Dual Quaternion Formulations

Updated 2 March 2026

Dual Quaternion Formulations are an extension of quaternion algebra using a dual unit to efficiently encode both rotation and translation in 3D space.
They provide a unified framework for modeling rigid-body motions, enabling advanced techniques in robot kinematics, SLAM, and pose optimization.
The approach underpins spectral theory, matrix decompositions, and variational integration, offering robust computational tools for control and simulation.

A dual quaternion is an algebraic object formed by extending the quaternion algebra with a dual unit $\epsilon$ satisfying $\epsilon^2=0$ . Dual quaternion formulations underpin a unified algebraic framework for modeling, analyzing, and optimizing rigid-body motions in three dimensions, combining rotation and translation into a single entity. These formulations have become foundational in the analysis and design of robot kinematics, pose estimation (SLAM), variational integration, signal processing, and dual quaternion linear algebra. The structure of dual quaternions and the associated function and optimization theory have catalyzed a broad spectrum of computational and theoretical advances, including spectral theory, generalized inverses, and dual quaternion matrix decompositions.

1. Algebraic Structure of Dual Quaternions

A dual quaternion $\widehat Q$ is defined as $\widehat Q = Q + \epsilon Q'$ , where $Q,Q'\in\mathbb Q$ (standard quaternions), and $\epsilon^2=0$ (Qi, 2022). The standard part $Q$ encodes rotation, and the dual part $Q'$ encodes translation (or infinitesimal motion component). Operations are as follows:

Addition: $(Q+\epsilon Q') + (P+\epsilon P') = (Q+P) + \epsilon(Q'+P')$ .
Multiplication: $(Q+\epsilon Q')(P+\epsilon P') = QP + \epsilon(QP'+Q'P)$ .
Conjugation: $\widehat Q^* = Q^* + \epsilon Q'^*$ , with $Q^*$ the quaternion conjugate.
Norm: $\|\widehat Q\| = |Q| + \frac{Q Q'^* + Q' Q^*}{2|Q|}\,\epsilon$ for $Q\ne 0$ ; $|Q'|\epsilon$ otherwise.

Invertibility of $\widehat Q$ requires $Q\ne 0$ , with inverse $\widehat Q^{-1}=Q^{-1}-\epsilon Q^{-1}Q'Q^{-1}$ . The set of unit dual quaternions parameterizes rigid-body displacements: $Q Q^*=1$ and $Q Q'^*+Q' Q^*=0$ (Qi, 2022, Qi et al., 2021).

2. Standard Dual Quaternion Functions

A function $\widehat f:\widehat{\mathbb Q}^n\rightarrow \widehat{\mathbb R}$ is standard if its leading (standard) part depends only on the standard parts of its argument. Formally, $\widehat f(x+\epsilon x') = f(x) + \epsilon f_d(x,x')$ requires $f$ independent of $x'$ . Standard dual quaternion functions include (Qi, 2022):

Power: $\widehat x^k = x^k + k\epsilon x^{k-1} x'$
Magnitude: $\|\widehat x\|$ as above
Euclidean 2-norm: $\|\widehat{\mathbf{x}}\|_2$
$k$ th largest eigenvalue of Hermitian dual quaternion matrices
Logarithm and exponential (for unit dual quaternion functions)
Sum, product, minimum, maximum, and compositions of standard functions

This closure of the standard class under these operations is central for optimization.

3. Dual Quaternion Optimization Theory

The general equality-constrained dual quaternion optimization problem is:

$\min_{\widehat x\in\widehat{\mathbb Q}^n} \big\{ \widehat f(\widehat x)\,:\,\widehat h_j(\widehat x)=0,\,j=1,\ldots,m \big\}$

with $\widehat f,\widehat h_j:\widehat{\mathbb Q}^n\to\widehat{\mathbb R}$ (Qi, 2022). The minimizer $\widehat x^*=x^*+\epsilon x'^*$ is computed by a two-stage reduction:

Stage I: Minimize $f(x)$ subject to $h_j(x)=0$ and auxiliary constraints. The optimum is $L^I$ .
Stage II: Minimize $f_d(x,x')$ subject to $f(x)=L^I$ , $h_j(x)=0$ , and additional constraints.

If all constraints and the objective are standard, Stage I depends only on $x$ , decoupling the optimization into classical quaternion least-squares for the standard parts and dual parts. This structure provides KKT-type necessary conditions analogous to standard smooth optimization (Qi, 2022).

4. Linear Algebra and Spectral Theory of Dual Quaternion Matrices

Central results include:

Three tiers of right-linear independence: LIN (linear), APP-LIN (appreciable), and weak LIN, with LIN $\Rightarrow$ APP-LIN $\Rightarrow$ weak LIN (Ling et al., 2022).
Hermitian Dual Quaternion Matrices: Spectral theorem—every Hermitian matrix has real dual-number eigenvalues, diagonalizable by a unitary transformation.
Minimax Principle: The Courant–Fischer dual quaternion analogue gives Rayleigh quotient variational characterizations of eigenvalues:

$\lambda_k = \min_{\dim S=k}\,\max_{x\in S\setminus\{0\}} R(x) = \max_{\dim T=m-k+1}\,\min_{x\in T\setminus\{0\}} R(x)\,,$

with $R(x) = \dfrac{x^*Ax}{x^*x}$ (Ling et al., 2022).

SVD of Dual Quaternion Matrices: Every $A\in \mathbb{DQ}^{m\times n}$ admits $A = U[\Sigma_r\,\,0\;\;0\,\,0]V^*$ with $U,V$ unitary, $\Sigma_r$ diagonal with $\sigma_i\in\mathbb D^{++}$ (Ling et al., 2022).
Generalized inverses (Moore-Penrose): $G$ is a generalized inverse if it satisfies the four Penrose equations; uniqueness is ensured under additional rank conditions (Ling et al., 2022, Ling et al., 2 Nov 2025, Xie et al., 2023).
Householder, QR, CS, and GSVD decompositions: Analogues exist, respecting dual-number and quaternionic structure (Ling et al., 2 Nov 2025).

The dual complex adjoint matrix $J(Q)$ provides a linearization in dual-complex variables, preserving multiplication and conjugation structure, facilitating spectral calculation, solution of matrix equations, and acceleration of iterative algorithms (e.g., Rayleigh quotient iteration) (Chen et al., 2024).

5. Applications to Rigid-Body Kinematics and Robotics

Dual quaternion formulations unify translation and rotation for efficient and compact representation of rigid motions (Qi, 2022, Xu et al., 2016):

Hand-Eye Calibration: The problem $A_i X = X B_i$ admits a dual quaternion least-squares formulation with standard constraints. Unit dual quaternion parametrization reduces the problem to two coupled or sequential quaternion subproblems (Qi, 2022, Qi, 2022).
Simultaneous Localization and Mapping (SLAM): Nodes are poses $\widehat x_i$ , edges are relative measurements $\widehat q_{ij}$ , with objectives of dual quaternion 2-norm type, solved by standard DQO reduction (Qi, 2022, Qi, 2022).
SE(3) Synchronization: Spectral methods on Hermitian dual quaternion matrices yield global pose estimates, refined by dual quaternion generalized power algorithms with provable linear error contraction (Zhao et al., 30 Jan 2026).
Rigid Body Simulation: Dual quaternion variational integrators preserve symplectic structure, energy, and momentum in direct all-6-DOF simulations, outperforming separate translation/rotation integrators (Xu et al., 2016, Silva et al., 2020).
Mobile Manipulator Dynamics: Dual quaternion Newton–Euler and Gauss’s Principle formulations handle arbitrary joint types and constraints, generalizing classic rigid-body dynamics (Silva et al., 2020).

6. Signal Processing and Functional Analysis

Dual-quaternion-valued functions enable the processing of multicomponent signals, where multidimensional data is naturally embedded in the dual quaternion algebra (Kenwright, 2023):

Dual-Quaternion Fourier Transform (DQFT): Extends classical Fourier transform theory, preserving non-commutativity, allowing left- and right-sided DQFT/IDQFT, and supporting convolutions, shift, and modulation theorems (Kenwright, 2023).
Applications: Unified non-linear filtering, compression, and analysis of multichannel data; direct handling of dual-quaternion-valued PDEs in robotics, imaging, and physics.

7. Theory of Dual Quaternion Norms and Metric Structure

Rigorous dual quaternion norm definitions underlie dual quaternion vector and matrix analysis (Qi et al., 2021):

Dual number absolute value: $|q|=|q_{\text{st}}| + \operatorname{sgn}(q_{\text{st}}) q_I \epsilon$ for appreciable $q$ ; $|q_I|\epsilon$ if $q_{\text{st}}=0$ .
Dual quaternion magnitude: As above, combining quaternion norm and dual part correction.
Vector norms: 1-norm, 2-norm, and $\infty$ -norm are extended, preserving key properties (nonnegativity, triangle inequality, scaling).
Metric and topological structure: The framework allows defining metrics, convergence, and stability analysis in dual quaternion vector spaces.

The integration of these norm properties with matrix spectral theory supplies foundational tools for variational analysis, least-squares optimization, and system stability verification.

The dual quaternion formalism integrates algebraic, geometric, and analytical methods, supporting a broad variety of foundational and computational advances in rigid-body kinematics, robotics, signal processing, and numerical linear algebra (Qi, 2022, Ling et al., 2022, Chen et al., 2024, Ling et al., 2 Nov 2025, Xu et al., 2016, Zhao et al., 30 Jan 2026, Kenwright, 2023, Qi, 2022, Qi et al., 2021, Silva et al., 2020, Xie et al., 2023).