Abstract-In this paper, we extend the concept of control barrier functions, developed initially for continuous time systems, to the discrete-time domain. We demonstrate safety-critical control for nonlinear discrete-time systems with applications to 3D bipedal robot navigation. Particularly, we mathematically analyze two different formulations of control barrier functions, based on their continuous-time counterparts, and demonstrate how these can be applied to discrete-time systems. We show that the resulting formulation is a nonlinear program in contrast to the quadratic program for continuous-time systems and under certain conditions, the nonlinear program can be formulated as a quadratically constrained quadratic program. Furthermore, using the developed concept of discrete control barrier functions, we present a novel control method to address the problem of navigation of a high-dimensional bipedal robot through environments with moving obstacles that present time-varying safety-critical constraints.

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This paper reviews control strategies employed in the injection-molding process. For clarity, the controlled variables have been categorized into all-phase control, phasedependent control, and cycle-to-cycle control. All-phase control includes variables that must be monitored and controlled at all times; i.e., in all the phases. Control of variables that are triggered during a specific phase are discussed under phase-dependent control. In cycle-to-cycle control, previous data are used to predict future trends and take appropriate corrective actions. The cyclic, dynamic, and unsteady state nature of the injection-molding process is discussed with respect to the conventional proportional-integral (PI) and proportional-integral-derivative (PID) controllers as well as the more advanced control schemes such as self-tuning control, optimal control, and statistical process control. Suggestions involving specific advanced control schemes and recommendations for future research in injection-molding process control also are made.

Abstract-An inspiration for developing a bipedal walking system is the ability to navigate rough terrain with discrete footholds like stepping stones. In this paper, we present a novel methodology to overcome the problem of dynamic walking over stepping stones with significant random changes to step length and step height at each step. Using a 2-step gait optimization, we not only consider the desired location of the next footstep but also the current configuration of the robot, thereby resolving the problem of step transition when we switch between different walking gaits. We then use gait interpolation to generate the desired walking gait in real-time. We demonstrate the method on a planar dynamical walking model of ATRIAS, an underactuated bipedal robot walking over a randomly generated stepping stones with step length and step height changing in the range of [30:80] (cm) and [-30:30] (cm) respectively. Experimental validation on the real robot was also successful for the problem of dynamic walking on stepping stones with step lengths varied within [23:78] (cm) and average walking speed of 0.6 (m/s).