Advanced State Variable Representations

This lesson expands the original overview into a sequence of small derivations. Read the geometry section first unless you already know right perturbations on SE(3). Read the time section second. The laboratory controls are embedded in the slides; no separate launch is needed. Read Notes opens a scrollable companion containing every equation, the default diagrams, and these explanations. All demonstrations are deliberately small, deterministic teaching examples, not performance benchmarks.

A robot can use a perfectly valid SO(3) rotation at every stored timestamp and still have an inappropriate interpolation model between timestamps. Conversely, a smooth polynomial in matrix entries can leave the rotation group. We therefore separate geometric representation from temporal representation, then combine them near the end. The displayed measurement function is a generic conceptual model, not a claim that every sensor has the same residual.

The pose maps coordinates from body B into world W. Thus multiplying an increment on the right means that the increment is expressed relative to the current body frame. Multiplying on the left means world-frame coordinates, after the proper adjoint conversion. A translation-first twist is required for the particular SE(3) wedge and adjoint matrices later in the lesson. The same letter R is commonly used for rotations and measurement covariances, so each equation states which one it means. Noise spectral density and finite-interval covariance have different units.

SO means special orthogonal: orthogonality plus determinant +1 excludes reflections. SE means special Euclidean: rotations combined with translations. The identity, inverse, and product remain in the same set. SO(2) is the best entry point because its first column lies exactly on a unit circle. That drawing does not identify SO(3) with a sphere. In 3D the order of finite rotations matters.

Differentiate the two coordinates of p with respect to theta. Their dot product is cos(theta)(-sin(theta)) + sin(theta)cos(theta) = 0. Since both p and v have unit norm, expanding the squared norm of p + delta v gives 1 + delta squared. This proves exactly why the tangent step is only a local approximation. The circle is an exact representation of SO(2) through the first column of its rotation matrix. A generic manifold may use a retraction other than the Lie exponential.

At the default increment 0.7 rad, the linear endpoint has norm sqrt(1.49), approximately 1.221, while the exponential endpoint has norm one. At zero increment the two endpoints coincide. Changing the current angle moves the tangent line along with the point. The two operations agree to first order in the increment, but not for a finite step.

Lowercase exp and log act on matrices. Capital Exp and Log include wedge and vee so their arguments or outputs are minimal vectors. The identity log branch works near the current estimate, not as a globally unique coordinate chart for every possible rotation. The local difference matches the right-update convention: bar R Exp(delta) = R implies delta = Log(bar R transpose R), within a consistent branch.

The skew matrix satisfies A cubed = -alpha squared A, which reduces the exponential series to a combination of I, A, and A squared. This yields Rodrigues’ formula. The limiting values define the formula continuously at zero. The tests check orthogonality and the SE(3) constructions at small, moderate, and near-pi angles; they do not confuse exact mathematical preservation with literally zero floating-point error.

Integrating a constant body twist while the body rotates produces a curved world-frame translation. The matrix exponential encodes that coupling. Pure translation, phi = 0, is the special case where the translational coordinate equals the resulting displacement. The appearance of J_l in the SE(3) closed form is compatible with our right composition T Exp(xi). Which side the update uses and which differential map occurs inside the closed form are separate concepts.

One shortcut matrix has orthogonal but non-unit columns, each of length sqrt(1 + delta squared). Its determinant is 1 + delta squared. Repeating it multiplies the scale each time. Normalizing the result could remove scale in this special planar example, but it would not remove the accumulated angular bias. The next lab visualizes both effects and distinguishes approximation error from machine precision.

The exact and shortcut trajectories start at the identity. Both basis columns are drawn, so the diagram can reveal scale and non-orthogonality rather than only a heading. For this particular Euler shortcut the columns remain perpendicular but become too long; the Frobenius orthogonality error measures this failure of unit length. The observed behavior is derived on the previous slide.

Expand Exp(xi) to first order inside the right update. A body point becomes R(a_i + rho + phi cross a_i) + t. Because phi cross a_i = -[a_i]_cross phi, the 3D Jacobian is [R, -R[a_i]_cross]. In 2D phi is a scalar and the angular column is R J a_i. We define residual as prediction minus measurement, so the normal-equation right-hand side has a minus sign. The numerical test perturbs the pose on the same side when checking this derivative.

Four non-collinear body points are transformed by the fixed synthetic ground truth: 35 degrees rotation and translation (0.8, 0.5) meters. The initial translation is (-0.5, -0.4). Measurements are noiseless here to make the convergence mechanism easy to inspect. This is a small least-squares example with fixed associations, not a claim that arbitrary nonconvex SLAM problems converge globally. The chart shows half the squared residual norm.

The adjoint identity maps each right perturbation to a left perturbation at the same fixed nominal pose. Because that coordinate mapping is linear, its covariance transformation is the standard linear covariance transformation. The resulting group-valued distribution is a pushforward of the local Gaussian; its density with respect to a group volume measure is not simply an unconstrained Gaussian density in the matrix entries. A valid Exp map does not make a single Gaussian a good approximation to multiple separated orientation hypotheses.

Conjugate the exponential by T: T exp(xi wedge) T inverse = exp(T xi wedge T inverse). Applying vee gives the adjoint. This derivation fixes the sign of the translation cross-product term. In 2D, using translation-first [rho_x, rho_y, omega], the adjoint is [[R, -Jt], [0, 1]]. The demonstration uses that exact planar counterpart and actual coordinate-frame endpoints rather than freehand arrows.

The demo applies a body-coordinate increment (0.7 meters, 0.2 meters, delta radians). The correct world-coordinate increment is computed by Ad(T). Both constructions yield the same homogeneous matrix up to floating-point precision. The coral comparison deliberately applies the unconverted vector on the left. All displayed frame axes are generated by transforming local basis vectors; no handedness is invented by the renderer.

The first pair maps an additive change in exponential coordinates into a small group increment on a chosen side. The second pair is the corresponding local inverse operation. It is incorrect to change the product order but keep an unspecified J unchanged. These are approximations in the small increment d, not exact finite rotation-vector addition rules. The tests compare the two differential identities against finite perturbations in the declared convention.

Differentiate R(t) = Exp(phi(t)) using the previous slide over a small time dt. The induced left increment is J_l(phi) phi_dot dt, which is the spatial angular velocity times dt. The right increment gives the body angular velocity. This is important when deriving IMU residuals or placing a stochastic prior on local pose coordinates: a coordinate derivative is not automatically the physical twist.

Representation variables, observation times, and output-query times need not coincide. A method can exploit this separation to avoid adding a full pose variable for every high-rate measurement, but it must correctly model how those measurements depend on the support states. This lesson distinguishes an exact all-observation-time GP teaching model from approximate support-only likelihood constructions.

The three time lists answer different engineering questions. Splines often choose regular knots. SDE-derived GPs may use irregular support times. A query does not necessarily add an optimization variable: the conditional distribution can be computed from nearby support states after inference. The demo later uses all observation times in the Markov chain so that the asynchronous likelihood is exact and easy to audit.

Knots are the nondecreasing scalar breakpoints u_i. Control coefficients c_i may be scalar positions, vectors, or, after a different construction, group-valued controls. In an ordinary vector spline, the basis itself gives the derivative with respect to each coefficient. For a nonlinear sensor residual, multiply that derivative by the sensor Jacobian. Compact support is the origin of the local factor connectivity.

The demonstration uses nine coefficients and an open-uniform clamped knot vector on [0, 6] seconds. Its first and last knots are repeated degree + 1 times. Zero-length intervals are handled by the zero-term rule. At the final endpoint the limiting basis places all weight on the last coefficient. At a linear-spline interior knot the derivative is not unique; the implementation reports the right derivative there and a left derivative at the final endpoint. Cubic derivatives agree across simple interior knots.

The selected coefficient has a coral basis function and a shaded support interval on the trajectory plot. The dashed trajectory is the reference with zero displacement; the teal trajectory is the modified result. Their difference at time t is exactly the displacement times the selected basis weight. The query value and velocity are computed by the basis and its analytic derivative, not by a hand-drawn interpolant.

Minimizing the squared whitened measurement residual plus the squared coefficient norm gives the left normal equations. The induced prior covariance of x is Psi Psi transpose. The inverse-free posterior mean follows from Gaussian conditioning and remains meaningful when the finite basis spans only a lower-dimensional state subspace. Writing K inverse without checking rank is not legitimate. A generic GP kernel is not guaranteed to have sparse precision; the SDE construction provides additional structure.

White noise is a generalized process; the SDE notation is shorthand for an integrated stochastic model. Constant velocity refers to the drift or expected motion, not a claim that velocity is constant in every realization. A random-walk mean is piecewise linear between observation-support times, while a constant-velocity posterior position mean has cubic pieces. The sample-path regularity statements are different and should not be inferred by looking only at the mean curve.

For the constant-velocity state, A has a one in its upper-right entry and zeros elsewhere; L = [0, 1] transpose. The matrix exponential is [[1, h], [0, 1]]. Integrating the outer product [u, 1] [u, 1] transpose from zero to h gives h cubed/3, h squared/2, and h. This is an exact discretization of this linear SDE for any positive interval. The process factor depends on actual elapsed time, including irregular gaps.

Stack the process residuals in a block-bidiagonal operator B and stack their covariances with the initial-state covariance in a block-diagonal matrix Q. Then the prior precision is B transpose Q inverse B. This avoids overloading the symbol Phi as both a continuous state transition and a stacked matrix. The demo adds scalar position measurements at states in the chain, so their information contributions are local. Its posterior precision is therefore block tridiagonal; a general SLAM posterior need not be.

The process-noise cross covariance between time a and the end of the interval h is Q(a) F(h-a) transpose. Gaussian conditioning gives Psi. Subtracting the part of the endpoint prediction already explained by x_k gives Lambda. A query can be evaluated without refitting the entire trajectory. However, a likelihood at an interior measurement time must not silently replace a stochastic bridge by its deterministic mean. In the exact demo every observation time is in the state grid, so query intervals contain no observations except their possible endpoints.

Apply the law of total covariance. First average the conditional bridge covariance, which is independent of the endpoint values in this linear model. Then add the covariance of the conditional mean induced by the jointly uncertain endpoints. Omitting their cross covariance is also wrong. For a random walk with q = 1 and a = h/2 = 0.5 seconds in a one-second interval, the bridge variance is 0.25 square meters even when both endpoints are known perfectly.

The 1 term in both kernels comes from initial position variance. In the CV kernel, s times t comes from independent initial velocity variance one. The integrated process covariance contributes the remaining expression. The tests compare this independently assembled kernel regression with the Markov-information solution and bridge interpolation. This demonstrates mathematical sparsity without pretending that the small JavaScript implementation uses a production sparse Cholesky solver.

The band includes support uncertainty plus the conditional bridge term. At a support time the bridge contribution vanishes, but the posterior support variance generally remains positive. The zero bridge term does not imply a perfectly known state. A larger process spectral density allows more deviation from the drift prior; larger measurement standard deviation decreases trust in observations. These knobs have different meanings. Switching motion-prior families also changes the physical units of q.

The endpoint checks follow directly from Exp(0) = I and Exp(Log(T0 inverse T1)) = T0 inverse T1. This is a constant-body-twist path when u is linear in time. The left and right expressions describe the same path under compatible logarithm choices, but raw coordinates on the two sides are different. Do not label this automatically the shortest SE(3) path: a distance or metric would first have to be specified.

Expand the vector expression and collect the coefficient of each c_i to recover the ordinary spline basis. Replacing vector increments by relative group logarithms and addition by ordered group multiplication defines a cumulative Lie-group spline. It is not an equality obtained by treating noncommuting group elements as vectors. The usual compact representation telescopes factors whose cumulative weights equal one, while weights equal zero contribute identity factors. Cubic control poses generally are not interpolated by the resulting curve.

At a planar 180-degree endpoint rotation R1 = -I and R0 = I. The entrywise midpoint is zero, which has determinant zero and cannot be a rotation. A separate translation interpolation plus a rotation-group interpolation remains a valid rigid pose at all times; it simply imposes a different motion model from the SE(3) exponential. The same distinction applies to the SE(2) demonstration.

The start pose is the identity; the end translation is (2, 1) meters. The rotation angle is adjustable. The teal origin traces the constant-twist trajectory; the blue origin follows a straight translation. The coral entrywise blend shares that translation but has an invalid rotation block in general. At fraction zero and one all methods meet their endpoints. The example uses a selected principal planar Log branch, not a globally unique interpolation through all possible rotations.

The reference pose T_k is fixed when defining the chart inside an interval. Differentiating T_k Exp(xi_k) gives the SE(3) right-Jacobian relation between coordinate rate and body twist. The 3-by-3 SO(3) Jacobian earlier is not a substitute for the 6-by-6 SE(3) Jacobian here. A random-walk local-coordinate bridge has a linear conditional mean in xi, recovering the two-pose group interpolation under fixed endpoints, but mapped samples and uncertainty remain group-valued and nonlinear. The scalar GP lab does not claim to implement this full pose-valued estimator.

This worked example is derived directly by differentiating the Euclidean norm and applying the spline chain rule. It assumes line of sight, a known anchor, synchronized time, and a point sensor located at the position represented by p. A lever arm would make orientation enter as well. In a radio-SLAM formulation the anchor or reflecting features may be unknown map variables, and association must be modeled. Temporal representation controls which trajectory variables are involved; it does not replace those physical modeling decisions.

The contrast is a modeling viewpoint, not a rigid separation between mathematical families. Gaussian priors on spline coefficients create covariance kernels. Some SDE priors produce spline-like posterior means. Sparse temporal support does not guarantee a globally sparse reduced SLAM system after map or velocity variables are marginalized. Decide based on the motion assumptions, needed derivatives, uncertainty treatment, and computational structure.

The repository tests separate model identities from user-interface behavior. The math suite uses independent finite-difference and dense-kernel comparisons where possible. Browser tests inspect generated math, control responses, slide navigation, focus, print fallbacks, and responsive layouts. Unit tests increase confidence in the modeled equations and numerical examples; they are not a formal proof of every possible nonlinear estimator or a validation of an unspecified physical sensor system.

The mean pose is held fixed for the covariance-side conversion. Exp/Log formulas assume compatible branches where needed; the group exponential is not globally injective. The SO(3) angular-velocity equations use three-vectors and 3-by-3 Jacobians. Their SE(3) counterparts use six-vectors and 6-by-6 Jacobians. Keep this sheet beside an implementation rather than copying an isolated formula without its conventions.

All bridge matrices are evaluated within a bracket of length h at offset a. At exact support times use the support mean and covariance, or take the limiting formula; do not invert a zero-length process covariance. The posterior interpolation statement assumes the chosen support chain and likelihood account for all intervening observations. For a nonlinear pose process these vector Gaussian formulas belong inside the declared local approximation, not directly on raw rotation-matrix entries.

Return to the tangent experiment to remember why local coordinates are not the global state. Return to the spline experiment to see why one coefficient has only a local influence. Return to the GP experiment to see why an interpolated posterior mean is not the entire posterior distribution. The key practical payoff is a representation that is valid in geometry and explicit about its temporal assumptions.

The handbook chapter supplies the overall organization. Micro Lie theory supplies explicit group and Jacobian conventions. The B-spline documentation and cumulative Lie-spline paper supply the basis and ordered-product constructions. Barfoot et al. supplies SDE-based sparse GP inference. Dong et al. supplies a local pose-group GP construction. The 2026 handbook year is the citation supplied by the public-release repository, not an independent assertion about final publication status.