How to Enable Zero-Gravity Teleoperating a Franka Robot?

A Quick Review of the PD Controller

In a previous post, I built an intuition for PID controller at Understanding PD Controllers in Robotics and Robot Learning. In joint space, a PD controller can be expressed as:

where

• $\tau$: the command sent to the actuator (pulse-width modulation), simplified here as torque.
• $q$: the current actual joint position.
• $q^d$: the desired joint position (target).
• $(q^d - q)$: the position error.
• $\dot{q}$: the current actual joint velocity.
• $\dot{q}^d$: the desired joint velocity (target).
• $(\dot{q}^d - \dot{q})$: the velocity error.
• $k_p$: the proportional gain (think of this as stiffness or a “spring”).
• $k_d$: the derivative gain (think of this as damping or “friction”).

The Zero-Gravity Configuration

How can we make the Franka feel like it’s floating in zero gravity? The key idea is to command the motors to only counteract gravity, without trying to hold a specific position.

It turns out we only need to modify a few lines in the ros2_controller (ros2_control) configuration:

task:
  k_pos_x: 0.0
  k_pos_y: 0.0
  k_pos_z: 0.0

I originally found this trick in CRISP - Compliant ROS2 Controllers for Learning-Based Manipulation Policies. I didn’t fully grasp it at first, but once I expanded the math, it made perfect sense. Intuitively, it looks like this:

It essentially says:

• ⭐️Regardless of the current joint positions($q$) or joint velocities($\dot{q}$), the controller will not apply torque to move the arm.
• ⭐️Regardless of the desired joint positions ($q^d$) or desired joint velocities ($\dot{q}^d$), the controller ignores them.

The robot simply stays there… unaffected by the world around it… It exerts no effort to move…

The Complete Controller Equation

The previous section builds up an intuition for how setting $k_p=0$ and $k_d=0$ leads to a “floating” state. However, it hides some underlying complexity.

First, when I physically move the robot, I am interacting with it in Cartesian space (task space). Therefore, the controller should technically be formulated in task space

rather than joint space:

Additionally, the task torque $\tau_{\text{task}}$ is just one part of the total commanded torque. There are several other components acting behind the scenes to keep the robot stable and safe. The complete equation looks like this:

where $\tau_d \in \mathbb{R}^{n}$ is the desired joint torque vector sent to the actuators, and $n = 7$ since the Franka Research 3 has 7 joints.

In our “floating” scenario, we simply set the gains of $\tau_{\text{task}}$ to $0$. The remaining $\tau$ components are responsible for maintaining the robot in “zero gravity”.

Remark

Unless otherwise stated, the mathematical notation in the remaining sections follows Russ Tedrake’s conventions in Robotic Manipulation.

Task-Space Impedance Torque: $\tau_{\text{task}}$

This can be expressed either in standard form:

or in Khatib’s operational-space form:

where

• $q \in \mathbb{R}^n$ denotes joint positions. It is often passed through EMA filter. A statistics tool that is used in quantitative trading quite a lot.
• $\dot{q} \in \mathbb{R}^n$ denotes joint velocities.
• $J(q) \in \mathbb{R}^{6 \times n}$ denotes Jacobian matrix mapping joint velocities $\dot{q}$ to end-effector spatial velocities $\dot{x} = J\dot{q}$. Its transpose $J^T$ maps Cartesian wrenches to joint torques.
• $K_x \in \mathbb{R}^{6 \times 6}$ denotes stiffness matrix where the diagonal entries are the spring constants in task space: $K_x = \text{diag}(k_{p_x}, k_{p_y}, k_{p_z}, k_{r_x}, k_{r_y}, k_{r_z})$. These are intentionally set to 0 in “floating” mode.
• $D_x \in \mathbb{R}^{6 \times 6}$ denotes damping matrix. The diagonal entries are damping constants: $D_x = \text{diag}(d_{p_x},\ldots,d_{r_z})$. They often defaults to $d_i = 2\sqrt{k_i}$ (critical damping).
• $e \in \mathbb{R}^6$ denotes the task-space error.
• $\Lambda_x \in \mathbb{R}^{6 \times 6}$ denotes operational-space inertia, defined as $\Lambda_x = (J M^{-1} J^T)^{-1}$ .

Error computation: $e$

The spatial error $e$ can be calculated either in the local end-effector frame $E$:

or in the world frame $W$:

Remark

The notation here is rigorous to avoid ambiguity.

• ${^W p^D} \in \mathbb{R}^3$ denotes the desired position of target frame $D$ measured from world frame $W$, expressed in $W$.
• ${^W p^E} \in \mathbb{R}^3$ denotes the current end-effector position measured from $W$, expressed in $W$
• ${^W R^D} \in SO(3)$ denotes the desired orientation of $D$ measured from $W$, the $SO(3)$ refers to special orthogonal group.
• ${^E R^W} = ({^W R^E})^T$ denotes the current inverse rotation of the end-effector.
• $\log_3(\cdot)$ denotes a logarithmic map such that $SO(3) \to \mathbb{R}^3$, which extracts the axis-angle rotation error vector.

Nullspace Torque: $\tau_{\text{null}}$

The nullspace projector $N(q) \in \mathbb{R}^{n \times n}$ projects torques into a space that does not affect the end-effector’s Cartesian motion. It has three common variants:

• Dynamic: $N^E = I_n - J^T \bar{J}^T$, where $\bar{J} = M^{-1} J^T \Lambda_x$ is the dynamically consistent pseudoinverse.
• Kinematic: $N^E = I_n - J^\dagger J$, where $J^\dagger$ is the standard Moore-Penrose pseudoinverse.
• None: $N^E = I_n$ (direct joint space control with no projection).

Gravity Compensation: $\tau_g$

The $g(q) \in \mathbb{R}^n$ is the generalized gravity vector (often computed via RNEA in Pinocchio). It provides the exact torque required at each joint to counteract the gravitational forces on all links.

Tip

This is the critical term that actually keeps the robot “floating” when task stiffness is zero.

Coriolis Compensation: $\tau_{\text{cor}}$

The Coriolis matrix $C(q,\dot{q}) \in \mathbb{R}^{n \times n}$ captures the velocity-dependent forces arising from the joint coupling.

Warning

Without this term, a fast-moving robot would “drift” because centrifugal and Coriolis forces would go uncompensated.

Friction Compensation: $\tau_{\text{fric}}$

This is sigmoid-based model provides a smooth approximation of Coulomb friction and viscosity, where

• $a_{1,i}$ is the friction amplitude
• $a_{2,i}$ is the sigmoid slope
• $a_{3,i}$ is the velocity dead-zone shift

Joint Limit Repulsion: $\tau_{\text{lim}}$

This creates a virtual “wall” that linearly ramps up repulsion torque as a joint approaches physical limits($q_i^{\min}, q_i^{\max}$) where

• $\delta_{\text{safe}}$ denotes the safe range threshold, e.g. 0.3 rad.
• $\tau_{\max}$ denotes the maximum repulsion torque, e.g. 5.0 Nm.

Remark

See Understanding Franka Robot Control Parameters for more on the limits of a Franka Research 3.

External Wrench: $\tau_{\text{ext}}$

where

represents external forces and torques applied to the end-effector.

Post-Processing and Safety

Before sending commands to the motors, two safety filters are applied to $\tau_d$:

1. Torque Rate Saturation: Limits $\|\tau_d - \tau_d^{\text{prev}}\|$ per timestep to prevent sudden, discontinuous torque spikes.
2. Exponential Moving Average (EMA): $\tau_d \leftarrow \alpha \, \tau_d^{\text{prev}} + (1 - \alpha) \, \tau_d$ to smooth the output signal over time.

The Manipulator Equation

All of these model-based terms ($M$, $C$, $\tau_g$) stems directly from the fundamental manipulator equation of motion:

References

• 📄A unified approach for motion and force control of robot manipulators: The operational space formulation by the famous Oussama Khatib
• Introduction to Robotics: Mechanics and Control
• Robotic Manipulation: Perception, Planning, and Control (Course Notes for MIT 6.421)