SANM: Sliced Adaptive-Neuro Mapping

• SANM is an online learning module that slices a high-dimensional disturbance-to-error mapping into multiple low-dimensional submappings, reducing underfitting in quadrotor control.
• It employs shallow radial basis function neural networks and Lyapunov-based adaptive laws to ensure stability and real-time compensation on both SO(3) and SE(3) formulations.
• The method integrates as a feedforward compensator within geometric controllers, enabling efficient, interpretable, and PE-free online adaptation on embedded microcontrollers.

Searching arXiv for the SANM-related papers and closely related context. Sliced Adaptive-Neuro Mapping (SANM) is an online learning module for disturbance and uncertainty identification in quadrotor geometric control. In the formulation introduced as a module instance of Dimension-Decomposed Learning (DiD-L), SANM addresses underfitting in high-dimensional online identification by axially “slicing” a high-dimensional nonlinear mapping into multiple low-dimensional submappings handled by shallow neural networks and Lyapunov-derived adaptive laws (Gao et al., 20 Aug 2025). The method is developed for attitude control on $SO(3)$ and extended to full quadrotor control on $SE(3)$, where it serves as a feedforward compensator embedded in geometric controllers and is designed to operate without offline pre-training and without a persistent excitation (PE) condition (Gao, 3 Oct 2025).

1. Terminological scope and placement within Dimension-Decomposed Learning

SANM is presented as the concrete online learning mechanism within the broader DiD-L framework (Gao et al., 20 Aug 2025). In that framework, the disturbance-identification problem is interpreted through the geometry of quadrotor tracking errors and then decomposed along coordinate axes into smaller learning tasks. The method is described as $SO(3)$-aware, interpretable, lightweight, online and adaptive, PE-free, and embedded-friendly, with reported operation at $400$ Hz on STM32-class microcontrollers (Gao et al., 20 Aug 2025).

The central premise is that direct online approximation of a high-dimensional disturbance-to-error map with a shallow neural network is prone to underfitting (Gao, 3 Oct 2025). SANM responds by replacing one entangled identification problem with several axis-wise subproblems. In the attitude-control formulation, the original six-dimensional target feature is decomposed into six one-dimensional features, yielding six slices; operationally, the network is split by body axis $j=1,2,3$, with each slice learning scalar quantities along the corresponding axis (Gao et al., 20 Aug 2025). In the full $SE(3)$ formulation, the decomposition is further articulated as $12$ one-dimensional slices organized into three decoupled axis-wise identification problems (Gao, 3 Oct 2025).

A recurrent source of ambiguity is that the acronym “SANM” is also used for “Symbolic Asymptotic Numerical Solver,” a continuation-based nonlinear solver developed for mesh deformation and unrelated to quadrotor control (Jia, 2021). In the control literature considered here, SANM specifically denotes Sliced Adaptive-Neuro Mapping (Gao et al., 20 Aug 2025).

2. Problem formulation on $SO(3)$ and $SE(3)$

In the attitude-control setting, quadrotor rotational kinematics and dynamics are written as

$\dot R = R[\Omega]_\times, \tag{1}$

$SE(3)$0

where $SE(3)$1, $SE(3)$2, $SE(3)$3, $SE(3)$4 is the inertia tensor, and $SE(3)$5 is a rotational disturbance term (Gao et al., 20 Aug 2025). The analysis distinguishes a known-inertia case, in which $SE(3)$6 is the unknown time-varying rotational disturbance, and an unknown-inertia case, in which the internal inertial term is absorbed into a generalized disturbance: $SE(3)$7

The rotational tracking errors are defined by Lie-algebra-induced coordinates,

$SE(3)$8

with

$SE(3)$9

These are the state variables SANM is designed to regulate through online identification and compensation (Gao et al., 20 Aug 2025).

The $SO(3)$0 extension introduces the full tracking-error vector

$SO(3)$1

together with the desired wrench

$SO(3)$2

pseudo-parameters

$SO(3)$3

and disturbance feature vector

$SO(3)$4

The corresponding nonlinear identification map is written as

$SO(3)$5

and SANM is built by introducing a local pseudo-inverse and then slicing that inverse map into low-dimensional submappings (Gao, 3 Oct 2025).

This setup suggests that SANM is not merely an add-on estimator but a structured identification layer tightly coupled to the geometry of the closed-loop control problem.

3. Slicing principle and low-dimensional submappings

The defining operation of SANM is the axial decomposition of a high-dimensional mapping into low-dimensional slices. In the attitude-control formulation, the paper first writes an unknown high-dimensional mapping from control moment, inertia, and disturbances to the rotational error state as

$SO(3)$6

Because the inverse relation is not globally bijective, a local pseudo-inverse is assumed on a compact set: $SO(3)$7 where

$SO(3)$8

SANM then slices this mapping into low-dimensional submappings, essentially one for each axis (Gao et al., 20 Aug 2025).

In the full $SO(3)$9 treatment, the decomposition is made explicit at the slice level. The reciprocal pseudo-parameter vector is defined as

$400$0

and the disturbance feature vector as

$400$1

The full inverse map is then sliced into $400$2 one-dimensional slices. For each axis $400$3,

$400$4

where

$400$5

Thus, each axis-wise problem depends only on a translational error pair, a rotational error pair, and the corresponding force and moment channels (Gao, 3 Oct 2025).

The stated rationale is threefold. First, underfitting is reduced because each network approximates a simpler low-dimensional map rather than a single difficult high-dimensional map (Gao et al., 20 Aug 2025). Second, interpretability is improved because each slice is associated with a specific coordinate direction and can be analyzed independently (Gao et al., 20 Aug 2025). Third, the decomposition is computationally economical enough for embedded deployment (Gao et al., 20 Aug 2025). A plausible implication is that the method uses coordinate-structured inductive bias rather than representational scale as its primary mechanism for improving online approximation.

4. Neural approximators and adaptive laws

For disturbance approximation, SANM uses shallow radial basis function neural networks. In the $400$6 attitude formulation, each axis $400$7 employs a single-hidden-layer RBF network with $400$8 neurons: $400$9 where $j=1,2,3$0 is the slice input, $j=1,2,3$1 is the Gaussian basis vector, $j=1,2,3$2 is the ideal weight vector, and $j=1,2,3$3 is the bounded approximation error (Gao et al., 20 Aug 2025). The Gaussian activation is

$j=1,2,3$4

with center $j=1,2,3$5 and width $j=1,2,3$6, and the online estimate is

$j=1,2,3$7

The associated weight update law is

$j=1,2,3$8

where $j=1,2,3$9 is the learning-rate matrix or scalar (Gao et al., 20 Aug 2025).

The $SE(3)$0 extension uses slice-wise shallow RBF networks with a $SE(3)$1-$SE(3)$2-$SE(3)$3 structure: $SE(3)$4 where $SE(3)$5, $SE(3)$6, $SE(3)$7 are bounded weights, $SE(3)$8 is the Gaussian basis vector, and $SE(3)$9 is an arbitrarily small approximation error (Gao, 3 Oct 2025). The hidden units are

$12$0

and the estimated disturbances are

$12$1

SANM also adapts physical pseudo-parameters online. In the attitude formulation, the inertia estimate uses reciprocal-form errors

$12$2

with bounded update laws of the form

$12$3

where the slice-wise expression depends on $12$4, moment components, and a scaling based on $12$5 (Gao et al., 20 Aug 2025).

In the $12$6 extension, the reciprocal errors are

$12$7

and the mass and inertia updates are given axis-wise by bounded reciprocal-form laws involving $12$8, $12$9, $SO(3)$0, $SO(3)$1, and $SO(3)$2 (Gao, 3 Oct 2025). The neural weight updates are

$SO(3)$3

$SO(3)$4

and are emphasized as online adaptive laws that do not require pre-training or PE (Gao, 3 Oct 2025).

5. Controller integration in geometric quadrotor control

SANM is integrated as a compensatory module within geometric control laws rather than as a standalone policy. In the $SO(3)$5 attitude setting, the desired moment for axis $SO(3)$6 is

$SO(3)$7

with positive gains $SO(3)$8 (Gao et al., 20 Aug 2025). When $SO(3)$9 is known, the inertial coupling term can be compensated explicitly; when $SE(3)$0 is unknown, that contribution is absorbed into the disturbance and learned online by SANM (Gao et al., 20 Aug 2025).

The $SE(3)$1 controller retains the standard geometric structure. The tracking errors are

$SE(3)$2

$SE(3)$3

and the desired attitude is constructed from the desired force direction and heading: $SE(3)$4 with

$SE(3)$5

For each axis $SE(3)$6, the desired force and moment components are defined as

$SE(3)$7

$SE(3)$8

with optional nominal cancellation of $SE(3)$9 in the known-inertia case (Gao, 3 Oct 2025). The resulting $\dot R = R[\Omega]_\times, \tag{1}$0 are then mapped to rotor thrusts and speeds through standard X-configuration allocation, including

$\dot R = R[\Omega]_\times, \tag{1}$1

This controller placement is significant because the learning module acts on explicitly defined disturbance channels tied to the geometric tracking errors, rather than replacing the underlying geometric control law.

6. Stability analysis and theoretical guarantees

A central feature of SANM is that the online learning laws are embedded in Lyapunov analysis. For rotational dynamics, the attitude error function is

$\dot R = R[\Omega]_\times, \tag{1}$2

The $\dot R = R[\Omega]_\times, \tag{1}$3 paper introduces the Lyapunov candidate

$\dot R = R[\Omega]_\times, \tag{1}$4

and then augments it with inertia and neural-network estimation-error terms (Gao et al., 20 Aug 2025). The resulting derivative satisfies

$\dot R = R[\Omega]_\times, \tag{1}$5

which yields

$\dot R = R[\Omega]_\times, \tag{1}$6

for

$\dot R = R[\Omega]_\times, \tag{1}$7

Under an initial condition in an almost-global domain $\dot R = R[\Omega]_\times, \tag{1}$8, the rotational error converges exponentially to an arbitrarily small ball,

$\dot R = R[\Omega]_\times, \tag{1}$9

with the almost-global qualifier attributed to the topological obstruction on $SE(3)$00 and the exclusion of the usual $SE(3)$01 singular configurations (Gao et al., 20 Aug 2025).

The $SE(3)$02 paper states the main stability claim as near-exponential stability (NES): the full closed-loop state converges exponentially to an arbitrarily small neighborhood of zero (Gao, 3 Oct 2025). The rotational component uses

$SE(3)$03

where

$SE(3)$04

with quadratic bounds

$SE(3)$05

for

$SE(3)$06

For sufficiently small $SE(3)$07,

$SE(3)$08

the associated matrices are positive definite (Gao, 3 Oct 2025).

The composite full-state Lyapunov function includes translational tracking, rotational tracking, parameter-estimation errors, and neural-weight estimation errors. Its derivative is upper bounded by

$SE(3)$09

where

$SE(3)$10

If a coupling term

$SE(3)$11

is sufficiently small, then $SE(3)$12, leading to

$SE(3)$13

and consequently

$SE(3)$14

The paper summarizes this as two propositions: almost-global NES for rotational dynamics on $SE(3)$15, and local NES for the full $SE(3)$16 quadrotor dynamics (Gao, 3 Oct 2025).

These results are framed under explicit assumptions, including local pseudo-invertibility on a compact set, bounded approximation error, boundedness of the error trajectory, and sufficiently small cross-term gains (Gao et al., 20 Aug 2025).

7. Computational profile, experiments, and relation to prior methods

The reported computational motivation for SANM is that online learning in flight control must remain lightweight. The $SE(3)$17 paper states that in experiments each of the three RBF slices uses only $SE(3)$18 neurons, so the learning module requires only $SE(3)$19 Gaussian evaluations per control cycle (Gao et al., 20 Aug 2025). The same paper reports simulation in Gazebo Harmonic / ArduPilot SITL with a $SE(3)$20 Hz control loop and time-varying disturbance generated by a suspended payload, using parameters such as $SE(3)$21 and $SE(3)$22 (Gao et al., 20 Aug 2025). The real-world implementation is reported on an STM32H750 ARM processor with BMI270 IMU, ArduPilot 4.6 firmware, C++ implementation, $SE(3)$23 Hz online learning on the FCU, and a Raspberry Pi 5 used for ROS2 and motion-capture communication while the learning and control loop itself runs on the STM32-class FCU (Gao et al., 20 Aug 2025).

The $SE(3)$24 paper emphasizes the same qualitative advantages: no offline pre-training, no PE condition, no explicit analytic disturbance model, no exact knowledge of disturbance structure, and, in the unknown-inertia case, no need to know $SE(3)$25 (Gao, 3 Oct 2025). Its novelty claim is tied to the decomposition of a difficult $SE(3)$26-dimensional identification problem into small $SE(3)$27D-to-$SE(3)$28D learning problems handled by shallow RBF networks and adaptive laws (Gao, 3 Oct 2025).

Relative to prior methods, the papers position SANM against robust adaptive geometric control without neural learning, neural-network-based geometric adaptive control, MLP-based controllers lacking Lyapunov interpretability, and deep learning approaches characterized as more computationally expensive and less interpretable (Gao et al., 20 Aug 2025). The stated comparative advantages are provable stability, online learning, low computational cost, interpretability, and embedded real-time feasibility (Gao et al., 20 Aug 2025).

A common misconception is to interpret SANM as simply “a neural network controller.” The control papers describe something more specific: a structured identification-and-compensation architecture in which shallow axis-wise RBF estimators, reciprocal parameter adaptation, and Lyapunov-certified update laws are embedded into geometric controllers on $SE(3)$29 and $SE(3)$30 (Gao et al., 20 Aug 2025). Another possible misunderstanding is to conflate SANM with the unrelated nonlinear solver SANM from graphics and numerical continuation; despite the identical acronym, that method belongs to asymptotic numerical continuation rather than adaptive quadrotor control (Jia, 2021).

Taken together, the published formulation presents SANM as a coordinate-structured online identification mechanism for quadrotor geometric control, developed to mitigate underfitting in shallow online approximators while preserving Lyapunov-based stability analysis and microcontroller-level deployability (Gao, 3 Oct 2025).