Admittance Control in Robotics

• Admittance control is a model-based methodology that translates external forces into motion using a virtual mass–damper–spring system.
• It underpins applications in teleoperation, surgical robotics, and human–robot collaboration by enabling compliant, adaptive interactions.
• Robust performance relies on advanced sensor fusion, rigorous parameter tuning, and constraint handling to maintain stability and safety.

Admittance control is a model-based interaction control methodology in which an external force signal modulates the motion of a mechanical system following a prescribed dynamic law, typically a virtual mass–damper–spring system. Within the control loop, admittance control transforms measured forces into motion commands (velocity, position, or acceleration), enabling compliant robotic behavior in environments that demand adaptability, safety, and precision. This concept is foundational for advanced human–robot interaction, teleoperation, manipulation, and medical robotics, providing robust force-to-motion coupling and facilitating rich forms of physical collaboration.

1. Mathematical Foundations and Core Formulation

Admittance controllers impose a virtual dynamic relationship between the measured interaction force $F_{\mathrm{ext}}(t)$ and the commanded robot motion $x(t)$, generally following:

$$M_d \, \ddot{x}(t) + B_d \, \dot{x}(t) + K_d \, x(t) = F_{\mathrm{ext}}(t) \tag{1}$$

where $M_d$, $B_d$, $K_d$ are the virtual inertia, damping, and stiffness matrices, respectively, and $x(t)$ denotes the output motion variable (typically Cartesian position or joint coordinates) (Nasiri et al., 17 Jul 2024).

Specialized versions adjust order and structure. For example, velocity-admittance laws drop the inertia ($M_d = 0$), yielding:

$$\dot{x}(t) = K_{\mathrm{adm}} (I - \Omega) f_{e} \tag{2}$$

Here, $K_{\mathrm{adm}}$ is a scalar or matrix gain, $(I-\Omega)$ projects forces into subspaces orthogonal to task constraints (e.g., tool shaft directions in surgical teleoperation) (Nasiri et al., 17 Jul 2024, Nasiri et al., 5 Apr 2024). This facilitates safe, responsive, and constraint-respecting motion, crucial when tool shafts must pivot through a remote center (RCM).

State-space and discrete-time representations are prevalent for digital implementation, allowing robust integration, filtering, and adaptation:

$$p[k+1] = p[k] + T_s \cdot K_{\mathrm{adm}\perp} \cdot f_{e}[k]$$

Admittance models are orthogonal to impedance control, which conversely regulates force output in response to commanded motion.

2. Sensory Integration, Constraint Handling, and Task Mapping

Modern implementations feature hybrid sensor fusion and constraint-based task allocation. Integrated force/torque sensors (F/T) at the robot base or tool provide six-axis wrenches, which—after frame transformation, bias and gravity compensation—yield interaction estimates at operational constraints (e.g., trocar port in laparoscopic surgery) (Nasiri et al., 17 Jul 2024, Nasiri et al., 5 Apr 2024).

Constraint handling commonly uses augmented Jacobians, null-space projections, and real-time redundancy resolution:

• The instrument’s desired twist and RCM velocity are stacked into an augmented command, $\xi_{aug}$.
• The full system Jacobian, $J_{total}$, encodes all kinematic relations.
• Inverse kinematic solvers use minimum-norm and null-space motions to solve $J_{total}\dot{q}_{aug} = \xi_{aug}$, maintaining fixed constraints and operational flexibility.

A projection matrix $\Omega$ defined by the instrument axis decouples compliance normal to the insertion direction, critical for minimizing lateral tissue force (Nasiri et al., 17 Jul 2024, Nasiri et al., 5 Apr 2024).

3. Parameter Selection, Adaptation, and Stability

Effective admittance hinges on expertly chosen dynamic parameters:

• Admittance gain ($K_{\mathrm{adm}}$): Initial values, typically of $\sim 0.1$–$0.25$ (m/s)/N, grant gentle compliance and are incrementally increased for responsiveness but must be capped to avoid excitation of high-frequency or unmodeled robot dynamics.
• Bandwidth constraints: The product $K_{\mathrm{adm}}\|\ I−\Omega\|$ and sensor bandwidth must not excite closed-loop instabilities; recommended bandwidths for force sensing and motion control are 500 Hz and 1 kHz, respectively (Nasiri et al., 17 Jul 2024).
• Null-space weighting and redundancy parameters: Parameters such as $\lambda$ (RCM interpolation) are held near nominal values via null-space strategies to avoid deviation from constraint pivots.

Parameter adaptation schemes, including energy-tank-based passivity conditions (Landi et al., 2017), evolve admittance matrices in response to detected deviations using residuals such as $$\psi(t) = \|F_{\mathrm{ext}} - M_d \ddot{x} - D_d \dot{x}\|$$, blending mass and damping over adaptation windows to preclude loss of passivity. Stability is assured if instantaneous passivity ($\dot{M}_d - 2D_d \preceq 0$) or sufficient energy storage is maintained; updating the inertia/damping ratio preserves interaction “feel.”

4. Safety, Compliance, and Constraint Projection

Safety features are integrated via control barrier functions (CBFs), saturation, and workspace monitoring:

• Exponential CBFs augmented with QP filters modify external forces when interaction would drive the system into unsafe regions, solving for minimally adjusted safe forces that enforce set invariance (Sun et al., 2022).
• Saturation strategies cap reference positions to prevent robot-environment contact forces exceeding designer-specified maxima, employing environment stiffness estimates for fine tuning (Pagliara et al., 11 Apr 2024).
• Switched admittance models enforce safety at workspace boundaries by automatically stiffening mass/damping/stiffness when the virtual trajectory approaches limits (Paul et al., 2022).
• Barrier potentials encoded as artificial spatial repulsion enforce strict avoidance of sensitive regions, maintaining passivity even under aggressive human manipulation (Kastritsi et al., 2022).

These mechanisms guarantee strict position or force constraints, with underlying theory and simulation confirming zero violation of forbidden regions during complex collaborative or teleoperated tasks.

5. Experimental Performance Across Applications

Empirical results from teleoperated MIS platforms, collaborative manipulation, contact-rich industrial tasks, and haptic teleoperation validate the admittance framework:

• Robot-assisted MIS: Circular tracking (radius 100 mm) achieves RMS positional errors of 1.8 mm; thread-passing tasks yield 100% success across 20 trials; rise time to lateral force steps is 40–50 ms without overshoot for a $K_{\mathrm{adm}}=0.25$ (m/s)/N (Nasiri et al., 17 Jul 2024).
• Physical human–robot interaction: Adaptive admittance suppresses oscillations within 0.3–0.4 s post deviation, never losing passivity (Landi et al., 2017). CBF-QP compensators maintain safe workspace boundaries under variable human interaction (Sun et al., 2022).
• Teleoperation and haptics: Decoupled force rendering via motion-error-based feedback yields natural, accurate, and safe handwriting in virtual environments; saturation protects fragile tools and improves accuracy, with mean force discrepancies reduced to 1.4 N (Pagliara et al., 11 Apr 2024).
• Contact-rich manipulation: Combining admittance with iterative learning attains >98% RMSE reduction over fixed-gain control across diverse tasks (button pressing, knob twisting) (Zhou et al., 25 Mar 2024).
• Human–robot co-carrying: Admittance-generated conflict-aware trajectory references, subsequent time-varying PVFC torques, and fractional energy compensation minimize human interaction force (mean ~0.25 N) and assure finite-time kinetic energy convergence (Trong et al., 31 Jul 2024).

6. Advanced and Adaptive Variants

Recent research extends admittance control with sophisticated adaptation, learning, and discrete-time robustification:

• Iterative Learning Control: Admittance gains are refined trial-to-trial via error feedback and pseudo-inverse stochastic gradient descents, delivering plug-and-play generalizability (Zhou et al., 25 Mar 2024).
• Fixed-time Integral Sliding Mode: Enforces global fixed-time convergence even under uncertainties, combining backstepping and non-singular surfaces to suppress chatter and guarantee compliance within theoretical time bounds (Sun et al., 2022).
• Set-valued and implicit Euler discrete-time schemes: Address impact-contact and actuator saturation, ensuring finite-time force control without chattering via differential-algebraic inclusion, multi-variable super-twisting algorithms, and projection-based torque limiting (Li et al., 28 Sep 2024).
• Mass-adaptive admittance: Integrates online payload estimation and virtual excitation force compensation for robotic manipulators, mitigating end-effector sag and maintaining stability under unknown mass conditions (Gholampour et al., 22 Apr 2025).
• Asymmetric stiffness: Allows extended compliance shaping via non-conservative “curl” components, with root-locus analytic conditions ensuring stability (Tsuji et al., 2023).

7. Practical Guidelines and Implementation Considerations

Design and tuning recommendations, system architecture, and real-time filtering are well-documented across domains:

• Gain tuning must balance compliance (low stiffness) against task accuracy and prevent excitation of unmodeled dynamics; critical damping relationships $b=2\sqrt{mk}$ are preferred for oscillation-free behavior.
• Projection matrices $(I-\Omega)$ are universally required for axial decoupling (RCM, shaft-constrained tools, floating bases).
• Sampling rates of 1 kHz for controller loops and force-sensor bandwidths of 500 Hz or greater are recommended for stability and transparency.
• Low-pass filtering of F/T data (cutoff $\sim$200 Hz) and tight integration with redundant kinematic chains and null-space regulation are necessary for constraint satisfaction in high-DOF systems (Nasiri et al., 17 Jul 2024, Nasiri et al., 5 Apr 2024).
• Safety is maintained via dynamic CBFs, QP force compensation, and energy-based passivity adaptation, with saturation and barrier schemes providing hard caps on physical interactions.

• "Teleoperation in Robot-assisted MIS with Adaptive RCM via Admittance Control" (Nasiri et al., 17 Jul 2024)
• "Adaptive Admittance Control for Safety-Critical Physical Human Robot Collaboration" (Sun et al., 2022)
• "Safe haptic teleoperations of admittance controlled robots with virtualization of the force feedback" (Pagliara et al., 11 Apr 2024)
• "Admittance Control for Adaptive Remote Center of Motion in Robotic Laparoscopic Surgery" (Nasiri et al., 5 Apr 2024)
• "Admittance Control Parameter Adaptation for Physical Human-Robot Interaction" (Landi et al., 2017)
• "Safe Human Robot-Interaction using Switched Model Reference Admittance Control" (Paul et al., 2022)
• "Stability analysis of admittance control using asymmetric stiffness matrix" (Tsuji et al., 2023)
• "A passive admittance controller to enforce Remote Center of Motion and Tool Spatial constraints with application in hands-on surgical procedures" (Kastritsi et al., 2022)
• "Fixed-time Integral Sliding Mode Control for Admittance Control of a Robot Manipulator" (Sun et al., 2022)
• "Active Admittance Control with Iterative Learning for General-Purpose Contact-Rich Manipulation" (Zhou et al., 25 Mar 2024)
• "A Passivity-based Nonlinear Admittance Control with Application to Powered Upper-limb Control under Unknown Environmental Interactions" (Kim et al., 2019)
• "Collaborative Robot Arm Inserting Nasopharyngeal Swabs with Admittance Control" (Lee et al., 21 Aug 2024)
• "Implicit Euler Discrete-Time Set-Valued Admittance Control for Impact-Contact Force Control" (Li et al., 28 Sep 2024)
• "Mass-Adaptive Admittance Control for Robotic Manipulators" (Gholampour et al., 22 Apr 2025)
• "Admittance Control-based Floating Base Reaction Mitigation for Limbed Climbing Robots" (Imai et al., 20 Sep 2024)
• "A Cooperation Control Framework Based on Admittance Control and Time-varying Passive Velocity Field Control for Human--Robot Co-carrying Tasks" (Trong et al., 31 Jul 2024)