Humanoid ankle linkages: parallel and serial designs

By Tolerance Labs engineering team · Updated 18 October 2026

Many humanoids drive the ankle with two actuators on the shank and a pair of rods to the foot. The layout saves mass at the foot, but it couples pitch and roll, changes the torque available across the range, and adds joints that wear. This guide covers what to check when you buy actuators for one.

For the reasons to move ankle actuators up the leg, and the three common layouts, see humanoid robot actuator layout. This guide goes into the parallel linkage in more depth.

Parallel and serial ankles

In a serial ankle, one actuator drives pitch and a second, carried on the pitch stage, drives roll. Each actuator owns one axis, whether it sits at the ankle or near the knee with a rod.

In a parallel ankle, two actuators on the shank each push or pull one rod. The rods attach to the foot on either side of the ankle’s center. Moving both rods the same way pitches the foot; moving them in opposite directions rolls it. Both actuators work on both axes at once.

How torque maps through the linkage

A simple model shows the main effect. Let both rods attach to the foot a distance a in front of or behind the pitch axis, and a distance b to either side of the roll axis. For small angles, the two rod forces F₁ and F₂ give:

• Pitch torque: τ_pitch = a × (F₁ + F₂).
• Roll torque: τ_roll = b × (F₁ − F₂).
• Solved for the rod forces: F₁ = (τ_pitch ÷ a + τ_roll ÷ b) ÷ 2 and F₂ = (τ_pitch ÷ a − τ_roll ÷ b) ÷ 2.

These relations are the transpose of the linkage Jacobian, the matrix that maps rod speeds to ankle angular speeds. In a real ankle the rods are not parallel to the shank and the lever arms change with angle, so compute the Jacobian from the CAD model at each pose. The small-angle form is enough to size the actuators.

Worked example: a 50 kg humanoid stands on one foot at push-off, with the center of pressure 0.12 m ahead of the ankle. Pitch torque: 50 × 9.81 × 0.12 ≈ 58.9 N·m. Add 10 N·m of roll torque for balance. Take a = 0.06 m and b = 0.04 m.

• τ_pitch ÷ a = 58.9 ÷ 0.06 ≈ 982 N. τ_roll ÷ b = 10 ÷ 0.04 = 250 N.
• F₁ = (982 + 250) ÷ 2 = 616 N. F₂ = (982 − 250) ÷ 2 = 366 N.
• If each actuator can push 800 N, the ankle reaches 2 × 800 × 0.06 = 96 N·m of pitch with no roll, or 2 × 800 × 0.04 = 64 N·m of roll with no pitch.
• With rotary actuators driving the rods through 0.03 m cranks, the more loaded actuator needs 616 × 0.03 ≈ 18.5 N·m with its crank at right angles to the rod.

The two motors share pitch, which is why each can be smaller than a single pitch actuator. But a hard roll command during push-off loads one actuator far more than the other, so size each actuator for the worst combination, not for pitch alone. The available torques form a diamond, not a rectangle: |τ_pitch ÷ a| + |τ_roll ÷ b| ≤ 2 × F_max. To estimate pitch torque for other poses, follow sizing joint torque for a legged robot.

Range of motion limits

Stroke is shared the same way torque is. To first order, rod 1 moves a × θ_pitch + b × θ_roll and rod 2 moves a × θ_pitch − b × θ_roll, with angles in radians.

• With ±20 mm of usable stroke per actuator, pitch alone reaches 0.020 ÷ 0.06 ≈ 0.33 rad, about 19°.
• Roll alone reaches 0.020 ÷ 0.04 = 0.5 rad, about 29°.
• At 10° of pitch, 0.175 rad, the rods have used 0.06 × 0.175 ≈ 10.5 mm, which leaves 9.5 mm for roll: 0.0095 ÷ 0.04 ≈ 0.24 rad, about 14°.

Torque also falls toward the ends of the range. The lever arm shrinks roughly as a × cos θ, so at 20° of pitch it is 0.94 × a, 6% less. A rotary crank loses more as it approaches alignment with its rod, where its effective radius drops toward zero. Check the actuator force or torque at the worst corner of the range, and keep cranks well away from that aligned position.

Backlash and stiffness of rod ends

Each rod has a ball joint or rod end at each end, and each has some clearance. That play adds directly to the ankle’s lost motion. If each rod end has 0.02 mm of radial play, a rod has 0.04 mm, and pitch play is about 0.04 ÷ 60 ≈ 0.00067 rad, about 0.04°. After wear opens each end to 0.1 mm, pitch play grows to 0.2 ÷ 60 ≈ 0.0033 rad, about 0.19°, which moves the toe, 0.15 m away, by about 0.5 mm.

Stiffness follows the same geometry. A steel rod 6 mm in diameter and 0.30 m long has an axial stiffness of E × A ÷ L = 200 × 10⁹ × 28.3 × 10⁻⁶ ÷ 0.30 ≈ 18.9 × 10⁶ N/m. Two rods at a = 0.06 m give a pitch stiffness of 2 × k × a² = 2 × 18.9 × 10⁶ × 0.0036 ≈ 136,000 N·m/rad. The rod is rarely the soft part: rod ends, the screw and nut, and the actuator’s bearings are in series with it and often dominate. If the whole path is 5 × 10⁶ N/m per rod, pitch stiffness falls to 36,000 N·m/rad, and 58.9 N·m deflects the ankle by about 1.6 mrad.

• Use preloaded or low-clearance spherical joints, and check play at incoming inspection.
• Load rods only along their axis. Ball joints at both ends keep bending out of the actuator’s screw.
• Plan to replace rod ends as wear items, and measure play on a schedule.

Backlash and torsional stiffness explains how play and compliance limit control bandwidth.

Calibrating a coupled linkage

Neither actuator’s zero means anything on its own, because each affects both axes. An offset in one rod tilts the foot in pitch and roll at once. With the example geometry, a 1 mm error in rod 1 alone gives a × θ_pitch = 0.5 mm and b × θ_roll = 0.5 mm, so pitch is off by 0.5 ÷ 60 ≈ 8.3 mrad, about 0.48°, and roll by 0.5 ÷ 40 = 12.5 mrad, about 0.72°.

• Clamp the foot flat in a fixture at a known pose, record both actuator positions and store them as the zero.
• Check at several other poses with an inclinometer or the foot IMU, and fit the link lengths if the errors grow with angle.
• Run the inverse kinematics in the controller each cycle, numerically or from a lookup table, since the full spatial linkage has no simple closed form.
• Consider an absolute encoder on the ankle axes themselves, which removes most of the dependence on rod geometry.

The general procedure is in calibrating a robot joint.

Checklist

• Choose serial or parallel from the mass you can accept at the foot and the range of motion you need.
• Compute rod forces from the worst combination of pitch and roll torque, not from pitch alone.
• Check actuator force or crank torque at the worst corner of the range of motion.
• Check the shared stroke: the pitch range left at full roll, and the roll range left at full pitch.
• Specify low-play rod ends, load rods axially, and treat rod ends as wear parts.
• Estimate stiffness of the whole path: rod, rod ends, screw and bearings.
• Plan a fixture calibration and a check at several poses.

Browse humanoid and legged robot joints and joint modules, or see complete humanoid robots.