Andrew Long, PhD

Movements can be learned implicitly in response to new environmental demands or explicitly through instruction and strategy. The former is often studied in an environment that perturbs movement so that people learn to correct the errors and store a new motor pattern. Here, we demonstrate in human walking that implicit learning of foot placement occurs even when an explicit strategy is used to block changes in foot placement during the learning process. We studied people learning a new walking… Show more

Movements can be learned implicitly in response to new environmental demands or explicitly through instruction and strategy. The former is often studied in an environment that perturbs movement so that people learn to correct the errors and store a new motor pattern. Here, we demonstrate in human walking that implicit learning of foot placement occurs even when an explicit strategy is used to block changes in foot placement during the learning process. We studied people learning a new walking pattern on a split-belt treadmill with and without an explicit strategy through instruction on where to step. When there is no instruction, subjects implicitly learn to place one foot in front of the other to minimize step-length asymmetry during split-belt walking, and the learned pattern is maintained when the belts are returned to the same speed, i.e., postlearning. When instruction is provided, we block expression of the new foot-placement pattern that would otherwise naturally develop from adaptation. Despite this appearance of no learning in foot placement, subjects show similar postlearning effects as those who were not given any instruction. Thus locomotor adaptation is not dependent on a change in action during learning but instead can be driven entirely by an unexpressed internal recalibration of the desired movement. Show less

In human motor learning, it is thought that the more information we have about our errors, the faster we learn. Here, we show that additional error information can lead to improved motor performance without any concomitant improvement in learning. We studied split-belt treadmill walking that drives people to learn a new gait pattern using sensory prediction errors detected by proprioceptive feedback. When we also provided visual error feedback, participants acquired the new walking pattern far… Show more

In human motor learning, it is thought that the more information we have about our errors, the faster we learn. Here, we show that additional error information can lead to improved motor performance without any concomitant improvement in learning. We studied split-belt treadmill walking that drives people to learn a new gait pattern using sensory prediction errors detected by proprioceptive feedback. When we also provided visual error feedback, participants acquired the new walking pattern far more rapidly and showed accelerated restoration of the normal walking pattern during washout. However, when the visual error feedback was removed during either learning or washout, errors reappeared with performance immediately returning to the level expected based on proprioceptive learning alone. These findings support a model with two mechanisms: a dual-rate adaptation process that learns invariantly from sensory prediction error detected by proprioception and a visual-feedback-dependent process that monitors learning and corrects residual errors but shows no learning itself. We show that our voluntary correction model accurately predicted behavior in multiple situations where visual feedback was used to change acquisition of new walking patterns while the underlying learning was unaffected. The computational and behavioral framework proposed here suggests that parallel learning and error correction systems allow us to rapidly satisfy task demands without necessarily committing to learning, as the relative permanence of learning may be inappropriate or inefficient when facing environments that are liable to change. Show less

Walking is highly adaptable to new demands and environments. We have previously studied adaptation of locomotor patterns via a split-belt treadmill, where subjects learn to walk with one foot moving faster than the other. Subjects learn to adapt their walking pattern by changing the location (spatial) and time (temporal) of foot placement. Here we asked whether we can induce adaptation of a specific walking pattern when one limb does not “walk” but instead marches in place (i.e.… Show more

Walking is highly adaptable to new demands and environments. We have previously studied adaptation of locomotor patterns via a split-belt treadmill, where subjects learn to walk with one foot moving faster than the other. Subjects learn to adapt their walking pattern by changing the location (spatial) and time (temporal) of foot placement. Here we asked whether we can induce adaptation of a specific walking pattern when one limb does not “walk” but instead marches in place (i.e., marching-walking hybrid). The marching leg's movement is limited during the stance phase, and thus certain sensory signals important for walking may be reduced. We hypothesized that this would produce a spatial-temporal strategy different from that of normal split-belt adaptation. Healthy subjects performed two experiments to determine whether they could adapt their spatial-temporal pattern of step lengths during the marching-walking hybrid and whether the learning transfers to over ground walking. Results showed that the hybrid group did adapt their step lengths, but the time course of adaptation and deadaption was slower than that for the split-belt group. We also observed that the hybrid group utilized a mostly spatial strategy whereas the split-belt group utilized both spatial and temporal strategies. Surprisingly, we found no significant difference between the hybrid and split-belt groups in over ground transfer. Moreover, the hybrid group retained more of the learned pattern when they returned to the treadmill. These findings suggest that physical rehabilitation with this marching-walking paradigm on conventional treadmills may produce changes in symmetry comparable to what is observed during split-belt training. Show less

Background. Step length asymmetry (SLA) is a common hallmark of gait poststroke. Though conventionally viewed as a spatial deficit, SLA can result from differences in where the feet are placed relative to the body (step position strategy), the timing between foot strikes (step time strategy), or the velocity of the body relative to the feet (step velocity strategy). Objective. The goal of this study was to characterize the relative contributions of each of these strategies to SLA. Methods. We… Show more

Background. Step length asymmetry (SLA) is a common hallmark of gait poststroke. Though conventionally viewed as a spatial deficit, SLA can result from differences in where the feet are placed relative to the body (step position strategy), the timing between foot strikes (step time strategy), or the velocity of the body relative to the feet (step velocity strategy). Objective. The goal of this study was to characterize the relative contributions of each of these strategies to SLA. Methods. We developed an analytical model that parses SLA into independent step position, step time, and step velocity contributions. This model was validated by reproducing SLA values for 25 healthy participants when their natural symmetric gait was perturbed on a split-belt treadmill moving at either a 2:1 or 3:1 belt-speed ratio. We then applied the validated model to quantify step position, step time, and step velocity contributions to SLA in 15 stroke survivors while walking at their self-selected speed. Results. SLA was predicted precisely by summing the derived contributions, regardless of the belt-speed ratio. Although the contributions to SLA varied considerably across our sample of stroke survivors, the step position contribution tended to oppose the other 2—possibly as an attempt to minimize overall SLA. Conclusions. Our results suggest that changes in where the feet are placed or changes in interlimb timing could be used as compensatory strategies to reduce overall SLA in stroke survivors. These results may allow clinicians and researchers to identify patient-specific gait abnormalities and personalize their therapeutic approaches accordingly. Show less

Many biological phenomena such as locomotion, circadian cycles and breathing are rhythmic in nature and can be modelled as rhythmic dynamical systems. Dynamical systems modelling often involves neglecting certain characteristics of a physical system as a modelling convenience. For example, human locomotion is frequently treated as symmetric about the sagittal plane. In this work, we test this assumption by examining human walking dynamics around the steady state (limit-cycle). Here, we adapt… Show more

Many biological phenomena such as locomotion, circadian cycles and breathing are rhythmic in nature and can be modelled as rhythmic dynamical systems. Dynamical systems modelling often involves neglecting certain characteristics of a physical system as a modelling convenience. For example, human locomotion is frequently treated as symmetric about the sagittal plane. In this work, we test this assumption by examining human walking dynamics around the steady state (limit-cycle). Here, we adapt statistical cross-validation in order to examine whether there are statistically significant asymmetries and, even if so, test the consequences of assuming bilateral symmetry anyway. Indeed, we identify significant asymmetries in the dynamics of human walking, but nevertheless show that ignoring these asymmetries results in a more consistent and predictive model. In general, neglecting evident characteristics of a system can be more than a modelling convenience—it can produce a better model. Show less

The ParkourBot climbs in a planar reduced-gravity vertical chute by leaping back and forth between the chute's two parallel walls. The ParkourBot is comprised of a body with two springy legs and its controls consist of leg angles at touchdown and the energy stored in them. During flight, the robot stores elastic potential energy in its springy legs and then converts this potential energy in to kinetic energy at touchdown, when it “kicks off” a wall. This paper describes the ParkourBot's… Show more

The ParkourBot climbs in a planar reduced-gravity vertical chute by leaping back and forth between the chute's two parallel walls. The ParkourBot is comprised of a body with two springy legs and its controls consist of leg angles at touchdown and the energy stored in them. During flight, the robot stores elastic potential energy in its springy legs and then converts this potential energy in to kinetic energy at touchdown, when it “kicks off” a wall. This paper describes the ParkourBot's mechanical design, modeling, and open-loop climbing experiments. The mechanical design makes use of the BowLeg, previously used for hopping on a flat ground. We introduce two models of the BowLeg ParkourBot: one is based on a nonzero stance duration using the spring-loaded inverted pendulum model, and the other is a simplified model (the simplest parkour model, or SPM) obtained as the leg stiffness approaches infinity and the stance time approaches zero. The SPM approximation provides the advantage of closed-form calculations. Finally, predictions of the models are validated by experiments in open-loop climbing in a reduced-gravity planar environment provided by an air table. Show less

The ParkourBot is a planar dynamic climbing biped robot and is capable of dynamically locomoting between two \vertical" walls in a reduced-gravity planar environment of an inclined air table. During
ight, springy legs are compressed to store elastic energy. At impact, this potential energy is quickly converted to kinetic energy. In this paper, we demonstrate that simple feedback control strategies with a high-speed vision system can be used to control the ParkourBot to bounce in place… Show more

The ParkourBot is a planar dynamic climbing biped robot and is capable of dynamically locomoting between two \vertical" walls in a reduced-gravity planar environment of an inclined air table. During
ight, springy legs are compressed to store elastic energy. At impact, this potential energy is quickly converted to kinetic energy. In this paper, we demonstrate that simple feedback control strategies with a high-speed vision system can be used to control the ParkourBot to bounce in place, climb up and climb down. Show less

Distributions in position and orientation are central to many problems in robot localization. To increase efficiency, a majority of algorithms for planar mobile robots use Gaussians defined on positional Cartesian coordinates and heading. However, the distribution of poses for a noisy two-wheeled robot
moving in the plane has been observed by many to be a “banana-shaped” distribution, which is clearly not Gaussian/normal in these coordinates. As uncertainty increases, many localization… Show more

Distributions in position and orientation are central to many problems in robot localization. To increase efficiency, a majority of algorithms for planar mobile robots use Gaussians defined on positional Cartesian coordinates and heading. However, the distribution of poses for a noisy two-wheeled robot
moving in the plane has been observed by many to be a “banana-shaped” distribution, which is clearly not Gaussian/normal in these coordinates. As uncertainty increases, many localization algorithms therefore become “inconsistent” due to the normality assumption breaking down. We observe that this is because the combination of Cartesian coordinates and heading is not the most appropriate set of coordinates to use, and that the banana distribution can be described in closed form as a Gaussian in an
alternative set of coordinates via the so-called exponential map.
With this formulation, we can derive closed-form expressions for propagating the mean and covariance of the Gaussian in these exponential coordinates for a differential-drive car moving along a trajectory constructed from sections of straight segments and arcs of constant curvature. In addition, we detail how to fuse two or more Gaussians in exponential coordinates together with given relative pose measurements between robots moving in formation. These propagation and fusion formulas utilized here reduce uncertainty in localization better than when using traditional methods. We demonstrate with numerical examples dramatic improvements in the estimated pose of three robots moving in formation when compared to classical Cartesian coordinate- based Gaussian fusion methods. Show less

Many robotics applications involve motion planning with uncertainty. In this paper, we focus on path planning for planar systems by optimizing the probability of successfully arriving at a goal. We approach this problem with a modified version of the Path-of-Probability (POP) algorithm.We extend the POP algorithm to allow for a moving target and to optimize the number of steps to reach the goal. One
tool that we develop in this paper to increase efficiency of the POP algorithm is a second… Show more

Many robotics applications involve motion planning with uncertainty. In this paper, we focus on path planning for planar systems by optimizing the probability of successfully arriving at a goal. We approach this problem with a modified version of the Path-of-Probability (POP) algorithm.We extend the POP algorithm to allow for a moving target and to optimize the number of steps to reach the goal. One
tool that we develop in this paper to increase efficiency of the POP algorithm is a second order closed-form uncertainty propagation formula. This formula is utilized to quickly propagate the mean and covariance of nonparametrized distributions for planar systems. The modified POP algorithm is demonstrated on a simple rolling disc example with a moving goal. Show less

Hybrid dynamical systems with impacts typically have controls that can influence the time of the impact as well as the result of the impact. The leg angle of a hopping robot is an example of an impact control because it can influence when the impact occurs and the direction of the impulse. This
paper provides a method for computing an explicit expression for the first derivative of a cost function encoding a desired trajectory. The first derivative can be used with standard optimization… Show more

Hybrid dynamical systems with impacts typically have controls that can influence the time of the impact as well as the result of the impact. The leg angle of a hopping robot is an example of an impact control because it can influence when the impact occurs and the direction of the impulse. This
paper provides a method for computing an explicit expression for the first derivative of a cost function encoding a desired trajectory. The first derivative can be used with standard optimization algorithms to find the optimal impact controls for motion planning of hybrid dynamical systems with impacts.
The resulting derivation is implemented for a simplified model of a dynamic climbing robot. Show less

Several different mechanical models of double-helical nucleic-acid structures that have been presented in the literature are reviewed here together with a new analysis method that provides a reconciliation between these disparate models. In all cases, terminology and basic results from the theory of Lie groups are used to describe rigid-body motions in a coordinate-free way, and when necessary, coordinates are introduced in a way in which simple equations result. We consider double-helical DNAs… Show more

Several different mechanical models of double-helical nucleic-acid structures that have been presented in the literature are reviewed here together with a new analysis method that provides a reconciliation between these disparate models. In all cases, terminology and basic results from the theory of Lie groups are used to describe rigid-body motions in a coordinate-free way, and when necessary, coordinates are introduced in a way in which simple equations result. We consider double-helical DNAs and RNAs which, in their unstressed referential state, have backbones that are either straight, slightly precurved, or bent by the action of a protein or other bound molecule. At the coarsest level, we consider worm-like chains with anisotropic bending stiffness. Then, we show how bi-rod models converge to this for sufficiently long filament lengths. At a finer level, we examine elastic networks of rigid bases and show how these relate to the coarser models. Finally, we show how results from molecular dynamics simulation at full atomic resolution (which is the finest scale considered here) and AFM experimental measurements (which is at the coarsest scale) relate to these models. Show less

We describe and experimentally validate the Simplest Parkour Model (SPM) for the ParkourBot, a planar dynamic climbing robot equipped with two springy BowLegs. By controlling the leg angles and injected energy at impact, the ParkourBot is capable of climbing up and down in a rigid chute on an inclined air table. The SPM consists of a point mass and two massless legs. The legs are assumed to be infinitely stiff, resulting in an instantaneous stance phase and a closed-form solution of the hybrid… Show more

We describe and experimentally validate the Simplest Parkour Model (SPM) for the ParkourBot, a planar dynamic climbing robot equipped with two springy BowLegs. By controlling the leg angles and injected energy at impact, the ParkourBot is capable of climbing up and down in a rigid chute on an inclined air table. The SPM consists of a point mass and two massless legs. The legs are assumed to be infinitely stiff, resulting in an instantaneous stance phase and a closed-form solution of the hybrid dynamics. In this paper, we show that the SPM is a good predictor of the actual experimental behavior. Using the SPM we compute the fixed points, stability and basins of attraction of period-1 limit cycles. Show less