Virtual Constraints and Hybrid Zero Dynamics for Realizing Underactuated Bipedal Locomotion

Underactuation is ubiquitous in human locomotion and should be ubiquitous in bipedal robotic locomotion as well. This chapter presents a coherent theory for the design of feedback controllers that achieve stable walking gaits in underactuated bipedal robots. Two fundamental tools are introduced, virtual constraints and hybrid zero dynamics. Virtual constraints are relations on the state variables of a mechanical model that are imposed through a time-invariant feedback controller. One of their roles is to synchronize the robot's joints to an internal gait phasing variable. A second role is to induce a low dimensional system, the zero dynamics, that captures the underactuated aspects of a robot's model, without any approximations. To enhance intuition, the relation between physical constraints and virtual constraints is first established. From here, the hybrid zero dynamics of an underactuated bipedal model is developed, and its fundamental role in the design of asymptotically stable walking motions is established. The chapter includes numerous references to robots on which the highlighted techniques have been implemented.Comment: 17 pages, 4 figures, bookchapte

Dynamic Walking: Toward Agile and Efficient Bipedal Robots

Dynamic walking on bipedal robots has evolved from an idea in science fiction to a practical reality. This is due to continued progress in three key areas: a mathematical understanding of locomotion, the computational ability to encode this mathematics through optimization, and the hardware capable of realizing this understanding in practice. In this context, this review article outlines the end-to-end process of methods which have proven effective in the literature for achieving dynamic walking on bipedal robots. We begin by introducing mathematical models of locomotion, from reduced order models that capture essential walking behaviors to hybrid dynamical systems that encode the full order continuous dynamics along with discrete footstrike dynamics. These models form the basis for gait generation via (nonlinear) optimization problems. Finally, models and their generated gaits merge in the context of real-time control, wherein walking behaviors are translated to hardware. The concepts presented are illustrated throughout in simulation, and experimental instantiation on multiple walking platforms are highlighted to demonstrate the ability to realize dynamic walking on bipedal robots that is agile and efficient

Master of Science

thesisThis thesis focuses on the design, modeling, fabrication, and testing of a ?ying and walking robot, called the Dynamic Underactuated Flying-Walking (DUCK) robot. The DUCK robot combines a high-mobility ?ying platform, such as a quadcopter (quadrotor helicopter), with passive-dynamic legs to create a versatile system that can ?y and walk. One of the advantages of using passive-dynamic legs for walking is that additional actuators are not needed for terrestrial locomotion, therefore simplifying the design, reducing overall weight, and decreasing power consumption. First, a mathematical model is developed for the DUCK robot, where the modeling combines the passive-dynamic walking mechanism with the swinging mass of the aerial platform. Second, simulations based on the model are used to help guide the design of two prototype robots, speci?cally to tailor the shape of the feet and the dimensions of the passive-dynamic walking mechanism. Third, an energy analysis is performed to compare the performances between ?ying and walking. More specifically, simulation results show that continuous active walking has a comparable energy efficiency to that of flying for the two prototype designs. For design Version 1, it is estimated that the robot is able to walk up to 1600 meters on a 30kJ battery (standard Li-Po battery) with a cost of transport of 1.0, while the robot can potentially fly up to 1800 meters horizontally with the weight of its legs and up to 2300 meters without the weight of its legs. Design Version 2 is estimated to be able to walk up to 4600 meters on a 30kJ battery with a cost of transport of .50, while it could fly up to 2600 meters with the weight of its legs or 4300 meters without its legs. The cost of transport of flying is estimated to be .89 in all scenarios. Finally, experimental results demonstrate the feasibility of combining an aerial platform with passive-dynamic legs to create an effective flying and walking robot. Two modes of walking are experimentally demonstrated: (1) passive walking down inclined surfaces for low-energy terrestrial locomotion and (2) active (powered) walking leveraging the capabilities of the flying platform, where thrust from the quadcopter's rotors enables the DUCK robot to walk on flat surfaces or up inclined surfaces

Hybrid Zero Dynamics of Planar Biped Walkers

Planar, underactuated, biped walkers form an important domain of applications for hybrid dynamical systems. This paper presents the design of exponentially stable walking controllers for general planar bipedal systems that have one degree-of-freedom greater than the number of available actuators. The within-step control action creates an attracting invariant set—a two-dimensional zero dynamics submanifold of the full hybrid model—whose restriction dynamics admits a scalar linear time-invariant return map. Exponentially stable periodic orbits of the zero dynamics correspond to exponentially stabilizable orbits of the full model. A convenient parameterization of the hybrid zero dynamics is imposed through the choice of a class of output functions. Parameter optimization is used to tune the hybrid zero dynamics in order to achieve closed-loop, exponentially stable walking with low energy consumption, while meeting natural kinematic and dynamic constraints. The general theory developed in the paper is illustrated on a five link walker, consisting of a torso and two legs with knees

Energy Shaping of Underactuated Systems via Interconnection and Damping Assignment Passivity-Based Control with Applications to Planar Biped Robots

The sought goal of this thesis is to show that total energy shaping is an effective and versatile tool to control underactuated mechanical systems. The performance of several approaches, rooted in the port-Hamiltonian formalism, are analyzed while tackling distinct control problems: i) equilibrium stabilization; ii) gait generation; iii) gait robustication. Firstly, a constructive solution to deal with interconnection and damping assignment passivity-based control (IDA-PBC) for underactuated two-degree-of-freedom mechanical systems is proposed. This strategy does not involve the resolution of any partial differential equation, since explicit solutions are given, while no singularities depending on generalized momenta are introduced by the controller. The methodology is applied to the stabilization of a translational oscillator with a rotational actuator system, as well as, to the gait generation for an underactuated compass-like biped robot (CBR). Then, the problem of gait generation is addressed using dissipative forces in the controller. In this sense, three distinct controllers are presented, namely simultaneous interconnection and damping assignment passivity-based control with dissipative forces, energy pumping-and-damping passivity-based control (EPD-PBC), and energy pumping-or-damping control. Finally, EPD-PBC is used to increase the robustness of the gait exhibited by the CBR over uncertainties on the initial conditions. The passivity of the system is exploited, as well as, its hybrid nature (using the hybrid zero dynamics method) to carry out the stability analysis. Besides, such an approach is applied to new gaits that are generated using IDA-PBC. Numerical case studies, comparisons, and critical discussions evaluate the performance of the proposed approaches

Master of Science

thesisFor those who have suffered stroke or spinal cord injury, rehabilitation is often the answer for improving gait function. Rehabilitative exercises, which often focus on the legs and deemphasize the role of the upper limbs, are done to help stimulate muscles and exploit neuroplasticity for the diminished functions. However, it has been shown that upper limb muscle activity can induce lower limb muscle activity. It has also been shown that proper arm swing is necessary during gait for balance. This thesis presents the design concept and fabricated prototype of a device that swings the arms during gait rehabilitation. The device is low-powered, lightweight, wearable, and capable of assisting the user's arm swing in the sagittal plane and has unhindered kinematics in the remaining unactuated degrees of freedom. The design comprises three key subassemblies: a backpack frame, an underactuated arm-swing mechanism, and a power train to transfer and amplify motor torques to the arm-swing mechanism. Tests are performed to validate the shoulder-angle prediction equations based on the noncollocated motor-angle sensor measurements, to validate the device's ability to provide adequate torque to generate arm-swing in a passive user, and to investigate whether or not the user's active involvement can be observed by examining motor torque or shoulder angles. The results show that the device does provide sufficient torque to move the arms with a factor of safety, but that the model-based shoulder-angle estimates obtained from the motor measurements have nonnegligible error with the current prototype. It is recommended that a Proportional-Derivative (PD) controller with high PD gains be used with the device because of its low root mean square (RMS) tracking error, shoulder-angle amplitude creation, and ability to diagnose user-assistance level (i.e., is the user passive or actively assisting arm swing) online by observing shoulder-angle amplitudes and peak motor torques