Force / Impedance Control

Deep Dive

Force / Impedance Control is a family of control strategies that regulate the interaction between a robot and its environment rather than just its position. This matters because pure position control becomes dangerous the moment a robot touches something: it will fight any obstacle to reach its target, generating large forces from tiny errors.

Tip

Position control asks “where should the arm be?” Impedance control asks “how should the arm feel when pushed?” — it turns the robot into a programmable spring-damper whose stiffness and softness you choose.

The problem with stiff position control

A position controller tries to hit a target pose no matter what. During contact, a 1 mm position error against a rigid surface can demand enormous force, because the controller assumes free space. That is fine for moving through air, but unsafe for assembly (peg-in-hole jams), polishing (the tool gouges), or working near people (a stiff arm does not yield).

Impedance control law

Instead of commanding a rigid position, impedance control commands a force that depends on how far and how fast the robot deviates from a desired motion:

F = K·(x_d − x) + D·(ẋ_d − ẋ)

Naming every symbol in words first:

• x_d — desired position (where you want the end effector).
• x — actual position (where it is).
• ẋ_d, ẋ — desired and actual velocity.
• K — virtual stiffness: how hard the robot springs back toward x_d. Low K = soft/compliant; high K = stiff.
• D — virtual damping: how much the robot resists velocity, suppressing bounce.

The robot behaves like a tunable mass-spring-damper attached to its target.

Impedance vs admittance

• Impedance: measure motion, command force. Best for backdrivable, torque-controlled robots (e.g. Franka, many cobots).
• Admittance: measure force, command motion. Best for stiff, position-controlled robots — a force sensor decides how far to move.

Mapping task forces to joints — the Jacobian

A desired task-space force F becomes joint torques through the transpose of the Jacobian:

τ = Jᵀ·F

Hybrid force/position control

Often you want force in some directions and position in others. Pressing a sponge along a table: control downward force to a setpoint while controlling position as you sweep along the surface.

   POSITION CONTROL (stiff)        IMPEDANCE CONTROL (springy)

   target ●                        target ●
          │  rigid link                   /\/\/\  virtual spring+damper
          │                                │
   ext →  ●  ← arm refuses to move  ext →  ●  ← arm yields, then pushes back
   force  ▲     (huge contact force) force ╲      (gentle, bounded force)
          │  fights the push              softly settles to equilibrium

Prerequisites

PID Control The feedback loop that tracks errors — impedance gains play a role analogous to P (stiffness) and D (damping).

Jacobian Maps task-space forces to joint torques via τ = Jᵀ·F.

Worked example: 1-D impedance

A gripper is pushed 2 cm past its target. We compute the restoring force.

• Desired position x_d = 0.10 m
• Actual position x = 0.12 m (pushed 2 cm beyond target)
• Stiffness K = 500 N/m
• Damping D = 20 N·s/m
• Actual velocity ẋ = 0.05 m/s, desired ẋ_d = 0

F = K·(x_d − x) + D·(ẋ_d − ẋ)
  = 500·(0.10 − 0.12) + 20·(0 − 0.05)
  = 500·(−0.02)       + 20·(−0.05)
  = −10               + (−1)
  = −11 N

The negative sign means the force points back toward the target (−x direction): the robot springs back gently with 11 N, not the unbounded force a stiff position controller would apply.

def impedance_force(x_d, x, xdot_d, xdot, K, D):
    return K * (x_d - x) + D * (xdot_d - xdot)

F = impedance_force(0.10, 0.12, 0.0, 0.05, K=500, D=20)
print(F)  # -11.0  (N, pushing back toward x_d)

Caution

Many robot-learning policies output target positions. A compliant low-level controller underneath is what keeps execution safe when the policy is wrong — without it, a confident bad prediction becomes a hard collision.

Related Terms

Robot Dynamics Mass, inertia, and torques — the physics impedance control shapes.

End Effector The contact point where interaction forces are felt and regulated.

Jacobian The bridge between task-space forces and joint torques.

Model Predictive Control An alternative that optimizes contact-rich motion over a horizon.

Sources

• Hogan, N. “Impedance Control: An Approach to Manipulation, Parts I–III.” ASME Journal of Dynamic Systems, Measurement, and Control, 1985. (Foundational impedance control paper.)
• Siciliano, B., Sciavicco, L., Villani, L., Oriolo, G. Robotics: Modelling, Planning and Control, Springer, 2009 — Ch. 9, Force Control.
• Raibert, M. & Craig, J. “Hybrid Position/Force Control of Manipulators.” ASME Journal of Dynamic Systems, Measurement, and Control, 1981.
• Franka Emika / libfranka Cartesian impedance control example, documentation accessed 2026-06-01.