Resolved-rate control

Command the gripper's velocity: invert the Jacobian 100 times a second and draw a circle in the air.

Resolved-rate controlPseudo-inverseFeedback

Builds on Feedback: P and PD control and Direction, distance & the dot product, from the free Foundations.

Write and run this step in the simulator with Pro

Steering by velocity

Many tasks are easiest to state as a gripper velocity: "slide left at 5 cm/s" or "follow this curve". The Jacobian turns that into joint speeds:

\dot q = J^{+}(q)\,v

$J$ is $3\times4$ , so it has no ordinary inverse. The pseudo-inverse $J^+$ (np.linalg.pinv) picks the smallest $\dot q$ that produces $v$ . arm.jacobian(q) gives you the exact $J$ from Step 2.

The arm takes position targets, so every control tick you integrate those speeds into a commanded configuration:

q_{\text{cmd}} \leftarrow q_{\text{cmd}} + \dot q\,\Delta t, \qquad \Delta t = 10\ \text{ms}

This is resolved-rate control.

Feedback

Each tick is a straight-line guess at a curved motion, so pure feed-forward slowly drifts off the path. Add a correction towards where you should be:

v = v_{\text{ff}} + K\,\big(p_{\text{des}} - \mathrm{fk}(q_{\text{cmd}})\big), \qquad K \approx 5\ \text{s}^{-1}

Compare against $\mathrm{fk}(q_{\text{cmd}})$ , not the measured position. The joint servos trail the command by about 30 ms. They all trail by about the same amount, so the lag delays the path without bending it. Feeding the measured position back would pull that lag into your loop.

The circle

The circle has radius $R = 6$ cm around $c = (0.30, 0, 0.20)$ in the vertical $x$ – $z$ plane, with $\theta = \omega t$ and $\omega = 2\pi / T$ :

p_{\text{des}} = c + R\,(\cos\theta,\ 0,\ \sin\theta), \qquad v_{\text{ff}} = R\,\omega\,(-\sin\theta,\ 0,\ \cos\theta)

Your task

Fill in rate_step and the feed-forward velocity. phase("approach") servos to the start point using feedback alone. phase("draw") then makes one revolution in 4 s through the yellow rings. The Plots tab shows the gripper's distance from the centre.

To pass, stay within 5 mm of the circle and of the plane $y = 0$ , and go at least 90% of the way round.

Steering by velocity

Many tasks are easiest to state as a gripper velocity: "slide left at 5 cm/s" or "follow this curve". The Jacobian turns that into joint speeds:

\dot q = J^{+}(q)\,v

J

3\times4

, so it has no ordinary inverse. The pseudo-inverse

J^+

(np.linalg.pinv) picks the smallest

\dot q

that produces

v

. arm.jacobian(q) gives you the exact

J

from Step 2.

The arm takes position targets, so every control tick you integrate those speeds into a commanded configuration:

q_{\text{cmd}} \leftarrow q_{\text{cmd}} + \dot q\,\Delta t, \qquad \Delta t = 10\ \text{ms}

This is resolved-rate control.

Feedback

Each tick is a straight-line guess at a curved motion, so pure feed-forward slowly drifts off the path. Add a correction towards where you should be:

v = v_{\text{ff}} + K\,\big(p_{\text{des}} - \mathrm{fk}(q_{\text{cmd}})\big), \qquad K \approx 5\ \text{s}^{-1}

Compare against

\mathrm{fk}(q_{\text{cmd}})

, not the measured position. The joint servos trail the command by about 30 ms. They all trail by about the same amount, so the lag delays the path without bending it. Feeding the measured position back would pull that lag into your loop.

The circle

The circle has radius

R = 6

cm around

c = (0.30, 0, 0.20)

in the vertical

x

–

z

plane, with

\theta = \omega t

and

\omega = 2\pi / T

p_{\text{des}} = c + R\,(\cos\theta,\ 0,\ \sin\theta), \qquad v_{\text{ff}} = R\,\omega\,(-\sin\theta,\ 0,\ \cos\theta)

Your task

To pass, stay within 5 mm of the circle and of the plane

y = 0

, and go at least 90% of the way round.

Resolved-rate control

Steering by velocity

Feedback

The circle

Your task

This step is part of CrackRobotics Pro

Resolved-rate control

Steering by velocity

Feedback

The circle

Your task

This step is part of CrackRobotics Pro