This paper presents a novel approach to fall prediction for bipedal robots, specifically targeting the detection of potential falls while standing caused by abrupt, incipient, and intermittent faults. Leveraging a 1D convolutional neural network (CNN), our method aims to maximize lead time for fall prediction while minimizing false positive rates. The proposed algorithm uniquely integrates the detection of various fault types and estimates the lead time for potential falls. Our contributions include the development of an algorithm capable of detecting abrupt, incipient, and intermittent faults in full-sized robots, its implementation using both simulation and hardware data for a humanoid robot, and a method for estimating lead time. Evaluation metrics, including false positive rate, lead time, and response time, demonstrate the efficacy of our approach. Particularly, our model achieves impressive lead times and response times across different fault scenarios with a false positive rate of 0. The findings of this study hold significant implications for enhancing the safety and reliability of bipedal robotic systems.

This work explores an innovative algorithm designed to enhance the mobility of underactuated bipedal robots across challenging terrains, especially when navigating through spaces with constrained opportunities for foot support, like steps or stairs. By combining ankle torque with a refined angular momentum-based linear inverted pendulum model (ALIP), our method allows variability in the robot’s center of mass height. We employ a dual-strategy controller that merges virtual constraints for precise motion regulation across essential degrees of freedom with an ALIP-centric model predictive control (MPC) framework, aimed at enforcing gait stability. The effectiveness of our feedback design is demonstrated through its application on the Cassie bipedal robot, which features 20 degrees of freedom. Key to our implementation is the development of tailored nominal trajectories and an optimized MPC that reduces the execution time to under 500 microseconds—and, hence, is compatible with Cassie’s controller update frequency. This paper not only showcases the successful hardware deployment but also demonstrates a new capability, a bipedal robot using a moving walkway.

Multi-objective or multi-destination path planning is crucial for mobile robotics applications such as mobility as a service, robotics inspection, and electric vehicle charging for long trips. This work proposes an anytime iterative system to concurrently solve the multi-objective path planning problem and determine the visiting order of destinations. The system is comprised of an anytime informable multi-objective and multi-directional RRT*algorithm to form a simple connected graph, and a solver that consists of an enhanced cheapest insertion algorithm and a genetic algorithm to solve approximately the relaxed traveling salesman problem in polynomial time. Moreover, a list of waypoints is often provided for robotics inspection and vehicle routing so that the robot can preferentially visit certain equipment or areas of interest. We show that the proposed system can inherently incorporate such knowledge to navigate challenging topology. The proposed anytime system is evaluated on large and complex graphs built for real-world driving applications. C++ implementations are available at: https://github.com/UMich-BipedLab/IMOMD-RRTStar.

For bipedal humanoid robots to successfully operate in the real world, they must be competent at simultaneously executing multiple motion tasks while reacting to unforeseen external disturbances in real-time. We propose Kinodynamic Fabrics as an approach for the specification, solution and simultaneous execution of multiple motion tasks in real-time while being reactive to dynamism in the environment. Kinodynamic Fabrics allows for the specification of prioritized motion tasks as forced spectral semi-sprays and solves for desired robot joint accelerations at real-time frequencies. We evaluate the capabilities of Kinodynamic fabrics on diverse physically-challenging whole-body control tasks with a bipedal humanoid robot both in simulation and in the real-world. Kinodynamic Fabrics outperforms the state-of-the-art Quadratic Program based whole-body controller on a variety of whole-body control tasks on run-time and reactivity metrics in our experiments. Our open-source implementation of Kinodynamic Fabrics as well as robot demonstration videos can be found at this url: https://adubredu.github.io/kinofabs

A new control paradigm using angular momentum and foot placement as state variables in the linear inverted pendulum model has expanded the realm of possibilities for the control of bipedal robots. This new paradigm, known as the ALIP model, has shown effectiveness in cases where a robot's center of mass height can be assumed to be constant or near constant as well as in cases where there are no non-kinematic restrictions on foot placement. Walking up and down stairs violates both of these assumptions, where center of mass height varies significantly within a step and the geometry of the stairs restrict the effectiveness of foot placement. In this paper, we explore a variation of the ALIP model that allows the length of the virtual pendulum formed by the robot's stance foot and center of mass to follow smooth trajectories during a step. We couple this model with a control strategy constructed from a novel combination of virtual constraint-based control and a model predictive control algorithm to stabilize a stair climbing gait that does not soley rely on foot placement. Simulations on a 20-degree of freedom model of the Cassie biped in the SimMechanics simulation environment show that the controller is able to achieve periodic gait.

This paper presents a reactive planning system that allows a Cassie-series bipedal robot to avoid multiple non-overlapping obstacles via a single, continuously differentiable control barrier function (CBF). The overall system detects an individual obstacle via a height map derived from a LiDAR point cloud and computes an elliptical outer approximation, which is then turned into a CBF. The QP-CLF-CBF formalism developed by Ames et al. is applied to ensure that safe trajectories are generated. Safe planning in environments with multiple obstacles is demonstrated both in simulation and experimentally on the Cassie biped.

In this article, we present a novel and flexible multitask multilayer Bayesian mapping framework with readily extendable attribute layers. The proposed framework goes beyond modern metric-semantic maps to provide even richer environmental information for robots in a single mapping formalism while exploiting intralayer and interlayer correlations. It removes the need for a robot to access and process information from many separate maps when performing a complex task, advancing the way robots interact with their environments. To this end, we design a multitask deep neural network with attention mechanisms as our front-end to provide heterogeneous observations for multiple map layers simultaneously. Our back-end runs a scalable closed-form Bayesian inference with only logarithmic time complexity. We apply the framework to build a dense robotic map, including metric-semantic occupancy and traversability layers. Traversability ground truth labels are automatically generated from exteroceptive sensory data in a self-supervised manner. We present extensive experimental results on publicly available datasets and data collected by a three-dimensional bipedal robot platform and show reliable mapping performance in different environments. Finally, we also discuss how the current framework can be extended to incorporate more information, such as friction, signal strength, temperature, and physical quantity concentration using Gaussian map layers. The software for reproducing the presented results or running on customized data is made publicly available.

We propose and experimentally demonstrate a reactive planning system for bipedal robots on unexplored, challenging terrain. The system includes: a multilayer local map for assessing traversability; an anytime omnidirectional control Lyapunov function for use with a rapidly exploring random tree star (RRT*) that generates a vector field for specifying motion between nodes; a subgoal finder when the final goal is outside of the current map; and a finite-state machine to handle high-level mission decisions. The system also includes a reactive thread that copes with robot deviations via a vector field, defined by a closed-loop feedback policy. The vector field provides real-time control commands to the robot's gait controller as a function of instantaneous robot pose. The system is evaluated on various challenging outdoor terrains and cluttered indoor scenes in both simulation and experiment on Cassie Blue, a bipedal robot with 20 degrees of freedom. All implementations are coded in C++ with the robot operating system and are available athttps://github.com/UMich-BipedLab/CLF_reactive_planning_system.

This paper presents a gait controller for bipedal robots to achieve highly agile walking over various terrains given local slope and friction cone information. Without these considerations, untimely impacts can cause a robot to trip and inadequate tangential reaction forces at the stance foot can cause slippages. We address these challenges by combining, in a novel manner, a model based on an Angular Momentum Linear Inverted Pendulum (ALIP) and a Model Predictive Control (MPC) foot placement planner that is executed by the method of virtual constraints. The process starts with abstracting from the full dynamics of a Cassie 3D bipedal robot, an exact low-dimensional representation of its center of mass dynamics, parameterized by angular momentum. Under a piecewise planar terrain assumption and the elimination of terms for the angular momentum about the robot's center of mass, the centroidal dynamics about the contact point become linear and have dimension four. Importantly, we include the intra-step dynamics at uniformly-spaced intervals in the MPC formulation so that realistic workspace constraints on the robot's evolution can be imposed from step-to-step. The output of the low-dimensional MPC controller is directly implemented on a high-dimensional Cassie robot through the method of virtual constraints. In experiments, we validate the performance of our control strategy for the robot on a variety of surfaces with varied inclinations and textures.

Ultimately, feedback control is about making adjustments using current state information in order to meet an objective in the future. In the control of bipedal locomotion, linear velocity of the center of mass has been widely accepted as the primary variable around which feedback control objectives are formulated. In this paper, we argue that it is easier to predict the one-step ahead evolution of angular momentum about the contact point than it is to make a similar prediction for linear velocity, and hence it provides a superior quantity for feedback control. So as not to confuse the benefits of predicting angular momentum with any other control design decisions, we reformulate the standard LIP model in terms of angular momentum and show how to regulate swing foot touchdown position at the end of the current step so as to meet an angular momentum objective at the end of the next step. We implement the resulting feedback controller on the 20 degree-of-freedom bipedal robot, Cassie Blue, where each leg accounts for nearly one-third of the robot’s total mass of 32 Kg. Under this controller, the robot achieves fast walking, rapid turning while walking, large disturbance rejection, and locomotion on rough terrain.

This paper reports on a novel nonparametric rigid point cloud registration framework, Semantic Continuous Visual Odometry (CVO), that jointly integrates geometric and semantic measurements such as color or semantic labels into the alignment process and does not require explicit data association. The point clouds are represented as nonparametric functions in a reproducible kernel Hilbert space. The alignment problem is formulated as maximizing the inner product between two functions, essentially a sum of weighted kernels, each of which exploits the local geometric and semantic features. As a result of the continuous models, analytical gradients can be computed, and a local solution can be obtained by optimization over the rigid body transformation group. Besides, we present a new point cloud alignment metric that is intrinsic to the proposed framework and takes into account geometric and semantic information. The evaluations using publicly available stereo and RGB-D datasets show that the proposed method outperforms state-of-the-art outdoor and indoor frame-to-frame registration methods. An open-source GPU implementation is also provided.

One of the big attractions of low-dimensional models for gait design has been the ability to compute solutions rapidly, whereas one of their drawbacks has been the difficulty in mapping the solutions back to the target robot. This paper presents a set of tools for rapidly determining solutions for “humanoids” without removing or lumping degrees of freedom. The main tools are: (1) C-FROST, an open-source C++ interface for FROST, a direct collocation optimization tool; and (2) multi-threading. The results will be illustrated on a 20-DoF floating-base model for a Cassie-series bipedal robot through numerical optimization and physical experiments.

This paper presents the first experimental results of crutch-less dynamic walking with paraplegics on a lower-body exoskeleton: ATALANTE, designed by the French start-up company Wandercraft. The methodology used to achieve these results is based on the partial hybrid zero dynamics (PHZD) framework for formally generating stable walking gaits. A direct collocation optimization formulation is used to provide fast and efficient generation of gaits tailored to each patient. These gaits are then implemented on the exoskeleton for three paraplegics. The end result is dynamically stable walking in an exoskeleton without the need for crutches. After a short period of tuning by the engineers and practice by the subjects, each subject was able to dynamically walk across a room of about 10 m up to a speed of 0.15 m/s (0.5 km/h) without the need for crutches or any other kind of assistance.

State-of-the-art robotic perception systems have achieved sufficiently good performance using Inertial Measurement Units (IMUs), cameras, and nonlinear optimization techniques, that they are now being deployed as technologies. However, many of these methods rely significantly on vision and often fail when visual tracking is lost due to lighting or scarcity of features. This paper presents a state-estimation technique for legged robots that takes into account the robot's kinematic model as well as its contact with the environment. We introduce forward kinematic factors and preintegrated contact factors into a factor graph framework that can be incrementally solved in real-time. The forward kinematic factor relates the robot's base pose to a contact frame through noisy encoder measurements. The preintegrated contact factor provides odometry measurements of this contact frame while accounting for possible foot slippage. Together, the two developed factors constrain the graph optimization problem allowing the robot's trajectory to be estimated. The paper evaluates the method using simulated and real sensory IMU and kinematic data from experiments with a Cassie-series robot designed by Agility Robotics. These preliminary experiments show that using the proposed method in addition to IMU decreases drift and improves localization accuracy, suggesting that its use can enable successful recovery from a loss of visual tracking.

The factor graph framework is a convenient modeling technique for robotic state estimation where states are represented as nodes, and measurements are modeled as factors. When designing a sensor fusion framework for legged robots, one often has access to visual, inertial, joint encoder, and contact sensors. While visual-inertial odometry has been studied extensively in this framework, the addition of a preintegrated contact factor for legged robots has been only recently proposed. This allowed for integration of encoder and contact measurements into existing factor graphs, however, new nodes had to be added to the graph every time contact was made or broken. In this work, to cope with the problem of switching contact frames, we propose a hybrid contact preintegration theory that allows contact information to be integrated through an arbitrary number of contact switches. The proposed hybrid modeling approach reduces the number of required variables in the nonlinear optimization problem by only requiring new states to be added alongside camera or selected keyframes. This method is evaluated using real experimental data collected from a Cassie-series robot where the trajectory of the robot produced by a motion capture system is used as a proxy for ground truth. The evaluation shows that inclusion of the proposed preintegrated hybrid contact factor alongside visual-inertial navigation systems improves estimation accuracy as well as robustness to vision failure, while its generalization makes it more accessible for legged platforms.

This paper develops a control approach with correctness guarantees for the simultaneous operation of lane keeping and adaptive cruise control. The safety specifications for these driver assistance modules are expressed in terms of set invariance. Control barrier functions (CBFs) are used to design a family of control solutions that guarantee the forward invariance of a set, which implies satisfaction of the safety specifications. The CBFs are synthesized through a combination of sum-of-squares program and physics-based modeling and optimization. A real-time quadratic program is posed to combine the CBFs with the performance-based controllers, which can be either expressed as control Lyapunov function conditions or as black-box legacy controllers. In both cases, the resulting feedback control guarantees the safety of the composed driver assistance modules in a formally correct manner. Importantly, the quadratic program admits a closed-form solution that can be easily implemented. The effectiveness of the control approach is demonstrated by simulations in the industry-standard vehicle simulator Carsim.

Supervised learning is used to build a control policy for robust, stable, dynamic walking of an underactuated bipedal robot. The training and testing sets consist of controllers based on a full dynamic model, virtual constraints, and parameter optimization to meet torque limits, friction cone, and environmental conditions. The controllers are designed to induce locally exponentially stable periodic walking gaits at various speeds, both forward and backward, and for various constant ground slopes. They are also designed to induce aperiodic gaits that transition among a subset of the periodic gaits in a fixed number of steps. In experiments, the learned policy allows a 3D bipedal robot to recover from a significant kick. It also enables the robot to walk down a 22 degree slope and walk on sinusoidally varying terrain, all without using a camera. During the development of these results, it is demonstrated that supervised learning of locally exponentially stable controllers can result in a loss of stability and a means to avoid this is suggested.

Bipedal robots are prime examples of complex cyber-physical systems (CPS). They exhibit many of the features that make the design and verification of CPS so difficult: hybrid dynamics, large continuous dynamics in each mode (e.g., 10 or more state variables), and nontrivial specifications involving nonlinear constraints on the state variables. In this paper, we propose a two-step approach to formally synthesize control software for bipedal robots so as to enforce specifications by design and thereby generate physically realizable stable walking. In the first step, we design outputs and classical controllers driving these outputs to zero. The resulting controlled system evolves on a lower dimensional manifold and is described by the hybrid zero dynamics governing the remaining degrees of freedom. In the second step, we construct an abstraction of the hybrid zero dynamics that is used to synthesize a controller enforcing the desired specifications to be satisfied on the full order model. Our two step approach is a systematic way to mitigate the curse of dimensionality that hampers the applicability of formal synthesis techniques to complex CPS. Our results are illustrated with simulations showing how the synthesized controller enforces all the desired specifications and offers improved performance with respect to a controller that was utilized to obtain walking experimentally on the bipedal robot AMBER 2.

This paper reports on an underactuated 3D bipedal robot with passive feet that can start from a quiet standing position, initiate a walking gait, and traverse the length of the laboratory (approximately 10 m) at a speed of roughly 1 m/s. The controller was developed using the method of virtual constraints, a control design method first used on the planar point-feet robots Rabbit and MABEL. For the preliminary experiments reported here, virtual constraints were experimentally tuned to achieve robust planar walking and then 3D walking. A key feature of the controller leading to successful 3D walking is the particular choice of virtual constraints in the lateral plane, which implement a lateral balance control strategy similar to SIMBICON. To our knowledge, MARLO is the most highly underactuated bipedal robot to walk unassisted in 3D.

Models of robotic bipedal walking are hybrid, with a differential equation that describes the stance phase and a discrete map describing the impact event, that is, the nonstance leg contacting the walking surface. The feedback controllers for these systems can be hybrid as well, including both continuous and discrete (event-based) actions. This paper concentrates on the event-based portion of the feedback design problem for 3-D bipedal walking. The results are developed in the context of robustly stabilizing periodic orbits for a simulation model of ATRIAS 2.1, which is a highly underactuated 3-D bipedal robot with series-compliant actuators and point feet, against external disturbances as well as parametric and nonparametric uncertainty. It is shown that left–right symmetry of the model can be used to both simplify and improve the design of event-based controllers. Here, the event-based control is developed on the basis of the Poincaré map, linear matrix inequalities and robust optimal control. The results are illustrated by designing a controller that enhances the lateral stability of ATRIAS 2.1.

This paper presents a feedback controller that allows MABEL, which is a kneed planar bipedal robot with 1-m-long legs, to accommodate terrain that presents large unexpected increases and decreases in height. The robot is provided no information regarding where the change in terrain height occurs and by how much. A finite-state machine is designed that manages transitions among controllers for flat-ground walking, stepping-up and -down, and a trip reflex. If the robot completes a step, the depth of a step-down or the height of a step-up can be immediately estimated at impact from the lengths of the legs and the angles of the robot’s joints. The change in height can be used to invoke a proper control response. On the other hand, if the swing leg impacts an obstacle during a step, or has a premature impact with the ground, a trip reflex is triggered on the basis of specially designed contact switches on the robot’s shins, contact switches at the end of each leg, and the current configuration of the robot. The design of each control mode and the transition conditions among them are presented. This paper concludes with experimental results of MABEL (blindly) accommodating various types of platforms, including ascent of a 12.5-cm-high platform, stepping-off an 18.5-cm-high platform, and walking over a platform with multiple ascending and descending steps.

This paper presents a feedback controller that allows MABEL, a kneed, planar bipedal robot, with 1 m-long legs, to accommodate an abrupt 20 cm decrease in ground height. The robot is provided information on neither where the step down occurs, nor by how much. After the robot has stepped off a raised platform, however, the height of the platform can be estimated from the lengths of the legs and the angles of the robot's joints. A real-time control strategy is implemented that uses this on-line estimate of step-down height to switch from a baseline controller, that is designed for flat-ground walking, to a second controller, that is designed to attenuate torso oscillation resulting from the step-down disturbance. After one step, the baseline controller is re-applied. The control strategy is developed on a simplified-design model of the robot and then verified on a more realistic model before being evaluated experimentally. The paper concludes with experimental results showing MABEL (blindly) stepping off a 20 cm high platform.

This paper presents a control design based on the method of virtual constraints and hybrid zero dynamics to achieve stable running on MABEL, a planar biped with compliance. In particular, a time-invariant feedback controller is designed such that the closed-loop system not only respects the natural compliance of the open-loop system, but also enables active force control within the compliant hybrid zero dynamics and results in exponentially stable running gaits. The compliant-hybrid-zero-dynamics-based controller with active force control is implemented experimentally and shown to realize stable running gaits on MABEL at an average speed of 1.95 m/s (4.4 mph) and a peak speed of 3.06 m/s (6.8 mph). The obtained gait has flight phases upto 39% of the gait, and an estimated ground clearance of 7.5 – 10 cm.

This paper focuses on steering a 3D robot while walking on a flat surface. A hybrid feedback controller designed in for stable walking along a straight line is modified so that it is capable of adjusting the net yaw rotation of the robot over a step in order to steer the robot along paths with mild curvature. The controller is designed on the basis of a single pre-defined trajectory for periodic walking along a straight line. In order to illustrate the role of internal/external (i.e., medial/lateral) rotation at the hip in achieving curved walking motions, the performance of two robots, one with internal/external rotation and one without, is compared.

A hybrid controller that induces stable running gaits on a monopedal robot is developed. The robot features a rigid leg with a revolute knee and a heavy torso with center of mass located far from the hip. The torso houses a novel powertrain that provides series compliance in the compression direction of the leg. The proposed control law is developed within the hybrid zero dynamics framework and it acts on two levels. On the first level, continuous within-stride control asymptotically imposes (virtual) holonomic constraints reducing the dynamics of the robot to a lower-dimensional hybrid subsystem. On the second level, event-based control stabilizes the resulting hybrid subsystem. The controller achieves the dual objectives of working harmoniously with the system's natural dynamics and inducing provably exponentially stable running motions, while all relevant physical constraints are respected.

This paper presents three feedback controllers that achieve an asymptotically stable, periodic, and fast walking gait for a 3-D bipedal robot consisting of a torso, revolute knees, and passive (unactuated) point feet. The walking surface is assumed to be rigid and flat; the contact between the robot and the walking surface is assumed to inhibit yaw rotation. The studied robot has 8 DOF in the single support phase and six actuators. In addition to the reduced number of actuators, the interest of studying robots with point feet is that the feedback control solution must explicitly account for the robot's natural dynamics in order to achieve balance while walking. We use an extension of the method of virtual constraints and hybrid zero dynamics (HZD), a very successful method for planar bipeds, in order to simultaneously compute a periodic orbit and an autonomous feedback controller that realizes the orbit, for a 3-D (spatial) bipedal walking robot. This method allows the computations for the controller design and the periodic orbit to be carried out on a 2-DOF subsystem of the 8-DOF robot model. The stability of the walking gait under closed-loop control is evaluated with the linearization of the restricted Poincare map of the HZD. Most periodic walking gaits for this robot are unstable when the controlled outputs are selected to be the actuated coordinates. Three strategies are explored to produce stable walking. The first strategy consists of imposing a stability condition during the search of a periodic gait by optimization. The second strategy uses an event-based controller to modify the eigenvalues of the (linearized) Poincare map. In the third approach, the effect of output selection on the zero dynamics is discussed and a pertinent choice of outputs is proposed, leading to stabilization without the use of a supplemental event-based controller.

Consider a biped evolving in the sagittal plane. The unexpected rotation of the supporting foot can be avoided by controlling the zero moment point (ZMP). The objective of this study is to propose and analyze a control strategy for simultaneously regulating the position of the ZMP and the joints of the robot. If the tracking requirements were posed in the time domain, the problem would be underactuated in the sense that the number of inputs would be less than the number of outputs. To get around this issue, the proposed controller is based on a path-following control strategy, previously developed for dealing with the underactuation present in planar robots without actuated ankles. In particular, the control law is defined in such a way that only the kinematic evolution of the robot's state is regulated, but not its temporal evolution. The asymptotic temporal evolution of the robot is completely defined through a one degree-of-freedom subsystem of the closed-loop model. Since the ZMP is controlled, bipedal walking that includes a prescribed rotation of the foot about the toe can also be considered. Simple analytical conditions are deduced that guarantee the existence of a periodic motion and the convergence toward this motion.

Consider a biped evolving in the sagittal plane. The unexpected rotation of the supporting foot can be avoided by controlling the zero moment point or ZMP. The objective of this study is to propose and analyze a control strategy for simultaneously regulating the position of the ZMP and the joints of the robot. If the tracking requirements were posed in the time domain, the problem would be underactuated in the sense that the number of inputs would be less than the number of outputs. To get around this issue, the proposed controller is based on a path-following control strategy previously developed for dealing with the underactuation present in planar robots without actuated ankles. In particular, the control law is defined in such a way that only the kinematic evolution of the robot's state is regulated, but not its temporal evolution. The asymptotic temporal evolution of the robot is completely defined through a one degree of freedom subsystem of the closed-loop model. Simple analytical conditions, which guarantee the existence of a periodic motion and the convergence towards this motion, are deduced.

Two feedback controllers that induce stable running gaits on a three-degree-of-freedom asymmetric hopper, termed the asymmetric spring loaded inverted pendulum (ASLIP), see Fig. 1, are compared in terms of their steady-state and transient behaviors. In each case, feedback is used to create a lower-dimensional hybrid subsystem that determines the existence and stability properties of periodic motions of the full-dimensional closed-loop system. The first controller creates a one degree-of-freedom subsystem through imposing two suitably selected (virtual) holonomic constraints on the configuration variables of the ASLIP. The second controller asymptotically imposes a single (virtual) holonomic constraint to create a two-degree-of-freedom subsystem that is diffeomorphic to a standard spring loaded inverted pendulum (SLIP). The two controllers induce identical steady-state behaviors. Under transient conditions, however, the underlying compliant nature of the SLIP allows significantly larger disturbances to be accommodated, with less actuator effort, and without violation of the unilateral constraints between the leg end and the ground.

Numerous robotic tasks associated with underactuation have been studied in the literature. For a large number of these in the plane, the mechanical models have a cyclic variable, the cyclic variable is unactuated, and all shape variables are independently actuated. This paper formulates and solves two control problems for this class of models. If the generalized momentum conjugate to the cyclic variable is conserved, a set of that outputs is defined. If the generalized momentum conjugate to the cyclic variable is not conserved, a feedback that asymptotically stabilizes an equilibrium is given. The results are illustrated on a ballistic nip motion and on a balancing task.

This paper presents the experimental implementation and validation of a framework for the systematic design, analysis, and performance enhancement of controllers that induce stable walking in N-link underactuated planar biped robots. Controllers designed via this framework act by enforcing virtual constraints - holonomic constraints imposed via feedback - on the robot's configuration. The stability properties of the resulting walking motions may be easily analyzed in terms of a two-dimensional sub-dynamic of the full walking model. The experimental validation is performed on RABBIT, a 5-link prototype constructed by the French project Commande de Robots a Pattes of the CNRS-GdR Automatique.

The primary goal of this paper is to demonstrate a means to prove asymptotically stable walking in an underactuated, planar, five-link biped robot model. The analysis assumes a rigid contact model when the swing leg impacts the ground and an instantaneous double support phase. The specific robot model analyzed corresponds to a prototype under development by the Centre National de la Recherche Scientifique (CNRS), Paris, France. A secondary goal of the paper is to establish the viability of the theoretically motivated control law. This is explored in a number of ways. First, it is shown how known time trajectories, such as those determined on the basis of walking with minimal energy consumption, can be incorporated into the proposed controller structure. Secondly, various perturbations to the walking motion are introduced to verify disturbance rejection capability. Finally, the controller is demonstrated on a detailed simulator for the prototype which includes torque limits and a compliant model of the walking surface, and thus a noninstantaneous double support phase.

Closed-loop, asymptotically stable walking motions are designed for a 5-link, planar bipedal robot model with one degree of underactuation. Parameter optimization is applied to the hybrid zero dynamics, a 1-DOF invariant subdynamics of the full robot model, in order to create asymptotically stable orbits. Tuning the dynamics of this 1-DOF subsystem via optimization is interesting because asymptotically stable orbits of the zero dynamics correspond to asymptotically stabilizable orbits of the full hybrid model of the walker. The optimization process uses a sequential quadratic programming (SQP) algorithm and is able to satisfy kinematic and dynamic constraints while approximately minimizing energy consumption and ensuring stability. This is in contrast with traditional approaches to the design of walking controllers where approximately optimal walking (time-) trajectories are derived and then enforced on the robot using a trajectory tracking controller.