Improving horizontal plane locomotion via leg angle control

The lateral leg spring model has been shown to accurately represent horizontal plane locomotion characteristics of sprawled posture insects such as the cockroach Blaberus discoidalis. While passively stable periodic gaits result from employing a constant leg touch-down angle for this model, utilizing a similar protocol for a point mass model of locomotion in three dimensions produces only unstable periodic gaits. In this work, we return to the horizontal plane model and develop a simple control law that prescribes variations in the leg touch-down angle in response to external perturbations. The resulting control law applies control once per stance phase, at the instant of leg touch-down, and depends upon previous leg angles defined in the body reference frame. As a result, our control action is consistent with the neural activity evidenced by B. discoidalis during locomotion over flat and rough terrain, and utilizes variables easily sensed by insect mechanoreceptors. Application of control in the lateral leg spring model is shown to improve stability of periodic gaits, enable stabilization of previously unstable periodic gaits, and maintain or improve the basin of stability of periodic gaits. The magnitude of leg touch-down angle variations utilized during stabilization appear consistent with the natural variations evidenced by single legs during locomotion over flat terrain.     

A hexapedal jointed-leg model for insect locomotion in the horizontal plane

We develop a simple model for insect locomotion in the horizontal (ground) plane. As in earlier work by Seipel et al. (Biol Cybern 91(0):76–90, 2004) we employ six actuated legs that also contain passive springs, but the legs, with “hip” and ‘knee’ joints, better represent insect morphology. Actuation is provided via preferred angle inputs at each joint, corresponding to zero torques in the hip and knee springs. The inputs are determined from estimates of foot forces in the cockroach Blaberus discoidalis via an inverse problem. The head–thorax–body is modeled as a single rigid body, and leg masses, inertia and joint dissipation are ignored. The resulting three degree-of-freedom dynamical system, subject to feedforward joint inputs, exhibits stable periodic gaits that compare well with observations over the insect’s typical speed range. The model’s response to impulsive perturbations also matches that of freely-running cockroaches (Jindrich and Full, J Exp Biol 205:2803–2823, 2002), and stability is maintained in the face of random foot touchdowns representative of real insects. We believe that this model will allow incorporation of realistic muscle models driven by a central pattern generator in place of the joint actuators, and that it will ultimately permit the study of proprioceptive feedback pathways involving leg force and joint angle sensing.     

Dynamics and stability of insect locomotion: a hexapedal model for horizontal plane motions

We develop a simple hexapedal model for the dynamics of insect locomotion in the horizontal plane. Each leg is a linear spring endowed with two inputs, controlling force-free length and hip position, in a stereotypical feedforward pattern. These represent, in a simplified manner, the effects of neurally activated muscles in the animal and are determined from measured foot force and kinematic body data for cockroaches. We solve the three-degree-of-freedom Newtonian equations for coupled translation-yawing motions in response to the inputs and determine branches of periodic gaits over the animals typical speed range. We demonstrate a close quantitative match to experiments and find both stable and unstable motions, depending upon input protocols.Our hexapedal model highlights the importance of stability in evaluating effective locomotor performance and in particular suggests that sprawled-posture runners with large lateral and opposing leg forces can be stable in the horizontal plane over a range of speeds, with minimalsensory feedback from the environment. Fore–aft force patterns characteristic of upright-posture runners can cause instability in the model. We find that stability can constrain fundamental gait parameters: our model is stable only when stride length and frequency match the patterns measured in the animal. Stability is not compromised by large joint moments during running because ground reaction forces tend to align along the leg and be directed toward the center of mass. Legs radiating in all directions and capable of generating large moments may allow very rapid turning and extraordinary maneuvers. Our results further weaken the hypothesis that polypedal, sprawled-posture locomotion with large lateral and opposing leg forces is less effective than upright posture running with fewer legs.     

Dynamics and stability of legged locomotion in the horizontal plane: a test case using insects

 Motivated by experimental studies of insects, we propose a model for legged locomotion in the horizontal plane. A three-degree-of freedom, energetically conservative, rigid-body model with a pair of compliant virtual legs in intermittent contact with the ground allows us to study how dynamics depends on parameters such as mass, moment of inertia, leg stiffness, and length. We find periodic gaits, and show that mechanics alone can confer asymptotic stability of relative heading and body angular velocity. We discuss the relevance of our idealized models to experiments and simulations on insect running, showing that their gait and force characteristics match observations reasonably well. We perform parameter studies and suggest that our model is relevant to the understanding of locomotion dynamics across species. Received: 17 April 2001 / Accepted in revised form: 20 November 2001     

Mechanical models for insect locomotion: dynamics and stability in the horizontal plane I. Theory

We study the dynamics and stability of legged locomotion in the horizontal plane. Motivated by experimental studies of insects, we develop two- and three-degree-of freedom rigid body models with pairs of ‘virtual’ elastic legs in intermittent contact with the ground. We focus on conservative compliant-legged models, but we also consider prescribed forces, prescribed leg displacements, and combined strategies. The resulting mechanical systems exhibit periodic gaits whose stability characteristics are due to intermittent foot contact, and are largely determined by geometrical criteria. Most strikingly, we show that mechanics alone can confer asymptotic stability in heading and body orientation. In a companion paper, we apply our results to rapidly running cockroaches. Received: 6 September 1999 / Accepted in revised form: 8 May 2000     

A model for insect locomotion in the horizontal plane: Feedforward activation of fast muscles, stability, and robustness

We develop a neuromechanical model for running insects that includes a simplified hexapedal leg geometry with agonist-antagonist muscle pairs actuating each leg joint. Restricting to dynamics in the horizontal plane and neglecting leg masses, we reduce the model to three degrees of freedom describing translational and yawing motions of the body. Muscles are driven by stylized action potentials characteristic of fast motoneurons, and modeled using an activation function and nonlinear length and shortening velocity dependence. Parameter values are based on measurements from depressor muscles and observations of kinematics and dynamics of the cockroach Blaberus discoidalis; in particular, motoneuronal inputs and muscle force levels are chosen to approximately achieve joint torques that are consistent with measured ground reaction forces. We show that the model has stable double-tripod gaits over the animal's speed range, that its dynamics at preferred speeds matches those observed, and that it maintains stable gaits, with low frequency yaw deviations, when subject to random perturbations in foot touchdown and lift-off timing and action potential input timing. We explain this in terms of the low-dimensional dynamics.     

Dynamics and stability of lateral plane locomotion on inclines

An actuated, lateral leg spring model is developed to investigate lateral plane locomotion dynamics and stability on inclines. A single actuation input, the force-free leg length, is varied in a feedforward fashion to explicitly and implicitly match prescribed lateral and fore-aft force profiles, respectively. Forward dynamic simulations incorporating the prescribed leg actuation are employed to identify periodic orbits for gaits in which the leg acts to either push the body away from or pull the body towards the foot placement point. Gait stability and robustness to external perturbation are found to vary significantly as a function of slope and velocity for each type of leg function. Results of these analyses suggest that the switch in leg function from pushing to pulling is governed by gait robustness, and occurs at increasing inclines for increasing velocities.     

The precedence effect for signals moving in the horizontal plane

The precedence effect refers to a group of auditory phenomena related to the ability to locate sound sources in reverberant environments. In the present study, this phenomenon was investigated using two moving signals. The first signal was direct (lead) and the other was delayed (lag). The motion of the sound source was created by successive switching of ten loudspeakers. The continuity of the motion was created by simultaneously attenuating the stimulus in the previous loudspeaker and enhancing it in the next one. The length of the path of the lead and lag was 34°. The lead moved from 34° to 0° (to the right) and the lag moved –52° to –86° (to the left). The duration of the lead and the lag was 1 s. Lead–lag delays ranged from 1 to 40 ms. Subjects had to indicate the location of the lag. The results indicate that the lead signal dominated in the sound localization at short delay durations (up to 18 ms). In spite of the instructions, all the subjects pointed at the lead, which suggests that they perceived the lag in this location. Two distinct sounds were perceived at the longest delays. The mean echo threshold and its standard deviation in eight subjects was 9.6 ± 4.5 ms.     

Effect of audiovisual training on monaural spatial hearing in horizontal plane

The article aims to test the hypothesis that audiovisual integration can improve spatial hearing in monaural conditions when interaural difference cues are not available. We trained one group of subjects with an audiovisual task, where a flash was presented in parallel with the sound and another group in an auditory task, where only sound from different spatial locations was presented. To check whether the observed audiovisual effect was similar to feedback, the third group was trained using the visual feedback paradigm. Training sessions were administered once per day, for 5 days. The performance level in each group was compared for auditory only stimulation on the first and the last day of practice. Improvement after audiovisual training was several times higher than after auditory practice. The group trained with visual feedback demonstrated a different effect of training with the improvement smaller than the group with audiovisual training. We conclude that cross-modal facilitation is highly important to improve spatial hearing in monaural conditions and may be applied to the rehabilitation of patients with unilateral deafness and after unilateral cochlear implantation.     

Cloud cover and horizontal plane eye damaging solar UV exposures

The spectral UV and the cloud cover were measured at intervals of 5 min with an integrated cloud and spectral UV measurement system at a sub-tropical Southern Hemisphere site for a 6-month period and solar zenith angle (SZA) range of 4.7° to approximately 80°. The solar UV spectra were recorded between 280 nm and 400 nm in 0.5 nm increments and weighted with the action spectra for photokeratitis and cataracts in order to investigate the effect of cloud cover on the horizontal plane biologically damaging UV irradiances for cataracts (UVBE_cat) and photokeratitis (UVBE_pker). Eighty five percent of the recorded spectra produced a measured irradiance to a cloud free irradiance ratio of 0.6 and higher while 76% produced a ratio of 0.8 and higher. Empirical non-linear expressions as a function of SZA have been developed for all sky conditions to allow the evaluation of the biologically damaging UV irradiances for photokeratitis and cataracts from a knowledge of the unweighted UV irradiances.     

A dynamical model for reflex activated head movements in the horizontal plane

 We present a controls systems model of horizontal-plane head movements during perturbations of the trunk, which for the first time interfaces a model of the human head with neural feedback controllers representing the vestibulocollic (VCR) and the cervicocollic (CCR) reflexes. This model is homeomorphic such that model structure and parameters are drawn directly from anthropomorphic, biomechanical and physiological studies. Using control theory we analyzed the system model in the time and frequency domains, simulating neck movement responses to input perturbations of the trunk. Without reflex control, the head and neck system produced a second-order underdamped response with a 5.2 dB resonant peak at 2.1 Hz. Adding the CCR component to the system dampened the response by approximately 7%. Adding the VCR component dampened head oscillations by 75%. The VCR also improved low-frequency compensation by increasing the gain and phase lag, creating a phase minimum at 0.1 Hz and a phase peak at 1.1 Hz. Combining all three components (mechanics, VCR and CCR) linearly in the head and neck system reduced the amplitude of the resonant peak to 1.1 dB and increased the resonant frequency to 2.9 Hz. The closed loop results closely fit human data, and explain quantitatively the characteristic phase peak often observed. Received: 15 April 1996 / Accepted in revised form: 1 July 1996     

Comparing inclined locomotion in a ground-living and a climbing ant species: sagittal plane kinematics

Formicine ants are able to detect slopes in the substrates they crawl on. It was assumed that hair fields between the main segments of the body and between the proximal leg segments contribute to graviception which triggers a change of posture in response to substrate slopes. The sagittal kinematics of two ant species were investigated and compared on different slopes. Cataglyphis fortis, a North African desert ant, is well known for its extraordinary sense of orientation in texturally almost uniform habitats, while Formica pratensis, a common central-European species, primarily uses landmarks and pheromone traces for orientation. A comparison of these two species reveals differences in postural adaptations during inclined locomotion. Only minor slope-dependent angular adjustments were observed. The largest is a 25° head rotation for Cataglyphis, even if the slope is changed by 150°, suggesting dramatic changes in the field of vision. The trunk’s pitch adjustment towards the increasing slope is low in both species. On all slopes Cataglyphis achieves higher running speeds than Formica and displays greater slope-dependent variation in body height. This indicates different strategies for coping with changing slopes. These specific aspects have to be reflected in the ants’ respective mode of slope perception.     

Information transmission in non-visual fingertip matching along a horizontal track in the median plane

Non-visually triggered arm movements over a horizontal table at shoulder height were analysed by an Information Theory approach according to a method suggested by Sakitt et al. (1983) and Sakitt (1980). The movement track was along the subject's median line and was indicated by a vertical metal ridge fixed to the table. The observer passively moved the subject's left index finger along the left side of the ridge to the target position. The blindfolded subject then had to move his right index finger along the right side of the ridge to match the left finger position. Direct contact between the two fingers was prevented by the ridge. We compared our results, which involve the transmission of information through the arm and shoulder joints of both arms, whith those of Sakitt et al. which involved just one elbow joint. We supplemented our experimental results with simulations and show that the value for the transmitted information, obtained using the method of analysis suggested by Sakitt et al., is very dependent upon the number of trials, and number and spacing of the targets. Sakitt et al. suggest that the Information Theory approach permits easy comparison between different tasks and different observers. Our results suggest that comparisons should be made with caution.     

Experimental and modelling investigation of learning a fast elbow flexion in the horizontal plane

Changes in the kinematic and electromyographic characteristics that occur while learning to move as fast as possible have been studied experimentally. Experimental investigation of what happens to the individual motor units (MUs) is more difficult. Access to each MU is impossible, and the recruitment and force developing properties of all individual MUs cannot be known. Thus, what is currently known about MU firing is based on experiments that have recorded relatively few MUs compared to what exists in the entire muscle. A recently developed muscle model (Raikova and Aladjov, 2002, J. Biomechanics, 35, 1123-1135) composed of MUs with different properties can be used for such investigation. The process of learning fast elbow flexion in the horizontal plane was simulated and the results were compared with experimentally measured data. Comparing the simulation results of the very first trial of a particular subject with those of the last trail (at the end of the learning process), it can be concluded that the speed of limb motion and muscle forces increase initially as a result of the more synchronous MUs activation and the increase of firing rate of active MUs. Further improvement necessitated an appreciable reduction in the motor task requirements (i.e. less muscle force and less MUs' activity) set in the computational algorithm by optimization criteria. This forced the next process-inclusion of additional MUs.     

Presenting joint kinematics of human locomotion using phase plane portraits and Poincaré maps

Additional graphical tools are needed to better visualize the joint kinematics of human locomotion. Standard plots in which the joint displacements are plotted against time or percent gait cycle do not provide sufficient information about the dynamics of the system. In this article, a study based on the two graphical tools of nonlinear dynamics to visualize the steady-state kinematics of human gait is presented. An experimental setup was developed to acquire the necessary data for application of the techniques. Twenty young adults, whose medical histories are free of gait pathology, were tested. Computerized electrogoniometers and foot switches were used to measure the kinematic data of the lower extremities and capture four instants of the gait cycle: heel strike, foot flat, heel off, and toe off. Phase plane portraits of each joint were constructed for the sagittal plane by plotting angular velocity against angular displacement. Poincaré maps were obtained by periodically sampling the joint profiles at toe off and plotting the ith iterate against the (i + 1)th one. Phase plane portraits are useful in monitoring the variations of joint velocity and position on the same graph in a more compact form. Poincaré maps are effective in differentiating steady gait from transient locomotion.     

Influence of recirculation over COD and N-NH4 removals from landfill leachate by horizontal flow constructed treatment wetland

Abstract

Treatment of landfill leachate is a challenge due to its complex chemical composition and high recalcitrance and because of high costs for conventional wastewater treatment. In our study, leachate from the Quitaúna Landfill, Sao Paulo Metropolitan Region, Brazil, was treated at a laboratory scale with a horizontal subsurface flow constructed treatment wetland (HF-CTW) operating under a recirculation regime. Two units planted with Heliconia psittacorum (HP) and Cyperus papyrus (CP), and one unplanted control unit were assessed. With a recirculation regime over 21?days, the planted units removed 40% of chemical oxygen demand (COD) while the control unit removed only 29%. True color removal efficiencies were 2, 22, and 23% for the control, HP, and CP HF-CTWs, respectively. The ammonium nitrogen removal efficiencies for a 21-day hydraulic retention time (HRT) were 63–81% for planted units and 72% for the control. The increase of the HRT from 7 to 21?days led to the enhancement of ammonium nitrogen removal but did not affect the COD and total nitrogen removals. This phenomenon is a consequence of leachate’s low biodegradability. The present study shows the importance of the HRT and plant presence for landfill leachate treatment using HF-CTWs.