Robert D. Gregg

Virtual Constraint Control of a Powered Prosthetic Leg: From Simulation to Experiments with Transfemoral Amputees.

Recent powered (or robotic) prosthetic legs independently control different joints and time periods of the gait cycle, resulting in control parameters and switching rules that can be difficult to tune by clinicians. This challenge might be addressed by a unifying control model used by recent bipedal robots, in which virtual constraints define joint patterns as functions of a monotonic variable that continuously represents the gait cycle phase. In the first application of virtual constraints to amputee locomotion, this paper derives exact and approximate control laws for a partial feedback linearization to enforce virtual constraints on a prosthetic leg. We then encode a human-inspired invariance property called effective shape into virtual constraints for the stance period. After simulating the robustness of the partial feedback linearization to clinically meaningful conditions, we experimentally implement this control strategy on a powered transfemoral leg. We report the results of three amputee subjects walking overground and at variable cadences on a treadmill, demonstrating the clinical viability of this novel control approach.

Reduction-based Control of Three-dimensional Bipedal Walking Robots

In this paper we develop the concept of reduction-based control, which is founded on a controlled form of geometric reduction known as functional Routhian reduction. We prove a geometric property of general serial-chain robots termed recursive cyclicity, identifying the inherent robot symmetries that we exploit with the Subrobot Theorem. This shows that any serial-chain robot can be decomposed for arbitrarily lower-dimensional analysis and control. We apply this method to construct stable directional three-dimensional walking gaits for a four-degree-of-freedom hipped bipedal robot. The controlled reduction decouples the biped’s sagittal-plane motion from the yaw and lean modes, and on the sagittal subsystem we use passivity-based control to produce known planar limit cycles on flat ground. The unstable yaw and lean modes are separately controlled to 2-periodic orbits through their shaped momenta. We numerically verify the existence of stable 2-periodic straight-walking limit cycles and demonstrate turning capabilities for the controlled biped.

The Difference Between Stiffness and Quasi-Stiffness in the Context of Biomechanical Modeling

The ankle contributes the majority of mechanical power during walking and is a frequently studied joint in biomechanics. Specifically, researchers have extensively investigated the torque-angle relationship for the ankle during dynamic tasks, such as walking and running. The slope of this relationship has been termed the “quasi-stiffness.” However, over time, researchers have begun to interchange the concepts of quasi-stiffness and stiffness. This is an especially important distinction as researchers currently begin to investigate the appropriate control systems for recently developed powered prosthetic legs. The quasi-stiffness and stiffness are distinct concepts in the context of powered joints, and are equivalent in the context of passive joints. The purpose of this paper is to demonstrate the difference between the stiffness and quasi-stiffness using a simple impedance-controlled inverted pendulum model and a more sophisticated biped walking model, each with the ability to modify the trajectory of an impedance controllers equilibrium angle position. In both cases, stiffness values are specified by the controller and the quasi-stiffness are shown during a single step. Both models have widely varying quasi-stiffness but each have a single stiffness value. Therefore, from this simple modeling approach, the differences and similarities between these two concepts are elucidated.

Stably Extending Two-Dimensional Bipedal Walking to Three Dimensions

In this paper we develop a feedback control law that results in stable walking gaits on flat ground for a three-dimensional bipedal robotic walker given stable walking gaits for a two-dimensional bipedal robotic walker. This is achieved by combining disparate techniques that have been employed in the bipedal robotic community: controlled symmetries, geometric reduction and hybrid zero dynamics. Controlled symmetries are utilized to obtain stable walking gaits for a two-dimensional bipedal robot walking on flat ground. These are related to walking gaits for a three-dimensional (hipless) bipedal robot through the use of geometric reduction. Finally, these walking gaits in three dimensions are made stable through the use of hybrid zero dynamics.

A geometric approach to three-dimensional hipped bipedal robotic walking

This paper presents a control law that results in stable walking for a three-dimensional bipedal robot with a hip. To obtain this control law, we utilize techniques from geometric reduction, and specifically a variant of Routhian reduction termed functional Routhian reduction, to effectively decouple the dynamics of the three-dimensional biped into its sagittal and lateral components. Motivated by the decoupling afforded by functional Routhian reduction, the control law we present is obtained by combining three separate control laws: the first shapes the potential energy of the sagittal dynamics of the biped to obtain stable walking gaits when it is constrained to the sagittal plane, the second shapes the total energy of the walker so that functional Routhian reduction can be applied to decoupling the dynamics of the walker for certain initial conditions, and the third utilizes an output zeroing controller to stabilize to the surface defining these initial conditions. We numerically verify that this method results in stable walking, and we discuss certain attributes of this walking gait.

Towards Biomimetic Virtual Constraint Control of a Powered Prosthetic Leg

This brief presents a novel control strategy for a powered prosthetic ankle based on a biomimetic virtual constraint. We first derive a kinematic constraint for the “effective shape” of the human ankle-foot complex during locomotion. This shape characterizes ankle motion as a function of the center of pressure (COP)-the point on the foot sole where the resultant ground reaction force is imparted. Since the COP moves monotonically from heel to toe during steady walking, we adopt the COP as a mechanical representation of the gait cycle phase in an autonomous feedback controller. We show that our kinematic constraint can be enforced as a virtual constraint by an output linearizing controller that uses only feedback available to sensors onboard a prosthetic leg. Using simulations of a passive walking model with feet, we show that this novel controller enforces exactly the desired effective shape, whereas a standard impedance (i.e., proportional-derivative) controller cannot. This brief provides a single, biomimetic control law for the entire single-support period during robot-assisted locomotion.

Control and Planning of 3-D Dynamic Walking With Asymptotically Stable Gait Primitives

In this paper, we present a hierarchical framework that enables motion planning for asymptotically stable 3-D bipedal walking in the same way that planning is already possible for zero moment point walking. This framework is based on the construction of asymptotically stable gait primitives for a class of hybrid dynamical systems with impacts. Each primitive corresponds to an asymptotically stable hybrid limit cycle that admits rules a priori for sequential composition with other primitives, reducing a high-dimensional feedback motion planning problem into a low-dimensional discrete tree search. As a constructive example, we develop this planning framework for the 3-D compass-gait biped, where each primitive corresponds to walking along an arc of constant curvature for a fixed number of steps. We apply a discrete search algorithm to plan a sequence of these primitives, taking the 3-D biped stably from start to goal in workspaces with obstacles. We finally show how this framework generalizes to more complex models by planning walking paths for an underactuated five-link biped.

Evidence for a Time-Invariant Phase Variable in Human Ankle Control

Human locomotion is a rhythmic task in which patterns of muscle activity are modulated by state-dependent feedback to accommodate perturbations. Two popular theories have been proposed for the underlying embodiment of phase in the human pattern generator: a time-dependent internal representation or a time-invariant feedback representation (i.e., reflex mechanisms). In either case the neuromuscular system must update or represent the phase of locomotor patterns based on the system state, which can include measurements of hundreds of variables. However, a much simpler representation of phase has emerged in recent designs for legged robots, which control joint patterns as functions of a single monotonic mechanical variable, termed a phase variable. We propose that human joint patterns may similarly depend on a physical phase variable, specifically the heel-to-toe movement of the Center of Pressure under the foot. We found that when the ankle is unexpectedly rotated to a position it would have encountered later in the step, the Center of Pressure also shifts forward to the corresponding later position, and the remaining portion of the gait pattern ensues. This phase shift suggests that the progression of the stance ankle is controlled by a biomechanical phase variable, motivating future investigations of phase variables in human locomotor control.

On the Mechanics of Functional Asymmetry in Bipedal Walking

This paper uses two symmetrical models, the passive compass-gait biped and a five-link 3-D biped, to computationally investigate the cause and function of gait asymmetry. We show that for a range of slope angles during passive 2-D walking and mass distributions during controlled 3-D walking, these models have asymmetric walking patterns between the left and right legs due to the phenomenon of spontaneous symmetry-breaking. In both cases a stable asymmetric family of gaits emerges from a symmetric family of gaits as the total energy increases (e.g., fast speeds). The ground reaction forces of each leg reflect different roles, roughly corresponding to support, propulsion, and motion control as proposed by the hypothesis of functional asymmetry in able-bodied human walking. These results suggest that body mechanics, independent of neurophysiological mechanisms such as leg dominance, may contribute to able-bodied gait asymmetry.

Experimental effective shape control of a powered transfemoral prosthesis

This paper presents the design and experimental implementation of a novel feedback control strategy that regulates effective shape on a powered transfemoral prosthesis. The human effective shape is the effective geometry to which the biological leg conforms - through movement of ground reaction forces and leg joints - during the stance period of gait. Able-bodied humans regulate effective shapes to be invariant across conditions such as heel height, walking speed, and body weight, so this measure has proven to be a very useful tool for the alignment and design of passive prostheses. However, leg joints must be actively controlled to assume different effective shapes that are unique to tasks such as standing, walking, and stair climbing. Using our previous simulation studies as a starting point, we model and control the effective shape as a virtual kinematic constraint on the powered Vanderbilt prosthetic leg with a custom instrumented foot. An able-bodied subject used a by-pass adapter to walk on the controlled leg over ground and over a treadmill. These preliminary experiments demonstrate, for the first time, that effective shape (or virtual constraints in general) can be used to control a powered prosthetic leg.