Understanding Franka Robot Control Parameters

🗺Big Picture

Think of the relationship between a driver(us) and a car(Franka).

Why do we need to know this parameters as a developer?

As developers, we are the drivers planning a route from point $A$ to point $B$ . We must plan a smooth path that doesn’t require slamming on the gas, jerky braking, or taking a sharp turn at high speed.

Why do we need to care about the robot’s limits?

You treat your car well, right? You should treat your robot the same way. However, the Franka robot is more conservative. It constantly checks: “Can my tires🛞 handle this turn at this speed?” If you try to move too fast or too suddenly, the system will intervene. The Franka controller will “abort” and throw an error if you planned an impossible “route”.

✒Notation

Let’s define the notation used in the Franka Control Interface Specification and Robot Limits.

q

$q$ refers to joint positions, sometimes called “robot configuration”. It is a $7$ -vector containing the positions of all joints, measured in radian.

Tip

When you saw $q$ , you are working in “joint space”.

p

$p$ refers to position vector.

Tip

Usually, this describes the end-effector position in Cartesian coordinates.

🧠Intuition

The $q$ and $p$ are “static”. On the contrary, their “rate of change”(derivative) are:

velocity, 1st derivative
acceleration, 2nd derivative
jerk, 3rd derivative

Imagine you’re in a car🚗.

Velocity is your speed.
Acceleration is you pushing the gas pedal.
Jerk is how ==suddenly== you slam your foot on the gas or brake pedal.

Then the constraints on these 3 are:

(Velocity) $\dot{p},\dot{q}$ : are you commanding the robot moving too fast?
(Acceleration) $\ddot{p},\ddot{q}$ : are you commanding the robot to speed up or slow down too hard?
(Jerk) $\dddot{p},\dddot{q}$ : are you commanding the robot to move too suddenly?

Joint trajectory requirements

Necessary conditions

For a command to be accepted, these must be true:

$q_{min} < q_c < q_{max}$
$-\dot{q}_{max} < \dot{q}_c < \dot{q}_{max}$
$-\ddot{q}_{max} < \ddot{q}_c < \ddot{q}_{max}$
$-\dddot{q}_{max} < \dddot{q}_c < \dddot{q}_{max}$

Remark

The subscript $_c$ from the $\dot{q}_c,\ddot{q}_c,\dddot{q}_c$ denotes the commanded velocity/acceleration/jerk of one specific joint.

Remark

The actual $\min$ and $\max$ values refer to Limits for Franka Research 3.

Recommended conditions

Tip

Recommended conditions are about smoothness.

$-{\tau_j}_{max} < {\tau_j}_d < {\tau_j}_{max}$
$-\dot{\tau_j}_{max} < \dot{\tau_j}_d < \dot{\tau_j}_{max}$

Remark

The $\tau$ refers to torque.

The $_j$ refers to a specific joint.

The $_d$ refers to desired torque you are asking for.

The amount of torque your command requests from a joint motor should stay below its recommended maximum.”

Warning

Asking for a very sudden change in torque ( $\dot{\tau_d}$ is high) is like yanking a handbrake instead of smoothly pressing a brake pedal. It will “jolt” the robot’s gears, causing vibration.

At the beginning of the trajectory, the following conditions should be fulfilled:

$q = q_c$
$\dot{q}_{c} = 0$
$\ddot{q}_{c} = 0$

At the end of the trajectory, the following conditions should be fulfilled:

$\dot{q}_{c} = 0$
$\ddot{q}_{c} = 0$

Remark

These conditions are crucial for trajectory generation. If you write your own trajectory generator rather instead of using a framework like MoveIt2, take these conditions seriously.

Tip

The figure in Modern Robotics: Mechanics, Planning, and Control Chapter 9 really illustrates the perfectly.

Let’s interpret it.

$q = q_c$ : the first commanded joint positions must match the robot’s current measured joint positions.

$\dot{q} = 0$ : the robot’s speed should be zero at the start and end of the motion.

$\ddot{q}=0$ : the robot should not be accelerating at the very start and very end.

Cartesian trajectory requirements

Necessary conditions

$T$ must be a proper homogeneous transformation matrix
$-\dot{p}_{max} < \dot{p_c} < \dot{p}_{max}$ (Cartesian velocity)
$-\ddot{p}_{max} < \ddot{p_c} < \ddot{p}_{max}$ (Cartesian acceleration)
$-\dddot{p}_{max} < \dddot{p_c} < \dddot{p}_{max}$ (Cartesian jerk)

Remark

The matrix $T$ (e.g., O_T_EE in libfranka: C++ library for Franka Robotics research robots) represents the end-effector’s pose. The O_T_EE reads aloud “origin transform to end effector”.
$\mathbf{T} = \begin{bmatrix}r_{11} & r_{12} & r_{13} & x \\ r_{21} & r_{22} & r_{23} & y \\ r_{31} & r_{32} & r_{33} & z \\ 0 & 0 & 0 & 1 \end{bmatrix}$

the last row is [0 0 0 1]

the rotation part(top left 3x3) should be orthogonal matrix.

Remark

The $p$ denotes the position ( $x,y,z$ ) of the end effector.

Remark

The $_c$ from the $\dot{p}_c,\ddot{p}_c,\dddot{p}_c$ denotes the commanded velocity/acceleration/jerk.

Conditions derived from inverse kinematics:

$q_{min} < q_c < q_{max}$
$-\dot{q}_{max} < \dot{q_c} < \dot{q}_{max}$
$-\ddot{q}_{max} < \ddot{q_c} < \ddot{q}_{max}$

Recommended conditions

Conditions derived from inverse kinematics:

$-{\tau_j}_{max} < {\tau_j}_d < {\tau_j}_{max}$
$-\dot{\tau_j}_{max} < \dot{{\tau_j}_d} < \dot{\tau_j}_{max}$

At the beginning of the trajectory, the following conditions should be fulfilled:

${}^OT_{EE} = {{}^OT_{EE}}_c$
$\dot{p}_{c} = 0$ (Cartesian velocity)
$\ddot{p}_{c} = 0$ (Cartesian acceleration)

Remark

the ${}^OT_{EE}$ is the robot’s current, measured pose (the 4x4 matrix).

the ${{}^OT_{EE}}_c$ is the commanded pose your algorithm is sending at the very first step (time = 0).

This is crucial. If the commanded starting pose ${{}^OT_{EE}}_c$ is 10cm away from the current pose, the robot would have to instantly jump (teleport) 10cm. This is physically impossible, probably violating the necessary condition.

At the end of the trajectory, the following conditions should be fulfilled:

$\dot{p}_{c} = 0$ (Cartesian velocity)
$\ddot{p}_{c} = 0$ (Cartesian acceleration)

Remark

These are all about smoothness.

Controller requirements

Necessary conditions

$-\dot{\tau_j}_{max} < \dot{{\tau_j}_d} < \dot{\tau_j}_{max}$

Recommended conditions

$-{\tau_j}_{max} < {\tau_j}_d < {\tau_j}_{max}$

At the beginning of the trajectory, the following conditions should be fulfilled:

${\tau_j}_{d} = 0$

Limits for Franka Research 3

Limits in the Cartesian space are as follows:

Name	Translation	Rotation	Elbow
$\dot{p}_{max}$	$3.0\frac{\text{m}}{\text{s}}$	$2.5\frac{\text{rad}}{\text{s}}$	$2.620\frac{\text{rad}}{\text{s}}$
$\ddot{p}_{max}$	$9.0\frac{\text{m}}{\text{s}^2}$	$17.0 \frac{\text{rad}}{\text{s}^2}$	$10.0 \frac{\text{rad}}{\text{s}^2}$
$\dddot{p}_{max}$	$4500.0\frac{\text{m}}{\text{s}^3}$	$8500.0\frac{\text{rad}}{\text{s}^3}$	$5000.0\frac{\text{rad}}{\text{s}^3}$

Remark

A limit of $3.0\frac{m}{s}$ means the end-effector’s speed in any direction ( $x, y, z$ ) cannot exceed 3 meters per second.
The $3.0$ is a peak speed limit. The speed of a trajectory isn’t simple as time = distance / velocity. Think of driving a car for 100 meters. You speed limit is 3m/s, but you can’t drive a car all the way 3m/s. Your speed will be 0m/s at the start and the end.
What really matters to our trajectory generation is: This limit sets a minimum possible time for the motion.

if we tell the robot to move 1.5 meters in 0.5 seconds, it would require the speed to be 3m/s for the whole time, this is impossible! The acceleration and jerk limit must be invalid.

Remark

The rotation limits apply to the angular velocity, acceleration, and jerk of the end-effector’s orientation.

Joint space limits are:

Name	Joint 1	Joint 2	Joint 3	Joint 4	Joint 5	Joint 6	Joint 7	Unit
$q_{max}$	2.9007	1.8361	2.9007	-0.1169	2.8763	4.6216	3.0508	$\text{rad}$
$q_{min}$	-2.9007	-1.8361	-2.9007	-3.0770	-2.8763	0.4398	-3.0508	$\text{rad}$
$\dot{q}_{max}$	2.62	2.62	2.62	2.62	5.26	4.18	5.26	$\frac{\text{rad}}{\text{s}}$
$\ddot{q}_{max}$	10	10	10	10	10	10	10	$\frac{\text{rad}}{\text{s}^2}$
$\dddot{q}_{max}$	5000	5000	5000	5000	5000	5000	5000	$\frac{\text{rad}}{\text{s}^3}$
${\tau_j}_{max}$	87	87	87	87	12	12	12	$\text{Nm}$
$\dot{\tau_j}_{max}$	1000	1000	1000	1000	1000	1000	1000	$\frac{\text{Nm}}{\text{s}}$
$\dot{q}_{offset}$	0.6599	0.2517	0.2000	0.3533	0.5757	0.4878	0.4628	$\frac{\text{rad}}{\text{s}}$
$\ddot{q}_{dec}$	6.0	2.585	3.5	4.0	17.0	5.5	17.0	$\frac{\text{rad}}{\text{s}^2}$

Remark

the $\ddot{q}_{dec}$ denotes the Maximum Deceleration Limits, the “braking power”. This value allows the robot to calculate its “stopping distance” (in this case, “stopping angle”). By knowing this guaranteed “braking” power, the controller can figure out when it needs to start telling the joint to slow down so it doesn’t slam into its physical limit ( $q_{max}$ ).

the $\dot{q}_{offset}$ denotes velocity offset, a small “buffer” velocity. A technical tuning parameter. It’s used in that square-root formula to help shape the “ramp” of the slowing-down curve. It helps ensure the velocity limit starts to decrease smoothly before it gets critically close to the joint limit, rather than all at once.

Remark

Note that the maximum joint velocity $\dot{q}_{max}$ depends on both the joint position and the direction of motion. The position based joint velocity limits are computed by:
$\begin{aligned} \dot{q_i} (q_i)_{max} &= \min(\dot{q}_{max,i}, \max(0, \; -\dot{q}_{offset,i} + \sqrt{\max(0, \; 2 \cdot \ddot{q}_{dec,i} \cdot (q_{max,i} - q_i))})) \\ \dot{q_i} (q_i)_{min} &= \max(\dot{q}_{min,i}, \min(0, \; \dot{q}_{offset,i} - \sqrt{\max(0, \; 2 \cdot \ddot{q}_{dec,i} \cdot (-q_{min,i} + q_i))})) \end{aligned}$
where:

$\dot{q_i} (q_i)_{max}$ is the maximum joint velocity for joint $i$ at position $q_i$ .

$\dot{q_i} (q_i)_{min}$ is the minimum joint velocity for joint $i$ at position $q_i$ .

$\dot{q}_{max,i}$ is the maximum joint velocity for joint $i$ from the table above.

$\dot{q}_{min,i}$ is the minimum joint velocity for joint $i$ . (simply $-\dot{q}_{max,i}$ )

$\dot{q}_{offset,i}$ is the velocity offset for joint $i$ from the table above.

$\ddot{q}_{dec,i}$ is the maximum deceleration limits for joint $i$ from the table above.

$q_{max,i}$ is the maximum joint position for joint $i$ from the table above.

$q_{min,i}$ is the minimum joint position for joint $i$ from the table above.

$q_i$ is the current joint position of joint $i$ .

Tip

Since we have this contradiction:

most motion planners cannot represent variable velocity limits and instead rely on constant limits,. e.g. pantor/ruckig: Motion Generation for Robots and Machines. Real-time. Jerk-constrained. Time-optimal.,

Franka’s joint velocity limit are variable limit!

What should we do then?

use the above formula. (libfranka has a wrapper for us.)

use the following heuristic table

Heuristic Optimal Limits

Name	Joint 1	Joint 2	Joint 3	Joint 4	Joint 5	Joint 6	Joint 7	Unit
$q_{max}$	2.3476	1.5454	2.4937	-0.4226	2.5100	4.2841	2.7045	$\text{rad}$
$q_{min}$	-2.3476	-1.5454	-2.4937	-2.7714	-2.5100	0.7773	-2.7045	$\text{rad}$
$\dot{q}_{max}$	2	1	1.5	1.25	3	1.5	3	$\frac{\text{rad}}{\text{s}}$
$\ddot{q}_{max}$	10	10	10	10	10	10	10	$\frac{\text{rad}}{\text{s}^2}$
$\dddot{q}_{max}$	5000	5000	5000	5000	5000	5000	5000	$\frac{\text{rad}}{\text{s}^3}$

Compared to the “theoretical” limits, these values are more conservative. I use them quite a lot.