(Georgia Institute of Technology, 2015)
Kolathaya, Shishir; Ames, Aaron D.
Model-based nonlinear controllers like feedback linearization and control Lyapunov functions are highly sensitive to the model parameters of the robot. This paper addresses the problem of realizing these controllers in a particular class of hybrid models-systems with impulse
effects-through a parameter sensitivity measure. This measure quantifies the sensitivity of a given model-based controller to parameter uncertainty along a particular trajectory. By using this measure, output boundedness of the controller (computed torque+PD) will be analyzed.
Given outputs that characterize the control objectives, i.e., the goal is to drive these outputs
to zero, we consider Lyapunov functions obtained from these outputs. The main result of this paper establishes the ultimate boundedness of the output dynamics in terms of this measure
via these Lyapunov functions under the assumption of stable hybrid zero dynamics. This is demonstrated in simulation on a 5-DOF underactuated bipedal robot.
(Georgia Institute of Technology, 2015)
Xu, Xiangru; Tabuada, Paulo; Grizzle, Jessy W.; Ames, Aaron D.
Barrier functions (also called certificates) have been an important tool for the verification of hybrid systems, and have also played important roles in optimization and multi-objective
control. The extension of a barrier function to a controlled system results in a control
barrier function. This can be thought of as being analogous to how Sontag extended Lyapunov
functions to control Lypaunov functions in order to enable controller synthesis for stabilization
tasks. A control barrier function enables controller synthesis for safety requirements specified
by forward invariance of a set using a Lyapunov-like condition. This paper develops several
important extensions to the notion of a control barrier function. The first involves robustness
under perturbations to the vector field defining the system. Input-to-State stability conditions
are given that provide for forward invariance, when disturbances are present, of a "relaxation"
of set rendered invariant without disturbances. A control barrier function can be combined with
a control Lyapunov function in a quadratic program to achieve a control objective subject to
safety guarantees. The second result of the paper gives conditions for the control law obtained
by solving the quadratic program to be Lipschitz continuous and therefore to gives rise to
well-defined solutions of the resulting closed-loop system.