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Knee damper and spring
The damper resists flexion speed; the spring pushes toward extension.
Walking
Hip motion is a minimum-jerk swing from toe-off to heel strike.
Prosthesis and body
Mass, centre of mass and segment lengths set the shank natural period.
Layers
Model checks
The science
Model. The thigh (with the socket) is driven by the hip: its angle th(t) follows a smooth minimum-jerk curve from toe-off to heel strike, so the hip is treated as a prescribed motion (the person's muscles). The shank and foot form a rigid body of mass m, with its centre of mass a distance d below the knee and moment of inertia Ic = m (0.3 L2)2. With phi the knee flexion angle and alpha = th - phi the absolute shank angle, the shank equation about the knee is
I_t alpha'' = - m g d sin(alpha) - m d L1 ( th'' cos(phi) - th'2 sin(phi) ) + tau_k, I_t = I_c + m d2The first term is gravity, the second is the inertial force from the accelerating knee (the hip is assumed to move at constant velocity, so only the thigh's rotation accelerates the knee). The knee torque is
tau_k = c phi' + k phi (+ stiff stops at phi = 0 and 130 deg)c is the hydraulic or pneumatic swing-phase damper, k the extension-assist spring. Integration is 4th-order Runge-Kutta with a 2.5 ms step. Toe clearance is the height of the lowest foot point above a floor that lies one leg length (straight leg + ankle height: the hip is at its highest in mid-stance, so this is the most generous floor) below the hip.
Natural period. With the hip locked and no damping the shank is a physical pendulum with period T = 2 pi sqrt(I_t / (m g d)); compare it with the swing time. When swing time is much shorter than half this period the shank lags (high flexion), when it is longer the shank follows the thigh.
Independent verification. (1) the small-swing period of the simulated shank against the closed form; (2) energy conservation of the undamped 55 degree swing over 20 s (relative drift ~1e-11); (3) convergence of the peak knee flexion between 2.5 ms and 0.5 ms steps; (4) the hip curve reaches its end angle with zero velocity.
Honest simplifications.
- Only the swing phase of the prosthetic leg is simulated dynamically. Stance is drawn as a straight leg with the foot planted on a treadmill-like floor: it is a visual aid, not a stance-phase model; hip height, pelvic tilt and vaulting by the sound leg are not modelled.
- The hip motion, initial knee angle and speed at toe-off, segment lengths, mass and damper values are typical or illustrative numbers, not patient data; real prosthetic fitting depends on the user, the knee mechanism and clinical tuning. This is an educational model, not a prescription tool.
- The damper is linear (c phi'); real hydraulic knees use speed-dependent, often phase-dependent resistance. The ankle is rigid.
What to try
- Slide the knee damping from 0 to 12 and watch the toe trail turn red when the toe drops below the floor line.
- Press "Auto-tune damping" to find the best safe damper for the current walking speed.
- Switch to fast and slow walking and compare peak flexion with the shank natural period (hip locked).
- The grey ghost leg is the same swing with a free knee (c = 0, k = 0).