Jessy W. Grizzle

Power management strategy for a parallel hybrid electric truck

Hybrid vehicle techniques have been widely studied recently because of their potential to significantly improve the fuel economy and drivability of future ground vehicles. Due to the dual-power-source nature of these vehicles, control strategies based on engineering intuition frequently fail to fully explore the potential of these advanced vehicles. In this paper, we present a procedure for the design of a near-optimal power management strategy. The design procedure starts by defining a cost function, such as minimizing a combination of fuel consumption and selected emission species over a driving cycle. Dynamic programming (DP) is then utilized to find the optimal control actions including the gear-shifting sequence and the power split between the engine and motor while subject to a battery SOC-sustaining constraint. Through analysis of the behavior of DP control actions, near-optimal rules are extracted, which, unlike DP control signals, are implementable. The performance of this power management control strategy is studied by using the hybrid vehicle model HE-VESIM developed at the Automotive Research Center of the University of Michigan. A tradeoff study between fuel economy and emissions was performed. It was found that significant emission reduction could be achieved at the expense of a small increase in fuel consumption.

Hybrid zero dynamics of planar biped walkers

Planar, underactuated, biped walkers form an important domain of applications for hybrid dynamical systems. This paper presents the design of exponentially stable walking controllers for general planar bipedal systems that have one degree-of-freedom greater than the number of available actuators. The within-step control action creates an attracting invariant set - a two-dimensional zero dynamics submanifold of the full hybrid model

RABBIT: a testbed for advanced control theory

whose restriction dynamics admits a scalar linear time-invariant return map. Exponentially stable periodic orbits of the zero dynamics correspond to exponentially stabilizable orbits of the full model. A convenient parameterization of the hybrid zero dynamics is imposed through the choice of a class of output functions. Parameter optimization is used to tune the hybrid zero dynamics in order to achieve closed-loop, exponentially stable walking with low energy consumption, while meeting natural kinematic and dynamic constraints. The general theory developed in the paper is illustrated on a five link walker, consisting of a torso and two legs with knees.

Observer design for nonlinear systems with discrete-time measurements

Describes the design, construction and control of an experimental bipedal robot platform for the study of walking.

Stable walking of a 7-DOF biped robot

This paper focuses on the development of asymptotic observers for nonlinear discrete-time systems. It is argued that instead of trying to imitate the linear observer theory, the problem of constructing a nonlinear observer can be more fruitfully studied in the context of solving simultaneous nonlinear equations. In particular, it is shown that the discrete Newton method, properly interpreted, yields an asymptotic observer for a large class of discrete-time systems, while the continuous Newton method may be employed to obtain a global observer. Furthermore, it is analyzed how the use of Broydens method in the observer structure affects the observers performance and its computational complexity. An example illustrates some aspects of the proposed methods; moreover, it serves to show that these methods apply equally well to discrete-time systems and to continuous-time systems with sampled outputs. >

A Compliant Hybrid Zero Dynamics Controller for Stable, Efficient and Fast Bipedal Walking on MABEL

The primary goal of this paper is to demonstrate a means to prove asymptotically stable walking in an underactuated, planar, five-link biped robot model. The analysis assumes a rigid contact model when the swing leg impacts the ground and an instantaneous double support phase. The specific robot model analyzed corresponds to a prototype under development by the Centre National de la Recherche Scientifique (CNRS), Paris, France. A secondary goal of the paper is to establish the viability of the theoretically motivated control law. This is explored in a number of ways. First, it is shown how known time trajectories, such as those determined on the basis of walking with minimal energy consumption, can be incorporated into the proposed controller structure. Secondly, various perturbations to the walking motion are introduced to verify disturbance rejection capability. Finally, the controller is demonstrated on a detailed simulator for the prototype which includes torque limits and a compliant model of the walking surface, and thus a noninstantaneous double support phase.

The Extended Kalman Filter as a Local Asymptotic Observer for Nonlinear Discrete-Time Systems

The planar bipedal testbed MABEL contains springs in its drivetrain for the purpose of enhancing both energy efficiency and agility of dynamic locomotion. While the potential energetic benefits of springs are well documented in the literature, feedback control designs that effectively realize this potential are lacking. In this paper, we extend and apply the methods of virtual constraints and hybrid zero dynamics, originally developed for rigid robots with a single degree of underactuation, to MABEL, a bipedal walker with a novel compliant transmission and multiple degrees of underactuation. A time-invariant feedback controller is designed such that the closed-loop system respects the natural compliance of the open-loop system and realizes exponentially stable walking gaits. Five experiments are presented that highlight different aspects of MABEL and the feedback design method, ranging from basic elements such as stable walking and robustness under perturbations, to energy efficiency and a walking speed of 1.5 m s−1 (3.4 mph). The experiments also compare two feedback implementations of the virtual constraints, one based on PD control of Westervelt et al., and a second that implements a full hybrid zero dynamics controller. On MABEL, the full hybrid zero dynamics controller yields a much more faithful realization of the desired virtual constraints and was instrumental in achieving more rapid walking.

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

The convergence aspects of the extended Kalman filter, when used as a deterministic observer for a nonlinear discrete-time system, are analyzed. The case of systems with nonlinear output maps as well as with linear maps is treated and the conditions needed to ensure the uniform boundedness of certain Riccati equations are related to the observability properties of the underlying nonlinear system. Furthermore, we show the convergence of the filter without any a priori boundedness assumptions on the error covariances as long as the states stay within a convex compact domain.

Rank invariants of nonlinear systems

This paper presents three feedback controllers that achieve an asymptotically stable, periodic, and fast walking gait for a 3-D bipedal robot consisting of a torso, revolute knees, and passive (unactuated) point feet. The walking surface is assumed to be rigid and flat; 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 six actuators. In addition to the reduced number of actuators, the interest of studying robots with point feet is that the feedback control solution must explicitly account for the robots natural dynamics in order to achieve balance while walking. We use an extension of the method of virtual constraints and hybrid zero dynamics (HZD), a very successful method for planar bipeds, in order to simultaneously compute a periodic orbit and an autonomous feedback controller that realizes the orbit, for a 3-D (spatial) bipedal walking robot. This method allows the computations for the controller design and the periodic orbit 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 Poincare map of the HZD. Most periodic walking gaits for this robot are unstable when the controlled outputs are selected to be the actuated coordinates. Three strategies are explored to produce stable walking. 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 to modify the eigenvalues of the (linearized) Poincare map. In the third approach, the effect of output selection on the zero dynamics is discussed and a pertinent choice of outputs is proposed, leading to stabilization without the use of a supplemental event-based controller.

Nonlinear control of mechanical systems with an unactuated cyclic variable

A linear algebraic framework for the analysis of rank properties of nonlinear systems is introduced. This framework gives a high-level interpretation of several existing algorithms built around the recursive computation of certain algebraic ranks associated with right-invertibility, left-invertibility, and dynamic decoupling. Furthermore, it can be used to establish links between these algorithms and the differential algebraic approach, as well as to solve some static and dynamic noninteracting control problems.