Force–length–velocity

A muscle’s force isn’t just “how hard it’s trying”. For the same activation $a$, the force is

$$F = a\, F_{\max}\, f_L(\ell)\, f_V(v) + F_{\text{passive}}(\ell)$$

• Length, $f_L$. Each fibre has an optimal length where its force is highest. Shorter or longer than that, it makes less: a bell-shaped curve, falling to about half at the ends of its working range.
• Velocity, $f_V$. Shortening fast, a muscle makes less force, down to zero at its top shortening speed. Being stretched while switched on (an eccentric contraction), it makes more than its isometric force, up to about 1.4–1.8 times. That’s why lowering a heavy weight is easier than lifting it.
• Passive force. Stretched past its optimal length, a muscle resists like a rubber band even when it’s switched off.

In this project

The eccentric part is what braked the muscle fighter’s punches in Muscles instead of motors. During a jab the opposing muscles (shoulder back, elbow bend) are being stretched, and if they are half switched on, as they were, they resist harder than they could pull. Forced open loop, the jab’s fist reached 8.6 m/s with them off and 3.0 m/s with them at half excitation. MuJoCo’s muscle model has all three terms; the stage 1 muscles give 0.5–1.0 of their peak over each joint’s range.