Task-agnostic controllers for partial-assist lower-limb exoskeletons aim to reliably mimic biological torque while seamlessly adapting to changing movement patterns. However, current approaches relying on hidden state estimators or neural networks lack explainability and safety guarantees, while force amplification methods risk instability with an inherent trade-off between sensitivity and robustness to control inputs. Energy shaping control uses a kinematic model-based framework to provide predictable, stable assistance, though its traditional passive form limits biomimetic performance. Previous work relaxed the strict passivity requirements to improve biomimicry but reduced the stability guarantees. This paper presents an optimization-based extension of the energy-shaping control framework that combines the stability benefits of energy shaping with the intuitive biomimicry of force amplification. Our framework enables controlled trade-offs between sensitivity to changing human impedance and high performance through adjustable cost contributions of force amplification and model-based terms. We provide theoretical guarantees of closed-loop stability to an invariant set under human joint impedance control, supported by empirical validation of stability characteristics of an ankle exoskeleton under varying controller passivity constraints. A study of ten able-bodied participants using bilateral ankle exoskeletons demonstrates that the biomimetic controller reduced biological ankle torque by 19.1% across various activities of daily life.

For powered prosthetic legs to be viable in everyday situations, they require an activity classification system that is not only accurate but also straightforward to understand and use. However, incorporating the numerous activity modes in real-world ambulation often requires high-dimensional feature spaces and restrictions on the leg leading each transition. This article addresses these challenges by delegating sit/stand transitions and variable-incline walking to the mid-level controller, effectively reducing the classification space to four states with easily distinguishable features. We implement simple heuristic rules for both prosthetic-led and intact-led (i.e., ambilateral) transitions, using lower-limb kinematic features, ground contact and inclination, and environmental distance from an ultrasonic sensor. Two transfemoral amputee subjects using a powered knee-ankle prosthesis demonstrated an ambilateral transition accuracy of 99.2% under both self-paced and rapid-paced/fatiguing conditions, with a 100% recovery rate due to backup logic or user-cued resets. The incline estimator enabled the prosthesis to continuously adapt between level and inclined surfaces without explicit classification. These results and an outdoor multi-terrain demonstration indicate that simple and straightforward transition logic can enable powered prosthetic legs to be used reliably across a broad array of daily activities.

Research in powered prosthesis control has explored the use of impedance-based control algorithms due to their biomimetic capabilities and intuitive structure. Modern impedance controllers feature parameters that smoothly vary over gait phase and task according to a data-driven model. However, these recent efforts only use continuous impedance control during stance and instead utilize discrete transition logic to switch to kinematic control during swing, necessitating two separate models for the different parts of the stride. In contrast, this paper presents a controller that uses smooth impedance parameter trajectories throughout the gait, unifying the stance and swing periods under a single, continuous model. Furthermore, this paper proposes a basis model to represent intertask relationships in the impedance parameters-a strategy that has previously been shown to improve model accuracy over classic linear interpolation methods. In the proposed controller, a weighted sum of Fourier series is used to model the impedance parameters of each joint as continuous functions of gait cycle progression and task. Fourier series coefficients are determined via convex optimization such that the controller best reproduces the joint torques and kinematics in a reference able-bodied dataset. Experiments with a powered knee-ankle prosthesis show that this simpler, unified model produces competitive results when compared to a more complex hybrid impedance-kinematic model over varying walking speeds and inclines.

Many powered prosthetic devices use load cells to detect ground interaction forces and gait events. These sensors introduce additional weight and cost in the device. Recent proprioceptive actuators enable an algebraic relationship between actuator torques and ground contact forces. This paper presents a proprioceptive force sensing paradigm which estimates ground reaction forces as a solution to detect gait events without a load cell. A floating body dynamic model is obtained with constraints at the center of pressure representing foot-ground interaction. Constraint forces are derived to estimate ground reaction forces and subsequently timing of gait events. A treadmill experiment is conducted with a powered knee-ankle prosthesis used by an able-bodied subject walking at various speeds and slopes. Results show accurate gait event timing, with pooled data showing heel strike detection lagging by only 6.7 ± 7.2 ms and toe off detection leading by 30.4 ± 11.0 ms compared to values obtained from the load cell. These results establish proof of concept for predicting gait events without a load cell in powered prostheses with proprioceptive actuators.

Robotic knee-ankle prostheses have often fallen short relative to passive microprocessor prostheses in time-based clinical outcome tests. User ambulation endurance is an alternative clinical outcome metric that may better highlight the benefits of robotic prostheses. However, previous studies were unable to show endurance benefits due to inaccurate high-level classification, discretized mid-level control, and insufficiently difficult ambulation tasks. In this case study, we present a phase-based mid-level prosthesis controller which yields biomimetic joint kinematics and kinetics that adjust to suit a continuum of tasks. We enrolled an individual with an above-knee amputation and challenged him to perform repeated, rapid laps of a circuit comprising activities of daily living with both his passive prosthesis and a robotic prosthesis. The participant demonstrated improved endurance with the robotic prosthesis and our mid-level controller compared to his passive prosthesis, completing over twice as many total laps before fatigue and muscle discomfort required him to stop. We also show that time-based outcome metrics fail to capture this endurance improvement, suggesting that alternative metrics related to endurance and fatigue may better highlight the clinical benefits of robotic prostheses.

One of the primary benefits of emerging powered prosthetic legs is their ability to facilitate step-over-step stair ascent by providing positive mechanical work. Existing control methods typically have distinct steady-state activity modes for walking and stair ascent, where activity transitions involve discretely switching between controllers and often must be initiated with a particular leg. However, these discrete transitions do not necessarily replicate able-bodied joint biomechanics, which have been shown to continuously adjust over a transition stride. This paper presents a phase-based kinematic controller for a powered knee-ankle prosthesis that enables continuous, biomimetic transitions between walking and stair ascent. The controller tracks joint angles from a data-driven kinematic model that continuously interpolates between the steady-state kinematic models, and it allows both the prosthetic and intact leg to lead the transitions. Results from experiments with two transfemoral amputee participants indicate that knee and ankle kinematics smoothly transition between walking and stair ascent, with comparable or lower root mean square errors compared to variations from able-bodied data.

Emerging partial-assistance exoskeletons can enhance able-bodied performance and aid people with patho-logical gait or age-related immobility. However, every person walks differently, which makes it difficult to directly compute assistance torques from joint kinematics. Gait-state estimation-based controllers use phase (normalized stride time) and task variables (e.g., stride length and ground inclination) to parameterize the joint torques. Using kinematic models that depend on the gait-state, prior work has used an Extended Kalman filter (EKF) to estimate the gait-state online. However, this EKF suffered from kinematic errors since it used a subject-independent measurement model, and it is still unknown how personalization of this measurement model would reduce gait-state tracking error. This paper quantifies how much gait-state tracking improvement a personalized measurement model can have over a subject-independent measurement model when using an EKF-based gait-state estimator. Since the EKF performance depends on the measurement model covariance matrix, we tested on multiple different tuning parameters. Across reasonable values of tuning parameters that resulted in good performance, personalization improved estimation error on average by 8.5 ± 13.8% for phase (mean ± standard deviation), 27.2$\pm \ 8.1{\%}$for stride length, and 10.5$\pm \ 13.5{\%}$for ground inclination. These findings support the hypothesis that personalization of the measurement model significantly improves gait-state estimation performance in EKF based gait-state tracking$(P \ll 0.05)$, which could ultimately enable reliable responses to faster human gait changes.

Robotic ankle exoskeletons have been shown to reduce human effort during walking. However, existing ankle exoskeleton control approaches are limited in their ability to apply biomimetic torque across diverse tasks outside of the controlled lab environment. Energy shaping control can provide task-invariant assistance without estimating the user's state, classifying task, or reproducing pre-defined torque trajectories. In previous work, we showed that an optimally task-invariant energy shaping controller implemented on a knee-ankle exoskeleton reduced the effort of certain muscles for a range of tasks. In this paper, we extend this approach to the sensor suite available at the ankle and present its implementation on a commercially-available, bilateral ankle exoskeleton. An experiment with three healthy subjects walking on a circuit and on a treadmill showed that the controller can approximate biomimetic profiles for varying terrains and task transitions without classifying tasks or switching control modes.

Most impedance-based walking controllers for powered knee–ankle prostheses use a finite state machine with dozens of user-specific parameters that require manual tuning by technical experts. These parameters are only appropriate near the task (e.g., walking speed and incline) at which they were tuned, necessitating many different parameter sets for variable-task walking. In contrast, this article presents a data-driven, phase-based controller for variable-task walking that uses continuously variable impedance control during stance and kinematic control during swing to enable biomimetic locomotion. After generating a data-driven model of variable joint impedance with convex optimization, we implement a novel task-invariant phase variable and real-time estimates of speed and incline to enable autonomous task adaptation. Experiments with above-knee amputee participants ($N=2$) show that our data-driven controller 1) features highly linear phase estimates and accurate task estimates, 2) produces biomimetic kinematic and kinetic trends as task varies, leading to low errors relative to able-bodied references, and 3) produces biomimetic joint work and cadence trends as task varies. We show that the presented controller meets and often exceeds the performance of a benchmark finite state machine controller for our two participants, without requiring manual impedance tuning.

Positive biomechanical outcomes have been reported with lower-limb exoskeletons in laboratory settings, but these devices have difficulty delivering appropriate assistance in synchrony with human gait as the task or rate of phase progression change in real-world environments. This paper presents a controller for an ankle exoskeleton that uses a data-driven kinematic model to continuously estimate the phase, phase rate, stride length, and ground incline states during locomotion, which enables the real-time adaptation of torque assistance to match human torques observed in a multi-activity database of 10 able-bodied subjects. We demonstrate in live experiments with a new cohort of 10 able-bodied participants that the controller yields phase estimates comparable to the state of the art, while also estimating task variables with similar accuracy to recent machine learning approaches. The implemented controller successfully adapts its assistance in response to changing phase and task variables, both during controlled treadmill trials (N=10, phase RMSE: 4.8 ± 2.4%) and a real-world stress test with extremely uneven terrain (N=1, phase RMSE: 4.8 ± 2.7%).

Passive prostheses cannot provide the net positive work required at the knee and ankle for step-over stair ascent. Powered prostheses can provide this net positive work, but user synchronization of joint motion and power input are critical to enabling natural stair ascent gaits. In this work, we build on previous phase variable-based control methods for walking and propose a stair ascent controller driven by the motion of the user's residual thigh. We use reference kinematics from an able-bodied dataset to produce knee and ankle joint trajectories parameterized by gait phase. We redefine the gait cycle to begin at the point of maximum hip flexion instead of heel strike to improve the phase estimate. Able-bodied bypass adapter experiments demonstrate that the phase variable controller replicates normative able-bodied kinematic trajectories with a root mean squared error of 12.66° and 2.64° for the knee and ankle, respectively. The knee and ankle joints provided on average 0.39 J/kg and 0.21 J/kg per stride, compared to the normative averages of 0.34 J/kg and 0.21 J/kg, respectively. Thus, this controller allows powered knee-ankle prostheses to perform net positive mechanical work to assist stair ascent.

Although the average healthy adult transitions from sit to stand over 60 times per day, most research on powered prosthesis control has only focused on walking. In this paper, we present a data-driven controller that enables sitting, standing, and walking with minimal tuning. Our controller comprises two high level modes of sit/stand and walking, and we develop heuristic biomechanical rules to control transitions. We use a phase variable based on the user's thigh angle to parameterize both walking and sit/stand motions, and use variable impedance control during ground contact and position control during swing. We extend previous work on data-driven optimization of continuous impedance parameter functions to design the sit/stand control mode using able-bodied data. Experiments with a powered knee-ankle prosthesis used by a participant with above-knee amputation demonstrate promise in clinical outcomes, as well as trade-offs between our minimal-tuning approach and accommodation of user preferences. Specifically, our controller enabled the participant to complete the sit/stand task 20% faster and reduced average asymmetry by half compared to his everyday passive prosthesis. The controller also facilitated a timed up and go test involving sitting, standing, walking, and turning, with only a mild (10%) decrease in speed compared to the everyday prosthesis. Our sit/stand/walk controller enables multiple activities of daily life with minimal tuning and mode switching.

Most controllers for lower-limb robotic prostheses require individually tuned parameter sets for every combination of speed and incline that the device is designed for. Because ambulation occurs over a continuum of speeds and inclines, this design paradigm requires tuning of a potentially prohibitively large number of parameters. This limitation motivates an alternative control framework that enables walking over a range of speeds and inclines while requiring only a limited number of tunable parameters. In this work, we present the implementation of a continuously varying kinematic controller on a custom powered knee-ankle prosthesis. The controller uses a phase variable derived from the residual thigh angle, along with real-time estimates of ground inclination and walking speed, to compute the appropriate knee and ankle joint angles from a continuous model of able-bodied kinematic data. We modify an existing phase variable architecture to allow for changes in speeds and inclines, quantify the closed-loop accuracy of the speed and incline estimation algorithms for various references, and experimentally validate the controller by observing that it replicates kinematic trends seen in able-bodied gait as speed and incline vary.

Natural dynamics, nonlinear optimization, and, more recently, convex optimization are available methods for stiffness design of energy-efficient series elastic actuators. Natural dynamics and general nonlinear optimization only work for a limited set of load kinetics and kinematics, cannot guarantee convergence to a global optimum, or depend on initial conditions to the numerical solver. Convex programs alleviate these limitations and allow a global solution in polynomial time, which is useful when the space of optimization variables grows (e.g., when designing optimal nonlinear springs or co-designing spring, controller, and reference trajectories). Our previous work introduced the stiffness design of series elastic actuators via convex optimization when the transmission dynamics are negligible, which is an assumption that applies mostly in theory or when the actuator uses a direct or quasi-direct drive. In this work, we extend our analysis to include friction at the transmission. Coulomb friction at the transmission results in a non-convex expression for the energy dissipated as heat, but we illustrate a convex approximation for stiffness design. We experimentally validated our framework using a series elastic actuator with specifications similar to the knee joint of the Open Source Leg, an open-source robotic knee-ankle prosthesis.

We present the design of a powered knee-ankle prosthetic leg, which implements high-torque actuators with low-reduction transmissions. The transmission coupled with a high-torque and low-speed motor creates an actuator with low mechanical impedance and high backdrivability. This style of actuation presents several possible benefits over modern actuation styles in emerging robotic prosthetic legs, which include free-swinging knee motion, compliance with the ground, negligible unmodeled actuator dynamics, less acoustic noise, and power regeneration. Benchtop tests establish that both joints can be backdriven by small torques (~1-3 Nm) and confirm the small reflected inertia. Impedance control tests prove that the intrinsic impedance and unmodeled dynamics of the actuator are sufficiently small to control joint impedance without torque feedback or lengthy tuning trials. Walking experiments validate performance under the designed loading conditions with minimal tuning. Lastly, the regenerative abilities, low friction, and small reflected inertia of the presented actuators reduced power consumption and acoustic noise compared to state-of-art powered legs.

This paper describes the design of a powered knee- and-ankle transfemoral prosthetic leg, which implements high torque density actuators with low-reduction transmissions. The low reduction of the transmission coupled with a high-torque and low-speed motor creates an actuator with low mechanical impedance and high backdrivability. This style of actuation presents several possible benefits over modern actuation styles implemented in emerging robotic prosthetic legs. Such benefits include free-swinging knee motion, compliance with the ground, negligible unmodeled actuator dynamics, and greater potential for power regeneration. Benchtop validation experiments were conducted to verify some of these benefits. Backdrive and free-swinging knee tests confirm that both joints can be backdriven by small torques (~3 Nm). Bandwidth tests reveal that the actuator is capable of achieving frequencies required for walking and running. Lastly, open-loop impedance control tests prove that the intrinsic impedance and unmodeled dynamics of the actuator are sufficiently small to control joint impedance without torque feedback.

Although there has been recent progress in control of multi-joint prosthetic legs for periodic tasks such as walking, volitional control of these systems for non-periodic maneuvers is still an open problem. In this paper, we develop a new controller that is capable of both periodic walking and common volitional leg motions based on a piecewise holonomic phase variable through a finite state machine. The phase variable is constructed by measuring the thigh angle, and the transitions in the finite state machine are formulated through sensing foot contact together with attributes of a nominal reference gait trajectory. The controller was implemented on a powered knee-ankle prosthesis and tested with a transfemoral amputee subject, who successfully performed a wide range of periodic and non-periodic tasks, including low- and high-speed walking, quick start and stop, backward walking, walking over obstacles, and kicking a soccer ball. The proposed approach is expected to provide better understanding of volitional motions and lead to more reliable control of multi-joint prostheses for a wider range of tasks.

Control systems for powered prosthetic legs typically divide the gait cycle into several periods with distinct controllers, resulting in dozens of control parameters that must be tuned across users and activities. To address this challenge, this paper presents a control approach that unifies the gait cycle of a powered knee-ankle prosthesis using a continuous, user-synchronized sense of phase. Virtual constraints characterize the desired periodic joint trajectories as functions of a phase variable across the entire stride. The phase variable is computed from residual thigh motion, giving the amputee control over the timing of the prosthetic joint patterns. This continuous sense of phase enabled three transfemoral amputee subjects to walk at speeds from 0.67 to 1.21 m/s and slopes from -2.5 to +9.0 deg. Virtual constraints based on task-specific kinematics facilitated normative adjustments in joint work across walking speeds. A fixed set of control gains generalized across these activities and users, which minimized the configuration time of the prosthesis.

This paper presents the mechatronic design and experimental validation of a novel powered knee-ankle orthosis for testing torque-driven rehabilitation control strategies. The modular actuator of the orthosis is designed with a torque dense motor and a custom low-ratio transmission (24:1) to provide mechanical transparency to the user, allowing them to actively contribute to their joint kinematics during gait training. The 4.88 kg orthosis utilizes frameless components and light materials, such as aluminum alloy and carbon fiber, to reduce its mass. A human subject experiment demonstrates accurate torque control with high output torque during stance and low backdrive torque during swing at fast walking speeds. This work shows that backdrivability, precise torque control, high torque output, and light weight can be achieved in a powered orthosis without the high cost and complexity of variable transmissions, clutches, and/or series elastic components.

Traditional control methodologies of rehabilitation orthoses/exoskeletons aim at replicating normal kinematics and thus fall into the category of kinematic control. This control paradigm depends on pre-defined reference trajectories, which can be difficult to adjust between different locomotor tasks and human subjects. An alternative control category, kinetic control, enforces kinetic goals (e.g., torques or energy) instead of kinematic trajectories, which could provide a flexible learning environment for the user while freeing up therapists to make corrections. We propose that the theory of underactuated potential energy shaping, which falls into the category of kinetic control, could be used to generate virtual body-weight support for stroke gait rehabilitation. After deriving the nonlinear control law and simulating it on a human-like biped model, we implemented this controller on a powered ankle-foot orthosis that was designed specifically for testing torque control strategies. Experimental results with an able-bodied human subject demonstrate the feasibility of the control approach for both positive and negative virtual body-weight augmentation.

This paper presents the experimental validation of a novel control strategy that unifies the entire gait cycle of a powered knee-ankle prosthetic leg without the need to switch between controllers for different periods of gait. Current control methods divide the gait cycle into several sequential periods each with independent controllers, resulting in many patient-specific control parameters and switching rules that must be tuned for a specific walking speed. The single controller presented is speed-invariant with a minimal number of control parameters to be tuned. A single, periodic virtual constraint is derived that exactly characterizes the desired actuated joint motion as a function of a mechanical phase variable across walking cycles. A single sensor was used to compute a phase variable related to the residual thigh angle's phase plane, which was recently shown to robustly represent the phase of non-steady human gait. This phase variable allows the prosthesis to synchronize naturally with the human user for intuitive, biomimetic behavior. A custom powered knee-ankle prosthesis was designed and built to implement the control strategy and validate its performance. A human subject experiment was conducted across multiple walking speeds (1 to 3 miles/hour) in a continuous sequence with the single phase-based controller, demonstrating its adaptability to the user's intended speed.

Predictive simulations of human walking could be used to investigate a wide range of questions. Promising moderately complex models have been developed using the robotics control technique hybrid zero dynamics (HZD). Existing simulations of human walking only consider the mean motion, so they cannot be used to investigate fall risk, which is correlated with variability. This work determines how to incorporate human-like variability into an HZD-based healthy human model to generate a more realistic gait. The key challenge is determining how to combine the existing mathematical description of variability with the dynamic model so that the biped is still able to walk without falling. To do so, the commanded motion is augmented with a sinusoidal variability function and a polynomial correction function. The variability function captures the variation in joint angles while the correction function prevents the variability function from growing uncontrollably. The necessity of the correction function and the improvements with a reduction of stance ankle variability are demonstrated via simulations. The variability in temporal measures is shown to be similar to experimental values.

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.

Recent powered (or robotic) prosthetic legs independently control different joints and time periods of the gait cycle, resulting in control parameters and switching rules that can be difficult to tune by clinicians. This challenge might be addressed by a unifying control model used by recent bipedal robots, in which virtual constraints define joint patterns as functions of a monotonic variable that continuously represents the gait cycle phase. In the first application of virtual constraints to amputee locomotion, this paper derives exact and approximate control laws for a partial feedback linearization to enforce virtual constraints on a prosthetic leg. We then encode a human-inspired invariance property called effective shape into virtual constraints for the stance period. After simulating the robustness of the partial feedback linearization to clinically meaningful conditions, we experimentally implement this control strategy on a powered transfemoral leg. We report the results of three amputee subjects walking overground and at variable cadences on a treadmill, demonstrating the clinical viability of this novel control approach.

This paper presents a hybrid mechanical model for the Gibbot, a robot that dynamically locomotes along a vertical wall in a manner analogous to gibbons swinging between branches in the forest canopy. We focus on one particular gait, continuous-contact brachiation, which always has one handhold in contact with the wall. We use zero-cost, unstable solutions corresponding to horizontal brachiation, originally found by Gomes and Ruina, as templates to generate open-loop stable gaits in arbitrary directions. The first case considered is passive brachiation down a shallow slope, roughly corresponding to upside-down locomotion of the well-studied compass-gait biped. We then consider underactuated brachiation with a constant forcing term at the elbow to produce open-loop stable descending and ascending gaits.

In this paper, we present a hierarchical framework that enables motion planning for asymptotically stable 3-D bipedal walking in the same way that planning is already possible for zero moment point walking. This framework is based on the construction of asymptotically stable gait primitives for a class of hybrid dynamical systems with impacts. Each primitive corresponds to an asymptotically stable hybrid limit cycle that admits rules a priori for sequential composition with other primitives, reducing a high-dimensional feedback motion planning problem into a low-dimensional discrete tree search. As a constructive example, we develop this planning framework for the 3-D compass-gait biped, where each primitive corresponds to walking along an arc of constant curvature for a fixed number of steps. We apply a discrete search algorithm to plan a sequence of these primitives, taking the 3-D biped stably from start to goal in workspaces with obstacles. We finally show how this framework generalizes to more complex models by planning walking paths for an underactuated five-link biped.

This paper applies geometric reduction-based control to derive a set of asymptotically stable dynamic walking gaits for a 3-D bipedal robot, each corresponding to walking along a nominal arc of constant curvature for a fixed number of steps. We show that any such set of asymptotically stable gait primitives may be composed in arbitrary order without causing the robot to fall, so any walking path that is a sequence of these gaits may be followed by the robot. This result enables motion planning for bipedal dynamic walkers, which are fast and energetically efficient, in a similar manner to what is already possible for biped locomotion based on Zero Moment Point (ZMP) equilibrium constraints.

The planar compass-gait biped has been extensively studied in the dynamic walking community, motivated by the gravity-based pendular efficiencies of human walking. These results can be extended to three dimensions using controlled geometric reduction for open-chain robots, by which stable 3-D walking gaits are built from known sagittal-plane limit cycles. We apply this method to the standard and with-torso compass-gait (hipless) bipeds, showing straight-ahead walking gaits (i.e., stable 1-step periodic limit cycles) as well as h-step turning in full circles (i.e., stable h-periodic limit cycles). These constant-curvature maneuvers are composed of stable 1-periodic turning gaits modulo heading change, demonstrating two types of gaits for directional dynamic walking in three dimensions.