Barbell Squat 3D

What is measured. A joint moment is the net turning effect of every muscle, tendon and ligament crossing a joint. It cannot be measured directly, so biomechanists compute it with inverse dynamics: record the motion, assume the body is a chain of rigid segments with known masses, and ask which moment each joint must supply for the segments to move that way.

Body model. Four segments in the sagittal plane: foot (fixed on the floor), shank, thigh and trunk (trunk + head + arms). Masses, centre-of-mass positions and radii of gyration are the Winter (2009) anthropometric fractions: thigh 10.0 %, shank 4.65 %, foot 1.45 % of body mass per leg; trunk, head and arms 67.8 %; segment lengths 0.245, 0.246 and 0.288 of body height for thigh, shank and hip-to-shoulder. Both legs are lumped, so moments are reported per leg (half of the total).

Kinematics. The knee flexion angle follows a minimum-jerk curve s(x) = 10x³ − 15x&sup4; + 6x&sup5; through descent, pause and ascent. The shank leans forward by 0.85 tanh(a k / 0.85) (a = ankle mobility, k = knee flexion), and the trunk lean β is found every instant by bisection so that the combined centre of mass of lifter and bar stays over the mid-foot, which is the balance condition. With a heavy bar the bar itself ends up close to the mid-foot line, which is why coaches say "keep the bar over the mid-foot":

where (a, b) places the bar on the trunk (high bar, low bar, front rack). Velocities and accelerations are central finite differences of these analytic positions.

Newton-Euler chain (top-down). For each segment, summed forces and moments about its centre of mass give the unknown reaction at its lower joint:

starting from the bar (force on the lifter −m_b(a_bar − g)) down through trunk, thigh and shank to the ankle. The foot is at rest, so its equilibrium gives the ground reaction F_g = F_ankle + m_foot g and the centre of pressure where F_g acts. Reported moments are the extensor moments: hip and knee extension and ankle plantarflexion.

Three independent checks run in the page. (1) The ground force and centre of pressure found from the segment chain must equal those from the single whole-body balance ΣF = Σ m a and ΣM = Σ(r×m a + Iα), which uses none of the joint moments. A sign error in any joint equation breaks it. (2) The power of the joint moments, M_hip(ω_trunk − ω_thigh) + M_knee(ω_thigh − ω_shank) + M_ankleω_shank, must equal dE/dt of the kinetic plus potential energy of the whole system, computed separately. (3) Held motionless at the bottom, the hip moment must equal the hand-statics value g(m_bΔx_bar + m_trunkΔx_trunk) and the floor must carry exactly body plus bar weight.

What it shows. Moving the bar from the trapezius to the rear deltoids (low bar) leans the trunk forward and shifts load from the knee to the hip; a long femur does the same; poor ankle mobility pushes the trunk forward too. A fast descent produces a force peak above 2 x weight at the turnaround although the lifter never leaves the floor.

Honest simplifications. Planar and symmetric (no side-to-side or twisting motion); rigid segments (no spine flexion, the trunk is one piece); the same anthropometric fractions for everyone; the foot never lifts or slides; no co-contraction, tendon elasticity or muscle force-length limits; moments are net values, so the real muscle forces (and joint compression) are larger than the moments alone suggest. It is an educational model, not a coaching or medical tool. Barbell geometry and plate colours follow common gym conventions.

Pick a preset, press Pause and scrub the depth by hand (tick Manual depth), or let the rep play. Try: (1) Low-bar 160 kg versus Front squat: watch the hip moment fall as the trunk becomes upright. (2) Long femurs, stiff ankles: the trunk must lean to keep the bar over the mid-foot. (3) Explosive: the ground force at the turnaround. (4) Pause squat: moments stay constant while the bar is still.