Unit Dual Quaternion Directed Graph (UDQDG)

• UDQDG is a directed graph where each arc carries a unit dual quaternion encoding both rotation and translation for rigid-body poses.
• It enforces formation consistency through cycle-consistency and factorization conditions, ensuring global realizability of desired configurations.
• The model extends classical Laplacian theory by integrating full pose constraints, enabling robust convergence analysis for multi-agent formation control.

A Unit Dual Quaternion Directed Graph (UDQDG) is a weighted directed graph in which each arc carries a unit dual quaternion encoding a desired relative rigid-body pose. In the formation-control interpretation, the vertex set indexes 3-D rigid bodies, the directed edges encode sensing or communication relations, and each label $\varphi(i,j)=\hat q_{d_{ij}}\in\hat{\mathbb U}$ specifies the desired relative configuration from agent $i$ to agent $j$. The construct was introduced as the graph-theoretic object that carries full relative pose information—rotation and translation simultaneously—and underlies a dual-quaternion Laplacian theory for directed multi-agent formation control (Cui et al., 14 Aug 2025, Qi et al., 2024).

1. Algebraic model and graph semantics

The underlying directed graph is $G=(V,E)$, and the UDQDG is written either as

$\Phi=(G,\varphi)$

or, equivalently, as

$\Phi=(G,\hat{\mathbb U},\varphi),$

where $\hat{\mathbb U}$ denotes the set of unit dual quaternions. For each arc $(i,j)\in E$, the edge label is a unit dual quaternion

$\varphi(i,j)=\hat q_{d_{ij}},$

interpreted as the desired relative pose from rigid body $i$ to rigid body $i$0 (Cui et al., 14 Aug 2025, Qi et al., 2024).

The algebraic carrier of these edge labels is the unit dual quaternion. A dual quaternion has the form

$i$1

with $i$2 and $i$3. It is unit iff

$i$4

The cited papers use the standard rigid-pose forms

$i$5

and

$i$6

so that the standard part encodes attitude and the dual part encodes translation coupled to that attitude (Cui et al., 14 Aug 2025, Qi et al., 2024).

This graph model differs structurally from an ordinary weighted digraph because the edge weights are not scalars measuring influence or coupling strength; each edge weight is a full rigid transformation constraint. It also differs from quaternion-only graph formulations because quaternion weights encode attitude only, whereas unit dual quaternion weights encode both rotation and translation in one algebraic object. If the reciprocity condition

$i$7

holds on opposite arcs, the structure reduces to the special symmetric unit-gain case called a dual quaternion unit gain graph (DQUGG) (Cui et al., 14 Aug 2025).

2. Reasonableness, balancedness, and realizability

A UDQDG edge-label scheme is not automatically realizable. The central consistency condition is that the desired relative configuration scheme $i$8 is reasonable iff there exists a desired formation vector

$i$9

such that, for every $j$0,

$j$1

This is the defining factorization of each edge label into absolute node poses, and it is the exact sense in which the labels represent a globally compatible rigid formation (Cui et al., 14 Aug 2025, Qi et al., 2024).

In the more general graph-theoretic language of unit weighted directed graphs, the same property is called balanced. For a UDQDG whose underlying digraph has a directed spanning tree, reasonableness is equivalent to a directed cycle-consistency condition: for every cycle

$j$2

one must have

$j$3

where the factor on each step is $j$4 on a forward arc and the conjugate $j$5 when the reverse arc is traversed. This is the exact directed analogue of cycle neutrality: the net rigid motion around every closed loop must be identity (Qi et al., 2024).

The realizations of a reasonable scheme are unique only up to a common rigid motion. One result characterizes the full solution set as

$j$6

while the formation-control analysis states convergence to the desired relative configuration modulo a right-multiplicative constant,

$j$7

This suggests the standard global-pose ambiguity of relative-pose problems: the formation shape is fixed, while the absolute placement of the entire formation remains free (Qi et al., 2024, Cui et al., 14 Aug 2025).

A common misconception is that assigning unit dual quaternion labels to edges is sufficient to specify a formation. The cited theory rules this out: labels must satisfy the realizability factorization or, equivalently, the cycle-consistency criterion. When both directions of an interaction are present, the additional symmetry

$j$8

is necessary (Qi et al., 2024).

3. Dual-quaternion Laplacian and spectral structure

For a UDQDG $j$9, the dual quaternion adjacency matrix is

$G=(V,E)$0

the out-degree matrix is $G=(V,E)$1, and the dual quaternion Laplacian is

$G=(V,E)$2

Entrywise, row $G=(V,E)$3 has diagonal term $G=(V,E)$4, off-diagonal term $G=(V,E)$5 on each outgoing arc, and zero otherwise. This is the direct extension of the classical directed Laplacian $G=(V,E)$6 from scalar to unit-dual-quaternion edge weights (Cui et al., 14 Aug 2025, Qi et al., 2024).

The fundamental structural identity is the similarity relation

$G=(V,E)$7

or, equivalently,

$G=(V,E)$8

where $G=(V,E)$9 is the ordinary Laplacian of the underlying directed graph and

$\Phi=(G,\varphi)$0

This holds exactly when the desired relative configuration scheme is reasonable. In effect, the desired formation is embedded into the Laplacian by diagonal conjugation with the node poses. Because dual quaternion multiplication is noncommutative, the order in this identity is essential (Cui et al., 14 Aug 2025, Qi et al., 2024).

The nullspace reflects formation invariance. In the reasonable case, the desired formation vector lies in the kernel: $\Phi=(G,\varphi)$1 If the underlying digraph has a directed spanning tree, the ordinary Laplacian has a simple zero eigenvalue with eigenvector $\Phi=(G,\varphi)$2, and all other eigenvalues lie in the open right half-plane; this transfers, through the similarity relation, to the formation-control analysis of the UDQDG Laplacian (Cui et al., 14 Aug 2025, Qi et al., 2024).

A related but narrower spectral theory exists for reciprocal-edge dual quaternion unit gain graphs, where every undirected edge is represented in both orientations and opposite orientations carry inverse gains. In that setting the adjacency and Laplacian matrices are dual Hermitian, interlacing theorems hold, and the adjacency spectral radius satisfies

$\Phi=(G,\varphi)$3

with equality iff the gain graph is balanced. For cycles, the adjacency and Laplacian eigenvalues admit closed forms in terms of the total cycle gain. This reciprocal-edge framework is not the same as an arbitrary directed UDQDG, but it supplies the spectral vocabulary for balanced unit-dual-quaternion gain structures (Cui et al., 2024).

4. Formation-control law and convergence theory

The primary control application of a UDQDG is integrated position-and-attitude formation control for multiple 3-D rigid bodies. The continuous-time law is

$\Phi=(G,\varphi)$4

with equivalent agent-wise form

$\Phi=(G,\varphi)$5

Here each state $\Phi=(G,\varphi)$6 is a unit dual quaternion, so the same Laplacian feedback acts simultaneously on translational and rotational errors; no separate graph-level position controller and attitude controller are required (Cui et al., 14 Aug 2025).

The convergence theorem assumes: the interaction graph is fixed; $\Phi=(G,\varphi)$7 contains a directed spanning tree; the desired relative configuration scheme is reasonable; and the preconditioner has the form

$\Phi=(G,\varphi)$8

with $\Phi=(G,\varphi)$9 positive diagonal. Under these conditions,

$\Phi=(G,\hat{\mathbb U},\varphi),$0

globally and asymptotically. If the initial state lies in the admissible dual quaternion space, then

$\Phi=(G,\hat{\mathbb U},\varphi),$1

converges $\Phi=(G,\hat{\mathbb U},\varphi),$2-linearly to zero with rate

$\Phi=(G,\hat{\mathbb U},\varphi),$3

where $\Phi=(G,\hat{\mathbb U},\varphi),$4 is the real part of the second smallest eigenvalue of $\Phi=(G,\hat{\mathbb U},\varphi),$5 (Cui et al., 14 Aug 2025).

The rate parameter $\Phi=(G,\hat{\mathbb U},\varphi),$6 plays the role of an algebraic-connectivity analogue for the directed formation problem. The theorem is phrased in terms of the classical scaled Laplacian $\Phi=(G,\hat{\mathbb U},\varphi),$7, not directly in terms of $\Phi=(G,\hat{\mathbb U},\varphi),$8, because the UDQDG Laplacian is related to $\Phi=(G,\hat{\mathbb U},\varphi),$9 by diagonal similarity. The proof yields the bound

$\hat{\mathbb U}$0

with a polynomial prefactor determined by the largest Jordan block size $\hat{\mathbb U}$1; in the undirected connected case the Laplacian is Hermitian and diagonalizable, so $\hat{\mathbb U}$2 and the prefactor reduces to $\hat{\mathbb U}$3 (Cui et al., 14 Aug 2025).

The conceptual advance relative to classical consensus is explicit. In the scalar law $\hat{\mathbb U}$4, all states converge to a common value. In the UDQDG law, scalar weights are replaced by desired relative poses $\hat{\mathbb U}$5, so the asymptotic configuration satisfies prescribed edgewise rigid-motion constraints rather than identical states. Earlier dual-quaternion consensus work already established pose consensus over directed graphs with directed spanning trees and showed how logarithmic maps can support decentralized formation control of mobile manipulators, but the UDQDG Laplacian embeds arbitrary desired relative configurations directly in the graph weights (Savino et al., 2018, Cui et al., 14 Aug 2025).

5. Projection, discrete iteration, and balance testing

Because the continuous-time law evolves in a linear ambient space, an Euler-type discretization does not automatically preserve the unit dual quaternion manifold. The proposed discrete implementation is therefore the projected iteration

$\hat{\mathbb U}$6

where projection is applied elementwise. For $\hat{\mathbb U}$7, the componentwise projector is

$\hat{\mathbb U}$8

The stopping criterion is based on the $\hat{\mathbb U}$9-norm,

$(i,j)\in E$0

and the iteration is terminated when $(i,j)\in E$1 is below a tolerance (Cui et al., 14 Aug 2025).

Projection is not a trivial normalization step, because feasibility couples the primal and dual parts through

$(i,j)\in E$2

A separate study of projection onto the unit dual quaternion set under the $(i,j)\in E$3-norm gives a complete case-based characterization of the metric projection, proves existence of a global minimizer, derives the KKT system, and provides a practical algorithm that returns the closest feasible unit dual quaternion rather than merely a feasible normalization. That work is directly relevant to projected UDQDG algorithms that evolve states in $(i,j)\in E$4 and then require exact restoration of rigid-motion feasibility (Li et al., 23 Oct 2025).

For balance verification, the UDQDG literature proposes two methods. The direct method solves

$(i,j)\in E$5

after decomposing $(i,j)\in E$6 and $(i,j)\in E$7 into the coupled quaternion systems

$(i,j)\in E$8

with unit-dual-quaternion compatibility constraints. The unit gain graph method converts the directed unit weighted graph to a bidirected unit gain graph with reciprocal weights, then solves the corresponding Hermitian problem. In numerical experiments on balanced unit dual quaternion directed cycles, the similarity residual

$(i,j)\in E$9

is used to certify balance, and a graph is declared balanced if $\varphi(i,j)=\hat q_{d_{ij}},$0 (Qi et al., 2024).

6. Neighboring formulations and scope of the concept

UDQDG is one member of a broader family of dual-quaternion graph formalisms. A closely related line parameterizes each pose by a 6D motion vector $\varphi(i,j)=\hat q_{d_{ij}},$1, uses the UDQ operator

$\varphi(i,j)=\hat q_{d_{ij}},$2

and defines edge residuals by

$\varphi(i,j)=\hat q_{d_{ij}},$3

This yields an unconstrained optimization view of dual-quaternion pose graphs, but it does not replace the multiplicative graph semantics: composition still occurs in unit dual quaternion space, not by adding motion vectors (Qi, 2022).

Another neighboring framework formulates SLAM directly as an equality-constrained standard dual quaternion optimization problem with node states $\varphi(i,j)=\hat q_{d_{ij}},$4, relative measurements $\varphi(i,j)=\hat q_{d_{ij}},$5, predicted relative poses $\varphi(i,j)=\hat q_{d_{ij}},$6, edge errors

$\varphi(i,j)=\hat q_{d_{ij}},$7

and objective

$\varphi(i,j)=\hat q_{d_{ij}},$8

That theory shows that equality-constrained dual quaternion optimization can be reduced to two quaternion optimization problems, which is structurally different from the UDQDG Laplacian route but still graph-based (Qi, 2022).

Distributed localization of visual sensor networks supplies another interpretation: a connected graph of cameras with unit-dual-quaternion node states, relative-pose measurements on edges, local cost functions assembled from neighbor residuals, steepest-descent updates in $\varphi(i,j)=\hat q_{d_{ij}},$9, and a normalization step enforcing

$i$0

This is a graph optimization on $i$1, but its primary object is a residual sum rather than a dual-quaternion Laplacian carrying prescribed desired edge labels (Varotto et al., 2022).

Planar unit dual quaternion pose graph optimization provides a lower-dimensional template. There the graph is directed, edge residuals are

$i$2

and the optimization is performed intrinsically on the product manifold by a Riemannian trust-region method with convergence to first-order stationary points. The state space is planar rather than full 3D, but the directed residual semantics and manifold viewpoint transfer directly (Warke et al., 2024).

Synchronization over $i$3 offers a further operator-based alternative. There, pairwise relative measurements are assembled into a Hermitian dual quaternion measurement matrix $i$4, a spectral initializer is computed by a dual quaternion power method, and a dual quaternion generalized power method enforces feasibility through per-iteration projection: $i$5 This framework supplies recovery guarantees and a Hermitian measurement-operator viewpoint, but it does not define a dual-quaternion Laplacian for arbitrary directed graphs (Zhao et al., 30 Jan 2026).

An alternative representation, augmented unit quaternions, retains quaternion-plus-translation pose content while using $i$6 real variables and only the unit-quaternion sphere constraint, thereby removing the dual-quaternion orthogonality constraint. That formulation is a representation-level alternative to unit-dual-quaternion graph models rather than a reformulation of UDQDG itself (Qi et al., 2023).

Taken together, these neighboring approaches delimit the scope of UDQDG. UDQDG is specifically the graph-theoretic formalism in which directed edges are labeled by desired relative unit dual quaternions and the resulting dual-quaternion Laplacian becomes the main algebraic operator. Residual-based pose-graph optimization, manifold trust-region methods, and Hermitian synchronization all work with the same underlying rigid-motion algebra, but they emphasize different operators, different constraints, and different notions of consistency.