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Jumping in frogs: assessing the design of the skeletal system by anatomically realistic modeling and forward dynamic simulation

William J. Kargo^*, Frank Nelson and Lawrence C. Rome

^* Present address: Neurosciences Institute, 10640 John Jay Hopkins Drive, San Diego, CA 92121, USA
Present address: Department of Zoology, 3029 Cordley Hall, Oregon State University, Convallis, OR 97331-2914, USA
Department of Biology, University of Pennsylvania, Philadelphia, PA 19104, USA

View larger version (33K): [in a new window]   Fig. 1. Joint kinematics (A) and joint torque patterns (B) during a maximal-effort jump in Rana pipiens. (A) The joint angle changes during the take-off phase of jumping (when the feet are in contact with the ground) are shown. In each panel, the y axis (joint angle) has the same range of 160°. (B) The net torques due to the combination of active muscle forces, passive forces in connective tissues and forces arising from interaction between the metatarsal segment and the ground (see Materials and methods). In each panel, the y axis (torque) has a range of 0.8 N cm. The joint degrees of freedom (DOFs) illustrated are: extensor DOF of the hip (Hip ext.), extensor DOF of the knee (Knee ext.), extensor DOF of the ankle (Ankle ext.), adduction DOF of the hip (Hip add.), external rotation DOF of the hip (Hip rot.), adduction DOF of the knee (Knee add.), external rotation DOF of the knee (Knee rot.) and the iliosacral joint.

View larger version (23K): [in a new window]   Fig. 2. The bone segments (right) of Rana pipiens and the local coordinate frames (LCFs) (left) attached to each bone segment in SIMM software. The bone segments are positioned in the `reference position', a position in which all the bones rest in a single, horizontal plane. In the reference position, the z axis of the LCF points out of the page. The orientation of the x and y axes for each segment are shown with different colors (orange, metatarsophalangeals, M; yellow, astragalus—calcaneus, A; green, tibiofibula, T; red, femur, F; light blue, pelvis, P; purple, vertebral column, V; dark blue, skull, S). The LCF for the urostyle overlaps that of V and is not shown. The LCFs for the forelimb bones (humerus, H; radius, R; hand, Ha) are not shown.

View larger version (59K): [in a new window]   Fig. 3. The ranges of motion and passive kinematics for the hindlimb and iliosacral joints of Rana pipiens. The location of the instantaneous center of rotation was determined about each joint axis. The white arcs overlying the joint images represent the range of motion about each joint axis. Red dots represent the locations of the instantaneous centers of rotation measured over this range of motion. The dotted lines show the x, y and z axis. The left column shows the ranges of motion and kinematics for the hip joint: top panel, flexion—extension of the femur F relative to the pelvis P; middle panel, abduction—adduction of the femur; bottom panel, external—internal rotation of the femur. The hip kinematics corresponded most closely to the kinematics of a ball-and-socket joint. The second column shows the ranges of motion and kinematics for the knee joint: flexion—extension of the tibiofibula T relative to the femur, abduction—adduction of the tibiofibula and external—internal rotation of the tibiofibula (Tibfib). Flexion—extension kinematics at the knee corresponded most closely to the kinematics of a rolling joint, while the kinematics about the other axes corresponded more closely to the kinematics typical of hinge joints. The top panels of the third and fourth columns show the ranges of motion and kinematics for ankle flexion—extension (rotation of the astragalus segment A relative to tibiofibula), and tarsometatarsal flexion—extension (rotation of the metatarsals M relative to the tarsals). The ranges of motion (ROMs) about the other axes of these two joints were minimal (<20°). Flexion—extension kinematics at the ankle corresponded most closely to the kinematics of a rolling joint. Tarsometatarsal kinematics was represented in the model as a hinge joint (i.e. a single instantaneous center of rotation throughout the range of motion). The bottom right panel shows the ranges of motion and kinematics for the iliosacral joint (flexion—extension of the vertebral column V relative to the pelvis; U, urostyle). The kinematics at this joint corresponded most closely to a gliding joint. The inset shows a diagram of the sacral diapophysis, which is the transverse process of the sacrum that forms a joint with the iliac process of the pelvis.

View larger version (40K): [in a new window]   Fig. 4. The body segments of the frog were modeled as geometric primitives of uniform density. To the left is a scanned image of the whole frog body. To the right are the geometric solids used to approximate the inertial properties of the skull, trunk, pelvis, thigh and calf segments (stadium solids; see Materials and methods), the astragalus segment (cylinder) and the foot segment (cone). The dimensions, mass, averaged density and estimated inertias for each segment are shown in Table 1.

View larger version (29K): [in a new window]   Fig. 5. The four frog models on which forward dynamic simulations of jumping were performed. Here, we assume that the hindlimbs are symmetrical with respect to jumping. Hence model 1 had five rotational degrees of freedom (DOFs). These DOFs are flexion—extension at the iliosacral (1), hip (2), knee (3), ankle (4) and tarsometatarsal (5) joints. Model 2 had seven rotational DOFs. The two extra DOFs compared with model 1 (6 and 7, shown in red) are abduction—adduction and external—internal rotation at the hip. These DOFs permitted the plane of the hindlimb to be rotated under the body and at different angles relative to the ground. Model 3 had eight rotational DOFs. The extra DOF compared with model 2 (8, shown in red) is external—internal rotation at the knee. This DOF permitted the distal limb, consisting of the tibiofibula, astragalus segment and foot, to be rotated further under the body. Model 4 had nine rotational DOFs. The extra DOF compared with model 3 (9, shown in red) is flexion—extension at the metatarsophalangeal joint. This DOF permitted the frog to move its center of mass longer distances during the ground-contact phase of the jump and to achieve higher take-off velocities

View larger version (36K): [in a new window]   Fig. 6. Jumping performance of model 1. (A) Model 1 did not permit rotations other than flexion—extension at the hindlimb joints. In a normal starting position (shown in Fig. 7A), jump distance was very short compared with the real frog (blue versus black recordings in D). Hence, to assess better its jumping potential, model 1 was placed in an unnatural starting position in which the plane of the hindlimbs and the long axis of the pelvis were oriented at 42° to the ground. The purple, orange and green arrows represent the ground reaction forces (GRFs) at the starting position that were produced by a unit extensor torque (1 N m) about the hip, knee and ankle joints, respectively. GRFs are in normalized units (i.e. N per N m of torque), so a torque value of 0.009 N m at the hip will produce 0.15 N of GRF (i.e. 0.009 N mx15 N N-1 m-1). At the starting position, a unit hip extensor torque produced the largest propulsive GRF. (B) The path of the center of mass (COM) of the frog during the ground-contact phase of the jump for 100 simulation runs in which the magnitudes of the extensor torques driving each relaxed DOF were randomly varied. The red path in B—D represents the simulation run in which the actual torques produced by the real frog were used to drive the model. The blue path represents a simulation run in which model 1 was placed at a more natural starting position in which the pelvis was oriented at 15° to the ground. (C) The vertical VV and horizontal VH velocity of the COM for the red and blue runs did not match the velocity of the real frog (black lines). (D) The predicted jump distances for the red and blue runs were shorter than those for the real frog. (E) The vertical and horizontal velocities were tightly correlated (r2=0.97, P<0.001) during simulations, signifying that take-off angles were the same for each run and equal to the angle of pelvis tilt. This occurs because the vectors of GRFs for a given torque are in the same direction for each joint (see A). (F) Accordingly, the magnitudes of vertical and horizontal velocities were tightly correlated to GRF (r2=0.90, P<0.01 for vertical and r2=0.81, P<0.01 for horizontal velocities).

View larger version (46K): [in a new window]   Fig. 7. Jumping performance of model 2. (A) Model 2 was placed in a normal starting position. The colored arrows represent ground reaction forces (GRFs) as in Fig. 6. In addition, the GRF per unit N m of torque is shown for hip external rotation (yellow) and hip adduction (blue). (B) The path of the center of mass (COM) of the frog during the ground-contact phase for 500 simulation runs in which the magnitudes of hindlimb torques were randomly varied. A large range of take-off angles was produced from a single starting position. The blue path in B—D represents the simulation run in which the actual torques produced by the real frog were used to drive the relaxed degrees of freedom. The red path represents a simulation run in which hip external rotation was increased fourfold compared with that produced by the real frog during a jump. (C) The vertical VV and horizontal VH velocities of the COM for the red simulation run matched those of the real frog (black lines) better than the blue run. However, this required an unphysiological level of external rotation torque. (D) The predicted jump distances for the red and blue runs were smaller than those for the real frog. (E) Unlike model 1, the vertical and horizontal velocities for each simulation run were not correlated with one another (i.e. take-off angle varied from trial to trial). This was because individual torque components produced different ratios of vertical to horizontal GRF (see arrows in A). (F) The magnitudes of the hip (HE) and ankle extensor (AE) torques were significantly (P<0.01; r2=0.69 and r2=0.63, respectively) correlated with variations in the peak horizontal velocity among the simulation runs. Only the magnitude of the hip external rotation (HR) torque was significantly (P<0.01, r2=0.59) correlated with variations in the peak vertical velocity.

View larger version (55K): [in a new window]   Fig. 8. Internal rotation of the tibiofibula at the starting jump position enhances the vertical component of the ground reaction force (GRF). Left column, model 3; right column, model 2; bottom panels, position of models at the start (0 ms) of the jumping simulation; top panels, position of models and orientation of the GRF (red arrow) 30 ms into the simulation. Model 3 had an extra degree of freedom about the knee compared with model 2, wherein the tibiofibula (the bone colored pink on the right side of model 3) was internally rotated about its long axis. By bringing the foot under the frog's body, this internal rotation increased the vertical component of the GRF relative to the horizontal component during the early portion of the jumping simulation. The GRF shown for both models was calculated in response to the same extensor torque pattern applied about the hip, knee and ankle joints.

View larger version (45K): [in a new window]   Fig. 9. Jumping performance of models 3 and 4. (A) Models 3 and 4 were both placed in the normal starting position. The colored arrows represent the ground reaction forces (GRFs) as in Figs 6 and 7 for both models. Note that the GRF generated by internal rotation at the knee is mostly lateral in direction (i.e. out of the page) and hence is not shown. (B) The path of the center of mass (COM) of model 3 during the ground-contact phase for 500 simulation runs in which the magnitudes of hindlimb torques were randomly varied. The red path in B-D represents the simulation run in which the actual torques produced by the real frog were used to drive the degrees of freedom (DOFs) in model 3. The blue path represents the simulation run in which the same torque pattern was used to drive model 4. (C) The vertical VV and horizontal VH velocities of the COM for the red simulation run matched those of the real frog (black lines) over the first 70ms. At this time, model 3 was maximally extended and the simulation ended. The vertical and horizontal velocities of the COM of model 4 more closely matched those of the real frog over the entire 90ms take-off phase (i.e. addition of the distal joint allowed model 4 to extend further during the remaining 15ms of the jump). (D) The predicted jump distance for model 3 was less than that of the real frog. However, the predicted jump distance for model 4 closely approximated that of the real frog. (E) As in model 2, vertical and horizontal velocities in model 3 were not correlated. (F) The magnitude of only the hip extensor (HE) torque was significantly (P<0.01, r2=0.71) correlated with variations in the peak horizontal velocity among the simulation runs in model 3. No single torque component was significantly correlated with variations in vertical velocity. In trials in which the ankle extensor (AE) torque was greater than 0.3 N cm (boxed region in the VV versus AE torque graph), the time (T) taken for the ankle to extend past 90° was significantly (r2=0.61, P<0.05) correlated with variations in vertical velocity (right panel). The later the ankle extended during the ground-contact phase, the larger the vertical velocity.

View larger version (25K): [in a new window]   Fig. 10. Comparison of the joint kinematics of model 3 (red lines), model 4 (blue lines) and experimental frogs (black line represents data from one frog). The forward dynamics of models 3 and 4 were driven with the joint torque pattern estimated from the kinematics of experimental frogs. The hindlimb joint angles of both models closely corresponded to the experimental data for the first 70ms of each simulation run. After 60-70ms, the metatarsal joint (Meta) of experimental frogs begins to extend (lower right panel). Model 3 did not capture this reversal of tarsometatarsal joint motion because the metatarsal—phalangeal segment was fixed to the ground. Model 4, which allowed passive rotation of the metatarsal segment above the ground (i.e. no active torques were applied about the tarsometatarsal joint), did capture this kinematic effect. ab—add, abduction—adduction; Ext, external.

View larger version (26K): [in a new window]   Fig. 11. The sensitivity of jumping performance to variations in the magnitudes of the iliosacral extensor torque and the hindlimb torques examined using model 4. The torque pattern in Fig. 9 was used to drive the forward dynamics, and individual torques were scaled in amplitude by a factor of 0.80-1.20. HR, hip external rotation torque; HA, hip adduction torque; HE, hip extensor torque; KE, knee extensor torque; AE, ankle extensor torque; IE, iliosacral extensor torque. The take-off angle (trajectory of the center of mass, COM) was most sensitive to variations in the amplitude of the hip external rotation torque (i.e. the range of take-off angles was largest when HR was scaled from 0.80 to 1.20 of its base value). The horizontal velocity VH was most sensitive to variations in the hip extensor torque. The vertical velocity VV was most sensitive to variations in the hip external rotation torque and the knee extensor torque.