Chapter 3: Stabilize the Last Meter of Contact — Impedance, Admittance, and Force Control

Overview

A collision-free path and a time-parameterized trajectory from Chapter 2 do not complete insertion, wiping, or surface following. At first contact, a small pose error can become a large interaction wrench. This chapter designs the execution layer that owns that boundary. The central decision is not a controller label. It is a contract stating who receives which reference, computes on which clock and rate, observes which sensors and limits, and relinquishes authority under which failure.

Evidence-status notice: the current Chapter 3 packet contains no classical control source with an inspected exact-primary-body locator. The equations and workflow below are therefore explicit S12 engineering synthesis and teaching models, not universal performance claims or hardware-gain recommendations. Statements marked remain pending until a fact checker reopens the cited primary body. Numerical stability bounds, vendor modes, force limits, and abort behavior must come from the selected platform's current official documentation and measured tests.

After reading this chapter... - You can separate position, velocity, and torque authority from impedance, admittance, and force-loop responsibilities. - You can write a spring–damper interaction model in joint or operational space and audit units and frames. - You can narrow controller choices from actuation authority and force observability. - You can inject delay, filtering, saturation, and stiff contact into a simulation and retain evidence. - You can specify independent abort, watchdog, and fallback behavior for a reduced-energy trial.

The practical question is: after Chapter 2's free-space trajectory reaches the approach pose for a peg-like insertion, what downstream interfaces and validation gates let the robot recover alignment under lateral error and contact uncertainty without leaving the approved force envelope?

1. Draw a contact controller by boundaries, not names

In this mental model, a learned policy, planner, and operator are all bounded reference proposers. None owns torque or the safety state.

task / policy / planner
  pose·twist·wrench·stiffness proposal + stamp + validity + horizon
                         ↓
reference arbiter / feasibility projection
  frame·unit·age·workspace·joint·collision·force envelope
                         ↓
contact controller
  impedance | admittance | force | hybrid position-force
                         ↓
vendor real-time controller / drive
                         ↓
robot ↔ environment
  joint state · TCP pose/twist · wrench · tactile/contact status

independent supervisor ─ watchdog · energy/force/rate limits · retreat/stop

ros2_control or a vendor bridge may carry references and state, but do not assume a ROS 2 process owns every motor-servo cycle. The first implementation artifact is an owner/rate/deadline/failure matrix. Populate rate from controller configuration and timestamp traces, not from a preferred number in prose.

Each packet needs a frame_id, units, monotonic stamp, sequence, validity flag, mode epoch, and source owner. A wrench is ambiguous without the point and frame in which it is expressed. Save raw and filtered wrench streams together so filter delay remains observable.

2. Position, velocity, and torque are different authorities

A position controller accepts q_d or x_d; a velocity controller accepts a desired joint rate or Cartesian twist; a torque controller accepts \tau_d. The interface name does not reveal the full loop. A vendor may run a fast torque/current loop below a position interface. A torque interface may still leave gravity compensation, friction compensation, torque-rate limits, and protective functions inside the platform.

Before changing modes, answer five questions:

1. Which physical quantity may the user process command directly?
2. Where does the inner loop tracking that command execute?
3. Are measured joint torque and external wrench sensed or estimated?
4. Which process and model revision own gravity, payload, and tool compensation?
5. What output follows command timeout, mode switching, and a protective stop?

This chapter uses impedance control as a design view that establishes a target dynamic relation between motion error and interaction wrench, rather than mere pose-trajectory tracking. Hogan (1985) and Ott et al. (2008) are canonical lineage candidates for this definition, but the current ledger lacks exact body locators. Do not promote this sentence to a source-verified claim until primary reopening.

3. Impedance: design the relation from motion error to wrench

A one-axis target interaction can be written as a virtual mass–spring–damper:

\[

M_d(\ddot{x}-\ddot{x}_d)+D_d(\dot{x}-\dot{x}_d)+K_d(x-x_d)=F_{ext}.

\]

With e=x_d-x, a simplified form that does not explicitly realize virtual inertia proposes

\[

F_{cmd}=K_de+D_d\dot{e}+F_{ff}.

\]

In multi-axis operational space, x includes position and orientation, while K_d and D_d are matrices expressed in a named frame. If the robot exposes torque authority, one candidate mapping is

\[

\tau_{cmd}=J(q)^TF_{cmd}+\tau_{null}+\tau_{model}.

\]

Here J^T maps a task wrench toward joint torque. Null-space action and model compensation require their own limit, collision, and model-validity checks. This equation does not authorize implementation. First verify whether the robot exposes that authority, how often the Jacobian and payload model update, and what remains of the target relation after saturation.

An arithmetic-only 1D example

In a verification simulator, let K_d=200\,\mathrm{N/m}, D_d=20\,\mathrm{N\,s/m}, e=0.002\,\mathrm{m}, \dot{e}=-0.01\,\mathrm{m/s}, and F_{ff}=0. The proposed wrench is then $0.2\,\mathrm{N}$. This arithmetic only shows spring and damping terms reinforcing or opposing one another. It is not a recommended gain or safety value for any arm, tool, or environment. Hardware validation must include identified inertia, inner-loop behavior, sensor noise, sampled delay, environment stiffness, and platform limits.

Large free-space errors can produce large wrench proposals. Bound wrench and wrench rate, and make the reference continuous through contact transitions. If an integral term is present, prevent windup under saturation and prove reset or bumpless transfer when changing mode.

4. Admittance: convert measured wrench into a motion reference

An admittance loop feeds measured interaction wrench into virtual dynamics and computes a motion correction. A teaching model for one axis is

\[

M_a\ddot{x}_c+D_a\dot{x}_c+K_ax_c=F_{meas}-F_d.

\]

The integrated x_c or \dot{x}_c becomes an offset to the downstream position or velocity reference. This structure can be considered when torque-level authority is unavailable but a stable position/velocity interface exists. It creates another risk: sensor noise, bias, and filter delay are integrated into motion. Bound correction velocity, acceleration, displacement, workspace, and accumulated energy explicitly.

Do not declare impedance or admittance universally superior. Seraji (1994), Calanca et al. (2016), and Keemink et al. (2018) are candidates for the comparison that actuation, inner-loop authority, sensor location, commanded interface, and environment govern the choice. The ledger marks this comparison high-risk/unverified; reopen the chosen robot's primary sections and official manual before turning it into hardware guidance.

5. Operational-space force and hybrid position-force control

An operational-space controller reasons in task coordinates about motion and wrench. A geometric map alone is not enough: singularity, redundancy, null-space motion, joint limits, and model uncertainty stay visible. If the surface unit normal n is known, a synthesis may use S_f=nn^T for a force-controlled subspace and S_p=I-S_f for its motion-controlled complement:

\[

F_{task}=S_pF_{motion}+S_fF_{force}.

\]

The description of hybrid position-force control as a partition of task directions aligns with the canonical lineage of Raibert and Craig (1981) and Khatib (1987). The primary bodies still need reopening, including this chapter's implementation conclusion that a valid contact frame and explicit contact-enter/contact-exit transitions are required.

For peg-like insertion, position tracking may own approach, then axial force or compliant motion may take over after first contact while lateral and rotational directions remain available for alignment recovery. The harder problem is not the selection matrix but a wrong contact frame. A tilted normal estimate or a wrench expressed about the wrong point can push tangentially.

Separate the transition states:

APPROACH → CONTACT_CANDIDATE → CONTACT_CONFIRMED → ALIGN/INSERT → SEATED | RETREAT

Confirm contact from several advancing timestamps, expected motion reduction, wrench direction, and sensor health—not one threshold sample. Threshold and dwell values are platform-specific and intentionally absent here.

6. Treat delay, sampling, filtering, and saturation as dynamics

A plausible continuous-time equation still becomes sample-and-hold execution with transport delay, timestamp uncertainty, filter phase lag, quantization, and saturation. Stiffer environments turn small penetration into larger wrench, so these effects are not mere software plumbing.

One energy bookkeeping view compares supplied port energy

\[

E_{in}(t)=\int_0^t F(\tau)^Tv(\tau)\,d\tau

\]

with changes in controller or storage energy. Passivity means that a system does not generate arbitrary energy under a specific port model and initial condition. It does not mean collision-free or safety-certified. Any bound for a sampled controller must preserve source assumptions and be rederived or tested for the selected hardware.

Colgate and Schenkel (1994), Lawrence (1993), and Ryu et al. (2004) are primary candidates for the proposition that delay, sample period, filtering, saturation, and environment stiffness can change contact stability and transparency, but the ledger contains no exact locators. Here these variables are validation hypotheses that must be swept and logged, not a transferred numerical bound.

Failure symptoms and diagnosis order

Do not tune gains first. Diagnose in this order: clock/stamp → frame/unit → sensor health/bias → owner/mode → saturation/limit → delay/filter → model/contact → gain. Gain changes can hide errors in every earlier layer.

7. A minimal reproducible contact experiment

This is interface validation synthesized for S12, not benchmark reproduction.

7.1 Simulation fixture

• Use a 6/7-DoF arm, parallel gripper, rigid peg, and a slightly larger hole or planar fixture.
• Let Chapter 2's planner own motion only to the pre-contact approach pose.
• Permit exactly one controller to own the command interface during contact.
• Log simulator ground-truth contact wrench and the noisy/filtered wrench seen by the controller.
• Version controller configuration, robot model, timestep, contact parameters, seed, and code hash.

7.2 Promotion gates

1. No-contact replay: replay trajectory and timestamps with the controller inactive; frames, units, and clocks must match.
2. Free-space controller: verify reference tracking and saturation without a contact object.
3. Soft simulated contact: at low initial energy, inspect contact transition, wrench direction, and retreat.
4. Fault injection: inject stale wrench, frozen timestamps, filter delay, packet loss, wrong normal, saturation, and controller crash one at a time.
5. Parameter sweep: vary environment stiffness, delay, filtering, and initial offset as independent axes, not only controller gain.
6. Held-out regression: freeze configuration and repeat with unused initial offsets and seeds.

7.3 A worked insertion trace

When the planner supplies APPROACH, the arbiter checks stamp and frame and approves only a reference inside the workspace envelope. At CONTACT_CANDIDATE, it reduces axial motion and checks wrench direction. After CONTACT_CONFIRMED, it permits lateral compliance or alignment correction while bounding axial wrench and correction displacement. If wrench age misses its deadline or the normal component reverses unexpectedly, the system does not continue collecting success evidence; it enters RETREAT. Version the seating criterion as a combination of insertion depth, velocity, wrench pattern, and gripper/task state rather than one pose.

The important record is not simply “success.” Preserve per-phase references, realized pose/twist, raw and filtered wrench, energy bookkeeping, command saturation, sample age, watchdog transition, abort reason, and recovery outcome.

This simulation and the following hardware runbook are S12 synthesis with Todorov et al. (2012), Hogan (1985), and ros2_control documentation as reopening candidates. They validate an interface rather than reproduce a source benchmark. Simulator success does not establish real contact fidelity or hardware safety.

7.4 A tuning trace that exposes interface faults before gains

Do not begin a tuning session by searching for gains that “look good.” The first trace replays references, state, and wrench packets while controller output is disconnected from the robot. Expand every packet into frame, unit, monotonic timestamp, sequence, validity, and mode epoch. Reconstruct the same synthetic contact event on the proposal, sensor, controller, and supervisor clocks. If a wrench frame is ambiguous, timestamps move backward, or two processes appear to own the mode, gains are not yet experimental variables. Repair the contract, assign a new configuration revision, and repeat the trace from the beginning.

The second trace runs in free space with a small bounded reference change. Plot position error, proposed wrench or motion correction, saturation state, and realized motion on a shared time axis. Persistent clipping is not automatically a request for lower stiffness. Isolate whether the reference envelope, unit conversion, arbiter, controller, or downstream platform limit produced it. Replay the same input with the contact controller disabled, enabled, and enabled behind the arbiter's reduced envelope. This comparison assigns responsibility without treating any one layer as an unexplained black box.

Only the third trace introduces a simulated fixture. Enable one direction and one contact transition at a time. Hold controller parameters fixed while varying initial offset, an environment parameter, sensor filtering, and transport delay as separate axes. A quiet nominal case is not sufficient when a declared fault injection produces oscillation, repeated saturation, an expired sensor packet, or an energy-accounting anomaly. Retain the supervisor transition for every injection. If the energy trace cannot be explained from signs, frames, storage state, and reset behavior, label the passivity assessment UNRESOLVED; do not convert an attractive plot into a stability statement.

The final trace is a paired comparison that changes exactly one controller parameter. Keep model revision, seed, initial condition, filter, delay, reference, and abort policy fixed. Compare observed peak wrench, pose and twist error, saturation duration, sample age, retreat cause, and recovery outcome together. A change that improves tracking but consumes force margin or causes a later retreat is a task-dependent trade-off, not an unqualified better gain. Preserve rejected configurations and their NO-GO reason so a later engineer does not repeat them without context.

This sequence turns the tuning log into an input contract for the next run. Give every trace a configuration hash, start and end timestamps, mode transitions, the reason for each change, and a reviewer verdict. Do not save only a screenshot; retain the raw event log and script revision that regenerate it. The hardware gate accepts neither the prettiest trace nor the largest apparent margin. It accepts a configuration whose nominal, boundary, and fault cases are reproducible, whose owners and clocks are known, and whose unresolved fields are empty. If any required field remains unknown, the valid outcome is a simulation checkpoint, not a reduced-energy hardware trial.

8. Bounded hardware validation

Do not copy gains out of simulation. First inspect current official documentation for the selected robot's command modes, update behavior, force sensing, payload/tool model, command timeout, software limits, protective functions, and emergency procedure. Assign separate operator and safety-observer roles and verify the physical E-stop and retreat path.

1. With no motion authority, validate sensor bias, frames, and timestamps.
2. Without a fixture, test mode transition and watchdog inside reduced workspace, speed, and acceleration envelopes.
3. Use a compliant dummy fixture and low initial energy with one enabled axis.
4. Test wrench-sensor disconnect, stale command, controller crash, and limit approach beginning with methods that create no motion.
5. Approve one contact direction at a time while the observer retains abort authority.
6. Expand axis, offset, or object only after preserving PASS evidence and NO-GO causes.

Software clamping or a collision checker is not a safety-rated stop. This chapter does not set a force limit, safe speed, payload, or retreat trajectory. No unapproved value belongs in the hardware configuration.

9. Contact authority remains when learned policies arrive

A learned policy may propose pose, twist, wrench, or stiffness references, while explicit downstream components retain contact stabilization and abort limits. That is the S12 architecture contract. Force-aware policy candidates such as DexForce and Reactive Diffusion have unresolved exact method and experiment locators, so this chapter makes no capability claim from them. Terry's DexForce notes are reader cross-links only: Korean and English.

A policy packet needs at least reference_type, frame, stamp, horizon, validity, an uncertainty representation, the limits assumed during training, and calibration/schema versions. The arbiter rejects stale chunks, commands in unknown frames, infeasible poses, abrupt stiffness changes, and force-envelope violations. Rejection leads to an approved HOLD/RETREAT/FALLBACK, not repetition of the last action.

A bounded Codex implementation prompt

Goal
  Implement a simulation-only contact-controller harness for the tabletop insertion task.

Context
  The Chapter 2 planner ends at a pre-contact pose. The robot/controller model,
  frames, clocks, command interfaces, and logger schema are supplied as versioned files.

Constraints
  Do not command hardware. Do not invent vendor modes, gains, force limits, or safety ratings.
  Keep policy/planner references separate from contact-loop and supervisor authority.
  Log raw/filtered wrench, proposal/realized command, stamps, mode epoch, saturation,
  watchdog state, and abort reason. Permit exactly one command owner.

Done when
  No-contact, free-space, soft-contact, delay/filter/saturation, stale-sensor,
  wrong-frame, controller-crash, and retreat tests are reproducible from a pinned config;
  every test has PASS/NO-GO evidence and a machine-readable event trace.

Safety
  Simulation only. Hardware backend must remain disabled and fail closed.
  Any missing frame, clock, limit, sensor health, or owner field is a hard error.

10. Artifact and metric checklist, then the handoff to data

Required artifacts

• Controller owner/rate/deadline/failure matrix
• Robot, tool, and contact-frame revisions plus transform record
• Raw/filtered sensor schema and clock-synchronization note
• Versioned impedance/admittance/hybrid configuration
• Saturation, anti-windup, mode-transition, and watchdog test logs
• Simulation model/contact parameters, seed, and code hash
• Fault-injection matrix and state-transition traces
• Bounded-hardware checklist with approver and NO-GO record
• Unresolved-primary list for exact locators ch03-c01 through ch03-c06

Metrics to report

• Per-phase sample age, jitter, gaps, and missed cycles
• Difference between proposed and realized reference plus saturation duration
• Observed pose/twist/wrench error traces and peak/settling measurements
• Contact/retreat transitions, abort/intervention count, and reason
• Energy-bookkeeping trace and controller internal state
• Sensor bias/drift, frame/calibration epoch, and data loss

These are not numbers for ranking different robots, fixtures, or simulators. They are regression evidence that preserves the task, controller, model, initial condition, and abort policy.

The interface handed to Chapter 4 is this log schema. A teleoperation demonstration cannot retain only cameras and actions. Contact phase, raw/filtered wrench, controller mode, proposed and realized action, saturation, watchdog, intervention, retreat, and calibration epoch need one reconstructable timeline. A VLA or world model in #S13 must obey the same lower interface. It may propose a task or skill reference, but torque and safety authority remain with independent controllers and supervisors.

Limitations and pending verification

• Classical contact-control sources have canonical identities but no exact-primary-body locators in the current packet.
• Vendor-specific impedance, admittance, and force modes, supported rates, limits, and abort behavior remain unresolved.
• A simple spring–damper and rigid-contact simulation omits gearbox dynamics, flex, friction, tactile richness, and contact geometry.
• Passivity is a property of a defined port and model; it does not automatically establish task success or human safety.
• Learned force-aware methods are interface candidates, not evidence of performance or hardware readiness in this chapter.

References

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24. Wenzhen Yuan et al. (2017). GelSight: High-Resolution Robot Tactile Sensors for Estimating Geometry and Force. Annotated primary reading. DOI: 10.3390/s17122762. — Role: primary reading for controller assumptions in contact, compliance, and passivity. Limitation: Bulky form factor (camera + optics) limits hand integration
25. Mike Lambeta et al. (2020). DIGIT: A Novel Design for a Low-Cost Compact High-Resolution Tactile Sensor with Application to In-Hand Manipulation. Annotated primary reading. arXiv:2005.14679. — Role: primary reading for controller assumptions in contact, compliance, and passivity. Limitation: Still bulky relative to a human fingerpad; mounting on small fingers difficult
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27. Taylor Howell et al. (2022). Predictive Sampling: Real-time Behaviour Synthesis with MuJoCo. Annotated primary reading. arXiv:2212.00541. — Role: primary reading for controller assumptions in contact, compliance, and passivity. Limitation: Simulation throughput or benchmark success does not establish real contact fidelity or hardware safety.
28. Shaoxiong Wang et al. (2022). TACTO: A Fast, Flexible, and Open-Source Simulator for High-Resolution Vision-Based Tactile Sensors. Annotated primary reading. arXiv:2012.08456. — Role: primary reading for controller assumptions in contact, compliance, and passivity. Limitation: Approximate shading; no true elastomer mechanics simulation
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30. Michael Posa, Cecilia Cantu & Russ Tedrake (2014). Planning Through Contact: A Unifying Approach to Manipulation. Annotated primary reading. DOI: 10.1177/0278364913506757. — Role: primary evidence for the modeling boundary of optimizing contact modes and dynamics together. Limitation: contact assumptions and evaluation conditions are source-specific and must be reopened before transfer to current hardware.
31. Jee-Hwan Ryu, Dong-Soo Kwon & Blake Hannaford (2004). Stable Haptic Interaction with Human and Virtual Environment Using a Passivity Observer and Controller. Annotated primary reading. DOI: 10.1109/TRA.2004.824689. — Role: primary evidence for adding energy monitoring through a passivity observer/controller in a contact loop. Limitation: delay, sampling, saturation, filtering, environment stiffness, and hardware interfaces bound the reported stability.