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TABLE S1. Description of the Mathematical Expressions Parameter Description Parameter Description 𝑲𝒆 Hip quasi-stiffness in extension stage ⃗𝑼 ⃗𝒇 Angular momentum of foot 𝑲𝒇 Hip quasi-stiffness in flexion stage ⃗𝑼 ⃗𝒔 Angular momentum of shank 𝜽𝒆 Hip excursion in extension stage ⃗⃗ 𝒕 𝑼 Angular momentum of thigh 𝜽𝒇 Hip excursion in flexion stage [𝑰𝒇 ] Matrix of moment of inertia of foot 𝑾 Body weight [𝑰𝒔 ] Matrix of moment of inertia of shank 𝑽 Gait speed [𝑰𝒕 ] Matrix of moment of inertia of thigh 𝑯 Body height ⃗𝑴 ⃗⃗ 𝒕𝑷 Thigh proximal moment in global coordinate system (hip moment) ⃗𝑴 ⃗⃗ 𝑮 Ground reaction moment ⃗𝑴 ⃗⃗ 𝒕𝑫 Thigh distal moment in global coordinate system ⃗𝑮 𝑭 Ground reaction force ⃗⃗⃗ 𝒔𝑷 𝑴 Shank proximal moment in global coordinate system (knee moment) ⃗ 𝒓 Vector from toe to center of pressure ⃗⃗⃗ 𝒔𝑫 𝑴 Shank distal moment in global coordinate system 𝑳𝒇 Foot length ⃗⃗⃗ 𝒇𝑷 𝑴 Foot proximal moment in global coordinate system (ankle moment) 𝑳𝒔 Shank length ⃗𝑴 ⃗⃗ 𝒇𝑫 Foot distal moment in global coordinate system 𝑳𝒕 Thigh length ⃗⃗⃗ 𝒕𝒑 𝑴 𝒆̅𝒀 Unit vector along foot segment ⃗⃗⃗ 𝒕𝒅 𝑴 𝒆̅𝒔𝒀 Unit vector along shank segment ⃗⃗⃗ 𝒔𝒑 𝑴 𝒆̅𝒕𝒀 Unit vector along thigh segment ⃗⃗⃗ 𝒔𝒅 𝑴 𝒎𝒇 Foot mass ⃗𝑴 ⃗⃗ 𝒇𝒑 𝒎𝒔 Shank mass ⃗𝑴 ⃗⃗ 𝒇 𝒅 Thigh proximal moment in shank anatomical coordinate system Thigh distal moment in shank anatomical coordinate system Shank proximal moment in shank anatomical coordinate system Shank distal moment in shank anatomical coordinate system Foot proximal moment in shank anatomical coordinate system Foot distal moment in shank anatomical coordinate system 𝒎𝒕 Thigh mass ⃗⃗ 𝒕𝑷 𝑹 Thigh proximal force in global coordinate system ⃗𝒇 𝒂 Foot acceleration ⃗⃗ 𝒕𝑫 𝑹 Thigh distal force in global coordinate system ⃗𝒔 𝒂 Shank acceleration ⃗⃗ 𝒔𝑷 𝑹 Shank proximal force in global coordinate system ⃗𝒕 𝒂 Thigh acceleration ⃗⃗ 𝒔𝑫 𝑹 Shank distal force in global coordinate system 𝒈 Magnitude of acceleration due to gravity ⃗𝑹 ⃗ 𝒇𝑷 Foot proximal force in global coordinate system 𝒇 𝑳𝒑 Distance between center of mass of foot to ankle ⃗⃗ 𝒇𝑫 𝑹 Foot distal force in global coordinate system 𝑳𝒔𝒑 Distance between center of mass of shank to knee ⃗⃗ 𝒕𝒑 𝑹 𝑳𝒕𝒑 Distance between center of mass of thigh to hip ⃗⃗ 𝒕𝒅 𝑹 Transformation matrix from anatomical coordinate frame of thigh to global coordinate frame Transformation matrix from global coordinate frame to anatomical coordinate frame of thigh Transformation matrix from anatomical coordinate frame of shank to global coordinate frame Transformation matrix from global coordinate frame to anatomical coordinate frame of shank Transformation matrix from anatomical coordinate frame of foot to global coordinate frame ⃗𝑹 ⃗ 𝒔𝒑 Thigh proximal force in shank anatomical coordinate system Thigh distal force in shank anatomical coordinate system Shank proximal force in shank anatomical coordinate system Shank distal force in shank anatomical coordinate system Shank proximal force in shank anatomical coordinate system Shank distal force in shank anatomical coordinate system 𝒇 [𝑨𝑮]𝒕 [𝑮𝑨]𝒕 [𝑨𝑮]𝒔 [𝑮𝑨]𝒔 [𝑨𝑮]𝒇 ⃗𝑹 ⃗ 𝒔𝒅 ⃗𝑹 ⃗ 𝒇𝒑 ⃗⃗ 𝒇 𝑹 𝒅 𝑿−𝒀−𝒁 Global coordinate system Transformation matrix from global coordinate frame to anatomical coordinate frame of foot 𝒙𝒕 − 𝒚𝒕 − 𝒛𝒕 Anatomical coordinate system of thigh ⃗𝝎 ⃗⃗ 𝒕 Angular velocity of thigh 𝒙𝒔 − 𝒚𝒔 − 𝒛𝒔 Anatomical coordinate system of shank ⃗⃗⃗ 𝝎̇𝒕 Angular acceleration of thigh 𝒙𝒇 − 𝒚𝒇 − 𝒛𝒇 Anatomical coordinate system of foot ⃗⃗⃗ 𝒔 𝝎 Angular velocity of shank 𝑴𝒁𝑯 Hip moment on the sagittal plane ⃗⃗⃗ 𝝎̇𝒔 Angular acceleration of shank ⃗𝒇 𝒅 Vector connecting center of mass of foot to toe ⃗⃗⃗ 𝒇 𝝎 Angular velocity of foot ⃗𝒇 𝒑 Vector connecting center of mass of foot to ankle ⃗⃗⃗ 𝝎̇𝒇 Angular acceleration of foot ⃗𝒅𝒔 Vector connecting center of mass of shank to ankle 𝑪𝑶𝑴𝒕 Center of mass of thigh ⃗𝒔 𝒑 Vector connecting center of mass of shank to knee 𝑪𝑶𝑴𝒔 Center of mass of shank ⃗𝒕 𝒅 Vector connecting center of mass of thigh to knee 𝑪𝑶𝑴𝒇 Center of mass of foot ⃗𝒕 𝒑 Vector connecting center of mass of thigh to hip ⃗⃗⃗ 𝒕 𝑴 Any moment applied on thigh ⃗𝒕 𝑭 Any force applied on thigh ⃗𝑴 ⃗⃗ 𝒔 Any moment applied on shank ⃗𝑭𝒔 Any force applied on shank ⃗⃗⃗ 𝒇 𝑴 Any moment applied on foot ⃗𝒇 𝑭 Any force applied on foot 𝑭𝒓 Froude number for walking 𝒆̅𝒀 Unit vector vertical to the ground and along 𝑌 [𝑮𝑨]𝒇
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