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torque produced by the PM, IL, and rectus femoris as the product of the PCSA and the MAL. The indicators were calculated for individual muscles, and the relative contribution was calculated as the ratios of indicators to the total hip flexion torque for the three muscles at various hip joint angles from extension to flexion (0°-90°).Estimation of the relative rotation speed by contrac︲tion of the flexorsAs an indicator of rotation speed by contraction of the PM, IL, and rectus femoris, the change of flexion angle (Δα) due to the 1% shortening of the muscle fiber (ΔFL) was calculated from the muscle FL, pennation angle, and MAL (R) by the following equations under the specific condition that each muscle has the same muscle contraction speed:For the rectus femoris, the MAL was represented by the length of the vertical line from the hip joint center (black dot) to the vector from the origin (red dot) to the insertion (blue dot) in all angle regions. For the psoas major (PM), PM had two heads, and the origins were represented by the T12 vertebral body for the superficial head (PM-I) and by the L1 transverse process for the deep head (PM-II). The MAL was represented by the radius of the femoral head for PM-I up to 50° and for PM-II up to 60°, and by the length of the vertical line from the hip joint center to the vector from the origins (red dot) to the insertion in larger joint angles. The iliacus (IL) had a broad origin region on the iliac fossa, which were represented by 4 points (green dot, yellow dot, red dot, and light blue dot) on the iliac crest including the anterior superior iliac spine (IL-I), the midpoint between the former and the apex of the iliac crest (IL-II), the apex of the iliac crest (IL-III), and the posterior superior iliac spine (IL-IV). The MAL was represented by the radius of the femoral head for IL-I in joint angles up to 10°, for IL-II up to 30°, for IL-III and IL-IV up to 60°, and by the length of the vertical line from the hip joint center to the vector from the origins to the insertion in larger joint angles.Figure 1　Measurement of moment arm length (MAL) of the three muscles under different hip joint angles.ΔFL = FL × 0.01, andΔα = arctan (ΔFL × cos θ / R) (equation 1).The equations were derived from the following geometrical reasoning. The shortening of the muscle length (ΔML) and the ΔFL were correlated with the following equation:ΔML = ΔFL × cos θThe ΔML was thought to flex the joint by the angle change Δα with the equation:ΔML = R × tan ΔαThe two equations gave an equation from which equation (1) was derived:tan Δα = ΔFL × cos θ / R355