the initial velocity 𝑛=2 −1 For the case of ,from(50), it is shown that the as 𝛾(0)̇ = log(𝐴 𝐵). Let the geodesic curve 𝛾(𝑡) be 𝑘 Riemannian mean 𝐴2 of 𝑁 given matrices {𝐵2} in UP(2) is −1 their arithmetic mean; that is, 𝛾 (𝑡) =𝐴exp (𝑡 log (𝐴 𝐵)) ∈ UP (𝑛) (43)

𝛾(0) = 𝐴, 𝛾(1) =𝐵 𝛾󸀠(0) = (𝐴−1𝐵) ∈ (𝑛) 1 𝑁 with ,and log up . 𝐴 = ∑𝐵𝑘. 𝐴 𝐵 2 𝑁 2 (51) Then, the midpoint of and is given by 𝑘=1

1 −1 𝐴∘𝐵=𝐴exp ( log (𝐴 𝐵)) , (44) 2 Next, for 𝑛=3, we obtain the Riemannian mean on UP(3) (𝐻(3)) as follows. and from (11) the geodesic distance 𝑑(𝐴, 𝐵) canbecomputed 𝑘 explicitly by Theorem 6. Given 𝑁 matrices {𝐵3} on the Heisenberg group 󵄩 󵄩 𝐻(3) by 𝑑 (𝐴,) 𝐵 = 󵄩 (𝐴−1𝐵)󵄩 . 󵄩log 󵄩𝐹 (45) 1𝑏𝑘 𝑏𝑘 4.2. Algorithm for the Riemannian Mean on UP(𝑛). For 𝑁 12 13 1 2 𝑁 given points 𝐵 ,𝐵 ,...,𝐵 in UP(𝑛), 𝐿 denotes the cost 𝑘 𝑘 𝐵3 =(01𝑏), (52) function of the minimization problem (13); that is, 23 00 1 𝑁 1 𝑘 2 min 𝐿 (𝐴) = min ∑𝑑(𝐵 ,𝐴) . (46) 𝐴∈ (𝑛) 𝐴∈ (𝑛) 2𝑁 UP UP 𝑘=1 where 𝑘=1,2,...,𝑁, then, one has the Riemannian mean 𝐴3 Following [22, 23], it has been shown that the Jacobi field is on the Heisenberg group 𝐻(3) such that equal to zero at the Riemannian mean. The Jacobi field for the Riemannian mean is equal to the sum of tangent vectors to all 1 1 𝑏 𝑏 − (𝑏 ,𝑏 ) geodesics (from mean to each point). Noticing the fact that 12 13 2 cov 12 23 the geodesic between two points 𝐴 and 𝐵 has already been 𝐴3 =(01 𝑏 ), (53) given by (43),wecanthencomputetheJacobifieldatpoint 23 𝐴 𝑁 𝐵𝑘 𝑡=0 to points (at )suchthat 00 1 −1 𝑘 𝑡 −1 𝑘 𝛾𝑘 (𝑡) =𝐴(𝐴 𝐵 ) =𝐴exp (𝑡 log (𝐴 𝐵 )) , 𝑁 𝑘 󵄨 where 𝑏𝑖𝑗 := (1/𝑁) ∑𝑘=1 𝑏𝑖𝑗 , 𝑖, 𝑗 = 1, 2, 3 (𝑖 <𝑗),andcov(𝑏12, 𝑑𝛾 (𝑡)󵄨 (47) 𝑘 󵄨 =𝐴 (𝐴−1𝐵𝑘), 𝑘=1,...,𝑁. 𝑁 𝑘 𝑘 󵄨 log 𝑏23):=(1/𝑁)∑𝑘=1(𝑏12 −𝑏12)(𝑏23 −𝑏23). 𝑑𝑡 󵄨𝑡=0 𝐴 Then, we suppose that the summation of all these vectors Proof. First, let us denote the Riemannian mean 3 by should be equal to zero; that is,

𝑁 󵄨 𝑁 1𝑎12 𝑎13 𝑑𝛾𝑘 (𝑡)󵄨 −1 𝑘 𝐿 = ∑ 󵄨 =𝐴∑ (𝐴 𝐵 )=0, 𝐴3 =(01𝑎23). (54) 𝐴 𝑑𝑡 󵄨 log (48) 𝑘=1 󵄨𝑡=0 𝑘=1 00 1 The Scientific World Journal 7

𝑘 Then, note that, for the given matrices {𝐵3} on 𝐻(3),their B2 Riemannian mean 𝐴3 has to satisfy (50), so we get the 20 following solutions: 15 1 𝑁 𝑎 = ∑𝑏𝑘 , 12 𝑁 12 10 𝑘=1 B1 23 𝑁 a 5 1 𝑘 𝑎23 = ∑𝑏 , (55) 𝑁 23 0 𝑘=1

3 1 𝑁 −5 B 𝑎 = ∑ (𝑎 −𝑏𝑘 )(𝑎 −𝑏𝑘 ). 0 13 𝑁 12 12 23 23 𝑘=1 5 0 𝑁 a 10 10 12 Asamatterofconvenience,supposingthat𝑏𝑖𝑗 := (1/𝑁)𝑘=1 ∑ 20 15 30 𝑘 𝑁 40 a13 𝑏𝑖𝑗 , 𝑖, 𝑗 = 1, 2, 3 (𝑖 <𝑗),andcov(𝑏12,𝑏23):=(1/𝑁)∑𝑘=1(𝑏12 − 𝑘 𝑘 𝑏12)(𝑏23 −𝑏23),weshowthat(54)isvalid. Figure 3: The geodesic triangle on 𝐻(3). This completes the proof of Theorem 6. More generally, while 𝑛>1, we can get the Riemannian which means that (58)isvalidand𝐴𝑛−1 satisfies the equation mean on UP(𝑛) given by the following theorem. 𝑁 𝑛 𝑚−1 𝑘 (−1) 𝑚 Theorem 7. Take 𝑛>1.For𝑁 given matrices {𝐵 } in UP(𝑛), ∑ ∑ (𝐴−1 𝐵𝑘 −𝐼) =0. 𝑛 𝑚 𝑛−1 𝑛−1 (61) one assumes that they are in the form of 𝑘=1𝑚=1 𝐵𝑘 𝑏𝑘 𝐵𝑘 =( 𝑛−1 𝑛−1) Moreover, from (41), we have that 𝑛 01 (56) −1 𝑘 log (𝐴𝑛−1𝐵𝑛−1) =0. (62) 𝑘 𝑘 𝑛−1 with 𝐵𝑛−1 ∈UP(𝑛−1)and 𝑏𝑛−1 ∈ R ; then, the Riemannian 𝑘 𝐴 𝑁 𝐵 Furthermore, it is shown that 𝐴𝑛−1 is the Riemannian mean mean 𝑛 of the matrices 𝑛 is given by 𝑘 of {𝐵 }.Atlast,wewrite𝐴𝑛−1 as 𝐴𝑛−1,sotheproofof 𝐴 𝑎 𝑛−1 𝐴 =( 𝑛−1 𝑛−1), Theorem 7 is completed. 𝑛 01 (57) Asshownabove,wegivetheiterativeformulaforcom- 𝐴 {𝐵𝑘 } 𝑎 where 𝑛−1 is the Riemannian mean of 𝑛−1 and 𝑛−1 is given puting the Riemannian mean for any dimension 𝑛>1.Either by the formula that (51)or(54)canbechosenastheinitialformula. −1 𝑁 𝑛−1 𝑚 𝑚 (−1) −1 𝑘 𝑎 = 𝐴 (∑ ∑ (𝐴 𝐵 −𝐼) ) 4.3. Simulations on 𝐻(3). In this section, we take two 𝑛−1 𝑛−1 𝑚+1 𝑛−1 𝑛−1 𝑘=1𝑚=0 examplestoillustratetheresultsabouttheRiemannianmean (58) 𝐻(3) 𝑁 𝑛−1 𝑚 on the Heisenberg group , which is the 3-dimensional (−1) −1 𝑚 −1 ×(∑ ∑ (𝐴 𝐵𝑘 −𝐼) 𝐴 𝑏𝑘 ). space. 𝑚+1 𝑛−1 𝑛−1 𝑛−1 𝑛−1 𝑘=1𝑚=0 Example 8. Consider the Riemannian mean of three points 1 2 3 Proof. For simplicity of exposition, we suppose that the 𝐵 , 𝐵 , 𝐵 on the Heisenberg group 𝐻(3).Using(43), we Riemannian mean 𝐴𝑛 is the block matrix in the form of can get the geodesics of three points on 𝐻(3),whichforma 𝐴 𝑎 geodesic triangle. In Figure 3, all of the curves are geodesics. 𝐴 =( 𝑛−1 𝑛−1) 𝑛 01 (59) Moreover, as shown in Figure 4, the midpoint of each geodesic is easy to be obtained by (44). Thus, each centerline 𝑘 𝑛−1 with 𝐴𝑛−1 ∈ UP(𝑛 − 1) and 𝑎𝑛−1 ∈ R . Since the Rie- connects a vertex to the midpoint of its opposing side. On 𝑘 𝐻(3), these centerlines always meet in a single point which mannian mean 𝐴𝑛 of the 𝑁 matrices {𝐵𝑛} should satisfy (50), we substitute the block matrix forms (59)and(57)into is coincident with the Riemannian mean computed by (54), (50). Then, we obtain the following matrix equation for the denoted by a red dot as shown in Figure 4. Riemannian mean 𝐴𝑛: 1 2 3 4 Example 9. Given four points 𝐵 , 𝐵 , 𝐵 , 𝐵 on the Heisen- 𝑁 𝑛 (−1)𝑚−1 𝐻(3) ∑ ∑ berg group , we can get a geodesic tetrahedron from 𝑘=1𝑚=1 𝑚 (43)(seeFigure 5), where all curves are geodesics. Moreover, similar to Example 8, the Riemannian means of three vertexes −1 𝑘 𝑚 −1 𝑘 𝑚−1 −1 𝑘 ×((𝐴𝑛−1𝐵𝑛−1 −𝐼) (𝐴𝑛−1𝐵𝑛−1 −𝐼) 𝐴𝑛−1 (𝑏𝑛−1 −𝑎𝑛−1))=0, on each curved face are obtained, denoted by red circles 00 (see Figure 6). Then, we plot each centerline which connects (60) a vertex to the Riemannian mean of its opposing side. It 8 The Scientific World Journal

B2 20 B2

15

10 10 B1 23 a 5 1 4 5 B B 0 20 23 a 3 −5 B 0 15 0

13 5 10 a 0 −5 B3 10 10 a 20 6 12 30 4 2 5 15 40 a13 0 −2 a12 −4 Figure 4: The Riemannian mean of three points. Figure 6: The Riemannian mean of four points.

Conflict of Interests

B2 The authors declare that there is no conflict of interests regarding the publication of this paper.

10 Acknowledgments 4 B The authors wish to express their appreciation to the review- 5 B1 20 ers for their helpful suggestions which greatly improved the 23

a presentation of this paper. This paper is supported by the 0 15 National Natural Science Foundation of China (nos. 61179031 and 10932002).

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