Multistage Insertion Formulation Overview

• Multistage Insertion Formulation is a modeling framework that decomposes insertion tasks into ordered stages with stage-dependent variables, constraints, and decision rules.
• It enables diverse applications by structuring robotic assembly, stochastic programming, and sequential systems through explicit stage interfaces and tailored execution policies.
• The framework enhances system performance and reliability by allowing nuanced control, precise recourse actions, and improved interaction between interdependent stages.

Multistage insertion formulation denotes a family of modeling patterns in which insertion is not treated as a single undifferentiated act but as an ordered process with stage-dependent variables, constraints, controllers, or recourse actions. In the cited literature, the term appears in several distinct senses: robotic peg-and-socket insertion formulated as pose estimation plus staged execution, goal-conditioned multistep insertion policies, two-stage reach-to-insert pipelines, mathematically rigorous multistage decompositions in stochastic programming, reverse-time insertion in sequence generation, stage-chain epidemic models in which incidence is inserted into the first infectious compartment, and reinsertion recourse in disrupted production sequences (Li et al., 22 May 2025, Spector et al., 2022, Meng et al., 6 May 2026, Dommel et al., 2021, Patel et al., 8 Jun 2026, Tverskoi et al., 1 Jul 2026, Yilmazlar et al., 2023).

1. Scope and recurrent formal structure

In the cited works, the phrase does not denote a single canonical formalism. Instead, it names a recurring structural idea: an insertion-related process is decomposed into multiple stages, and those stages are coupled either by geometry, information, or state evolution.

Domain	Multistage object	Representative formal element
Robotic assembly	staged correction, approach, contact, or handoff	$p_g=p+\Delta p$, $P_{next}$, $g_1,\dots,g_K$, $I_{\mathrm{ins}}(\mathbf p)$
Stochastic programming	stagewise value insertion or bounded plan revision	$U \lhd X$, $v_t,V_t$, $\delta_{nd}$, $r_{it}$, $X^K$
Other sequential systems	token insertion, stage-chain progression, reinsertion recourse	$\operatorname{ins}(x,i,a)$, $P_{next}$0, $P_{next}$1

Across these formulations, the stage interface is explicit. In robotic assembly, the interface is often a target pose, a goal image, or a switching region. In stochastic programming, it is a conditional value object, a nodewise distribution choice, or a revision counter. In other sequential systems, it is a transition kernel, a stage-chain compartment, or a reinsertion decision variable. This suggests that “multistage insertion formulation” is best understood as a structural descriptor for how insertion is represented, rather than as the name of a single algorithm or theorem.

2. Robotic assembly: from delta-pose prediction to stage-conditioned insertion

In robotic peg-and-socket assembly, the most explicit formulation is EasyInsert, which casts insertion as relative pose prediction plus staged execution rather than direct low-level action prediction (Li et al., 22 May 2025). The training set is

$P_{next}$2

with dual wrist-camera RGB observations $P_{next}$3 and a 4-DoF relative pose label $P_{next}$4. Under the assumptions that the socket maintains a horizontal position on the working surface and that the plug’s insertion axis aligns with the gripper’s operational axis in orientation, the current pose is $P_{next}$5, the predicted target is

$P_{next}$6

and the supervision label is generated from a reference inserted pose $P_{next}$7 by

$P_{next}$8

Execution is explicitly hierarchical. Coarse alignment moves to

$P_{next}$9

fine adjustment uses

$g_1,\dots,g_K$0

and the close-contact phase adds random lateral perturbations,

$g_1,\dots,g_K$1

The stage switch is based on whether horizontal and angular deviations exceed a predefined threshold $g_1,\dots,g_K$2, and the resulting target is sent to a motion planner in a closed loop at 15 Hz, with model queries every 0.5–1.5 seconds. A single ResNet18-based diffusion model predicts $g_1,\dots,g_K$3; the staged hierarchy is not learned as separate policies. In real-world experiments, with just 5 hours of training data, EasyInsert achieves over 90% success in zero-shot insertion for 13 out of 15 unseen novel objects, and with one human demonstration plus 4 minutes of automatically collected data for fine-tuning, it reaches over 90% success rate for all 15 objects (Li et al., 22 May 2025).

InsertionNet 2.0 uses a different multistage abstraction. It extends a supervised residual insertion policy into a sequence of stage goals $g_1,\dots,g_K$4, where each stage is represented by saved goal images from two wrist-mounted cameras (Spector et al., 2022). In the 3-step key experiment, the stages are: key inserted, lock turned, and lock turned back. The observation is

$g_1,\dots,g_K$5

with stacked stereo RGB input $g_1,\dots,g_K$6, force $g_1,\dots,g_K$7, and moment $g_1,\dots,g_K$8. Training jointly uses

$g_1,\dots,g_K$9

The execution update is

$I_{\mathrm{ins}}(\mathbf p)$0

with the residual phase restricting learned motion to lateral and angular correction while $I_{\mathrm{ins}}(\mathbf p)$1 is handled separately. Stage transitions are procedural: a similarity function decides when the current stage image has been reached, and control then switches to the next saved stage image. The paper reports above $I_{\mathrm{ins}}(\mathbf p)$2 success on 200 trials in 16 real-life insertion tasks, $I_{\mathrm{ins}}(\mathbf p)$3 success on the 3-step key-lock task, and $I_{\mathrm{ins}}(\mathbf p)$4 one-shot success on an unseen insertion task using only a saved final-state image (Spector et al., 2022).

The earlier InsertionNet paper uses a hybrid but less formal multistage structure (Spector et al., 2021). It separates a localization process from a residual insertion process. The learned policy

$I_{\mathrm{ins}}(\mathbf p)$5

maps a 45° wrist-camera image and a 6D wrench to a corrective end-effector displacement and is trained with

$I_{\mathrm{ins}}(\mathbf p)$6

The online update is

$I_{\mathrm{ins}}(\mathbf p)$7

Backward learning generates supervision from perturbations around a known inserted pose using $I_{\mathrm{ins}}(\mathbf p)$8 and $I_{\mathrm{ins}}(\mathbf p)$9. The paper reports $U \lhd X$0 average success over insertion tasks, $U \lhd X$1 over threading tasks, and two end-to-end assembly tasks with multi-insertion objectives (Spector et al., 2021).

A more explicitly staged two-policy design appears in “From Reach to Insert” (Meng et al., 6 May 2026). The reach stage is a vision-based diffusion policy trained by imitation learning and producing a short-horizon action sequence

$U \lhd X$2

with $U \lhd X$3. The insert stage is an SAC policy over a POMDP with binary sparse reward

$U \lhd X$4

The two stages are hard-switched by end-effector position: $U \lhd X$5 The insertion critic uses tactile change

$U \lhd X$6

and the paper reports $U \lhd X$7 average success at $U \lhd X$8 clearance, with a $U \lhd X$9 reduction in maximum interaction force and a $v_t,V_t$0 reduction in torque (Meng et al., 6 May 2026).

3. Geometric and hybrid insertion primitives beyond peg-in-hole assembly

A distinct robotic meaning of multistage insertion is given by rope-through-loop knotting. “A Virtual Magnetic Field Formulation” models the target loop $v_t,V_t$1 as an infinitely narrow current-carrying conductor and computes a motion direction from the Biot–Savart field (Marzinotto et al., 2016): $v_t,V_t$2

$v_t,V_t$3

This is a position-based controller with termination conditions, and the stopping rule is that insertion is deemed complete when the magnetic field intensity ceases to increase and starts to decrease. The primitive has an internal approach–traverse–stop structure, but the broader knotting system is explicitly a hybrid system with basic knotting actions: Grasp Rope, Release Rope, Twist Rope, Turn Base, and Insertion. The paper also gives a Behavior Tree realization, reports $v_t,V_t$4 field computation for a loop discretized into $v_t,V_t$5 segments, and states that in 42,000 simulated insertion attempts under random deformation only 14 failed insertions occurred, all at the highest noise setting (Marzinotto et al., 2016).

A more optimization-centered use of stage structure appears in “Task Planning for Multiple Item Insertion using ADMM” (Zheng, 2023). Here the environment is a 2D bookshelf, the state is indexed by stage $v_t,V_t$6, and each item $v_t,V_t$7 has pose variables $v_t,V_t$8 and $v_t,V_t$9. The model combines stage-indexed rigid-body geometry, separating-plane non-overlap, static equilibrium, and binary contact/collision variables $\delta_{nd}$0 and $\delta_{nd}$1. The objective penalizes both deviation from original poses and inter-stage motion. The resulting problem is formulated as an MINLP and solved by ADMM through a consensus split

$\delta_{nd}$2

The MIP subproblem keeps binary variables and mixed-integer constraints, while the NLP subproblem keeps the nonlinear physics and relaxes binaries to $\delta_{nd}$3. The paper is explicit that its actual experiments solve one additional insertion episode rather than a full long-horizon sequence of item insertions; nevertheless, the stage index $\delta_{nd}$4 is already present in the formulation. On 400 random bookshelf problems, the reported success rates are $\delta_{nd}$5, $\delta_{nd}$6, and $\delta_{nd}$7 for 4, 6, and 8 total items, respectively, while hardware success is about $\delta_{nd}$8 (Zheng, 2023).

4. Stochastic programming: inserting stages, distributions, and revisions

In stochastic programming, multistage insertion formulation has a foundational rather than geometric meaning. “Foundations of Multistage Stochastic Programming” replaces the usual informal nested pathwise statement by a rigorous formulation in which controls are stochastic processes adapted to the information generated by the uncertainty process (Dommel et al., 2021). The base problem is

$\delta_{nd}$9

where $r_{it}$0 means each control $r_{it}$1 is measurable with respect to $r_{it}$2. The paper’s stagewise value objects are

$r_{it}$3

$r_{it}$4

with Bellman-type relations defined through conditional expectations and conditional essential infima rather than pointwise conditioning on sample paths. The paper explicitly states that it does not use the word “insertion”; the closest construct is the pair $r_{it}$5, which inserts a mathematically rigorous stagewise decomposition into the optimization problem.

“A Node Formulation for Multistage Stochastic Programs with Endogenous Uncertainty” makes the stage insertion more literal by allowing decisions at node $r_{it}$6 to select one distribution $r_{it}$7 in a multi-distribution scenario tree (Pantuso, 2021). The model uses node decisions $r_{it}$8, binary distribution selectors $r_{it}$9, and continuation values $X^K$0, with the defining constraints

$X^K$1

$X^K$2

The selected $X^K$3 activates a child-node set $X^K$4 and the associated probabilities, so future branching law is decision-dependent. This node formulation avoids explicit non-anticipativity constraints because decisions are indexed directly by nodes rather than by full scenarios.

In distributionally robust multistage optimization, stage insertion appears as recursive robustification. “Mathematical Foundations of Distributionally Robust Multistage Optimization” distinguishes a conditional distributionally robust functional

$X^K$5

from a nested/composite risk functional formed by recursively composing stagewise conditionals (Pichler et al., 2021). In the product-space setting, the recursive stage insertion is

$X^K$6

and this yields Bellman equations of the form

$X^K$7

The paper’s main caution is that conditional distributionally robust functionals and their nested analogues are generally different.

Two later papers turn stage insertion into explicit revision-stage selection. “Adaptive Multistage Stochastic Programming” allows at most $X^K$8 revisions for each state variable and models revision timing either by selected stages $X^K$9 or by cumulative revision counters $\operatorname{ins}(x,i,a)$0 (Kayacık et al., 2024). The linearized commitment logic is

$\operatorname{ins}(x,i,a)$1

together with

$\operatorname{ins}(x,i,a)$2

The paper proves

$\operatorname{ins}(x,i,a)$3

and develops reduced NAC formulations and a master–subproblem decomposition. “K-revision multistage stochastic programming” instead defines a future plan $\operatorname{ins}(x,i,a)$4 at each node and counts a revision whenever that continuation plan differs from the inherited one (Wang et al., 17 Jan 2026). A plan is $\operatorname{ins}(x,i,a)$5-revisable if

$\operatorname{ins}(x,i,a)$6

for every scenario $\operatorname{ins}(x,i,a)$7, and the direct formulation uses variables $\operatorname{ins}(x,i,a)$8, $\operatorname{ins}(x,i,a)$9, and $P_{next}$00 with

$P_{next}$01

The paper also gives a combinatorial characterization: a policy is not $P_{next}$02-revisable if and only if there exists an $P_{next}$03-inconsistent ELBE subtree of height $P_{next}$04. This turns bounded revision into a forbidden embedded-subtree condition and underlies the ST and STDP formulations (Wang et al., 17 Jan 2026).

5. Other stage-structured insertions: sequence generation, epidemics, and production reinsertion

In insertion language modeling, the stage object is neither a robot pose nor a value function but a variable-length sequence. “A Continuous-Time Markov Chain Framework for Insertion LLMs” represents a state as $P_{next}$05 and defines one-token insertion by

$P_{next}$06

The forward process is deletion, the reverse process is insertion, and reverse-time denoising becomes a multistage insertion process. In the joint model, the reverse rate is

$P_{next}$07

whereas the independent model uses per-gap rates and supports multiple insertions in parallel within a reverse step. The number and granularity of insertion stages are controlled by continuous time and its discretization (Patel et al., 8 Jun 2026).

In epidemic modeling, the insertion is into a stage chain. “Approximating Peak Prevalence in Multistage SIR Epidemics” writes the deterministic $P_{next}$08 model as

$P_{next}$09

so new infections are inserted into $P_{next}$10 at incidence

$P_{next}$11

and then progress through

$P_{next}$12

Under Erlang scaling $P_{next}$13, the stage-expanded system converges to a delay formulation in which prevalence and a weighted stage functional become unweighted and triangularly weighted moving averages of incidence (Tverskoi et al., 1 Jul 2026).

In production planning, insertion takes the form of reinsertion recourse. “Mixed-model Sequencing with Reinsertion of Failed Vehicles” is a bi-objective two-stage stochastic program in which the first stage chooses a launch sequence and the second stage reinsert failed vehicles after failures are realized (Yilmazlar et al., 2023). The recourse insertion variable is

$P_{next}$14

where $P_{next}$15 means failed vehicle $P_{next}$16 is inserted at position $P_{next}$17, including a dummy terminal position for “not reinserted this horizon.” The model includes readiness constraints, a spacing restriction

$P_{next}$18

and a waiting-time term

$P_{next}$19

The paper is explicit that the optimization is formally two-stage, even though the operational interpretation is dynamic reinsertion as suitable positions occur (Yilmazlar et al., 2023).

6. Interpretive issues, common misconceptions, and limitations

Several recurring limitations clarify what multistage insertion formulation does and does not mean in practice. In robotic assembly, the stage logic is often hand-designed rather than inferred. EasyInsert uses a predefined threshold $P_{next}$20 for coarse-versus-fine switching, but the paper does not provide its numerical value; it also assumes an obstacle-free region between plug and socket and handles close contact through random perturbation rather than force-guided insertion (Li et al., 22 May 2025). InsertionNet 2.0 uses saved stage images and “our similarity function to switch between steps,” but does not formulate stage discovery as an HMM, an explicit finite-state automaton, or RL options with learned termination; it also reports that the relation policy is less robust than the delta policy to obstacles and strong grasping changes (Spector et al., 2022). “From Reach to Insert” likewise uses a manually phase-switched policy based on whether $P_{next}$21, not a learned switch or a tactile transition detector, and it trains a separate reach policy for each geometry (Meng et al., 6 May 2026).

A second misconception is that the presence of a stage index automatically implies a fully developed long-horizon multistage problem. The bookshelf ADMM paper has a stage index $P_{next}$22, but the implemented optimization is best understood as one additional insertion episode in a 2D quasi-static shelf model, with long-horizon multiple-item insertion left to future work (Zheng, 2023). The mixed-model sequencing paper discusses reinsertion dynamically, but its mathematical model is a bi-objective two-stage stochastic program rather than a multistage stochastic program with repeated revelation and nonanticipativity over multiple recourse dates (Yilmazlar et al., 2023).

A third misconception is that informal nested stagewise notation is mathematically harmless. The foundational stochastic-programming papers make the opposite point. “Foundations of Multistage Stochastic Programming” argues that pathwise nested problem statements suppress measurability, conditioning-on-null-set, and essential-infimum issues, and replaces them with adapted stochastic processes and measurable intermediate value functions (Dommel et al., 2021). “Mathematical Foundations of Distributionally Robust Multistage Optimization” further shows that conditional distributionally robust functionals and nested/composite risk functionals are generally different objects, so stage insertion through conditional evaluation must be specified with care (Pichler et al., 2021).

Taken together, these works indicate that multistage insertion formulation is a structural family rather than a single theory. In one regime it means staged geometric execution around contact; in another it means nodewise insertion of distributions, risk functionals, or continuation plans; in still others it means reverse-time token insertion, linear-chain stage progression, or second-stage reinsertion recourse. The unifying principle is explicit stage coupling, but the mathematical substance of the stages depends entirely on the domain and on what is being inserted.