TECHNICAL ANALYSIS v2 HYUNDAI × BOSTON DYNAMICS DOCUMENT UM-KICK-02

Atlas Unseen Moves: Kick Mechanics

A physics-based reading of an anatomically anomalous free kick.

Abstract

At the moment of strike, Atlas's body is rotated 66° from the goal direction — body forward azimuth 26°, line to goal 92°. The ball departs at 81° azimuth, which is 55° off body forward — an off-axis strike that loads angular momentum into the ball. Over the following 1.05 s of flight the ball arcs to a peak lateral position of x = +2.4 m (well wide of the goal frame) before the Magnus force curves it back through a 1.8-metre lateral arc and into the post. The body-vs-kick mismatch and the resulting Magnus curve are the central demonstration: a configuration human anatomy does not support, resolved by aerodynamic spin into a goal the body is not pointing toward.

CINE · 03 Atlas free-kick — full take · 120 fps · f001 → f815

Source · Atlas(Walk_Twist_Kick)_CINE_03.mov

§1 · CONFIGURATION

The setup

Atlas's body is rotated 66° from the goal direction. The ball must curve through space to reach a goal the body is not pointing toward.

§2 · POWER + SPIN

Impulse transfer

Force, energy, and spin are delivered in a single 8.3-ms frame at the foot–ball boundary.

§3 · TRAJECTORY

The curve

Sidespin produces a 2.4-metre Magnus arc — the ball travels wide of goal, then bends back into the post.

§1The Setup — Body Faces Away From GoalPILLAR I · CONFIGURATION

At the moment of strike, Atlas's body is rotated 66° from the line to goal. The pelvis forward axis points at 26° azimuth; the goal lies at 92°. The ball departs at 81° — 55° off body forward — then arcs through 2.4 m of lateral overshoot before the Magnus deflection curves it back to the post. The setup is the wow: the body never points where the ball ends up.

FIG · TOP-DOWN GEOMETRY Atlas's body axis is 90° to the kick line. Atlas and ball both lie on the world X axis; the kick fires perpendicular along the Y axis toward goal.

Top-down geometry at strike — Fig. 1 · The setup Top-down geometry at f592. Atlas faces **26°** while the goal lies at **92°**; the ball launches at **81°** and curves through a **2.4-metre lateral overshoot** on its way to the post. Measured from Atlas (Walk_Twist_Kick)_v03.blend at scene FPS 120.

§1.1Striking surface

Atlas's flat rectangular foot enables the off-axis strike. Where a human curved instep would constrain contact along the body forward direction (Lees & Nolan 1998), the flat plate can deliver impulse at any angle to body orientation — including the 55° off-axis angle measured here. The geometry of the striking surface is what makes the setup possible.

Foot length

28cm

Heel to toe, Atlas D1 nominal. Rigid sole.

Foot width

10cm

Outside-of-foot contact area for redirection.

Pelvis yaw

26°

Body forward azimuth at f592.

Foot velocity azimuth

46.7°

Pre-contact (f591) plane of the strike.

Ball launch azimuth

81°

Post-impact direction of momentum.

Contact angle

34°

Foot motion vs ball departure. The geometry that generates spin.

Fig. 2 · Kick phases Standard soccer kick phases (Approach → Plant → Loading → Strike → Follow-through). The Atlas sequence follows the same five biomechanical phases as documented in human kickers, but the contact event (P4) substitutes a flat plate for the curved instep. Adapted from Lees, A. & Nolan, L. (1998). The biomechanics of soccer: A review. *J Sports Sci* 16(3):211–234.

FIG. 1 · THE SETUP — BODY FACES AWAY FROM GOAL Wide hero at strike (f592). Body forward at 26°, line to goal at 92° — an off-axis configuration of 66°.

§2Impulse TransferPILLAR II · POWER + SPIN

Across a single animation frame — 8.33 ms — the foot delivers the energy and angular momentum that will define the next 1.05 seconds of flight. The strike is hard, off-centre, and rotational. The off-axis impulse — body pointing at 26°, ball launched at 81° — is the mechanism that imparts spin. A flat plate striking a sphere off-centre generates angular momentum about a near-vertical axis.

Body-to-kick mismatch

55°

Body forward (26°) vs ball launch (81°). The mechanism that loads spin.

Foot impact velocity

14.81m/s

Pre-contact (f591). Peak: 31.77 m/s at f588.

Average contact force

1364N

J/Δt over the 8.33-ms contact frame. ≈ 1.36 kN.

Ball launch velocity

26.43m/s

Peak in flight: 32.39 m/s (117 km/h).

Energy to ball

150J

95% of foot KE lost in the 8.33-ms contact window.

Implied spin rate

6.4rev/s

≈ 384 rpm. High-spin free-kick regime.

Magnus effect pressure differential — Fig. 3 · Magnus effect Pressure differential around a spinning ball produces a lateral force F_M perpendicular to the flight direction.

Linear impulse

$$\vec{J} = \vec{F}\,\Delta t = m_\text{ball}\,\vec{v}_\text{ball}$$

J = 0.43 · 26.43 = 11.37 N·s

Average contact force

$$F_\text{avg} = \dfrac{J}{\Delta t}$$

F = 11.37 ÷ 8.33 ms = 1364 N ≈ 1.36 kN

Magnus force

$$\vec{F}_\text{M} = \tfrac{1}{2}\,\rho\,A\,C_L(\text{Sp})\,v^2\,\hat{n}$$

F_M ≈ 0.73 N · C_L = 0.62 · Sp = ωr/v

Implied spin

$$\omega = \dfrac{\text{Sp}\,\bar{v}}{r}$$

ω ≈ 6.4 rev/s ≈ 384 rpm

§3The Curve — Ball Flight Through Space and BackPILLAR III · TRAJECTORY

The ball does not travel toward the goal. It travels past it, then curves back. Peak lateral displacement reaches x = +2.40 m at t = 0.40 s — well outside the goal frame — before the Magnus force pulls it back to x = 0 at the post. A spin-free projectile from the same launch state would cross the goal plane at x = 2.57 m, missing by the full width of the goal.

Lateral curl magnitude

1.82m

From peak X back to goal centre. The signature of spin.

Peak lateral position

2.40m

At t = 0.40 s (f640). Outside the 7.3 m goal frame.

No-spin miss

2.6m

Where the ball would cross y = 11 m under pure ballistics.

Peak height

2.80m

Apex Z at f640. Clears the 1.8 m wall by 1.0 m.

Flight time

1.05s

Strike (f592) → settled in goal (f815).

Launch elevation

30.3°

Above horizontal, measured at f598.

FIG. 3 · THE CURVE — BALL FLIGHT TO POST 3/4 side-rear view. Ball departs the contact, peaks at +2.4 m of lateral offset, then curls 1.82 m back across the goal mouth.

Magnus deflection in free kick — Fig. 4 · Magnus deflection Magnus deflection in a right-foot free kick. Outside-of-foot spin imparts Magnus force that curves the ball toward goal.

Side view · YZ plane

Gravity arc to apex, fall into goal. Dashed = no-spin reference.

measured no-spin wall / goal

Top view · XY plane · the curve

Out to peak X = +2.40 m, then Magnus deflection bends the ball back to the post.

measured (curving) no-spin (straight)

Perspective · oblique 3D

3/4 view of the full curving trajectory. Measured (red) vs no-spin straight line.

The Atlas demonstration is a small-scale instance of the spinning-ball spiral described by Dupeux et al. (2010). As the ball decelerates from 32.39 m/s through the curl-back phase, its Magnus radius tightens — so the trajectory bends harder near the end of flight, finding the post from outside the goal frame. Dupeux, G., Le Goff, A., Quéré, D., Clanet, C. (2010). The spinning ball spiral. New J Phys 12, 093004.

§4MethodologyDATA + ASSUMPTIONS

Source data: Blender 5.1 animation, 120 fps. World-space coordinates sampled from the rig pose at each frame; the ball trajectory is keyed with 9 control points and BÉzier (auto-clamped) interpolation.
Mass & size: m_foot = 1.5 kg (Atlas D1 distal-leg nominal). m_ball = 0.43 kg, r = 0.11 m (FIFA Law 2, size 5 midpoint).
Contact window: Δt = 1/120 s = 8.33 ms. Single-frame analog at scene FPS.
Velocity: Three-frame centered difference: v(f) = (p(f+1) − p(f−1)) · FPS / 2.
Magnus model: F_M = ½ρAC_Lv². Lift coefficient backed out of the observed lateral acceleration during the curl-back phase. Spin parameter Sp = ωr/v with C_L ≈ Sp in the linear regime; implied ω = 6.4 rev/s.
No-spin reference: Drag-free projectile, a = (0, 0, −9.81) m/s², seeded with the post-impact state at f598.
Bones queried: head.nky · base · r_foot_IK_ctrl · l_foot_IK_ctrl
Preservation: Source .blend opened in --background mode and never written. MD5 hash unchanged before and after extraction.