Impedance control for assembly

Position-control a peg into a hole and one of two things happens: it slides in, or it jams and you crush the peg. Real assembly needs the arm to be soft along the insertion axis — to feel its way in. The Control track covered impedance generally; this lesson covers the manipulation-specific tricks that turn 70% peg-in-hole into 99%.

The classic peg-in-hole setup

Hole has clearance \delta (e.g., 0.1 mm). Peg approaches from above. Three error sources:

• Position error: peg is not exactly above the hole. Sub-millimeter at best from vision.
• Orientation error: peg axis not perfectly vertical. Sub-degree at best.
• Hole position uncertainty: the workpiece may have shifted.

Pure position control: command peg straight down. If position error > clearance, peg jams against the hole edge and the arm exerts huge force fighting it.

Impedance control: command peg straight down with low stiffness in x/y/yaw. Peg encounters the hole edge; the soft directions yield; the peg slides into alignment as it descends. Same task, no force.

Compliance directions per phase

Phase	Stiff axes	Compliant axes
Approach (free space)	All	None — track the path
Search (above hole)	Z (down with constant force)	X, Y (slide along surface)
Insertion	Z (force-controlled descent)	X, Y, roll, pitch (align to hole)
Seated check	All (verify position)	None

The classical insertion stiffness is ~50 N/m laterally — soft enough for a 0.5 mm misalignment to require only 25 mN of correction force, well below what would damage anything.

Search strategies

If the hole position is uncertain, the peg has to find it. Common search patterns:

• Spiral search: peg presses down with constant force; the controller commands a spiral pattern of small lateral motions in x-y. When the peg drops (z-acceleration spike), the hole was found.
• Lissajous search: orthogonal sinusoidal motions in x and y. Faster coverage of the search region.
• Tilt-and-slide: rotate the peg slightly, slide it along the surface. Catches edges that pure translation misses.
• Random walk: small random displacements with rejection on z-rise. Robust but slow.

The choice depends on uncertainty distribution. Tight uncertainty: small spiral. Wide: large Lissajous.

Force-event detection

The hole drop event (peg's z dropping suddenly) is detected by:

• Z-position discontinuity: if the commanded z-velocity is constant but the measured z drops faster, the peg fell into the hole.
• Force-direction change: while sliding on the surface, normal force is +Z. Inside the hole, normal force comes from the side walls.
• Joint torque pattern: distinctive pattern when the peg seats vs jams.

Production code combines all three. False-positive rate matters — a misdetected "found the hole" leads to push too hard and damage the part.

Why impedance beats pure motion

Naive approach: vision pose → motion plan → execute. Failure mode: vision is off by 0.5 mm; clearance is 0.1 mm; arm fights the hole edge with full positional stiffness. Result: jam, error, retry.

Impedance approach: vision gives an approximate pose; impedance lets the arm yield to mismatch; the peg slides into the hole even when vision was wrong. The control system absorbs the perception error.

This is the difference between a 70% pick-and-place pipeline and a 99% one. The impedance layer is doing the same thing a human's wrist does when fitting a bolt — soft enough to feel, stiff enough to push.

Combining with vision

Even with impedance, vision matters: the search region is bounded by perception accuracy. Too wide a search → minutes per insertion; too narrow → misses the hole. The 2026 production stack:

1. Vision: detect hole position with ~2 mm 95% confidence radius.
2. Approach: arm moves to "above hole" pose with high stiffness.
3. Search: spiral covering 4 mm radius (2× vision uncertainty).
4. Insertion: impedance for 5 mm of descent, then verify seating.
5. Tactile verification: F/T or fingertip sensor confirms expected force pattern.

Hardware needs

• Torque-controlled arm: Franka Panda, KUKA LBR iiwa, UR e-series with FT300, or any cobot with a wrist F/T sensor. Hobby servos can't do this.
• Wrist F/T sensor: 6-axis, 100+ Hz update rate. ATI, OptoForce, or recent open-source designs.
• Compliance from the gripper itself: a passive compliant gripper (ones with foam pads or springs) augments active impedance for free.

Learning-based approaches

2024+ research uses reinforcement learning or imitation learning to discover impedance + search strategies that outperform hand-designed ones. The classical recipe still works as a baseline; learned policies improve on it for specific tasks (oddly-shaped pegs, multi-stage assembly).

Key references:

• HIL-SERL — real-world RL with human-in-the-loop, hits 100% on contact-rich tasks in ~1 hour of demos.
• Diffusion policies for assembly — multimodal trajectories cover variation in part presentation.

Common production gotchas

• Sensor zero drift: F/T sensors drift over hours. Re-zero in free space before every assembly cycle.
• Mass compensation: the F/T sensor reads the weight of the gripper plus part. Subtract.
• Stiffness too high during search: arm thinks it found the hole at every bump. Lower lateral stiffness; raise force-event threshold.
• Insertion velocity too high: peg jams or chamfers itself. Slow descent (1–5 mm/s) is the production norm.

Exercise

In MuJoCo, simulate a peg-in-hole with 0.5 mm clearance, 5° initial alignment error, 2 mm position error. First try pure position control: watch the peg jam or miss. Then implement Cartesian impedance with low x/y stiffness. Same initial conditions: peg slides in. The before/after video is the most convincing demo of impedance control.

Next

Non-prehensile manipulation — moving objects without grasping. Pushing, sliding, flipping, and the math of robot billiards.