Worked solutionOne step at a time
Count the joints
Each joint is one number the controller is free to choose.
Count what the task pins down
Holding position and rotation takes six numbers, which leaves one: the elbow swing in the reel.
Now drop rotation from the task
If only the hand's position matters, the task pins down three numbers instead of six.
Answer: C, four
7 − 3 = 4. The arm can swing its elbow and also turn its hand three ways, all without moving the point it is holding. This count holds away from singularities, where the arm loses a direction of motion.
What the simulation showedMotion that the hand never sees
The reel runs a Franka Panda in MuJoCo with its hand held at a fixed pose.
- The seven joints all move while the hand stays where it is. That motion lives in the null space of the Jacobian.
- The same freedom is then used for something useful: the elbow moves out of the way of an obstacle while the hand keeps doing its job.
Wrong answersWhy each one is tempting
| Pick | What went wrong |
|---|---|
| A · 0 | Assumed a six-joint arm, or that any task uses every joint. |
| B · 1 | Answered for the full pose and missed that the task changed. |
| D · 7 | Counted the joints and forgot to subtract the task. |
Follow-ups an interviewer asks nextTry these before opening them
Same question for a six-joint arm that only has to hold position?
6 − 3 = 3. Six-joint arms are redundant too when the task is smaller than a full pose, which is common in spraying, welding and pointing a camera.
What do you use the spare dimensions for?
Keeping the elbow clear of obstacles, staying away from joint limits and singularities, and holding a posture that leaves the arm ready for its next move.
How do you command motion that will not disturb the hand?
Project it through the null space: q̇ = J⁺ẋ + (I − J⁺J)·q̇₀. The first term does the task. The second takes any joint motion you like, q̇₀, and removes the part that would move the hand.
What happens to the count at a singularity?
The Jacobian loses rank, so the null space gains a dimension. That is not good news: it means one direction of hand motion can no longer be produced at all.
