Dual Quaternion Skinning

• Dual Quaternion Skinning is a technique that integrates rotation and translation into a unified framework, providing precise control over articulated motion.
• It employs methods like SEP(LERP), DLB, and ScLERP to balance computational efficiency, visual quality, and stability in animation pipelines.
• The KenLERP hybrid method enhances flexibility by blending coupled and decoupled interpolation techniques to mitigate artifacts in skinning implementations.

Dual quaternion skinning is a technique for encoding and interpolating rigid body transformations—specifically, rotation and translation—within a unified algebraic framework. The method leverages dual quaternions to represent transformations compactly, offering a powerful tool for computer graphics tasks such as skeletal animation and geometric deformation. Dual quaternion skinning addresses the need for interpolants that simultaneously capture coupled rotational and translational degrees of freedom, in contrast to matrix or ordinary quaternion approaches that treat them separately or incur undesirable artifacts (Kenwright, 2023).

1. Dual-Quaternion Structure and Fundamentals

A dual quaternion $\hat{q}$ is defined as

$\hat{q} = q_r + \epsilon\,q_d$

where $q_r$ is the real quaternion (encoding rotation), $q_d$ is the dual quaternion component (encoding translation), and $\epsilon$ is the dual unit ($\epsilon^2 = 0$). The encoding supports the following conversions:

• From (rotation, translation) to dual quaternion:

Given rotation $q_r = [w_r, v_r]$ and translation $t = (t_x, t_y, t_z)$,

$$q_t = [0, t_x, t_y, t_z]\ q_d = \frac{1}{2}(q_r \otimes q_t)\ \hat{q} = q_r + \epsilon q_d$$

• From dual quaternion to (rotation, translation):

For unit $\hat{q}$,

$$\text{rotation}: \quad q = q_r\ \text{translation}: \quad t = \text{vector part of }\, [2 q_d \otimes q_r^*]$$

Basic arithmetic leverages quaternion addition, scalar multiplication, Hamilton product (denoted $\otimes$), and conjugation. The norm is defined as $$\|\hat{q}\| = \sqrt{\left((\hat{q} \otimes \hat{q}^*)_\text{real part}\right)}$$ (Kenwright, 2023).

2. Interpolation Strategies for Dual-Quaternion Skinning

Efficient skinning and deformation require the definition of interpolation schemes over dual quaternions. Several methodologies have been formalized:

2.1 SEP(LERP): Decoupled Interpolation

• Translation is linearly interpolated (LERP) in $\mathbb{R}^3$.
• Rotation uses quaternion SLERP.
• The result is recombined into a dual quaternion by reconstructing the dual part from the interpolated rotation and translation.
• This approach is computationally inexpensive and stable, but as translation and rotation are uncoupled, motion may exhibit unnatural pivoting (Kenwright, 2023).

2.2 DLB: Dual-Quaternion Linear Blending

• Performs direct linear blending of the eight dual quaternion scalars:

$$\hat{q}_\text{lin} = (1 - t)\hat{q}_A + t\hat{q}_B\ \hat{q}(t) = \frac{\hat{q}_\text{lin}}{\|\hat{q}_\text{lin}\|}$$

• DLB is computationally efficient, but does not precisely follow coupled screw motion, potentially leading to visual artifacts like "kinking" for large rotations (Kenwright, 2023).

2.3 ScLERP: Screw-Linear Interpolation

• Executes interpolation along the shortest path "screw motion" in 6-DoF space:

$$\Delta = \hat{q}_A^* \otimes \hat{q}_B \ \Delta^t = \exp(t\log\Delta) \ \hat{q}(t) = \hat{q}_A \otimes \Delta^t$$

• A closed-form is given via dual-angles $(\theta + \epsilon d)$ and dual-axis $\hat{v}$:

$$\Delta = \cos\left(\frac{\theta + \epsilon d}{2}\right) + \hat{v} \sin\left(\frac{\theta + \epsilon d}{2}\right)\ \Delta^t = \cos\left(\frac{t(\theta + \epsilon d)}{2}\right) + \hat{v} \sin\left(\frac{t(\theta + \epsilon d)}{2}\right)$$

• This method achieves visually uniform, constant-speed interpolation in the full transformation space with no need for post-normalization when both endpoints are unit dual quaternions (Kenwright, 2023).

3. Practical Trade-Offs and Best Practices

A comparison of methods reveals the following:

Method	Cost	Visual Quality	Stability
SEP(LERP)	Low (LERP + SLERP)	Decoupled; can look unnatural	Stable; independent transformations
DLB	Very low (add, norm)	Inexpensive; "kinking" with large rot.	Requires renormalization
ScLERP	Higher (dual trig)	True screw motion; uniform speed	Uniform; dual trig precision needed

• Shortest-path enforcement: Always use the shortest quaternion path by flipping the sign of $q_{r,A}$ or $q_{r,B}$ if $q_{r,A} \cdot q_{r,B} < 0$.
• Renormalization: Essential after DLB to avoid drift.
• ScLERP implementation: Prefer closed-form dual-angle formulas to circumvent iterative $\exp$/$\log$ computations.

These trade-offs enable practitioners to select a method aligned with computational, aesthetic, or control imperatives (Kenwright, 2023).

4. Hybrid Interpolation: KenLERP

KenLERP introduces a bias parameter $\beta \in \mathbb{R}$ to blend between the fully coupled screw motion (ScLERP) and the decoupled SEP(LERP):

1. Obtain decomposed results from ScLERP $(q^c_r, t^c)$ and SEP(LERP) $(q^d_r, t^d)$.
2. Blend with coupling bias $\beta$:

$$\hat{t} = (1 - \beta) t^d + \beta t^c \ \hat{q} = \mathrm{SLERP}(q^d_r, q^c_r, \beta)$$

1. Rebuild dual quaternion:

$$q_t = [0, \hat{t}]\ q_d = \frac{1}{2}(\hat{q} \otimes q_t)\ \hat{q}(t) = \hat{q} + \epsilon\,q_d$$

• $\beta=0$ gives fully decoupled interpolation; $\beta=1$ is fully coupled screw-motion; $\beta>1$ exaggerates coupling for an "over-damped" look (Kenwright, 2023).

5. Skinning Implementation Considerations

• Bone representation: Store each bone transformation as a unit dual quaternion.
• Per-vertex skinning: Weight and sum per-bone dual quaternions via DLB or KenLERP, followed by renormalization.
• Artifact mitigation: The bias $\beta$ in KenLERP may be tuned per-bone or per-vertex to address twist or pivot artifacts.
• Transformation application: Always renormalize and transform points via

$x' = \hat{q} \otimes x \otimes \hat{q}^*$

using the conjugate variant specified: keep the real part, negate the dual.

By employing these protocols and selecting among the interpolation and blending methods above, it is feasible to perform robust, efficient, and visually coherent dual quaternion skinning for articulated animation and geometric deformation tasks (Kenwright, 2023).