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Jumping in frogs: assessing the design of the skeletal system by anatomically realistic modeling and forward dynamic simulation
* 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
Author for correspondence (e-mail:
lrome{at}sas.upenn.edu
)
Accepted 25 March 2002
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Summary |
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 TOP Summary IntroductionMaterials and methodsResultsDiscussionReferences |
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Key words: frog, jumping, Rana pipiens, modelling, behaviour degrees of freedom, skeleton, joint, torque
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Introduction |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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,
1998). An ultimate goal of
this integrative approach is to understand enough about the molecular and
macroscopic components of muscular systems so that a comprehensive model can
be developed that would enable us to predict how alterations in one parameter
(e.g. crossbridge detachment rate) will affect motor performance. With the recent development of new biophysical and whole-animal techniques, we are for the first time in the position where molecular properties can be related to whole-animal function in a quantitative manner. To proceed to this new level, it is important to have an animal and behavioral model in which (i) muscle length changes and the recruitment pattern of the responsible fiber types can be determined, (ii) the overall body biomechanics are well defined and (iii) the molecular and biophysical properties of the fiber types are measurable.
The frog Rana pipiens presents a superb model in all these
respects. Although different fiber types in frogs are not anatomically
separated as in fish (Rome et al.,
1984), the extensor muscles used for jumping are quite homogeneous
in fiber type and mechanical properties
(Lutz et al., 1998). In
addition, there is compelling evidence that during maximal-distance jumping
all the extensor muscle fibers are maximally activated
(Hirano and Rome, 1984; Lutz
and Rome, 1994,
1996a). Thus, the extensor
muscles of a jumping frog behave similarly to an isolated muscle experiment in
which the fiber (or bundle pure in fiber type) is maximally activated by
direct electrical stimulation. This represents a tremendous simplification in
terms of modeling. Further, frog muscle fibers are amenable to all
physiological and biophysical techniques. Finally, because of the large muscle
strains compared with cyclical locomotory movements such as running and
swimming, the muscle length changes and overall body mechanics during the
one-shot ballistic jump of frogs can be relatively easily quantified
(Calow and Alexander, 1973;
Hirano and Rome, 1984;
Marsh, 1994;
Marsh and John-Alder, 1994;
Peplowsiki and Marsh, 1997).
Still, a significant obstacle to integrating from muscle function to
locomotion is that the musculoskeletal system of any animal is complex.
Previously, we conducted experiments on the semimembranosous muscle of frog
and tried to relate its mechanical performance to overall jumping performance
(Lutz and Rome, 1994,
1996a,
b). However, frog hindlimbs
have in excess of 15 muscles that contribute to overall performance, and these
muscles may perform different types of contraction
(Mai and Lieber, 1990;
Olson and Marsh, 1998;
Gillis and Biewener, 2000).
Thus, it is difficult to predict whole-animal movements from the mechanics of
a single (or even a few) muscles. Musculoskeletal modeling can be an enormous
help by keeping track of the forces generated by multiple muscles, so that the
net action of all the muscles can be determined. In addition to muscle
function, modeling can provide insight into how other physical components
(e.g. joints, ligaments, bones and segment mass distributions) affect the
transformation of neuromotor commands into limb and body motions
(Crago, 2000;
Dhaherlab et al., 2000; Pandy
and Sasaki, 2001; Yeadon,
1990).
In this study, we developed a skeletal model of the frog that contained the
bones, joints and segment masses and moments of inertia as a first step
towards creating an integrative musculoskeletal model. In addition to
measuring and describing the anatomical features of the frog skeleton, we used
the model along with a reverse-engineering approach to test important aspects
of the design and function of the skeletal system of frogs. Frog jumps differ
from those of humans and other mammals in several important ways. In frogs,
the hindlimb bones do not lie in a single plane throughout the jump, and
hindlimb joint rotations other than extension are prominent
(Lombard and Abbot, 1906;
Gans and Parsons, 1966).
Further, two joints (the tarsometatarsal and iliosacral), which are nearly
fixed in humans, are flexible in proficient jumpers such as Rana
pipiens, and they may contribute greatly to performance
(Emerson and de Jongh,
1980).
We tested the importance of the extra joints and degrees of freedom using our model. We performed a series of forward dynamic simulations of jumping while varying the number of joints and degrees of freedom in different configurations of the model. We compared simulated jumping performance with the jumping performance of real frogs. Further, because the ability to alter the jumping trajectory may be important in the frog's behavioral repertoire, we also tested how these additional joints and degrees of freedom create opportunities to produce a wide range of jumping trajectories.
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Materials and methods |
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,
1996a,
b) was analyzed. The film
contained orthogonal views of jumps (top and side views), and corrections were
made, as detailed by Lutz and Rome
(1996a), for parallax errors
that occur with a 45° mirror. From the films, we determined the trajectory
of the frog's center of mass (COM), which was located near the center of the
abdominalthoracic segment, the three-dimensional joint angles at the hip and
knee and the one-dimensional joint angles (flexion—extension) about the
iliosacral, ankle, tarsometatarsal and metatarsophalangeal joints
(Fig. 1). We followed a
procedure detailed by Vaughan et al.
(1996) for calculating
three-dimensional joint angles at the hip and knee. This procedure is detailed
in Electronic Appendix 1. In short, three markers were digitized on the
pelvis, thigh and calf segments. An orthogonal x, y, z reference
system was embedded in each of these segments based on the locations of the
markers. The angular orientation of the segments was determined in
three-dimensional space, and the orientation of one segment was determined
relative to another (e.g. the thigh relative to the pelvis and the calf
relative to the thigh). Five jumps in three different frogs were examined in
this way.
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To find the torques produced during jumping
(Fig. 1), we performed an
inverse dynamic analysis. The joint velocities and accelerations were
estimated using a difference equation in which the difference between data
points was 5 ms (i.e. 200 frames s-1). The time series of joint
angles, joint velocities and joint accelerations were input to SIMM (Software
for Interactive Musculoskeletal Modeling, Motion Analysis Corporation, Santa
Rosa, CA, USA), which is a graphical modeling environment, together with the
estimated inertial parameters of the frog body segments (e.g. center of mass
location, mass and inertia tensor, see Segmental inertial
measurements). Dynamics Pipeline Software (Motion Analysis Corporation,
Santa Rosa, CA, USA) was then used to connect the SIMM motion file to SD/Fast
(Symbolic Dynamics, Inc., Mountain View, CA, USA). The SD/Fast software then
solved the following inverse dynamic equation for the system (in 1 ms time
steps):
 | (1) |
and
are the first and second derivatives,
respectively, of q, T(q,) is the vector
of joint torque inputs (due to muscle activation) that is driving joint
motion, G(q) and V(q,) are
vectors of gravity- and motion-dependent terms and I is the system
mass matrix. SD/Fast used Kane's method to compute
T(q,) required to produce the
body-segment rotations measured from kinematic analyses. In performing these
calculations, the metatarsal segment was assumed to be rigidly fixed to the
ground to avoid having to supply the ground reaction forces to the inverse
dynamics solver. Joint torque values were computed for a total of five jumps
by three different frogs.
Bone scanning
The bones of the frog Rana pipiens (Schreber) were scanned using a
three-dimensional laser scanner (resolution 50µm) manufactured by Cyberware
(Cyberware Inc., Monterey, CA, USA) and controlled by a Silicon Graphics
O2 UNIX computer. An average-sized frog, 28 g mass and with an
extended hindlimb length of approximately 90 mm, was killed with an overdose
of Tricaine (Sigma Chemical Co.) and pithed in accordance with IACUC
procedures. Excess muscle, organs and connective tissues were dissected from
the skeleton, but all tissues surrounding the joints were left intact to
ensure proper joint motion. The intact skeleton of the frog was placed on a
rotating stage, and the scanner was initiated to move in the horizontal
direction to obtain one surface scan of the skeleton. The stage was rotated by
10°, and a second surface scan of the skeleton was taken. The skeleton was
scanned and rotated 36 times (i.e. in 10° increments) to obtain a complete
three-dimensional scan. The skeleton was then placed on the rotating stage in
a different orientation and a second three-dimensional scan was obtained. This
was performed five times to obtain five complete scans. The scans were merged
into a single three-dimensional image of the skeleton using software from
Cyberware. Individual bone segments were then disarticulated, and the
remaining skeletal complex was scanned using the above procedure. All the
removed bone segments were individually scanned as well. This procedure was
used so that the relative positioning between bone segments was maintained in
the graphical modeling environment (see below). For example, the femur and
tibiofibula, which are connected at the knee joint, were scanned together with
connective tissues intact and then individually scanned after disarticulating
the two bones. The individual scans were then correctly positioned relative to
each other by matching their orientations to an overlaid scan of the entire
bone complex.
The three-dimensional images of the individual bone segments were converted into bone files by a utility program in SIMM 2.2. The bone files, which list the polygons and polygon coordinates that compose the three-dimensional image, were then imported into SIMM, where the correct orientation between bones was maintained.
Establishment of local coordinate frames
In SIMM, the individual bone segments were positioned in a configuration
that served as an arbitrary starting point or reference anatomical position.
In this configuration, all the bones rested in a horizontal plane (see
Fig. 2). A local coordinate
frame was attached to the following bone segments: femur, tibiofibula,
astragalus—calcaneus segment, metatarsophalangeal segment, pelvis,
urostyle, vertebral column (all nine vertebrae considered as a single rigid
segment) and skull.
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The orientation and the origin of the local coordinate frames (LCFs) were established as follows. The pelvis LCF was oriented such that the x axis pointed from the central acetabulum of the right hip joint through the central acetabulum of the left hip joint. The z axis was orthogonal to the x axis and pointed dorsally in the reference configuration (i.e. out of the page when looking down on the frog). The y axis was determined by the right-hand rule and pointed caudally along the long axis of the pelvis. The origin of the pelvis LCF was positioned mid-way between the centers of the right and left acetabula.
The LCF for both the right and left femora was oriented such that the x axis was parallel to the long axis of the femur and pointed to the frog's left when in the reference position. The z axis was orthogonal to the x axis and pointed dorsally in the reference position. The femur y axis was determined by the right-hand rule and pointed caudally in the reference position. The origin of the femur LCF was positioned at the instantaneous center of femoral rotation relative to the pelvis (see Joint kinematics: descriptions, measurements and modeling). This position was located approximately 1.5 mm from the most central, proximal point of the femur and within the femoral head. The LCFs for the tibiofibula, astragalus—calcaneus and metatarsophalangeal segments were oriented in a manner similar to that of the femur LCF, i.e. the x axis for each LCF was parallel to the long axis of the bone segment, the z axis pointed dorsally in the reference configuration and the y axis was determined by the right-hand rule. The origin of each of these segments' LCFs was positioned to intersect with the most proximal, central point of the respective bone segment.
The origin of the vertebral segment's LCF was positioned at the most
caudal, central tip of the sacrum. The sacrum is the most caudal vertebra next
to the elongated urostyle, and its transverse processes form a joint with the
most rostral tips of the iliac crest
(Emerson and de Jongh, 1980).
In the reference configuration, the z axis of the vertebral segment
pointed dorsally, the x axis pointed to the left of the frog and the
y axis pointed caudally. The origin of the skull's LCF was positioned
at a central point within the foramen magnum at the level of the skull's
attachment to the first vertebra. The axes were oriented similarly to that of
the vertebral segment's axes. Finally, the origin of the urostyle's LCF was
positioned at the most rostral, central tip of the urostyle, where it
articulated with the sacrum. In this report, we do not discuss LCFs for the
forelimb bones and for the clavicle—scapula—sternum segment.
Joint kinematics: descriptions, measurements and modeling
A joint specifies the displacements that relate the position and
orientation of a moving bone segment relative to a reference or fixed bone
segment. In the frog model, the following joints were defined: hip joints,
displacement of the femur relative to the pelvis; knee joints, displacement of
the tibiofibula relative to the femur; ankle joints, displacement of the
astragalus—calcaneus segment relative to the tibiofibula;
tarsometatarsal joints, displacement of the metatarsophalangeal segment
relative to the astragalus segment; iliosacral joint, displacement of the
vertebral segment relative to the pelvis; and sacro-urostyle joint,
displacement of the urostyle relative to the vertebral segment. The forelimb
joints were ignored, and the joint between the first vertebra and skull was
fixed such that the angle between their respective y axes was
0°.
We used a custom-made jig apparatus (see
Lutz and Rome, 1996a) to
measure the kinematics of a moving joint member with respect to a fixed joint
member. For each joint examined, the fixed and mobile bone segments were
removed from frogs as a single unit. Major limb muscles were removed from the
bone segments, but small muscles, ligaments and other connective tissues
surrounding the joint capsule were left intact. The fixed and mobile members
were rigidly secured to the stationary and moving arms of the jig,
respectively, by Mizzy low-heat compound. For the hip, the pelvis was fixed
and the femur was mobile. For the knee, the femur was fixed and the
tibiofibula was mobile. For the ankle, the tibiofibula was fixed and the
astragalus—calcaneus segment was mobile. For the tarsometatarsal joint,
the astragalus was fixed and the metatarsal segment was mobile. For the
iliosacral joint, the pelvis was fixed and the vertebral column was mobile.
The jig permitted 180° of rotation and unopposed translation of the mobile
member relative to the fixed member within a single plane of motion. A digital
camera (Nikon Coolpix 990, 1.8 megapixels) was positioned orthogonal to this
plane of motion, 1.83 m from the approximate center of the joint. The
horizontal and vertical dimensions of the digital image were calibrated by
placing rulers in the view of the camera along both dimensions.
The joint members were placed in the reference position in the jig (reference position shown in Fig. 2), and the mobile member was first rotated about its z axis. Rotation about the z axis is the primary range of motion in the frog hindlimb joints and was referred to here as flexion—extension. The top row of Fig. 3 shows the flexion—extension ranges of motion for the hip, knee, ankle and tarsometatarsal joints. Counterclockwise rotation of the left femur about its z axis was termed hip extension and clockwise rotation was termed hip flexion (opposite convention for the right hip). Counterclockwise rotation of the left tibiofibula was termed knee flexion and clockwise rotation was termed knee extension (opposite for the right knee). Counterclockwise rotation of the left astragalus segment about its z axis was termed ankle extension and clockwise rotation was termed ankle flexion (opposite for the right ankle). The flexion—extension angle for each joint was the angle between the x axis of the moving segment and the x axis of the fixed segment (dotted line in top row of Fig. 3). Each hindlimb joint was rotated through a 160° range of flexion—extension, and a digital image was captured at each 10° increment.
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After measuring flexion—extension kinematics at the hindlimb joints, the joint members were re-positioned in the jig and placed in the reference configuration. The moving member was then rotated about its y axis. Rotation about the y axis of a hindlimb bone was referred to here as abduction—adduction. The second row of Fig. 3 shows the abduction—adduction ranges for the hip and knee. The ankle and tarsometatarsal joints had small (<20°) ranges of abduction—adduction and are not shown. Counterclockwise rotation of the left femur and tibiofibula about the respective y axes was termed adduction and clockwise rotation was termed abduction (opposite convention for the right hindlimb). The abduction angle was the angle between the z axis of the moving member in the reference position (dotted line in the second row of Fig. 3) and the z'-axis in the rotated position. The femur was rotated through an abduction—adduction range of 120°, and the tibiofibula was rotated through a range of 60°, each in 10° increments.
After measuring abduction—adduction kinematics at the hindlimb joints, the joint members were re-positioned in the jig and placed in the reference configuration. The moving member was then rotated about its long axis (x axis) using the jig's second, independent axis of rotation. Rotation about the long axis of a hindlimb bone is referred to here as external—internal rotation. The third row of Fig. 3 shows the external—internal ranges of motion for the hip and knee. The ankle and tarsometatarsal joints had small (<15°) ranges of external—internal rotation and so are not shown. When viewed proximally to distally down the shaft of the moving bone (as in Fig. 3), counterclockwise rotation about the long axis was termed internal rotation and clockwise rotation was termed external rotation. The external—internal rotation angle was the angle between the y axis of the moving segment in the reference position (dotted line in third row of Fig. 3) and the y'-axis in its rotated position. The femur was rotated through a range of 100°, and the tibiofibula was rotated through a range of 60°, each in 10° increments.
To measure the kinematics of the iliosacral joint, the pelvis was secured
to the fixed arm of the jig and the vertebral column was secured to the moving
arm. The vertebral column was rotated through a 100° range of motion about
its x axis, and images were captured every 10°. When viewed from
the frog's right side (as in the lower right panel of
Fig. 3), counterclockwise
rotation of the vertebral segment was termed vertebral extension and clockwise
rotation was termed flexion. Rotations about the other axes of the vertebral
segment are minimal in the frog (Emerson
and de Jongh, 1980), so these were not measured. Iliosacral joint
images were captured in four frogs, hip joint images in eight frogs, knee
images in six frogs and ankle images in five frogs.
The images were analyzed to determine the locations of the instantaneous
centers of rotation about each joint axis examined (see
Lieber and Boakes, 1988). To
minimize the errors associated with determining the instantaneous center of
rotation, extended wires (4 cm in length) were placed into the moving segment
before the joint images were captured. One wire was placed along the long axis
of the bone and a second wire was placed perpendicular to the long axis.
Markers (1 mm2) were then placed at the tips of each wire. The
marker positions (A and B) at each successive joint position
were digitized in Matlab. The location of the instantaneous center of rotation
was determined to be the intersection point of the perpendicular bisectors of
vector AnAn+1 and vector
BnBn+1, where n refers to the position
number and n+1 is the position resulting from a 10° rotation
(Kinzel and Gutkowski,
1983).
The joint images were analyzed to determine the locations of the LCFs for
the fixed and moving segments. The origins of the LCFs were marked on both
segments using small dots of paint (approximately 0.50 mm2). The
dot locations were digitized at successive rotation angles (10°
increments). The location of the moving segment's origin was subtracted from
the location of the fixed segment's origin at each rotation angle. Thus, for
each joint axis, the x and y locations of the moving
segment's LCF relative to the fixed segment's LCF were described as a function
of the rotation angle 
. This information was used to model the
appropriate kinematic functions in SIMM. In SIMM, three kinematic functions
were specified for each joint, one for each joint axis. So, for example,
translation between the femur and pelvis in the plane of hip extension was
specified as a function of the hip extension angle. SIMM smoothly interpolates
between the discretely specified variables using a natural cubic spline.
For three-dimensional rotations, the order of rotations about the specified
axes is important and must be specified for a unique description of joint
motion, i.e. the rotations are not commutative
(Kinzel and Gutkowski, 1983).
In our joint definitions, we specified the order of rotations to be rotation
about the z axis, x axis and then y axis of the
proximal segment's LCF. Rotation about the z axis, i.e.
flexion—extension, is the primary range of motion in the hindlimb, so
this was chosen as the first rotational component in each joint. We found that
changing the order of rotations had no discernible effect on the dynamic
behavior of frog models examined (see Forward dynamic modeling).
Segmental inertial measurements
The mass, moment of inertia and center of mass were determined for each of
the hindlimb and trunk segments. These measurements were then entered into a
segment description file for inputting to SIMM. The segment mass and moments
of inertia were determined in four frogs that had similar segment lengths (an
extended hindlimb length of approximately 90 mm, see
Table 1) and total mass (28 g)
to the frog that was laser-scanned. Each frog was killed and frozen in the
reference position. The body was cut into a number of segments; care was taken
to make the cuts at similar positions and orientations in each frog. The
segments included the thigh, calf, astragalus, foot (both metatarsals and
phalanges included), a pelvic segment, which spanned from the most caudal
aspect of the pelvis (ischium) to the most rostral tip of the iliac crest, an
abdominal-thoracic (trunk) segment, which spanned from the tip of the iliac
crest to the base of the skull, and the skull (see
Fig. 4). These segments
contained muscle, skin, tendon, organs and bone. Because these tissues have
slightly different densities, an average density was measured for each
segment. To do this, each segment was weighed to determine its mass
(M) and then lowered into a water-filled graduated cylinder and its
volume (v) was determined by weighing displaced water. The average
density (
) was then calculated as described in Nigg
(1999) as:
 | (2) |
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The moments of inertia for each segment were calculated as described by
Yeadon (1990) based on a
simplifying assumption that segment density was uniform and equal to the
averaged density. Each segment was represented as a geometric solid of uniform
density. We modeled the thigh, calf, pelvic, abdominal-thoracic and skull
segments as stadium solids (see Yeadon,
1990). A stadium solid is an elongated geometric solid bounded by
parallel stadia (i.e. a rectangle with an adjoining semicircle at each end of
its width) on its two ends. The stadium dimensions were estimated by measuring
several parameters of the frog segments. These parameters included the
perimeter, width and depth of the segment ends (i.e. the bounding stadia) and
the segment length (i.e. distance between the stadia). The astragalus segment
was modeled as a cylinder, the foot segment was modeled as a cone and the
specific dimensions for each were measured (see Electronic Appendix 2 for
calculation of moments of inertia for each segment).
Forward dynamic modeling
In this study, we used forward dynamic simulations to test how different
degrees of freedom in the hindlimb joints of the frog affect jumping
performance. Forward dynamic simulations were performed using the Dynamics
Pipeline software, which works by connecting the skeletal model in SIMM to
SD/Fast. SD/Fast computes and solves the equations of motion for the model
when given a set of forces or torques acting on the skeletal system. A
separate equation of motion is solved for each degree of freedom and is of the
general form described for other rigid-body, musculoskeletal models
(Crago, 2000;
Zajac, 1993):
 | (3) |
) is the vector of joint
moments due to muscle forces,
TP(q,) is a vector of passive
moments due to stretching of connective tissues about the joints and
TE(q,) is a vector of moments
that arise from interactions with the environment. In this study, we excluded
submodels of the muscles and neural control to focus solely on the joint
degrees of freedom that are critical for jumping performance. Therefore, to
drive the motion of the model, we specified a pattern of joint torque inputs
instead of specifying a muscle activation pattern. Thus,
TM(q,) from equation 3 was
replaced with user-defined pattern of torque inputs, TI. In
addition to simplifying the control input, we assumed the contributions of
passive structures TP(q,) to
be negligible, so this term was removed from equation 3. A series of progressively higher-dimensional models was constructed in which a kinematic degree of freedom (DOF) that was constrained in one model was relaxed in a subsequent model. The four models are described in the Results and shown schematically in Fig. 5. We used two strategies to examine the dynamic behavior of the frog models. In the first strategy, we wanted to explore the range of dynamic behaviors that the model was capable of producing. To do this, we applied unit torque steps about each relaxed, rotational DOF in the model to drive its motion. The torque steps were 80 ms in duration and applied synchronously about each joint. A vector of random numbers was generated before each simulation run to scale the magnitude of the applied torque steps. The scalars ranged from 0 to 0.009 N m for the hip extensor torque, from -0.004 to 0.004 N m for the hip external (-) or internal (+) rotation torque, from -0.004 to 0.004 N m for the hip adduction (-) or abduction (+) torque, from 0 to 0.007 N m for the knee extensor torque and from 0 to 0.007 N m for the ankle extensor torque. We set the maximum value for the extensor scalars (i.e. hip, knee and ankle extensor torques) to be the peak torque that the real frog produces during a representative, maximal-distance jump (see Kinematics and inverse dynamic analyses of frog jumping). For the other scalars, we chose an intermediate range of values in which both directions of torque (e.g. hip abduction and hip adduction) could be produced. For each model, 1000 simulations was run with different randomized scaling factors. We determined the trajectory of the COM, the take-off angle and the joint angles for each simulation run.
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The second strategy to examine the dynamic behavior of the frog models was simply to use the torque values produced by the real frog to drive the motion of the models. If we could not produce a maximal-distance jump in the model under study, then clearly something was lacking in the model.
Several assumptions were made in all the jumping simulations. In models
1-3, the right and left foot segments (metatarsals and phalanges) were fixed
to the ground. In model 4, only the phalanges were fixed to the ground. By
modeling the foot—ground contact as a jointed connection, ground
reaction forces were automatically included in the model rather than having to
supply them explicitly (Nigg,
1999). However, because each frog model was connected to the
ground, jumping distance had to be estimated. Jump distance was calculated as
the sum of the horizontal displacement of the COM during the take-off and
aerial phases of the jump. The horizontal displacement during the aerial phase
was estimated using ballistics equations described by Hirano and Rome
(1984).
A second assumption we made in each simulation was that the forelimb
segments could be removed without any effect on jumping performance. The
forelimb segments are not likely to contribute much, if any, power to the jump
(Calow and Alexander, 1973;
Hirano and Rome, 1984;
Peters et al., 1996;
Marsh, 1994). Also, the small
mass of the forelimb segments (approximately 5% of total body mass) is likely
to have a negligible effect on the trajectory of the center of mass. We also
assumed that the atlanto-occipital joint and intervertebral joints did not
contribute significantly to jumping, and these joints were therefore held
rigid in each model. Finally, we assumed that the iliosacral joint was a
revolute joint in each model. The digitized measurements provide evidence that
this joint may be a gliding joint, in which trunk translation and rotation are
independent of one another (see Behavior and modeling of the ankle,
tarsometatarsal, metatarsophalangeal and iliosacral joints in Results).
However, gliding joints are computationally difficult to model, and others
have hypothesized that translation of the trunk (relative to the pelvis) may
be important only during swimming and in frogs specialized for swimming
(Emerson and de Jongh,
1980).
Static analysis of force transmission
Measurements of the ground reaction force (GRF) can be used to predict the
trajectory of the frog's COM using relatively simple ballistics equations
(Hirano and Rome, 1984;
Marsh, 1994). It is unclear
whether and how the frog actively varies the GRF to generate different
trajectories and take-off angles. If the goal is to produce a maximal-distance
jump, the frog should generate GRFs that are oriented at approximately 42°
to the ground (Hirano and Rome,
1984). However, if the goal is to jump over an obstacle or to
generate low take-off angles (i.e. high accelerations), the frog must adjust
the GRF to higher or lower angles, respectively. The degrees of freedom in the
hindlimb models and the associated starting configuration might limit this
ability. To examine the range of force directions that each model can produce,
we calculated the Jacobian matrix for each model in its starting
configuration, which was determined from video analysis of jumping frogs (see
Kinematic and inverse dynamic analyses of jumping frogs). The
transpose of the Jacobian matrix relates the joint torques to the GRF by the
following:
 | (4) |
is an n-dimensional joint torque vector, F is an
m-dimensional end-effector output force and JT is
the transpose of the Jacobian matrix. J is an mxn
matrix, where m denotes the degrees of freedom of the end-effector
space and n denotes the number of actuated joint variables
(calculation of J is shown in Electronic Appendix 3). For each of the
models, the GRF during simulation runs was calculated at the starting
configuration of the limb. The velocity and joint angles of the ensuing,
dynamic jumps were then calculated. The GRFs at the starting limb
configuration were related to the trajectory of the frog models using linear
regression techniques.
Sensitivity analysis
We examined how sensitive jumping performance was to variations in the
magnitude of individual joint torques. The torque pattern that was estimated
using an inverse dynamic analysis of jumping was systematically modified by
scaling the magnitude of each torque (e.g. the hip extensor torque) to 80-120%
of its base value. Each torque component was individually examined in this
way, including the iliosacral extensor torque. The sensitivity
SP of the vertical and horizontal velocities of the COM
and the sensitivity of takeoff angles in response to a change in the torque
magnitude about a single axis was determined as:
 | (5) |
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Results |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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; Lutz and
Rome, 1996a; Peters et al.,
1996). In addition, we found that flexion occurred about the
tarsometatarsal joint for the first 60 ms and extension occurred about this
joint for the last 15-20 ms. We also found some degree of rotation about the
secondary degrees of freedom at the hip and knee joints. For the jump shown in
Fig. 1A, there was a moderate
amount of external rotation (range 30°) and abduction (range 25°)
about the hip joint. These joint motions acted to bring the femur into the
same plane as the long axis of the pelvis. We also found moderate degrees of
internal rotation (range 30°) and abduction (range 20°) about the knee
joint. These joint motions acted similarly to bring the tibiofibula into the
same plane as the femur and long axis of the pelvis. On the basis of the kinematics of the analyzed jumps and the measured inertial parameters of the hindlimb and axial segments (Table 1), we estimated the net torques produced about each of the degrees of freedom during jumping. We used an inverse dynamic analysis and assumed the metatarsal segment to be rigidly fixed to the ground (see Materials and methods). The net torques about the iliosacral, hip, knee and ankle joints, which correspond to the jump shown in Fig. 1A, are presented in Fig. 1B. Net torques at each joint varied with time. Extensor torques about the iliosacral, hip, knee and ankle joints peaked at successively later times into the jump (15, 40, 50 and 70 ms, respectively) and this temporal staggering was consistent for each jump analyzed. The peak magnitude of the extensor torque was larger about the hip than about the knee and ankle joints (0.85±0.02 N cm, 0.7±0.04 N cm and 0.7±0.04 N cm, respectively; means ± S.E.M., N=5 jumps) and relatively smaller about secondary degrees of freedom at the hip and knee. For some jumps (data not shown), hip adduction torques were more significant (e.g. peak of 0.65 N cm; peak magnitude 0.4±0.05 N cm, mean ± S.E.M., N=5 jumps). The finding that extension ranges of motion and extensor torques were larger than motion and torques about the other degrees of freedom indicates that most of the joint work was performed by hip, knee and ankle extension.
Bones and segment properties of Rana pipiens
The bone segments that were laser-scanned and used to construct the
skeletal models of Rana pipiens are shown in
Fig. 2. The mass, moment of
inertia, center of mass and geometric dimensions were determined for each of
the hindlimb and trunk segments (Fig.
4), and the mean values from 4 frogs are shown in
Table 1.
Behavior and model of the hip joint
A goal of this study was to determine the importance of joint degrees of
freedom to jumping performance. It was first necessary to measure the behavior
and degrees of freedom of each joint in the real frog and then use this
information to model the appropriate behavior of the virtual joints.
Flexion—extension is the primary range of motion at the hindlimb
joints and represents rotation of a bone segment about the z axis of
its LCF. The locations of the instantaneous centers of rotation for each joint
during flexion—extension are shown as a collection of red dots in the
top row of Fig. 3. The
instantaneous centers of hip extension tended to cluster into a single,
circumscribed region (area 1.6±0.19 mm2, mean ±
S.E.M., N=8) located within the femoral head and approximately 1.5 mm
from its most proximal point. This tight clustering indicated that the
location of the instantaneous center was approximately constant throughout the
range of passively applied hip extension. This was in agreement with
previously reported data (Lieber and
Shoemaker, 1992). Thus, we modeled the virtual hip joint in the
extension—flexion plane as a revolute joint in which the position of the
instantaneous center of rotation was fixed. The location of the instantaneous
center of rotation was positioned 1.5 mm along the long axis of the femur from
its most proximal point.
The rotational DOFs of the femur about its x and y axes represented hip external—internal rotation and hip abduction—adduction, respectively. The sequence of instantaneous centers of rotation for both external—internal rotation and abduction—adduction (first column, bottom two panels of Fig. 3) tended to cluster into a single, circumscribed region (areas 1.1±0.26 mm2 and 1.9±0.3 mm2, respectively, means ± S.E.M., N=8 frogs). For abduction—adduction, the instantaneous centers clustered in a position located approximately 1.5-2 mm from the femur's most proximal point. For external—internal rotation, the instantaneous centers clustered at a position near the center of the acetabulum. The tight clustering indicated that the location of the instantaneous center of rotation about each joint axis was approximately constant throughout the passively applied range of motion. Thus, the hip joint could be modeled as a gimbal joint, which consists of three independent revolute joints. The intersection of the instantaneous centers of rotation for each revolute joint was positioned 1.5 mm along the long axis of the femur, from its most proximal point, and at the level of the central acetabulum.
Behavior and modeling of the knee joint
For the majority of frogs examined (four out of six) the
flexion—extension kinematics at the knee conformed most closely to a
rolling joint. As shown in Fig.
3 (top row, second panel from left), the positions of the
instantaneous centers of rotation for the knee traversed a curve that
approximately traced the joint surface of the proximal bone. The instantaneous
center of rotation was located at one end of this curve at the extreme range
of flexion and `rolled' to the other end of the curve, along the surface of
the proximal bone, as the moving segment was extended. Thus, we modeled the
flexion—extension of the virtual knee as a rolling joint so that the
tibiofibula segments smoothly traversed an arc of 70° along the surfaces
of the distal femur.
The rotational DOF of the tibiofibula about its x axis was termed knee external—internal rotation. The range of knee external—internal rotation was approximately 60° (±30° from the reference configuration) before significant torsion of connective tissues surrounding the knee joint was noticed. The locations of the instantaneous centers of rotation tended to cluster into a single, circumscribed region located at the level of the mid-tibial crest (second column, bottom panel of Fig. 3). Thus, we modeled the knee joint as a type of universal or Hooke's joint, which consists of two independent joints. Knee flexion and extension occurred about a rolling joint, and knee external—internal rotation occurred about a revolute joint whose instantaneous center of rotation was located at the instantaneous center for knee flexion. That is, as the instantaneous center for knee flexion traversed the surface of the distal femur, the instantaneous center for external rotation was carried along with it. The measured range of knee adduction, i.e. rotation about the y axis, was 45-50° (see second column, middle panel in Fig. 3).
Behavior and modeling of the ankle, tarsometatarsal,
metatarsophalangeal and iliosacral joints
For the majority of frogs examined (three out of five), the
flexion—extension kinematics at the ankle conformed most closely to a
rolling joint. As shown in Fig.
3 (top row, third panel from left), the positions of the
instantaneous centers of rotation for the ankle traversed a curve that
approximately traced the joint surface of the proximal bone. The location of
the instantaneous center of rotation was located at one end of this curve at
the extreme range of flexion and `rolled' to the other end of the curve, along
the surface of the proximal bone, as the moving segment was extended. Thus, we
modeled the virtual ankle joints as a rolling joint so that astragalus
segments smoothly traversed a 90° arc along the surfaces of the
tibiofibula.
The tarsometatarsal joint was modeled as revolute joint. The instantaneous center of rotation was positioned at the point of contact between the distal end of the astragalus segment and the proximal metatarsals (see location of red dot in Fig. 3, top row, right panel).
Because of the difficulty in accurately measuring kinematics about this small and delicate metatarsophalangeal joint in the jig (the ends of the two bones were less than 1 mm in diameter), we simply modeled this joint as a revolute joint. The position of the instantaneous center of rotation was placed at the point of intersection between the bone segments.
The measurement of iliosacral kinematics is shown in Fig. 3 (bottom right panel). Flexion—extension of the vertebral segment occurred about its x axis. We examined iliosacral kinematics in four frogs. The instantaneous center of vertebral rotation did not follow a consistent path among these frogs. This variability might be due to the fact that the iliosacral joint is to some extent a true gliding joint. In a gliding joint, the x- and y-translations and rotations within the plane are independent of each other. To avoid the complexities associated with modeling such a joint, we approximated the iliosacral joint as a revolute joint, in which the center of rotation was located at the contact point between the tip of the iliac crest and the transverse processes of the sacrum. The measured range of motion was 90° (30° extended relative to the reference position and 60° flexed).
Four models of jumping frogs
A schematic diagram of the kinematic degrees of freedom making up the four
skeletal models is shown in Fig.
5.
Model 1: planar hindlimb model
For simplicity in modeling, it has sometimes been assumed that the hindlimb
joints of frogs extend within a single plane during jumping and that other
DOFs at the hip and knee could be ignored
(Alexander, 1995). In model 1,
we assessed this possibility by constraining all the hindlimb joints to only
flex and extend. The initial flexion angles at the start of the simulations
were determined for the hip, knee, ankle, tarsometatarsal and iliosacral
joints (see Kinematic and inverse dynamic analyses of jumping frogs).
If the pelvis of model 1 was positioned at 15-20° to the ground, similar
to the real frog, then the plane in which the hindlimb was oriented would also
be at 20° to the ground. Extension of the hindlimb within this plane would
necessarily lead to a low take-off angle and, hence, a short jump distance
(blue trace in Fig. 6D).
Take-off angles of 42° are necessary for maximal-distance jumping. Thus,
to test whether model 1 could in theory permit maximal-distance jumping, it
was necessary to invoke an unphysiological starting position in which the
pelvis was tilted at 42° to the ground and the hindlimbs rested in an
unnatural starting position (see Fig.
6A).
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We tested whether model 1 could reproduce maximal-distance jumping. To do this, we used the torque values generated by the real frog to drive the forward dynamics of model 1. Only the hindlimb `extensor' torques and the iliosacral torque were used to drive the model dynamics (i.e. the other hindlimb DOFs were fixed, and torques about these DOFs were therefore zero). When model 1 was placed in a physiological starting position in which the pelvis was oriented at 15° to the ground, the take-off angle was also 15° and the jump distance was only 0.380 m (blue lines in Fig. 6B-D). Even when unphysiological starting positions were used (pelvis tilted at 42° to the ground; red lines in Fig. 6B-D), the jump distance was only 70% of that obtained by the real frog. The black lines in Fig. 6C,D represent the trajectory of a real frog jumping at 25°C. The peak total velocity (i.e. the vector sum of the vertical VV and horizontal VH velocities) of the 42° run was 1.83 m s-1 compared with 2.33 m s-1 for the real frog, and the trajectory of the COM during the ground-contact phase resulted in a predicted jump distance of 0.552 m compared with 0.704 m for the real frog. The inability to produce both maximal-distance jumping and a range of take-off angles suggests that additional DOFs and joints are critical for jumping.
Model 2: three-DOF hip joint
In the actual frog, the hip joint is not constrained to only extend during
jumping; other DOFs at the hip might be critical for jumping performance.
Model 2 captures the three-dimensional properties of the hip by adding the
external—internal rotation and abduction—adduction DOFs. The
remaining hindlimb joints were constrained to only flex and extend. The hip
was positioned in its initial configuration as determined from the kinematic
analysis: flexed by 32°, adducted by 18° and internally rotated by
15°. The knee and ankle were initially flexed by 155° and 150°,
respectively.
We first examined the range of dynamic behaviors that model 2 could produce. To do this, a batch of simulations was run in which we randomly varied the magnitude of the torque steps applied about each rotational DOF. Trajectories of the virtual frog's COM are shown in Fig. 7B. Both internal and external rotation torques were applied about the femur's x axis, and both abduction and adduction torques were applied about the femur's y axis. Five hundred trajectories are shown starting from the onset of the torque steps for a period of 95 ms. What is most evident from Fig. 7B is that a large range of take-off angles was produced in this model compared with model 1. The take-off angles ranged from 0 to 90° relative to the ground. The peak vertical and horizontal velocities of the COM showed no significant correlation among simulation runs (see Fig. 7E) because, unlike model 1, the individual hindlimb torques produced different ratios of horizontal to vertical GRF.
We examined the GRFs produced by each hindlimb torque at the starting limb configuration. Fig. 7A shows the GRF vectors produced by a unit hip extensor (purple), knee extensor (orange), ankle extensor (green), hip external rotation (yellow) and hip adduction (blue) torque. The GRF vectors are based on unit torque inputs, but it is important to keep in mind that these vectors will be scaled by the actual torque values shown in Fig. 1 (e.g. the GRF due to a unit hip extensor torque is 16.93 N N-1 m-1 and thus the GRF due to 0.009 N m of hip extensor torque is 0.15 N). A unit hip extensor torque produced a large horizontal and smaller vertical force (ratio 15.5:6.8). A unit ankle extensor torque produced a similar ratio of horizontal to vertical force (13.2:8.0). A unit knee extensor torque produced a relatively small horizontal force (2.1 N) and a vertical force (7.2 N) comparable with that produced by hip and ankle extensor torques. The knee extensor torque produced a very large lateral force (18.2 N) compared with the lateral forces produced by the hip (-5.1 N; negative values represent medially directed forces) and ankle extensor unit torques (5.9 N). Both hip external rotation and hip adduction unit torques produced relatively large vertical forces (11.1 and 9.3 N, respectively), but horizontal forces that opposed forward translation (-8.0 and -9.1 N, respectively).
On the basis of the static descriptions of torque transmission, we predicted that hip and ankle extensor torques should accelerate the COM most strongly in the horizontal direction and that hip external rotation and adduction torques should accelerate the COM most strongly in the vertical direction. This relationship was in fact observed (see Fig. 7F). The magnitude of both the hip and ankle extensor torques was significantly (P<0.01) correlated with the peak horizontal velocity of the COM (r2=0.69 and r2=0.63, respectively). The magnitude of the hip external rotation torque was significantly correlated with the peak vertical velocity (P<0.01, r2=0.59). The hip adduction torque did not show a significant correlation with peak vertical velocity. Thus, increasing the external rotation torque will produce higher take-off angles and lower acceleration take-offs, and increasing ankle and hip extensor torques will produce lower take-off angles and higher acceleration take-offs. However, it is important to keep in mind that the majority of the jumping muscles are biarticular, and independent regulation of hindlimb torques may not be possible in the real frog.
We tested whether model 2 produced maximal-distance jumping when the real jumping torques (shown in Fig. 1) were used to drive its forward dynamics. To our surprise, we found that model 2 did not produce maximal-distance jumping. Instead, the take-off angle was approximately 13° and the vertical velocity was only 0.4 m s-1 (blue trajectories in Fig. 7B-D). This resulted in a predicted jump distance of 0.370 m (Fig. 7D). If we increased the hip external rotation torque by four times that observed in the real frog, model 2 produced jumps that more closely resembled maximal-distance jumps (red trajectory in Fig. 7B-D). That we could not produce maximal-distance jumping in model 2 using physiological estimates of hindlimb torque values suggested that additional DOFs must be added to the frog model.
Model 3: two-DOF knee joint
The frog knee joint exhibits an overflexion mechanism in which the calf is
rotated along its long axis and carried over the dorsal aspect of the thigh in
the extreme ranges of knee flexion
(Lombard and Abbot, 1906).
This over-flexion mechanism may enhance the jumping performance of the model.
Thus, we added this DOF at the knee joint in model 3. The knee was then
internally rotated by 30°, the estimated rotation angle at the starting
position of the jump (see Materials and methods). As shown in
Fig. 8, this rotation brought
the foot more underneath the body and more within the sagittal plane compared
with model 2 and, thereby, increased the vertical component of the GRF. As
described above, there is an additional DOF in the knee in the
adduction—abduction plane. Preliminary simulations showed that this DOF
had little effect on jumping performance and thus, for computational
simplicity, we fixed this DOF so that the adduction angle was constant at
90°C. The remaining joint angles were the same as the initial angles in
model 2.
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We first examined the range of dynamic behaviors that model 3 could produce by randomly varying the magnitude of torque steps applied about each rotational DOF. The trajectories of the virtual frog's COM, which were generated by driving the forward dynamics of the model with randomized torque steps, are shown in Fig. 9B. Both internal and external rotation torques and both abduction and adduction torques were applied at the hip. Five hundred trajectories are shown starting from the onset of the torque steps for a period of 95 ms. Similar to model 2, we found that model 3 produced a large range of take-off angles (0-90°) from a single starting position. However, unlike model 2, we found that model 3 produced near-maximal-distance jumping using physiological estimates of hindlimb torque values.
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When the hindlimb torques computed in the real frog were used to drive the forward dynamics of model 3, the simulated jump closely matched that of the real frog. Fig. 9C shows the horizontal and vertical velocity of the COM of model 3 (red lines) compared with the real frog (black lines). Fig. 10 shows the hindlimb joint angles of model 3 (red lines) compared with the real frog (black lines). The trajectory of the COM and the hindlimb joint angles were very similar for the first 70 ms of the jump. After that time, the hindlimb of model 3 was maximally extended and the simulation was terminated. This early termination was due to the fact that the metatarsophalangeal segment of model 3 was rigidly secured to the ground. Thus, unlike the real frog, the tarsometatarsal joint of model 3 did not extend during the last 10-15 ms of the jump (this joint flexes during the first 60-70 ms of the jump; see Fig. 10). The jump of model 3 had a predicted distance of 0.612 m compared with 0.704m in the real frog (Fig. 9D). Before examining how adding the metatarsophalangeal joint enhances jumping performance, we first examined in more detail why model 3 produced a much better jump than model 2.
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We examined how the GRFs produced by the individual hindlimb torques were different in model 3 compared with model 2. The GRF vectors produced by unit torque inputs are shown in Fig. 9A (purple, hip extensor; green, ankle extensor; orange, knee extensor; yellow, hip external rotation; blue, hip adduction). The GRFs produced by hip extensor, knee extensor, hip external rotation and hip adduction torques were similar to the GRFs produced by the same torques in model 2, i.e. the ratio of vertical to horizontal to lateral force for each torque was similar in both models. However, an ankle extensor torque in model 2 produced GRFs that were dramatically different from those produced in model 3. A unit ankle extensor torque produced a vertical force that was twice that produced in model 2 (15.8 N compared with 8.0 N). Thus, internal rotation of the tibiofibula (Fig. 8, left panel) allowed the ankle torque to produce a GRF with a larger vertical than horizontal component (15.8:5.0). In terms of absolute values, the real torque pattern in the frog produced a total GRF at the starting limb position in which the vertical component was 0.99 N for one hindlimb (1.98 N for both limbs) and the horizontal component was 1.01 N (2.02 N for both limbs). The same torque pattern
