Robot and interface specifications

Realtime control commands sent to the robot should fulfill recommended and necessary conditions. Recommended conditions should be fulfilled to ensure optimal operation of the robot. If necessary conditions are not met then the motion will be aborted.

The final robot trajectory is the result of processing the user-specified trajectory ensuring that recommended conditions are fulfilled. As long as necessary conditions are met, the robot will try to follow the user-provided trajectory but it will only match the final trajectory if it also fulfills recommended conditions. If the necessary conditions are violated, an error will abort the motion: if, for instance, the first point of the user defined joint trajectory is very different from robot start position (\(q(t=0) \neq q_c(t=0)\)) a start_pose_invalid error will abort the motion.

Values for the constants used in the equations below are shown in the Limits for Panda and Limits for Franka Research 3 section.

Joint trajectory requirements

Necessary conditions

  • \(q_{min} < q_c < q_{max}\)

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

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

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

Cartesian trajectory requirements

Necessary conditions

  • \(T\) is 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:

  • \(q_{min} < q_c < q_{max}\)

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

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

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)

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

  • \(\dot{p}_{c} = 0\) (Cartesian velocity)

  • \(\ddot{p}_{c} = 0\) (Cartesian acceleration)

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 Panda

Limits in the Cartesian space are as follows:

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 are:

Name

Joint 1

Joint 2

Joint 3

Joint 4

Joint 5

Joint 6

Joint 7

Unit

\(q_{max}\)

2.8973

1.7628

2.8973

-0.0698

2.8973

3.7525

2.8973

\(\text{rad}\)

\(q_{min}\)

-2.8973

-1.7628

-2.8973

-3.0718

-2.8973

-0.0175

-2.8973

\(\text{rad}\)

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

2.1750

2.1750

2.1750

2.1750

2.6100

2.6100

2.6100

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

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

15

7.5

10

12.5

15

20

20

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

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

7500

3750

5000

6250

7500

10000

10000

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

\({\tau_j}_{max}\)

87

87

87

87

12

12

12

\(\text{Nm}\)

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

1000

1000

1000

1000

1000

1000

1000

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

The arm can reach its maximum extension when joint 4 has angle \(q_{elbow-flip}\), where \(q_{elbow-flip} = -0.467002423653011\:rad\). This parameter is used to determine the flip direction of the elbow.

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

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

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 are:

Name

Joint 1

Joint 2

Joint 3

Joint 4

Joint 5

Joint 6

Joint 7

Unit

\(q_{max}\)

2.7437

1.7837

2.9007

-0.1518

2.8065

4.5169

3.0159

\(\text{rad}\)

\(q_{min}\)

-2.7437

-1.7837

-2.9007

-3.0421

-2.8065

0.5445

-3.0159

\(\text{rad}\)

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

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

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

The arm can reach its maximum extension when joint 4 has angle \(q_{elbow-flip}\), where \(q_{elbow-flip} = -0.467002423653011\:rad\). This parameter is used to determine the flip direction of the elbow.

Important

Note that the maximum joint velocity depends on the joint position. The maximum and minimum joint velocities at a certain joint position are calculated as:

_images/pbv_equations_max.svg

Maximum velocities

_images/pbv_equations_min.svg

Minimum velocities

Visualization of the joint limits of FR3
_images/pbv_limits_j1.svg

Velocity limits of Joint 1

_images/pbv_limits_j2.svg

Velocity limits of Joint 2

_images/pbv_limits_j3.svg

Velocity limits of Joint 3

_images/pbv_limits_j4.svg

Velocity limits of Joint 4

_images/pbv_limits_j5.svg

Velocity limits of Joint 5

_images/pbv_limits_j6.svg

Velocity limits of Joint 6

_images/pbv_limits_j7.svg

Velocity limits of Joint 7

As most motion planners can only deal with fixed velocity limits (rectangular limits), we are providing here a suggestion on which values to use for them.

Name

Joint 1

Joint 2

Joint 3

Joint 4

Joint 5

Joint 6

Joint 7

Unit

\(q_{max}\)

2.3093

1.5133

2.4937

-0.4461

2.4800

4.2094

2.6895

\(\text{rad}\)

\(q_{min}\)

-2.3093

-1.5133

-2.4937

-2.7478

-2.4800

0.8521

-2.6895

\(\text{rad}\)

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

2

1

1.5

1.25

3

1.5

3

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

These limits are only a suggestion, you are free to define your own rectangles within the specification. However, these are the values that are used in the rate limiter and in the URDF inside franka_ros.

Denavit–Hartenberg parameters

The Denavit–Hartenberg parameters for the Panda’s kinematic chain are derived following Craig’s convention and are as follows:

_images/dh-diagram.png

Panda’s 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

\({}^0T_{1}\) is the transformation matrix which describes the position and orientation of frame 1 in frame 0. A kinematic chain can be calculated like the following: \({}^0T_{2} = {}^0T_{1} * {}^1T_{2}\)