Comparative analysis of Dipodomys species indicates that kangaroo rat hindlimb anatomy is adapted for rapid evasive leaping.

Body size is a key factor that influences antipredator behavior. For animals that rely on jumping to escape from predators, there is a theoretical trade-off between jump distance and acceleration as body size changes at both the inter- and intraspecific levels. Assuming geometric similarity, acceleration will decrease with increasing body size due to a smaller increase in muscle cross-sectional area than body mass. Smaller animals will likely have a similar jump distance as larger animals due to their shorter limbs and faster accelerations. Therefore, in order to maintain acceleration in a jump across different body sizes, hind limbs must be disproportionately bigger for larger animals. We explored this prediction using four species of kangaroo rats (Dipodomys spp.), a genus of bipedal rodent with similar morphology across a range of body sizes (40-150 g). Kangaroo rat jump performance was measured by simulating snake strikes to free-ranging individuals. Additionally, morphological measurements of hind limb muscles and segment lengths were obtained from thawed frozen specimens. Overall, jump acceleration was constant across body sizes and jump distance increased with increasing size. Additionally, kangaroo rat hind limb muscle mass and cross-sectional area scaled with positive allometry. Ankle extensor tendon cross-sectional area also scaled with positive allometry. Hind limb segment length scaled isometrically, with the exception of the metatarsals, which scaled with negative allometry. Overall, these findings support the hypothesis that kangaroo rat hind limbs are built to maintain jump acceleration rather than jump distance. Selective pressure from single-strike predators, such as snakes and owls, likely drives this relationship.

Elastic energy storage across speeds during steady-state hopping of desert kangaroo rats (Dipodomys deserti).

Small bipedal hoppers, including kangaroo rats, are not thought to benefit from substantial elastic energy storage and return during hopping. However, recent species-specific material properties research suggests that, despite relative thickness, the ankle extensor tendons of these small hoppers are considerably more compliant than had been assumed. With faster locomotor speeds demanding higher forces, a lower tendon stiffness suggests greater tendon deformation and thus a greater potential for elastic energy storage and return with increasing speed. Using the elastic modulus values specific to kangaroo rat tendons, we sought to determine how much elastic energy is stored and returned during hopping across a range of speeds. In vivo techniques were used to record tendon force in the ankle extensors during steady-speed hopping. Our data support the hypothesis that the ankle extensor tendons of kangaroo rats store and return elastic energy in relation to hopping speed, storing more at faster speeds. Despite storing comparatively less elastic energy than larger hoppers, this relationship between speed and energy storage offers novel evidence of a functionally similar energy storage mechanism, operating irrespective of body size or tendon thickness, across the distal muscle-tendon units of both small and large bipedal hoppers.

Instrument-Assisted Soft Tissue Mobilization Forces Applied by Trained Clinicians During a Simulated Treatment.

CONTEXT: Instrument-assisted Soft Tissue Mobilization (IASTM) is a therapeutic intervention used by clinicians to identify and treat myofascial dysfunction or pathology. However, little is known about the amount of force used by clinicians during an IASTM treatment and how it compares to reports of force in the current literature. OBJECTIVE: To quantify the range of force applied by trained clinicians during a simulated IASTM treatment scenario. DESIGN: Experimental. SETTING: University research laboratory. PARTICIPANTS: Eleven licensed clinicians (physical therapist = 2, chiropractor = 2, and athletic trainer = 7) with professional IASTM training participated in the study. The participants reported a range of credentialed experience from 1 to 15 years (mean = 7 [4.7] y; median = 6 y). INTERVENTION: Participants performed 15 one-handed unidirectional sweeping strokes with each of the 5 instruments for a total of 75 data points each. Force data were collected from a force plate with an attached skin simulant during a hypothetical treatment scenario. MAIN OUTCOME MEASURES: Peak force and average forces for individual strokes across all instruments were identified. Averages for these forces were calculated for all participants combined, as well as for individual participants. RESULTS: The average of peak forces produced by our sample of trained clinicians was 6.7 N and the average mean forces was 4.5 N. Across individual clinicians, average peak forces ranged from 2.6 to 14.0 N, and average mean forces ranged from 1.6 to 10.0 N. CONCLUSIONS: The clinicians in our study produced a broad range of IASTM forces. The observed forces in our study were similar to those reported in prior research examining an IASTM treatment to the gastrocnemius of healthy individuals and greater than what has been reported as effective in treating delayed onset muscle soreness. Our data can be used by researchers examining clinically relevant IASTM treatment force on patient outcomes.

Plantar flexor muscles of kangaroo rats (Dipodomys deserti) shorten at a velocity to produce optimal power during jumping.

The musculotendon work contributions across all joints during jumping by kangaroo rats are not well understood. Namely, measures of external joint work do not provide information on the contributions from individual muscles or in-series elastic structures. In this study, we examined the functional roles of a major ankle extensor muscle, the lateral gastrocnemius (LG), and a major knee extensor muscle, the vastus lateralis (VL), through in vivo sonomicrometry and electromyography techniques, during vertical jumping by kangaroo rats. Our data showed that both muscles increased shortening and activity with higher jumps. We found that knee angular velocity and VL muscle shortening velocity were coupled in time. In contrast, the ankle angular velocity and LG muscle shortening velocity were decoupled, and rapid joint extension near the end of the jump produced high power outputs at the ankle joint. Further, the decoupling of muscle and joint kinematics allowed the LG muscle to prolong the period of shortening velocity near optimal velocity, which likely enabled the muscle to sustain maximal power generation. These observations were consistent with an LG tendon that is much more compliant than that of the VL.

Functional morphology of the ankle extensor muscle-tendon units in the springhare Pedetes capensis shows convergent evolution with macropods for bipedal hopping locomotion.

This study assesses the functional morphology of the ankle extensor muscle-tendon units of the springhare Pedetes capensis, an African bipedal hopping rodent, to test for convergent evolution with the Australian bipedal hopping macropods. We dissect and measure the gastrocnemius, soleus, plantaris, and flexor digitorum longus in 10 adult springhares and compare them against similar-sized macropods using phylogenetically informed scaling analyses. We show that springhares align reasonably well with macropod predictions, being statistically indistinguishable with respect to the ankle extensor mean weighted muscle moment arm (1.63 vs. 1.65 cm, respectively), total muscle mass (41.1 vs. 29.2 g), total muscle physiological cross-sectional area (22.9 vs. 19.3 cm2 ), mean peak tendon stress (26.2 vs. 35.2 MPa), mean tendon safety factor (4.7 vs. 3.6), and total tendon strain energy return capacity (1.81 vs. 1.82 J). However, total tendon cross-sectional area is significantly larger in springhares than predicted for a similar-sized macropod (0.26 vs. 0.17 cm2 , respectively), primarily due to a greater plantaris tendon thickness (0.084 vs. 0.048 cm2 ), and secondarily because the soleus muscle-tendon unit is present in springhares but is vestigial in macropods. The overall similarities between springhares and macropods indicate that evolution has favored comparable lower hindlimb body plans for bipedal hopping locomotion in the two groups of mammals that last shared a common ancestor ~160 million years ago. The springhare's relatively thick plantaris tendon may facilitate rapid transfer of force from muscle to skeleton, enabling fast and accelerative hopping, which could help to outpace and outmaneuver predators.

Estimation of the force-velocity properties of individual muscles from measurement of the combined plantarflexor properties.

The force-velocity (F-V) properties of isolated muscles or muscle fibers have been well studied in humans and other animals. However, determining properties of individual muscles in vivo remains a challenge because muscles usually function within a synergistic group. Modeling has been used to estimate the properties of an individual muscle from the experimental measurement of the muscle group properties. While this approach can be valuable, the models and the associated predictions are difficult to validate. In this study, we measured the in situ F-V properties of the maximally activated kangaroo rat plantarflexor group and used two different assumptions and associated models to estimate the properties of the individual plantarflexors. The first model (Mdl1) assumed that the percent contributions of individual muscles to group force and power were based upon the muscles' cross-sectional area and were constant across the different isotonic loads applied to the muscle group. The second model (Mdl2) assumed that the F-V properties of the fibers within each muscle were identical, but because of differences in muscle architecture, the muscles' contributions to the group properties changed with isotonic load. We compared the two model predictions with independent estimates of the muscles' contributions based upon sonomicrometry measurements of muscle length. We found that predictions from Mdl2 were not significantly different from sonomicrometry-based estimates while those from Mdl1 were significantly different. The results of this study show that incorporating appropriate fiber properties and muscle architecture is necessary to parse the individual muscles' contributions to the group F-V properties.

Yank: the time derivative of force is an important biomechanical variable in sensorimotor systems.

The derivative of force with respect to time does not have a standard term in physics. As a consequence, the quantity has been given a variety of names, the most closely related being 'rate of force development'. The lack of a proper name has made it difficult to understand how different structures and processes within the sensorimotor system respond to and shape the dynamics of force generation, which is critical for survival in many species. We advocate that ∂[Formula: see text]/∂t be termed 'yank', a term that has previously been informally used and never formally defined. Our aim in this Commentary is to establish the significance of yank in how biological motor systems are organized, evolve and adapt. Further, by defining the quantity in mathematical terms, several measurement variables that are commonly reported can be clarified and unified. In this Commentary, we first detail the many types of motor function that are affected by the magnitude of yank generation, especially those related to time-constrained activities. These activities include escape, prey capture and postural responses to perturbations. Next, we describe the multi-scale structures and processes of the musculoskeletal system that influence yank and can be modified to increase yank generation. Lastly, we highlight recent studies showing that yank is represented in the sensory feedback system, and discuss how this information is used to enhance postural stability and facilitate recovery from postural perturbations. Overall, we promote an increased consideration of yank in studying biological motor and sensory systems.

Exploring Bipedal Hopping through Computational Evolution.

Bipedal hopping is an efficient form of locomotion, yet it remains relatively rare in the natural world. Previous research has suggested that the tail balances the angular momentum of the legs to produce steady state bipedal hopping. In this study, we employ a 3D physics simulation engine to optimize gaits for an animat whose control and morphological characteristics are subject to computational evolution, which emulates properties of natural evolution. Results indicate that the order of gene fixation during the evolutionary process influences whether a bipedal hopping or quadrupedal bounding gait emerges. Furthermore, we found that in the most effective bipedal hoppers the tail balances the angular momentum of the torso, rather than the legs as previously thought. Finally, there appears to be a specific range of tail masses, as a proportion of total body mass, wherein the most effective bipedal hoppers evolve.

Why do mammals hop? Understanding the ecology, biomechanics and evolution of bipedal hopping.

Bipedal hopping is a specialized mode of locomotion that has arisen independently in at least five groups of mammals. We review the evolutionary origins of these groups, examine three of the most prominent hypotheses for why bipedal hopping may have arisen, and discuss how this unique mode of locomotion influences the behavior and ecology of modern species. While all bipedal hoppers share generally similar body plans, differences in underlying musculoskeletal anatomy influence what performance benefits each group may derive from this mode of locomotion. Based on a review of the literature, we conclude that the most likely reason that bipedal hopping evolved is associated with predator avoidance by relatively small species in forested environments. Yet, the morphological specializations associated with this mode of locomotion have facilitated the secondary acquisition of performance characteristics that enable these species to be highly successful in ecologically demanding environments such as deserts. We refute many long-held misunderstandings about the origins of bipedal hopping and identify potential areas of research that would advance the understanding of this mode of locomotion.

Jumping mechanics of desert kangaroo rats.

Kangaroo rats are small bipedal desert rodents that use erratic vertical jumps to escape predator strikes. In this study we examined how individual hind limb joints of desert kangaroo rats (Dipodomys deserti) power vertical jumps across a range of heights. We hypothesized that increases in net work would be equally divided across hind limb joints with increases in jump height. To test this hypothesis, we used an inverse dynamics analysis to quantify the mechanical output from the hind limb joints of kangaroo rats jumping vertically over a wide range of heights. The kangaroo rats in this study reached maximal jump heights up to ∼9-times hip height. Net joint work increased significantly with jump height at the hip, knee and ankle, and decreased significantly at the metatarsal-phalangeal joint. The increase in net work generated by each joint was not proportional across joints but was dominated by the ankle, which ranged from contributing 56% of the work done on the center of mass at low jumps to 70% during the highest jumps. Therefore, the results of this study did not support our hypothesis. However, using an anatomical model, we estimated that a substantial proportion of the work delivered at the ankle (48%) was transferred from proximal muscles via the biarticular ankle extensors.

Grizzly bear (Ursus arctos horribilis) locomotion: forelimb joint mechanics across speed in the sagittal and frontal planes.

The majority of terrestrial locomotion studies have focused on parasagittal motion and paid less attention to forces or movement in the frontal plane. Our previous research has shown that grizzly bears produce higher medial ground reaction forces (lateral pushing from the animal) than would be expected for an upright mammal, suggesting frontal plane movement may be an important aspect of their locomotion. To examine this, we conducted an inverse dynamics analysis in the sagittal and frontal planes, using ground reaction forces and position data from three high-speed cameras of four adult female grizzly bears. Over the speed range collected, the bears used walks, running walks and canters. The scapulohumeral joint, wrist and the limb overall absorb energy (average total net work of the forelimb joints, -0.97 W kg-1). The scapulohumeral joint, elbow and total net work of the forelimb joints have negative relationships with speed, resulting in more energy absorbed by the forelimb at higher speeds (running walks and canters). The net joint moment and power curves maintain similar patterns across speed as in previously studied species, suggesting grizzly bears maintain similar joint dynamics to other mammalian quadrupeds. There is no significant relationship with net work and speed at any joint in the frontal plane. The total net work of the forelimb joints in the frontal plane was not significantly different from zero, suggesting that, despite the high medial ground reaction forces, the forelimb acts as a strut in that plane.

Grizzly bear (Ursus arctos horribilis) locomotion: gaits and ground reaction forces.

Locomotion of plantigrade generalists has been relatively little studied compared with more specialised postures even though plantigrady is ancestral among quadrupeds. Bears (Ursidae) are a representative family for plantigrade carnivorans, they have the majority of the morphological characteristics identified for plantigrade species, and they have the full range of generalist behaviours. This study compared the locomotion of adult grizzly bears (Ursus arctos horribilis Linnaeus 1758), including stride parameters, gaits and analysis of three-dimensional ground reaction forces, with that of previously studied quadrupeds. At slow to moderate speeds, grizzly bears use walks, running walks and canters. Vertical ground reaction forces demonstrated the typical M-shaped curve for walks; however, this was significantly more pronounced in the hindlimb. The rate of force development was also significantly higher for the hindlimbs than for the forelimbs at all speeds. Mediolateral forces were significantly higher than would be expected for a large erect mammal, almost to the extent of a sprawling crocodilian. There may be morphological or energetic explanations for the use of the running walk rather than the trot. The high medial forces (produced from a lateral push by the animal) could be caused by frontal plane movement of the carpus and elbow by bears. Overall, while grizzly bears share some similarities with large cursorial species, their locomotor kinetics have unique characteristics. Additional studies are needed to determine whether these characters are a feature of all bears or plantigrade species.

From Behavior to Bio-Inspiration: Aerial Reorientation and Multi-Plane Stability in Kangaroo Rats, Computational Models, and Robots.

Tails play essential roles in functions related to locomotor stability and maneuverability among terrestrial and arboreal animals. In kangaroo rats, bipedal hopping rodents, tails are used as effective inertial appendages for stability in hopping, but also facilitate stability and maneuverability during predator escape leaps. The complexity of tail functionality shows great potential for bio-inspiration and robotic device design, as maneuvering is accomplished by a long and light-weight inertial appendage. To (1) further understand the mechanics of how kangaroo rats use their tails during aerial maneuvers and (2) explore if we can achieve this behavior with a simplified tail-like appendage (i.e., template), we combined quantified animal observations, computational simulations, and experiments with a two degrees of freedom (2-DoF) tailed robot. We used video data from free-ranging kangaroo rats escaping from a simulated predator and analyzed body and tail motion for the airborne phase. To explain tail contributions to body orientation (i.e., spatial reorientation), we built a mid-air kangaroo rat computational model and demonstrated that the three-dimensional body orientation of the model can be controlled by a simplified 2-DoF tail with a nonlinear control strategy. Resulting simulated trajectories show movement patterns similar to those observed in kangaroo rats. Our robot experiments show that a lightweight tail can generate a large yaw displacement and stabilize pitch and roll angles to zero simultaneously. Our work contributes to better understanding of the form-function relationship of the kangaroo rat tail and lays out an important foundation for bio-inspiration in robotic devices that have lightweight tail-like appendages for mid-air maneuvering.

Springing into action: Comparing escape responses between bipedal and quadrupedal rodents.

Predation is a fundamental selective pressure on animal morphology, as morphology is directly linked with physical performance and evasion. Bipedal heteromyid rodents, which are characterized by unique morphological traits such as enlarged hindlimbs, appear to be more successful than sympatric quadrupedal rodents at escaping predators such as snakes and owls, but no studies have directly compared the escape performance of bipedal and quadrupedal rodents. We used simulated predator attacks to compare the evasive jumping ability of bipedal kangaroo rats (Dipodomys) to that of three quadrupedal rodent groups-pocket mice (Chaetodipus), woodrats (Neotoma), and ground squirrels (Otospermophilus). Jumping performance of pocket mice was remarkably similar to that of kangaroo rats, which may be driven by their shared anatomical features (such as enlarged hindlimb muscles) and facilitated by their relatively small body size. Woodrats and ground squirrels, in contrast, almost never jumped as a startle response, and they took longer to perform evasive escape maneuvers than the heteromyid species (kangaroo rats and pocket mice). Among the heteromyids, take-off velocity was the only jump performance metric that differed significantly between species. These results support the idea that bipedal body plans facilitate vertical leaping in larger-bodied rodents as a means of predator escape and that vertical leaping likely translates to better evasion success.

Effects of a Total Motion Release (TMR®) Protocol for the Single Leg Squat on Asymmetrical Movement Patterns.

BACKGROUND: Improving single leg squat (SLS) movement symmetry may benefit rehabilitation protocols. The Total Motion Release® (TMR®) protocol has been theorized to evaluate and improve patient-perceived movement asymmetries. HYPOTHESIS/PURPOSE: The purpose of this study was to evaluate whether perceived asymmetries identified by a TMR® scoring protocol were related to biomechanical asymmetries and whether improving perceived asymmetries influenced movement mechanics. It was hypothesized that participants with perceived asymmetries would also present with biomechanical asymmetries. A secondary hypothesis was that participants would reduce their perceived asymmetries after performing the TMR® protocol and subsequently have greater biomechanical symmetry. STUDY DESIGN: Descriptive Cohort (Laboratory Study). METHODS: Twenty participants (10 female, 10 male) with self-identified bilateral differences of 10 points or greater on the TMR® scoring scale were recruited for the study. The non-preferred side was defined as the side that scored higher. 3Dimensional motion capture was used to bilaterally assess baseline SLS depth as well as hip, knee, and ankle kinematics and kinetics. For the TMR® protocol, sets of 10 SLSs were performed on the preferred leg until their perceived asymmetries were resolved (i.e., both sides scored equally), or four sets had been completed. Kinematics and kinetics were collected immediately after the intervention and after a 10-minute rest period. RESULTS: Participants had biomechanical asymmetries at baseline for knee flexion, ankle flexion, and knee moments. Following the intervention, participants had reduced TMR® scores on the non-preferred leg, and this coincided with increased knee joint moments on that side. Although perceived asymmetries were resolved after the intervention, kinematic and kinetic asymmetries at the knee and ankle were still present. CONCLUSIONS: A TMR® intervention could benefit rehabilitation protocols by reducing factors of dysfunction and increasing the ability of patients to load the non-preferred knee. Further investigations are necessary to elucidate the importance of asymmetrical movement patterns. LEVEL OF EVIDENCE: 3b.

Collision-based mechanics of bipedal hopping.

The muscle work required to sustain steady-speed locomotion depends largely upon the mechanical energy needed to redirect the centre of mass and the degree to which this energy can be stored and returned elastically. Previous studies have found that large bipedal hoppers can elastically store and return a large fraction of the energy required to hop, whereas small bipedal hoppers can only elastically store and return a relatively small fraction. Here, we consider the extent to which large and small bipedal hoppers (tammar wallabies, approx. 7 kg, and desert kangaroo rats, approx. 0.1 kg) reduce the mechanical energy needed to redirect the centre of mass by reducing collisions. We hypothesize that kangaroo rats will reduce collisions to a greater extent than wallabies since kangaroo rats cannot elastically store and return as high a fraction of the mechanical energy of hopping as wallabies. We find that kangaroo rats use a significantly smaller collision angle than wallabies by employing ground reaction force vectors that are more vertical and center of mass velocity vectors that are more horizontal and thereby reduce their mechanical cost of transport. A collision-based approach paired with tendon morphometry may reveal this effect more generally among bipedal runners and quadrupedal trotters.

Differences in lower extremity joint stiffness during drop jump between healthy males and females.

The primary purpose of this study was to examine sex differences in lower extremity joint stiffness during vertical drop jump performance. A secondary purpose was to examine the potential influence of sex on the relationship between joint stiffness and jump performance. Thirty healthy and active individuals performed 15-drop jumps from 30 and 60 cm boxes. Hip, knee, and ankle joint stiffnesses were calculated for subphases of landing using a 2nd order polynomial regression model. Males had greater hip stiffness during the loading phase in drop jumps from both box heights than females' drop jump from 60 cm box. Also, males had a greater ground reaction force at the end of eccentric phase, net jump impulse, and jump height regardless of box height. The 60 cm box height increased knee stiffness during the loading phase, but reduced hip stiffness during the loading phase and knee and ankle stiffness during the absorption phase regardless of sex. Joint stiffnesses significantly predicted drop jump height for females (p < .001, r2 = 0.579), but not for males (p = .609, r2 = -0.053). These results suggest that females may have different strategies to maximize drop jump height as compared to males.

Examining movement asymmetries during three single leg tasks using interlimb and single subject approaches.

PURPOSE: s: To examine whether healthy individuals displayed asymmetric trunk and lower extremity kinematics in the frontal and sagittal planes using both interlimb and single subject models. METHODS: Trunk, pelvis, and lower extremity kinematic waveforms were analyzed bilaterally during the single leg squat (SLS), forward step down (FSD), and lateral step down (LSD). Participants identified task specific preferred and non-preferred legs based on perceived stability for interlimb analyses. Movement patterns were also analyzed with a single subject approach that included Fisher's exact tests to assess whether asymmetries were related to the task. RESULTS: Participants were found to have increased pelvic drop on the non-preferred leg during the LSD from 41 to 77% of the movement (p = 0.01). No other bilateral differences were found for interlimb analyses. Single subject analyses indicated that no task had a greater probability of finding or not finding asymmetries. Associations were found between the FSD and SLS for frontal plane hip (p < 0.01) and knee motion (p < 0.01). CONCLUSIONS: Interlimb analyses can be influenced by intraparticipant movement variability between preferred and non-preferred legs. Movement asymmetries during single leg weightbearing are likely task dependent and a battery of tests is necessary for assessing bilateral differences.

Comparison between the kinematics for kangaroo rat hopping on a solid versus sand surface.

In their natural habitats, animals move on a variety of substrates, ranging from solid surfaces to those that yield and flow (e.g. sand). These substrates impose different mechanical demands on the musculoskeletal system and may therefore elicit different locomotion patterns. The goal of this study is to compare bipedal hopping by desert kangaroo rats (Dipodomys deserti) on a solid versus granular substrate under speed-controlled conditions. To accomplish this goal, we developed a rotary treadmill, which is able to have different substrates or uneven surfaces. We video recorded six kangaroo rats hopping on a solid surface versus sand at the same speed (1.8 m s-1) and quantified the differences in the hopping kinematics between the two substrates. We found no significant differences in the hop period, hop length or duty cycle, showing that the gross kinematics on the two substrates were similar. This similarity was surprising given that sand is a substrate that absorbs mechanical energy. Measurements of the penetration resistance of the sand showed that the combination of the sand properties, toe-print area and kangaroo rat weight was probably the reason for the similarity.

Application of Polynomial Regression Model for Joint Stiffness.

Quasi-stiffness (joint stiffness) is often used to characterize leg properties during athletic and other activities and has been reported by a single slope of angle-moment curve. However, the joint angle-moment relationship of some relationship are not effectively represented by a simple linear regression model. Thus, the purpose of this analysis was to investigate the benefits of utilizing a 2nd order polynomial regression (quadratic) model as compared to the linear model when calculating lower extremity joint stiffness incorporating subdivided eccentric phases. Thirty healthy and active college students performed 15 drop jumps from a 30-cm platform. The eccentric phase was identified as the time from initial foot contact (IC) to the lowest vertical position of the center of mass and subdivided into the loading and attenuation phases, separated by the peak vertical ground reaction force. Lower extremity joint stiffnesses (hip, knee, and ankle) for the loading and attenuation phases were calculated using a linear and quadratic model. Multiple 2 by 2 repeated measures ANOVAs were performed. In the post-hoc analyses, the quadratic model had greater goodness-of-fit (r 2 and RMSE) than the linear model (p < .05) for all joints. The quadratic model revealed differences between the loading and attenuation phases for both hip (p = .001) and knee stiffness (p < .001). These results suggest that the quadratic model is more representative of the angle-moment relationship while subdividing the eccentric phase of a drop jump into the loading and attenuation phases.