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Force Control

• Force Control
  • Prerequisites
  • General Motivation
  • Course Content
    • Interaction control overview
      • Mathematical Foundation
    • Active interaction control
      • Indirect Force Control
        • Mechanical Impedance - the shared foundation
        • Impedance control
        • Admittance control
      • Direct Force Control
        • Hybrid Force/Motion Control
    • Programming exercise
      • Setup your environment
      • Exercise 1: Baseline Passive Behavior
    • Summary exercise
    • Do you want to implement a real project ?
  • Credits
  • Ressources

Prerequisites

To be able to fully appreciate content of this lesson, it is expected that one has a solid understanding of:

• Linear Algebra and multivariable calculus
• Classical mechanics

It is also expected that one has followed all lessons from [Chapter 1]:Basics of Motion Control(https://ras-university.github.io/robotic-courses/docs/chap1_basic_motion_ctrl/), and in particular, is familiar with the following concepts:

• Control theory:
  • First- and second-order system response
  • Proportional, integral and derivative control
  • Transfer functions and feedback
• Basic Robotics
  • Joint-space vs. task-space
  • Kinematics and dynamics

General Motivation

_{Northwestern Robotics (2018) Modern Robotics, Chapter 11.1: Control System Overview. YouTube video, 16 March. Available at: https://www.youtube.com/watch?v=mGuDXlZEoSc}

_{Lynch, K.M. and Park, F.C. (2017) Modern Robotics: Mechanics, Planning, and Control. Cambridge: Cambridge University Press.}

In motion control problems, the robot’s objective is to follow a predefined trajectory as accurately as possible regardless of contact with the environment. This is suitable for free-space movements where external forces are negligible or undesirable. While the premise of motion control might be basic in nature, it is a fundamental part of any higher-level robot manipulation.

However, motion control alone is not sufficient when a robot physically interacts with its environment. Indeed unregulated contact can cause slippage, loss of contact, damage and excessive force. This is where force control becomes essential: it ensures that the robot applies and regulates the desired amount of force during contact, making the interaction both safe and effective. A force control strategy modifies the robot’s joint positions or torques to account for interaction forces at the end-effector.

This is why, in the past decade, research in robot force control has increased significantly, with applications across medical, industrial and service robotics. In industrial robotics, typical tasks that involves robot interaction with an environment such as polishing, cutting, scraping, pick-and-place, welding, etc. can benefit from the use of different methods of force control strategies. Another particularly impactful area is physical human-robot interaction, where the robot must respond to human-applied forces and work in synchrony. In these scenarios, force control enables safe, adaptive and cooperative behavior in shared tasks.

Course Content

Interaction control overview

While motion control focuses on following a desired trajectory regardless of external contact, force control aims to regulate how much force is exchanged between the robot and its environment. This raises a fundamental question: how does the robot respond to forces during contact? There are two broad paradigms for addressing this:

• Passive interaction control: The trajectory of the end-effector is driven by the interaction forces due to the inherent nature or compliance of the robot (i.e., internally, such as joints, servo, joints, etc.). In passive control, the end-effector’s motion naturally deflects under force, as in soft robots. But, this lacks flexibility (every specific task might require a special end-effector to be designed and it can also have position and orientation deviations)​ and high contact forces could occur because there is no force measurement.

• Active interaction control: It relies on sensors (e.g., force/torque sensors) and/or feedback controllers to measure interaction forces and adjust the robot’s commands accordingly—whether by modifying its trajectory or the way it manipulates objects. This approach enables real-time reactions to contact, offering high flexibility and accuracy. However, it comes with added complexity and limitations in speed. To achieve effective task execution and robust disturbance rejection, active control is typically combined with some degree of passive compliance. Active strategies can be futer divided into indirect methods (such as admittance and impedance control) and direct force control techniques (such as hybrid force/motion control).

Mathematical Foundation

Note: The following builds upon the equation derived in the Dynamics chapter of this course.

$$ \boldsymbol{\tau} = \boldsymbol{M}(\boldsymbol{\theta}) \ddot{\boldsymbol{\theta}} + \boldsymbol{C}(\boldsymbol{\theta}, \dot{\boldsymbol{\theta}}) \dot{\boldsymbol{\theta}} + \boldsymbol{g}(\boldsymbol{\theta}) + \boldsymbol{J}^T(\boldsymbol{\theta}) \boldsymbol{F}_{\text{ee}} $$

Ensure that the meaning of each term is clearly understood before engaging in this section.

_{Northwestern Robotics (2018) Modern Robotics, Chapter 11.5: Force Control. YouTube video, 16 March. Available at: https://www.youtube.com/watch?v=M1U629sREiY}

_{Lynch, K.M. and Park, F.C. (2017) Modern Robotics: Mechanics, Planning, and Control. Cambridge: Cambridge University Press.}

At a quasi-static level (i.e., for slow or stationary motions $\ddot{\boldsymbol{\theta}} = 0$ and $\dot{\boldsymbol{\theta}} = 0$), the joint torques $\boldsymbol{\tau}$ are related to the end-effector contact force $\boldsymbol{F_{\text{tip}}}$ by the following fundamental equation:

$$ \boldsymbol{\tau} = \boldsymbol{g}(\boldsymbol{\theta}) + \boldsymbol{J}^T(\boldsymbol{\theta}) \boldsymbol{F_{\text{tip}}} $$

where:

• $\boldsymbol{\tau} \in \mathbb{R}^m$ is the vector of joint torques (units: $\mathrm{Nm}$), where $m$ is the number of robot joints.
• $\boldsymbol{g}(\boldsymbol{\theta}) \in \mathbb{R}^m$ is the vector of joint torques required to counteract gravity at configuration $\boldsymbol{\theta}$ (units: $\mathrm{Nm}$).
• $\boldsymbol{J}(\boldsymbol{\theta}) \in \mathbb{R}^{n \times m}$ is the manipulator Jacobian, mapping joint velocities (units: $\mathrm{rad/s}$) to end-effector velocities (units: $\mathrm{m/s}$ and/or $\mathrm{rad/s}$), where:
• $\boldsymbol{F_{\text{tip}}} \in \mathbb{R}^n$ is the force and moment (wrench) applied at the end-effector (units: $\mathrm{N}$ for forces, $\mathrm{Nm}$ for moments).

Note that through the whole course:

• $n$ is the number of task-space dimensions (e.g., $n=6$ in 3D Cartesian space, $n=2$ in planar tasks or $n=3$ if the rotation around the $z$-axis is considered)
• $m$ is the number of robot joints.

This relationship holds for both types of interaction control, but the way it is used differs:

• In passive interaction control, $\boldsymbol{F_{\text{tip}}}$ is imposed by the environment, and the resulting motion depends on the robot’s intrinsic compliance.
• In active interaction control, this equation becomes a control law: we specify a desired contact force $\boldsymbol{F_d}$ and compute the torque needed to apply it:

$$ \boldsymbol{\tau} = \tilde{\boldsymbol{g}}(\boldsymbol{\theta}) + \boldsymbol{J}^T(\boldsymbol{\theta}) \boldsymbol{F_d} $$

where:

• $\tilde{\boldsymbol{g}}(\boldsymbol{\theta}) \in \mathbb{R}^n$ is a model-based estimate of gravitational torques.

This corresponds to open-loop force control: it assumes the robot’s model and the environment are accurate. However, because models are never perfect, this method is sensitive to disturbances and uncertainties.

To improve robustness, feedback is introduced — typically using a PI (Proportional-Integral) controller:

$$ \boldsymbol{\tau} = \tilde{\boldsymbol{g}}(\boldsymbol{\theta}) + \boldsymbol{J}^T(\boldsymbol{\theta}) \left( \boldsymbol{F_d} + \boldsymbol{K_p} \boldsymbol{F_e} + \boldsymbol{K_i} \int \boldsymbol{F_e}\, dt \right) $$

where:

• $\boldsymbol{K_p} \in \mathbb{R}^{n\times n}$ is the proportional gain matrix
• $\boldsymbol{K_i} \in \mathbb{R}^{n\times n}$ is the integral gain matrix
• $\boldsymbol{F_e} = \boldsymbol{F_d} - \boldsymbol{F_{\text{tip}}} \in \mathbb{R}^n$ is the force error (units: $\mathrm{N}$ for forces, $\mathrm{Nm}$ for moments).

This feedback formulation allows the robot to adjust its torques based on measured force errors, improving stability and accuracy even as the environment changes.

Active interaction control

Active interaction control strategies can be grouped into two categories: those performing indirect force control and those performing direct force control. The main difference is that indirect methods regulate interaction forces through motion behaviors without explicitly closing a force feedback loop whereas direct force control explicitly commands and tracks contact forces through sensor-based feedback.

Indirect Force Control

Indirect force control strategies achieve force regulation by modulating the robot’s motion response rather than directly commanding forces. The robot behaves like a virtual mechanical system (typically a mass-spring-damper) so that when it is pushed, it responds with motion that generates restoring forces, just like a compliant structure would.

In this approach, we are not controlling nor tracking the force directly. Instead, we are controlling how the robot moves when it encounters a force, which in turn determines the contact force. This behavior is what enables impedance and admittance control.

Imagine pressing on the end-effector of a robot. With only knowledge of the robot’s own parameters - i.e. not the properties of the environment, we can design a controller so that the robot responds as if you were pushing against a virtual spring-mass-damper system, not because it’s tracking a specific force, but because its motion is governed by a virtual dynamic behavior that generates restoring forces naturally. Rather than rigidly following a trajectory, the controller modulates the system’s virtual dynamics (e.g., stiffness, damping) and lets the compliant behavior manage contact.

What does it really mean for a robot to “behave like a spring-mass-damper” ?

_{Gesta, A. (2022) POLAR SkillShare 4 – Impedance Control for Robotic Rehabilitation. YouTube video, 29 July. Available at: https://www.youtube.com/watch?v=Vz5c3il0Dys}

This short video by Polytech Montréal gives a concise and intuitive introduction to these concepts. It explores the dynamic relationship between force and motion, and shows how this relationship — known as mechanical impedance — underlies both impedance and admittance control. If you’re unfamiliar with these terms, watch this first — it sets the stage for what follows.

Mechanical Impedance - the shared foundation

At the core of both impedance and admittance control lies the idea of mechanical impedance. It describes how a physical system resists motion when subjected to a force. Unlike static stiffness (which links force to displacement), impedance captures dynamic behavior, i.e. how velocity (or acceleration) influences the force a system generates or absorbs.

Mathematically, impedance $\boldsymbol{Z}$ defines a dynamic relationship between the force $\boldsymbol{F}$ and the velocity $\boldsymbol{v}$:

$$ \boldsymbol{F} = \boldsymbol{Z} \, \boldsymbol{v} $$

Depending on the desired dynamic behavior, $\boldsymbol{Z}(s)$ may represent different physical components:

• Inertial response (mass): $\boldsymbol{F} = \boldsymbol{M} \boldsymbol{a}$

• Elastic response (Hooke’s law): $\boldsymbol{F} = \boldsymbol{K} \boldsymbol{x}$

• Damping behavior: $\boldsymbol{F} = -\boldsymbol{B} \boldsymbol{v}$

In practice, robots are modeled as a combination of these components.

What’s important is that we’re not just controlling position or force directly, we’re shaping how the robot behaves when interacting with the environment. By making the robot behave like a particular mechanical system (e.g., a stiff spring, a soft damper, or a heavy object), we get force behavior for free as a result of motion.

From this shared foundation, two distinct approaches emerge:

• In impedance control, we specify the mechanical impedance and generate force in response to motion.
• In admittance control, we specify the inverse behavior — the mechanical admittance — and generate motion in response to force.

Impedance control

Impedance control is a strategy where the robot is made to replicate the behaviour of a mechanical system, usually a combination of mass, spring, and damper, designed to resist motion when subjected to a force. We do not directly command the contact force, but rather define how the robot should respond when force is applied to it by dynamically adjusting the virtual inertia, damping and stiffness of the robot

This desired behavior is formalized by the following second-order differential equation:

$$ \boldsymbol{M_d} \ddot{\boldsymbol{x}} + \boldsymbol{B_d} \dot{\boldsymbol{x}} + \boldsymbol{K_d}(\boldsymbol{x} - \boldsymbol{x_r}) = \boldsymbol{f_{\text{ext}}} $$

where:

• $\boldsymbol{x} \in \mathbb{R}^n$ is the end-effector position (units: $\mathrm{m}$),
• $\boldsymbol{x_r} \in \mathbb{R}^n$ is the reference trajectory (units: $\mathrm{m}$),
• $\boldsymbol{M_d} \in \mathbb{R}^{n \times n}$ is the desired virtual mass matrix (units: $\mathrm{kg}$),
• $\boldsymbol{B_d} \in \mathbb{R}^{n \times n}$ is the desired virtual damping matrix (units: $\mathrm{Ns/m}$),
• $\boldsymbol{K_d} \in \mathbb{R}^{n \times n}$ is the desired virtual stiffness matrix (units: $\mathrm{N/m}$),
• $\boldsymbol{f_{\text{ext}}} \in \mathbb{R}^n$ is the external force exerted by the environment (units: $\mathrm{N}$).

This equation describes how the robot should react to forces. For example:

• A high stiffness ($\boldsymbol{K_d}$) makes the robot stiffer, resisting deviations from $\boldsymbol{x_r}$.
• A high damping ($\boldsymbol{B_d}$) smooths out motion and reduces vibrations.
• A high inertia ($\boldsymbol{M_d}$) makes the robot less sensitive to sudden external disturbances.

The system doesn’t track force, the force arises naturally through this interaction law. Hence, by controlling the impedance of the robot we can manipulate the behavior of the end-effector depending on the interaction with the environment. For example, we can set very low impedance (or high compliance) and make it act like a very loose spring which is useful in physical human-robot interaction or we can set the stiffness very high where the robot would only move to the desired position without much oscillation which is important for precise industrial tasks.

Stiffness control : a subset of impedance control

Through impedance control, it is possible to achieve a desired dynamic behaviour. A subset of this task is to achieve a static behaviour. Instead of specifying how the robot responds dynamically (with mass, damping, and stiffness), we only define a static relationship between the deviation in position/orientation and the force exerted on the environment. This is done by acting on the elements of the stiffness K in the impedance model while ignoring inertial and damping terms → Stiffness control

The control law focuses on maintaining a desired position while allowing some compliance, without needing explicit force sensing.

• High stiffness $\boldsymbol{K_d}$ → accurate position tracking, but less compliant
• Low stiffness $\boldsymbol{K_d}$ → more compliant, but allows larger motion deviations under external forces.

This trade-off allows us to limit contact forces and moments, even without a force/torque sensor — simply by choosing the right stiffness. However, in the presence of disturbances (e.g. joint friction), using too low a stiffness may cause the end-effector to deviate significantly from the desired position, even without external contact.

_{ARMLab CU-ICAR (2014) Planar Cable Robot with Variable Stiffness. YouTube video, 15 September. Available at https://www.youtube.com/watch?v=pXH7rwrzh6s}

_{Zhou, X., Jun, S. and Krovi, V. (2014) ‘Planar Cable Robot with Variable Stiffness’, Proceedings of the 2014 International Symposium on Experimental Robotics, Marrakech/Essaouira, Morocco, 15–18 June.}

This video illustrates the concept of stiffness control. Observe how a system with high stiffness tracks its desired position more accurately and reacts faster to disturbances, while a compliant (low-stiffness) system shows larger deviations upon collision with an external object. The graph highlights how stiffness influences both the speed and magnitude of the response, an essential trade-off in designing compliant behaviors without direct force sensing.

But how is impedance control actually implemented in a robot? How do we compute the joint torques τ that produce a response matching this desired impedance?

To implement impedance control, we must first express the robot’s true dynamics in task space, where the desired mechanical behavior is defined.

• Joint-space dynamics:

$$ \boldsymbol{\tau} = \boldsymbol{M}(\boldsymbol{\theta}) \ddot{\boldsymbol{\theta}} + \boldsymbol{C}(\boldsymbol{\theta}, \dot{\boldsymbol{\theta}}) \dot{\boldsymbol{\theta}} + \boldsymbol{g}(\boldsymbol{\theta}) + \boldsymbol{J}^T(\boldsymbol{\theta}) \boldsymbol{f}_{\text{ee}} $$

• Task-space dynamics: We can express the robot’s motion in task space using the kinematic relationships:

• $\boldsymbol{\dot{x}} = \boldsymbol{J}(\boldsymbol{\theta}) \dot{\boldsymbol{\theta}}$
• $\boldsymbol{\ddot{x}} = \boldsymbol{J}(\boldsymbol{\theta}) \ddot{\boldsymbol{\theta}} + \dot{\boldsymbol{J}}(\boldsymbol{\theta}, \dot{\boldsymbol{\theta}}) \dot{\boldsymbol{\theta}}$

The task-space dynamics are then given by: $$ \boldsymbol{\Lambda}(\boldsymbol{\theta}) \ddot{\boldsymbol{x}} + \boldsymbol{\eta}(\boldsymbol{\theta}, \dot{\boldsymbol{x}}) = \boldsymbol{F}_{\text{ee}} $$

where:

• $\boldsymbol{\Lambda}(\boldsymbol{\theta}) = \left( \boldsymbol{J}(\boldsymbol{\theta}) \boldsymbol{M}^{-1}(\boldsymbol{\theta}) \boldsymbol{J}^T(\boldsymbol{\theta}) \right)^{-1}$ is the task-space inertia matrix,
• $\boldsymbol{\eta}(\boldsymbol{\theta}, \dot{\boldsymbol{x}})$ represents Coriolis, centrifugal, and gravity effects in task space,
• $\boldsymbol{F}_{\text{ee}}$ is the force applied at the end-effector.

---

The key idea to compute the joint torques $\boldsymbol{\tau}$ inducing a response matching this desired impedance is to design a control law that cancels the robot’s internal dynamics and replaces them with the behavior of a virtual mass-spring-damper system.

To do this, we define a desired force at the end-effector:

$$ \boldsymbol{F}_{\text{ee}_d} = \boldsymbol{M_d} \ddot{\boldsymbol{x}} + \boldsymbol{B_d} \dot{\boldsymbol{x}} + \boldsymbol{K_d}(\boldsymbol{x} - \boldsymbol{x_r}) $$

We, then, use an internal model of the robot to cancel its actual task-space dynamics and inject this desired behavior.

What is the internal model of a robot?

In the context of impedance control, the internal model refers to the controller’s estimate of the robot’s dynamics, which is used to predict and compensate for the robot’s natural behavior. It is not the actual dynamics of the robot, but rather an approximation based on known or identified parameters.

The internal model typically includes:

• An estimate of the task-space inertia matrix: $\tilde{\boldsymbol{\Lambda}}(\boldsymbol{\theta})$
• An estimate of the Coriolis, centrifugal, and gravitational effects: $\tilde{\boldsymbol{\eta}}(\boldsymbol{\theta}, \dot{\boldsymbol{x}})$

These are computed using internal approximations of:

• The manipulator Jacobian: $\tilde{\boldsymbol{J}}(\boldsymbol{\theta})$
• The joint-space dynamics: $\tilde{\boldsymbol{M}}(\boldsymbol{\theta})$, $\tilde{\boldsymbol{C}}(\boldsymbol{\theta}, \dot{\boldsymbol{\theta}})$, $\tilde{\boldsymbol{g}}(\boldsymbol{\theta})$

---

The impedance-control algorithm in task-space:

\[ \boldsymbol{\tau} = \boldsymbol{J}^T(\boldsymbol{\theta}) \left( \underbrace{\tilde{\boldsymbol{\Lambda}}(\boldsymbol{\theta}) \ddot{\boldsymbol{x}} + \tilde{\boldsymbol{\eta}}(\boldsymbol{\theta}, \dot{\boldsymbol{x}})}_{\text{internal model of robot dynamics}} - \underbrace{\boldsymbol{M_d} \ddot{\boldsymbol{x}} + \boldsymbol{B_d} \dot{\boldsymbol{x}} + \boldsymbol{K_d}(\boldsymbol{x} - \boldsymbol{x_r})}_{\text{desired impedance behavior}} \right) \]

where:

• $\boldsymbol{\tau} \in \mathbb{R}^m$ is the commanded joint torque (units: $\mathrm{Nm}$),
• $\boldsymbol{J}(\boldsymbol{\theta}) \in \mathbb{R}^{n\times m}$ is the manipulator Jacobian,
• $\tilde{\boldsymbol{\Lambda}}(\boldsymbol{\theta})$ is the estimated mass matrix in task space,
• $\tilde{\boldsymbol{\eta}}(\boldsymbol{\theta}, \dot{\boldsymbol{x}})$ represents Coriolis, centrifugal, and gravity terms,
• $\boldsymbol{M_d}$, $\boldsymbol{B_d}$, $\boldsymbol{K_d}$ are the desired virtual impedance parameters.

This means that the robot behaves as if it has the mechanical properties we choose — even if it doesn’t physically have them. For instance, you can make a light robot behave like a massive, heavily damped system, enhancing safety and predictability during contact. This control strategy can be visualized as compensating for the robot’s true dynamics while injecting the desired mechanical impedance behavior.

The figure below illustrates this structure at the system level:

The impedance controller uses the current motion state of the end-effector (i.e., measured position and velocity) together with the desired motion trajectory to compute a virtual force based on the programmed impedance behavior. This force is then mapped to joint torques using the Jacobian transpose and applied to the manipulator. Optionally, a force sensor may be used to enhance performance or enable force feedback.

Admittance control

It is conceptually the dual of impedance. In impedance control, we define how much force the robot should apply in response to a deviation in motion. In admittance control, it’s the opposite: we measure an external force and compute how much the robot should move to accommodate that force. Rather than generating force commands from motion errors, admittance control takes a force input and outputs a position or velocity adjustment. It creates a compliant behavior by letting the robot “yield” in a controlled way. Practically, the robot is typically position-controlled at its core, but an outer loop takes the force error and computes a small shift in the commanded position (or trajectory) to relieve or accommodate that force​. For instance, if a force of 10N is pushing the robot off its path, an admittance controller might say “yield by 1 mm” (depending on a compliance setting) – effectively, the robot moves slightly until the force reduces.

Admittance control often involves two loops:

• an outer force loop that modifies the target position based on force input.
• an inner high-bandwidth control loop that ensures accurate tracking of the compliant position or velocity reference.

Outer Loop: Virtual Mechanical Behavior

The outer loop defines a second-order mechanical behavior that governs how the compliant trajectory $\boldsymbol{x_c}$ evolves in response to the external force $\boldsymbol{f_e}$:

$$ \boldsymbol{M_d}(\ddot{\boldsymbol{x}}_c - \ddot{\boldsymbol{x}}_r) + \boldsymbol{B_d}(\dot{\boldsymbol{x}}_c - \dot{\boldsymbol{x}}_r) + \boldsymbol{K_d}(\boldsymbol{x}_c - \boldsymbol{x}_r) = \boldsymbol{f_e} $$

where:

• $\boldsymbol{x_c} \in \mathbb{R}^n$: compliant position generated by the controller (units: $\mathrm{m}$),
• $\boldsymbol{x_r} \in \mathbb{R}^n$: desired nominal trajectory (units: $\mathrm{m}$),
• $\boldsymbol{f_e} \in \mathbb{R}^n$: external force measured at the end-effector (units: $\mathrm{N}$),
• $\boldsymbol{M_d} \in \mathbb{R}^{n \times n}$: desired virtual inertia matrix (units: $\mathrm{kg}$),
• $\boldsymbol{B_d} \in \mathbb{R}^{n \times n}$: desired virtual damping matrix (units: $\mathrm{Ns/m}$),
• $\boldsymbol{K_d} \in \mathbb{R}^{n \times n}$: desired virtual stiffness matrix (units: $\mathrm{N/m}$).

In many implementations, the reference trajectory is slowly varying or static, so $\dot{\boldsymbol{x}}_r = \boldsymbol{0}$ and $\ddot{\boldsymbol{x}}_r = \boldsymbol{0}$, simplifying the equation to:

$$ \boldsymbol{M_d} \ddot{\boldsymbol{x}}_c + \boldsymbol{B_d} \dot{\boldsymbol{x}}_c + \boldsymbol{K_d}(\boldsymbol{x}_c - \boldsymbol{x}_r) = \boldsymbol{f_e} $$

This dynamic relationship defines how the commanded position $\boldsymbol{x}_c$ is adjusted in response to the external force $\boldsymbol{f_e}$.

Inner Loop: Tracking the Compliant Motion

The inner controller drives the robot to track $\boldsymbol{x}_c$ using a standard feedback law, for instance:

$$ \boldsymbol{F} = \boldsymbol{K_p}(\boldsymbol{x}_c - \boldsymbol{x}) + \boldsymbol{K_d}(\dot{\boldsymbol{x}}_c - \dot{\boldsymbol{x}}) $$

where:

• $\boldsymbol{x} \in \mathbb{R}^n$: actual end-effector position (units: $\mathrm{m}$),
• $\dot{\boldsymbol{x}} \in \mathbb{R}^n$: actual end-effector velocity (units: $\mathrm{m/s}$),
• $\boldsymbol{K_p} \in \mathbb{R}^{n \times n}$: proportional gain matrix (units: $\mathrm{N/m}$),
• $\boldsymbol{K_d} \in \mathbb{R}^{n \times n}$: derivative gain matrix (units: $\mathrm{Ns/m}$),
• $\boldsymbol{F} \in \mathbb{R}^n$: commanded force or torque to the low-level controller (units: $\mathrm{N}$ or $\mathrm{Nm}$).

This control law ensures that the robot tracks the compliant reference $\boldsymbol{x}_c$ while respecting the dynamic behavior imposed by the admittance model.

Both impedance and admittance achieve force indirectly by shaping how the robot responds to contact. They can be used to make a robot compliant without sacrificing stability. However, they have different practical considerations: for example, admittance control assumes a very stiff inner position control (common in industrial arms) and adjusts commands externally, which works well when you can trust the position control to execute quickly​. Impedance control embeds compliance in the low-level control (joint torques), which can be more direct but may face stability issues if the environment is very stiff and the controller is also stiff​. In summary, indirect force control lets the robot simulate a mechanical behavior (softness or stiffness) to handle contact forces gracefully, rather than directly pushing or pulling with a specified force.

Deeper in the theory: Sapienza University course - Impedance control

_{De Luca, A. (2020) Robotics 2 – Impedance Control. YouTube video, 6 May. Sapienza University of Rome. Available at:(https://www.youtube.com/watch?v=IolG5V_skv8)}

This lecture from Sapienza University of Rome provides a thorough and rigorous look at impedance control. It dives into the mathematical foundations and practical considerations behind the method, making it especially useful if you’re aiming to understand how impedance control is derived and implemented at a deeper level. Recommended for students who are already comfortable with dynamics, control theory and robotic modeling.

Direct Force Control

Direct force control uses explicit feedback from the interaction with the environment to regulate the forces applied by the robot. Based on this contact force, the controller adjusts the robot’s joint torques or positions to achieve a desired force profile necessary to complete a task.

Consider a 1-DOF, linear, position-controlled robot as shown below. The goal is to control the contact force $f$ between the robot’s end-effector and a wall so that it tracks a desired force $f_d$.

To achieve this, we can use a Proportional-Derivative (PD) control law as below to convert the error $f_e$ to a position error $x_e$. Note that the command sent to the robot is a computed desired position $x_{com} = x + x_e$ and its internal position controller takes care of reaching it:

Here’s how it behaves:

• When there is no contact, the force/torque sensor measures zero contact force: $f = 0$, so the error $f_e = f_{d} - f = f_d > 0$.
• Assuming stationary initial conditions, this creates a position error $x_e = x_com - x> 0$, which leads to $x_com > x$ which in turn makes the robot move right toward the wall.
• Upon contact between the robot end-effector and the wall, the sensed force becomes positive but still smaller than $f_d$ assuming “soft” collision. Therefore, initially, one still has $x_{com} > x = x_{wall},$ meaning that the robot is commanded to penetrate the wall.
• Finally, interactions between the PD controller and the stiffness of the end-effector, of the wall and of the robot’s internal position controller (characterized in part by its own PID gains) brings the system to a steady state where:
  • $f_e = 0$
  • $x_{com} > x_{wall}$ — i.e. the robot slightly compresses the contact, holding a constant desired force.

This is a basic example. In more complex tasks, we often need to control motion along some axes and force along others, depending on environmental constraints. This leads naturally to hybrid force/motion control, which generates position commands in unconstrained directions and force commands in constrained ones.

Hybrid Force/Motion Control

_{Magic Marks (2021) Architecture of Hybrid Position/Force Control System | Industrial Automation & Robotics. YouTube video, 22 March. Available at: https://www.youtube.com/watch?v=BXu9C3joUSk}

This very short video gives a very good intuition of hybrid force/motion control

The aim of hybrid force/motion control is to split up simultaneous control of both end-effector motion and contact forces into two separate decoupled but coordinated subproblems

The idea is to decouple all 6 degrees of freedom of the task (3 translations and 3 rotations) into two orthogonal subspaces and apply either a motion based control or a force based control onto each of the axes. Unconstrained (free) axes are controlled in position while constrained axes are controlled by applying a constant force

Consider the illustrative scenario below, where a robot end-effector slides on a table surface. Here, the vertical direction (Z-axis) demands force control to maintain consistent contact force, while the horizontal direction (X-axis) requires position control to follow a prescribed trajectory. Such scenarios frequently occur in grinding or polishing tasks.

How to Choose the Constraints: Natural vs Artificial

In hybrid force/motion control, one of the most important design decisions is how to split the task space into motion-controlled and force-controlled directions. This is not arbitrary — it should be based on both the task requirements and the physical properties of the robot’s environment. We make this decision by distinguishing between:

• Natural constraints : limitations imposed by the environment that restrict motion or wrench. These are not imposed by the robot, but arise from rigid contact (e.g. a table blocks downward motion, or a peg in a hole restricts lateral translation).
• Artificial constraints : constraints imposed by the controller. These are where we actively control either position or force, depending on what the task requires.

In theory:

• Force control should be applied along directions that are naturally constrained (e.g. control the contact force along the surface normal).
• Motion control should be applied along directions that are unconstrained by the environment (e.g. free tangential movement).

To formalize this, we introduce a binary selection vector:

$$ \boldsymbol{s} \in \{0, 1\}^n $$ where $s_j = 1$ means that force control is applied along the $j$-th axis, and $s_j = 0$ means motion control is applied.

The hybrid control objective is expressed as:

$$ s_j (F_j - F_{d,j}) + (1 - s_j)(x_j - x_{d,j}) = 0 $$

This means:

• If $s_j = 1$ (force-controlled axis), enforce the desired force $F_{d,j}$, and let position adjust freely.
• If $s_j = 0$ (motion-controlled axis), track the desired position $x_{d,j}$, and allow the force to vary naturally.

We can write the same equation more compactly in vector form:

$$ \boldsymbol{s} \odot (\boldsymbol{F} - \boldsymbol{F_d}) + (\boldsymbol{1} - \boldsymbol{s}) \odot (\boldsymbol{x} - \boldsymbol{x_d}) = \boldsymbol{0} $$

where:

• $\boldsymbol{x} \in \mathbb{R}^n$: actual end-effector position (units: $\mathrm{m}$),
• $\boldsymbol{x_d} \in \mathbb{R}^n$: desired position trajectory (units: $\mathrm{m}$),
• $\boldsymbol{F} \in \mathbb{R}^n$: measured contact wrench (units: $\mathrm{N}$ or $\mathrm{Nm}$),
• $\boldsymbol{F_d} \in \mathbb{R}^n$: desired contact wrench (units: $\mathrm{N}$ or $\mathrm{Nm}$),
• $\boldsymbol{s} \in {0,1}^n$: selection vector indicating which directions are force-controlled,
• $\odot$: Hadamard (element-wise) product.

This structure allows the robot to behave appropriately in tasks with partial constraint — controlling contact forces where needed, while allowing motion freedom in unconstrained directions.

_{Northwestern Robotics (2018) Modern Robotics, Chapter 11.6: Hybrid Motion-Force Control. YouTube video, 16 March. Available at: https://www.youtube.com/watch?v=UR0GpaaBVKk.}

_{Lynch, K.M. and Park, F.C. (2017) Modern Robotics: Mechanics, Planning, and Control. Cambridge: Cambridge University Press.}

This short video provides a mathematical explanation of hybrid force/motion control, expanding on the concepts introduced in this section. It details how task space is split into force- and motion-controlled directions using a selection matrix, and presents the hybrid control law in formal terms. This video offers a more theoretical perspective, complementing the intuitive examples and experiments discussed earlier.

Conceptual Exercise

Drag each axis into the correct constraint category.

Notation: Each term refers to a motion or force component in the contact frame c:

• k̇ – linear velocity along axis k (x, y, or z)
• ωk – angular velocity about axis k
• fk – force along axis k
• μk – moment (torque) about axis k

Sliding on a Flat Surface

ẋ

ẏ

ż

ω_x

ω_y

ω_z

f_x

f_y

f_z

μ_x

μ_y

μ_z

🟦 Natural Constraints

🟨 Artificial Constraints

Check Answer

Insertion of a peg in a hole

ẋ

ẏ

ż

ω_x

ω_y

ω_z

f_x

f_y

f_z

μ_x

μ_y

μ_z

🟦 Natural Constraints

🟨 Artificial Constraints

Check Answer

Crank Rotation

ẋ

ẏ

ż

ω_x

ω_y

ω_z

f_x

f_y

f_z

μ_x

μ_y

μ_z

🟦 Natural Constraints

🟨 Artificial Constraints

Check Answer

Deeper in the theory: Indian Institue of Science (Bengaluru) course

_{NPTEL – Indian Institute of Science, Bengaluru (2021) lec36 Force control of manipulators, Hybrid position/force control of manipulators. YouTube video, 4 January. Available at: https://www.youtube.com/watch?v=TyzTkIbWPyQ}

This lecture dives deeper into the theoretical formulation behind hybrid force/motion control. It builds on the ideas introduced in Chapter 2.2 and 2.2.1, detailing how tasks are decomposed using selection matrices, and how controllers are structured to enforce force or motion objectives in orthogonal subspaces. The video covers the mathematical models, control laws, and stability considerations involved—making it an excellent resource for students already comfortable with robot dynamics and control theory, and who want to understand the full structure of hybrid controllers from the ground up.

Programming exercise

Now that you’ve explored the theory behind interaction control, it’s time to bring it to life in simulation. You’ll get hands-on experience applying active force control to a simplified robot model — and see how your controller responds to real-time contact dynamics. (Please refer to the Install Webots section if you haven’t installed it yet.)

Setup your environment

Head to the following page:

🔗 Boom Monopod Simulation

There, you’ll find a description of the setup, including:

• A Webots .wbt world file
• A controller implemented in Python
• Instructions to run the simulation

The system consists of a 1-DOF leg mounted on a boom that moves only vertically. Your task will be to implement a control law that regulates the contact force applied against the ground.

Exercise 1: Baseline Passive Behavior

Goal: Understand passive compliance in action.
Task: Examine the given code and answer the following questions:

• What’s the role of neutral?

  Answer

  neutral defines the default compression (spring rest length offset) for the leg. It’s like a target for the uncompressed length — ensuring that during liftoff, a small preload is added to generate thrust.

• Try varying thrust:

  What happens ?

  • Higher thrust → higher jump when leaving the ground.
  • Lower thrust → makes the monopod barely lift or not lift off at all.

  Why ?
  Hint: You can add display the values of neutral and leg_length in poll_control function so that you better grasp what happens in each phase of the movement

  In the poll_control function, the spring force applied to the leg is computed as: $$spring\text{_}force=−120 × (leg\text{_}length − neutral)$$ where leg_length is the measured extension of the leg, and neutral is either zero or a positive thrust value. During ground contact and while the robot is descending, the controller sets neutral equal to thrust, which is a small positive offset. Since the leg length is negative during ground contact (the robot is compressed against the ground), increasing neutral makesleg_length − neutral more negative. Because the spring force formula multiplies this quantity by a negative spring constant, the resulting spring force becomes more positive. In other words, the actuator generates a stronger upward pushing force. This upward force helps the robot resist ground penetration and store additional energy for a potential rebound or jump. The greater the thrust value (neutral offset), the larger the upward force produced during contact. Thus, adjusting neutral provides a way to control how forcefully the robot pushes against the ground.

• What happens if b_y is 0? Why?

  Answer

  When b_y == 0, the foot is not in contact with the ground. Thus, neutral = 0, and no additional thrust force is applied — only passive spring behavior (no active force).

• In the poll_control function, locate the line where the spring force is computed. Try varying the spring stiffness (the numerical value 120):

  What happens ?

  • Higher stiffness → stronger upward force during contact → robot jumps faster and higher, but rebounds can become too violent or unstable.
  • Lower stiffness → softer upward force → robot lands more gently but struggles to jump or rebound effectively.

• How does changing stiffness interact with thrust (neutral)?

  Answer

  Thrust (neutral) shifts the spring force curve upward, while stiffness controls the slope (how sharply force changes with leg deformation). With a higher spring stiffness, even a small thrust offset causes a large increase in spring force, making the thrust effect much more powerful. With a lower spring stiffness, the thrust still shifts the neutral point, but the resulting upward force is smaller. Thus, the two parameters together shape how much lift and bounce the robot achieves.

• Which combination would you recommend for:
  • Soft landings with minimal bounce?
  • Maximizing jump height and quick rebound?

  Answer

  • For soft landings with minimal bounce, use low stiffness and small thrust. This combination produces gentle upward forces and high compliance.
  • For maximizing jump height and quick rebound, use high stiffness and a large thrust. This combination generates strong upward forces, stores more energy in the spring, and enables more powerful jumps — but at the cost of stability.

Summary exercise

Select the correct option in each cell, then click Check Answers.

Do you want to implement a real project ?

For those interested in applying the concepts introduced in this course, the 2-Linkages Robotic Arm Hybrid Position/Force Control project provides an example of a system simple enough to be designed and implemented independently. It illustrates how hybrid position/force control can be realized on a basic two-joint robotic arm, offering a concrete starting point for translating theoretical knowledge into hands-on experimentation.

Credits

This course was created by Salim Boussofara and Aude Billard, and funded by IEEE RAS and EPFL.

It makes use of selected material from:

• Brandberg, E., Engelking, P., Jiang, Y., Kumar, N., Mbagna-Nanko, R., Narasimhan, R., Rai, A., Shaik, S., Varikuti, V.R.R. and Yu, M. (n.d.) Advanced Robotics for Manufacturing. [online] Available under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License at: https://opentextbooks.clemson.edu/me8930/chapter/force-control-of-a-manipulator/
• Lynch, K.M. and Park, F.C. (2017) Modern Robotics: Mechanics, Planning, and Control. Cambridge: Cambridge University Press.

It has also been inspired by:

• Villani, L. and De Schutter, J. (2016) Force Control. In: Siciliano, B., Khatib, O. (eds) Springer Handbook of Robotics. Springer Handbooks. Springer, Cham. https://doi.org/10.1007/978-3-319-32552-1_9

Ressources

Want to learn more ? –> Free Online Courses

If you’re interested in a deeper exploration of force control, hybrid control, and interaction dynamics, check out this excellent university-level material:

• Chapter 2.12 of MIT’s Introduction to Robotics (2.12) —> A thorough breakdown of hybrid position/force control, compliance modeling, and the math behind force regulation during interaction

• Lecture 13 - MIT 6.881 (Robotic Manipulation), Fall 2021 - Force Control (part 2) A deep-dive lecture that walks through the same content with examples and derivations — perfect if you want to learn summarize all you have learnt in this course.

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