A wristed surgical instrument (e.g. Intuitive's EndoWrist) adds two extra rotational joints — pitch and yaw — right behind the jaws, driven remotely by four cables running the length of the shaft. Each cable wraps a small capstan pulley of radius r, so a joint rotation of θ pays out one cable and pulls in its antagonist by the same arc length:
cable travel Δs = r · θ (capstan relation)
tip orientation R = R_z(roll) · R_y(pitch) · R_z(yaw)
tip position p = p_wrist + R · [L_jaw, 0, 0]ᵀ
Stacking pitch and yaw at the wrist — on top of shaft roll and insertion, which the arm already provides — gives the instrument tip 7 independent degrees of freedom, one more than an unconstrained human wrist+hand (6), because the wrist joints keep working no matter how the shaft itself is oriented.
- Pitch / yaw sliders — command the two wrist joints; the model applies the capstan relation to report how far each cable pair must travel to reach that angle.
- Roll slider — spins the whole shaft about its own axis, which a rigid straight tool can also do — it doesn't add dexterity at the tip, only reorients whatever pitch/yaw the wrist already has.
- Tip deflection δ — the total angle between the jaw axis and the shaft axis, computed as the angle of the composed rotation R. A rigid (non-wristed) instrument is stuck at δ = 0° whenever it needs to approach tissue side-on.
- Dexterity vs. rigid tool — the extra reach angle the wrist buys back that a straight instrument physically cannot produce without repositioning the whole arm.
Real-world relevance: this pitch-yaw-roll-grip wrist is what lets a robotic instrument suture at steep angles inside a confined cavity that a straight laparoscopic tool — locked to whatever angle the trocar happens to present — cannot reach at all.