Why this note exists
A tendon is not a rigid link. It wraps around pulleys and guides, its tension drops across every contact, and the moment arm it produces is a function of joint angle — not a constant. Most quick models quietly assume otherwise, and then the controller spends its life compensating for the assumption.
These are my working notes on the static side of that problem.
The pieces I keep coming back to
- Moment arm as a function of configuration. For a cable crossing joint $i$, the torque is $r_i(q)\,T$, where $r_i$ changes as the cable wraps.
- Capstan friction. Tension across a wrap follows $T_{\text{out}} = T_{\text{in}}\,e^{-\mu\theta}$; a hand with several guides can lose a large fraction of the commanded force before it reaches the fingertip.
- Routing as a design variable. Where the cable enters and how many guides it crosses changes both the moment arm profile and the friction loss — the two cannot be tuned independently.
Open questions
- When is a constant-moment-arm approximation acceptable for grasp force control?
- How much hysteresis is needed before a rate-independent friction model stops being useful?
Status
Notes in progress; no polished write-up yet.