Admittance Control

Admittance control implements an outer-loop controller that maps measured or estimated contact forces into a commanded motion. This is defined by a target mechanical admittance, which is the inverse of impedance. The core equation is:

Δx_desired = Admittance_Matrix * F_measured

Where Δx_desired is the resulting position or velocity adjustment and F_measured is the external force/torque vector. This creates a virtual mass-spring-damper system that the environment interacts with, making the robot inherently compliant.

By design, admittance control makes a robot inherently compliant and safe for physical human-robot interaction (pHRI). When an unexpected force is detected (e.g., a collision or a human push), the controller does not resist; it yields and moves according to the programmed admittance. This contrasts with stiff position controllers that fight disturbances, which can be dangerous. This characteristic is critical for:

• Collaborative robotics (cobots) working alongside humans.
• Assembly tasks requiring part insertion or alignment.
• Medical and rehabilitation robotics where patient safety is paramount.

The admittance controller generates a desired motion trajectory (position or velocity), which is then tracked by a fast, high-gain inner-loop controller. This inner loop is typically a standard PID or model-based position/velocity controller. The architecture is:

1. Sense: Measure external force/torque via a wrist-mounted F/T sensor or joint torque sensing.
2. Map: Compute desired motion adjustment using the admittance law.
3. Track: The inner loop drives the actuators to precisely achieve this new motion reference. This separation allows the system to combine the soft, compliant behavior of the outer loop with the precise tracking and disturbance rejection of the inner loop.

Admittance and impedance control are dual concepts but have distinct implementations:

• Admittance Control (Force-In, Motion-Out): Measures force, commands motion. Requires an accurate, high-bandwidth inner motion loop. Better suited for interactions with stiff environments (like a wall) because it directly regulates the motion response to force.
• Impedance Control (Motion-In, Force-Out): Commands a motion, but modulates torque to achieve a desired force/position relationship. Often implemented via torque-controlled actuators. Can become unstable in contact with very stiff environments.

In practice, admittance control is often implemented on traditional position-controlled industrial robots by adding a force sensor, while impedance control is native to torque-controlled robots like the KUKA LBR iiwa or many research platforms.

Performance is critically dependent on high-fidelity, low-noise force/torque (F/T) sensing. The controller's stability and responsiveness are tied to the sensor's:

• Bandwidth: Must be higher than the desired admittance control frequency.
• Signal-to-Noise Ratio: Noise directly injects spurious motion commands.
• Location: Typically a 6-axis F/T sensor mounted at the wrist, between the last joint and the end-effector, to measure all external contact forces. Alternatives include:
  • Joint torque sensing (e.g., via motor current or strain gauges).
  • External observers that estimate contact forces from motor currents and acceleration, though these are less direct.

A canonical industrial application demonstrating admittance control's strength. A rigid position controller would jam a peg into a hole's edge. With admittance control:

1. Upon initial misaligned contact, the F/T sensor detects lateral forces.
2. The admittance law translates these forces into a commanded lateral velocity for the end-effector.
3. The robot 'floats' along the edge of the hole until the forces drop (indicating alignment), then commands a downward motion to insert the peg. This force-guided search strategy elegantly solves a complex contact-rich task without explicit geometric models of the parts, showcasing the power of compliant, reactive behavior.

Admittance control is not a universal solution. Its performance is constrained by several factors, leading to hybrid implementations.

• Inherent Limitation: It is a motion-generating strategy. If the robot's actuators are saturated or hit a joint limit, it cannot generate the commanded compliant motion, potentially leading to high, uncontrolled contact forces.
• Stability in Hard Contact: Can become unstable during interactions with very stiff environments due to sensor noise and discrete-time delay in the force-to-motion loop.
• Hybrid Approach: Many modern systems use parallel impedance-admittance control or switch between strategies based on task phase. For example, use admittance for free motion and guided insertion, then switch to impedance for maintaining steady contact force.
• Requires Force Sensing: Depends entirely on accurate, low-latency force/torque measurement.

Impedance control is the conceptual dual to admittance control. Instead of mapping force to motion, it regulates the dynamic relationship between a robot's end-effector position and the resulting contact force. The controller makes the robot behave like a programmable mass-spring-damper system, defining a target mechanical impedance. This is ideal for tasks requiring consistent interaction stiffness, such as precision assembly or maintaining contact with a surface.

• Key Difference: Impedance control accepts a motion command and outputs a force. Admittance control accepts a force measurement and outputs a motion command.
• Implementation: Often implemented via torque control at the joint level.

Force control is a broader category of strategies where the primary control objective is to regulate the interaction force between a robot and its environment. Admittance control is one method to achieve this. Other methods include direct force feedback control, which uses a force/torque sensor to close a loop around the measured force error.

• Applications: Polishing, deburring, peg-in-hole insertion, and any task where maintaining a specific contact force is more critical than tracking a precise position trajectory.
• Challenge: Requires high-bandwidth force sensing and careful stability analysis due to environmental stiffness.

Series Elastic Actuation (SEA) is a hardware design paradigm that enables high-fidelity force control and is highly compatible with admittance control. A compliant elastic element (e.g., a spring) is placed in series between the motor and the output link.

• Benefits for Admittance Control: The spring provides intrinsic mechanical compliance, protects the motor from impacts, and allows for accurate force measurement via spring deflection. This simplifies implementing the desired force-to-motion relationship.
• Trade-off: Introduces additional dynamics (resonance) that must be managed by the controller.

Whole-Body Control (WBC) is a hierarchical control framework for complex robots like humanoids. It coordinates all degrees of freedom to execute multiple tasks (e.g., balance, foot placement, arm manipulation) simultaneously while respecting physical constraints like torque limits and contact forces.

• Integration with Admittance: An admittance control law can be formulated as one of the tasks within a WBC stack. For example, the primary task could be maintaining balance via centroidal dynamics, while a secondary, lower-priority task implements admittance for compliant arm interaction.
• Mathematical Foundation: Often solved as a Quadratic Program (QP) that optimizes joint accelerations or torques.

For legged robots, admittance control is critically applied at the feet to manage balance. The Ground Reaction Force (GRF) is the force vector from the ground. The Center of Pressure (CoP) is the point where the GRF's tangential moment is zero.

• Admittance for Balance: A legged robot can use an admittance controller to adjust its foot position or body posture based on the measured GRF or CoP error, providing compliant stability on uneven terrain.
• Dynamic Stability: By regulating the CoP within the support polygon, the robot can absorb pushes and adapt to slopes.

Hybrid force/position control is a strategy that explicitly decouples control directions into force-controlled and position-controlled subspaces. For example, when inserting a peg into a hole, position is controlled along the axis of the hole, while force is controlled in the radial directions to avoid jamming.

• Relation to Admittance: Admittance control can be seen as a unified, continuous approach to hybrid control. Instead of a binary switch, it defines a continuous admittance transfer function that blends force and position objectives across all directions based on the desired dynamic behavior.