Control Interface Specification and Robot Limits

Overview

Realtime control commands sent to the robot must satisfy a set of recommended and necessary conditions. Recommended conditions ensure optimal and stable behavior, while necessary conditions are strict safety and feasibility requirements. Violating any necessary condition results in an immediate motion abort.

The executed robot trajectory is based on the user-defined trajectory but adjusted to ensure that recommended conditions are respected. As long as all necessary conditions are satisfied, the robot will follow the commanded trajectory; however, it will only match it exactly if the recommended conditions are also fulfilled.

If a necessary condition is violated, the system raises an error and stops the motion. For example, if the first point of a user-provided joint trajectory differs too much from the robot’s actual start configuration (\(q(t=0) \neq q_c(t=0)\)), a start_pose_invalid error is triggered.

All constants referenced in the equations below are provided in the Limits for Franka Emika Robot (FER) and Limits for Franka Research 3 (FR3) sections.

Control Modes

Joint Space Control Requirements

Necessary Conditions

  • \(q_{min} < q_c < q_{max}\)

  • \(-\dot{q}_{max} < \dot{q}_c < \dot{q}_{max}\)

  • \(-\ddot{q}_{max} < \ddot{q}_c < \ddot{q}_{max}\)

  • \(-\dddot{q}_{max} < \dddot{q}_c < \dddot{q}_{max}\)

Torque Control Requirements

Necessary Conditions

  • \(-\dot{\tau_j}_{max} < \dot{{\tau_j}_d} < \dot{\tau_j}_{max}\)

Cartesian Space Control Requirements

Necessary Conditions

  • \(T\) is a proper 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)

Conditions derived from inverse kinematics:

  • \(q_{min} < q_c < q_{max}\)

  • \(-\dot{q}_{max} < \dot{q}_c < \dot{q}_{max}\)

  • \(-\ddot{q}_{max} < \ddot{q}_c < \ddot{q}_{max}\)

System Limits for Supported Robot Models

Limits for Franka Research 3 (FR3)

Cartesian-space limits:

Name

Translation

Rotation

Elbow

\(\dot{p}_{max}\)

3.0 \(\frac{\text{m}}{\text{s}}\)

2.5 \(\frac{\text{rad}}{\text{s}}\)

2.620 \(\frac{rad}{\text{s}}\)

\(\ddot{p}_{max}\)

9.0 \(\frac{\text{m}}{\text{s}^2}\)

17.0 \(\frac{\text{rad}}{\text{s}^2}\)

10.0 \(\frac{rad}{\text{s}^2}\)

\(\dddot{p}_{max}\)

4500.0 \(\frac{\text{m}}{\text{s}^3}\)

8500.0 \(\frac{\text{rad}}{\text{s}^3}\)

5000.0 \(\frac{rad}{\text{s}^3}\)

Joint-space limits:

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

\(\frac{\text{rad}}{\text{s}^2}\)

The arm reaches its maximum extension when joint 4 is at \(q_{elbow-flip} = -0.467002423653011\:rad\).

Position-Based Velocity Limits

Important

The maximum joint velocity depends on both the joint position and the direction of motion. The position-dependent velocity limits can be queried via:

Robot::getUpperJointVelocityLimits(
    const std::array<double, 7UL>& joint_positions
);
Robot::getLowerJointVelocityLimits(
    const std::array<double, 7UL>& joint_positions
);

Position-based limits are computed as:

\[\dot{q_i}(q_i)_{max} = \min( \dot{q}_{max,i}, \max( 0, -\dot{q}_{offset,i} + \sqrt{\max(0, 2 \cdot \ddot{q}_{dec,i} (q_{max,i} - q_i))} ) )\]
\[\dot{q_i}(q_i)_{min} = \max( \dot{q}_{min,i}, \min( 0, \dot{q}_{offset,i} - \sqrt{\max(0, 2 \cdot \ddot{q}_{dec,i} (-q_{min,i} + q_i))} ) )\]

These limits ensure safety and may be more restrictive than the nominal hardware capabilities.

Users may choose their own joint ranges and velocity limits depending on the requirements of their motion generator.

_images/pbv_limits_generic.svg

Position-based velocity limits.

Suggested rectangular position–velocity 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

\(\frac{\text{rad}}{\text{s}}\)

Limits for Franka Emika Robot (FER)

Cartesian-space limits:

Name

Translation

Rotation

Elbow

\(\dot{p}_{max}\)

1.7 \(\frac{\text{m}}{\text{s}}\)

2.5 \(\frac{\text{rad}}{\text{s}}\)

2.1750 \(\frac{rad}{\text{s}}\)

\(\ddot{p}_{max}\)

13.0 \(\frac{\text{m}}{\text{s}^2}\)

25.0 \(\frac{\text{rad}}{\text{s}^2}\)

10.0 \(\frac{rad}{\text{s}^2}\)

\(\dddot{p}_{max}\)

6500.0 \(\frac{\text{m}}{\text{s}^3}\)

12500.0 \(\frac{\text{rad}}{\text{s}^3}\)

5000.0 \(\frac{rad}{\text{s}^3}\)

Joint-space limits:

The arm reaches its maximum extension when joint 4 is at \(q_{elbow-flip} = -0.467002423653011\:rad\). This parameter determines the elbow flip direction.

Kinematic Configuration

Denavit–Hartenberg Parameters

The Denavit–Hartenberg parameters for the Franka Research 3 kinematic chain (following Craig’s convention) are shown below:

_images/dh-diagram-frankarobotics.png

Franka Research 3 kinematic chain.

Joint

\(a\;(\text{m})\)

\(d\;(\text{m})\)

\(\alpha\;(\text{rad})\)

\(\theta\;(\text{rad})\)

Joint 1

0

0.333

0

\(\theta_1\)

Joint 2

0

0

\(-\frac{\pi}{2}\)

\(\theta_2\)

Joint 3

0

0.316

\(\frac{\pi}{2}\)

\(\theta_3\)

Joint 4

0.0825

0

\(\frac{\pi}{2}\)

\(\theta_4\)

Joint 5

-0.0825

0.384

\(-\frac{\pi}{2}\)

\(\theta_5\)

Joint 6

0

0

\(\frac{\pi}{2}\)

\(\theta_6\)

Joint 7

0.088

0

\(\frac{\pi}{2}\)

\(\theta_7\)

Flange

0

0.107

0

0

Note

\({}^0T_{1}\) describes the pose of frame 1 in frame 0.

The full kinematic chain is computed as:

\({}^0T_{2} = {}^0T_{1} \cdot {}^1T_{2} \cdot {}^2T_{n}\)