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.     

Intermanual coordination: from behavioural principles to neural-network interactions

Locomotion in vertebrates and invertebrates has a long history in research as the most prominent example of interlimb coordination. However, the evolution towards upright stance and gait has paved the way for a bewildering variety of functions in which the upper limbs interact with each other in a context-specific manner. The neural basis of these bimanual interactions has been investigated in recent years on different scales, ranging from the single-cell level to the analysis of neuronal assemblies. Although the prevailing viewpoint has been to assign bimanual coordination to a single brain locus, more recent evidence points to a distributed network that governs the processes of neural synchronization and desynchronization that underlie the rich variety of coordinated functions. The distributed nature of this network accounts for disruptions of interlimb coordination across various movement disorders.     

Limb kinematics during locomotion in the two-toed sloth (Choloepus didactylus,Xenarthra) and its implications for the evolution of the sloth locomotor apparatus

In order to gain insight into the function of the extant sloth locomotion and its evolution, we conducted a detailed videoradiographic analysis of two-toed sloth locomotion (Xenarthra: Choloepus didactylus). Both unrestrained as well as steady-state locomotion was analyzed. Spatio-temporal gait parameters, data on interlimb coordination, and limb kinematics are reported. Two-toed sloths displayed great variability in spatio-temporal gait parameters over the observed range of speeds. They increase speed by decreasing the durations of contact and swing phases, as well as by increasing step length. Gait utilization also varies with no strict gait sequence or interlimb timing evident in slow movements, but a tendency to employ diagonal sequence, diagonal couplet gaits in fast movements. In contrast, limb kinematics were highly conserved with respect to ‘normal’ pronograde locomotion. Limb element and joint angles at touch down and lift off, element and joint excursions, and contribution to body progression of individual elements are similar to those reported for non-cursorial mammals of small to medium size. Hands and feet are specialized to maintain firm connection to supports, and do not contribute to step length or progression. In so doing, the tarsometatarsus lost its role as an individual propulsive element during the evolution of suspensory locomotion. Conservative kinematic behavior of the remaining limb elements does not preclude that muscle recruitment and neuromuscular control for limb pro- and retraction are also conserved. The observed kinematic patterns of two-toed sloths improve our understanding of the convergent evolution of quadrupedal suspensory posture and locomotion in the two extant sloth lineages.     

Are interlimb interactions disturbed in patients with Parkinson’s disease? A study under unloading conditions

During natural human locomotion, neural connections are activated that are typical of regulation of the quadrupedal walking. The interaction between the neural networks generating rhythmic movements of the upper and lower limbs depends on tonic state of each of these networks regulated by motor signals from the brain. Distortion of these signals in patients with Parkinson’s disease (PD) may lead to disruption of the interlimb interactions. We examined the effect of movements of the limbs of one girdle on the parameters of the motor activity of another limb girdle at their joint cyclic movements under the conditions of arm and leg unloading in 17 patients with PD and 16 healthy subjects. We have shown that, in patients, the effect of voluntary and passive movements of arms, as well as the active movement of the distal parts of arms, on the voluntary movement of legs is weak, while in healthy subjects, the effect of arm movements on the parameters of voluntary stepping is significant. The effect of arm movements on the activation of the involuntary stepping by vibrational stimulation of-legs in patients was absent, while in healthy subjects, the motor activity of arms increased the possibility of involuntary rhythmic movements activation. Differences in the effect of leg movements on the rhythmic movements of arms were found in both patients and healthy subjects. The interlimb interaction appeared after drug administration. However, the effect of the drug was not sufficient for the recovery of normal state of the neural networks in patients. In PD patients, neural networks generating stepping rhythm have an increased tonic activity, which prevents the activation and appearance of involuntary rhythmic movements facilitating the effects of arms on legs.     

Muscle contributions to support during gait in an individual with post-stroke hemiparesis

Walking requires coordination of muscles to support the body during single stance. Impaired ability to coordinate muscles following stroke frequently compromises walking performance and results in extremely low walking speeds. Slow gait in post-stroke hemiparesis is further complicated by asymmetries in lower limb muscle excitations. The objectives of the current study were: (1) to compare the muscle coordination patterns of an individual with flexed stance limb posture secondary to post-stroke hemiparesis with that of healthy adults walking very slowly, and (2) to identify how paretic and non-paretic muscles provide support of the body center of mass in this individual. Simulations were generated based on the kinematics and kinetics of a stroke survivor walking at his self-selected speed (0.3 m/s) and of three speed-matched, healthy older individuals. For each simulation, muscle forces were perturbed to determine the muscles contributing most to body weight support (i.e., height of the center of mass during midstance). Differences in muscle excitations and midstance body configuration caused paretic and non-paretic ankle plantarflexors to contribute less to midstance support than in healthy slow gait. Excitation of paretic ankle dorsiflexors and knee flexors during stance opposed support and necessitated compensation by knee and hip extensors. During gait for an individual with post-stroke hemiparesis, adequate body weight support is provided via reorganized muscle coordination patterns of the paretic and non-paretic lower limbs relative to healthy slow gait.     

Human crawling performance and technique revealed by inertial measurement units

Human crawling performance and technique are of broad interest to roboticists, biomechanists, and military personnel. This study explores the variables that define crawling performance in the context of an outdoor obstacle course used by military organizations worldwide to evaluate the effects of load and personal equipment on warfighter performance. Crawling kinematics, measured from four body-worn inertial measurement units (IMUs) attached to the upper arms and thighs, are recorded for thirty-three participants. The IMU data is distilled to four metrics of crawling performance; namely, crawl speed, crawl stride time, ipsilateral limb coordination, and contralateral limb coordination. We hypothesize that higher performance (as identified by higher crawl speeds) is associated with more coordinated limbs and lower stride times. A cluster analysis groups participants into high and low performers exhibiting statistically significant differences across the four performance metrics. In particular, high performers exhibit superior limb coordination associated with a “diagonal gait” in which contralateral limbs move largely in-phase to produce faster crawl speeds and shorter crawl stride times. In contrast, low performers crawl at slower speeds with longer crawl stride times and less limb coordination. Beyond these conclusions, a major contribution of this study is a method for deploying wearable IMUs to study crawling in contextually relevant (i.e. non-laboratory) environments.     

Spatio-temporal gait characteristics during transitions from trot to canter in horses

Gaits can be defined based upon specific interlimb coordination patterns characteristic of a limited range of speeds, with one or more defining variables changing discontinuously at a transition. With changing speed, horses perform a repertoire of gaits (walk, trot, canter and gallop), with transitions between them. Knowledge of the series of kinematic events necessary to realize a gait is essential for understanding the proximate mechanisms as well as the control underlying gait transitions. We studied the kinematics of the actual transition from trot to canter in miniature horses. The kinematics were characterized at three different levels: the whole-body level, the spatio-temporal level of the foot falls and the level of basic limb kinematics. This concept represents a hierarchy: the horse's center of mass (COM) moves forward by means of the coordinated action of the limbs and changes in the latter are the result of alterations in the basic limb kinematics. Early and short placement of the fore limb was observed before the dissociation of the footfalls of one of the diagonal limb pairs when entering the canter. Dissociation coincided with increased amplitude and wavelength of the oscillations of the trunk in the sagittal plane. The increased amplitude cannot be explained solely by the passive effects of acceleration or by neck and head movements which are inconsistent with the timing of the transition. We propose that the transition is initiated by the fore limb followed by subsequent changes in the hind limbs in a series of kinematic events that take about 2.5 strides to complete.     

Control of prosthetic gait

Recent advances in two types of prosthetic gait are particularly noteworthy, namely work on limb and neural prostheses. Current work on artificial limbs has been oriented towards improving devices, with commercialization as the driving force. Progress has been made in understanding how the compliant properties of the foot, ankle and knee joints of artificial legs affect the energetics and kinematics of gait. Work is continuing on automated systems for fabricating sockets with improved fit to increase the comfort of artificial limbs. Neural prostheses use electrical stimulation to activate paralyzed muscle: Advances have been made in understanding how to model the patterns of neural prosthetic gait and how neural prostheses respond to disturbances. Work in real-time control of stimulated muscle has progressed in the area of system identification and in using natural sensors for feedback signals. There still remains a wide gap, however, between able-bodied gait and the gait that can be achieved using current neural prosthesis systems.     

Leg and Joint Stiffness in Children with Spastic Diplegic Cerebral Palsy during Level Walking

Individual joint deviations are often identified in the analysis of cerebral palsy (CP) gait. However, knowledge is limited as to how these deviations affect the control of the locomotor system as a whole when striving to meet the demands of walking. The current study aimed to bridge the gap by describing the control of the locomotor system in children with diplegic CP in terms of their leg stiffness, both skeletal and muscular components, and associated joint stiffness during gait. Twelve children with spastic diplegia CP and 12 healthy controls walked at a self-selected pace in a gait laboratory while their kinematic and forceplate data were measured and analyzed during loading response, mid-stance, terminal stance and pre-swing. For calculating the leg stiffness, each of the lower limbs was modeled as a non-linear spring, connecting the hip joint center and the corresponding center of pressure, with varying stiffness that was calculated as the slope (gradient) of the axial force vs. the deformation curve. The leg stiffness was further decomposed into skeletal and muscular components considering the alignment of the lower limb. The ankle, knee and hip of the limb were modeled as revolute joints with torsional springs whose stiffness was calculated as the slope of the moment vs. the angle curve of the joint. Independent t-tests were performed for between-group comparisons of all the variables. The CP group significantly decreased the leg stiffness but increased the joint stiffness during stance phase, except during terminal stance where the leg stiffness was increased. They appeared to rely more on muscular contributions to achieve the required leg stiffness, increasing the muscular demands in maintaining the body posture against collapse. Leg stiffness plays a critical role in modulating the kinematics and kinetics of the locomotor system during gait in the diplegic CP.     

Trunk's Natural Inclination Influences Stance Limb Kinetics,but Not Body Kinematics,during Gait Initiation in Able Men

The imposing mass of the trunk in relation to the whole body has an important impact on human motion. The objective of this study is to determine the influence of trunk''s natural inclination - forward (FW) or backward (BW) with respect to the vertical - on body kinematics and stance limb kinetics during gait initiation.Twenty-five healthy males were divided based on their natural trunk inclination (FW or BW) during gait initiation. Instantaneous speed was calculated at the center of mass at the first heel strike. The antero-posterior impulse was calculated by integrating the antero-posterior ground reaction force in time. Ankle, knee, hip and thoraco-lumbar (L5) moments were calculated using inverse dynamics and only peaks of the joint moments were analyzed. Among all the investigated parameters, only joint moments present significant differences between the two groups. The knee extensor moment is 1.4 times higher (P<0.001) for the BW group, before the heel contact. At the hip, although the BW group displays a flexor moment 2.4 times higher (P<0.001) before the swing limb''s heel-off, the FW group displays an extensor moment 3.1 times higher (P<0.01) during the swing phase. The three L5 extensor peaks after the toe-off are respectively 1.7 (P<0.001), 1.4 (P<0.001) and 1.7 (P<0.01) times higher for the FW group. The main results support the idea that the patterns described during steady-state gait are already observable during gait initiation. This study also provides reference data to further investigate stance limb kinetics in specific or pathologic populations during gait initiation. It will be of particular interest for elderly people, knowing that this population displays atypical trunk postures and present a high risk of falling during this forward stepping.     

Kinematics of lower limbs during walking are emulated by springy walking model with a compliantly connected,off-centered curvy foot

The dynamics of the center of mass (CoM) during walking and running at various gait conditions are well described by the mechanics of a simple passive spring loaded inverted pendulum (SLIP). Due to its simplicity, however, the current form of the SLIP model is limited at providing any further information about multi-segmental lower limbs that generate oscillatory CoM behaviors and their corresponding ground reaction forces. Considering that the dynamics of the CoM are simply achieved by mass-spring mechanics, we wondered whether any of the multi-joint motions could be demonstrated by simple mechanics. In this study, we expand a SLIP model of human locomotion with an off-centered curvy foot connected to the leg by a springy segment that emulates the asymmetric kinematics and kinetics of the ankle joint. The passive dynamics of the proposed expansion of the SLIP model demonstrated the empirical data of ground reaction forces, center of mass trajectories, ankle joint kinematics and corresponding ankle joint torque at various gait speeds. From the mechanically simulated trajectories of the ankle joint and CoM, the motion of lower-limb segments, such as thigh and shank angles, could be estimated from inverse kinematics. The estimation of lower limb kinematics showed a qualitative match with empirical data of walking at various speeds. The representability of passive compliant mechanics for the kinetics of the CoM and ankle joint and lower limb joint kinematics implies that the coordination of multi-joint lower limbs during gait can be understood with a mechanical framework.     

Light Accelerates Morphogenesis and Acquisition of Interlimb Stepping in Chick Embryos

Chicks are bipedal precocious vertebrates that achieve adaptive locomotor skill within hours after hatching. Development of limb movement has been extensively studied in the chicken embryo, but few studies have focused on the preparations leading to precocious locomotor skill. Chicks typically hatch after 21 days of incubation, and recent studies provided evidence that the neural circuits for intralimb control of stepping are established between embryonic days (E) 18–20. It has also been shown that variations in light exposure during embryogenesis can accelerate or delay the onset of hatching and walking by 1 to 2 days. Our earlier work revealed that despite these differences in time to hatch, chicks incubated in different light conditions achieved similar locomotor skill on the day of hatching. Results suggested to us that light exposure during incubation may have accelerated development of locomotor circuits in register with earlier hatching. Thus, in this study, embryos were incubated in 1 of 3 light conditions to determine if development of interlimb coordination at a common time point, 19 days of incubation, varied with light exposure during embryogenesis. Leg muscle activity was recorded bilaterally and burst analyses were performed for sequences of spontaneous locomotor-related activity in one or more ankle muscles to quantify the extent of interlimb coordination in ovo. We report findings indicating that the extent of interlimb coordination varied with light exposure, and left-right alternating steps were a more reliable attribute of interlimb coordination for embryos incubated in constant bright light. We provide evidence that morphological development of the leg varied with light exposure. Based on these findings, we propose that light can accelerate the development of interlimb coordination in register with earlier hatching. Our results lead us to further propose that alternating left-right stepping is the default pattern of interlimb coordination produced by locomotor circuits during embryogenesis.     

A Behavioral Mechanism of How Increases in Leg Strength Improve Old Adults’ Gait Speed

We examined a behavioral mechanism of how increases in leg strength improve healthy old adults’ gait speed. Leg press strength training improved maximal leg press load 40% (p = 0.001) and isometric strength in 5 group of leg muscles 32% (p = 0.001) in a randomly allocated intervention group of healthy old adults (age 74, n = 15) but not in no-exercise control group (age 74, n = 8). Gait speed increased similarly in the training (9.9%) and control (8.6%) groups (time main effect, p = 0.001). However, in the training group only, in line with the concept of biomechanical plasticity of aging gait, hip extensors and ankle plantarflexors became the only significant predictors of self-selected and maximal gait speed. The study provides the first behavioral evidence regarding a mechanism of how increases in leg strength improve healthy old adults’ gait speed.     

Noisy Interlimb Coordination Can Be a Main Cause of Freezing of Gait in Patients with Little to No Parkinsonism

Freezing of gait in patients with Parkinson’s disease is associated with several factors, including interlimb incoordination and impaired gait cycle regulation. Gait analysis in patients with Parkinson’s disease is confounded by parkinsonian symptoms such as rigidity. To understand the mechanisms underlying freezing of gait, we compared gait patterns during straight walking between 9 patients with freezing of gait but little to no parkinsonism (freezing patients) and 11 patients with Parkinson’s disease (non-freezing patients). Wireless sensors were used to detect foot contact and toe-off events, and the step phase of each foot contact was calculated by defining one stride cycle of the other leg as 360°. Phase-resetting analysis was performed, whereby the relation between the step phase of one leg and the subsequent phase change in the following step of the other leg was quantified using regression analysis. A small slope of the regression line indicates a forceful correction (phase reset) at every step of the deviation of step phase from the equilibrium phase, usually at around 180°. The slope of this relation was smaller in freezing patients than in non-freezing patients, but the slope exhibited larger step-to-step variability. This indicates that freezing patients executed a forceful but noisy correction of the deviation of step phase, whereas non-freezing patients made a gradual correction of the deviation. Moreover, freezing patients tended to show more variable step phase and stride time than non-freezing patients. Dynamics of a model of two coupled oscillators interacting through a phase resetting mechanism were examined, and indicated that the deterioration of phase reset by noise provoked variability in step phase and stride time. That is, interlimb coordination can affect regulation of the gait cycle. These results suggest that noisy interlimb coordination, which probably caused forceful corrections of step phase deviation, can be a cause of freezing of gait.     

The effect of uphill and downhill walking on gait parameters: A self-paced treadmill study

It has been shown that gait parameters vary systematically with the slope of the surface when walking uphill (UH) or downhill (DH) (Andriacchi et al., 1977; Crowe et al., 1996; Kawamura et al., 1991; Kirtley et al., 1985; McIntosh et al., 2006; Sun et al., 1996). However, gait trials performed on inclined surfaces have been subject to certain technical limitations including using fixed speed treadmills (TMs) or, alternatively, sampling only a few gait cycles on inclined ramps. Further, prior work has not analyzed upper body kinematics. This study aims to investigate effects of slope on gait parameters using a self-paced TM (SPTM) which facilitates more natural walking, including measuring upper body kinematics and gait coordination parameters.Gait of 11 young healthy participants was sampled during walking in steady state speed. Measurements were made at slopes of +10°, 0° and −10°. Force plates and a motion capture system were used to reconstruct twenty spatiotemporal gait parameters. For validation, previously described parameters were compared with the literature, and novel parameters measuring upper body kinematics and bilateral gait coordination were also analyzed.Results showed that most lower and upper body gait parameters were affected by walking slope angle. Specifically, UH walking had a higher impact on gait kinematics than DH walking. However, gait coordination parameters were not affected by walking slope, suggesting that gait asymmetry, left-right coordination and gait variability are robust characteristics of walking. The findings of the study are discussed in reference to a potential combined effect of slope and gait speed. Follow-up studies are needed to explore the relative effects of each of these factors.     

Interlimb coordination,gait, and neural control of quadrupedalism in chimpanzees

Interlimb coordination is directly relevant to the understanding of the neural control of locomotion, but few studies addressing this topic for nonhuman primates are available, and no data exist for any hominoid other than humans. As a follow-up to Jungers and Anapol's ([1985] Am. J. Phys. Anthropol. 67:89–97) analysis on a lemur and talapoin monkey, we describe here the patterns of interlimb coordination in two chimpanzees as revealed by electromyography. Like the lemur and talapoin monkey, ipsilateral limb coupling in chimpanzees is characterized by variability about preferred modes within individual gaits. During symmetrical gaits, limb coupling patterns in the chimpanzee are also influenced by kinematic differences in hindlimb placement (“overstriding”). These observations reflect the neurological constraints placed on locomotion but also emphasize the overall flexibility of locomotor neural mechanisms. Interlimb coordination patterns are also species-specific, exhibiting significant differences among primate taxa and between primates and cats. Interspecific differences may be suggestive of phylogenetic divergence in the basic mechanisms for neural control of locomotion, but do not preclude morphological explanations for observed differences in interlimb coordination across species. Am J Phys Anthropol 102:177–186, 1997 © 1997 Wiley-Liss, Inc.     

Gait Pattern Differences between Children with Mild Scoliosis and Children with Unilateral Cerebral Palsy

This study was conducted to investigate the effects of asymmetrical body posture alone, i.e., the effects seen in children with mild scoliosis, vs. the effects of body posture control impairment, i.e., those seen in children with unilateral cerebral palsy on gait patterns. Three-dimensional instrumented gait analysis (3DGA) was conducted in 45 children with hemiplegia and 51 children with mild scoliosis. All the children were able to walk without assistance devices. A set of 35 selected spatiotemporal gait and kinematics parameters were evaluated when subjects walked on a treadmill. A cluster analysis revealed 3 different gait patterns: a scoliotic gait pattern and 2 different hemiplegic gait patterns. The results showed that the discrepancy in gait patterns was not simply a lower limb kinematic deviation in the sagittal plane, as expected. Additional altered kinematics, such as pelvic misorientation in the coronal plane in both the stance and swing phases and inadequate stance phase hip ad/abduction, which resulted from postural pattern features, were distinguished between the 3 gait patterns. Our study provides evidence for a strong correlation between postural and gait patterns in children with unilateral cerebral palsy. Information on differences in gait patterns may be used to improve the guidelines for early therapy for children with hemiplegia before abnormal gait patterns are fully established. The gait pathology characteristic of scoliotic children is a potential new direction for treating scoliosis that complements the standard posture and walking control therapy exercises with the use of biofeedback.     

Spirometry in Healthy Subjects: Do Technical Details of the Test Procedure Affect the Results?

Introduction

Spirometry should follow strict quality criteria. The American Thoracic Society (ATS) recommends the use of a noseclip; however there are controversies about its need. ATS also indicates that tests should be done in the sitting position, but there are no recommendations neither about position of the upper limbs and lower limbs nor about who should hold the mouthpiece while performing the maneuvers: evaluated subject or evaluator.

Objectives

To compare noseclip use or not, different upper and lower limbs positions and who holds the mouthpiece, verifying if these technical details affect spirometric results in healthy adults.

Methods

One hundred and three healthy individuals (41 men; age: 47 [33–58] years; normal lung function: FEV₁/FVC = 83±5, FEV₁ = 94 [88–104]%predicted, FVC = 92 [84–102]%predicted) underwent a protocol consisting of four spirometric comparative analysis in the sitting position: 1) maximum voluntary ventilation (MVV) with vs without noseclip; 2) FVC performed with vs without upper limbs support; 3) FVC performed with lower limbs crossed vs lower limbs in neutral position; 4) FVC, slow vital capacity and MVV comparing the evaluated subject holding the mouthpiece vs evaluator holding it.

Results

Different spirometric variables presented statistically significant difference (p<0.05) when analysing the four comparisons; however, none of them showed any variation larger than those considered as acceptable according to the ATS reproducibility criteria.

Conclusions

There was no relevant variation in spirometric results when analyzing technical details such as noseclip use during MVV, upper and lower limb positions and who holds the mouthpiece when performing the tests in healthy adults.     

The effect of a novel gait retraining device on lower limb kinematics and muscle activation in healthy adults

The Re-Link Trainer (RLT) is a modified walking frame with a linkage system designed to apply a non-individualized kinematic constraint to normalize gait trajectory of the left limb. The premise behind the RLT is that a user’s lower limb is constrained into a physiologically normal gait pattern, ideally generating symmetry across gait cycle parameters and kinematics. This pilot study investigated adaptations in the natural gait pattern of healthy adults when using the RLT compared to normal overground walking. Bilateral lower limb kinematic and electromyography data were collected while participants walked overground at a self-selected speed, followed by walking in the RLT. A series of 2-way analyses of variance examined between-limb and between-condition differences. Peak hip extension and knee flexion were reduced bilaterally when walking in the RLT. Left peak hip extension occurred earlier in the gait cycle when using the RLT, but later for the right limb. Peak hip flexion was significantly increased and occurred earlier for the constrained limb, while peak plantarflexion was significantly reduced. Peak knee flexion and plantarflexion in the right limb occurred later when using the RLT. Significant bilateral reductions in peak electromyography amplitude were evident when walking in the RLT, along with a significant shift in when peak muscle activity was occurring. These findings suggest that the RLT does impose a significant constraint, but generates asymmetries in lower limb kinematics and muscle activity patterns. The large interindividual variation suggests users may utilize differing motor strategies to adapt their gait pattern to the imposed constraint.