Restricted Discrete Invariance and Self-Synchronization For Stable Walking of Bipedal Robots

Models of bipedal locomotion are hybrid, with a continuous component often generated by a Lagrangian plus actuators, and a discrete component where leg transfer takes place. The discrete component typically consists of a locally embedded co-dimension one submanifold in the continuous state space of the robot, called the switching surface, and a reset map that provides a new initial condition when a solution of the continuous component intersects the switching surface. The aim of this paper is to identify a low-dimensional submanifold of the switching surface, which, when it can be rendered invariant by the closed-loop dynamics, leads to asymptotically stable periodic gaits. The paper begins this process by studying the well-known 3D Linear Inverted Pendulum (LIP) model, where analytical results are much easier to obtain. A key contribution here is the notion of \textit{self-synchronization}, which refers to the periods of the pendular motions in the sagittal and frontal planes tending to a common period. The notion of invariance resulting from the study of the 3D LIP model is then extended to a 9-DOF 3D biped. A numerical study is performed to illustrate that asymptotically stable walking may be obtained.Comment: Conferenc

An Inverse Dynamics Approach to Control Lyapunov Functions

With the goal of moving towards implementation of increasingly dynamic behaviors on underactuated systems, this paper presents an optimization-based approach for solving full-body dynamics based controllers on underactuated bipedal robots. The primary focus of this paper is on the development of an alternative approach to the implementation of controllers utilizing control Lyapunov function based quadratic programs. This approach utilizes many of the desirable aspects from successful inverse dynamics based controllers in the literature, while also incorporating a variant of control Lyapunov functions that renders better convergence in the context of tracking outputs. The principal benefits of this formulation include a greater ability to add costs which regulate the resulting behavior of the robot. In addition, the model error-prone inertia matrix is used only once, in a non-inverted form. The result is a successful demonstration of the controller for walking in simulation, and applied on hardware in real-time for dynamic crouching

On Centroidal Dynamics and Integrability of Average Angular Velocity

In the literature on robotics and multibody dynamics, the concept of average angular velocity has received considerable attention in recent years. We address the question of whether the average angular velocity defines an orientation framethat depends only on the current robot configuration and provide a simple algebraic condition to check whether this holds. In the language of geometric mechanics, this condition corresponds to requiring the flatness of the mechanical connection associated to the robotic system. Here, however, we provide both a reinterpretation and a proof of this result accessible to readers with a background in rigid body kinematics and multibody dynamics but not necessarily acquainted with differential geometry, still providing precise links to the geometric mechanics literature. This should help spreading the algebraic condition beyond the scope of geometric mechanics,contributing to a proper utilization and understanding of the concept of average angular velocity.Comment: 8 pages, accepted for IEEE Robotics and Automation Letters (RA-L

Stabilization of Exoskeletons through Active Ankle Compensation

This paper presents an active stabilization method for a fully actuated lower-limb exoskeleton. The method was tested on the exoskeleton ATALANTE, which was designed and built by the French start-up company Wandercraft. The main objective of this paper is to present a practical method of realizing more robust walking on hardware through active ankle compensation. The nominal gait was generated through the hybrid zero dynamic framework. The ankles are individually controlled to establish three main directives; (1) keeping the non-stance foot parallel to the ground, (2) maintaining rigid contact between the stance foot and the ground, and (3) closing the loop on pelvis orientation to achieve better tracking. Each individual component of this method was demonstrated separately to show each component's contribution to stability. The results showed that the ankle controller was able to experimentally maintain static balance in the sagittal plane while the exoskeleton was balanced on one leg, even when disturbed. The entire ankle controller was then also demonstrated on crutch-less dynamic walking. During testing, an anatomically correct manikin was placed in the exoskeleton, in lieu of a paraplegic patient. The pitch of the pelvis of the exoskeleton-manikin system was shown to track the gait trajectory better when ankle compensation was used. Overall, active ankle compensation was demonstrated experimentally to improve balance in the sagittal plane of the exoskeleton manikin system and points to an improved practical approach for stable walking

Orbit Characterization, Stabilization and Composition on 3D Underactuated Bipedal Walking via Hybrid Passive Linear Inverted Pendulum Model

A Hybrid passive Linear Inverted Pendulum (H-LIP) model is proposed for characterizing, stabilizing and composing periodic orbits for 3D underactuated bipedal walking. Specifically, Period-l (P1) and Period -2 (P2) orbits are geometrically characterized in the state space of the H-LIP. Stepping controllers are designed for global stabilization of the orbits. Valid ranges of the gains and their optimality are derived. The optimal stepping controller is used to create and stabilize the walking of bipedal robots. An actuated Spring-loaded Inverted Pendulum (aSLIP) model and the underactuated robot Cassie are used for illustration. Both the aSLIP walking with PI or P2 orbits and the Cassie walking with all 3D compositions of the PI and P2 orbits can be smoothly generated and stabilized from a stepping-in-place motion. This approach provides a perspective and a methodology towards continuous gait generation and stabilization for 3D underactuated walking robots

First Steps Towards Full Model Based Motion Planning and Control of Quadrupeds: A Hybrid Zero Dynamics Approach

The hybrid zero dynamics (HZD) approach has become a powerful tool for the gait planning and control of bipedal robots. This paper aims to extend the HZD methods to address walking, ambling and trotting behaviors on a quadrupedal robot. We present a framework that systematically generates a wide range of optimal trajectories and then provably stabilizes them for the full-order, nonlinear and hybrid dynamical models of quadrupedal locomotion. The gait planning is addressed through a scalable nonlinear programming using direct collocation and HZD. The controller synthesis for the exponential stability is then achieved through the Poincaré sections analysis. In particular, we employ an iterative optimization algorithm involving linear and bilinear matrix inequalities (LMIs and BMIs) to design HZD-based controllers that guarantee the exponential stability of the fixed points for the Poincaré return map. The power of the framework is demonstrated through gait generation and HZD-based controller synthesis for an advanced quadruped robot, —Vision 60, with 36 state variables and 12 control inputs. The numerical simulations as well as real world experiments confirm the validity of the proposed framework

Predicting the post-impact velocity of a robotic arm

Starting from the recorded dynamic response of a 7DOF torque-controlled robot while intentionally impacting a rigid surface, we investigate the possibility of predicting the post-impact robot velocity from the ante-impact velocity and configuration. The velocity prediction is obtained by means of an impact map, derived using the framework of nonsmooth mechanics, that makes use of the known rigid-body robot model and the assumption of a frictionless inelastic impact. The main contribution is proposing a methodology that allows for a meaningful quantitative comparison between the recorded post-impact data, that exhibits a damped oscillatory response after the impact, and the post-impact velocity prediction derived via the rigid-body robot model, that presents no oscillations. The results of this approach are promising and the recorded impact data (18 experiments) is made publicly available, together with the numerical routines employed to generate the quantitative comparison, to further stimulate research in this field