Rhythmic Insertion Tasks in Robotics

• Rhythmic Insertion Tasks (RIT) are robotic manipulation tasks characterized by periodic motion during contact to overcome misalignment, slip, and jamming.
• They integrate continuous feedback from tactile, proprioceptive, or visual sensors with adaptive control methods such as extremum seeking and reinforcement learning.
• RIT enable both one-time precise insertions and robust, repeated-cycle operations in tasks like key insertion and wrench–nut screwing.

Searching arXiv for the cited Rhythmic Insertion Tasks papers and topic context. Searching arXiv for "(Burner et al., 2024)". Rhythmic Insertion Tasks (RIT) are contact-rich manipulation tasks that exploit periodic motion under contact to resolve misalignment, stick–slip, and jamming, using tactile and/or proprioceptive feedback to adapt while maintaining safe contact. In the current arXiv literature considered here, RIT encompass both direct insertion under pose uncertainty and repeated insertion cycles in which alignment, seating, rotation, and reset recur rhythmically. Two explicit realizations illustrate the breadth of the category: a model-free Extremum Seeking Control (ESC) law for tactile key insertion that wiggles all six end-effector pose degrees of freedom at distinct frequencies (Burner et al., 2024), and a sim-to-real framework for wrench-based nut screwing that combines an object-centric reinforcement learning (RL) insertion policy with failure forecasting and lift-and-retry recovery (Liu et al., 9 Jul 2025).

1. Definition and task class

RIT are defined by the use of periodic motion during insertion or seating under contact. The central premise is that rhythmic excitation makes contact informative, relieves wedging, and supports online adaptation in settings where geometry, friction, and pose error interact nonlinearly. The task class includes direct insertion, such as inserting rigid keys into various locks under pose uncertainty, and repeated insertion tasks, such as seating a wrench on a nut, rotating the nut, resetting, and repeating the cycle.

The direct insertion formulation emphasizes human-like wiggling. Humans often wiggle during insertion, using touch and proprioception to feel constraints and relieve jamming. The tactile insertion system formalizes this behavior with a model-free ESC law that applies periodic perturbations to the 6-DOF pose and demodulates tactile feedback to estimate a descent direction of a performance objective (Burner et al., 2024).

The repeated insertion formulation emphasizes long-horizon consistency. In wrench-based nut screwing, RIT are characterized by rhythmic repetition of phases—align, insert, rotate, reset, then repeat—and by the requirement that millimeter-level accuracy be sustained across many cycles because small errors compound over time. In that formulation, the insertion event is the achievement of a desired tool-to-object relative pose in $SE(3)$, followed by a rotation of at least $60^\circ$ counter-clockwise and a reset to nominal orientation (Liu et al., 9 Jul 2025).

A concise way to view the task class is that RIT combine three ingredients: periodic contact excitation, feedback about insertion state, and an adaptation mechanism that converts repeated contact into progress. This suggests that RIT are not limited to a single sensing modality or controller family; rather, they are organized by the functional role of rhythmic motion in contact-rich insertion.

2. Core mechanisms of rhythmic insertion

The first core mechanism is periodic motion itself. In the tactile key-in-lock system, the robot sinusoidally wiggles all six end-effector pose degrees of freedom at distinct frequencies. The periodic excitation addresses three common RIT challenges: multi-DOF sinusoidal wiggling explores local directions that reduce contact strain and move the tip toward the keyhole; rhythmic modulation helps overcome static friction and prevents wedging by redistributing contact forces; and tactile feedback from GelSight provides a strain-like signal that encodes the state of contact (Burner et al., 2024).

The second core mechanism is feedback-driven adaptation. In the ESC formulation, rhythmic motion is not merely superimposed on insertion; it is converted into an online optimizer of a scalar objective that balances insertion progress and tactile strain. In the wrench–nut formulation, the learned insertion policy operates in an object-centric coordinate frame, while a failure forecasting module estimates the probability that the ongoing insertion attempt will succeed within a future window and triggers recovery when that probability falls below a threshold (Liu et al., 9 Jul 2025).

The third core mechanism is rhythmic task structuring. The tactile key-in-lock system uses continuous multi-frequency dithers during insertion. The wrench–nut system uses a discrete rhythmic scheduler with the sequence align $\rightarrow$ insert $\rightarrow$ rotate $\rightarrow$ reset. In both cases, periodicity is functional rather than decorative: it either probes local contact geometry continuously or structures repeated contact episodes so that errors can be detected and corrected before they accumulate.

Mechanism	Tactile key insertion	Wrench–nut screwing
Periodic component	Sinusoidal perturbations in all 6 DOFs	Repeated align, insert, rotate, reset cycle
Feedback source	GelSight Mini strain-like signal and insertion depth	RGB-D 6D pose tracking and failure forecasting
Adaptation mechanism	Extremum-seeking demodulation and pose update	Object-centric RL policy plus recovery

A common misconception is that RIT are equivalent to applying a fixed oscillation. The available formulations instead embed rhythmic motion inside a closed-loop adaptation process. This suggests that rhythmicity is the probing and recovery substrate, while performance depends on how sensing, representation, and control exploit that substrate.

3. Extremum-seeking controlled wiggling for tactile insertion

In “Extremum Seeking Controlled Wiggling for Tactile Insertion” (Burner et al., 2024), RIT are instantiated as model-free tactile key insertion on a UR10 industrial manipulator. The robot controls the 6-DOF tip pose of the key via tool-space servoing. The key head is grasped between two GelSight Mini tactile sensors; one is powered off to act as a compliant barrier and the other is powered on to provide tactile imagery. The sensor’s internal camera observes gel pad deformation, and a Lucas–Kanade homography tracker registers each frame to the first frame over a patch with $10\%$ margins.

The strain-like tactile signal is defined from the Euclidean norm of the 2D displacements of the four tracked patch corners relative to their initial locations:

$$L_{\mathrm{strain}}(t)=\sqrt{\sum_{i=1}^{4}\|p_i(t)-p_i(t_0)\|^2},$$

with $L_{\mathrm{strain}}$ reported as $0$ for values below $3$ pixels. Frames are processed at $60^\circ$0–$60^\circ$1 Hz due to sensor and interface constraints; the experiments operate effectively at $60^\circ$2 Hz tactile feedback. Insertion progress is measured from the robot’s $60^\circ$3 coordinate, where $60^\circ$4 is the initial tip position and $60^\circ$5 is the keyhole depth. The insertion loss is

$60^\circ$6

and insertion success is declared when $60^\circ$7 mm.

The scalar objective minimized by ESC balances insertion progress and tactile strain:

$60^\circ$8

with $60^\circ$9. The estimated parameter vector is the 6-DOF key tip pose,

$\rightarrow$0

where $\rightarrow$1 are intrinsic Euler angles $\rightarrow$2. The applied parameters are the estimate plus sinusoidal dithers,

$\rightarrow$3

with translation amplitudes $\rightarrow$4 mm, rotation amplitudes $\rightarrow$5 deg, and dither frequencies $\rightarrow$6 Hz. The per-DOF periodic signals are $\rightarrow$7, enabling multi-parameter gradient estimation via frequency separation.

ESC processes the objective through a standard pipeline: high-pass filter $\rightarrow$8 with a first-order HPF ($\rightarrow$9 Hz cutoff) to remove the DC component; demodulate by elementwise multiplication with $\rightarrow$0; low-pass filter with a first-order LPF ($\rightarrow$1 Hz cutoff) to extract correlation terms proportional to the local gradient; and integrate to update $\rightarrow$2 with diagonal gains $\rightarrow$3. The continuous-time update law is

$\rightarrow$4

The algorithmic loop is explicit. It initializes $\rightarrow$5, the desired depth $\rightarrow$6, the ESC parameters $\rightarrow$7, the filters, and $\rightarrow$8. At the tactile update rate of approximately $\rightarrow$9 Hz, it acquires a GelSight frame, computes $\rightarrow$0, reads robot $\rightarrow$1, computes $\rightarrow$2, forms $\rightarrow$3, high-pass filters and demodulates the objective, low-pass filters the result, updates $\rightarrow$4, forms $\rightarrow$5, and commands the UR10 to $\rightarrow$6. Termination occurs either when $\rightarrow$7 mm or when safety or time limits are reached.

The experimental evaluation covers four lock/key types: L1 cylindrical pin–tumbler, L2 dimpled cam, L3 tubular cam, and L4 disc–detainer padlock. Keyhole depths were set to $\rightarrow$8 mm (L1), $\rightarrow$9 mm (L2), $10\%$0 mm (L3), and $10\%$1–$10\%$2 mm (L4 depending on trial set). Initial translation perturbations in $10\%$3 and $10\%$4 were $10\%$5 mm and $10\%$6 mm, and rotations about $10\%$7, $10\%$8, and $10\%$9 were perturbed within $$L_{\mathrm{strain}}(t)=\sqrt{\sum_{i=1}^{4}\|p_i(t)-p_i(t_0)\|^2},$$0 and $$L_{\mathrm{strain}}(t)=\sqrt{\sum_{i=1}^{4}\|p_i(t)-p_i(t_0)\|^2},$$1. Over $$L_{\mathrm{strain}}(t)=\sqrt{\sum_{i=1}^{4}\|p_i(t)-p_i(t_0)\|^2},$$2 deterministic single-parameter perturbation trials, $$L_{\mathrm{strain}}(t)=\sqrt{\sum_{i=1}^{4}\|p_i(t)-p_i(t_0)\|^2},$$3 succeeded with mean insertion time $$L_{\mathrm{strain}}(t)=\sqrt{\sum_{i=1}^{4}\|p_i(t)-p_i(t_0)\|^2},$$4 s. Over $$L_{\mathrm{strain}}(t)=\sqrt{\sum_{i=1}^{4}\|p_i(t)-p_i(t_0)\|^2},$$5 randomly initialized trials with joint random perturbations in both translation and rotation, $$L_{\mathrm{strain}}(t)=\sqrt{\sum_{i=1}^{4}\|p_i(t)-p_i(t_0)\|^2},$$6 succeeded with mean insertion time $$L_{\mathrm{strain}}(t)=\sqrt{\sum_{i=1}^{4}\|p_i(t)-p_i(t_0)\|^2},$$7 s. L4 was easiest, with deterministic success of $$L_{\mathrm{strain}}(t)=\sqrt{\sum_{i=1}^{4}\|p_i(t)-p_i(t_0)\|^2},$$8 at approximately $$L_{\mathrm{strain}}(t)=\sqrt{\sum_{i=1}^{4}\|p_i(t)-p_i(t_0)\|^2},$$9 s mean time and random success of $L_{\mathrm{strain}}$0 at approximately $L_{\mathrm{strain}}$1 s mean time, whereas L3 was hardest due to orientation sensitivity and flat face geometry that offers little contact guidance.

These results establish a textbook RIT realization: periodic motion is realized as sinusoidal perturbations in all six DOFs; feedback-driven adaptation is realized by demodulating tactile strain and insertion depth; and task objective alignment is realized by minimizing strain while maximizing insertion depth. The formulation is model-free and uses the same ESC parameters across four different lock geometries without tuning, except for specifying the key length or depth.

4. Object-centric sim-to-real rhythmic insertion with failure forecasting

In “Failure Forecasting Boosts Robustness of Sim2Real Rhythmic Insertion Policies” (Liu et al., 9 Jul 2025), RIT are studied through wrench-based nut screwing on a bolt. The task is formalized by a desired tool-to-object relative pose $L_{\mathrm{strain}}$2. At each timestep, the robot estimates camera-frame poses and seeks a time $L_{\mathrm{strain}}$3 such that

$L_{\mathrm{strain}}$4

This expresses precise alignment in $L_{\mathrm{strain}}$5 and defines the insertion event as seating the wrench head on the nut. In the rhythmic cycle used experimentally, the align phase places the wrench head approximately $L_{\mathrm{strain}}$6 cm above the nut and oriented to match the nut’s tilt, the insert phase seats the wrench at the desired relative pose, the rotate phase turns the nut at least $L_{\mathrm{strain}}$7 counter-clockwise, and the reset phase lifts the wrench and returns it to nominal orientation.

A central technical choice is the object-centric coordinate-frame representation. Rather than expressing the end-effector pose in the robot or world frame, the policy uses the wrench pose in the nut’s coordinate frame. With homogeneous transforms $L_{\mathrm{strain}}$8,

$L_{\mathrm{strain}}$9

and the wrench pose in the nut frame can be written as

$0$0

The goal condition becomes $0$1. The paper attributes a large sim-to-real gain to this choice because it removes nuisance variability due to robot base or world calibration errors and variations in wrench grasp or bolt and nut placements, so long as the initial tool-object relative pose lies in the training distribution.

The insertion policy observes the current tool-to-object relative pose and the goal,

$0$2

and outputs an action

$0$3

applied via a task-space controller. Training uses PPO in Isaac Gym. Rewards and curriculum follow IndustReal’s signed-distance field design and sampling-based curriculum. Initial tool-object relative position is randomized within $0$4 cm in $0$5–$0$6 and $0$7 cm in $0$8, with relative yaw sampled in $0$9. Observation noise of xyz $3$0 mm and yaw $3$1 is injected to make simulation more difficult than reality. The learned policy handles align and insert; rotate and reset are predefined open-loop programs.

The failure forecasting module estimates the probability that the insertion attempt will succeed within a future window:

$3$2

Recovery is triggered when $3$3. Three models are reported. The Time-Only empirical model assumes $3$4 depends only on $3$5, with $3$6 working well empirically. The Moving-Window survival model uses observation history $3$7 and time $3$8 as features, predicts Weibull parameters $3$9 with a 3-layer MLP, and defines

$60^\circ$00

so that

$60^\circ$01

It is trained by maximizing the Weibull log-likelihood of success times while treating failures as censored. During evaluation, Optuna-selected $60^\circ$02 and $60^\circ$03. The Full-Trajectory success classifier uses a 2-layer MLP with $60^\circ$04 and is preferred in real experiments because it does not depend on exact time-step alignment.

When the failure forecaster indicates $60^\circ$05, the recovery policy $60^\circ$06 lifts and retries. Recovery plans a path back to a pre-insertion pose by interpolating $60^\circ$07 sub-goals with the $60^\circ$08 distance between consecutive sub-goals bounded by the insertion policy’s step size. The controller executes $60^\circ$09 consecutive recovery steps in simulation and $60^\circ$10 in real experiments, then returns control to the insertion policy. The pre-insertion alignment step and a small lift of approximately $60^\circ$11 cm mitigate frictional stick, rotation misalignment, and poor initializations.

The physical system uses a KUKA iiwa 14 with a Robotiq 3-finger gripper, an Intel RealSense D435 RGB-D camera, and FoundationPose for unified 6D pose estimation and tracking of the wrench and nut. Objects are 3D printed wrench, nut, and bolt sets of sizes $60^\circ$12–$60^\circ$13. The object-centric policy and forecasting models are trained only on size-$60^\circ$14 in simulation, then tested on sizes $60^\circ$15–$60^\circ$16 in simulation and sizes $60^\circ$17, $60^\circ$18, and $60^\circ$19 in reality.

5. Empirical behavior, robustness, and generalization

The two reported RIT systems expose different empirical regimes. The tactile ESC system operates at a low tactile feedback rate of $60^\circ$20 Hz and exhibits slow, roughly linear convergence, yielding multi-minute insertions. The wrench–nut system operates in a vision-based sim-to-real loop and is evaluated both on one-time insertions and on repeated cycles, where robustness is measured not only by single-attempt success but also by the ability to sustain many consecutive rounds (Burner et al., 2024, Liu et al., 9 Jul 2025).

For tactile key insertion, the headline results are $60^\circ$21 success over $60^\circ$22 deterministic single-parameter perturbation trials with mean insertion time $60^\circ$23 s, and $60^\circ$24 success over $60^\circ$25 randomly initialized trials with mean insertion time $60^\circ$26 s. Sensitivity summaries show higher success on smaller perturbations and longer times for larger ones, consistent with ESC capture ranges and the non-convexity of lock faces. Convex lock faces, especially the disc–detainer configuration L4, favor ESC, while small convex regions and flat faces reduce guidance and increase local minima.

For wrench–nut screwing in simulation, low-friction results at size-$60^\circ$27 show $60^\circ$28 success for the robot/world-frame IndustReal baseline and $60^\circ$29 for the Object-Centric policy without recovery, with steps reduced from $60^\circ$30 to $60^\circ$31. With recovery and forecasting, Moving-Window achieves up to $60^\circ$32 on size-$60^\circ$33, $60^\circ$34 on size-$60^\circ$35, $60^\circ$36 on size-$60^\circ$37, $60^\circ$38 on size-$60^\circ$39, and $60^\circ$40 on size-$60^\circ$41. Under high friction, IndustReal drops to $60^\circ$42 on size-$60^\circ$43, Object-Centric without recovery reaches $60^\circ$44, and Full-Trajectory achieves $60^\circ$45 on size-$60^\circ$46, $60^\circ$47 on size-$60^\circ$48, $60^\circ$49 on size-$60^\circ$50, $60^\circ$51 on size-$60^\circ$52, and $60^\circ$53 on size-$60^\circ$54.

Real-world single-time insertion results further sharpen the contrast among representations and recovery schemes. Over $60^\circ$55 trials per method and size, IndustReal achieves $60^\circ$56 success on size-$60^\circ$57, $60^\circ$58 on size-$60^\circ$59, and $60^\circ$60 on size-$60^\circ$61. Object-Centric without recovery achieves $60^\circ$62, $60^\circ$63, and $60^\circ$64 respectively. Time-Only recovery reaches $60^\circ$65 on size-$60^\circ$66 with $60^\circ$67 steps and $60^\circ$68 resets, $60^\circ$69 on size-$60^\circ$70 with $60^\circ$71 steps and $60^\circ$72 resets, and $60^\circ$73 on size-$60^\circ$74 with $60^\circ$75 steps and $60^\circ$76 resets. Full-Trajectory recovery reaches $60^\circ$77 on size-$60^\circ$78 with $60^\circ$79 steps and $60^\circ$80 resets, $60^\circ$81 on size-$60^\circ$82 with $60^\circ$83 steps and $60^\circ$84 resets, and $60^\circ$85 on size-$60^\circ$86 with $60^\circ$87 steps and $60^\circ$88 resets.

For long-horizon rhythmic insertion, the wrench–nut paper reports that recovery allows the system to sustain many more consecutive rounds, consistent with the geometric-distribution analysis

$60^\circ$89

For size-$60^\circ$90, $60^\circ$91 without recovery implies $60^\circ$92, while $60^\circ$93 with Full-Trajectory implies $60^\circ$94. The object-centric policy also generalizes zero-shot across unseen part sizes: trained only on size-$60^\circ$95, it is tested on sizes $60^\circ$96–$60^\circ$97 in simulation and on sizes $60^\circ$98, $60^\circ$99, and $\rightarrow$00 in reality.

Taken together, the empirical record suggests two distinct robustness routes within RIT. One route is model-free extremum seeking over tactile contact cues in a low-bandwidth but geometry-agnostic setting. The other is object-centric visuomotor control with explicit failure anticipation and reset in a repeated-cycle setting.

6. Limitations, misconceptions, and research directions

The tactile ESC formulation has several explicit limitations. Insertion is slow because low tactile sampling at $\rightarrow$01 Hz and first-order filtering limit gradient estimation speed and therefore the rate at which $\rightarrow$02 can be adapted. ESC is most effective near convex optima; raised rims, sharp edges, and flat faces cause local minima and wedging. Reported failure modes include getting stuck on edges, orientation errors—especially about the insertion axis for tubular locks—and exceeding strain limits. The approach also relies on high-quality tactile imagery and compliant pads, so damage to the pads affects performance (Burner et al., 2024).

The wrench–nut formulation has a different dependency structure. Accurate 6D pose tracking through FoundationPose and the RealSense D435 is central, and large tracking failures would degrade performance. Simulation contact and friction modeling can deviate from reality; friction-induced rotation is identified as a major failure mode. The Moving-Window forecaster depends on time alignment and control frequency and did not directly transfer because of real–sim frequency mismatch. The classifier avoids this issue but still relies on the same policy distribution. Recovery is intentionally simple—lift and retry—and may be insufficient for more complex contact-rich assemblies (Liu et al., 9 Jul 2025).

No baseline such as non-wiggling or fixed oscillation without ESC is reported for the tactile insertion system. The paper therefore frames its deterministic perturbation sweeps as a sensitivity study rather than a comparative ablation. In the wrench–nut system, the principal comparative axis is representation and recovery: robot/world-frame IndustReal versus object-centric policy, with and without failure forecasting and retry. A plausible implication is that RIT should not be evaluated solely by one-time insertion success, because the repeated-cycle regime exposes compounding-error phenomena that single-attempt benchmarks can hide.

Several research directions are explicitly suggested. For tactile ESC, promising directions include reshaping the objective for faster convergence, adding priors such as lockhole location or contact geometry to expand the convex region and avoid local minima, using impedance control to enhance compliance and improve strain sensing, and developing learning-informed ESC that learns task-specific periodic perturbations beyond pure sinusoids or augments the cost with learned contact features. For wrench–nut RIT, future work includes incorporating tactile sensing to detect seating and friction stick/slip, improving simulation contact models, adapting survival models to variable control frequency, and exploring end-to-end visuomotor policies once robust sim-to-real bridges are available.

More broadly, RIT should not be reduced to either tactile wiggling or vision-based repetition alone. The present literature shows two complementary formulations: continuous multi-DOF rhythmic excitation converted into a gradient estimate, and discrete rhythmic execution stabilized by object-centric state representation and failure-triggered recovery. This suggests that the unifying concept is the use of periodic structure to make insertion state observable and correctable under uncertainty.