Dual Quaternion Guidance (DQG)

• Dual Quaternion Guidance (DQG) is a unified geometric framework for 3D rigid-body motion that represents rotation and translation as one algebraic object.
• It integrates control, estimation, and formation strategies using dual-quaternion kinematics and graph-based methods for precise pose tracking and coordination.
• DQG offers practical benefits like avoiding Euler singularities, efficient projection, and robust interpolation, making it ideal for real-time robotics and learning-based tracking.

Dual Quaternion Guidance (DQG) can be understood as a family of guidance, control, estimation, and trajectory-generation methods for 3D rigid bodies in which position and attitude are represented by unit dual quaternions and manipulated directly through dual-quaternion kinematics, graph operators, or optimization procedures. Across recent literature, this viewpoint appears in multi-agent formation control, pose-following, virtual-structure control, SE(3) synchronization, coordinate transformation, and learning-based tracking, with the common feature that rigid motion is handled as a single algebraic object rather than as separate rotation and translation variables (Cui et al., 14 Aug 2025, Arrizabalaga et al., 2023, Giribet et al., 7 Apr 2025).

1. Scope and conceptual identity

In the cited literature, DQG is not a single standardized algorithm but a coherent geometric paradigm. In multi-agent formation control, it denotes Laplacian-based coordination laws on unit dual quaternion directed graphs (UDQDGs), where desired relative poses appear as edge weights and the closed-loop objective is a prescribed relative formation rather than mere consensus (Cui et al., 14 Aug 2025). In virtual-structure control, it denotes a cluster-space controller in which a robot formation is treated as a single rigid body whose pose is a unit dual quaternion and whose geometry variables are regulated alongside pose (Giribet et al., 7 Apr 2025). In pose-following, it denotes a path-following law on $SE(3)$ in which the full pose error is expressed as a dual quaternion and the path parameter is itself controlled (Arrizabalaga et al., 2023).

This suggests that DQG is best regarded as an umbrella term for guidance strategies on $SE(3)$ formulated natively in dual quaternion algebra. Within that umbrella, the same representation supports at least five technical roles. It acts as a state variable for rigid-body pose, as an error variable for feedback design, as an edge weight in graph-based coordination, as an optimization variable in synchronization and calibration, and as a latent or equivariant representation in learning-based rigid-motion modeling (Zhao et al., 30 Jan 2026, Bektaş, 2024, Vieira et al., 2023).

A central historical point is the relation to consensus. The dual-quaternion formation law of (Cui et al., 14 Aug 2025) reduces to the classical Olfati-Saber and Murray consensus law $\dot{x}(t) = -Lx(t)$ when desired relative configurations are trivial identities and states are real, and it reduces to Savino et al.’s dual quaternion consensus protocol when all agents are required to converge to the same pose. DQG therefore generalizes consensus from identical-state agreement to arbitrary desired relative configurations in full pose space (Cui et al., 14 Aug 2025).

2. Algebraic and kinematic foundations

The common algebraic object is the dual quaternion

$\hat q = q_s + q_d \epsilon,$

where $q_s,q_d\in\mathbb Q$ and $\epsilon^2=0$. A unit dual quaternion satisfies

$$|\hat q| = 1 \quad\Longleftrightarrow\quad |q_s| = 1,\qquad q_s^* q_d + q_d^* q_s = 0,$$

and the unit set is written as

$$\hat{\mathbb U} := \left\{\hat q = q_s + q_d \epsilon \in \hat{\mathbb Q}\,\middle|\, |q_s|=1,\ q_s^*q_d + q_d^*q_s = 0 \right\}.$$

Equivalent formulations also appear in real-vector form as

$$\Omega := \Big\{ (\overrightarrow{\tilde{q}_s},\overrightarrow{\tilde{q}_d})\in\mathbb{R}^8 \;\Big|\; \|\overrightarrow{\tilde{q}_s}\|_2^2 = 1,\ \overrightarrow{\tilde{q}_d}^\top \overrightarrow{\tilde{q}_s}=0 \Big\},$$

which is the unit dual quaternion manifold used in projection algorithms (Cui et al., 14 Aug 2025, Li et al., 23 Oct 2025).

Several cited works write the rigid-body pose in equivalent unit dual quaternion forms. One standard expression is

$\hat q = q_s + \frac{1}{2} p^w q_s\,\epsilon,$

where $SE(3)$0 is a unit quaternion for attitude and $SE(3)$1 is a vector quaternion for translation. Another is

$SE(3)$2

with $SE(3)$3 and $SE(3)$4. A third equivalent form used in $SE(3)$5 synchronization is

$SE(3)$6

All three encode the same geometric content: rotation in the standard part and translation in the dual part (Cui et al., 14 Aug 2025, Giribet et al., 7 Apr 2025, Zhao et al., 30 Jan 2026).

The kinematic equation that turns this representation into a guidance framework is the dual-quaternion twist relation

$SE(3)$7

with the twist written in one cited formulation as

$SE(3)$8

For pose-following, the dual twist is also written as

$SE(3)$9

with pose kinematics

$\dot{x}(t) = -Lx(t)$0

These equations place DQG directly on $\dot{x}(t) = -Lx(t)$1 rather than on separate translational and rotational state spaces (Giribet et al., 7 Apr 2025, Arrizabalaga et al., 2023).

Error representation follows the same pattern. For tracking or following, the pose error is expressed as a relative dual quaternion such as

$\dot{x}(t) = -Lx(t)$2

or, in multi-agent settings, as relative products $\dot{x}(t) = -Lx(t)$3. This unified error variable is the basis for geometric feedback, Laplacian coupling, and synchronization objectives (Arrizabalaga et al., 2023, Cui et al., 14 Aug 2025).

3. Graph-based guidance and formation control

The graph-theoretic core of DQG is the unit dual quaternion directed graph

$\dot{x}(t) = -Lx(t)$4

where $\dot{x}(t) = -Lx(t)$5 is a directed interaction graph and each edge $\dot{x}(t) = -Lx(t)$6 carries a desired relative configuration

$\dot{x}(t) = -Lx(t)$7

The associated UDQDG Laplacian is

$\dot{x}(t) = -Lx(t)$8

with dual quaternion adjacency entries $\dot{x}(t) = -Lx(t)$9 on edges and zero otherwise. This Laplacian is the direct analogue of the scalar graph Laplacian, but its off-diagonal entries are rigid motions rather than real scalars (Qi et al., 2024, Cui et al., 14 Aug 2025).

A desired relative configuration scheme is reasonable, or balanced, when there exists a formation vector $\hat q = q_s + q_d \epsilon,$0 such that

$\hat q = q_s + q_d \epsilon,$1

On a directed graph with a directed spanning tree, this is equivalent to a cycle condition: for every cycle, the product of appropriately oriented edge weights equals the identity. It is also equivalent to a Laplacian similarity relation with the real Laplacian of the underlying graph,

$\hat q = q_s + q_d \epsilon,$2

depending on the convention used for $\hat q = q_s + q_d \epsilon,$3. This is the structural statement that a balanced dual quaternion formation problem is, after a diagonal dual quaternion change of coordinates, a classical Laplacian problem (Qi et al., 2024, Cui et al., 14 Aug 2025).

The formation-control law developed in (Cui et al., 14 Aug 2025) is

$\hat q = q_s + q_d \epsilon,$4

with $\hat q = q_s + q_d \epsilon,$5 and $\hat q = q_s + q_d \epsilon,$6 a positive diagonal matrix. In component form,

$\hat q = q_s + q_d \epsilon,$7

Under a directed spanning tree and a reasonable desired relative configuration scheme, the closed-loop system globally and asymptotically converges to the desired relative formation up to a right-multiplicative constant dual quaternion, and the convergence is $\hat q = q_s + q_d \epsilon,$8-linear with rate driven by the second smallest eigenvalue of the real matrix $\hat q = q_s + q_d \epsilon,$9 (Cui et al., 14 Aug 2025).

The same section of the literature clarifies a common misconception: DQG in formation control is not restricted to pose consensus. When $q_s,q_d\in\mathbb Q$0, the law reduces to consensus; when $q_s,q_d\in\mathbb Q$1 are distinct, agents converge to different positions and orientations that realize a prescribed formation. The graph-theoretic development of dual quaternion adjacency, logarithm adjacency, and dual quaternion Laplacian matrices provides the spectral machinery for that generalization, including Hermitian structure for certain adjacency models and positive semidefiniteness of dual quaternion Laplacians (Qi et al., 2022).

4. Estimation, synchronization, and manifold projection

A second major branch of DQG concerns estimation problems in which valid $q_s,q_d\in\mathbb Q$2 poses must be recovered or maintained under noise, discretization, or optimization. In SE(3) synchronization, absolute poses $q_s,q_d\in\mathbb Q$3 are estimated from noisy relative measurements by solving

$q_s,q_d\in\mathbb Q$4

equivalently

$q_s,q_d\in\mathbb Q$5

where $q_s,q_d\in\mathbb Q$6 is a Hermitian dual quaternion measurement matrix. The resulting two-stage pipeline uses a spectral initializer computed via the power method on the Hermitian dual quaternion measurement matrix, followed by a dual quaternion generalized power method (DQGPM) with per-iteration projection,

$q_s,q_d\in\mathbb Q$7

The reported guarantees are an estimation error bound for the spectral estimator and, for DQGPM, a finite-iteration error bound together with linear error contraction up to an explicit noise-dependent threshold (Zhao et al., 30 Jan 2026).

Projection is therefore not an auxiliary detail but a foundational operation. The projection problem under the $q_s,q_d\in\mathbb Q$8-norm is posed as

$q_s,q_d\in\mathbb Q$9

with

$\epsilon^2=0$0

Recent work studies this projection systematically, identifies several distinct cases based on the relationship between the standard and dual parts in vector form, and formulates the unit dual quaternion set as

$\epsilon^2=0$1

In discrete formation-control implementations, the same need appears in the projected iteration

$\epsilon^2=0$2

which preserves the unit dual quaternion constraint during numerical integration (Li et al., 23 Oct 2025, Cui et al., 14 Aug 2025).

A related estimation theme arises in symmetric similarity 3D coordinate transformation. There the dual quaternion algorithm is formulated as an EIV or CEIV problem with constraints

$\epsilon^2=0$3

and a modified misclosure vector is introduced to address severe ill-conditioning in the symmetric CEIV system. The paper’s principal methodological conclusion is that QA and DQA give identical transformation parameters and precision when properly modeled, and that claims that DQA yields more precise parameters than QA are not supported. This is an important corrective to any overgeneralized claim that dual quaternion formulations automatically improve estimation accuracy; the benefit may lie instead in unified representation, symmetry handling, or algorithmic convenience (Bektaş, 2024).

At the matrix-algebra level, the dual complex adjoint matrix provides another estimation and control tool. It maps dual quaternion matrices to structured dual complex matrices, preserves addition, multiplication, unitary and Hermitian structure, supports standard right eigenvalue analysis, gives a direct solution to the Hand-Eye calibration problem, and reduces dual quaternion linear systems to dual complex systems. For DQG, this supplies efficient eigenvalue computation and linear-system machinery for dual quaternion operators that arise in synchronization and formation guidance (Chen et al., 2024).

5. Tracking, virtual structures, learning, and trajectory generation

In virtual-structure control for UAV groups, the formation is treated as a single rigid body whose cluster pose is encoded by

$\epsilon^2=0$4

The controller separates a pose control module from a geometry-based adaptive strategy. For the pose subsystem, the commanded cluster angular and linear velocities are

$\epsilon^2=0$5

$\epsilon^2=0$6

with integral states governed by additional differential equations. Geometry variables such as distances $\epsilon^2=0$7 and angles $\epsilon^2=0$8 are regulated separately, and the resulting cluster-space twist is mapped to individual robot velocities by Jacobian-based relations. For 2R and 3R formations, gains are scheduled as functions of distance or inertia, so the controller explicitly adapts to formation geometry (Giribet et al., 7 Apr 2025).

Pose-following extends dual quaternion tracking by adding a path parameter $\epsilon^2=0$9 and controlling progression along a geometric reference with a moving frame. The full pose error is

$$|\hat q| = 1 \quad\Longleftrightarrow\quad |q_s| = 1,\qquad q_s^* q_d + q_d^* q_s = 0,$$0

with dual twist error

$$|\hat q| = 1 \quad\Longleftrightarrow\quad |q_s| = 1,\qquad q_s^* q_d + q_d^* q_s = 0,$$1

The feedback term is a Lie-group PD law,

$$|\hat q| = 1 \quad\Longleftrightarrow\quad |q_s| = 1,\qquad q_s^* q_d + q_d^* q_s = 0,$$2

combined with a feedforward cancellation term and a path-speed controller

$$|\hat q| = 1 \quad\Longleftrightarrow\quad |q_s| = 1,\qquad q_s^* q_d + q_d^* q_s = 0,$$3

The reported result is almost global asymptotic stability of pose-following, together with the practical advantage that progression along the path can slow down when the body is far from the reference pose and speed up when it is close (Arrizabalaga et al., 2023).

Learning-based DQG introduces online disturbance compensation while preserving the same geometric structure. In the velocity-level trajectory-tracking controller of (Nieto et al., 6 Jan 2026), pose is represented by a unit dual quaternion

$$|\hat q| = 1 \quad\Longleftrightarrow\quad |q_s| = 1,\qquad q_s^* q_d + q_d^* q_s = 0,$$4

and the nominal geometric feedback law is augmented with GP mean predictions for unknown rotational and translational disturbances. A Lyapunov-based analysis establishes probabilistic ultimate boundedness of the pose tracking error under bounded GP uncertainty. The simulation scenarios explicitly include correlated rotational and translational effects arising from magnetometer perturbations, which are precisely the type of coupled disturbances that motivate a dual quaternion representation (Nieto et al., 6 Jan 2026).

A further extension appears in rigid-motion learning. Dual quaternion models are translation and rotation equivariant and better learn object trajectories than real-valued or quaternion-only networks. On 3DPW, the reported metrics for Dual Quaternion CoRPoF are $$|\hat q| = 1 \quad\Longleftrightarrow\quad |q_s| = 1,\qquad q_s^* q_d + q_d^* q_s = 0,$$5 VIM, $$|\hat q| = 1 \quad\Longleftrightarrow\quad |q_s| = 1,\qquad q_s^* q_d + q_d^* q_s = 0,$$6 FDE, and $$|\hat q| = 1 \quad\Longleftrightarrow\quad |q_s| = 1,\qquad q_s^* q_d + q_d^* q_s = 0,$$7 validation loss, compared with $$|\hat q| = 1 \quad\Longleftrightarrow\quad |q_s| = 1,\qquad q_s^* q_d + q_d^* q_s = 0,$$8, $$|\hat q| = 1 \quad\Longleftrightarrow\quad |q_s| = 1,\qquad q_s^* q_d + q_d^* q_s = 0,$$9, and $$\hat{\mathbb U} := \left\{\hat q = q_s + q_d \epsilon \in \hat{\mathbb Q}\,\middle|\, |q_s|=1,\ q_s^*q_d + q_d^*q_s = 0 \right\}.$$0 for Quaternion CoRPoF and $$\hat{\mathbb U} := \left\{\hat q = q_s + q_d \epsilon \in \hat{\mathbb Q}\,\middle|\, |q_s|=1,\ q_s^*q_d + q_d^*q_s = 0 \right\}.$$1, $$\hat{\mathbb U} := \left\{\hat q = q_s + q_d \epsilon \in \hat{\mathbb Q}\,\middle|\, |q_s|=1,\ q_s^*q_d + q_d^*q_s = 0 \right\}.$$2, and $$\hat{\mathbb U} := \left\{\hat q = q_s + q_d \epsilon \in \hat{\mathbb Q}\,\middle|\, |q_s|=1,\ q_s^*q_d + q_d^*q_s = 0 \right\}.$$3 for real CoRPoF. This suggests that, within learning-based DQG, the algebra can serve not only for control but also for equivariant prediction of future rigid motion (Vieira et al., 2023).

6. Applications, implementation, advantages, and caveats

The application range explicitly discussed in the cited literature includes autonomous mobile robots, unmanned aerial vehicles, autonomous underwater vehicles, small satellite constellations, geodesy, hand-eye calibration, multi-scan point-set registration, and robot manipulation (Cui et al., 14 Aug 2025, Bektaş, 2024, Zhao et al., 30 Jan 2026). In software, a notable implementation platform is DQ Robotics, which exposes dual quaternion poses, geometric primitives, Jacobians, constrained kinematic controllers, and adjoint operations across MATLAB, Python, and C++. Reported average times for dual quaternion multiplication are $$\hat{\mathbb U} := \left\{\hat q = q_s + q_d \epsilon \in \hat{\mathbb Q}\,\middle|\, |q_s|=1,\ q_s^*q_d + q_d^*q_s = 0 \right\}.$$4 in C++, $$\hat{\mathbb U} := \left\{\hat q = q_s + q_d \epsilon \in \hat{\mathbb Q}\,\middle|\, |q_s|=1,\ q_s^*q_d + q_d^*q_s = 0 \right\}.$$5 in Python bindings, $$\hat{\mathbb U} := \left\{\hat q = q_s + q_d \epsilon \in \hat{\mathbb Q}\,\middle|\, |q_s|=1,\ q_s^*q_d + q_d^*q_s = 0 \right\}.$$6 in MATLAB, and $$\hat{\mathbb U} := \left\{\hat q = q_s + q_d \epsilon \in \hat{\mathbb Q}\,\middle|\, |q_s|=1,\ q_s^*q_d + q_d^*q_s = 0 \right\}.$$7 in a native Python implementation, which supports real-time deployment claims for DQ-based control loops (Adorno et al., 2019).

The technical advantages repeatedly emphasized are unified representation of rotation and translation, absence of Euler-angle singularities, scalar-like algebra compared with matrix formulations, natural relative pose encoding through multiplication and conjugation, and direct compatibility with screw-theoretic interpolation and twist propagation (Cui et al., 14 Aug 2025, Kenwright, 2023). These strengths explain why DQG appears in both graph-based formation problems and path-based tracking problems: in each case, the controller acts on one object that already lives on $$\hat{\mathbb U} := \left\{\hat q = q_s + q_d \epsilon \in \hat{\mathbb Q}\,\middle|\, |q_s|=1,\ q_s^*q_d + q_d^*q_s = 0 \right\}.$$8.

Trajectory generation introduces explicit design trade-offs. Interpolation methods include SEP(LERP), DLB, ScLERP, and KenLERP. SEP(LERP) decouples rotation and translation; DLB linearly blends dual quaternions and renormalizes; ScLERP performs screw linear interpolation

$$\hat{\mathbb U} := \left\{\hat q = q_s + q_d \epsilon \in \hat{\mathbb Q}\,\middle|\, |q_s|=1,\ q_s^*q_d + q_d^*q_s = 0 \right\}.$$9

and KenLERP introduces a bias parameter $$\Omega := \Big\{ (\overrightarrow{\tilde{q}_s},\overrightarrow{\tilde{q}_d})\in\mathbb{R}^8 \;\Big|\; \|\overrightarrow{\tilde{q}_s}\|_2^2 = 1,\ \overrightarrow{\tilde{q}_d}^\top \overrightarrow{\tilde{q}_s}=0 \Big\},$$0 to interpolate between fully coupled and fully decoupled behavior. A common misconception is that dual quaternion interpolation implies a single canonical motion law. The cited literature instead presents several options with different trade-offs between computational cost, aesthetic factors, and coupling dependency (Kenwright, 2023).

The main caveats are equally consistent across the literature. Dual quaternion multiplication is non-commutative; spectral theory for dual quaternion matrices is more intricate than for real or complex matrices; implementation often requires custom arithmetic, projection operators, or specialized adjoint constructions; and some estimation problems can be highly ill-conditioned without devices such as modified misclosure vectors or carefully chosen projection steps (Cui et al., 14 Aug 2025, Bektaş, 2024, Chen et al., 2024). Another objective caution is that dual quaternion formulations do not automatically dominate alternative representations in every metric: in symmetric similarity transformation, DQA and QA produce identical transformation parameters and covariance results when properly modeled (Bektaş, 2024).

Taken together, these results establish DQG as a mature geometric framework rather than a narrowly defined controller. Its unifying principle is that guidance, coordination, estimation, and interpolation for rigid-body motion are all performed directly in dual quaternion space, so that the algebra of the representation and the geometry of $$\Omega := \Big\{ (\overrightarrow{\tilde{q}_s},\overrightarrow{\tilde{q}_d})\in\mathbb{R}^8 \;\Big|\; \|\overrightarrow{\tilde{q}_s}\|_2^2 = 1,\ \overrightarrow{\tilde{q}_d}^\top \overrightarrow{\tilde{q}_s}=0 \Big\},$$1 remain aligned throughout the modeling and control pipeline.