Model Description

The model is planar (xy plane) and there are seven rigid bodies. All rotations are defined to be positive about the z axis for a child body relative to its parent body, e.g thigh relative to trunk. When all configuration variables are set to zero, the model is standing upright and the trunk’s hip joint is at the inertial origin with the human facing in the positive x direction.

Rigid bodies and constants

  • Trunk (A):

    • ma: mass [kg]

    • ia: moment of inertia about mass center wrt to the trunk reference frame [kg*m^2]

    • xa: local x location of mass center wrt to the hip joint [m]

    • ya: local y location of mass center wrt to the hip joint [m]

    • glut_r_A_origin_x: local right x location of origin wrt to the hip joint [m]

    • glut_r_A_origin_y: local right y location of origin wrt to the hip joint [m]

    • glut_r_A_middle_x: local right x location of middle wrt to the hip joint [m]

    • glut_r_A_middle_y: local right y location of middle wrt to the hip joint [m]

    • hams_r_A_origin_x: local right x location of origin wrt to the hip joint [m]

    • hams_r_A_origin_y: local right y location of origin wrt to the hip joint [m]

    • ilio_r_A_origin_x: local right x location of origin wrt to the hip joint [m]

    • ilio_r_A_origin_y: local right y location of origin wrt to the hip joint [m]

    • rect_r_A_origin_x: local right x location of origin wrt to the hip joint [m]

    • rect_r_A_origin_y: local right y location of origin wrt to the hip joint [m]

    • glut_l_A_origin_x: local left x location of origin wrt to the hip joint [m]

    • glut_l_A_origin_y: local left y location of origin wrt to the hip joint [m]

    • glut_l_A_middle_x: local left x location of middle wrt to the hip joint [m]

    • glut_l_A_middle_y: local left y location of middle wrt to the hip joint [m]

    • hams_l_A_origin_x: local left x location of origin wrt to the hip joint [m]

    • hams_l_A_origin_y: local left y location of origin wrt to the hip joint [m]

    • ilio_l_A_origin_x: local left x location of origin wrt to the hip joint [m]

    • ilio_l_A_origin_y: local left y location of origin wrt to the hip joint [m]

    • rect_l_A_origin_x: local left x location of origin wrt to the hip joint [m]

    • rect_l_A_origin_y: local left y location of origin wrt to the hip joint [m]

  • Right Thigh (B)

    • mb: mass [kg]

    • ib: moment of inertia about mass center wrt to right thigh reference frame [kg*m^2]

    • xb: local x location of mass center wrt to the hip joint [m]

    • yb: local y location of mass center wrt to the hip joint [m]

    • lb: joint to joint segment length [m]

    • gast_r_B_origin_x: local x location of origin wrt to the hip joint [m]

    • gast_r_B_origin_y: local y location of origin wrt to the hip joint [m]

    • glut_r_B_insert_x: local x location of insertion wrt to the hip joint [m]

    • glut_r_B_insert_y: local y location of insertion wrt to the hip joint [m]

    • ilio_r_B_insert_x: local x location of insertion wrt to the hip joint [m]

    • ilio_r_B_insert_y: local y location of insertion wrt to the hip joint [m]

    • rect_r_B_middle_x: local x location of middle wrt to the hip joint [m]

    • rect_r_B_middle_y: local y location of middle wrt to the hip joint [m]

    • rect_r_B_middle_r: radius of knee wrapping circle [m]

    • vast_r_B_middle_x: local x location of middle wrt to the hip joint [m]

    • vast_r_B_middle_y: local y location of middle wrt to the hip joint [m]

    • vast_r_B_middle_r: radius of knee wrapping circle [m]

    • vast_r_B_origin_x: local x location of origin wrt to the hip joint [m]

    • vast_r_B_origin_y: local y location of origin wrt to the hip joint [m]

  • Right Shank (C)

    • mc: mass [kg]

    • ic: moment of inertia about mass center wrt to right shank reference frame [kg*m^2]

    • xc: x location of mass center wrt to the knee joint [m]

    • yc: y location of mass center wrt to the knee joint [m]

    • lc: joint to joint segment length [m]

    • gast_r_C_middle_x: local x location of middle wrt to the knee joint [m]

    • gast_r_C_middle_y: local y location of middle wrt to the knee joint [m]

    • hams_r_C_insert_x: local x location of insertion wrt to the knee joint [m]

    • hams_r_C_insert_y: local y location of insertion wrt to the knee joint [m]

    • rect_r_C_insert_x: local x location of insertion wrt to the knee joint [m]

    • rect_r_C_insert_y: local y location of insertion wrt to the knee joint [m]

    • sole_r_C_origin_x: local x location of origin wrt to the knee joint [m]

    • sole_r_C_origin_y: local y location of origin wrt to the knee joint [m]

    • tibi_r_C_origin_x: local x location of origin wrt to the knee joint [m]

    • tibi_r_C_origin_y: local y location of origin wrt to the knee joint [m]

    • tibi_r_C_middle_x: local x location of middle wrt to the knee joint [m]

    • tibi_r_C_middle_y: local y location of middle wrt to the knee joint [m]

    • vast_r_C_insert_x: local x location of insertion wrt to the knee joint [m]

    • vast_r_C_insert_y: local y location of insertion wrt to the knee joint [m]

  • Right Foot (D)

    • md: mass [kg]

    • id: moment of inertia about mass center wrt to right foot reference frame [kg*m^2]

    • xd: local x location of mass center wrt to the ankle joint [m]

    • yd: local y location of mass center wrt to the ankle joint [m]

    • hxd: local x location of heel wrt to the ankle joint [m]

    • txd: local x location of toe wrt to the ankle joint [m]

    • fyd: local y location of heel and toe relative to ankle joint [m]

    • gast_r_D_insert_x: local x location of insertion wrt to the ankle joint [m]

    • gast_r_D_insert_y: local y location of insertion wrt to the ankle joint [m]

    • sole_r_D_insert_x: local x location of insertion wrt to the ankle joint [m]

    • sole_r_D_insert_y: local y location of insertion wrt to the ankle joint [m]

    • tibi_r_D_insert_x: local x location of insertion wrt to the ankle joint [m]

    • tibi_r_D_insert_y: local y location of insertion wrt to the ankle joint [m]

  • Left Thigh (E)

    • me: mass [kg]

    • ie: moment of inertia about mass center wrt to left thigh reference frame [kg*m^2]

    • xe: local x location of mass center wrt to the hip joint [m]

    • ye: local y location of mass center wrt to the hip joint [m]

    • le: joint to joint segment length [m]

    • gast_l_E_origin_x: local x location of origin wrt to the hip joint [m]

    • gast_l_E_origin_y: local y location of origin wrt to the hip joint [m]

    • glut_l_E_insert_x: local x location of insertion wrt to the hip joint [m]

    • glut_l_E_insert_y: local y location of insertion wrt to the hip joint [m]

    • ilio_l_E_insert_x: local x location of insertion wrt to the hip joint [m]

    • ilio_l_E_insert_y: local y location of insertion wrt to the hip joint [m]

    • rect_l_E_middle_x: local x location of middle wrt to the hip joint [m]

    • rect_l_E_middle_y: local y location of middle wrt to the hip joint [m]

    • rect_l_E_middle_r: radius of knee wrapping circle [m]

    • vast_l_E_middle_x: local x location of middle wrt to the hip joint [m]

    • vast_l_E_middle_y: local y location of middle wrt to the hip joint [m]

    • vast_l_E_middle_r: radius of knee wrapping circle [m]

    • vast_l_E_origin_x: local x location of origin wrt to the hip joint [m]

    • vast_l_E_origin_y: local y location of origin wrt to the hip joint [m]

  • Left Shank (F)

    • mf: mass [kg]

    • if: moment of inertia about mass center wrt to left shank reference frame [kg*m^2]

    • xf: x location of mass center wrt to the knee joint [m]

    • yf: y location of mass center wrt to the knee joint [m]

    • lf: joint to joint segment length [m]

    • gast_l_F_middle_x: local x location of middle wrt to the knee joint [m]

    • gast_l_F_middle_y: local y location of middle wrt to the knee joint [m]

    • hams_l_F_insert_x: local x location of insertion wrt to the knee joint [m]

    • hams_l_F_insert_y: local y location of insertion wrt to the knee joint [m]

    • rect_l_F_insert_x: local x location of insertion wrt to the knee joint [m]

    • rect_l_F_insert_y: local y location of insertion wrt to the knee joint [m]

    • sole_l_F_origin_x: local x location of origin wrt to the knee joint [m]

    • sole_l_F_origin_y: local y location of origin wrt to the knee joint [m]

    • tibi_l_F_origin_x: local x location of origin wrt to the knee joint [m]

    • tibi_l_F_origin_y: local y location of origin wrt to the knee joint [m]

    • tibi_l_F_middle_x: local x location of middle wrt to the knee joint [m]

    • tibi_l_F_middle_y: local y location of middle wrt to the knee joint [m]

    • vast_l_F_insert_x: local x location of insertion wrt to the knee joint [m]

    • vast_l_F_insert_y: local y location of insertion wrt to the knee joint [m]

  • Left Foot (G)

    • mg: mass [kg]

    • ig: moment of inertia about mass center wrt to left foot reference frame [kg*m^2]

    • xg: local x location of mass center wrt to the ankle joint [m]

    • yg: local y location of mass center wrt to the ankle joint [m]

    • hxg: local x location of heel wrt to the ankle joint [m]

    • txg: local x location of toe wrt to the ankle joint [m]

    • fyg: local y location of heel and toe relative to ankle joint [m]

    • gast_l_G_insert_x: local x location of insertion wrt to the ankle joint [m]

    • gast_l_G_insert_y: local y location of insertion wrt to the ankle joint [m]

    • sole_l_G_insert_x: local x location of insertion wrt to the ankle joint [m]

    • sole_l_G_insert_y: local y location of insertion wrt to the ankle joint [m]

    • tibi_l_G_insert_x: local x location of insertion wrt to the ankle joint [m]

    • tibi_l_G_insert_y: local y location of insertion wrt to the ankle joint [m]

  • Muscle Properties

    • Gastrocnemius

      • F_M_max_gast_r: maximum isometric force [N]

      • l_M_opt_gast_r: optimal fiber length [m]

      • l_T_slack_gast_r: tendon slack length [m]

      • F_M_max_gast_l: maximum isometric force [N]

      • l_M_opt_gast_l: optimal fiber length [m]

      • l_T_slack_gast_l: tendon slack length [m]

    • Gluteus maximus, medius, and minimus

      • F_M_max_glut_r: maximum isometric force [N]

      • l_M_opt_glut_r: optimal fiber length [m]

      • l_T_slack_glut_r: tendon slack length [m]

      • F_M_max_glut_l: maximum isometric force [N]

      • l_M_opt_glut_l: optimal fiber length [m]

      • l_T_slack_glut_l: tendon slack length [m]

    • Hamstrings

      • F_M_max_hams_r: maximum isometric force [N]

      • l_M_opt_hams_r: optimal fiber length [m]

      • l_T_slack_hams_r: tendon slack length [m]

      • F_M_max_hams_l: maximum isometric force [N]

      • l_M_opt_hams_l: optimal fiber length [m]

      • l_T_slack_hams_l: tendon slack length [m]

    • Ilioposas

      • F_M_max_ilio_r: maximum isometric force [N]

      • l_M_opt_ilio_r: optimal fiber length [m]

      • l_T_slack_ilio_r: tendon slack length [m]

      • F_M_max_ilio_l: maximum isometric force [N]

      • l_M_opt_ilio_l: optimal fiber length [m]

      • l_T_slack_ilio_l: tendon slack length [m]

    • Rectus femoris

      • F_M_max_rect_r: maximum isometric force [N]

      • l_M_opt_rect_r: optimal fiber length [m]

      • l_T_slack_rect_r: tendon slack length [m]

      • F_M_max_rect_l: maximum isometric force [N]

      • l_M_opt_rect_l: optimal fiber length [m]

      • l_T_slack_rect_l: tendon slack length [m]

    • Soleus

      • F_M_max_sole_r: maximum isometric force [N]

      • l_M_opt_sole_r: optimal fiber length [m]

      • l_T_slack_sole_r: tendon slack length [m]

      • F_M_max_sole_l: maximum isometric force [N]

      • l_M_opt_sole_l: optimal fiber length [m]

      • l_T_slack_sole_l: tendon slack length [m]

    • Tibialis anterior

      • F_M_max_tibi_r: maximum isometric force [N]

      • l_M_opt_tibi_r: optimal fiber length [m]

      • l_T_slack_tibi_r: tendon slack length [m]

      • F_M_max_tibi_l: maximum isometric force [N]

      • l_M_opt_tibi_l: optimal fiber length [m]

      • l_T_slack_tibi_l: tendon slack length [m]

    • Vastus intermedius, medialis, and lateralis

      • F_M_max_vast_r: maximum isometric force [N]

      • l_M_opt_vast_r: optimal fiber length [m]

      • l_T_slack_vast_r: tendon slack length [m]

      • F_M_max_vast_l: maximum isometric force [N]

      • l_M_opt_vast_l: optimal fiber length [m]

      • l_T_slack_vast_l: tendon slack length [m]

  • Other constants

    • kc: ground contact stiffness [N/m^3]

    • cc: ground contact damping [s/m]

    • mu: ground contact friction coefficient

    • vs: ground contact velocity constant [m/s]

    • g: acceleration due to gravity [m/s^2]

Generalized coordinates

  • qax, qay: location of trunk hip joint relative to inertial origin

  • qa: angle of trunk relative to inertial reference frame, qa=0 makes trunk standing upright and qa>0 leans trunk backwards

  • qb: angle of right thigh relative to trunk (hip), qb=0 makes thigh aligned with trunk and qb>0 abducts the hip

  • qc: angle of right shank relative to right thigh (knee), qc=0 makes shank aligned with thigh and qc>0 extends the knee

  • qd: angle of right foot relative to right shank (ankle), qd=0 makes foot 90 deg to shank and qd>0 dorsiflexes the foot

  • qe: angle of left thigh relative to trunk (hip), qe=0 makes thigh aligned with trunk and qe>0 abducts the hip

  • qf: angle of left shank relative to left thigh (knee), qf=0 makes shank aligned with thigh and qf>0 extends the knee

  • qg: angle of left foot relative to left shank (ankle), qg=0 makes foot 90 deg to shank and qg>0 dorsiflexes the foot

Muscle activation states

  • a_ilio_r: muscle activation

  • a_hams_r: muscle activation

  • a_glut_r: muscle activation

  • a_rect_r: muscle activation

  • a_vast_r: muscle activation

  • a_gast_r: muscle activation

  • a_sole_r: muscle activation

  • a_tibi_r: muscle activation

  • a_ilio_l: muscle activation

  • a_hams_l: muscle activation

  • a_glut_l: muscle activation

  • a_rect_l: muscle activation

  • a_vast_l: muscle activation

  • a_gast_l: muscle activation

  • a_sole_l: muscle activation

  • a_tibi_l: muscle activation

Specified inputs

  • Fax, Fay: “hand of god”, forces acting on the trunk mass center relative to inertial origin

  • Ta: “hand of god”, torque acting on trunk relative to inertial frame

  • Tb: hip joint torque, Tb>0 extends the hip

  • Tc: knee joint torque, Tc>0 abducts the knee

  • Td: ankle joint torque, Td>0 plantarflexes the foot

  • Te: hip joint torque, Te>0 extends the hip

  • Tf: knee joint torque, Tf>0 abducts the knee

  • Tg: ankle joint torque, Tg>0 plantarflexes the foot

  • v : treadmill belt speed

  • e_ilio_r: muscle excitation

  • e_hams_r: muscle excitation

  • e_glut_r: muscle excitation

  • e_rect_r: muscle excitation

  • e_vast_r: muscle excitation

  • e_gast_r: muscle excitation

  • e_sole_r: muscle excitation

  • e_tibi_r: muscle excitation

  • e_ilio_l: muscle excitation

  • e_hams_l: muscle excitation

  • e_glut_l: muscle excitation

  • e_rect_l: muscle excitation

  • e_vast_l: muscle excitation

  • e_gast_l: muscle excitation

  • e_sole_l: muscle excitation

  • e_tibi_l: muscle excitation