Understanding Admittance Control in Robotics

Introduction

While impedance control focuses on how a robot resists external forces, admittance control does the opposite — it defines how a robot yields to them.

Admittance control is essential for tasks that require compliance, adaptability, and safe human–robot interaction. Instead of commanding forces, the controller generates motion (position or velocity) in response to applied forces.

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What Is Admittance Control?

Admittance control regulates a robot’s motion based on measured external forces. Unlike impedance control, which outputs forces based on motion, admittance control outputs motion based on force.

💡 Analogy:
Imagine the robot’s end-effector as a balloon tied to a point. When you push on it, it moves smoothly and naturally. That behavior is admittance control.

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How Admittance Control Works

• The robot measures external forces acting on it (via force/torque sensors or motor torque estimation).
• A virtual dynamic model (mass–spring–damper) computes the desired motion.
• The robot’s low-level controller tracks that motion using position, velocity, or torque control.

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Full Admittance Control Model

The most general form of admittance control is defined by a virtual mass–spring–damper system:

Mx¨+Bx˙+Kx=Fext

Where:

• $x$ — position 
• $\dot{x}$ — velocity
• $\ddot{x}$ — acceleration
• $F_{ext}$ — external force
• $M$ — virtual inertia
• $B$ — virtual damping
• $K$ — virtual stiffness

This equation describes how the robot should move in response to applied forces.

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Simplified (Quasi-Static) Admittance Control

In many real-world robotic systems—especially collaborative or low-speed applications—the virtual inertia $M$ can be neglected. This results in a simplified, quasi-static admittance model:

Bx˙+Kx=Fext

This approximation is valid when:

• Motions are slow
• The inner position/velocity control loop is stiff
• High-frequency dynamics are not critical

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Velocity-Based Admittance Control

Solving the simplified model for velocity gives:

x˙desired=(1/B)(Fext−Kx)

Where:

• $\dot{x}_{desired}$ — commanded velocity
• $F_{ext}$ — measured external force
• $K$ — virtual stiffness
• $B$ — virtual damping

Tuning Guidelines

• Increase B → slower, more damped motion
• Increase K → stiffer, less compliant response

This form is commonly used when the robot tracks velocity commands.

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Position-Based Admittance Control

For systems with a strong position controller, desired position can be computed directly:

xdesired=(1/K)Fext−(B/K)x˙

This formulation expresses compliance explicitly and includes damping to smooth the response.

It is best suited for:

• Human-guided motion
• Teaching modes
• Low-speed interaction tasks

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Compliance: The Core Concept

Compliance defines how easily a robot moves in response to force:

C=1/K

• High compliance → large motion from small force
• Low compliance → stiff, resistant behavior

Admittance control directly adjusts compliance through the virtual stiffness parameter.

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Real-World Example

🫱 Human–Robot Collaboration

A robot arm holding a tray can use admittance control to adapt its position as a person places or removes objects. As the external force changes, the robot automatically moves to maintain balance while remaining compliant and safe.

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When to Use Admittance Control

Admittance control is ideal for tasks that require motion adaptation and safety:

• Human–robot collaboration
• Object handovers
• Surface following and contour tracking
• Soft or uncertain assembly
• Teaching-by-demonstration

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Benefits of Admittance Control

✅ High compliance and smooth interaction
✅ Safer behavior in human environments
✅ Natural response to unpredictable forces
✅ Intuitive tuning via stiffness and damping

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