Generalized Constant-Twist Prior

• Generalized constant-twist priors are Lie group-based motion models that enforce temporal constancy of body-frame twists.
• They employ continuous-time Gaussian-process formulations and discrete ternary factors to deliver sparse, efficient state estimation.
• On SE(3), these priors couple rotation and translation, improving observability under partial sensing and enabling higher-order motion generalizations.

Searching arXiv for the cited papers and closely related work on constant-twist / Lie-group trajectory priors. Generalized constant-twist motion priors are motion models on Lie groups that enforce approximate temporal constancy of twist, usually in a body-fixed frame, while respecting the nonlinear geometry of the state manifold. In the matrix Lie-group formulation, the state trajectory evolves on a group $G$, the instantaneous body-frame velocity is represented in the Lie algebra, and the prior is imposed either as a continuous-time Gaussian-process model with white noise on acceleration or as a discrete ternary factor that penalizes differences of successive time-normalized twists (Dong et al., 2017, Baxter et al., 23 Aug 2025). On $SE(3)$, this construction couples rotation and translation through the group exponential and Jacobian structure, which is the central reason it can regularize orientation even under position-only sensing (Baxter et al., 23 Aug 2025). Related work generalizes the constant-twist idea further by promoting constant body-centric acceleration via white-noise-on-jerk priors, thereby extending expressiveness beyond constant-velocity mean motion while preserving sparse estimation structure (Tang et al., 2018, Retan et al., 2022).

1. Conceptual definition and geometric setting

A generalized constant-twist motion prior is defined for trajectories $X(t)\in G$, where $G$ is a matrix Lie group with Lie algebra $g$, exponential and logarithm maps $\exp:g\rightarrow G$ and $\log:G\rightarrow g$, and local coordinates obtained through hat and vee operators (Dong et al., 2017). In the continuous-time construction, the body-frame twist is

$$\varpi(t) := (X(t)^{-1}\dot{X}(t))^\vee \in \mathbb{R}^N,$$

with left-invariant kinematics

$\dot{X}(t)=X(t)\,\hat{\varpi}(t).$

The prior assumes white noise on body-frame acceleration,

$$\dot{\varpi}(t)=w(t), \qquad w(t)\sim GP(0,Q_c\delta(t-t')),$$

so the prior mean encourages constant twist (Dong et al., 2017).

A discrete-time counterpart expresses the same idea directly on successive states $SE(3)$0. Using the right-minus convention,

$SE(3)$1

the generalized constant-twist prior enforces $SE(3)$2 through a ternary residual over three consecutive states (Baxter et al., 23 Aug 2025). This formulation compares consecutive algebra-valued velocities in a common tangent space at the identity and therefore avoids explicit transport under the chosen right-minus convention (Baxter et al., 23 Aug 2025).

This suggests two complementary interpretations. In continuous-time SLAM and STEAM, generalized constant twist is a Gaussian-process prior over trajectories on manifolds (Dong et al., 2017). In sparse factor-graph estimation for discrete target tracking, it is a ternary Lie-group factor that encodes the same kinematic regularity without introducing explicit velocity states (Baxter et al., 23 Aug 2025).

2. Continuous-time Gaussian-process formulation on matrix Lie groups

The continuous-time construction on arbitrary matrix Lie groups is built from local coordinates around each knot state $SE(3)$3,

$SE(3)$4

so that

$SE(3)$5

The key differential relation uses the right Jacobian $SE(3)$6,

$SE(3)$7

For small intervals, $SE(3)$8, giving the local approximation

$SE(3)$9

which converts the nonlinear group dynamics into a locally linear stochastic differential equation in tangent coordinates (Dong et al., 2017).

Defining the local Markov state

$X(t)\in G$0

the system becomes

$X(t)\in G$1

Because Gaussian processes induced by linear time-varying SDEs of this form have block-tridiagonal inverse covariance, the resulting prior factors are exactly sparse and connect only consecutive knot states (Dong et al., 2017).

The same paper gives the fundamental solution for constant $X(t)\in G$2,

$X(t)\in G$3

and the closed-form process covariance over $X(t)\in G$4,

$X(t)\in G$5

This is the Lie-group analogue of the vector-space white-noise-on-acceleration prior, recovered locally in tangent space (Dong et al., 2017).

3. Sparse factorization, residuals, and interpolation

The continuous-time GP prior induces a binary factor between adjacent knots $X(t)\in G$6 and $X(t)\in G$7 with residual

$X(t)\in G$8

weighted by $X(t)\in G$9, yielding

$G$0

In local $G$1-coordinates, the Jacobians are

$G$2

Mapped back to group variables, perturbations use $G$3, with residuals expressed through $G$4 and Jacobian terms involving $G$5 (Dong et al., 2017).

A central feature of this representation is $G$6 interpolation between neighboring knots. For $G$7,

$G$8

with

$G$9

The interpolated pose and body-frame velocity are then expressed by

$g$0

$g$1

where the block partitions of $g$2 and $g$3 are used exactly as given in the construction (Dong et al., 2017).

In SLAM, this interpolation machinery allows asynchronous measurements to be attached as factors on neighboring knots rather than requiring an instantiated state at every measurement time. The sum of GP prior factors contributes a block-tridiagonal information matrix, while landmark and sensor factors preserve the standard sparse structure (Dong et al., 2017).

4. Discrete ternary constant-twist factor on Lie groups

A more compact discrete formulation appears in relative navigation and target tracking on Lie groups. The prior is encoded as a ternary factor on $g$4 with residual

$g$5

This is the “difference-of-twists” form and directly enforces approximate constancy of algebra-valued velocity across adjacent intervals (Baxter et al., 23 Aug 2025).

The same work also presents an equivalent “predict-and-compare” form. Defining

$g$6

one predicts

$g$7

To first order, $g$8 and $g$9 coincide (Baxter et al., 23 Aug 2025).

For the difference-of-twists residual, the Jacobians under right-invariant local coordinates are given explicitly: $\exp:g\rightarrow G$0

$\exp:g\rightarrow G$1

$\exp:g\rightarrow G$2

where $\exp:g\rightarrow G$3 (Baxter et al., 23 Aug 2025). For the predict-and-compare implementation, the closed-form Jacobians involve $\exp:g\rightarrow G$4, $\exp:g\rightarrow G$5, and $\exp:g\rightarrow G$6, and are stated explicitly in the same source (Baxter et al., 23 Aug 2025).

This ternary construction differs structurally from the GP state-space formulation. It requires no extra velocity states, preserves sparsity through a minimal three-node factor, and can be inserted directly into factor-graph toolkits such as GTSAM (Baxter et al., 23 Aug 2025). A plausible implication is that it trades some of the continuous-time expressiveness of the GP approach for a simpler deployment path in discrete estimation problems.

5. Specialization to $\exp:g\rightarrow G$7 and body-frame coupling

On $\exp:g\rightarrow G$8, the twist is

$\exp:g\rightarrow G$9

with hat operator

$\log:G\rightarrow g$0

and left-invariant kinematics

$\log:G\rightarrow g$1

in the continuous-time GP formulation (Dong et al., 2017). The discrete tracking formulation uses the standard $\log:G\rightarrow g$2 exponential

$\log:G\rightarrow g$3

where

$\log:G\rightarrow g$4

and the logarithm

$\log:G\rightarrow g$5

for $\log:G\rightarrow g$6, degenerating to $\log:G\rightarrow g$7 for $\log:G\rightarrow g$8 using first-order limits (Baxter et al., 23 Aug 2025).

The geometric significance of generalized constant twist in $\log:G\rightarrow g$9 is the coupling between translation and rotation in the body frame. Because translation in $$\varpi(t) := (X(t)^{-1}\dot{X}(t))^\vee \in \mathbb{R}^N,$$0 depends on $$\varpi(t) := (X(t)^{-1}\dot{X}(t))^\vee \in \mathbb{R}^N,$$1, an angular component $$\varpi(t) := (X(t)^{-1}\dot{X}(t))^\vee \in \mathbb{R}^N,$$2 curves the translational trajectory. This is the mechanism by which position-only observations can indirectly regularize orientation: the optimizer favors orientations whose body-frame twist explains measured positions with a temporally consistent arc rather than allowing arbitrary yaw drift (Baxter et al., 23 Aug 2025).

Two deployment modes illustrate this property. In Mode (A), the target is represented entirely in $$\varpi(t) := (X(t)^{-1}\dot{X}(t))^\vee \in \mathbb{R}^N,$$3, and the constant-twist prior regularizes orientation even when only USBL positions are measured. In Mode (B), the target representation switches between $$\varpi(t) := (X(t)^{-1}\dot{X}(t))^\vee \in \mathbb{R}^N,$$4 and $$\varpi(t) := (X(t)^{-1}\dot{X}(t))^\vee \in \mathbb{R}^N,$$5 via boundary factors, so the same generalized constant-twist prior reduces to constant translational velocity in $$\varpi(t) := (X(t)^{-1}\dot{X}(t))^\vee \in \mathbb{R}^N,$$6 segments and resumes translation–rotation coupling in $$\varpi(t) := (X(t)^{-1}\dot{X}(t))^\vee \in \mathbb{R}^N,$$7 segments (Baxter et al., 23 Aug 2025).

The reported dynamic docking results quantify the effect. During USBL-only intervals, Mode A yields $$\varpi(t) := (X(t)^{-1}\dot{X}(t))^\vee \in \mathbb{R}^N,$$8 m mean$$\varpi(t) := (X(t)^{-1}\dot{X}(t))^\vee \in \mathbb{R}^N,$$9std position error, versus $\dot{X}(t)=X(t)\,\hat{\varpi}(t).$0 m for Mode B and $\dot{X}(t)=X(t)\,\hat{\varpi}(t).$1 m for raw USBL. During optical intervals both modes are approximately $\dot{X}(t)=X(t)\,\hat{\varpi}(t).$2 m, specifically $\dot{X}(t)=X(t)\,\hat{\varpi}(t).$3 for Mode A and $\dot{X}(t)=X(t)\,\hat{\varpi}(t).$4 for Mode B. Overall, Mode A achieves $\dot{X}(t)=X(t)\,\hat{\varpi}(t).$5 m versus $\dot{X}(t)=X(t)\,\hat{\varpi}(t).$6 m for Mode B (Baxter et al., 23 Aug 2025). Qualitatively, Mode A extrapolates arcs consistent with body-frame twist and maintains an orientation estimate through measurement gaps, whereas Mode B extrapolates straight lines in $\dot{X}(t)=X(t)\,\hat{\varpi}(t).$7 during position-only segments (Baxter et al., 23 Aug 2025).

6. Generalizations beyond constant velocity

The standard constant-twist prior is equivalent to a white-noise-on-acceleration prior whose mean encourages constant body-centric velocity. On $\dot{X}(t)=X(t)\,\hat{\varpi}(t).$8, this is the conventional STEAM prior and can be written as

$\dot{X}(t)=X(t)\,\hat{\varpi}(t).$9

with mean pose trajectory

$$\dot{\varpi}(t)=w(t), \qquad w(t)\sim GP(0,Q_c\delta(t-t')),$$0

up to the stated BCH approximation (Tang et al., 2018). The same work argues that this prior cannot sufficiently represent trajectory sections with non-zero acceleration and can bias posterior estimates (Tang et al., 2018).

The white-noise-on-jerk prior generalizes constant twist by making the prior mean constant acceleration rather than constant velocity. In the minimal form,

$$\dot{\varpi}(t)=w(t), \qquad w(t)\sim GP(0,Q_c\delta(t-t')),$$1

and in the full STEAM treatment the local state is

$$\dot{\varpi}(t)=w(t), \qquad w(t)\sim GP(0,Q_c\delta(t-t')),$$2

with corresponding three-level transition and covariance matrices that preserve exact sparsity (Tang et al., 2018). The paper reports that WNOJ adds $$\dot{\varpi}(t)=w(t), \qquad w(t)\sim GP(0,Q_c\delta(t-t')),$$3, increasing the state dimension and prior factor size, and yields approximately $$\dot{\varpi}(t)=w(t), \qquad w(t)\sim GP(0,Q_c\delta(t-t')),$$4 higher optimization cost, while preserving banded sparsity and linear-time scaling in the number of knots (Tang et al., 2018).

Empirically, WNOJ improves over WNOA across several datasets. On KITTI, overall error changes from $$\dot{\varpi}(t)=w(t), \qquad w(t)\sim GP(0,Q_c\delta(t-t')),$$5 to $$\dot{\varpi}(t)=w(t), \qquad w(t)\sim GP(0,Q_c\delta(t-t')),$$6 on train and from $$\dot{\varpi}(t)=w(t), \qquad w(t)\sim GP(0,Q_c\delta(t-t')),$$7 to $$\dot{\varpi}(t)=w(t), \qquad w(t)\sim GP(0,Q_c\delta(t-t')),$$8 on test. On the University of Toronto dataset, translational error is reduced from $$\dot{\varpi}(t)=w(t), \qquad w(t)\sim GP(0,Q_c\delta(t-t')),$$9 to $SE(3)$00, a $SE(3)$01 reduction; on Richmond Hill, error is reduced from $SE(3)$02 to $SE(3)$03, an $SE(3)$04 reduction (Tang et al., 2018).

A closely related constant-acceleration prior on $SE(3)$05 appears in radar odometry, where body-frame acceleration is explicitly estimated and the non-commutativity of $SE(3)$06 is handled by a third-order Magnus expansion (Retan et al., 2022). The nominal dynamics are

$SE(3)$07

with discrete mean pose update

$SE(3)$08

where $SE(3)$09 includes first-, second-, and third-order commutator terms (Retan et al., 2022). On approximately $SE(3)$10 km of real radar data over $SE(3)$11 sequences, constant acceleration with the polar model gives overall translational error $SE(3)$12 versus $SE(3)$13 for constant velocity with the Cartesian model, rotational error $SE(3)$14 deg/m versus $SE(3)$15, and runtime $SE(3)$16 ms per frame versus $SE(3)$17 ms (Retan et al., 2022).

These developments indicate that “generalized constant twist” can refer either to Lie-group generalization of constant body velocity (Dong et al., 2017, Baxter et al., 23 Aug 2025) or to higher-order priors that generalize the mean motion from constant twist to constant body acceleration (Tang et al., 2018, Retan et al., 2022). The latter interpretation is explicit in the WNOJ literature (Tang et al., 2018).

7. Related interpretations, limitations, and practical considerations

A distinct but conceptually related usage appears in swing–twist decomposition within Clifford algebra. There, a rotor $SE(3)$18 is decomposed into swing and twist relative to a chosen axis $SE(3)$19, with twist extracted by the projection

$SE(3)$20

or, for unit axis $SE(3)$21,

$SE(3)$22

where $SE(3)$23 (Dobrowolski, 2015). The same source proposes that these formulas can be used to design a generalized constant-twist motion prior for 3D motion by keeping the twist about a chosen axis constant while allowing swing to vary (Dobrowolski, 2015). This is not the matrix Lie-group GP or ternary-factor formulation, but it is a mathematically precise axis-constrained interpretation of constant twist.

Within estimation on Lie groups, several practical issues recur. In the GP construction, $SE(3)$24 controls the strength of the constant-twist assumption, and the knot spacing $SE(3)$25 must be small enough that $SE(3)$26 is a good local approximation (Dong et al., 2017). In the ternary factor, $SE(3)$27 is modeled as Gaussian covariance and is practically scaled with the forward interval, for example $SE(3)$28 with $SE(3)$29 (Baxter et al., 23 Aug 2025). The discrete formulation also notes failure modes: large motion changes between intervals are penalized, and long gaps with changing heading can accumulate orientation error; suggested mitigations include robust scaling $SE(3)$30, soft domain priors, and multi-horizon twist factors (Baxter et al., 23 Aug 2025).

Common comparisons to alternative priors are explicit. Euclidean constant-velocity or constant-acceleration priors in $SE(3)$31 are simple but ignore rotational dynamics and do not couple translation with rotation (Baxter et al., 23 Aug 2025). GP priors on Lie groups provide principled uncertainty, exact sparsity, and $SE(3)$32 interpolation, but are computationally heavier and may require maintaining velocity states (Baxter et al., 23 Aug 2025). WNOA and WNOJ on manifolds provide smoothness priors with different expressiveness–complexity trade-offs; WNOJ is preferred when sustained accelerations or motion distortion are important, while WNOA remains adequate for near-constant-velocity motion (Tang et al., 2018).

Taken together, the literature defines generalized constant-twist motion priors as a family of geometric motion regularizers that preserve Lie-group structure while encoding low-order temporal consistency. Their most characteristic property on $SE(3)$33 is body-frame translation–rotation coupling, which improves observability under partial sensing, while their principal axis of generalization is the move from constant velocity to constant acceleration through higher-order stochastic priors (Dong et al., 2017, Baxter et al., 23 Aug 2025, Tang et al., 2018, Retan et al., 2022).