Impedance control

Overview

Impedance control shapes how a robot responds to contact. Rather than only commanding a position, the controller gives the end effector a virtual spring and damper. A push from the environment then produces a force that follows from the chosen stiffness and damping 2610.05695 §2.2. Each card uses this idea differently. 2610.05695 §2.2 frames its reaching design as classical PD control in end-effector space. 2412.03619 §II-A uses a Cartesian mass-damper-spring law with an estimated robot model, so no force sensor is needed. 2412.13474 §5.1 uses a spring only as the execution layer of a planned trajectory, with stiffness set to zero when the robot should yield. 2509.17053 §4.1 runs a joint-space spring and damper at 2 kHz beneath a learned policy that updates its targets at 25 Hz.

The controller in 2610.05695 is a virtual model controller. Each spring-damper force is mapped to joint torques through the transpose of the Jacobian at its attachment point, and no inverse kinematics is used 2610.05695 eq. 1 2610.05695 §3. The card lists both impedance-control and virtual-model-control. The variants below are spring-damper designs, and the same mapping underlies the other cards except the joint-space one, which works on joint errors directly.

Variants

Spring-damper reaching in end-effector space

A spring of stiffness k and a damper of damping c join the end effector to a desired position. The force is k x + c \dot{x}, with x the error from end effector to target 2610.05695 eq. 2. Orientation uses rotational springs and dampers in the same way 2610.05695 §2.2. On hardware the paper uses 15 and 45 N/m stiffness with 1.5 Ns/m damping 2610.05695 §4.1. Card: Virtual model control for compliant reaching under uncertainties.

Moving reference

The target moves on a straight line from start to goal. A scalar ramps from zero to one in finite time, and the spring law is applied to the moving target 2610.05695 §2.2. The paper says this limits the force needed over long distances 2610.05695 §2.2. The card's experiments use 0.1 and 0.5 m/s reference speeds 2610.05695 §4.1.

Force-driven reaching

A virtual cart with mass and viscous friction is attached to the end effector through a spring and damper. A constant force pulls the cart along the path. If the end effector is blocked, the cart stops too, so the robot force is bounded by that pull 2610.05695 §2.2. Cart mass and friction are not stated, and the card reports no test result for this design.

Repulsive obstacle springs

Each obstacle connects to the end effector through a nonlinear spring and damper. Stiffness and damping grow as the separation shrinks, and forces from several obstacles add up 2610.05695 §2.3. The card gives one piecewise stiffness law, with a 0 N/m stiffness beyond 20 cm, that the paper uses in its experiments 2610.05695 §4.2. The damping law is not stated.

Model-based task-space impedance with a disturbance observer

The joint torque has two parts. A feedforward term uses estimated inertia, Coriolis and gravity terms to follow the desired motion. A feedback term applies the Cartesian spring and damper, mapped by the Jacobian transpose 2412.03619 eq. 3 2412.03619 eq. 7. For the rehabilitation robots the stiffness is 30 I on the master and 20 I on the second robot, and the damping is 2 times the square root of that stiffness, written 2√30 I and 2√20 I 2412.03619 tab. 2. A nonlinear disturbance observer lumps model error, friction and external forces into one term and cancels it 2412.03619 eq. 9 2412.03619 §II-C. The authors adapt the observer from an earlier design and give no proof in the card. Card: A Teleoperation System with Impedance Control and Disturbance Observer for Robot-Assisted Rehabilitation.

Bilateral teleoperation with a virtual spring on the master

The leader ("master") robot renders a force to the operator. That force is a virtual spring, mapped through the Jacobian transpose, driven by the position difference between the master and the second robot's end effector. No force sensor is used, and the second robot's position is sent back over UDP 2412.03619 eq. 8 2412.03619 §II-B. The card does not state the force-feedback stiffness 2412.03619 §II-B.

Free physical interaction mode with damping only

In free human interaction, the desired point is set to the current end-effector position. The torque then reduces to damping plus gravity compensation, so a person can move the arm freely 2412.03619 eq. 5 2412.03619 §II-A. The paper turns the disturbance observer off on the master in this mode 2412.03619 §V. A backdrivable haptic device does not need the damping term, which the paper notes 2412.03619 §II-A.

Trajectory tracked by a high-stiffness spring, with hand-over

A planner predicts the human's goal from the first 0.2 s of motion and outputs a Cartesian trajectory. The robot starts with zero translational stiffness. The trajectory is sent as the rest position of an impedance controller with stiffness [0, 0, 400] N/m and run without replanning 2412.13474 §5.1. In the handover task, the robot keeps 400 N/m until a predicted transition point and then sets it to 0 2412.13474 §5.2. The rotational stiffness is "kept high" but its value is not stated 2412.13474 §5.1. Card: Planning Human-Robot Co-manipulation with Human Motor Control Objectives and Multi-component Reaching Strategies.

Joint-space impedance under a learned policy, with estimated force

A policy follows the action chunking design of ACT and outputs 6-DOF end-effector targets at 25 Hz 2509.17053 §4.1. Each target is converted by real-time inverse kinematics to joint targets \hat q, whose method the paper does not state. A joint-space law then runs at 2 kHz 2509.17053 eq. 5 2509.17053 §4.1:

\tau_{cmd} = B(\hat v - v) + K(\hat q - q) + \hat\tau 2509.17053 eq. 5

The target velocity \hat v is zero, \hat\tau is a gravity and Coriolis feedforward refreshed at 250 Hz, and the gains K and B are fixed presets whose values are not stated 2509.17053 §4.1. The paper leaves variable impedance to future work 2509.17053 §6. The law has no inertia term, because the tasks are treated as quasi-static 2509.17053 §4.1.

Eq. 5 contains no force term. The estimated wrench reaches the arm only through the policy's targets, and the policy takes the wrench as an input. The wrench is the difference between measured joint torques and the torque a digital twin needs at zero acceleration, mapped through a Jacobian pseudo-inverse 2509.17053 eq. 6 2509.17053 §4.2. The paper gives no damping for the pseudo-inverse, so the estimate can grow near singularities 2509.17053 §4.2. Card: FILIC: Dual-Loop Force-Guided Imitation Learning with Impedance Torque Control for Contact-Rich Manipulation Tasks.

When to use which

Implementation sketch

The sketch follows the spring-damper core of the card's equations. The damping sign assumes a static target, which is our reading of eq. 2. The paper's notation for a moving target is unclear (see Contradictions). The model-based variant adds feedforward terms with estimated inertia, Coriolis and gravity 2412.03619 eq. 3, and the card's own pseudocode gives that form.

# One impedance step for a single end-effector spring-damper  [eq. 2]
# Input:  q, dq (joint state), x_d (target), k, c (gains)
# Output: tau (joint torques)

def impedance_step(q, dq, x_d, k, c):
    # Pose and Jacobian at the current joint state  [§3]
    x_ee, J = forward_kinematics(q)
    # End-effector velocity in task space  [§3]
    dx_ee = J @ dq
    # Error from end effector to target, and its rate  [eq. 2]
    err = x_d - x_ee
    d_err = -dx_ee  # static target only
    # Virtual spring-damper force  [eq. 2]
    f = k * err + c * d_err
    # Map the force to joint torques  [eq. 1]
    tau = J.T @ f
    return tau

Converting torque to motor current is a hardware step. The paper uses a linear approximation from the motor's performance graph 2610.05695 §3. It is not included here.

The joint-space variant 2509.17053 eq. 5 drops the Jacobian. Compute the torque as damping times the negative joint velocity, plus stiffness times the joint error to the IK target, plus the feedforward term. The IK step that produces the target is not stated in the paper, so it needs its own choice.

Contradictions

The cards disagree on sign conventions, on how the target is reached, and on gains. Each side is cited below, and neither is resolved here.

• The damping term for a moving target is ambiguous. eq. 2 uses the error rate, and the text does not say whether the target's velocity is included 2610.05695 §2.2.
• The card attributes the 5 cm error of the moving reference to low stiffness, as in the previous case. That previous case used 15 N/m, while the moving reference run used 45 N/m 2610.05695 §4.1.
• The stiffness and damping notation for obstacles is inconsistent between §2.3 and the text under it, and the unit of distance in the piecewise law is inferred to be centimetres 2610.05695 §4.2.
• Within the teleoperation card, the error terms have opposite signs in its impedance law and its control law. eq. 2 writes the errors as (ẋ − ẋd) and (x − xd), while eq. 3 writes them as (ẋd − ẋ) and (xd − x). The text does not reconcile them, and the card's pseudocode follows eq. 3 2412.03619 eq. 2 2412.03619 eq. 3.
• Across cards, the stiffness values are not comparable. 2610.05695 §4.1 uses 15 and 45 N/m on a humanoid arm, 2412.03619 tab. 2 uses 20 and 30 on two planar rehabilitation robots, and 2412.13474 §5.1 uses 400 N/m on a Franka arm. 2509.17053 §4.1 gives no gain values at all, and its law acts on joint errors rather than Cartesian ones. The tasks and robots differ, so the cards do not show which stiffness is better.
• The reaching card uses no inverse kinematics 2610.05695 §3. The FILIC card converts each policy target to joint targets with real-time inverse kinematics 2509.17053 §4.1. The two designs reach a goal in different ways, and neither card compares them.
• The teleoperation card's force-feedback gain is named but not valued 2412.03619 eq. 8, so its rendered force ranges cannot be reproduced from the card alone 2412.03619 §IV-D.

Papers

• Virtual model control for compliant reaching under uncertainties: spring-damper virtual models mapped to joint torques for reaching, moving-reference and obstacle designs, with a contact test on a humanoid arm.
• A Teleoperation System with Impedance Control and Disturbance Observer for Robot-Assisted Rehabilitation: bilateral teleoperation of two planar rehabilitation robots with impedance control and a nonlinear disturbance observer, with free and recorded motion modes; qualitative results.
• Planning Human-Robot Co-manipulation with Human Motor Control Objectives and Multi-component Reaching Strategies: a planner that predicts the human goal and hands over authority at a predicted point by setting an impedance stiffness to zero; one operator, qualitative results.
• FILIC: Dual-Loop Force-Guided Imitation Learning with Impedance Torque Control for Contact-Rich Manipulation Tasks: a 25 Hz learned policy over a 2 kHz joint-space impedance loop, with force estimated from joint torques and a digital twin; author-reported success gains, gains not stated.