Kaveh Akbari Hamed

Performance Analysis and Feedback Control of ATRIAS, A Three-Dimensional Bipedal Robot

This paper develops feedback controllers for walking in 3D, on level ground, with energy efficiency as the performance objective. Assume The Robot Is AS phere (ATRIAS) 2.1 is a new robot that has been designed for the study of 3D bipedal locomotion, with the aim of combining energy efficiency, speed, and robustness with respect to natural terrain variations in a single platform. The robot is highly underactuated, having 6 actuators and, in single support, 13 degrees of freedom. Its sagittal plane dynamics are designed to embody the spring loaded inverted pendulum (SLIP), which has been shown to provide a dynamic model of the body center of mass during steady running gaits of a wide diversity of terrestrial animals. A detailed dynamic model is used to optimize walking gaits with respect to the cost of mechanical transport (CMT), a dimensionless measure of energetic efficiency, for walking speeds ranging from 0.5 m=s ðÞ to 1.4 m=s

Preliminary walking experiments with underactuated 3D bipedal robot MARLO

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.

Event-Based Stabilization of Periodic Orbits for Underactuated 3-D Bipedal Robots With Left-Right Symmetry

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.

Continuous-time controllers for stabilizing periodic orbits of hybrid systems: Application to an underactuated 3D bipedal robot

This paper presents a systematic approach to exponentially stabilize periodic orbits in nonlinear systems with impulse effects, a special class of hybrid systems. Stabilization is achieved with a time invariant continuous-time controller. The presented method assumes a parametrized family of continuous-time controllers has been designed so that (1) a periodic orbit is induced, and (2) the orbit itself is invariant under the choice of parameters in the controllers. By investigating the properties of the Poincaré return map, a sensitivity analysis is presented that translates the stabilization problem into a set of Bilinear Matrix Inequalities (BMIs). A BMI optimization problem is set up to select the parameters of the continuous-time controller to achieve exponential stability. We illustrate the power of the approach by finding new stabilizing solutions for periodic orbits of an underactuated 3D bipedal robot.

Exponentially stabilizing continuous-time controllers for periodic orbits of hybrid systems

This paper presents a systematic approach for the design of continuous-time controllers to robustly and exponentially stabilize periodic orbits of hybrid dynamical systems arising from bipedal walking. A parameterized family of continuous-time controllers is assumed so that (1) a periodic orbit is induced for the hybrid system, and (2) the orbit is invariant under the choice of controller parameters. Properties of the Poincaré map and its first- and second-order derivatives are used to translate the problem of exponential stabilization of the periodic orbit into a set of bilinear matrix inequalities (BMIs). A BMI optimization problem is then set up to tune the parameters of the continuous-time controller so that the Jacobian of the Poincaré map has its eigenvalues in the unit circle. It is also shown how robustness against uncertainty in the switching condition of the hybrid system can be incorporated into the design problem. The power of this approach is illustrated by finding robust and stabilizing continuous-time feedback laws for walking gaits of two underactuated 3D bipedal robots.

Robust event-based stabilization of periodic orbits for hybrid systems: Application to an underactuated 3D bipedal robot

The first return map or Poincaré map can be viewed as a discrete-time dynamical system evolving on a hyper surface that is transversal to a periodic orbit; the hyper surface is called a Poincaré section. The Poincaré map is a standard tool for assessing the stability of periodic orbits in non-hybrid as well as hybrid systems. In addition, it can be used for stabilization of periodic orbits if the underlying dynamics of the system depends on a set of parameters that can be updated by a feedback law when trajectories cross the Poincaré section. This paper addresses an important practical obstacle that arises when designing feedback laws on the basis of the Jacobian linearization of the Poincaré map. In almost all practical cases, the Jacobians must be estimated numerically, and when the underlying dynamics presents a wide range of time scales, the numerical approximations of the first partial derivatives are sufficiently inaccurate that controller tuning is very difficult. Here, a robust control formalism is proposed whereby a convex set of approximations to the Jacobian linearization is systematically generated and a stabilizing controller is designed through two appropriate sets of linear matrix inequalities (LMIs). The result is illustrated on a walking gait of a 3D underactuated bipedal robot.

Experimental results for 3D bipedal robot walking based on systematic optimization of virtual constraints

Feedback control laws which create asymptotically stable periodic orbits for hybrid systems are an effective means for realizing dynamic legged locomotion in bipedal robots. To address the challenge of designing such control laws, we recently introduced a method to systematically select a stabilizing feedback control law from a parameterized family of feedback laws by solving an offline optimization problem. The method has been used elsewhere to design a stable gait based on virtual constraints, and its potential effectiveness was illustrated via simulation results. In this paper, we present the first experimental demonstration of a controller designed using this new offline optimization method. The new controller is compared with a nominal controller in experiments on MARLO, a 3D point-foot bipedal robot. Compared to the nominal controller, the optimized controller leads to improved lateral control and longer sustained walking.

Iterative Robust Stabilization Algorithm for Periodic Orbits of Hybrid Dynamical Systems: Application to Bipedal Running*

Abstract This paper presents a systematic numerical algorithm to design optimal Hoo continuous-time controllers to robustly stabilize periodic orbits for hybrid dynamical systems in the presence of discrete-time uncertainties. A parameterized set of closed-loop hybrid systems is assumed for which there exists a common periodic orbit. The algorithm is created based on an iterative sequence of optimization problems involving Bilinear and Linear Matrix Inequalities (BMIs and LMIs). At each iteration, the optimal %oo problem is translated into a BMI optimization problem which can be easily solved using available software packages. Some sufficient conditions for the convergence of the iterative algorithm are presented. The power of the algorithm is then demonstrated in designing robust stabilizing virtual constraints for running of a highly underactuated bipedal robot with 7 degrees of underactuation in the presence of impact model uncertainties.

Decentralized feedback controllers for exponential stabilization of hybrid periodic orbits: Application to robotic walking

This paper presents a systematic algorithm to design time-invariant decentralized feedback controllers to exponentially stabilize periodic orbits for a class of hybrid dynamical systems arising from bipedal walking. The algorithm assumes a class of parameterized and nonlinear decentralized feedback controllers which coordinate lower-dimensional hybrid subsystems based on a common phasing variable. The exponential stabilization problem is translated into an iterative sequence of optimization problems involving bilinear and linear matrix inequalities, which can be easily solved with available software packages. A set of sufficient conditions for the convergence of the iterative algorithm to a stabilizing decentralized feedback control solution is presented. The power of the algorithm is demonstrated by designing a set of local nonlinear controllers that cooperatively produce stable walking for a 3D autonomous biped with 9 degrees of freedom, 3 degrees of underactuation, and a decentralization scheme motivated by amputee locomotion with a transpelvic prosthetic leg.

Decentralized Feedback Controllers for Robust Stabilization of Periodic Orbits of Hybrid Systems: Application to Bipedal Walking

This paper presents a systematic algorithm to design time-invariant decentralized feedback controllers to exponentially and robustly stabilize periodic orbits for hybrid dynamical systems against possible uncertainties in discrete-time phases. The algorithm assumes a family of parameterized and decentralized nonlinear controllers to co-ordinate interconnected hybrid subsystems based on a common phasing variable. The exponential and