A Unified Fractal Impedance Control Framework

Adnan Takriti; Mahmoud Sadeq; Quentin Rouxel; Daniele Cafolla; Michael Mistry; Carlo Tiseo

cs.RONov 5, 2026

A Unified Fractal Impedance Control Framework

Authors: Adnan Takriti, Mahmoud Sadeq, Quentin Rouxel, Daniele Cafolla, Michael Mistry, Carlo Tiseo

Organizations: School of Engineering and Informatics, University of Sussex, Brighton, UK · Department of Mechanical and Automation Engineering, Chinese University of Hong Kong, HK · Department of Computer Science, Swansea University, Swansea, UK · School of Informatics, University of Edinburgh, Edinburgh, UK

Abstract

Physical human--robot interaction requires controllers capable of maintaining intrinsically bounded interactions with the environment. The Fractal Impedance Controller (FIC) has been proven to guarantee these safety characteristics by design. This paper introduces the robotic implementation and experimental validation of its most advanced architecture: the Harmonic-Passive Motion Paradigm (H-PMP), an evolution of the Passive Motion Paradigm designed to mimic human-like interaction. Furthermore, we present a unified framework for FIC deployment, supported by an open-source software suite featuring ROS2, MuJoCo, and physical hardware integrations. Through three experiments conducted on a Kinova Gen3 arm (trajectory tracking, a wall press with sudden loss of contact, and learning-by-demonstration) we compare the H-PMP against earlier FIC architectures. The results demonstrate that the H-PMP achieves a seamless trade-off between admittance and impedance behaviours. This makes it ideal for deployment in unstructured environments, enabling the robot to transition between interaction modes with no computational overhead.

Figures & tables

Panel of Figure 1 — Fig. 1: The experiments on the physical arm (Section V ). Left: the lemniscate of the trajectory tracking experiment (orange), drawn about the arm in its MuJoCo model, posed from a recorded H-PMP trial; the part behind the gripper is shown lighter. Middle: the 7-DOF Kinova Gen3 with its gripper closed, facing the wall plate on its quick-release mount (wall press and release). No force sensor is fitted: the controllers estimate the external wrench from the joint torques. Right: the operator teaches the arm a path by guiding it at the wrist (learning by demonstration).

Panel of Figure 2 — Fig. 2: The FIC Suite. FIC Studio acts on the robot only through ROS2 topics: commands (mode, targets, profile selection, parameter updates) and states (joint and end-effector state, wrench, controller internals). The robot node, fic_control , runs the task-space layer of the FIC library at $1\text{\,}\mathrm{kHz}$ and sends each tick to the joint inner loop, fic_inner_loop , which steps with MuJoCo in simulation or exchanges torques with the vendor driver through shared memory on the robot.

Panel of Figure 3 — Fig. 3: FIC Studio’s web interface (Kinova Gen3, simulation target). Top: the dashboard, with the robot in 3D and live plots of the end-effector position, the joint angles and the joint efforts. Bottom: the tuning panel where the user can update the controller effort profiles.

#	Name	Space
1	Impedance	Joint
2	Impedance with planner	Joint
3	Admittance	Joint
4	HapFIC	Joint
5	Impedance	Cartesian
6	Impedance with planner	Cartesian

TABLE I: FIC Studio Control Modes

Panel of Figure 5 — Fig. 4: The H-PMP control architecture comprises a task-space admittance controller and a joint-space HapFIC. Task-Space : The system can implement either a traditional admittance behaviour or a HapFIC by altering how the admittance modifies $x^{\mathrm{des}}$ to yield $x^{\mathrm{adm}}$ . This output is then passed to the Cartesian planner (Plan.) and a Task-Space FIC (FIC TS ), which consists of a 3D Cartesian controller and a quaternion controller. The resulting output wrench is transformed into the desired joint torques by pre-multiplying it by the Jacobian transpose ( $J^{T}$ ). Joint-Space : The torque error $\tau_{\mathrm{des}}-\tau+g(q)$ is processed by the dead-band ( $\mathrm{dz}^{h}$ ) described in Eq. ( 7 ) before being sent to the joint admittance block. The HapFIC algorithm (the joint-space HapFIC of Table I ) within the joint admittance generates an offset $\delta$ . This offset is added to the IK solution $q^{\mathrm{IK}}$ —which is computed using $x^{\mathrm{adm}}$ as its input—to determine the admittance-modified joint angles $q^{\mathrm{adm}}$ . These angles are subsequently sent as input to the Joint FICs (FIC JS ) to compute the FIC control torques $\tau^{J}$ . Finally, the gravity vector $g(q)$ is added to $\tau^{J}$ to generate the final actuator control torques $\tau_{c}$ .

Fig. 5: Trajectory tracking on the physical arm (

A=

0.10\text{,}\mathrm{m}$$ ,

20\text{\,}\mathrm{s}

period, one cycle per trial from the home pose,

n=10

per controller): commanded path (dotted), every trial, the mean path and the

90\text{\,}\mathrm{\%}

tube, in the colour of each controller. Each controller is shown in 3D with the horizontal plane below it: Cartesian impedance with planner and Cartesian impedance with joint-space HapFIC (top two rows), Task-Space HapFIC and H-PMP (bottom two rows).

Fig. 6: Trajectory tracking on the physical arm over time (the trials of Fig. 5 ): the

x

(red),

y

(green) and

z

(blue) position of every trial from the centre

c

of the lemniscate, commanded dotted, with the

5\text{\,}\mathrm{s}

hold at the centre after the cycle.

Controller	MAE	RMSE	Peak	Ori.	$x$ gain
	(mm)	(mm)	(mm)	( ∘ )
Cartesian impedance with planner	41.4	43.19 [42.65, 43.72]	69.4 [69.0, 69.8]	1.94	0.69
Cartesian impedance with joint-space HapFIC	11.8	12.56 [11.81, 13.65]	20.9 [20.3, 22.2]	1.41	0.94
Task-Space HapFIC	20.3	21.11 [21.00, 21.22]	32.2 [31.8, 32.4]	3.65	0.97
H-PMP	7.3	7.80 [7.06, 9.10]	16.9 [15.5, 18.8]	3.33	0.99

TABLE II: Trajectory Tracking on the Arm: Mean [Range of the Ten Trials]

Controller	$n$	$F_{\mathrm{hold}}$	shift	past
		(N)	(mm)	(mm)
Cart. imp. with joint-space HapFIC	2	16.9	0.0	68.7
Task-Space HapFIC	1	4.6	$-$ 50.0	36.3
H-PMP	2	12.6	$-$ 50.0	23.4

TABLE III: Wall Press and Release on the Arm, Expected Maximum Force

20\text{\,}\mathrm{N}

: Mean [Range]

Fig. 7: Wall press and release on the physical arm at an expected maximum force of

20\text{\,}\mathrm{N}

(first press of each controller; H-PMP and Cartesian impedance with joint-space HapFIC with the contact profile). Top: the distance from the wall (zero at the touch, negative past the face; the wall face dotted) and the commanded target

x^{\mathrm{des}}

(dash-dot), which ramps

100\text{\,}\mathrm{mm}

past the face and is then held; bottom: the contact force estimated from the joint torques, with the expected maximum force (dashed). The vertical dashed lines mark each controller’s release, in its colour.

Fig. 8: Learning by demonstration on the physical arm (one teaching per setup, by hand; bottom: the H-PMP with its joint-space HapFIC on the MCU): the taught path (wide, light), the replay (solid) and the replay with the hand push (dashed) in 3D, over the taught movement.

Fig. 9: Learning by demonstration on the physical arm, about the hand push of Fig. 8 . Top:

d

, the end-effector’s distance from the unpushed replay at the same replay time; bottom:

\lVert F\rVert

, the force on the end-effector during the pushed replay, estimated from the joint torques (

50\text{\,}\mathrm{ms}

mean); both against time from the start of the push (

t=0

). The push, located in the recorded force estimate, is shaded in the colour of its setup.

	Task-Space HapFIC	H-PMP	H-PMP (MCU)
Teaching speed, mean ( $\mathrm{mm}\text{\,}{\mathrm{s}}^{-1}$ )	150	144	75
Replay RMSE over the movement (mm)	55.7	40.7	42.4
Replay error, peak (mm)	119.0	92.8	87.6
Peak perturbation force $F_{\mathrm{p}}$ (N)	27.8	11.3	51.3
Peak deviation $d_{\max}$ (mm)	132	67	167
Residual deviation $d_{\mathrm{r}}$ (mm)	77	4.6	85

TABLE IV: Learning by Demonstration on the Arm (One Teaching per Setup)

Impedance Control

Impedance control lets robots adjust their stiffness and damping to manage forces during contact with people and objects.

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Jun 6, 2026Yongyan Cao, Jinshan TangModel Predictive Control Human-Robot Interaction
Voryx Robotics LLC, San Jose, CA 95136, USA · Department of Health Administration and Policy, George Mason University, Fairfax, VA, USA
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Multimodal Interaction Estimation for Contact-Aware Adaptive Impedance Control
Oct 16, 2026Mateusz Jaszczuk, Paris Perdikaris, Nadia FigueroaImpedance Control Contact Force Estimation
Department of Mechanical Engineering, University of Pennsylvania

Jun 6, 2026J/K move · Enter open · S save

Toward Interaction Dynamics: A Predictive Framework for Safe Physical Human Robot Interaction

Yongyan Cao, Jinshan Tang

Voryx Robotics LLC, San Jose, CA 95136, USA · Department of Health Administration and Policy, George Mason University, Fairfax, VA, USA

Model Predictive Control Human-Robot Interaction

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Physical human-robot interaction requires yielding transiently to contact yet recovering the commanded reference under sustained load. Finite-stiffness impedance control retains a static deflection there, while predictive alternatives typically optimize a nonlinear robot or impedance model online. Operational-space cancellation instead exposes a translational error double integrator with a fixed transition matrix and a configuration-scheduled input map, making interaction a predictive quantity rather than a property re-derived per configuration. We build on it a compact offset-free interaction-error MPC for torque-controlled manipulators: a force-domain random-walk state estimates persistent interaction and model error, and a 30-variable convex QP maps the correction through the current task inertia while constraining the applied joint torque. Conditional results establish impedance equivalence of the unconstrained passive feedback, offset-free regulation at feasible frozen configurations, and quadratic stabilizability of the scheduled backbone. In a 1kHz MuJoCo simulation of a 7-DOF Franka FR3, the estimator cuts steady-state error under a repeated 15N step from 2.77mm to 0.042mm when added to the otherwise identical 100Hz MPC. A stiffness-and-damping-calibrated impedance baseline attains 2.59mm but briefly saturates and needs 3.3x the peak positive joint power. Adding ideal measured-force cancellation to that baseline gives 1.39mm, so constant-load rejection is not unique to MPC; the sensorless controller still reaches 0.042mm in the moving task, a 65x reduction without force sensing and without the baseline's saturation or power cost. Demonstrated in simulation under a shared actuator budget, the contribution is an efficient operational-space realization complementing rather than replacing broader interaction-control architectures.

Abstract

Impedance Control

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