Touchdown Angle Research Articles

Dynamic characteristics and stability criterion of rotary galloping gait with an articulated passive spine joint

Based on the simplified planar quadruped model with a passive spine joint, this paper focuses on studying the effect of the spine joint on the dynamic performance and the stability of the rotary galloping gait. The stable, periodic, and symmetric rotary galloping gait is achieved by the particle swarm optimization method without any parameter symmetry for leg motion and without any limitation for spine motion. By analyzing the symmetric pattern of the rotary galloping gait, the stability criterion is derived: the closer to zero is the pitching angle at peak time of the center of mass, the better is the stability of the galloping gait. Then, we study the effect of rear leg touchdown angle and spine stiffness on the stability. Moreover, the unstable rotary galloping gait can be stabilized by adjusting one or more of the touchdown angle, spine stiffness, and leg stiffness. Furthermore, the effect of the spine stiffness on the key kinematic and kinetic properties is also studied for different initial values and mechanical parameters. The presented stability criterion and the achieved dynamic performance analysis results will be instructive for the design and control of the quadruped robot.

Quadrupedal running with a flexible torso: control and speed transitions with sums-of-squares verification

This paper studies quadrupedal bounding in the presence of flexible torso and compliant legs with non-trivial inertia, and it proposes a method for speed transitions by sequentially composing locally stable bounding motions corresponding to different speeds. First, periodic bounding motions are generated simply by positioning the legs during flight via suitable (virtual) holonomic constraints imposed on the leg angles; at this stage, no control effort is developed on the support legs, producing efficient, nearly passive, bounding gaits. The resulting motions are then stabilized by a hybrid control law which coordinates the movement of the torso and legs in continuous time, and updates the leg touchdown angles in an event-based fashion. Finally, through sums-of-squares programming, formally verified estimates of the domain of attraction of stable fixed points are employed to realize stable speed transitions by switching among different bounding gaits in a sequential fashion.

Control of a Cheetah Robot in Passive Bounding Gait

Passive dynamics is always one of research emphases of the legged robots. Studies have proved that cheetah robot could achieve stably passive bounding motion under proper initial conditions in the ideal case. However, the actual robot must have energy dissipation because of friction and collision compared with the theoretical model. This paper aims to propose a control method that can drive the cheetah robot running in passive bounding gait. First, a sagittal-plane model with a rigid torso and two compliant legs is introduced to capture the dynamics of robot bounding. Numerical return map studies of the bounding model reveal that there exists a large variety of passively cyclic bounding motions (fixed points). Based on the distribution law of fixed points, an open-loop control method including touchdown angle control strategy and leg length control strategy is put forward. At last, prototype of the cheetah robot is designed and manufactured, and locomotion experiment are carried out. The experiment results show that the cheetah robot can achieve a stable bounding motion at different speeds with the proposed control method.

A galloping quadruped model using left–right asymmetry in touchdown angles

Among quadrupedal gaits, the galloping gait has specific characteristics in terms of locomotor behavior. In particular, it shows a left–right asymmetry in gait parameters such as touchdown angle and the relative phase of limb movements. In addition, asymmetric gait parameters show a characteristic dependence on locomotion speed. There are two types of galloping gaits in quadruped animals: the transverse gallop, often observed in horses; and the rotary gallop, often observed in dogs and cheetahs. These two gaits have different footfall sequences. Although these specific characteristics in quadrupedal galloping gaits have been observed and described in detail, the underlying mechanisms remain unclear. In this paper, we use a simple physical model with a rigid body and four massless springs and incorporate the left–right asymmetry of touchdown angles. Our simulation results show that our model produces stable galloping gaits for certain combinations of model parameters and explains these specific characteristics observed in the quadrupedal galloping gait. The results are then evaluated in comparison with the measured data of quadruped animals and the gait mechanisms are clarified from the viewpoint of dynamics, such as the roles of the left–right touchdown angle difference in the generation of galloping gaits and energy transfer during one gait cycle to produce two different galloping gaits.

Running shoe crash-pad design alters shoe touchdown angles and ankle stability parameters during heel–toe running

Purpose: Running shoe crash-pads are commonly used to improve cushioning during heel–toe running. However, guidelines regarding shape and positioning of crash-pads are rare. This research examined the effect of systematically modified running shoe crash-pads on vertical ground reaction force characteristics and lower extremity kinematics during heel–toe running.Methods: Four custom-made running shoes featured crash-pad variations regarding horizontal extent and vertical position embedded within ethylene-vinyl acetate (EVA) midsoles. Crash-pads were located at the posterior-lateral midsole and had two sizes (narrow and wide) and two positions (top and bottom). Twenty-four runners performed five valid laboratory running trials per shoe while capturing ground reaction forces and lower extremity kinematics. Perception of cushioning, stability, and overall preference was assessed during outdoor loop running. Shoe variable means were compared for main (p < .05) and interaction (simple) effects (p < .05) by two within participants’ factors (size and position) repeated measures analyses of variance (ANOVA) and effect size estimation (pη2).Results: Rearfoot motion variables indicated biomechanical effects on running shoe stability induced by crash-pad size and position. Shock attenuation was marginally affected by shoe modifications and was not perceived by runners. Runners also did not perceive rearfoot stability alterations among shoes.Conclusion: Crash-pad modifications at the posterior-lateral region of EVA midsoles do not necessarily change substantially biomechanical and perceived running shoe cushioning properties. However, crash-pads applied in this research influenced rearfoot stability characteristics during running. In addition to dual density use at the medial shoe border, dual density use at the posterior-lateral midsole can provide opportunities to control rearfoot stability during heel–toe running.

Dynamic Imbalance Analysis and Stability Control of Galloping Gait for a Passive Quadruped Robot.

Some imbalance and balance postures of a passive quadruped robot with a simplified mathematical model are studied. Through analyzing the influence of the touchdown angle of the rear leg on the posture of the trunk during the flight phase, the stability criterion is concluded: the closer are the two moments which are the zero time of the pitching angle and the peak time of the center of mass, the better is the stability of the trunk posture during the flight phase. Additionally, the validity of the stability criterion is verified for the cat, greyhound, lion, racehorse, basset hound, and giraffe. Furthermore, the stability criterion is also applicable when the center of the mass of body is shifted. Based on the stability criterion, the necessary and sufficient condition of the galloping stability for the quadruped robot is proposed to attain a controlled thrust. The control strategy is designed by an optimization dichotomy algorithm for seeking the zero point of the balance condition. Through the control results, it is demonstrated that the imbalance posture of the trunk could be stabilized by adjusting the stiffness of four legs.

A template candidate for miniature legged robots in quasi-static motion

The paper introduces the Switching Four-bar Mechanism (sfm), a new low-dimensional kinematic abstraction for miniature legged robots, aimed at quasi-static motion planning in the horizontal plane. The model comprises a rigid torso and four rigid legs which engage in an alternating tetrapod gait. As the gait is executed, the torso and the legs form two switching four-bar linkages, parameterized by the leg touchdown and liftoff angles, as well as the leg angular velocity. We show that the sfm model captures on average experimentally observed behaviors of an eight-legged miniature robot crawling at low speeds quasi-statically. This work represents a first step toward a template that captures critical aspects of the kinematic behavior of miniature legged robots implementing quasi-static gaits. Such template can be used as a tool to facilitate motion planning tasks with such robots.

Hybrid control for SLIP-based robots running on unknown rough terrain

SUMMARYRapid and efficient dynamic stability control has been one of the important motivations in legged robot research, especially for legged robots running at high speed and/or on rough terrain. This paper presents a feasible control strategy, named Hybrid Feedback Control (HFC), for running systems based on the spring-loaded inverted pendulum principle (SLIP). The HFC strategy, which comprises two modules (i.e., touchdown angle control and energy compensation), predicts and regulates touchdown angle of the current cycle and need-to-complement energy input of the next cycle through hybrid feedback of flying apex state. This strategy can significantly reduce the computational complexity and enable the system to quickly converge to its control target, meeting the requirements of real-time control. Simulation experiments on various terrains were conducted to verify the adaptability of our HFC strategy. Results of these simulation experiments show that the approach herein can realize the periodical stability control of SLIP systems on different terrain conditions quickly and effectively.

Automated Transitions Between Walking and Running in Legged Robots

This paper addresses the synthesis of controllers for walk-to-run transition (WRT) and run-to-walk transition (RWT) in legged robots with adjustable leg compliance. Inspired by human kinematics, we propose a detailed procedure for the WRT and RWT in an adjustable-stiffness spring and mass model, and derive control parameters that ensure effective gait transitions. The WRT is achieved by modulating the compliance of the leg in a piecewise constant way during the transition stride. In the RWT, in addition to the leg compliance, we modulate the touchdown angle of attack. The merits of the approach proposed are demonstrated via a simulation that incorporates the control of walking, followed by a transition to running with first increasing and then decreasing speed, and finally a transition back to walking.

Compliant bipedal model with the center of pressure excursion associated with oscillatory behavior of the center of mass reproduces the human gait dynamics

Although the compliant bipedal model could reproduce qualitative ground reaction force (GRF) of human walking, the model with a fixed pivot showed overestimations in stance leg rotation and the ratio of horizontal to vertical GRF. The human walking data showed a continuous forward progression of the center of pressure (CoP) during the stance phase and the suspension of the CoP near the forefoot before the onset of step transition. To better describe human gait dynamics with a minimal expense of model complexity, we proposed a compliant bipedal model with the accelerated pivot which associated the CoP excursion with the oscillatory behavior of the center of mass (CoM) with the existing simulation parameter and leg stiffness. Owing to the pivot acceleration defined to emulate human CoP profile, the arrival of the CoP at the limit of the stance foot over the single stance duration initiated the step-to-step transition. The proposed model showed an improved match of walking data. As the forward motion of CoM during single stance was partly accounted by forward pivot translation, the previously overestimated rotation of the stance leg was reduced and the corresponding horizontal GRF became closer to human data. The walking solutions of the model ranged over higher speed ranges (~1.7m/s) than those of the fixed pivoted compliant bipedal model (~1.5m/s) and exhibited other gait parameters, such as touchdown angle, step length and step frequency, comparable to the experimental observations. The good matches between the model and experimental GRF data imply that the continuous pivot acceleration associated with CoM oscillatory behavior could serve as a useful framework of bipedal model.

Three-Dimensional Kinematics during the Take-Off Phase in Competitive Long Jumping

The purpose of the present study was to identify the relationships among selected kinematic variables that affect the take-off phase and performance in elite jumpers. The jump distance was found to be related to: I) the athlete's approach speed before the instant of touch down; and ii) the exchange in spatial velocity components at take-off, which results in a gain in maximum vertical velocity of the centre of mass (CM), favoured by the use of an optimum touch-down angle of the take-off leg, an active landing of the foot at touch-down, and a motion of the take-off leg during the compression phase that helps to manage the loss of horizontal velocity. Nonetheless, the results show that an adequate velocity transformation requires an adaptive technical model to help jumpers to build an efficient individual technical pattern.

Principal component based analysis of biomechanical inter-trial variability in individuals with chronic ankle instability

BackgroundBiomechanical variability during movement may influence joint stability in individuals with chronic ankle instability (CAI). The purpose of this study was to compare the kinematic and the kinetic inter-trial variability between healthy and CAI individuals. MethodsEleven individuals with CAI and 11 matched controls performed five repetitions of a single-leg landing task. Biomechanical data were collected from 100ms before to 200ms after touchdown, and were used to calculate touchdown angles, peak angles and moments at the ankle joint in the frontal and sagittal planes. In addition, principal component analyses were used to quantify kinematic and kinetic patterns in the same planes across the 300ms time window. Five trial averages and inter-trial variability were calculated for all variables for each subject. Independent t-tests were used to compare variables between groups. FindingsThe CAI group displayed greater inter-trial variability for principal component scores in the sagittal and frontal planes. The sagittal plane principal component captured a phase shift in plantar–flexion motion before touchdown, while the frontal plane principal component captured the general magnitude of motion during the entire movement. The CAI group therefore exhibited greater inter-trial variability in the sagittal plane before touchdown and in the frontal plane during the entire movement. InterpretationWhile average motions did not differ between groups, the CAI group displayed greater kinematic inter-trial variability when analyzed with the principal component analysis. More variable joint motions may indicate less dynamic stability in the CAI group, which may originate from greater ligamentous laxity or diminished neuromotor control.

A model-experiment comparison of system dynamics for human walking and running

The human musculo-skeletal system comprises high complexity which makes it difficult to identify underlying basic principles of bipedal locomotion. To tackle this challenge, a common approach is to strip away complexity and formulate a reductive model. With utter simplicity a bipedal spring-mass model gives good predictions of the human gait dynamics, however, it has not been fully investigated whether center of mass motion over time of walking and running is comparable between the model and the human body over a wide range of speed. To test the model׳s ability in this respect, we compare sagittal center of mass trajectories of model and human data for speeds ranging from 0.5m/s to 4m/s. For simulations, system parameters and initial conditions are extracted from experimental observations of 28 subjects. The leg parameters stiffness and length are extracted from functional fitting to the subjects׳ leg force–length curves. With small variations of the touch-down angle of the leg and the vertical position of the center of mass at apex, we find successful spring-mass simulations for moderate walking and medium running speeds. Predictions of the sagittal center of mass trajectories and ground reaction forces are good, but their amplitudes are overestimated, while contact time is underestimated. At faster walking speeds and slower running speeds we do not find successful model locomotion with the extent of allowed parameter variation. We conclude that the existing limitations may be improved by adding complexity to the model.

ACL Research Retreat V

ACL Research Retreat V

Modeling posture-dependent leg actuation in sagittal plane locomotion

The spring loaded inverted pendulum template has been shown to accurately model the steady locomotion dynamics of a variety of running animals, and has served as the inspiration for an entire class of dynamic running robots. While the template models the leg dynamics by an energy-conserving spring, insects and animals have structures that dissipate, store and produce energy during a stance phase. Recent investigations into the spring-like properties of limbs, as well as animal response to drop-step perturbations, suggest that animals use their legs to manage energy storage and dissipation, and that this management is important for gait stability. In this paper, we extend our previous analysis of control of the spring loaded inverted pendulum template via changes in the leg touch-down angle to include energy variations during the stance phase. Energy variations are incorporated through leg actuation that varies the force-free leg length during the stance phase, yet maintains qualitatively correct force and velocity profiles. In contrast to the partially asymptotically stable gaits identified in previous analyses, incorporating energy and leg angle variations in this manner produces complete asymptotic stability. Drop-step perturbation simulations reveal that the control strategy is rather robust, with gaits recovering from drops of up to 30% of the nominal hip height.

Leg recirculation in horizontal plane locomotion

A protocol prescribing leg motion during the swing phase is developed for the planar lateral leg spring model of locomotion. Inspired by experimental observations regarding insect leg function when running over rough terrain, the protocol prescribes the angular velocity of the swing-leg relative to the body in a feedforward manner, yielding natural variations in the leg touch-down angle in response to perturbations away from a periodic orbit. Analysis of the reduced order model reveals that periodic gait stability and robustness to external perturbations depends strongly upon the angular velocity of the leg at touch-down. While the leg angular velocity at touch-down provides control over gait stability and can be chosen to stabilize unstable gaits, the resulting basin of stability is much smaller than that observed for the original lateral leg spring model with a fixed leg touch-down angle. Comparisons to experimental leg angular velocity data for running cockroaches reveal that while the proposed protocol is qualitatively correct, smaller leg angular accelerations occur during the second half of the swing phase. Modifications made to the recirculation protocol to better match experimental observations yield large improvements in the basin of stability.

Lateral stability of the spring-mass hopper suggests a two-step control strategy for running

This paper investigates the control of running gaits in the context of a spring loaded inverted pendulum model in three dimensions. Specifically, it determines the minimal number of steps required for an animal to recover from a perturbation to a specified gait. The model has four control inputs per step: two touchdown angles (azimuth and elevation) and two spring constants (compression and decompression). By representing the locomotor movement as a discrete-time return map and using the implicit function theorem we show that the number of recovery steps needed following a perturbation depends upon the goals of the control mechanism. When the goal is to follow a straight line, two steps are necessary and sufficient for small lateral perturbations. Multistep control laws have a larger number of control inputs than outputs, so solutions of the control problem are not unique. Additional constraints, referred to here as synergies, are imposed to determine unique control inputs for perturbations. For some choices of synergies, two-step control can be expressed as two iterations of its first step policy and designed so that recovery occurs in just one step for all perturbations for which one-step recovery is possible.

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.

Speed Control for a Biped Robot

A biologically inspired model for a biped robot is developed. Each leg is modeled as a massless spring equipped with one radial and one angular actuator. The two legs in the model are attached to a point mass. The biped locomotion is modeled as a hybrid dynamic system that switches between four operation phases, with distinct dynamic behavior in each phase. A novel, simple yet robust control law that utilizes symmetry is developed to control the robot's speed, hopping height and balance during forward motion. The controller utilizes a feedforward artificial neural network, trained offline, to find approximate symmetric touchdown angles that are used by the control algorithm to determine actuator corrections. Simulation results show that the controller is able to maintain reference forward speed while maintaining balance during forward motion.

Determinants of difference in leg stiffness between endurance- and power-trained athletes

Understanding the leg and joint stiffness during human movement would provide important information that could be utilized for evaluating sports performance and for injury prevention. In the present study, we examined the determinants of the difference in the leg stiffness between the endurance-trained and power-trained athletes. Seven distance runners and seven power-trained athletes performed in-place hopping, matching metronome beats at 3.0 and 1.5 Hz. Leg and joint stiffness were calculated from kinetic and kinematics data. Electromyographic activity (EMG) was recorded from six leg muscles. At both hopping frequencies, the power-trained athletes demonstrated significantly higher leg stiffness than the distance runners. Hip, knee, and ankle joints were analyzed for stiffness and touchdown angles. Ankle stiffness was significantly greater in the power-trained athletes than the distance runners at 3.0 Hz as was knee stiffness at 1.5 Hz. There was no significant difference in touchdown angle between the DR and PT groups at either hopping frequencies. When significant difference in EMG activity existed between two groups, it was always greater in the distance runners than the power-trained athletes. These results suggest that (1) the difference in leg stiffness between endurance-trained and power-trained athletes is best attributed to increased joint stiffness, and (2) the difference in joint stiffness between the two groups may be attributed to a lack of similarity in the intrinsic stiffness of the muscle-tendon complex rather than in altered neural activity.