Patterns of mechanical energy change in tetrapod gait: pendula, springs and work

Kinematic and center of mass (CoM) mechanical variables used to define terrestrial gaits are compared for various tetrapod species. Kinematic variables (limb phase, duty factor) provide important timing information regarding the neural control and limb coordination of various gaits. Whereas, mechanical variables (potential and kinetic energy relative phase, %Recovery, %Congruity) provide insight into the underlying mechanisms that minimize muscle work and the metabolic cost of locomotion, and also influence neural control strategies. Two basic mechanisms identified by Cavagna et al. (1977. Am J Physiol 233:R243-R261) are used broadly by various bipedal and quadrupedal species. During walking, animals exchange CoM potential energy (PE) with kinetic energy (KE) via an inverted pendulum mechanism to reduce muscle work. During the stance period of running (including trotting, hopping and galloping) gaits, animals convert PE and KE into elastic strain energy in spring elements of the limbs and trunk and regain this energy later during limb support. The bouncing motion of the body on the support limb(s) is well represented by a simple mass-spring system. Limb spring compliance allows the storage and return of elastic energy to reduce muscle work. These two distinct patterns of CoM mechanical energy exchange are fairly well correlated with kinematic distinctions of limb movement patterns associated with gait change. However, in some cases such correlations can be misleading. When running (or trotting) at low speeds many animals lack an aerial period and have limb duty factors that exceed 0.5. Rather than interpreting this as a change of gait, the underlying mechanics of the body's CoM motion indicate no fundamental change in limb movement pattern or CoM dynamics has occurred. Nevertheless, the idealized, distinctive patterns of CoM energy fluctuation predicted by an inverted pendulum for walking and a bouncing mass spring for running are often not clear cut, especially for less cursorial species. When the kinematic and mechanical patterns of a broader diversity of quadrupeds and bipeds are compared, more complex patterns emerge, indicating that some animals may combine walking and running mechanics at intermediate speeds or at very large size. These models also ignore energy costs that are likely associated with the opposing action of limbs that have overlapping support times during walking. A recent model of terrestrial gait (Ruina et al., 2005. J Theor Biol, in press) that treats limb contact with the ground in terms of collisional energy loss indicates that considerable CoM energy can be conserved simply by matching the path of CoM motion perpendicular to limb ground force. This model, coupled with the earlier ones of pendular exchange during walking and mass-spring elastic energy savings during running, provides compelling argument for the view that the legged locomotion of quadrupeds and other terrestrial animals has generally evolved to minimize muscle work during steady level movement.     

BigDog-inspired studies in the locomotion of goats and dogs

Collision-based expenditure of mechanical energy and the compliance and geometry of the leg are fundamental, interrelated considerations in the mechanical design of legged runners. This article provides a basic context and rationale for experiments designed to inform each of these key areas in Boston Dynamic's BigDog robot. Although these principles have been investigated throughout the past few decades within different academic disciplines, BigDog required that they be considered together and in concert with an impressive set of control algorithms that are not discussed here. Although collision reduction is an important strategy for reducing mechanical cost of transport in the slowest and fastest quadrupedal gaits, walking and galloping, BigDog employed an intermediate-speed trotting gait without collision reduction. Trotting, instead, uses a spring-loaded inverted pendulum mechanism with potential for storage and return of elastic strain energy in appropriately compliant structures. Rather than tuning BigDog's built-in leg springs according to a spring-mass model-based virtual leg-spring constant , a much stiffer distal leg spring together with actuation of the adjacent joint provided good trotting dynamics and avoided functional limitations that might have been imposed by too much compliance in real-world terrain. Adjusting the directional compliance of the legs by adopting a knee-forward, elbow-back geometry led to more robust trotting dynamics by reducing perturbations about the pitch axis of the robot's center of mass (CoM). BigDog is the most successful large-scale, all-terrain trotting machine built to date and it continues to stimulate our understanding of legged locomotion in comparative biomechanics as well as in robotics.     

Gait Transitions in Human Infants: Coping with Extremes of Treadmill Speed

Spinal pattern generators in quadrupedal animals can coordinate different forms of locomotion, like trotting or galloping, by altering coordination between the limbs (interlimb coordination). In the human system, infants have been used to study the subcortical control of gait, since the cerebral cortex and corticospinal tract are immature early in life. Like other animals, human infants can modify interlimb coordination to jump or step. Do human infants possess functional neuronal circuitry necessary to modify coordination within a limb (intralimb coordination) in order to generate distinct forms of alternating bipedal gait, such as walking and running? We monitored twenty-eight infants (7–12 months) stepping on a treadmill at speeds ranging between 0.06–2.36 m/s, and seventeen adults (22–47 years) walking or running at speeds spanning the walk-to-run transition. Six of the adults were tested with body weight support to mimic the conditions of infant stepping. We found that infants could accommodate a wide range of speeds by altering stride length and frequency, similar to adults. Moreover, as the treadmill speed increased, we observed periods of flight during which neither foot was in ground contact in infants and in adults. However, while adults modified other aspects of intralimb coordination and the mechanics of progression to transition to a running gait, infants did not make comparable changes. The lack of evidence for distinct walking and running patterns in infants suggests that the expression of different functional, alternating gait patterns in humans may require neuromuscular maturation and a period of learning post-independent walking.     

Torsion and Antero-Posterior Bending in the In Vivo Human Tibia Loading Regimes during Walking and Running

Bending, in addition to compression, is recognized to be a common loading pattern in long bones in animals. However, due to the technical difficulty of measuring bone deformation in humans, our current understanding of bone loading patterns in humans is very limited. In the present study, we hypothesized that bending and torsion are important loading regimes in the human tibia. In vivo tibia segment deformation in humans was assessed during walking and running utilizing a novel optical approach. Results suggest that the proximal tibia primarily bends to the posterior (bending angle: 0.15°–1.30°) and medial aspect (bending angle: 0.38°–0.90°) and that it twists externally (torsion angle: 0.67°–1.66°) in relation to the distal tibia during the stance phase of overground walking at a speed between 2.5 and 6.1 km/h. Peak posterior bending and peak torsion occurred during the first and second half of stance phase, respectively. The peak-to-peak antero-posterior (AP) bending angles increased linearly with vertical ground reaction force and speed. Similarly, peak-to-peak torsion angles increased with the vertical free moment in four of the five test subjects and with the speed in three of the test subjects. There was no correlation between peak-to-peak medio-lateral (ML) bending angles and ground reaction force or speed. On the treadmill, peak-to-peak AP bending angles increased with walking and running speed, but peak-to-peak torsion angles and peak-to-peak ML bending angles remained constant during walking. Peak-to-peak AP bending angle during treadmill running was speed-dependent and larger than that observed during walking. In contrast, peak-to-peak tibia torsion angle was smaller during treadmill running than during walking. To conclude, bending and torsion of substantial magnitude were observed in the human tibia during walking and running. A systematic distribution of peak amplitude was found during the first and second parts of the stance phase.     

What are the relations between mechanics, gait parameters, and energetics in terrestrial locomotion?

Are the different energy-conserving mechanics (i.e., pendulum and spring) used in different gaits reflected in differences in energetics and/or stride parameters? The analysis included published data from several species and new data from horses. When changing from pendulum to spring mechanics, there is a change in the slope of metabolic rate (MR) vs. speed in all species, in birds and quadrupeds there is no step increase, and in humans there are conflicting reports. At the trot-gallop transition, where quadrupeds are hypothesized to change from spring mechanics to some combination of spring and pendulum mechanics, there is a change in slope of MR vs. speed in horses but not in other species. Stride frequency (SF) is a logarithmic function of walking speed in all species, a linear function of trotting/running speed, and nearly independent of speed in galloping. In humans and horses there is a discontinuity in SF at the walk-trot (run) transition but not in birds. The slope of time of contact vs. speed does not change with mechanics in most species, but it does in humans. In horses and humans, there is a discontinuity at the walk-trot (run) transition and data for other species do not permit generalization. Duty factor (DF) in humans is greater than 0.5 in walking (pendulum mechanics) and less than 0.5 when running (spring mechanics). However, this is not true in many species that have DF>0.5 at the lowest speeds where they use spring mechanics. When trotting at low speeds, horses use forelimb DF>0.5 and hind limb DF<0.5. Thus, it is confusing to distinguish between walking and running by DF.     

The biomechanics of skipping gaits: a third locomotion paradigm?

Skipping, a gait children display when they are about four- to five-years-old, is revealed to be more than a behavioural peculiarity. By means of metabolic and biomechanical measurements at several speeds, the relevance of skipping is shown to extend from links between bipedal and quadrupedal locomotion (namely galloping) to understanding why it could be a gait of choice in low-gravity conditions, and to some aspects of locomotion evolution (ground reaction forces of skipping seem to originate from pushing the walking gait to unnaturally high speeds). When the time-courses of mechanical energy and the horizontal ground reaction force are considered, a different locomotion paradigm emerges, enabling us to separate, among the bouncing gaits, the trot from the gallop (quadrupeds) and running from skipping (bipeds). The simultaneous use of pendulum-like and elastic mechanisms in skipping gaits, as shown by the energy curve analysis, helps us to understand the low cost of transport of galloping quadrupeds.     

Stride length and its determinants in humans, early hominids, primates, and mammals

Primate stride lengths during quadrupedal locomotion are very long when compared to those of nonprimate quadrupedal mammals at the speed of trot/gallop transition. These exceptional lengths are a consequence of the relatively long limbs of primates and the large angular excursions of their limbs during quadrupedalism. When quadrupedal primates employ bipedal gaits they exhibit much lower angular excursions. Consequently their bipedal stride lengths do not appear to be exceptional in length when compared to other mammals. Angular excursions of the lower limbs of modern humans are not exceptionally large. However, when running, humans exhibit relatively long periods of flight (i.e., they have low duty factors) when compared to other mammals including primates. Because of these long periods of flight and their relative long lower limbs, humans have running stride lengths that are at the lower end of the range of stride lengths of quadrupedal primates. The stride length of the Laetoli hominid trails are evaluated in this context.     

Ground reaction forces and center of mass mechanics of bipedal capuchin monkeys: Implications for the evolution of human bipedalism

Tufted capuchin monkeys are known to use both quadrupedalism and bipedalism in their natural environments. Although previous studies have investigated limb kinematics and metabolic costs, their ground reaction forces (GRFs) and center of mass (CoM) mechanics during two and four‐legged locomotion are unknown. Here, we determine the hind limb GRFs and CoM energy, work, and power during bipedalism and quadrupedalism over a range of speeds and gaits to investigate the effect of differential limb number on locomotor performance. Our results indicate that capuchin monkeys use a “grounded run” during bipedalism (0.83–1.43 ms^?1) and primarily ambling and galloping gaits during quadrupedalism (0.91–6.0 ms^?1). CoM energy recoveries are quite low during bipedalism (2–17%), and in general higher during quadrupedalism (4–72%). Consistent with this, hind limb vertical GRFs as well as CoM work, power, and collisional losses are higher in bipedalism than quadrupedalism. The positive CoM work is 2.04 ± 0.40 Jkg^?1 m^?1 (bipedalism) and 0.70 ± 0.29 Jkg^?1 m^?1 (quadrupedalism), which is within the range of published values for two and four‐legged terrestrial animals. The results of this study confirm that facultative bipedalism in capuchins and other nonhuman primates need not be restricted to a pendulum‐like walking gait, but rather can include running, albeit without an aerial phase. Based on these results and similar studies of other facultative bipeds, we suggest that important transitions in the evolution of hominin locomotor performance were the emergences of an obligate, pendulum‐like walking gait and a bouncy running gait that included a whole‐body aerial phase. Am J Phys Anthropol, 2013. © 2012 Wiley Periodicals, Inc.     

Optimal running speed and the evolution of hominin hunting strategies

Recent discussion of the selective pressures leading to the evolution of modern human postcranial morphology, seen as early as Homo erectus, has focused on the relative importance of walking versus running. Specifically, these conversations have centered on which gait may have been used by early Homo to acquire prey. An element of the debate is the widespread belief that quadrupeds are constrained to run at optimally efficient speeds within each gait, whereas humans are equally efficient at all running speeds. The belief in the lack of optimal running speeds in humans is based, however, on a number of early studies with experimental designs inadequate for the purpose of evaluating optimality. Here we measured the energetic cost of human running (n = 9) at six different speeds for five minutes at each speed, with careful replicates and controls. We then compared the fit of linear versus curvilinear models to the data within each subject. We found that individual humans do, in fact, have speeds at which running is significantly less costly than at other speeds (i.e., an optimal running speed). In addition, we demonstrate that the use of persistence hunting methods to gain access to prey at any running speed, even the optimum, would be extremely costly energetically, more so than a persistence hunt at optimal walking speed. We argue that neither extinct nor extant hominin populations are as flexible in the chosen speeds of persistence hunting pursuits as other researchers have suggested. Variations in the efficiency of human locomotion appear to be similar to those of terrestrial quadrupeds.     

Joint Loads in Marsupial Ankles Reflect Habitual Bipedalism versus Quadrupedalism

Joint surfaces of limb bones are loaded in compression by reaction forces generated from body weight and musculotendon complexes bridging them. In general, joints of eutherian mammals have regions of high radiodensity subchondral bone that are better at resisting compressive forces than low radiodensity subchondral bone. Identifying similar form-function relationships between subchondral radiodensity distribution and joint load distribution within the marsupial postcranium, in addition to providing a richer understanding of marsupial functional morphology, can serve as a phylogenetic control in evaluating analogous relationships within eutherian mammals. Where commonalities are established across phylogenetic borders, unifying principles in mammalian physiology, morphology, and behavior can be identified. Here, we assess subchondral radiodensity patterns in distal tibiae of several marsupial taxa characterized by different habitual activities (e.g., locomotion). Computed tomography scanning, maximum intensity projection maps, and pixel counting were used to quantify radiodensity in 41 distal tibiae of bipedal (5 species), arboreal quadrupedal (4 species), and terrestrial quadrupedal (5 species) marsupials. Bipeds (Macropus and Wallabia) exhibit more expansive areas of high radiodensity in the distal tibia than arboreal (Dendrolagus, Phascolarctos, and Trichosurus) or terrestrial quadrupeds (Sarcophilus, Thylacinus, Lasiorhinus, and Vombatus), which may reflect the former carrying body weight only through the hind limbs. Arboreal quadrupeds exhibit smallest areas of high radiodensity, though they differ non-significantly from terrestrial quadrupeds. This could indicate slightly more compliant gaits by arboreal quadrupeds compared to terrestrial quadrupeds. The observed radiodensity patterns in marsupial tibiae, though their statistical differences disappear when controlling for phylogeny, corroborate previously documented patterns in primates and xenarthrans, potentially reflecting inferred limb use during habitual activities such as locomotion. Despite the complex nature of factors contributing to joint loads, broad observance of these patterns across joints and across a variety of taxa suggests that subchondral radiodensity can be used as a unifying form-function principle within Mammalia.     

Interactions between locomotion and ventilation in tetrapods

Interactions between locomotion and ventilation have now been studied in several species of reptiles, birds and mammals, from a variety of perspectives. Among these perspectives are neural interactions of separate but linked central controllers; mechanical impacts of locomotion upon ventilatory pressures and flows; and the extent to which the latter may affect gas exchange and the energetics of exercise. A synchrony, i.e. 1:1 pattern of coordination, is observed in many running mammals once they achieve galloping speeds, as well as in flying bats, some flying birds and hopping marsupials. Other, non-1:1, patterns of coordination are seen in trotting and walking quadrupeds, as well as running bipedal humans and running and flying birds. There is evidence for an energetic advantage to coordination of locomotor and respiratory cycles for flying birds and running mammals. There is evidence for a mechanical constraint upon ventilation by locomotion for some reptiles (e.g. iguana), but not for others (e.g. varanids and crocodilians). In diving birds the impact of wing flapping or foot paddling on differential air sac pressures enhances gas exchange during the breath hold by improving diffusive and convective movement of air sac oxygen to parabronchi. This paper will review the current state of our knowledge of such influences of locomotion upon respiratory system function.     

The effects of natural substrate discontinuities on the quadrupedal gait kinematics of free‐ranging Saimiri sciureus

Wild primates encounter complex matrices of substrates that differ in size, orientation, height, and compliance, and often move on multiple, discontinuous substrates within a single bout of locomotion. Our current understanding of primate gait is limited by artificial laboratory settings in which primate quadrupedal gait has primarily been studied. This study analyzes wild Saimiri sciureus (common squirrel monkey) gait on discontinuous substrates to capture the realistic effects of the complex arboreal habitat on walking kinematics. We collected high‐speed video footage at Tiputini Biodiversity Station, Ecuador between August and October 2017. Overall, the squirrel monkeys used more asymmetrical walking gaits than symmetrical gaits, and specifically asymmetrical lateral sequence walking gaits when moving across discontinuous substrates. When individuals used symmetrical gaits, they used diagonal sequence gaits more than lateral sequence gaits. In addition, individuals were more likely to change their footfall sequence during strides on discontinuous substrates. Squirrel monkeys increased the time lag between touchdowns both of ipsilaterally paired limbs (pair lag) and of the paired forelimbs (forelimb lag) when walking across discontinuous substrates compared to continuous substrates. Results indicate that gait flexibility and the ability to alter footfall patterns during quadrupedal walking may be critical for primates to safely move in their complex arboreal habitats. Notably, wild squirrel monkey quadrupedalism is diverse and flexible with high proportions of asymmetrical walking. Studying kinematics in the wild is critical for understanding the complexity of primate quadrupedalism.     

Ability of the planar spring-mass model to predict mechanical parameters in running humans

The planar spring-mass model is a simple mathematical model of bouncing gaits, such as running, trotting and hopping. Although this model has been widely used in the study of locomotion, its accuracy in predicting locomotor mechanics has not been systematically quantified. We determined the percent error of the model in predicting 10 locomotor parameters in running humans by comparing the model predictions to experimental data from humans running in normal gravity and simulated reduced gravity. We tested the hypotheses that the model would overestimate horizontal impulse and the change in mechanical energy of the centre of mass (COM) during stance. The model provided good predictions of stance time, vertical impulse, contact length, duty factor, relative stride length and relative peak force. All predictions of these parameters were within 20% of measured values and at least 90% of predictions of each parameter were within 10% of measured values (median absolute errors: <7%). This suggests that the model incorporates all features of running humans that have a significant influence upon these six parameters. As simulated gravity level decreased, the magnitude of the errors in predicting each of these parameters either decreased or stayed constant, indicating that this is a good model of running in simulated reduced gravity. As hypothesised, horizontal impulse and change in mechanical energy of the COM during stance were overestimated (median absolute errors: 43.6% and 26.2%, respectively). Aerial time and peak vertical COM displacement during stance were also systematically overestimated (median absolute errors: 17.7% and 22.9%, respectively). Care should be taken to ensure that the model is used only to investigate parameters which it can predict accurately. It would be useful to extend this analysis to other species and gaits.     

The metabolic and mechanical costs of step time asymmetry in walking

Animals use both pendular and elastic mechanisms to minimize energy expenditure during terrestrial locomotion. Elastic gaits can be either bilaterally symmetric (e.g. run and trot) or asymmetric (e.g. skip, canter and gallop), yet only symmetric pendular gaits (e.g. walk) are observed in nature. Does minimizing metabolic and mechanical power constrain pendular gaits to temporal symmetry? We measured rates of metabolic energy expenditure and calculated mechanical power production while healthy humans walked symmetrically and asymmetrically at a range of step and stride times. We found that walking with a 42 per cent step time asymmetry required 80 per cent (2.5 W kg⁻¹) more metabolic power than preferred symmetric gait. Positive mechanical power production increased by 64 per cent (approx. 0.24 W kg⁻¹), paralleling the increases we observed in metabolic power. We found that when walking asymmetrically, subjects absorbed more power during double support than during symmetric walking and compensated by increasing power production during single support. Overall, we identify inherent metabolic and mechanical costs to gait asymmetry and find that symmetry is optimal in healthy human walking.     

Segment and joint angles of hind limb during bipedal and quadrupedal walking of the bonobo (Pan paniscus)

We describe segment angles (trunk, thigh, shank, and foot) and joint angles (hip, knee, and ankle) for the hind limbs of bonobos walking bipedally ("bent-hip bent-knee walking," 17 sequences) and quadrupedally (33 sequences). Data were based on video recordings (50 Hz) of nine subjects in a lateral view, walking at voluntary speed. The major differences between bipedal and quadrupedal walking are found in the trunk, thigh, and hip angles. During bipedal walking, the trunk is approximately 33-41 degrees more erect than during quadrupedal locomotion, although it is considerably more bent forward than in normal human locomotion. Moreover, during bipedal walking, the hip has a smaller range of motion (by 12 degrees ) and is more extended (by 20-35 degrees ) than during quadrupedal walking. In general, angle profiles in bonobos are much more variable than in humans. Intralimb phase relationships of subsequent joint angles show that hip-knee coordination is similar for bipedal and quadrupedal walking, and resembles the human pattern. The coordination between knee and ankle differs much more from the human pattern. Based on joint angles observed throughout stance phase and on the estimation of functional leg length, an efficient inverted pendulum mechanism is not expected in bonobos.     

Analysis of the gait generation principle by a simulated quadruped model with a CPG incorporating vestibular modulation

This study aims to understand the principles of gait generation in a quadrupedal model. It is difficult to determine the essence of gait generation simply by observation of the movement of complicated animals composed of brains, nerves, muscles, etc. Therefore, we build a planar quadruped model with simplified nervous system and mechanisms, in order to observe its gaits under simulation. The model is equipped with a mathematical central pattern generator (CPG), consisting of four coupled neural oscillators, basically producing a trot pattern. The model also contains sensory feedback to the CPG, measuring the body tilt (vestibular modulation). This spontaneously gives rise to an unprogrammed lateral walk at low speeds, a transverse gallop while running, in addition to trotting at a medium speed. This is because the body oscillation exhibits a double peak per leg frequency at low speeds, no peak (little oscillation) at medium speeds, and a single peak while running. The body oscillation autonomously adjusts the phase differences between the neural oscillators via the feedback. We assume that the oscillations of the four legs produced by the CPG and the body oscillation varying according to the current speed are synchronized along with the varied phase differences to keep balance during locomotion through postural adaptation via the vestibular modulation, resulting in each gait. We succeeded in determining a single simple principle that accounts for gait transition from walking to trotting to galloping, even without brain control, complicated leg mechanisms, or a flexible trunk.     

Morphology and the gradient of a symmetric potential predict gait transitions of dogs

Gaits and gait transitions play a central role in the movement of animals. Symmetry is thought to govern the structure of the nervous system, and constrain the limb motions of quadrupeds. We quantify the symmetry of dog gaits with respect to combinations of bilateral, fore–aft, and spatio-temporal symmetry groups. We tested the ability of symmetries to model motion capture data of dogs walking, trotting and transitioning between those gaits. Fully symmetric models performed comparably to asymmetric with only a \(22\%\) increase in the residual sum of squares and only one-quarter of the parameters. This required adding a spatio-temporal shift representing a lag between fore and hind limbs. Without this shift, the symmetric model residual sum of squares was \(1700\%\) larger. This shift is related to (linear regression, \(n=5\), \(p=0.0328\)) dog morphology. That this symmetry is respected throughout the gaits and transitions indicates that it generalizes outside a single gait. We propose that relative phasing of limb motions can be described by an interaction potential with a symmetric structure. This approach can be extended to the study of interaction of neurodynamic and kinematic variables, providing a system-level model that couples neuronal central pattern generator networks and mechanical models.     

Human Quadrupeds,Primate Quadrupedalism,and Uner Tan Syndrome

Since 2005, an extensive literature documents individuals from several families afflicted with “Uner Tan Syndrome (UTS),” a condition that in its most extreme form is characterized by cerebellar hypoplasia, loss of balance and coordination, impaired cognitive abilities, and habitual quadrupedal gait on hands and feet. Some researchers have interpreted habitual use of quadrupedalism by these individuals from an evolutionary perspective, suggesting that it represents an atavistic expression of our quadrupedal primate ancestry or “devolution.” In support of this idea, individuals with “UTS” are said to use diagonal sequence quadrupedalism, a type of quadrupedal gait that distinguishes primates from most other mammals. Although the use of primate-like quadrupedal gait in humans would not be sufficient to support the conclusion of evolutionary “reversal,” no quantitative gait analyses were presented to support this claim. Using standard gait analysis of 518 quadrupedal strides from video sequences of individuals with “UTS”, we found that these humans almost exclusively used lateral sequence–not diagonal sequence–quadrupedal gaits. The quadrupedal gait of these individuals has therefore been erroneously described as primate-like, further weakening the “devolution” hypothesis. In fact, the quadrupedalism exhibited by individuals with UTS resembles that of healthy adult humans asked to walk quadrupedally in an experimental setting. We conclude that quadrupedalism in healthy adults or those with a physical disability can be explained using biomechanical principles rather than evolutionary assumptions.