Stability of an underactuated bipedal gait

A self-excited biped walking mechanism consisting of two legs that are connected in series at the hip joint through a servomotor is studied as a cyclic system with collisions. A torque proportional to angle between the shank of the swinging leg and the vertical is seen to sustain a gait. Each leg has a thigh and a shank connected at a passive knee joint that has a knee stopper restricting hyperextension similar to the human knee. A mathematical model for the dynamics of the system including the impact equations is used to analyse the stability of the system through examination of phase plane plots. Attractor lines along which the system approaches stability have been identified. A leg length for optimal stability has been identified. The biological basis for the proposed system has been identified by comparison with human gait.     

Coupled oscillators utilised as gait rhythm generators of a two-legged walking machine

The gait of current two-legged walking machines differs from that of humans, although the kinematic structures of these machines' legs frequently imitate human limbs. This paper presents a method of generating the trajectories of hip and knee joint angles resulting in a gait pattern similar to that of a human. For this purpose the solutions of coupled van der Pol oscillator equations are utilised. There is much evidence that these equations can be treated as a good model of the central pattern generator generating functional (also locomotional) rhythms in living creatures. The oscillator equations are solved by numerical integration. The method of changing the type of gait by changing appropriate parameter values in the oscillator equations is presented (change of velocity and trajectory of leg-ends). The results obtained enable enhanced control of twolegged walking systems by including gait pattern generators which will assume a similar role to that of biological generators.     

Modeling of a bipedal locomotor using coupled nonlinear oscillators of Van der Pol

 Research to date points to an understanding of human biped locomotion that has been primarily experimental in nature largely due to the complexity of the process. In view of the new, exciting possibilities of programmed electrostimulation of artificial muscles to generate motion (locomotion), a critical study at the theoretical level is greatly warranted. There is strong evidence that many biological clocks consist of a population of mutually coupled oscillators [Pavlidis T (1973) Biological oscillators, Academic; Johnsson A (1978) Zur Biophysik biologischer Oszillatoren. In: Biophisik, Springer]. In this work, a form of bipedal locomotion is simulated by using mutually coupled nonlinear oscillators. A planar model, which includes three out of the six determinants of gait that characterize the human locomotion, was adopted. Received: 4 August 2002 / Accepted: 12 November 2002 / Published online: 7 March 2003 Correspondence to: M. S. Dutra (e-mail: max@serv.com.ufrj.br) Acknowledgements. The authors would like to express their gratitude to CNPq, CAPES, and FAPERJ for the financial support provided during the course of this research.     

A passive dynamic walking robot that has a deterministic nonlinear gait

There is a growing body of evidence that the step-to-step variations present in human walking are related to the biomechanics of the locomotive system. However, we still have limited understanding of what biomechanical variables influence the observed nonlinear gait variations. It is necessary to develop reliable models that closely resemble the nonlinear gait dynamics in order to advance our knowledge in this scientific field. Previously, Goswami et al. [1998. A study of the passive gait of a compass-like biped robot: symmetry and chaos. International Journal of Robotic Research 17(12)] and Garcia et al. [1998. The simplest walking model: stability, complexity, and scaling. Journal of Biomechanical Engineering 120(2), 281-288] have demonstrated that passive dynamic walking computer models can exhibit a cascade of bifurcations in their gait pattern that lead to a deterministic nonlinear gait pattern. These computer models suggest that the intrinsic mechanical dynamics may be at least partially responsible for the deterministic nonlinear gait pattern; however, this has not been shown for a physical walking robot. Here we use the largest Laypunov exponent and a surrogation analysis method to confirm and extend Garcia et al.'s and Goswami et al.'s original results to a physical passive dynamic walking robot. Experimental outcomes from our walking robot further support the notion that the deterministic nonlinear step-to-step variations present in gait may be partly governed by the intrinsic mechanical dynamics of the locomotive system. Furthermore the nonlinear analysis techniques used in this investigation offer novel methods for quantifying the nature of the step-to-step variations found in human and robotic gait.     

A rotation and translation invariant discrete saliency network

 We describe a neural network that enhances and completes salient closed contours in images. Our work is different from all previous work in three important ways. First, like the input provided to primary visual cortex (V1) by the lateral geniculate nucleus (LGN), the input to our computation is isotropic. That is, it is composed of spots, not edges. Second, our network computes a well-defined function of the input based on a distribution of closed contours characterized by a random process. Third, even though our computation is implemented in a discrete network, its output is invariant to continuous rotations and translations of the input image. Received: 11 July 2002 / Accepted in revised form: 25 October 2002 Acknowledgements. L.R.W. was supported in part by Los Alamos National Laboratory. J.W.Z. was supported in part by the Albuquerque High Performance Computing Center. We wish to thank Jonas August and Steve Zucker for their insightful comments. Correspondence to: L.R. Williams (e-mail: Williams@cs.unm.edu)     

Repetitive gait of passive bipedal mechanisms in a three-dimensional environment

The inherent dynamics of bipedal, kneed mechanisms are studied with emphasis on the existence and stability, of repetitive gait in a three-dimensional environment, in the absence of external, active control. The investigation is motivated by observations that sustained anthropomorphic locomotion is largely a consequence of geometric and inertial properties of the mechanism. While the modeling excludes active control, the energy dissipated in ground and knee collisions is continuously re-injected by considering gait down slight inclines. The paper describes the dependence of the resulting passive gait in vertically constrained and unconstrained mechanisms on model parameters, such as ground compliance and ground slope. We also show the possibility of achieving statically unstable gait with appropriate parameter choices.     

Identification of passive elastic joint moment-angle relationships in the lower extremity

The purpose of this study was to develop a method for identifying subject-specific passive elastic joint moment-angle relationships in the lower extremity, which could subsequently be used to estimate passive contributions to joint kinetics during gait. Twenty healthy young adults participated in the study. Subjects were positioned side-lying with their dominant limb supported on a table via low-friction carts. A physical therapist slowly manipulated the limb through full sagittal hip, knee, and ankle ranges of motion using two hand-held 3D load cells. Lower extremity kinematics, measured with a passive marker motion capture system, and load cell readings were used to compute joint angles and associated passive joint moments. We formulated a passive joint moment-angle model that included eight exponential functions to account for forces generated via the passive stretch of uni-articular structures and bi-articular muscles. Model parameters were estimated for individual subjects by minimizing the sum of squared errors between model predicted and experimentally measured moments. The model predictions closely replicated measured joint moments with average root-mean-squared errors of 2.5, 1.4, and 0.7 Nm about the hip, knee, and ankle respectively. We show that the models can be coupled with gait kinematics to estimate passive joint moments during walking. Passive hip moments were substantial from terminal stance through initial swing, with energy being stored as the hip extended and subsequently returned during pre- and initial swing. We conclude that the proposed methodology could provide quantitative insights into the potentially important role that passive mechanisms play in both normal and abnormal gait.     

Paired nonlinear behavior of active and passive joint torques associated with preparation for walk-to-run gait transition

This study investigated the lower extremity torque's active and passive features during the walk-to-run gait transition with continuously increased walking speed. Fourteen volunteers participated in the experiment. Kinematic and kinetic data were collected synchronously. Five strides leading up the gait transition were examined. Peaks of the passive (e.g., contact) and active (e.g., generalized muscle torques), along with net joint torque, and time to peak torques exhibited significant differences at the last stride before gait transition, compared to the first four strides, at the ankle, knee, and hip joints, respectively. Selected peak joint active and passive torques showed significant and opposite trends at critical events within a stride cycle: such ankle joint right after heel-contact, knee joint during weight acceptance, and both hip and knee joints right before toe-off. The magnitude and the corresponding time to active and passive peak torque changed in a nonlinear pattern before the transition from walk to run. The lower extremity segment-interaction during gait transition appeared to be an active reorganization exemplified by the interaction between the lower extremity's active and passive torque components.     

A mathematical model of adaptive behavior in quadruped locomotion

Locomotion involves repetitive movements and is often executed unconsciously and automatically. In order to achieve smooth locomotion, the coordination of the rhythms of all physical parts is important. Neurophysiological studies have revealed that basic rhythms are produced in the spinal network called, the central pattern generator (CPG), where some neural oscillators interact to self-organize coordinated rhythms. We present a model of the adaptation of locomotion patterns to a variable environment, and attempt to elucidate how the dynamics of locomotion pattern generation are adjusted by the environmental changes. Recent experimental results indicate that decerebrate cats have the ability to learn new gait patterns in a changed environment. In those experiments, a decerebrate cat was set on a treadmill consisting of three moving belts. This treadmill provides a periodic perturbation to each limb through variation of the speed of each belt. When the belt for the left forelimb is quickened, the decerebrate cat initially loses interlimb coordination and stability, but gradually recovers them and finally walks with a new gait. Based on the above biological facts, we propose a CPG model whose rhythmic pattern adapts to periodic perturbation from the variable environment. First, we design the oscillator interactions to generate a desired rhythmic pattern. In our model, oscillator interactions are regarded as the forces that generate the desired motion pattern. If the desired pattern has already been realized, then the interactions are equal to zero. However, this rhythmic pattern is not reproducible when there is an environmental change. Also, if we do not adjust the rhythmic dynamics, the oscillator interactions will not be zero. Therefore, in our adaptation rule, we adjust the memorized rhythmic pattern so as to minimize the oscillator interactions. This rule can describe the adaptive behavior of decerebrate cats well. Finally, we propose a mathematical framework of an adaptation in rhythmic motion. Our framework consists of three types of dynamics: environmental, rhythmic motion, and adaptation dynamics. We conclude that the time scale of adaptation dynamics should be much larger than that of rhythmic motion dynamics, and the repetition of rhythmic motions in a stable environment is important for the convergence of adaptation. Received: 10 July 1997 / Accepted in revised form: 13 March 1998     

Identification of the cable parameters in the somatic shunt model

 We show that the first five moments of the soma potential and soma current uniquely and stably determine the soma conductance and capacitance and the dendritic electrotonic length, conductance, and capacitance in the so-called somatic shunt model of the passive behavior of a neuron. We test our resulting input admittance algorithm on synthetic data and demonstrate the regularizing effect of knowledge of the ratio of soma to dendrite surface areas. Received: 9 June 1999 / Accepted in revised form: 24 January 2000     

The connectivity of the brain: multi-level quantitative analysis

 We develop a mathematical formalism or calculating connectivity volumes generated by specific topologies with various physical packing strategies. We consider four topologies (full, random, nearest-neighbor, and modular connectivity) and three physical models: (i) interior packing, where neurons and connection fibers are intermixed, (ii) sheeted packing where neurons are located on a sheet with fibers running underneath, and (iii) exterior packing where the neurons are located at the surfaces of a cube or sphere with fibers taking up the internal volume. By extensive cross-referencing of available human neuroanatomical data we produce a consistent set of parameters for the whole brain, the cerebral cortex, and the cerebellar cortex. By comparing these inferred values with those predicted by the expressions, we draw the following general conclusions for the human brain, cortex, and cerebellum: (i) Interior packing is less efficient than exterior packing (in a sphere). (ii) Fully and randomly connected topologies are extremely inefficient. More specifically we find evidence that different topologies and physical packing strategies might be used at different scales. (iii) For the human brain at a macrostructural level, modular topologies on an exterior sphere approach the data most closely. (iv) On a mesostructural level, laminarization and columnarization are evidence of the superior efficiency of organizing the wiring as sheets. (v) Within sheets, microstructures emerge in which interior models are shown to be the most efficient. With regard to interspecies similarities and differences we conjecture (vi) that the remarkable constancy of number of neurons per underlying square millimeter of cortex may be the result of evolution minimizing interneuron distance in grey matter, and (vii) that the topologies that best fit the human brain data should not be assumed to apply to other mammals, such as the mouse for which we show that a random topology may be feasible for the cortex. Received: 14 December 1994/Accepted in revised form: 23 May 1995     

Biomechanical mechanism of lateral trunk lean gait for knee osteoarthritis patients

The biomechanical mechanism of lateral trunk lean gait employed to reduce external knee adduction moment (KAM) for knee osteoarthritis (OA) patients is not well known. This mechanism may relate to the center of mass (COM) motion. Moreover, lateral trunk lean gait may affect motor control of the COM displacement. Uncontrolled manifold (UCM) analysis is an evaluation index used to understand motor control and variability of the motor task. Here we aimed to clarify the biomechanical mechanism to reduce KAM during lateral trunk lean gait and how motor variability controls the COM displacement. Twenty knee OA patients walked under two conditions: normal and lateral trunk lean gait conditions. UCM analysis was performed with respect to the COM displacement in the frontal plane. We also determined how the variability is structured with regards to the COM displacement as a performance variable. The peak KAM under lateral trunk lean gait was lower than that under normal gait. The reduced peak KAM observed was accompanied by medially shifted knee joint center, shortened distance of the center of pressure to knee joint center, and shortened distance of the knee–ground reaction force lever arm during the stance phase. Knee OA patients with lateral trunk lean gait could maintain kinematic synergy by utilizing greater segmental configuration variance to the performance variable. However, the COM displacement variability of lateral trunk lean gait was larger than that of normal gait. Our findings may provide clinical insights to effectively evaluate and prescribe gait modification training for knee OA patients.     

Coupling of posture and gait: mode locking and parametric excitation

 We investigate the temporal coordination of human gait and posture and infer the nature of their coupling. Participants viewed a sinusoidally oscillating visual display which induced medial-lateral postural sway during treadmill walking, while display frequency was varied (0.075–1.025 Hz). First, postural responses exhibited the usual low-pass characteristic but with an additional resonance peak near the preferred stride frequency, although shifted downward by 0.12 Hz; this provides evidence of a coupling from gait to posture. Second, the step cycle adapted to mode lock with the visual driver and postural sway, as well as displaying instances of intermittency (slipping in and out of phase) and quasiperiodicity (phase wandering); this provides evidence of a coupling from posture to gait. We observed a spectrum of integer mode locks, including a large 1:1 trapping region about the stride frequency and superharmonic entrainment (stride frequency > driver frequency) at lower driver frequencies. A coupled-oscillator model that incorporates a novel parametric coupling from posture to the gait “stiffness” term reproduces these features of the data, including the resonance peak shift. Biological coordination patterns may thus emerge naturally as properties of a system of appropriately coupled oscillators. Received: 23 June 1999 / Accepted in revised form: 10 January 2001     

Proximal placement of lateral thigh skin markers reduces soft tissue artefact during normal gait using the Conventional Gait Model

A primary source of measurement error in gait analysis is soft-tissue artefact. Hip and knee angle measurements, regularly used in clinical decision-making, are particularly prone to pervasive soft tissue on the femur. However, despite several studies of thigh marker artefact it remains unclear how lateral thigh marker height affects results using variants of the Conventional Gait Model. We compared Vicon Plug-in Gait hip and knee angle estimates during gait using a proximal and distal thigh marker placement for ten healthy subjects. Knee axes were estimated by optimizing thigh rotation offsets to minimize knee varus-valgus range during gait. Relative to the distal marker, the proximal marker produced 37% less varus-valgus range and 50% less hip rotation range (p < 0.001), suggesting that it produced less soft-tissue artefact in knee axis estimates. The thigh markers also produced different secondary effects on the knee centre estimate. Using whole gait cycle optimization, the distal marker showed greater minimum and maximum knee flexion (by 6° and 2° respectively) resulting in a 4° reduction in range. Mid-stance optimization reduced distal marker knee flexion by 5° throughout, but proximal marker results were negligibly affected. Based on an analysis of the Plug-in Gait knee axis definition, we show that the proximal marker reduced sensitivity to soft-tissue artefact by decreasing collinearity between the points defining the femoral frontal plane and reducing anteroposterior movement between the knee and thigh markers. This study suggests that a proximal thigh marker may be preferable when performing gait analysis using the Plug-in Gait model.     

Women with knee osteoarthritis have more pain and poorer function than men, but similar physical activity prior to total knee replacement

Background

Osteoarthritis of the knee is a major clinical problem affecting a greater proportion of women than men. Women generally report higher pain intensity at rest and greater perceived functional deficits than men. Women also perform worse than men on function measures such as the 6-minute walk and timed up and go tests. Differences in pain sensitivity, pain during function, psychosocial variables, and physical activity levels are unclear. Further the ability of various biopsychosocial variables to explain physical activity, function and pain is unknown.

Methods

This study examined differences in pain, pain sensitivity, function, psychosocial variables, and physical activity between women and men with knee osteoarthritis (N = 208) immediately prior to total knee arthroplasty. We assessed: (1) pain using self-report measures and a numerical rating scale at rest and during functional tasks, (2) pain sensitivity using quantitative sensory measures, (3) function with self-report measures and specific function tasks (timed walk, maximal active flexion and extension), (4) psychosocial measures (depression, anxiety, catastrophizing, and social support), and (5) physical activity using accelerometry. The ability of these mixed variables to explain physical activity, function and pain was assessed using regression analysis.

Results

Our findings showed significant differences on pain intensity, pain sensitivity, and function tasks, but not on psychosocial measures or physical activity. Women had significantly worse pain and more impaired function than men. Their levels of depression, anxiety, pain catastrophizing, social support, and physical activity, however, did not differ significantly. Factors explaining differences in (1) pain during movement (during gait speed test) were pain at rest, knee extension, state anxiety, and pressure pain threshold; (2) function (gait speed test) were sex, age, knee extension, knee flexion opioid medications, pain duration, pain catastrophizing, body mass index (BMI), and heat pain threshold; and (3) physical activity (average metabolic equivalent tasks (METS)/day) were BMI, age, Short-Form 36 (SF-36) Physical Function, Kellgren-Lawrence osteoarthritis grade, depression, and Knee Injury and Osteoarthritis Outcome Score (KOOS) pain subscale.

Conclusions

Women continue to be as physically active as men prior to total knee replacement even though they have significantly more pain, greater pain sensitivity, poorer perceived function, and more impairment on specific functional tasks.     

Activating the somatosensory system enhances net quadriceps moment during gait

Quadriceps muscle rehabilitation following knee injury or disease is often hampered by pain, proprioception deficits or instability associated with inhibition of quadriceps activation during walking. The cross-modal plasticity of the somatosensory system with common sensory pathways including pain, pressure and vibration offers a novel opportunity to enhance quadriceps function during walking. This study explores the effectiveness of an active knee brace that used intermittent cutaneous vibration during walking to enhance the peak knee flexion moment (KFM) during early stance phase as a surrogate for net quadriceps moment (balance between knee extensor and flexor muscle moments). The stimulus was turned on prior to heel strike and turned off at mid-stance of the gait cycle. Twenty-one subjects with knee pathologies known to inhibit quadriceps function were tested walking under three conditions: control (no brace), a passive brace, and an active brace. Findings show that compared to the control, subjects wearing an active brace during gait exhibited a significant (p < 0.001) increase in peak KFM and no significant difference when wearing a passive brace (p = 0.17). Furthermore, subjects with low KFM and knee flexion angle (KFA) in control exhibited the greatest increase in KFA at loading response in the active brace condition (R = 0.47, p < 0.05). Intermittent cutaneous stimulation during gait, therefore, provides an efficient method for increasing the KFM in patients with knee pathologies. This study’s results suggest that intermittent vibration stimulus can activate the cross-modalities of the somatosensory system in a manner that gates pain stimulus and possibly restores quadriceps function in patients with knee pain.     

Multicolor fluorescence in situ hybridization analysis of the spermatozoa of a male heterozygous for a reciprocal translocation t(11;22)(q23;q11)

A reciprocal translocation between chromosomes 11 and 22 is a site-specific translocation that has been seen in many families with no common ancestry. This translocation is of particular interest because balanced carriers have a 0.7–3.7% risk of having children with the supernumerary der(22), resulting from a 3:1 segregation. We have used a three color fluorescence in situ hybridization (FISH) with specific DNA probes to determine the chromosome segregation pattern of a male carrier of a translocation t(11;22)(q23;q11). The probes selected included a centromeric marker for chromosome 11, a marker closely linked to the centromere of chromosome 22, and a third probe distal to the translocation breakpoint of chromosome 22. The results showed that 3 : 1 segregation is preferential in this patient, with 40.1% of spermatozoa belonging to this segregation type. Alternate segregation followed with 27.4% of analyzed spermatozoa; 17.6% resulted from adjacent 1 and 12.5% resulted from adjacent 2 segregation. We detected 0.5% of presumably diploid spermatozoa. Complementary adjacent 1 products were observed at statistically different frequencies (P = 0.02). Complementary adjacent 2 products without recombination in the interstitial segments were also seen at different frequencies (P = 0.002). In 3 : 1 segregation, the products containing one chromosome were observed more frequently than those with three chromosomes (P = 0.0001). The 24,+der(22) gamete was seen more frequently than all of the other gametes combined which had 24 chromosomes resulting from 3 : 1 segregation. The results of this study demonstrate that in this t(11;22) carrier, 3 : 1 segregation is preferential but not exclusive. Received: 9 December 1998 / Accepted: 1 March 1999     

The passive elastic moment at the knee and its influence on human gait

The elastic component of the passive moment at the knee was measured in situ. The force needed to manually range the knee from approximately 90 degrees of flexion to full extension was measured. Hip and ankle angle were held fixed. The passive knee moment, computed from the force and knee angle data, was compared to the total knee moment required for normal gait. This comparison suggested that the passive moment can contribute a significant portion of the total joint moment during some phases of the gait cycle.     

Computational model of the cockroach escape behavior: winner and losers in a population code

 I present a comprehensive biologically oriented computational model to account for the escape response of the cockroach on the ground. This model is an expansion of previous work that accounted only for discriminating left from right wind directions [Ezrachi et al. (1999) Biol Cybern 81: 89–99]. The model is composed of computational elements describing the biological processes taking place in the various neurons and includes input which emulates empirical data. With this model it is possible to obtain escape behavior that resembles natural behavior. The model is used to address an ongoing debate as to whether the cockroach's turn direction is determined by computations carried out by the entire neuronal population (PC) or rather by a “winner-take-all” (WTA) mechanism. I suggest that the computation mechanism that underlies the cockroach escape response is composed of both PC and WTA principles. Based on the properties of the suggested new mechanism I denote it a “Darwinian population code.” Received: 26 March 2002 / Accepted in revised form: 24 June 2002 Acknowledgements. I thank H. Parnas for her advice and assistance, J. M. Camhi for helpful comments, and D. Lipson for developing the simulation tools. Correspondence to: E. A. Ezrachi (e-mail: erez@piano.ls.huji.ac.il, Tel.: +972-2-6585818, Fax: +972-2-6585569)     

A numerical method for renal models that represent tubules with abrupt changes in membrane properties

 The urine concentrating mechanism of mammals and birds depends on a counterflow configuration of thousands of nearly parallel tubules in the medulla of the kidney. Along the course of a renal tubule, cell type may change abruptly, resulting in abrupt changes in the physical characteristics and transmural transport properties of the tubule. A mathematical model that faithfully represents these abrupt changes will have jump discontinuities in model parameters. Without proper treatment, such discontinuities may cause unrealistic transmural fluxes and introduce suboptimal spatial convergence in the numerical solution to the model equations. In this study, we show how to treat discontinuous parameters in the context of a previously developed numerical method that is based on the semi-Lagrangian semi-implicit method and Newton's method. The numerical solutions have physically plausible fluxes at the discontinuities and the solutions converge at second order, as is appropriate for the method. Received: 13 November 2001 / Revised version: 28 June 2002 / Published online: 26 September 2002 This work was supported in part by the National Institutes of Health (National Institute of Diabetes and Digestive and Kidney Diseases, grant DK-42091.) Mathematics Subject Classification (2000): 65-04, 65M12, 65M25, 92-04, 92C35, 35-04, 35L45 Keywords or phrases: Mathematical models – Differential equations – Mathematical biology – Kidney – Renal medulla – Semi-Lagrangian semi-implicit