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\begin{document}
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\title{Power-Dual SEW Retargeting and Haptic-Energy Supervision for Heterogeneous 7-DoF Exoskeleton Teleoperation}


% author names and affiliations
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\author{\IEEEauthorblockN{Xiangtian Kuang}
\IEEEauthorblockA{College of Mechanical and Vehicle Engineering\\
Chongqing University\\
Chongqing, China 400714\\
Email: xiangtiankuang@gmail.com}}
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% As a general rule, do not put math, special symbols or citations
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\begin{abstract}
Motion retargeting and force reflection cannot be designed independently when
a wearable master and a remote robot have different link lengths, joint axes,
and reachable workspaces. This paper develops a coupled formulation for a
heterogeneous 7-DoF exoskeleton--robot pair. An analytic
Shoulder--Elbow--Wrist (SEW) construction first generates feasible Cartesian
elbow and wrist targets; a bounded, branch-seeded least-squares stage then
recovers the slave configuration. The local differential of this complete map
is evaluated by guarded central differences and its transpose defines the
virtual-work-consistent generalized-force map. Slave interaction torque is
obtained from a bias-corrected inverse-dynamics residual, while a damped
least-squares reconstruction provides an end-effector wrench for diagnostics
and Cartesian baselines. On the master side, the mapped feedback passes through
filtering, gain, rate, and torque limits before a final radial energy
projection. A one-step storage invariant is established for this haptic
feedback contribution; no claim is made for the independently added dynamics
compensation or for arbitrary delayed two-port stability. Because the custom
master exoskeleton is still under fabrication, this version reports no
physical performance result. Instead, it fixes a gated validation protocol
with independent pose, wrench, timing, energy, task, and human-subject
evidence, so that later claims can be evaluated without retrospective changes
to endpoints or exclusions.
\end{abstract}

\begin{IEEEkeywords}
Bilateral teleoperation; Upper-limb exoskeleton; Heterogeneous retargeting;
Shoulder--elbow--wrist angle; Interaction wrench estimation;
Energy-aware haptics; Human--robot interaction.
\end{IEEEkeywords}


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\section{Introduction}

Bilateral teleoperation enables a human operator to perform manipulation in
remote, hazardous, or inaccessible environments while retaining human
perception and decision-making in the control loop
~\cite{Rebelo2014Teleop,Jiang2019Telerehab}. Upper-limb exoskeletons
are attractive master interfaces because their joint sensing can capture
operator posture and their distributed actuators can return kinesthetic cues
~\cite{Gull2020ExoReview,Kim2017Harmony}. They are also relevant to emerging
robot-demonstration and manipulation-data pipelines, in which the fidelity and
continuity of the human motion representation directly affect the quality of
the commanded robot behavior~\cite{Wang2023Dexcap,Fang2023Airexo}.

The bilateral case becomes substantially more difficult when the wearable
master and remote robot are kinematically heterogeneous. Direct joint
mirroring is generally unavailable, while a task-space inverse-kinematics
solution may change redundancy branches, approach joint limits, or become
sensitive to initialization near singular configurations
~\cite{Sinha2019IK,Brahmi2019IK}. Moreover, a motion mapping alone
does not determine how a slave-side contact wrench should be reflected to the
operator. If motion scaling, reach saturation, filtering, actuator limits, and
communication effects are ignored in the force path, the resulting feedback
may be geometrically plausible but energetically inconsistent.

The force path introduces two additional difficulties. First, model-based
wrench reconstruction from joint torques is affected by friction, inertial
parameter error, acceleration estimation, torque bias, and Jacobian
conditioning~\cite{Mohammadi2013DO,Indri2020FrictionID}.
Second, active force reflection can deliver net energy to the operator,
particularly during stiff contact or in the presence of delay and
asynchronous measurements. Passivity-oriented and semi-active haptic devices
illustrate useful safety principles
~\cite{Porcini2020Passivity,Buongiorno2019Teleop}, but an energy claim must identify exactly
which torque contribution is supervised and which downstream commands remain
outside the accounting.

\vspace{1mm}
\noindent\textbf{Research question.}
This work asks the following overarching question:
\emph{How should motion retargeting, interaction-wrench reconstruction, and
force reflection be formulated together so that a heterogeneous 7-DoF
exoskeleton--robot pair preserves meaningful arm geometry while exposing and
bounding the energy released by its haptic feedback channel?}
The question is decomposed into three testable subquestions:

\begin{itemize}
    \item \textbf{RQ1 -- geometric retargeting:} Which SEW quantities should
    be transferred between dissimilar embodiments, and how should reach
    limits, redundancy branches, and geometric degeneracies be handled?

    \item \textbf{RQ2 -- wrench reconstruction:} Under what modeling,
    sensing, and conditioning assumptions can slave joint-torque residuals
    provide a useful six-dimensional interaction-wrench estimate?

    \item \textbf{RQ3 -- energy-aware force reflection:} How should the
    differential of the heterogeneous motion map, actuator limits, signal
    processing, and communication conditions enter the slave-to-master force
    map and its discrete-time energy budget?
\end{itemize}

\vspace{1mm}
\noindent\textbf{Contributions.}
The contributions of this work are threefold:

\begin{enumerate}
    \item \textbf{A branch-qualified motion/force contract.} We factor
    heterogeneous retargeting into an analytic SEW Cartesian target and a
    bounded embodiment-specific joint recovery, compute the differential of
    the complete map on a fixed local branch, and use its transpose to derive
    the associated generalized-force map. The same contract declares when
    clipping, fallback geometry, solver failure, or an active joint limit makes
    the differential unusable.

    \item \textbf{An implementation-explicit interaction pathway.} A
    bias-corrected inverse-dynamics residual supplies the generalized slave
    interaction torque used by the proposed force map. A frame-consistent
    damped least-squares solve separately reconstructs the end-effector wrench
    for interpretation and for a Cartesian-Jacobian baseline, avoiding an
    unreported wrench-to-torque round trip in the proposed pathway.

    \item \textbf{A final-stage haptic-energy projection with a bounded claim.}
    The complete feedback pipeline is specified as filter, gain, rate limit,
    torque saturation, and radial energy projection. We prove the resulting
    one-step storage invariant for the haptic contribution and provide a gated
    protocol that independently tests geometry, estimation, mapping, timing,
    energy, task performance, and human factors.
\end{enumerate}

The custom 7-DoF upper-limb exoskeleton described in this paper is the target
master platform for this framework. Because fabrication and physical
integration are ongoing, the present manuscript deliberately separates the
method formulation and experimental protocol from empirical outcome claims.

The remainder of the paper is organized as follows.
Section~\ref{sec:related_work} reviews heterogeneous retargeting, wrench
estimation, and energy-aware teleoperation.
Section~\ref{sec:problem} defines the system variables, mappings, and research
hypotheses. Section~\ref{sec:hardware} describes the master-exoskeleton
architecture and its measurable design requirements.
Section~\ref{sec:framework} presents the geometry- and energy-aware
teleoperation framework. Section~\ref{sec:experiments} specifies the planned
evaluation protocol, and Section~\ref{sec:results} reserves the reporting
structure for subsequently acquired data.
Section~\ref{sec:discussion} discusses limitations and validity threats, and
Section~\ref{sec:conclusion} concludes the paper.

\section{Related Work}
\label{sec:related_work}

Research on bilateral teleoperation spans posture retargeting, exoskeleton design, interaction force estimation, and passivity-based haptic rendering. Modern teleoperation and tactile-robotics frameworks emphasize bidirectional exchange of motion and force between a human and a remote environment, with a strong focus on safety and haptic awareness. This section reviews the most relevant lines of work and highlights the remaining gaps that motivate our approach.

\subsection{Isomorphic and Vision-Aided Teleoperation}
In isomorphic teleoperation, the master and slave share similar kinematic structures, so joint-space mirroring or simple task-space tracking is sufficient. Representative systems include rehabilitation and assistance-focused upper-limb exoskeletons~\cite{Rebelo2014Teleop} and proprioceptive master devices~\cite{Lee2014Teaching}. 

Recent low-cost teleoperation and data-collection platforms demonstrate the
feasibility of dexterous manipulation through several distinct interface
families. Vision- or tracking-driven systems include ACE, AnyTeleop, and
OpenTeach~\cite{Yang2024ACE,Qin2023AnyTeleop,Iyer2024OpenTeach}. They support
multiple embodiments and scalable demonstration collection, but pose
measurement and visual feedback do not by themselves define a
power-conjugate kinesthetic channel. Physical master platforms such as HOMIE,
Mobile ALOHA, and GELLO instead exploit an isomorphic cockpit or a
robot-matched low-cost controller
~\cite{HOMIE2025,Fu2024MobileALOHA,Wu2023Gello}. Their embodiment match
simplifies command generation, whereas it does not resolve force reflection
for a kinematically different wearable master. System-level analyses further
summarize challenges in exoskeleton-based teleoperation and rehabilitation
interfaces~\cite{Gull2020ExoReview}.

\subsection{Heterogeneous 7-DoF Arm Retargeting}
Retargeting between kinematically heterogeneous master--slave arms typically relies on numerical inverse kinematics (IK) with redundancy resolution or null-space shaping. Classical IK approaches for redundant 7-DoF manipulators~\cite{Sinha2019IK,Brahmi2019IK} have been widely applied in human--robot retargeting and humanoid control. Their continuity, convergence, and computational cost depend on the solver, initialization, constraints, and treatment of redundancy, particularly near joint or workspace boundaries.

Recent exoskeleton-based teleoperation systems explore posture-based
homo--hetero mapping, such as the upper-limb heteromorphic teleoperation
framework in~\cite{Cheng2024HomoHetero}. Geometric SEW constructions
provide an alternative representation of reach and arm-plane redundancy
~\cite{Elias2024SEW}.
However, an analytic SEW point construction does not by itself resolve
platform-specific joint recovery, joint limits, branch continuity, or the
dual force map induced by the retargeting.

\subsection{Force Estimation in Teleoperation and Exoskeletons}
Accurate estimation of the environment interaction wrench remains central for high-quality haptic feedback. Model-based residual-dynamics estimators and nonlinear disturbance observers have been applied to torque-controlled manipulators~\cite{Mohammadi2013DO}. However, the approach is sensitive to friction, modeling error, and torque noise. 

Exoskeleton-based teleoperation architectures often rely on direct F/T sensing or joint-based observers, such as force-sensitive exoskeletons for elderly care robotics~\cite{Toedtheide2023ForceSensitive} and lightweight tactile interfaces~\cite{Forouhar2024TactileExo}. These systems demonstrate feasibility but remain affected by compliance, transmission losses, and bias handling. Vision-only pose teleoperation~\cite{Li2020VisionTeleop} can avoid robot inverse-dynamics estimation, but its pose signal does not itself provide a power-conjugate wrench for kinesthetic feedback. These observations motivate configuration-stratified validation of residual estimation rather than a general robustness claim for lightweight hardware.

\subsection{Passivity-Based Haptic Rendering}
The transparency--stability trade-off in bilateral teleoperation has been well established, especially under communication delay and stiff contact. Passive and semi-active actuators, including MR clutches, offer hardware-level means to limit active energy injection~\cite{Pisetskiy2021MRclutch}. Passivity-oriented control has also been integrated into multi-DoF exoskeleton teleoperation~\cite{Buongiorno2019Teleop}.

The passivity observer/controller construction provides a foundational
sampled-data energy-accounting mechanism~\cite{Hannaford2002TDPC}.
Time-domain passivity control has also been applied to exoskeleton-based
bilateral teleoperation with independently passivated slave devices
~\cite{Porcini2020Passivity}. Multi-sensory control strategies further improve
dynamic transparency in upper-limb exoskeletons
~\cite{Zimmermann2020Transparency}. These methods establish the relevant
stability--transparency trade-off, while the force coordinate induced by a
heterogeneous motion map and the exact torque included in discrete energy
accounting still have to be declared for each implementation.

\subsection{Research Gap}
The literature reveals three coupled gaps that motivate the present work:

\begin{itemize}
    \item \textbf{Retargeting is often separated from force reflection.}
    Geometry-preserving motion mappings are commonly evaluated through pose
    error alone, without deriving the differential map required to relate
    generalized power between heterogeneous embodiments.

    \item \textbf{Wrench-estimation assumptions are insufficiently isolated.}
    Residual-dynamics estimators depend on model quality, torque sensing,
    friction compensation, acceleration estimation, and Jacobian conditioning;
    these effects require independent ground truth and paired evaluation.

    \item \textbf{Energy claims must be tied to a precisely defined port.}
    Filtering, scaling, saturation, timing, and communication can alter the
    torque and velocity used for power computation. An energy argument is
    therefore meaningful only when it identifies the exact controlled torque,
    every downstream operation, and the scope of the communication model.
\end{itemize}

To address these gaps, the present framework connects an SEW target
construction, platform-specific joint recovery, model-based residual wrench
estimation, and an energy-aware master torque supervisor through a common
problem formulation. The framework is evaluated only after its geometric,
estimation, timing, and energy assumptions have been tested separately.
No novelty is claimed for the SEW coordinate~\cite{Elias2024SEW}, residual
estimation~\cite{Mohammadi2013DO}, DLS inversion, or time-domain energy
accounting~\cite{Hannaford2002TDPC} in isolation. The claimed methodological
advance is their branch-qualified coupling: the differential is taken through
the complete constrained retargeting map; its transpose acts on the same
generalized slave interaction quantity; tracking, transport, shaping, and
post-allocation errors are separated; and the energy statement is attached to
the actual applied haptic increment rather than to an upstream wrench. This
coupling, and not a list of standard modules, defines the unit tested by the
protocol.

\begin{table*}[t]
\centering
\footnotesize
\renewcommand{\arraystretch}{1.18}
\begin{tabularx}{\textwidth}{p{2.8cm} X X X}
\toprule
\textbf{Open interface} & \textbf{Required mathematical contract}
& \textbf{Method element in this work} & \textbf{Evidence required before a claim}\\
\midrule
Heterogeneous motion
& Feasible targets, branch state, limits, and nonsmooth events
& SEW Cartesian construction plus bounded joint recovery
& Pose ground truth, failure/discontinuity rates, and timing against matched baselines\\
Motion-to-force coupling
& Differential of the complete fixed-branch map and its virtual-work dual
& Guarded finite-difference $A$ and generalized-force map $A^\top$
& Differential convergence, validity rate, and actual/reference power mismatch\\
Interaction reconstruction
& Torque residual, frame, reference point, spatial order, and observability
& Offline bias correction and DLS end-effector wrench reconstruction
& Independent six-axis wrench reference, singular-value stratification, and perturbations\\
Haptic energy
& Exact candidate/applied torque, velocity sample, timing, and downstream operations
& Final-stage radial projection of the shaped haptic contribution
& Independent energy reconstruction, intervention cost, and finite contact/network envelope\\
\bottomrule
\end{tabularx}
\caption{Claim-to-evidence structure used in place of categorical
``yes/no'' comparisons. Each row identifies an interface that must be defined
mathematically and tested independently before an end-to-end conclusion.}
\label{tab:claim_evidence}
\end{table*}

\section{System Overview and Problem Formulation}
\label{sec:problem}

\subsection{System Boundary and Notation}

The system comprises a human operator, a 7-DoF wearable master exoskeleton, a
communication channel, a 7-DoF robot arm, and a remote environment. Let
$q_m,\dot q_m\in\mathbb{R}^{7}$ denote the measured master joint position and
velocity, and let $q_s,\dot q_s\in\mathbb{R}^{7}$ denote the slave state. The
master and slave SEW points are denoted by
$p_{S_i},p_{E_i},p_{W_i}\in\mathbb{R}^{3}$, where $i\in\{m,s\}$.
Twists use the ordering $[v^\top\ \omega^\top]^\top$ and wrenches use
$[f^\top\ m^\top]^\top$, both at the end-effector point. The mathematical
formulation expresses their axes in a common task frame $\{C\}$. The current
URDF simulation realizes this as an identity alignment between the fixed
master and slave base axes; physical experiments require measured base-to-$C$
calibrations and a calibrated tool-center point.

The commanded slave configuration is written as
\begin{equation}
    q_s^{\mathrm{ref}}[k]
    = \mathcal{R}\!\left(q_m[k];b_k,\theta_R\right),
    \label{eq:retarget_map}
\end{equation}
where $b_k$ is the numerical branch seed and $\theta_R$ collects link lengths,
frame calibrations, tolerances, and joint limits. Thus, $\mathcal{R}$ is not
assumed to be a globally single-valued smooth map. To avoid conflating two
distinct operations, we factor it as
\begin{equation}
    y_s^{\mathrm{ref}} = \mathcal{G}(y_m;\theta_G),\qquad
    q_s^{\mathrm{ref}}
    = \mathcal{K}_s(y_s^{\mathrm{ref}};b_k,\theta_K).
    \label{eq:retarget_factorization}
\end{equation}
Here, $\mathcal{G}$ is the analytic Cartesian SEW target construction and
$\mathcal{K}_s$ is the embodiment-specific joint recovery. The SEW descriptor
$y_i$ contains reach direction, reach magnitude, swivel angle, and a terminal
orientation objective. This factorization permits the geometric construction
to be evaluated independently of the numerical or analytic solver used by a
particular slave arm.

\subsection{Differential Mapping and Power Objective}

At configurations where $\mathcal{R}$ is differentiable and $b_k$ identifies
one fixed branch, define the local retargeting differential
\begin{equation}
    A(q_m;b_k) =
    \frac{\partial\mathcal{R}(q_m;b_k,\theta_R)}{\partial q_m},
    \qquad
    \dot q_s^{\mathrm{map}} = A(q_m;b_k)\dot q_m.
    \label{eq:retarget_differential}
\end{equation}
Reach clipping, branch changes, and joint-limit projections can make this map
nonsmooth; those events are logged explicitly and are not covered by an
unqualified smoothness assumption. Let $\bar A[k]$ denote the most recent
valid differential, updated at 50\,Hz and held between updates. Two velocities
must be distinguished in the current multirate simulator. At every 500-Hz
haptic sample, the unmodified algebraic velocity
$\dot q_s^{\mathrm{map}}[k]=\bar A[k]\dot q_m[k]$ is recomputed for the
virtual-work contract. At a 50-Hz mapping instant $k_u$, the slave-controller
feedforward
\begin{equation}
    \dot q_s^{\mathrm{ctrl}}[k_u]
    =\mathrm{clip}_{0.8\dot q_{s,\max}}\!
      \left(\bar A[k_u]\dot q_m[k_u]\right)
    \label{eq:controller_reference_velocity}
\end{equation}
is computed once and held until the next update. Thus,
$\dot q_s^{\mathrm{map}}$, $\dot q_s^{\mathrm{ctrl}}$, and the actual
$\dot q_s$ are conceptually separate signals. The current run-level log stores
the actual velocity and the resulting algebraic power scalar but not both
reference vectors; explicit vector logging is a G0a prerequisite.

The present simulator colocates retargeting and slave control, so it has no
forward network delay. For the G3 network tests, mapping update $i$ creates the
forward packet
$u_f[i]=(q_s^{\mathrm{ref}}[i],\dot q_s^{\mathrm{ctrl}}[i],i)$, while the
corresponding $\bar A[i]$ is retained in the synchronized master log. If
$\kappa_f(k)$ is the most recent forward-packet index accepted by slave sample
$k$, the physical controller uses
\begin{equation}
 q_{s,\mathrm{net}}^{\mathrm{ref}}[k]
 =q_s^{\mathrm{ref}}[\kappa_f(k)],\qquad
 \dot q_{s,\mathrm{net}}^{\mathrm{ctrl}}[k]
 =\dot q_s^{\mathrm{ctrl}}[\kappa_f(k)].
 \label{eq:forward_held_reference}
\end{equation}
The slave does not require $A$ for tracking. Instead, each return packet echoes
the active forward-map identifier
$\mu(j)=\kappa_f(j)$ at its slave source sample $j$. This identifier binds a
returned residual and velocity to the exact held reference and historical
$\bar A[\mu(j)]$ that generated them.

Let $\hat\tau_{s,\mathrm{int}}$ be the estimated generalized slave interaction
torque under the environment-on-robot sign convention. Virtual work motivates
the corresponding environment-on-master reaction
\begin{equation}
    \tau_{m,\mathrm{dual}}
    = \sigma_f A(q_m;b_k)^\top\hat\tau_{s,\mathrm{int}},
    \label{eq:power_dual_map}
\end{equation}
where $\sigma_f>0$ is a declared force scale. No additional minus sign is
introduced after fixing this reaction convention. On a valid branch,
\begin{equation}
    \tau_{m,\mathrm{dual}}^\top\dot q_m
    =
    \sigma_f\hat\tau_{s,\mathrm{int}}^\top
    \dot q_s^{\mathrm{map}}.
    \label{eq:power_dual_identity}
\end{equation}
\begin{proposition}[Fixed-branch virtual-work dual]
If $\delta q_s=A\,\delta q_m$ on a certified local branch, then
$\tau_{m,\mathrm{dual}}=\sigma_fA^\top\tau_s$ is the unique generalized
master force satisfying
$\tau_{m,\mathrm{dual}}^\top\delta q_m
=\sigma_f\tau_s^\top\delta q_s$ for every $\delta q_m$.
\end{proposition}
\begin{IEEEproof}
Substitution gives
$\sigma_f\tau_s^\top\delta q_s
=\sigma_f\tau_s^\top A\delta q_m
=(\sigma_fA^\top\tau_s)^\top\delta q_m$. Equality for every virtual
displacement fixes the generalized-force coefficient uniquely.
\end{IEEEproof}
This is an algebraic identity, not actual bilateral power equality. Before
delay or master-side shaping, slave tracking error contributes
\begin{equation}
    \Delta P_{\mathrm{trk}}
    = \sigma_f\hat\tau_{s,\mathrm{int}}^\top
      \left(A\dot q_m-\dot q_s\right).
    \label{eq:tracking_power_error}
\end{equation}
Delay, filtering, rate limits, saturation, and energy projection introduce
additional measurable differences and are reported separately.

\subsection{Controlled Haptic Contribution and Storage}

All nominal filtering, gain, rate limiting, and haptic-torque saturation
precede the energy projection. Let $\tau_{h,\mathrm{app}}[k]$ denote the final
haptic feedback contribution returned by that projection. Using the convention
that positive power is delivered by this contribution to the operator,
\begin{equation}
    P_h[k]
    = \tau_{h,\mathrm{app}}[k]^\top\dot q_m[k],
    \label{eq:applied_port_power}
\end{equation}
and the unbounded discrete storage update is
\begin{equation}
    E^+[k+1] = E[k] - P_h[k]\Delta t_k,
    \label{eq:applied_port_energy}
\end{equation}
where $\Delta t_k=t_{k+1}-t_k>0$ is the measured interval of the applied
output sample. The fixed-rate simulation is the special case
$\Delta t_k=T_s$.
The complete motor command may additionally contain independently designed
master dynamics compensation,
$\tau_{m,\mathrm{cmd}}=\tau_{m,\mathrm{comp}}+\tau_{h,\mathrm{app}}$.
The present storage does not account for $\tau_{m,\mathrm{comp}}$ or the
mechanical energy of the exoskeleton. Consequently, the analytical claim is
limited to the haptic contribution. No later operation may modify
$\tau_{h,\mathrm{app}}$ except an emergency zero-output action. On physical
hardware this requirement must be enforced either by reserving verified
actuator headroom for the haptic component or by relocating the projection
and accounting to the actual post-allocation torque. A combined low-level
saturation that changes the haptic increment invalidates the stated invariant
unless that actual increment is reconstructed and used for accounting.
Communication-delay robustness and total human--device passivity are not
inferred from \eqref{eq:applied_port_energy}.

\subsection{Testable Hypotheses and Scope}

The evaluation is organized around four hypotheses:
\begin{itemize}
    \item \textbf{H1:} the SEW target construction reduces the locked
    trajectory-level composite rate of solver/tolerance failure or observable
    joint/swivel discontinuity relative to the primary IK baseline at
    comparable task-space accuracy;
    \item \textbf{H2:} on the preregistered noisy and near-singular subset,
    regularized residual-torque inversion yields lower force and moment RMSE
    than an undamped solve, with a measurable bias--variance trade-off;
    \item \textbf{H3:} for a matched wrench input, the differential-dual map
    $A^\top J_s^\top F$ reduces cross-embodiment power mismatch relative to
    the mapping-agnostic $J_m^\top F$ reflection;
    \item \textbf{H4:} when reconstructed from the actual post-allocation
    haptic increment, the final energy projection prevents lower-budget
    violations beyond numerical tolerance relative to projection bypass, with
    a measurable transparency cost over the tested envelope.
\end{itemize}
The virtual-work identity and the one-step storage inequality are deterministic
contracts verified at G0a; they are not treated as statistically discovered
effects.
These hypotheses deliberately avoid universal claims about arbitrary stiffness
or delay. Their acceptance criteria, statistical tests, and failure reporting
are defined before physical data are collected.

\begin{figure*}[!ht]
    \centering
    \includegraphics[width=0.32\linewidth]{assets/overview.jpg}%
    \includegraphics[width=0.32\linewidth]{assets/explain.jpg}%
    \includegraphics[width=0.32\linewidth]{assets/joint_distribution.jpg}
    \caption{Overall hardware design of the proposed 7-DoF upper-limb exoskeleton. 
    (Left) Intended wearable arrangement and adjustable human--exoskeleton interface. 
    (Middle) Joint arrangement and kinematic routing used in the current mechanical design. 
    (Right) Distribution of the seven rotational joints ($q_0$--$q_6$) across the shoulder, elbow, and wrist modules.}
    \label{fig:exo_overview}
\end{figure*}

\section{Target Master Platform and Design Requirements}
\label{sec:hardware}

Prior upper-limb exoskeletons have explored anatomically aligned actuation and precise joint-center matching to improve motion capture accuracy and wearer comfort. Representative shoulder--arm systems emphasize accurate glenohumeral alignment and low-distortion rotational layouts~\cite{Yan2014ShoulderExo}. Wrist and distal-arm devices similarly prioritize compactness and minimal reflected inertia to preserve natural hand motion~\cite{Buongiorno2018WRES}. 

The target master device is a custom 7-DoF upper-limb exoskeleton currently
under fabrication for bilateral teleoperation. Unlike a load-assistance
exoskeleton, its design objectives are posture-measurement fidelity,
backdrivability, bounded kinesthetic feedback, and compatibility with the SEW
representation in Section~\ref{sec:sew}. These objectives are not treated as
measured properties at the present stage. They are converted into explicit
characterization requirements in Section~\ref{sec:experiments}. An overview of
the current mechanical design is provided in Fig.~\ref{fig:exo_overview}.

\subsection{Three-DoF Orthogonal Shoulder Architecture}
The shoulder module design (Fig.~\ref{fig:shoulder_arch}) places three nominally
orthogonal rotational axes near the human glenohumeral joint, consistent with
prior shoulder-exoskeleton design principles~\cite{Yan2014ShoulderExo}.
Fabrication tolerance, soft-tissue motion, and subject-specific anatomy will
prevent perfect alignment; the resulting misalignment must therefore be
measured rather than assumed. The proposed actuator layout is:

\begin{itemize}
    \item \textbf{(1) Back-mounted flexion/extension axis.}  
    The first DoF actuator is placed behind the scapular region to keep bulk
    away from the upper arm and to approximate a flexion--extension axis.

    \item \textbf{(2) Lateral shoulder abduction axis.}  
    The second actuator is positioned laterally to approximate an
    abduction--adduction axis; alignment error and parasitic motion remain
    quantities to be measured.

    \item \textbf{(3) Humeral-roll ring gear.}  
    A lightweight annular ring gear encircles the upper arm and is driven by a
    compact motor to approximate a coaxial humeral-roll axis. Its reflected
    inertia and friction will be established experimentally rather than inferred
    from the layout.
\end{itemize}

\begin{figure}[t]
    \centering
    \includegraphics[width=0.75\linewidth]{assets/shoulder.jpg}
    \caption{Intended three-DoF shoulder architecture; physical joint-center
    alignment remains a characterization outcome.}
    \label{fig:shoulder_arch}
\end{figure}

\subsection{Forearm and Wrist Module with Symmetric 3-DoF Design}
The forearm--wrist assembly (Fig.~\ref{fig:wrist_module}) follows the same
geometric design objective. Three rotational DoFs---forearm roll, wrist pitch,
and wrist yaw---are realized through compact transmission mechanisms informed
by wrist-exoskeleton designs~\cite{Buongiorno2018WRES}. The present design does
not assume magnetorheological actuation. Actuator type, reduction ratio,
continuous torque, peak torque, friction, and torque bandwidth will be reported
from the fabricated system.

\begin{figure}[t]
    \centering
    \includegraphics[width=0.78\linewidth]{assets/wrist.jpg}
    \caption{Current symmetric three-DoF wrist-module design with nominal
    yaw--pitch--roll axes.}
    \label{fig:wrist_module}
\end{figure}

\subsection{Structure and Human-Fit Requirements}
A hybrid frame combining aluminum or composite links with polymer interface
components is intended to balance structural stiffness, distal mass, and
wearability. These choices create testable requirements rather than guaranteed
outcomes: structural deflection must be separated from encoder motion,
backdriving torque must be measured throughout the workspace, and fitting
adjustments must cover a declared anthropometric range. Comfort and alignment
will be assessed independently of task performance.

\subsection{Interface to the Teleoperation Framework}
The mechanical design exposes the following interfaces to the teleoperation
architecture:
\begin{itemize}
    \item joint positions and synchronized velocities for master-side SEW
    reconstruction;
    \item calibrated joint-torque commands and measured actuator states for
    kinesthetic feedback;
    \item declared torque, velocity, position, and thermal limits for nominal
    haptic shaping and independent low-level safety;
    \item hardware timestamps and safety-state logs for independent power and
    timing reconstruction.
\end{itemize}
The experiment protocol treats sensing accuracy, actuator bandwidth,
backdrivability, structural stiffness, alignment, and safety thresholds as
prerequisites for system-level evaluation.

% ---------------- Geometry- and Energy-Aware Teleoperation Framework ----------------

\section{Geometry- and Energy-Aware Teleoperation Framework}
\label{sec:framework}

The framework contains four stages. First, master joint measurements are
converted to a master SEW descriptor. Second, an
analytic geometric construction produces a feasible slave SEW target, after
which a bounded platform-specific solver recovers the slave joint reference
and a guarded finite difference estimates the differential $A$. Third, a
bias-corrected inverse-dynamics residual estimates generalized slave
interaction torque; a DLS solve provides a Cartesian wrench for diagnostics
and the direct-Jacobian baseline. Fourth, $A^\top$ maps the generalized
interaction to the master, followed in strict order by a torque filter, gain,
rate limit, saturation, and final haptic-energy projection. The current
repository implementation binds these stages to \texttt{master\_7dof.urdf} and
\texttt{real\_slave\_7dof.urdf}. Physical frame calibration, timing, and
behavior remain validation requirements in Section~\ref{sec:experiments}.

\begin{figure*}[t]
\centering
\begin{tikzpicture}[
    >=Latex,
    block/.style={draw, rounded corners, align=center, minimum height=8mm,
                  text width=25mm, font=\scriptsize},
    wide/.style={block, text width=31mm},
    line/.style={->, thick},
    base/.style={->, dashed, semithick}
]
\node[block] (master) {Master state\\$q_m,\dot q_m$};
\node[block, right=6mm of master] (sew) {Analytic SEW\\target $\mathcal G$};
\node[wide, right=6mm of sew] (recover) {Bounded recovery\\$\mathcal K_s(\cdot;b_k)$};
\node[block, right=6mm of recover] (slave) {Slave tracking\\and environment};

\node[wide, below=12mm of slave] (residual) {Bias-corrected residual\\
$\hat\tau_{s,\mathrm{int}}$};
\node[block, left=6mm of residual] (dual) {Power-dual map\\$A^\top$};
\node[wide, left=6mm of dual] (shape) {Filter $\rightarrow$ gain $\rightarrow$ rate\\
$\rightarrow$ saturation};
\node[wide, left=6mm of shape] (energy) {Final haptic-energy\\projection};
\node[block, left=6mm of energy] (output) {Applied haptic\\$\tau_{h,\mathrm{app}}$};

\node[block, below=9mm of residual] (dls) {DLS wrench\\${}^C\hat F_{\mathrm{int}}$};
\node[wide, left=6mm of dls] (direct) {Cartesian baseline\\
$({}^CJ_{E_m})^\top{}^C\hat F_{\mathrm{int}}$};

\draw[line] (master) -- (sew);
\draw[line] (sew) -- (recover);
\draw[line] (recover) -- node[above,font=\scriptsize] {$q_s^{\mathrm{ref}}$} (slave);
\draw[line] (slave) -- (residual);
\draw[line] (residual) -- (dual);
\draw[line] (dual) -- (shape);
\draw[line] (shape) -- (energy);
\draw[line] (energy) -- (output);
\draw[line] (recover.south) |- node[pos=0.35,left,font=\scriptsize]
    {guarded FD $A$} (dual.north);
\draw[base] (residual) -- (dls);
\draw[base] (dls) -- (direct);
\draw[base] (direct.west) -| node[pos=0.25,below,font=\scriptsize]
    {baseline only} (shape.south);
\end{tikzpicture}
\caption{Executable information flow. The proposed feedback path maps the
bias-corrected generalized residual directly through $A^\top$. DLS wrench
reconstruction feeds diagnostics and the dashed Cartesian baseline. All
nominal shaping precedes the final projection, which supervises only the
applied haptic contribution.}
\label{fig:method_architecture}
\end{figure*}

\subsection{Heterogeneous SEW Target Construction}
\label{sec:sew}

\begin{figure*}[t]
    \centering
    \subfloat[SEW master and slave geometry\label{fig:sew_geometry_all_a}]{
        \includegraphics[width=0.31\linewidth]{assets/sew_geometry.png}
    }\hfill
    \subfloat[Swivel angle definition\label{fig:sew_geometry_all_b}]{
        \includegraphics[width=0.31\linewidth]{assets/swivel_angle.png}
    }\hfill
    \subfloat[Two-sphere intersection for elbow\label{fig:sew_geometry_all_c}]{
        \includegraphics[width=0.31\linewidth]{assets/two_sphere_intersection.png}
    }
    \caption{Geometric interpretation of the proposed SEW retargeting:
    (a) master and slave SEW triangles remain geometrically consistent
    despite different link lengths;
    (b) the swivel angle $\phi_m$ is defined as the rotation from the
    reference normal $n_{\mathrm{ref}}$ to the upper-arm normal $n_m$
    around the reach direction $\hat{x}_m$;
    (c) the slave elbow lies on the intersection circle of two spheres
    centered at $S_s$ and $W_s^{\mathrm{ref}}$ with radii $L_1$ and $L_2$,
    with $\phi_m$ selecting a unique point on this circle.}
    \label{fig:sew_geometry_all}
\end{figure*}

Bilateral teleoperation between heterogeneous 7-DoF arms requires a posture
representation that does not assume identical joint coordinates. Motivated by
the conventional SEW redundancy coordinate for 7-DoF arms
~\cite{Elias2024SEW}, we use a Shoulder--Elbow--Wrist descriptor to construct a
feasible Cartesian target.
The target construction is analytic; recovery of a complete slave joint vector
is treated separately because it depends on the slave joint axes, limits, and
available kinematic solver. Geometry-preserving retargeting is also used in
vision-based pipelines, where pose or keypoint measurements may feed a broader
optimization-based robot mapping~\cite{Wu2023Gello,Rong2020FrankMocap}.

% ==========================================================
\subsubsection{SEW Descriptor and Validity Domain}
SEW makes the redundant arm-plane coordinate explicit, but it does not by
itself guarantee temporal continuity or remove algorithmic singularities
~\cite{Elias2024SEW}. Fig.~\ref{fig:sew_geometry_all} illustrates the
geometric structure used here. Let
$p_{S_m},p_{E_m},p_{W_m}\in\mathbb{R}^3$ be the master shoulder, elbow, and
wrist positions expressed in the common chest frame $\{C\}$.

The shoulder--wrist direction and reach are
\begin{equation}
    r_m = p_{W_m}-p_{S_m},\qquad
    d_m=\|r_m\|,\qquad
    \hat{x}_m = \frac{r_m}{\|r_m\|}.
\end{equation}

The master upper-arm plane normal is
\begin{equation}
    n_m=\frac{(p_{E_m}-p_{S_m})\times(p_{W_m}-p_{S_m})}
              {\|(p_{E_m}-p_{S_m})\times(p_{W_m}-p_{S_m})\|}.
\end{equation}
These normalized quantities are valid only when
\begin{equation}
    d_m>\epsilon_r,\qquad
    \left\|(p_{E_m}-p_{S_m})\times(p_{W_m}-p_{S_m})\right\|>\epsilon_n.
    \label{eq:sew_validity}
\end{equation}
A violation marks the pose invalid for differential feedback. The current
defaults are $\epsilon_r=10^{-9}$\,m and
$\epsilon_n=10^{-8}$\,m$^2$; these are numerical degeneracy guards, not
physical accuracy thresholds, and G0c includes threshold sensitivity.
A geometric placeholder may be returned for software continuity, but it is not
treated as a valid SEW measurement.

A chest-frame reference normal is obtained by projecting
the chest vertical axis $e_z=[0\ 0\ 1]^\top$ onto the plane
orthogonal to $\hat{x}_m$:
\begin{equation}
    n_{\mathrm{ref}}
    =\frac{ e_z-(e_z^\top\hat{x}_m)\hat{x}_m }
           { \| e_z-(e_z^\top\hat{x}_m)\hat{x}_m \| }.
\end{equation}
The implemented reference-axis fallback is activated when the norm of the
projected unit axis in the denominator is below $10^{-6}$.

The swivel angle is the signed rotation from $n_{\mathrm{ref}}$ to $n_m$
about~$\hat{x}_m$ (see Fig.~\ref{fig:sew_geometry_all}(b)):
\begin{equation}
    \phi_m=\mathrm{atan2}\!\left(
        \hat{x}_m^\top(n_{\mathrm{ref}}\!\times n_m),~~
        n_{\mathrm{ref}}^\top n_m
    \right),
\end{equation}
where the current realization uses the principal value. Because the
Rodrigues construction below is $2\pi$-periodic, this choice does not create a
Cartesian discontinuity solely at the angle wrap; branch and fallback
continuity are addressed separately.

% ==========================================================
\subsubsection{Closed-form Wrist and Elbow Reconstruction}

Let $p_{S_s}$ denote the slave shoulder position, and $L_1,L_2$ its upper-arm and
forearm link lengths. A feasible slave wrist distance must satisfy the triangle
inequality with a small safety margin $\varepsilon>0$:
\begin{equation}
    d_{\min}=|L_1-L_2|+\varepsilon,\qquad
    d_{\max}=L_1+L_2-\varepsilon.
\end{equation}
The present software uses $\varepsilon=10^{-3}$\,m; the reach margin and
degeneracy thresholds are varied in the same G0c sensitivity study.
We clip the master reach into this feasible interval:
\begin{equation}
    d_s=\mathrm{clip}(d_m;\,d_{\min},d_{\max}),\qquad
    p_{W_s}^{\mathrm{ref}}=p_{S_s}+d_s \hat{x}_m.
\end{equation}
This is the absolute-reach policy implemented in the current software.
Normalized reach and a calibrated affine reach scale are retained as
preregistered ablations because absolute transfer may overuse clipping when
master and slave link-length sums differ substantially.

The slave elbow $p_{E_s}$ lies on the circle defined by the intersection of two
spheres centered at $p_{S_s}$ and $p_{W_s}^{\mathrm{ref}}$:
\[
\|p_{E_s}-p_{S_s}\|=L_1,\qquad
\|p_{E_s}-p_{W_s}^{\mathrm{ref}}\|=L_2,
\]
as shown in Fig.~\ref{fig:sew_geometry_all}(c). Let
\[
d=\|p_{W_s}^{\mathrm{ref}}-p_{S_s}\|,\qquad
\hat{e}_1 = \frac{p_{W_s}^{\mathrm{ref}}-p_{S_s}}{d}.
\]

Using Rodrigues' rotation operator,
\begin{equation}
    \mathcal{R}(\hat{e}_1,\phi_m)
    =I+\sin\phi_m[\hat{e}_1]_\times
      +(1-\cos\phi_m)[\hat{e}_1]^2_\times,
\end{equation}
we form an orthonormal basis of the circle plane as
\begin{equation}
    \hat{e}_3=\mathcal{R}(\hat{e}_1,\phi_m)n_{\mathrm{ref}},\qquad
    \hat{e}_2=\frac{\hat{e}_1\times \hat{e}_3}
                   {\|\hat{e}_1\times\hat{e}_3\|}.
\end{equation}

From the cosine law,
\begin{equation}
    \cos\theta=\frac{L_1^2+d^2-L_2^2}{2L_1d},\qquad
    \sin\theta=\sqrt{1-\cos^2\theta},
\end{equation}
the elbow target is obtained as
\begin{equation}
    p_{E_s}^{\mathrm{ref}}
    =p_{S_s}+L_1(\cos\theta\,\hat{e}_1+\sin\theta\,\hat{e}_2).
    \label{eq:elbow_target}
\end{equation}

\begin{proposition}[Feasible SEW target]
For $L_1,L_2>0$, $0<\varepsilon<\min(L_1,L_2)$, and
$d_s\in[d_{\min},d_{\max}]$, the targets in
\eqref{eq:elbow_target} satisfy
$\|p_{E_s}^{\mathrm{ref}}-p_{S_s}\|=L_1$ and
$\|p_{E_s}^{\mathrm{ref}}-p_{W_s}^{\mathrm{ref}}\|=L_2$, provided the SEW
basis is valid.
\end{proposition}
\begin{IEEEproof}
The clipped interval strictly satisfies the two triangle inequalities.
The cosine-law expression therefore yields
$\cos\theta\in[-1,1]$. Orthonormality of
$(\hat e_1,\hat e_2)$ gives the first distance directly; substituting
$p_{W_s}^{\mathrm{ref}}=p_{S_s}+d_s\hat e_1$ and the cosine law gives the
second.
\end{IEEEproof}

This construction makes explicit how $p_{E_s}$ is selected on the SEW circle
by the chosen swivel branch. It provides an analytic Cartesian elbow target;
continuity of the resulting joint trajectory
still depends on reference-normal handling, numerical branch state, and the
platform-specific joint recovery described next.

% ==========================================================
\subsubsection{Temporal Continuity Requirements}

The current implementation obtains temporal branch preference by warm-starting
the bounded recovery from the preceding valid slave solution. It does not
maintain a separate swivel-angle filter, an unwrapped swivel state, or a
last-valid reference normal. Reach clipping, reference-axis fallback, a
degenerate master arm plane, and an active slave joint limit therefore mark the
sample as nonsmooth and invalidate $A$. Pose continuity and elbow-flip rate are
measured from trajectories rather than inferred from the geometry. Stateful
normal blending or angle unwrapping may be evaluated later, but is not part of
the method claimed here.

% ==========================================================
\subsubsection{Embodiment-Specific Joint Recovery}

The terminal orientation is transferred as
\begin{equation}
    R_{s,\mathrm{EE}}^{\mathrm{ref}}
    = R_{m,\mathrm{EE}}R_\Delta,
    \label{eq:orientation_transfer}
\end{equation}
where $R_\Delta$ is a fixed neutral-pose calibration. The present aligned-URDF
simulation uses $R_\Delta=I$ and the current code path hard-wires that identity;
it does not yet expose $R_\Delta$ as a mapper parameter. Ingestion of the
measured orientation offset and its regression test are G0a requirements before
hardware use.

Given the Cartesian targets and \eqref{eq:orientation_transfer}, the current
slave recovery solves the bounded residual problem
\begin{equation}
    q_s^{\mathrm{ref}}
    =\arg\min_{\underline q_s\le q_s\le\overline q_s}
      \left\|r_K(q_s)\right\|_2^2,
\label{eq:joint_recovery}
\end{equation}
\begin{equation}
r_K(q_s)=
\begin{bmatrix}
s_E\!\left(p_{E_s}(q_s)-p_{E_s}^{\mathrm{ref}}\right)\\
s_W\!\left(p_{W_s}(q_s)-p_{W_s}^{\mathrm{ref}}\right)\\
s_R\,\mathrm{Log}\!\left(R_{s,\mathrm{EE}}(q_s)^\top
R_{s,\mathrm{EE}}^{\mathrm{ref}}\right)\\
s_q\,\mathrm{wrap}(q_s-q_{\mathrm{seed}})
\end{bmatrix}.
\label{eq:joint_recovery_residual}
\end{equation}
The scale factors make position and angular residuals explicit. The current
software fixes $s_E=s_W=5$, $s_R=1$, and $s_q=10^{-5}$ in SI units and uses
the URDF lower and upper joint bounds; $\mathrm{wrap}$ acts componentwise on
$[-\pi,\pi)$. These values are implementation
parameters, not empirically optimal constants, and are included in the
sensitivity protocol.

A trust-region reflective least-squares solver is warm-started from the
previous solution during sequential operation. At initialization, a staged
shoulder--elbow--wrist seed and the joint-range midpoint are both evaluated,
and the lower task-residual solution is retained. The solver uses at most
400 function evaluations and
$\mathrm{ftol}=\mathrm{xtol}=\mathrm{gtol}=10^{-12}$. A recovery is accepted
only when the solver succeeds, all values are finite, the master geometry is
valid, no URDF limit is violated, both elbow and wrist errors are at most
$2\times10^{-4}$\,m, and the orientation error is at most
$2\times10^{-3}$\,rad. Accordingly, \emph{analytic} refers only to the
Cartesian SEW target, not the complete map
$q_m\mapsto q_s^{\mathrm{ref}}$.

% ==========================================================
\subsubsection{SEW Degenerate Cases and Fallback Construction}
\label{sec:sew_degenerate}

When $\hat{x}_m$ becomes nearly collinear with $e_z$, the projected reference
normal becomes ill-conditioned. Separately, a nearly zero reach or nearly
collinear shoulder--elbow--wrist points violates
\eqref{eq:sew_validity}. The latter two cases are invalid master geometry and
are never certified for differential feedback.

To keep the reference construction defined, we introduce a fallback
orthogonal direction.
Let $e_x = [1\ 0\ 0]^\top$ and $e_y = [0\ 1\ 0]^\top$. The deterministic
fallback selects the candidate with the larger projected norm,
\begin{equation}
    e_f=\arg\max_{e\in\{e_y,e_x\}}
    \left\|e-(e^\top\hat x_m)\hat x_m\right\|.
\end{equation}
It is then projected onto the plane orthogonal to $\hat{x}_m$:
\[
\tilde{n}_{\mathrm{ref}} = e_f - (e_f^\top \hat{x}_m)\hat{x}_m,\qquad
n_{\mathrm{ref}} = \frac{\tilde{n}_{\mathrm{ref}}}{\|\tilde{n}_{\mathrm{ref}}\|}.
\]
This construction keeps the reference normal defined when $\hat{x}_m$ is
nearly vertical, but activation is explicitly marked as nonsmooth. The current
implementation neither blends the threshold nor reuses a preceding normal.
Deliberate passages through this region are therefore retained in the failure
and continuity evaluation.

The SEW module therefore provides an analytic construction of feasible reach,
direction, and swivel targets. Full-map continuity, joint-limit feasibility,
and real-time performance remain properties to be established for the selected
slave-joint recovery implementation.

\subsubsection{Guarded Local Retargeting Differential}
\label{sec:retarget_differential}

The complete bounded map has no analytic Jacobian in the current
implementation. Let $q_s^0=\mathcal R(q_m;b_k)$ be a valid, smooth base
solution and $e_j$ the $j$th master-joint basis vector. Its local differential
is approximated by
\begin{equation}
    A_{:,j}\approx
    \frac{\mathrm{wrap}\!\left(
      \mathcal R(q_m+h e_j;q_s^0)
      -\mathcal R(q_m-h e_j;q_s^0)\right)}{2h},
    \label{eq:finite_difference_A}
\end{equation}
with $h=10^{-4}$\,rad. All positive and negative perturbations are initialized
from the same bounded base solution so that the calculation targets one local
branch.

A recovered pose is marked smooth only if it meets the stated task tolerances
and has no reach clipping, reference-axis fallback, degenerate master geometry,
joint-limit violation, or joint clearance below the configured
$10^{-4}$-rad limit margin. For column $j$, define
$d_j^+=\mathrm{wrap}(q_s^+-q_s^0)/h$ and
$d_j^-=\mathrm{wrap}(q_s^0-q_s^-)/h$. A column is certified only when both
perturbation solves succeed, all three poses are marked smooth, the reach-clip
region is unchanged, the largest wrapped joint jump from the base is no greater
than $0.25$\,rad, and
\begin{equation}
    \eta_j =
    \frac{\|d_j^+-d_j^-\|_2}
    {1+\max(\|d_j^+\|_2,\|d_j^-\|_2)}
    \le 0.05 .
    \label{eq:differential_consistency}
\end{equation}
The finite-difference step and all three guard thresholds are current software
defaults, not universal constants; G0c varies them in a locked convergence and
sensitivity study. The active-limit gate currently refers to recovered
\emph{slave} joints. The mapper does not yet reject a master perturbation
$q_m\pm he_j$ outside the master URDF limits; G0a must add that domain check and
mark the corresponding differential invalid rather than silently using an
out-of-domain central sample. A preregistered one-sided boundary rule may be
evaluated separately. The complete $A$ is valid only if all seven columns are
finite and certified. Invalid columns are represented as non-finite values for
diagnostics; they are never used for feedback. In the closed-loop software, a
failed pose update holds the preceding pose reference, whereas an invalid
differential sets the new mapped-feedback target and reference velocity to zero
so that the upstream rate limit returns the haptic output toward zero. One
differential evaluation requires one base recovery and fourteen perturbed
recoveries. The present convenience wrapper first performs a separate nominal
pose recovery and then calls that evaluator, for a total of two nominal and
fourteen perturbed solves per 50-Hz update; reusing the first nominal solution
is an implementation optimization still to be made. High-percentile runtime
and validity rate are therefore mandatory evaluation outcomes.


\subsection{Locked Slave Tracking Interface}
\label{sec:slave_tracking}

Slave tracking is not claimed as a contribution, but it is part of the
experimental plant and cannot be left implicit. The current numerical fixture
uses the fixed computed-torque law
\begin{equation}
 \begin{aligned}
 \tau_{s,\mathrm{ctrl}}
 &=\mathrm{sat}_{\alpha_s\tau_{s,\max}}\!\left[
 h_s+M_s a_s^{\mathrm{cmd}}\right],\\
 a_s^{\mathrm{cmd}}
 &=K_p\,\mathrm{diff}(q_s,q_s^{\mathrm{ref}})
 +K_d(\dot q_s^{\mathrm{ctrl}}-\dot q_s)+a_{\mathrm{lim}} .
 \end{aligned}
 \label{eq:slave_tracking_controller}
\end{equation}
where $a_{\mathrm{lim}}$ is a continuous acceleration-domain soft joint-limit
guard and $\alpha_s=0.70$ in the present simulation. The gain vectors, the
0.12-rad soft-limit buffer, the 500-Hz update, and all effort and numerical
acceleration limits are archived with the fixture configuration. Physical
hardware may require a different certified low-level controller; whichever
controller is selected will be locked and shared by every paired mapping and
haptic condition so that it is not an unreported experimental factor.

\subsection{Inverse-Dynamics Residual Wrench Estimation}
\label{sec:interaction}

The slave-side contact wrench is reconstructed from joint-torque measurements
without assuming that the residual is an unbiased measurement of contact.
The formulation follows model-based residual methods used for contact
detection and interaction estimation
~\cite{Mohammadi2013DO,DeLuca2006Collision,Wang2022InteractionTorque}.
Unlike a generalized-momentum observer, the present estimator explicitly uses
the instantaneous inverse-dynamics residual. Its performance therefore depends
on acceleration estimation, friction compensation, model quality, torque
sensing, and the spatial directions observable through the tool Jacobian.

% ==========================================================
\subsubsection{Joint-Space Inverse-Dynamics Residual}

Let $\tau_s^{\mathrm{meas}}$ denote a \emph{calibrated load-side-equivalent}
joint-torque signal, not an unqualified raw motor-current estimate. Under the
environment-on-robot convention used here, its required measurement balance is
\begin{equation}
    \tau_s^{\mathrm{meas}}
    =\tau_s^{\mathrm{dyn}}+\tau_{s,\mathrm{int}}+b_\tau+\eta_\tau,
    \label{eq:load_side_torque_balance}
\end{equation}
where $b_\tau$ and $\eta_\tau$ are bias and measurement noise. If the physical
drive reports actuator-on-link torque under the usual manipulator dynamics
$M\ddot q+h=\tau_{\mathrm{act}}+\tau_{\mathrm{ext}}$, its sign and
transmission mapping must first be converted to
\eqref{eq:load_side_torque_balance}; the conversion will be verified against
the independent wrist force/torque sensor.
The model-predicted torque from rigid-body dynamics is
\begin{equation}
    \tau_s^{\mathrm{dyn}}
    = M_s(q_s)\ddot q_s + h_s(q_s,\dot q_s)
      + \tau_{\mathrm{fric}}^{\mathrm{ff}},
\end{equation}
where $h_s$ contains Coriolis, centrifugal, and gravity terms and
$\tau_{\mathrm{fric}}^{\mathrm{ff}}$ is an optional identified friction
feedforward term. It is zero in the present rigid-body simulation and must not
be described as identified until physical calibration is complete.

We define the \emph{residual torque}
\begin{equation}
    r_s = \tau_s^{\mathrm{meas}} - \tau_s^{\mathrm{dyn}},
\end{equation}
which contains external joint torque together with modeling error, friction
error, sensor bias, and measurement noise. The estimate used by the current
formulation is
\begin{equation}
    \hat\tau_{s,\mathrm{int}} = r_s-\hat b_\tau,
    \label{eq:residual_torque_estimate}
\end{equation}
where $\hat b_\tau$ is identified from a declared, independently confirmed
no-contact calibration sequence. Static contact cannot be distinguished from
bias using low velocity alone; consequently, online bias updates are disabled
unless an independent contact gate is available. A true momentum-observer
extension may be introduced later, but it must then be implemented and
evaluated as a separate estimator rather than attributed to
\eqref{eq:residual_torque_estimate}.
For physical data, the causal acceleration estimator, its cutoff and order,
initialization, group delay, and timestamp alignment with torque measurements
will be fixed from calibration data and reported; the current rigid-body
simulation supplies acceleration directly and cannot validate that pipeline.

% ==========================================================
\subsubsection{Mapping to Chest-Frame Wrench}

Let $J_{E_s}^{\mathrm{LWA}}(q_s)\in\mathbb{R}^{6\times n_s}$ be the
local-world-aligned geometric Jacobian at the slave end-effector point. The
axes are rotated into $\{C\}$ without changing that reference point:
\begin{equation}
    {}^C\!J_{E_s}
    =\operatorname{blkdiag}(R_{CW},R_{CW})
      J_{E_s}^{\mathrm{LWA}}.
    \label{eq:chest_jacobian}
\end{equation}
No translational adjoint term appears because the twist and wrench remain
referenced at the same end-effector point. With the
$[\mathrm{linear};\mathrm{angular}]$ and $[\mathrm{force};\mathrm{moment}]$
orders, virtual work gives
\begin{equation}
    \tau_{s,\mathrm{int}}
    = \left({}^{C}\!J_{E_s}\right)^{\!\top} {}^{C}\!F_{\mathrm{int}},
\end{equation}
where ${}^{C}\!F_{\mathrm{int}}\in\mathbb{R}^6$ is the interaction wrench at
the slave end-effector. The implemented DLS estimate is the regularized
least-squares solution only after the translational and rotational coordinates
have been given a common physical scale. Let the characteristic length
$\ell_c>0$ be selected from calibration data and frozen before the locked
test set, and define
\begin{equation}
 \begin{aligned}
 S_{\ell}&=\operatorname{diag}(\ell_c^{-1}I_3,I_3),\\
 \widetilde J_s&=S_{\ell}\,{}^C\!J_{E_s},\\
 \widetilde F&=S_{\ell}^{-\top}{}^C\!F
 =[\ell_c f^\top\;m^\top]^\top .
 \end{aligned}
 \label{eq:spatial_scaling}
\end{equation}
Then $\widetilde J_s^\top\widetilde F
=({}^C\!J_{E_s})^\top{}^C\!F$, so the virtual-work pairing is unchanged
while all entries of $\widetilde F$ have torque units. The dimensionally
scaled DLS estimate is
\begin{equation}
    \widetilde F_{\mathrm{int}}^\star
    =\arg\min_{\widetilde F}
    \left\|\widetilde J_s^\top\widetilde F
    -\hat\tau_{s,\mathrm{int}}\right\|_2^2
    +\lambda_{\ell}^{\,2}\|\widetilde F\|_2^2,
    \label{eq:dls_objective}
\end{equation}
namely
\begin{equation}
    {}^C\!\hat F_{\mathrm{int}}
    =S_{\ell}^{\top}
    \left(\widetilde J_s\widetilde J_s^\top
    +\lambda_{\ell}^{\,2}I_6\right)^{-1}
    \widetilde J_s\hat\tau_{s,\mathrm{int}}.
    \label{eq:dls_wrench}
\end{equation}

For a singular value $\widetilde\sigma_i$ of $\widetilde J_s$, the
scaled residual-to-wrench gain is
$\widetilde\sigma_i/(\widetilde\sigma_i^2+\lambda_{\ell}^{\,2})$; damping
limits near-singular amplification but introduces projection bias. The
current simulation code predates \eqref{eq:spatial_scaling}: it applies an
unweighted six-dimensional solve to force and moment coordinates and uses the
numerical default $\lambda=10^{-3}$. Its reconstructed wrench and unscaled
condition number are therefore diagnostic only, because they depend on the
chosen length and unit convention. Before any physical accuracy claim, G0a
requires implementing \eqref{eq:spatial_scaling}--\eqref{eq:dls_wrench},
selecting $(\ell_c,\lambda_{\ell})$ only on the calibration set, and using the
same frozen scaling for singularity stratification and all DLS baselines. The
rotation in \eqref{eq:chest_jacobian} does not improve the scaled Jacobian
singular values.

% ==========================================================
\subsubsection{Signal Conditioning and Identifiability}

The current estimator stores the raw residual, bias-corrected residual, and
unfiltered DLS wrench. It does not apply a separate wrench filter. Nominal
low-pass filtering is instead applied once to the mapped master generalized
torque in Section~\ref{sec:haptics}. Acceleration-estimation filtering,
friction identification, and any future wrench filter must be calibrated and
logged separately. Force and moment errors are reported separately because
their units and sensor characteristics differ; a combined norm requires a
declared characteristic length.

% ==========================================================
\subsubsection{Estimator Scope}

The residual estimator is expected to degrade under inertial-parameter error,
unmodeled payload, drivetrain friction, acceleration noise, torque bias, and
poor Jacobian conditioning. Damping can trade variance for bias but cannot
recover wrench components that are not observable from joint torque. These
effects are therefore experimental factors, not assumed robustness
properties. An independent wrist force/torque sensor supplies ground truth,
and trials near singular configurations remain in the analysis.

The proposed differential-dual pathway applies $A^\top$ directly to the
bias-corrected generalized residual in
\eqref{eq:residual_torque_estimate}. The DLS wrench in
\eqref{eq:dls_wrench} is used for physical interpretation, comparison with
independent six-axis ground truth, and the direct
$({}^C\!J_{E_m})^\top{}^C\!\hat F_{\mathrm{int}}$ baseline. In general,
$({}^C\!J_{E_s})^\top{}^C\!\hat F_{\mathrm{int}}
\ne\hat\tau_{s,\mathrm{int}}$ because DLS introduces a regularized
projection; the method therefore does not silently map through this round
trip.



\subsection{Power-Dual Haptic Rendering and Feedback-Energy Supervision}
\label{sec:haptics}

Time-domain passivity and energy-supervision methods have been studied for
multi-DoF exoskeleton teleoperation
~\cite{Buongiorno2019Teleop,Porcini2020Passivity}. The present formulation
distinguishes whether the generalized-force map is dual to the heterogeneous
motion map from whether the shaped haptic contribution respects a discrete
energy budget. The latter is not a passivity claim for the total actuator port
or an arbitrary delayed bilateral network.

% ==========================================================
\subsubsection{Differential-Dual Generalized-Force Mapping}

For a valid held differential, let $\kappa_r(k)\le k$ denote the source index
of the most recent return-channel sample accepted at master sample $k$, and
let $\chi_r[k]\in\{0,1\}$ indicate that the held return sample has not exceeded
its timeout. The proposed current-map pathway applies the declared force scale
to that delayed, bias-corrected joint residual:
\begin{equation}
    \tau_{h,\mathrm{raw}}[k]
    =\chi_r[k]\sigma_f \bar A[k]^\top
    \hat\tau_{s,\mathrm{int}}[\kappa_r(k)].
    \label{eq:raw_master_torque}
\end{equation}
Equation~\eqref{eq:raw_master_torque} states the timestamp policy of the
current implementation: the most recently arrived slave residual is
re-expressed through the \emph{current} held map. It therefore satisfies a
receiver-sample algebraic duality with
$\dot q_s^{\mathrm{map}}[k]=\bar A[k]\dot q_m[k]$, but it is not a claim of
source-time power equality at configuration $\kappa_r(k)$. For the present
fixed-delay queue, $\kappa_r(k)=k-d$ with integer $d\ge0$. The protocol
separately reports the resulting transport/configuration error and compares
this policy with source-stamped
$\bar A[\mu(\kappa_r(k))]^\top
\hat\tau_{s,\mathrm{int}}[\kappa_r(k)]$.
Here $\mu(\kappa_r(k))$ is the active forward-map identifier echoed by that
return packet, as defined in \eqref{eq:forward_held_reference}; using
$\bar A[\kappa_r(k)]$ without this binding would be incorrect under asymmetric
forward delay.
The current implementation sets $\sigma_f=1$ and uses $\gamma$ below as the
single feedback-strength parameter; the two gains are not tuned
simultaneously. The present fixture uses $\gamma=0.50$ and $a_\tau=0.222$.
The constructor currently checks finiteness but does not reject every
out-of-contract lower-bound value; strict validation of
$\gamma\ge0$ and $0<a_\tau\le1$ is a G0a software requirement.
The positive sign follows the environment-on-device reaction convention
defined in Section~\ref{sec:problem}. The direct Cartesian baseline is
$\tau_{h,\mathrm{direct}}=({}^{C}\!J_{E_m})^\top
{}^{C}\!\hat F_{\mathrm{int}}$. Because DLS projects the residual, the
experiment reports $\|({}^{C}\!J_{E_s})^\top{}^{C}\!\hat
F_{\mathrm{int}}-\hat\tau_{s,\mathrm{int}}\|$ and includes a matched
$A^\top({}^{C}\!J_{E_s})^\top{}^{C}\!\hat F_{\mathrm{int}}$ ablation. This
separates the mapping comparison from DLS projection error.

% ==========================================================
\subsubsection{Strictly Ordered Candidate-Torque Shaping}

The raw mapping is filtered once in generalized master coordinates:
\begin{equation}
    \bar\tau_h[k]
    =(1-a_\tau)\bar\tau_h[k-1]
      +a_\tau\tau_{h,\mathrm{raw}}[k],
    \qquad 0<a_\tau\le1.
    \label{eq:haptic_filter}
\end{equation}
After a configured nonnegative nominal gain $\gamma$, the rate limit is
referenced to the previously applied haptic output and is followed by
per-joint saturation:
\begin{align}
    \Delta\tau_h[k]
    &=\mathrm{clip}_{\dot\tau_{\max}\Delta t_k}\!\left(
      \gamma\bar\tau_h[k]-\tau_{h,\mathrm{app}}[k-1]\right),\\
    \tau_{h,\mathrm{rate}}[k]
    &=\tau_{h,\mathrm{app}}[k-1]
      +\Delta\tau_h[k],\\
    \tau_{h,\mathrm{cand}}[k]
    &=\mathrm{sat}_{\tau_{\max}}\!\left(
      \tau_{h,\mathrm{rate}}[k]\right).
    \label{eq:candidate_torque}
\end{align}
Both operators act componentwise and symmetrically.
Thus, the fixed order is filter, gain, rate limit, torque saturation, and only
then energy projection. The candidate power uses the time-aligned master
velocity:
\begin{equation}
    P_{\mathrm{cand}}[k]
    = \tau_{h,\mathrm{cand}}[k]^\top\dot q_m[k].
    \label{eq:candidate_power}
\end{equation}
An invalid $A$ sets the new raw target to zero; the nominal filter and rate
limit then return the applied haptic output toward zero while storage accounting
continues on the actual output. In contrast, a non-finite torque or velocity,
or a nonpositive sample period, activates an immediate zero-output numerical
fail-safe and leaves filter state and storage unchanged. No operation modifies
the projected haptic contribution downstream, apart from an emergency
zero-output action.

% ==========================================================
\subsubsection{Discrete Haptic-Energy Projection}

Let $E[k]\in[E_{\min},E_{\max}]$ be the software energy budget. The final
radial factor, with $0<\bar\rho\le1$ and
$E_{\mathrm{av}}[k]=E[k]-E_{\min}$, is
\begin{equation}
    \rho[k]=
    \begin{cases}
        \bar\rho, & P_{\mathrm{cand}}[k]\le 0,\\[4pt]
        \displaystyle
        \min\!\left(\bar\rho,\;
        \max\!\left(0,\,
        \frac{E_{\mathrm{av}}[k]}
        {P_{\mathrm{cand}}[k]\Delta t_k}\right)\right),
        & P_{\mathrm{cand}}[k]>0 .
    \end{cases}
    \label{eq:energy_projection}
\end{equation}
The final haptic contribution and storage update are
\begin{equation}
    \tau_{h,\mathrm{app}}[k]
    =\rho[k]\tau_{h,\mathrm{cand}}[k],
    \label{eq:applied_torque}
\end{equation}
\begin{equation}
    E[k+1]=\mathrm{clip}\!\left(
      E[k]-\tau_{h,\mathrm{app}}[k]^\top
      \dot q_m[k]\Delta t_k,\,
      E_{\min},E_{\max}\right).
    \label{eq:bounded_energy_update}
\end{equation}
Charging above $E_{\max}$ is discarded as virtual dissipation. The numerical
lower clip is a finite-precision guard; the projection itself supplies the
lower-bound invariant. The initial value $E[0]\in[E_{\min},E_{\max}]$ and
$E_{\min}$, $E_{\max}$, $\bar\rho$, $a_\tau$, and $\gamma$ will be selected
from fixture-safe calibration runs and locked before confirmatory comparisons;
the present simulation values are software-test parameters rather than
validated human-use settings.

\begin{proposition}[One-step haptic-storage invariant]
Assume $E[k]\in[E_{\min},E_{\max}]$, $\Delta t_k>0$, finite
$\tau_{h,\mathrm{cand}}[k]$ and $\dot q_m[k]$, and no downstream modification
of $\tau_{h,\mathrm{app}}[k]$. Equations
\eqref{eq:energy_projection}--\eqref{eq:bounded_energy_update} imply first that
the \emph{unclipped} update in \eqref{eq:applied_port_energy} satisfies
$E^+[k+1]\ge E_{\min}$, and consequently
$E[k+1]\in[E_{\min},E_{\max}]$.
\end{proposition}
\begin{IEEEproof}
If $P_{\mathrm{cand}}\le0$, the applied haptic power is nonpositive and the
unclipped storage cannot decrease. If $P_{\mathrm{cand}}>0$,
\eqref{eq:energy_projection} gives
$\rho P_{\mathrm{cand}}\Delta t_k\le E[k]-E_{\min}$, so the unclipped update is at
least $E_{\min}$. The final clip enforces the upper capacity and preserves the
lower bound.
\end{IEEEproof}
Applying the proposition inductively gives the same storage bound over any
finite sequence of valid samples; this still does not establish closed-loop
stability. More specifically, define discarded charge
\begin{equation}
 D[k]=\max\!\left(0,E[k]-P_h[k]\Delta t_k-E_{\max}\right).
\end{equation}
Then, over $N$ valid samples,
\begin{equation}
 \sum_{k=0}^{N-1}P_h[k]\Delta t_k
 =E[0]-E[N]-\sum_{k=0}^{N-1}D[k]
 \le E[0]-E_{\min}.
 \label{eq:net_haptic_energy_bound}
\end{equation}
This bound concerns signed net released energy, not the cumulative positive
work after repeated recharge events.

% ==========================================================
\subsubsection{Scope of the Energy Claim}

Equations~\eqref{eq:energy_projection}--\eqref{eq:applied_torque} constrain
energy released by the sampled haptic feedback contribution under the stated
timing and sign conventions. They do not constrain independently added
dynamics compensation and do not establish stability of the human, robot,
environment, or communication channel as arbitrary dynamical systems. A
stronger delayed bilateral claim requires an explicit two-port treatment and
an analysis including sampling, filters, saturation, and delay.

% ==========================================================
\subsubsection{Executable Update and Logging Contract}

\begin{figure}[t]
\begin{algorithmic}[1]
\Require synchronized master/slave states, $\tau_s^{\mathrm{meas}}$, $E[k]$
\If{retargeting update is due}
    \State construct SEW target and solve bounded recovery
    \State compute guarded $A$ using \eqref{eq:finite_difference_A}
\EndIf
\State compute $\hat\tau_{s,\mathrm{int}}$ and diagnostic
${}^C\!\hat F_{\mathrm{int}}$
\State accept the declared return-channel sample and retain its source index
$\kappa_r(k)$ and echoed forward-map identifier $\mu(\kappa_r(k))$
\State set $\tau_{h,\mathrm{raw}}\gets
\chi_r[k]\sigma_f\bar A[k]^\top
\hat\tau_{s,\mathrm{int}}[\kappa_r(k)]$ if the held
differential and return channel are currently valid; otherwise $0$
\State apply \eqref{eq:haptic_filter}--\eqref{eq:candidate_torque}
\State project and account exactly once using
\eqref{eq:energy_projection}--\eqref{eq:bounded_energy_update}
\State output $\tau_{h,\mathrm{app}}$ unchanged
\end{algorithmic}
\caption{One executable update of the proposed motion, estimation, and haptic
path. Retargeting may run at a slower declared rate than the dynamics and
haptic loop; its output and validity flag are then held between updates.}
\label{alg:framework_update}
\end{figure}

The present simulation stores time series for:
\begin{itemize}
    \item the bias-corrected slave residual and DLS wrench;
    \item raw, candidate, and applied haptic torque, the post-update storage
    $E[k+1]$, and $\rho[k]$.
\end{itemize}
Retargeting success, differential validity, and fallback are currently stored
only as aggregate run-level rates or counts, as are rate-limit and
haptic-torque-saturation events. They have no per-sample reason codes. The
numerical supervisor likewise exposes a fail-safe flag internally, but the
current run-level simulator does not persist it. G0a requires per-sample flags,
reason codes, and the reference-vector logs identified above before such
evidence is reported. Physical tests additionally require
source timestamps, packet age, measured period and jitter, packet loss,
emergency-stop state, actuator derating/limit state, and the actual accepted
low-level torque. A quantity not recorded cannot support a timing or
energy-accounting claim.

% ==========================================================
The haptic module therefore defines a power-dual raw reaction and a
final-stage storage projection for the haptic contribution. Whether the
implementation improves transparency and remains usable over a finite contact
and network envelope is determined by the protocol below.


\section{Experimental Design and Prospective Validation Protocol}
\label{sec:experiments}

\noindent\textit{Protocol status:}
The experiments described in this section constitute a prospective validation
protocol. Physical data collection under this protocol has not yet begun, and
no experimental outcome is reported or implied. Numerical safety limits,
factor levels, and sample sizes that depend on fabricated hardware will be
fixed after calibration but before the corresponding locked outcomes are
inspected. Calibration and pilot data used for tuning will remain separate
from the confirmatory test set. The present repository supports deterministic
software checks and a closed-loop numerical smoke test (G0a--G0b below).
The multi-run simulation study (G0c) and all physical stages (G1--G4) remain
prospective; this distinction is maintained in every reported table and
figure.

\subsection{Evaluation Questions and Staged Design}

Table~\ref{tab:rq_traceability} prevents the hardware prerequisite,
deterministic contracts, empirical hypotheses, and end-to-end external
validation from being conflated.

\begin{table*}[t]
\centering
\footnotesize
\renewcommand{\arraystretch}{1.16}
\begin{tabularx}{\textwidth}{p{1.8cm} p{3.1cm} X X p{1.0cm}}
\toprule
\textbf{Role} & \textbf{Claim or hypothesis} & \textbf{Locked comparison}
& \textbf{Primary evidence} & \textbf{Gate}\\
\midrule
Prerequisite
& Master and slave hardware satisfy their declared operating envelopes
& Measured performance versus preregistered requirements
& Calibration, backdrive, bandwidth, latency, headroom, and safety limits
& G1\\
RQ1 / H1
& Branch-qualified SEW retargeting reduces the locked composite event rate
& SEW pathway versus matched bounded and task-priority IK
& Paired trajectory-level failure-or-discontinuity rate
& G0c/G2\\
RQ2 / H2
& Regularization improves noisy/near-singular wrench reconstruction
& DLS versus undamped solve, with bias/friction/model ablations
& Force RMSE and moment RMSE against an independent six-axis sensor
& G0c/G2\\
RQ3 / H3
& Differential-dual mapping reduces cross-embodiment power mismatch
& $A^\top J_s^\top F$ versus $J_m^\top F$ with the same $F$
& Paired $\epsilon_{P,c}^{\mathrm{act}}$ and transport-error decomposition
& G0c/G3\\
RQ3 / H4
& Actual-applied haptic supervision enforces its budget at a transparency cost
& Projection on/off plus a locked time-domain PO/PC baseline
& Paired floor deficit, pre-clip audit, and torque-level projection distortion
& G0a/G3\\
External validity
& Integrated usefulness after all component gates pass
& Proposed, PO/PC, and no-kinesthetic conditions
& Task success; human surface-force RMSE; safety events
& G4\\
\bottomrule
\end{tabularx}
\caption{Traceability from research questions to claims, comparisons, primary
evidence, and validation gates. Hardware characterization is a prerequisite,
not a separate algorithmic research claim.}
\label{tab:rq_traceability}
\end{table*}

All algorithm parameters will be tuned on a separate calibration set.
Trajectories, random seeds, failure definitions, exclusion rules, primary
outcomes, and statistical models will be frozen before evaluating the locked
test set. Compared methods will receive identical inputs, initial conditions,
joint and torque limits, network traces, and computing resources.

\begin{table*}[t]
\centering
\footnotesize
\renewcommand{\arraystretch}{1.18}
\begin{tabularx}{\textwidth}{p{1.0cm} p{3.0cm} X X}
\toprule
\textbf{Gate} & \textbf{Stage and purpose} & \textbf{Required evidence}
& \textbf{Exit criterion before advancing}\\
\midrule
G0a & Deterministic software contracts
& URDF/frame binding, bounded recovery, finite-difference checks,
DLS algebra, invalid-sample fail-safe, output ordering, and independent
energy reconstruction
& Versioned tests pass; no unexplained non-finite value; tolerances and
failure semantics frozen\\
G0b & Current closed-loop numerical smoke test
& Declared master/slave models, one scripted free-space/contact trajectory,
three implemented feedback paths, and invariant/event logging
& All runs terminate normally; algebraic and storage invariants satisfy
frozen numerical tolerances; no performance claim is made\\
G0c & Locked multi-run simulation study
& Paired trajectories and seeds, stratified kinematic cases, contact and
delay factors, robustness perturbations, matched-input baselines, and
per-sample protocol-state logs
& Runner, dataset, primary numerical endpoints, and analysis script frozen
before the locked batch\\
G1 & Hardware bench characterization
& Master sensing, torque/backdrive, bandwidth, limits, temperature,
deflection and emergency stop; slave tracking, TCP/F/T, residual, and
accepted-command calibration
& Documented safe operating envelope and automatic termination limits\\
G2 & Component validation
& SEW/solver comparison on a locked workspace and wrench estimation against
an independent six-axis sensor
& Primary component endpoints and failure definitions pass their
preregistered feasibility criteria\\
G3 & Non-human bilateral fixture
& Power mismatch, haptic storage, delay/contact factors, transparency,
oscillation, and all single-factor ablations
& No unresolved safety event; human-safe conditions selected without viewing
human outcomes\\
G4 & Integrated and human evaluation
& Locked manipulation tasks followed by ethics-approved, counterbalanced
within-participant testing
& Complete reporting of successes, failures, withdrawals, and protocol
deviations\\
\bottomrule
\end{tabularx}
\caption{Gated evidence plan. A later stage cannot be used to compensate for
an unresolved component or safety failure at an earlier gate.}
\label{tab:validation_gates}
\end{table*}

\subsection{Current-Code Simulation and Numerical Verification}
\label{sec:current_code_validation}

\subsubsection{G0a: deterministic software contracts}

The present regression suite exercises the declared URDF, joint, and frame
contract; bounded SEW recovery and invalid-domain handling; an independent
directional finite-difference check of $A$; the fixed-branch virtual-work
identity; DLS recovery of a synthetic known wrench; frame-mismatch rejection;
the filter--rate-limit--saturation--projection order; exact storage-floor
projection; the unilateral virtual-wall sign convention; and finite execution
of a short closed-loop run. These tests are treated as executable
specifications, not as samples in a statistical performance comparison.

Before the locked G0c batch, the suite and logger will additionally cover
calibrated $R_\Delta$ and TCP/F/T transforms, convergence over multiple
finite-difference steps, explicit master-domain checks, source-stamped held
differentials, independent packet age and timeout state, and per-sample
validity and failure-reason codes. The logger will separately persist
$\dot q_s^{\mathrm{map}}$, $\dot q_s^{\mathrm{ctrl}}$, actual $\dot q_s$,
$E[k]$, the unclipped $E^+[k+1]$, clipped $E[k+1]$, and the numerical
fail-safe state. Random seeds, solver tolerances, loop rates, delay queues, and
test configurations will be archived.

\subsubsection{G0b: current closed-loop smoke test}

The executable smoke test loads \texttt{master\_7dof.urdf} and
\texttt{real\_slave\_7dof.urdf}. It integrates both rigid-body models at
\SI{500}{\hertz}, updates and holds the retargeting differential at
\SI{50}{\hertz}, and applies the currently implemented fixed
\SI{80}{\milli\second} return delay. A computed-torque human proxy drives the
master and a computed-torque controller drives the slave through a scripted
approach--contact--return trajectory against a unilateral spring--damper wall.
The only three presently executable paired conditions are:
\begin{enumerate}
    \item $A(q_m)^\top\hat\tau_{s,\mathrm{int}}$ with final applied-port
    energy supervision;
    \item $J_m^\top{}^C\!\hat F_{\mathrm{int}}$ with the same filter, rate
    limit, saturation, and energy supervision;
    \item $A(q_m)^\top\hat\tau_{s,\mathrm{int}}$ with only the energy
    projection bypassed while all other guards remain active.
\end{enumerate}
The run stores joint/reference trajectories, synthetic residual and DLS-wrench
signals, raw/candidate/applied master torque, contact force and penetration,
power-identity errors, applied power, $\rho$, supervised and shadow storage,
tracking errors, mapping runtime, and aggregate mapping/limit events.
Consequently it can expose integration, sign, ordering, non-finite-state, and
storage-accounting defects.

It cannot establish physical wrench accuracy, transparency, human
performance, or stability over an operating envelope. The simulated
measurement is generated from the same nominal dynamics used by the plant,
with prescribed bias and noise; the virtual TCP is not yet a calibrated
physical TCP; and the present output contains one trajectory, one seed, one
contact impedance, and one fixed return delay. Moreover, condition 1 uses a
generalized residual whereas condition 2 uses a DLS wrench. Their contrast is
therefore an end-to-end feedback-chain smoke test, not the matched-input
mapping comparison required by H3.

\begin{table*}[t]
\centering
\footnotesize
\renewcommand{\arraystretch}{1.15}
\begin{tabularx}{\textwidth}{p{2.1cm} p{4.0cm} X X}
\toprule
\textbf{Subsystem} & \textbf{Available in the current repository}
& \textbf{Evidence it may support now}
& \textbf{Required before the next claim}\\
\midrule
Model contract
& Both seven-DoF URDFs, declared joint/frame names, limits, and
master/common/slave frame checks
& Reproducible numerical model binding
& Measured $R_\Delta$, master alignment, and slave TCP/F/T calibration\\
Retargeting
& Bounded SEW target/recovery, fallback logic, finite-difference $A$, and
aggregate success/validity counts
& Local algebra and deterministic regression checks
& Locked IK baselines, stratified trajectory generator, source stamps, and
per-sample failure reasons\\
Interaction estimate
& Bias-corrected generalized residual and fixed-damping DLS wrench with
synthetic bias/noise
& Estimator algebra under the shared nominal model
& Dimensionally scaled Jacobian, frozen damping, model/friction
perturbations, and independent six-axis ground truth\\
Haptic path
& $A^\top\hat\tau_s$, direct $J_m^\top\hat F$, output shaping, applied-port
energy projection, and shadow storage
& Virtual-work, execution-order, and storage-invariant checks
& Matched $A^\top J_s^\top\hat F$ baseline, locked PO/PC, source-stamped $A$,
and actuator-accepted torque audit\\
Closed loop and network
& One numerical contact trajectory, one seed, fixed return-delay queues, and
CSV/NPZ/JSON output
& G0b smoke testing only
& Multi-trajectory runner, asymmetric time-varying channels, jitter/loss and
timeouts, complete per-sample state, and independent physical logs\\
\bottomrule
\end{tabularx}
\caption{Code-to-evidence boundary. ``Available'' describes executable
capability, not a measured result. Items in the last column are completion
criteria rather than assumed properties.}
\label{tab:code_evidence_boundary}
\end{table*}

\subsubsection{G0c: locked multi-run simulation design}

G0c converts the single-run smoke test into a reproducible paired experiment
without elevating numerical output to hardware evidence. Four master-motion
families will be generated: free-space workspace sweeps, repeated
approach--contact--retreat motion, multi-axis contact probes, and trajectories
that deliberately approach joint limits, low manipulability, or SEW
reference-normal degeneracy. Every feedback method receives the same initial
state, trajectory identifier, random seed, and environment/network trace.

The core software-only development grid uses fixed return delays of
\SIlist{0;40;80;120}{\milli\second} and three virtual impedance pairs
$(K,B)=\{(200,15),(500,30),(800,45)\}$ in
\si{\newton\per\metre} and \si{\newton\second\per\metre}, respectively.
The three implemented conditions above are run first. After their interfaces
are frozen, the matched-input
$A^\top J_s^\top{}^C\!\hat F_{\mathrm{int}}$ condition and the locked PO/PC
baseline are added. A preliminary minimum of 20 paired
trajectory--seed realizations per core cell is used to estimate variability;
the final locked count is set by confidence-interval precision on the
trajectory-level primary endpoint, not by treating time samples as independent
replicates. The extended generator must make these distinct master
trajectories as well as distinct perturbation realizations; changing only the
current synthetic sensor-noise seed does not create an independent task
trajectory. These numerical factor levels are a development grid and do not
define safe hardware settings.

A separate one-factor-at-a-time robustness block varies measurement-noise and
bias magnitudes, finite-difference step, dynamics/inertial and friction-model
perturbations, and dynamics-to-mapping rate ratio. Jitter, packet loss,
asymmetric delay, out-of-order rejection, and timeout/recovery traces enter
only after the corresponding two-channel state machine and per-sample logger
pass G0a. This separation prevents an incomplete network mock-up from being
presented as delay robustness.

The G0c experimental unit is one complete trajectory--seed realization.
Deterministic primary gates are maximum virtual-work-identity error, maximum
independently reconstructed pre-clip storage deficit, and absence of
unexplained non-finite states. Paired trajectory-level numerical outcomes are
the composite SEW event $C_r$, mapping-valid fraction, runtime tail
percentiles, normalized power mismatch
\eqref{eq:normalized_actual_power_mismatch}, projection distortion
\eqref{eq:projection_distortion}, contact-force and motion-tracking errors,
limit/saturation events, and shadow-budget deficit
\eqref{eq:shadow_floor_deficit}. The present single-run artifacts are retained
as development records but are excluded from the locked batch and from every
confirmatory confidence interval.

\subsection{Instrumentation, Synchronization, and Safety}

The final manuscript will report the manufacturer, model, measurement range,
resolution, sampling rate, calibration method, and calibration date of each
external instrument. The minimum independent instrumentation is:
\begin{itemize}
    \item a calibrated six-axis force/torque sensor at the slave wrist;
    \item an external pose-reference system or metrology procedure for
    end-effector and exoskeleton alignment measurements;
    \item calibrated load cells or torque transducers for actuator and
    backdrivability characterization;
    \item a common clock or measured clock-offset process for master, slave,
    network, and reference sensors.
\end{itemize}

Raw signals will be recorded before filtering. Logged variables will include
joint position, velocity, measured and commanded torque, reference wrench,
estimated wrench, SEW state, mapping differential, raw/candidate/applied haptic
torque, independently added compensation torque, total low-level command,
the command accepted by the drive, active current/thermal derating limits,
storage level, projection factor, packet delay, controller period, and safety
events. Before G3, componentwise command allocation must demonstrate that the
declared haptic increment reaches the actuator unchanged; otherwise the
post-allocation increment, rather than the pre-allocation software value, will
be used in \eqref{eq:applied_port_energy}. The nominal loop rate will not be
inferred from a configuration value; the measured period distribution, missed
deadlines, and 99th-percentile jitter will be reported.

Joint position, velocity, torque, temperature, and workspace limits will be
derived from measured hardware ratings and a documented risk assessment.
High-stiffness, high-delay, and controller-ablation tests will first use a
rigidly mounted robotic input fixture or recorded-motion replay without a
person wearing the device. Automatic termination thresholds will be locked
before testing. Terminated trials will be retained and reported as safety
events rather than silently discarded.

\subsection{Hardware Characterization}
\label{sec:hardware_characterization}

Hardware characterization will precede teleoperation experiments. Each joint
of the wearable master will be evaluated in both motion directions at no fewer
than three representative arm configurations and with repeated trials per
condition. The slave bench will independently characterize its tracking,
residual-estimation, TCP/F/T, and command-acceptance chain rather than
inheriting the nominal URDF properties. The following properties will be
measured:
\begin{enumerate}
    \item range of motion, encoder resolution, absolute accuracy,
    repeatability, zero-offset drift, and human-joint alignment error;
    \item commanded-to-measured torque gain, linearity, hysteresis,
    continuous and peak torque, saturation, and thermal behavior;
    \item static breakaway torque, velocity-dependent friction, passive
    backdriving torque, and observable reflected inertia;
    \item closed-loop torque bandwidth and phase lag using a safely bounded
    chirp or multisine excitation;
    \item structural stiffness and end-point deflection under calibrated
    loads;
    \item total mass, moving mass by segment, fitting range, donning time, and
    emergency-stop latency;
    \item sensor-to-actuator latency, control-period jitter, and packet-loss
    behavior.
\end{enumerate}

Motor current alone will not be used as torque ground truth. Backdrivability
will be reported as a function of joint angle and velocity. Aggregate torque
norms will be accompanied by per-joint values and percentages of measured
continuous and peak ratings.

\subsection{SEW Retargeting Evaluation}
\label{sec:sew_protocol}

The complete retargeting pipeline will be compared with:
\begin{enumerate}
    \item anthropometrically scaled joint-space mapping;
    \item damped least-squares IK with warm starting and joint-limit
    regularization;
    \item task-priority IK with an explicit elbow-swivel or manipulability
    objective.
\end{enumerate}
These three baseline runners are not part of the present single-trajectory
simulator. They must share the same input and output contract, expose the same
validity/failure fields, and pass G0a before their parameters are frozen for
G0c and G2.
Baseline parameters will be selected on calibration trajectories under the same
joint limits, task weights, and compute-time budget as the proposed method.
The analytic SEW target construction and the platform-specific joint recovery
will also be timed and evaluated separately.

The locked configuration set will stratify samples across the nominal
workspace, reach boundaries, joint-limit neighborhoods, low-manipulability
postures, and deliberate reference-normal degeneracies. The trajectory set
will include workspace sweeps, repeated reach--retract motion, overhead motion,
and passages through fallback regions. Dataset sizes
$N_q$ and $N_{\mathrm{traj}}$ will be set by a precision analysis so that the
bootstrap 95\% confidence-interval half-width for the paired
composite-event-rate difference is below a preregistered tolerance
$\Delta_{\mathrm{SEW}}$. The tolerance and resulting sizes will be frozen
before executing the locked set.

Because master and slave link lengths differ, position error will be measured
relative to the feasible slave target rather than directly between physical
wrists:
\begin{equation}
    e_p=\|p_{W_s}-p_{W_s}^{\mathrm{ref}}\|.
\end{equation}
Additional outcomes are
\begin{align}
    e_{\mathrm{dir}}
    &=\cos^{-1}\!\left(
      \operatorname{clip}_{[-1,1]}(\hat{x}_s^\top\hat{x}_m)\right),\\
    e_\phi
    &=\left|\operatorname{wrap}_{[-\pi,\pi)}
      (\phi_s-\phi_m)\right|,\\
    e_R
    &=\cos^{-1}\!\left(
      \operatorname{clip}_{[-1,1]}\!\left[
      \frac{\operatorname{tr}((R_{s,\mathrm{EE}}^{\mathrm{ref}})^\top
      R_{s,\mathrm{EE}})-1}{2}\right]
      \right).
\end{align}
For locked trajectory $r$, let $v_k=1$ denote a valid bounded recovery,
$\Delta q_s^{\,w}[k]=\operatorname{wrap}(q_s[k]-q_s[k-1])$, and
$\Delta\phi_s^{\,w}[k]=
\operatorname{wrap}(\phi_s[k]-\phi_s[k-1])$. These increments are evaluated
between adjacent accepted 50-Hz retargeting updates. Define
\begin{align}
 F_r&=\mathbf{1}\!\left\{\exists k:
 v_k=0\ \vee\ e_p[k]>\bar e_p\right.\nonumber\\[-1mm]
 &\hspace{37mm}\left.\vee\ e_R[k]>\bar e_R\right\},\\
 D_r&=\mathbf{1}\!\left\{\exists k:
 |\Delta\phi_s^{\,w}[k]|>\bar\delta_\phi\right.\nonumber\\[-1mm]
 &\hspace{31mm}\left.\vee\
 \|\Delta q_s^{\,w}[k]\|_\infty>\bar\delta_q\right\},\\
 C_r&=\max(F_r,D_r).
 \label{eq:sew_composite_event}
\end{align}
The discontinuity thresholds apply only between adjacent, valid master
samples whose increment is below a frozen input-step threshold; commanded
resets and preregistered degeneracy transitions are labeled separately rather
than silently excluded. All tolerances in
\eqref{eq:sew_composite_event} are selected on calibration trajectories and
frozen before the locked set. Thus $D_r$, rather than an unobservable latent
branch label, is the operational definition of a branch-discontinuity event.
The single primary H1 endpoint is the paired
trajectory-level rate of $C_r=1$. Solver/tolerance failure $F_r$,
discontinuity $D_r$, elbow-position error, joint-limit violations,
reach-clipping rate, differential validity, trajectory jerk, and runtime are
secondary or diagnostic endpoints. Runtime on the deployed processor will be
reported using median, 95th-percentile, 99th-percentile, and maximum values. A
real-time claim will be made only if the locked timing criterion is satisfied.
Absolute, normalized, and calibrated reach-transfer policies are compared as
a separate ablation.

\subsection{Interaction-Wrench Estimation}
\label{sec:wrench_protocol}

The current shared-model simulation can verify residual/DLS algebra and
sensitivity to prescribed numerical perturbations, but it cannot supply
independent wrench ground truth or support H2. Physical H2 evidence begins
only after the following sensor and model-calibration protocol passes G1.

The wrist force/torque sensor will provide independent ground truth after tool
gravity, payload, coordinate-transform, and timestamp calibration. The test
matrix will contain:
\begin{enumerate}
    \item unloaded free-space motion for bias and false-contact assessment;
    \item quasi-static forces and moments in positive and negative directions
    along all six wrench axes;
    \item bounded multiaxial dynamic loading and contact transients;
    \item configurations stratified by the minimum singular value and
    condition number of the frozen scaled Jacobian $\widetilde J_s$;
    \item payload, inertial-parameter, friction, and torque-noise perturbations
    within the characterized uncertainty envelope.
\end{enumerate}
Each load, pose, direction, and motion condition will be repeated in randomized
order. The repeat count will be chosen so that the 95\% interval half-width for
each co-primary RMSE is below preregistered force and moment tolerances.

Comparisons will include an undamped residual pseudoinverse, fixed-damping
residual inversion, removal of identified friction compensation, and removal
of bias correction. Damping and all observer parameters will be selected only
from calibration data. If a wrench filter is added in an exploratory analysis,
it will be reported as a distinct, non-confirmatory method rather than folded
into the current estimator.

Force and moment will be analyzed separately. Outcomes will include RMSE, MAE,
signed bias, 95th-percentile absolute error, direction error, correlation,
frequency-response gain and phase, contact-onset latency, free-space
false-positive rate, and peak error. Error will also be plotted against
Jacobian singular value, speed, acceleration, payload perturbation, and torque
noise. Near-singular trials will not be removed or hidden inside a
workspace-wide average.
Force RMSE and moment RMSE over the locked matrix are the co-primary estimator
endpoints; the remaining quantities diagnose when and why errors occur.

\subsection{Mapping Consistency, Energy, Delay, and Transparency}
\label{sec:energy_protocol}

The non-human fixture study will compare:
\begin{enumerate}
    \item the differential-dual force map with haptic-energy projection;
    \item the same force map without the energy projection;
    \item a direct master-Jacobian-transpose wrench reflection with matched
    scale, shaping, and saturation;
    \item an $A^\top J_s^\top\hat F$ variant using the same DLS wrench as the
    direct baseline;
    \item a discrete-time passivity observer/controller (PO/PC) baseline
    following Hannaford and Ryu~\cite{Hannaford2002TDPC}, implemented at the
    same master haptic port.
\end{enumerate}
Only items 1--3 are executable in the current simulator, and even their
item~1--item~3 contrast is the end-to-end chain comparison identified in
Table~\ref{tab:code_evidence_boundary}. Item 4 is required before any pure
$A^\top$-versus-$J_m^\top$ mapping claim, and item 5 is required before the
fixture and human comparisons; both must pass G0a and the locked G0c runner
before G3.
The PO/PC baseline will receive the same raw mapping, upstream filter, nominal
gain, and actuator limits; only its passivity-intervention law differs, and its
parameters are selected from the calibration set. The primary H3 mapping
contrast is item 4 versus item 3 because both receive
the same reconstructed wrench; it will also be repeated offline with the
independent force/torque reference. The full proposed path in item 1 uses the
unprojected bias-corrected joint residual, whereas item 3 uses the DLS wrench.
Their comparison is therefore labeled an end-to-end chain comparison, not a
pure mapping ablation.
Signal filtering, torque limits, environment trajectories, and network traces
will be identical across paired comparisons. Single-factor ablations will
separately remove the generalized-torque filter, rate limiting, torque
saturation when fixture safety permits, and energy projection; multiple
elements will not be removed under one ablation label. ``Without energy
projection'' retains all other limits and is not described as an unprotected
controller.

The environment will include free space and preregistered stiffness and damping
levels within the verified fixture and actuator safety envelope. A network
emulator will impose constant, asymmetric, and time-varying delays, bounded
jitter, and packet loss. The matrix will include a zero-delay reference, the
measured nominal network condition, at least two fixed-delay levels, and one
recorded jitter/loss trace within the G3 safety envelope. Exact levels and
repeat counts will be locked before confirmatory execution.
The present implementation contains only a fixed return-delay queue; it does
not yet implement independent forward/return channels, packet sequencing,
jitter/loss injection, or timeout transitions. The semantics below are
therefore requirements to be implemented and deterministically tested before
G0c/G3, not descriptions of the current executable.

Network semantics are fixed independently of those traces. Let
$\kappa_f(k)$ and $\kappa_r(k)$ be the newest accepted source indices at slave
and master sample $k$ for the forward reference packet $u_f$ in
\eqref{eq:forward_held_reference} and the return residual packet,
respectively. The return packet contains the residual, reconstructed wrench,
$({}^{C}\!J_{E_s})^\top{}^C\!\hat F_{\mathrm{int}}$, actual slave velocity,
active forward-map identifier $\mu$, source timestamp, and sequence number;
this payload supports both direct and matched-wrench baselines.
Packets use synchronized clocks and monotonically increasing sequence numbers.
Duplicates, corrupted packets, and packets no newer than the last accepted
sequence number are discarded; no interpolation or retransmission is
introduced by the experiment. The last accepted sample is held until its
preregistered channel timeout. A forward timeout invokes the same bounded
zero-velocity safe hold in every condition. A return timeout sets the raw
haptic target to zero, after which the declared filter, rate limit, saturation,
and energy projection produce the controlled fade. This sets $\chi_r[k]=0$;
accepting a fresh in-time packet sets it to one. Every arrival, rejection, loss,
held-sample age, active $\mu$, and timeout transition is logged. Constant
return delay on the 500-Hz channel is the special case
$\kappa_r(k)=k-d$. Because forward packets exist only on the 50-Hz update set
$\mathcal I_f$, a constant forward delay $d_f$ is instead
\begin{equation}
 \kappa_f(k)=\max\{i\in\mathcal I_f:t_i^{f}+d_f\le t_k^{s}\},
\end{equation}
when that set is nonempty. Thus asymmetric and time-varying traces change the
held references and their echoed map identifiers without changing controller
semantics.

The fixed-branch algebraic identity will first be checked using
\begin{equation}
    e_P^{\mathrm{map}}[k]=
    \tau_{h,\mathrm{raw}}[k]^\top\dot q_m[k]
    -\chi_r[k]\sigma_f
     \hat\tau_{s,\mathrm{int}}[\kappa_r(k)]^\top
     \dot q_s^{\mathrm{map}}[k].
\end{equation}
where the residual in the second term is the same delayed sample
$\hat\tau_{s,\mathrm{int}}[\kappa_r(k)]$ used in
\eqref{eq:raw_master_torque}. This is a numerical contract check rather than a
performance endpoint. Errors
$\|\dot q_s^{\mathrm{ctrl}}-\dot q_s^{\mathrm{map}}\|$ and
$\|\dot q_s-\dot q_s^{\mathrm{ctrl}}\|$ separately quantify limiting/hold and
tracking effects. The current-map and source-stamped delay policies are
compared using
\begin{equation}
 e_{\tau,A}[k]=
 \chi_r[k]\sigma_f\left\|\left(\bar A[k]
 -\bar A[\mu(\kappa_r(k))]\right)^\top
 \hat\tau_{s,\mathrm{int}}[\kappa_r(k)]\right\|_2 .
\end{equation}
The primary H3 endpoint uses actual slave velocity and aligned source/arrival
timestamps. For condition $c$, let $\tau_{s,c}$ be the source generalized
torque actually supplied to its mapping: the bias-corrected residual for the
full proposed chain and
$({}^{C}\!J_{E_s})^\top{}^C\!\hat F_{\mathrm{int}}$ for the two matched-wrench
mapping conditions. Then
\begin{align}
    P_{m,c}^{\mathrm{raw}}[k]
    &=\tau_{m,c}^{\mathrm{raw}}[k]^\top\dot q_m[k],\\
    P_{s,c}[\kappa_r(k)]
    &=\chi_r[k]\tau_{s,c}[\kappa_r(k)]^\top
      \dot q_s[\kappa_r(k)].
\end{align}
\begin{equation}
    \epsilon_{P,c}^{\mathrm{act}}=
    \frac{\sum_k|P_{m,c}^{\mathrm{raw}}[k]
      -\sigma_fP_{s,c}[\kappa_r(k)]|\Delta t_k}
    {\frac{1}{2}\sum_k\left(
      |P_{m,c}^{\mathrm{raw}}[k]|
      +|\sigma_fP_{s,c}[\kappa_r(k)]|\right)\Delta t_k+\epsilon_P}.
    \label{eq:normalized_actual_power_mismatch}
\end{equation}
Here $\epsilon_P>0$ has energy units and is a locked numerical regularizer
used only for near-zero-power trials.
To prevent the stored lower clip from hiding an accounting defect, every
sample will independently reconstruct
\begin{align}
 E_{\mathrm{audit}}^+[k+1]
 &=E[k]-\tau_{h,\mathrm{app}}[k]^\top
 \dot q_m[k]\Delta t_k,\\
 \delta_E^-[k]
 &=\max(0,E_{\min}-E_{\mathrm{audit}}^+[k+1]).
\end{align}
The budget component of H4 is a deterministic gate: all independently
reconstructed samples must satisfy
$\delta_E^-[k]\le\bar\delta_E$, where $\bar\delta_E$ is the frozen numerical
roundoff tolerance. To make the projection-bypass comparison explicit, each
paired trajectory $r$ also carries an independently reconstructed shadow
budget for condition $c$:
\begin{align}
 E_{c,r}^{\mathrm{sh}}[0]&=E[0],\\
 E_{c,r}^{\mathrm{sh}}[k+1]
 &=\min\!\left(E_{\max},
 E_{c,r}^{\mathrm{sh}}[k]
 -P_{h,c}[k]\Delta t_k\right),\\
 B_{c,r}&=\max_k\max\!\left(
 0,E_{\min}-E_{c,r}^{\mathrm{sh}}[k]\right).
 \label{eq:shadow_floor_deficit}
\end{align}
For the confirmatory enforcement contrast, $c=\mathrm{proj}$ uses the actual
projected-run output, while $c=\mathrm{shadow}$ is an offline counterfactual
that substitutes that same run's logged $\tau_h^{\mathrm{cand}}$ for the
applied torque without feeding it back into the filter, rate limiter, robot, or
trajectory; explicitly,
$P_{h,\mathrm{proj}}=\tau_{h,\mathrm{app}}^\top\dot q_m$ and
$P_{h,\mathrm{shadow}}=(\tau_h^{\mathrm{cand}})^\top\dot q_m$ on the same
logged samples. This avoids falsely assuming that two live closed-loop
conditions retain an identical candidate sequence after intervention. The primary
floor-deficit reduction is
$\Delta B_r=B_{\mathrm{shadow},r}-B_{\mathrm{proj},r}$ on a
calibration-selected, energy-challenging fixture subset. The live projection-
bypass run remains an end-to-end secondary comparison with its own
condition-specific candidate and trajectory.

The single primary transparency-cost scalar is the trajectory-level,
torque-space projection distortion
\begin{equation}
 D_{\mathrm{proj},r}=
 \frac{\sum_k\|\tau_{h,\mathrm{app}}[k]
 -\tau_h^{\mathrm{cand}}[k]\|_2\Delta t_k}
 {\sum_k\|\tau_h^{\mathrm{cand}}[k]\|_2\Delta t_k+\epsilon_\tau},
 \label{eq:projection_distortion}
\end{equation}
where $\epsilon_\tau>0$ has units of \si{\newton\metre\second} and is frozen
for near-zero-torque trials. This metric quantifies transparency only at the
generalized-torque projection level. Lower is better, and an acceptable upper
limit $\bar D_{\mathrm{proj}}$ is set from calibration/fixture data before
confirmatory trials. H4 is accepted only if the deterministic audit passes,
the one-sided 95\% confidence interval for the paired mean
$\Delta B_r$ lies above a frozen nonnegative superiority margin, and the
one-sided 95\% upper confidence bound for mean $D_{\mathrm{proj},r}$ does not
exceed $\bar D_{\mathrm{proj}}$. Haptic-storage diagnostics will include
$\min_k(E_{\mathrm{audit}}^+[k]-E_{\min})$,
$\max_k\delta_E^-[k]$, discrepancy between the reconstructed pre-clip value
and the software pre-clip value, number and duration of violations beyond a
fixed numerical tolerance, and signed net released energy from
\eqref{eq:net_haptic_energy_bound}.
Secondary transparency outcomes will include motion-tracking error,
force-reflection gain and phase, contact-force error, time with
$\rho<\bar\rho$, and rendered-impedance error. Oscillation amplitude, settling
time, spectral energy, torque-limit events, safety stops, and divergent trials
will be reported as diagnostics.

Any passivity or stability statement will be limited to its analytical
assumptions and experimentally tested envelope. The absence of instability in
a finite dataset will not be described as proof for arbitrary stiffness or
delay.

\subsection{End-to-End Teleoperation Tasks}
\label{sec:end_to_end_protocol}

Integrated testing will progress through:
\begin{enumerate}
    \item three-dimensional free-space path tracking;
    \item force-regulated following of an instrumented surface;
    \item constrained insertion with randomized initial pose and clearance;
    \item manipulation of an instrumented compliant or fragile object.
\end{enumerate}
Targets, paths, object parameters, timeouts, and termination rules will be
sampled from a locked set. The complete method will be compared with no
kinesthetic feedback and the locked PO/PC baseline. Unsafe
unregulated conditions will remain fixture-only.

The primary integrated-task outcome is success probability under a
preregistered success definition. Completion time, Cartesian and force
tracking error, peak unintended contact force, contact impulse, unintended
contacts, insertion retries, and safety stops are secondary outcomes. A
timeout or safety stop counts as failure; completion time among successful
trials and failure probability are analyzed separately.

\subsection{Human-Subject Protocol}
\label{sec:user_protocol}

Participant recruitment will begin only after approval by the appropriate
institutional ethics committee. The approval identifier, informed-consent
procedure, inclusion and exclusion criteria, anthropometric fitting range,
compensation, and withdrawal procedure will be reported. The confirmatory
sample size will be determined by an a priori repeated-measures power analysis
for the primary contrast, with $\alpha=0.05$, power of at least 0.80,
anticipated attrition declared separately, and an effect-size assumption
supported by pilot data or relevant literature. The calculation and resulting
integer sample size will be preregistered before confirmatory recruitment.
Pilot participants used for tuning will not enter the confirmatory analysis.

A randomized, counterbalanced, within-participant design will compare:
\begin{enumerate}
    \item the proposed energy-supervised kinesthetic feedback;
    \item the locked PO/PC kinesthetic baseline;
    \item visual teleoperation without kinesthetic force feedback.
\end{enumerate}
Direct unregulated force reflection will not be applied to human participants.
Participants will receive standardized fitting, training, and
criterion-based familiarization. Condition and task order will be balanced,
and mandatory rest intervals will reduce fatigue.

The primary human-performance outcome is surface-following force RMSE; any
change requires a dated protocol amendment before confirmatory recruitment.
Secondary outcomes include success, completion time, path error, peak
unintended force, contact impulse, corrective motions, and safety
interventions. Subjective outcomes include NASA-TLX workload
~\cite{Hart1988TLX}, perceived force
fidelity, comfort, and preference. A separate psychophysical stiffness-
discrimination block may be included only if it is fixed before recruitment;
it will not replace the preregistered robotic-task endpoint.

\subsection{Statistical Analysis}
\label{sec:statistics_protocol}

The configuration or trajectory is the experimental unit for component tests;
the participant is the experimental unit for the human study. Repeated trials
from one participant will not be treated as independent human samples.
Algorithm comparisons on identical inputs will use paired analyses.

Component-level outcomes will be summarized using mean, standard deviation,
median, interquartile range, bootstrap 95\% confidence interval, and relevant
tail percentiles. Human continuous outcomes will be analyzed with
mixed-effects models containing feedback condition, task, delay when
applicable, and their preregistered interactions as fixed effects and
participant as a random effect. Binary success will use a logistic
mixed-effects model; ordinal responses will use an ordinal model when
appropriate.

Assumptions, residuals, and heteroscedasticity will be checked. Robust,
transformed, or preregistered nonparametric alternatives will be used when
needed. The four confirmatory algorithmic outcome families are fixed as:
H1, the single composite event $C_r$ in
\eqref{eq:sew_composite_event}; H2, force RMSE and moment RMSE as two
co-primary estimator endpoints; H3, normalized actual power mismatch
$\epsilon_{P,c}^{\mathrm{act}}$; and H4, the deterministic storage gate plus
paired floor deficit $\Delta B_r$ and the single torque-level transparency-cost
endpoint $D_{\mathrm{proj},r}$.

All confirmatory directions and margins are frozen from calibration and
engineering tolerances before the locked data are inspected. H1 uses the
matched task-priority IK as its primary comparator and passes when the
one-sided 95\% upper confidence bound for
$p(C_r=1)_{\mathrm{SEW}}-p(C_r=1)_{\mathrm{IK}}$ is below
$-\Delta_{\mathrm{H1}}$. H2 passes only when the corresponding upper bounds
for both
$\mathrm{RMSE}_{F,\mathrm{DLS}}-\mathrm{RMSE}_{F,\mathrm{undamped}}$ and
$\mathrm{RMSE}_{M,\mathrm{DLS}}-\mathrm{RMSE}_{M,\mathrm{undamped}}$ are below
their negative superiority margins. H3 passes when the upper bound for the
paired mean difference
$\epsilon_{P,A^\top J_s^\top F}^{\mathrm{act}}
-\epsilon_{P,J_m^\top F}^{\mathrm{act}}$ is below
$-\Delta_{\mathrm{H3}}$. H4 uses the three conjunctive rules stated after
\eqref{eq:projection_distortion}; failure of any one rejects H4. Every margin
is nonnegative and has zero as the fallback only when calibration cannot
justify a larger practically relevant value.

These are one-sided superiority or acceptability decisions; two-sided 95\%
intervals and effect sizes will still be reported. Additional pairwise method
comparisons will use Holm correction within the corresponding H1--H4 family.
Integrated-task and human outcomes form separately declared external-validity
families; all remaining measures are labeled secondary or diagnostic. Exact
$p$ values will be reported. Missing data, aborted trials, sensor dropouts,
outlier rules, and protocol deviations will be disclosed.

\subsection{Reproducibility and Data Availability}
\label{sec:reproducibility_protocol}

The replication package will contain the exact URDF and calibration files,
controller parameters, test trajectories, network-emulator traces, random
seeds, anonymized trial-level data, and analysis and figure-generation scripts.
The operating system, processor, real-time settings, software versions,
execution thread, and timing method will be reported. Raw and filtered signals
will remain distinguishable, and coordinate transforms, spatial-vector order,
units, and sign conventions will be machine-readable.

All parameter sweeps, failed trials, safety stops, and protocol deviations will
be retained. Each final table and figure will be generated from one versioned
analysis entry point, and the released code and data will be archived at an
immutable commit and DOI subject to participant privacy and hardware-vendor
restrictions.

\section{Results}
\label{sec:results}

No quantitative result is reported in the present working draft. After the
protocol has been executed, this section will report outcomes in the following
fixed order: (i) deterministic software contracts and the locked multi-run
simulation, explicitly labeled numerical, (ii) hardware characterization and
timing, (iii) component-level retargeting and wrench estimation,
(iv) non-human mapping-consistency, energy, and delay tests,
(v) end-to-end task performance, and (vi) human-subject outcomes. Failed and
terminated trials will be included according to the rules above.


\section{Discussion and Limitations}
\label{sec:discussion}

\subsection{Methodological Trade-offs}

\textbf{SEW degeneracy and branch continuity.}
The reference-normal construction becomes ill-conditioned when the
shoulder--wrist direction approaches the reference axis. A fallback direction
keeps the equations defined but does not automatically make the threshold
transition smooth. Similarly, an analytic Cartesian elbow target does not
guarantee a continuous slave joint trajectory. The current implementation uses
warm-started recovery and validity gates rather than swivel hysteresis or
normal blending; failure behavior must therefore be evaluated explicitly.

\textbf{Reach feasibility and differential validity.}
Reach clipping makes the target feasible but changes the local motion scale and
may introduce a nonsmooth boundary. The differential map in
\eqref{eq:retarget_differential} is therefore valid only on a fixed smooth
branch. Clipping and joint-limit events must be excluded from a smooth-map
proof or handled by a hybrid formulation, while remaining included in the
empirical failure analysis.

\textbf{Residual-wrench identifiability.}
The inverse-dynamics residual contains contact torque together with inertial,
friction, bias, and sensing errors. Damped inversion reduces amplification but
introduces bias and cannot recover unobservable wrench components. Independent
force/torque ground truth and configuration-stratified analysis are necessary
before describing the estimator as robust.

\textbf{Energy constraint versus bilateral stability.}
The energy projection constrains only the sampled haptic feedback contribution
under a specific timing and sign convention. Dynamics compensation, device
mechanics, the communication channel, and the slave port lie outside this
storage. A stronger total-port or bilateral claim requires an explicit
two-port treatment and a proof whose assumptions include filters, saturation,
sampling, compensation, and delay.

\textbf{Transparency versus intervention.}
Any reduction of candidate torque used to preserve an energy reserve alters the
rendered impedance. The relevant question is therefore not whether the method
is simultaneously ``fully transparent'' and constrained, but how much force
fidelity, phase, and task performance are lost for a measured reduction in
released energy or oscillatory behavior.

\subsection{Current Limitations and Validity Threats}

\begin{itemize}
    \item \textbf{Prototype status.} The master exoskeleton is under
    fabrication, so mass, torque capability, backdrivability, bandwidth,
    alignment, comfort, timing, and safety performance are not yet empirical
    contributions.

    \item \textbf{Implementation and calibration status.} The software core now
    includes bounded recovery, guarded $A$, the declared spatial-vector order,
    direct residual mapping, and final haptic-energy projection. The current
    simulation nevertheless assumes aligned base axes, a simulated TCP,
    synchronous clocks, fixed delay, and simplified rigid-body contact.
    Calibration ingestion for the master/common/slave transforms and
    $R_\Delta$ is not yet implemented; timestamps, torque calibration, and
    total-port energy measurements also remain pending.

    \item \textbf{Computational burden.} The current wrapper executes two
    nominal recoveries and fourteen perturbed recoveries per differential
    update. The present multirate implementation does not establish real-time
    feasibility on the target processor; validity rate and tail latency may
    require solve reuse, an analytic or differentiated map, or a reduced-order
    alternative.

    \item \textbf{Model dependence.} Wrench reconstruction depends on the
    slave rigid-body model, payload identification, friction compensation, and
    measured joint torque. Conclusions may not transfer to position-controlled
    robots without comparable sensing.

    \item \textbf{Finite evaluation envelope.} Contact stiffness, delay,
    jitter, packet loss, motion speed, subject anthropometry, and tasks will
    cover finite preregistered ranges. Claims must remain within those ranges.

    \item \textbf{Human-study generalizability.} A laboratory within-subject
    study cannot by itself establish long-term comfort, training effects, or
    performance in field environments. Participant diversity and device-fit
    exclusions must be disclosed.
\end{itemize}

These limitations are incorporated into the prospective protocol rather than
being inferred after observing favorable or unfavorable results.

\section{Conclusion}
\label{sec:conclusion}

This paper formulates a geometry- and energy-aware architecture for
heterogeneous 7-DoF exoskeleton teleoperation. The central distinction is
between an analytic SEW Cartesian target, platform-specific joint recovery, and
the differential map that defines a virtual-work-consistent force-reflection
objective. The proposed feedback path maps a bias-corrected generalized
slave residual directly through $A^\top$; DLS wrench reconstruction is retained
for physical interpretation and Cartesian baselines. Final-stage radial
projection supplies a one-step energy invariant for the shaped haptic
contribution, not for the complete bilateral system.

The present working draft intentionally makes no quantitative hardware,
real-time, accuracy, passivity, stability, transparency, or usability claim.
Instead, it fixes the evidence required to evaluate those claims: measured
exoskeleton characterization, paired retargeting baselines, independent
six-axis wrench ground truth, finite contact and network envelopes,
single-factor ablations, end-to-end tasks, and a powered human-subject study.
After the prototype and software implementation are complete, the protocol will
be executed without replacing failed trials or retrospectively changing the
primary outcomes. The abstract, results, discussion, and conclusion will then
be updated using measured effect sizes and uncertainty intervals.

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% references section

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% BibTeX documentation can be easily obtained at:
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\end{thebibliography}


% that's all folks
\end{document}
