Asymptotically Stable Walking of a Five-Link Underactuated 3D Bipedal Robot (1002.3258v1)

Abstract: This paper presents three feedback controllers that achieve an asymptotically stable, periodic, and fast walking gait for a 3D (spatial) bipedal robot consisting of a torso, two legs, and passive (unactuated) point feet. 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 6 actuators. The interest of studying robots with point feet is that the robot's natural dynamics must be explicitly taken into account to achieve balance while walking. We use an extension of the method of virtual constraints and hybrid zero dynamics, in order to simultaneously compute a periodic orbit and an autonomous feedback controller that realizes the orbit. This method allows the computations 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 Poincar\'e map of the hybrid zero dynamics. Three strategies are explored. 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. In the third approach, the effect of output selection is discussed and a pertinent choice of outputs is proposed, leading to stabilization without the use of a supplemental event-based controller.

• The paper introduces a reduced-dimension optimization using hybrid zero dynamics to design asymptotically stable gaits in a 3D five-link biped robot.
• It employs stability-constrained optimization and event-based feedback control to adjust eigenvalues and manage underactuation across sagittal and frontal planes.
• Numerical simulations and low-dimensional Poincaré maps validate the controllers, advancing dynamic stability in robotic locomotion with minimal foot support.

Asymptotically Stable Walking of a Five-Link Underactuated 3D Bipedal Robot

This paper examines the feedback control problem for achieving asymptotically stable, periodic walking in a three-dimensional (3D) bipedal robot model with five links. The biped includes a torso, knees modeled with revolute joints, and point feet, which are unactuated. The stabilization problem is notably challenging in this setting due to underactuation—a robot with eight degrees of freedom but only six actuators in its single support phase.

The paper's technical approach involves the application of virtual constraints and hybrid zero dynamics, strategies previously shown effective for planar bipedal robots. The task is extended to address the challenges of 3D modeling, requiring coordination in both the sagittal and frontal planes. This enhancement is non-trivial due to the increased complexity of unactuated 3D dynamics. The paper aims to discover periodic gaits using a reduced-dimension optimization problem carried out on a 2-DOF subsystem, addressing the impact and continuous dynamics of the gait cycle.

A primary contribution is the successful adaptation of the hybrid zero dynamics method to spatial robots, allowing the dynamics of an 8-DOF robot to be analyzed within a reduced 2-DOF framework. This reduction significantly impacts the feasibility of designing controllers for high-dimensional bipedal robots.

The specific challenges of achieving stable bipedal walking without large feet—which assist balance—are addressed with three feedback control strategies:

1. Stability-Constrained Optimization: Here, the design of periodic motions is integrated with stability conditions directly within an optimization framework. This enables the design of inherently stable gaits through calculated eigenvalue placements.
2. Event-Based Feedback Controller: This strategy modifies the eigenvalues of the linearized Poincaré map, designed to take corrective actions stride-to-stride. The strategy introduces extra robustness against disturbances and model uncertainties.
3. Selection of Controlled Outputs: A dynamic stabilization approach is explored through judicious selection of output variables that are linear combinations of the configuration variables. By altering the outputs, the system dynamics can inherently support stabilization without depending on additional event-based controllers.

The numerical simulations accompanying these methods provide evidence of the controllers' capability to achieve stable walking gaits under various conditions. Importantly, the controllers' effectiveness is verified through the computation of low-dimensional Poincaré maps and stability analysis via eigenvalues, showcasing the practicality of the reduced-order models for control design.

The implications of this work extend to the field of robotic locomotion over uneven and conforming surfaces, realistic humanoid robotics, and other advanced robotics applications where energy efficiency and dynamic stability are paramount. This paper propels forward the understanding of legged locomotion in robots without relying on simplifying assumptions like large foot areas or idealized contact conditions.

In the future, expanding these control techniques may involve incorporating dynamics that allow yaw rotations or extending the methodology to robots with mechanically complex feet. These developments could potentially enhance the stability and adaptability of bipedal locomotion systems in real-world applications.