 Geometric Control and Motion Planning for ThreeDimensional Bipedal Locomotion. Ph.D. Dissertation, Department of Electrical and Computer Engineering, University of Illinois at UrbanaChampaign, 2010.
(Abstract, PDF)
This thesis presents a hierarchical geometric control approach for fast and energetically efficient
bipedal dynamic walking in threedimensional (3D) space to enable motion planning applications
that have previously been limited to inefficient quasistatic walkers. In order to produce exponentially
stable hybrid limit cycles, we exploit system energetics, symmetry, and passivity through the
energyshaping method of controlled geometric reduction. This decouples a subsystem corresponding
to a lowerdimensional robot through a passivitybased feedback transformation of the system
Lagrangian into a special form of controlled Lagrangian with broken symmetry, which corresponds
to an equivalent closedloop Hamiltonian system with uppertriangular form. The first control
term reduces to mechanicallyrealizable passive feedback that establishes a functional momentum
conservation law that controls the "divided" cyclic variables to setpoints or periodic orbits. We
then prove extensive symmetries in the class of open kinematic chains to present the multistage
application of controlled reduction. A reductionbased control law is derived to construct straightahead
and turning gaits for a 4DOF and 5DOF hipped biped in 3D space, based on the existence
of stable hybrid limit cycles in the sagittal planeofmotion. Given such a set of asymptotically
stable gait primitives, a dynamic walker can be controlled as a discretetime switched system that
sequentially composes gait primitives from step to step. We derive "funneling" rules by which a
walking path that is a sequence of these gaits may be stably followed by the robot. The primitive
set generates a tree exploring the action space for feasible walking paths, where each primitive
corresponds to walking along a nominal arc of constant curvature. Therefore, dynamically stable
motion planning for dynamic walkers reduces to a discrete search problem, which we demonstrate
for 3D compassgait bipeds. After reflecting on several connections to human biomechanics, we
propose extensions of this energyshaping control paradigm to robotassisted locomotor therapies.
 Virtual Constraint Control of a Powered Prosthetic Leg: From Simulation to Experiments with Transfemoral Amputees.
R. Gregg T. Lenzi, L. Hargrove, and J. Sensinger. IEEE Transactions on Robotics, 30(6): 14551471, 2014, doi: 10.1109/TRO.2014.2361937.
(Abstract, PDF, Experiment Movie)
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 humaninspired
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.
 Evidence for a TimeInvariant Phase Variable in Human Ankle Control.
R. Gregg, E. Rouse, L. Hargrove, and J. Sensinger. PLoS ONE 9(2):e89163, 2014, doi: 10.1371/journal.pone.0089163.
(Abstract)
Human locomotion is a rhythmic task in which patterns of muscle activity are modulated by statedependent feedback to accommodate perturbations. Two popular theories have been proposed for the underlying embodiment of phase in the human pattern generator: a timedependent internal representation or a timeinvariant 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 heeltotoe 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.
 Towards Biomimetic Virtual Constraint Control of a Powered Prosthetic Leg.
R. Gregg and J. Sensinger. IEEE Transactions on Control System Technology, vol. 22, no. 1, 2014.
(Abstract, PDF)
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 anklefoot 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 exactly enforces the desired effective shape whereas a standard impedance (i.e., proportionalderivative) controller cannot.
This work provides a single, biomimetic control law for the entire singlesupport period during robotassisted locomotion.
 Controlled Reduction with Unactuated Cyclic Variables: Application to 3D Bipedal Walking with Passive Yaw Rotation.
R. Gregg and L. Righetti. IEEE Transactions on Automatic Control, vol. 58, no. 10, 2013.
(Abstract, PDF, StraightAhead Movie, Steering Movie)
This paper shows that viscous damping can shape momentum conservation laws in a manner that stabilizes yaw rotation and enables steering for underactuated 3D walking. We first show that unactuated cyclic variables can be controlled by passively shaped conservation laws given a stabilizing controller in the actuated coordinates. We then exploit this result to realize controlled geometric reduction with multiple unactuated cyclic variables. We apply this underactuated control strategy to a fivelink 3D biped to produce exponentially stable straightahead walking and steering in the presence of passive yawing.
 The Difference Between Mechanical Stiffness and QuasiStiffness in the Context of Biomechanical Modeling. E. Rouse, R. Gregg, L. Hargrove, and J. Sensinger. IEEE Transactions on Biomedical Engineering, vol. 60, no. 2, pp. 562568, 2013.
(Abstract)
The ankle contributes the majority of mechanical power during walking and is a frequently studied joint in biomechanics. Specifically, researchers have extensively investigated the torqueangle relationship for the ankle during dynamic tasks, such as walking and running. The slope of this relationship has been termed the “quasistiffness.” However, over time, researchers have begun to interchange the concepts of quasistiffness and mechanical stiffness. This is an especially important distinction as researchers currently begin to investigate the appropriate control systems for recently developed powered prosthetic legs.
The quasistiffness and mechanical 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 mechanical stiffness and quasistiffness 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 controller’s equilibrium angle position. In both cases, mechanical stiffness values are specified by the controller and the quasistiffness are shown during a single “step.” Both models have widely varying quasistiffness but each have a single mechanical stiffness value. Therefore, from this simple modeling approach, the differences and similarities between these two concepts are elucidated.
 On the Mechanics of Functional Asymmetry in Bipedal Walking.
R. Gregg, Y. Dhaher, A. Degani, and K. Lynch. IEEE Transactions on Biomedical Engineering, vol. 59, no. 5, pp. 13101318, 2012.
(Abstract, PDF)
This paper uses two symmetrical models, the passive compassgait biped and a fivelink 3D biped, to computationally investigate the cause and function of gait asymmetry. We show that for a range of slope angles during passive 2D walking and mass distributions during controlled 3D walking, these models have asymmetric walking patterns between the left and right legs due to the phenomenon of spontaneous symmetrybreaking. 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 ablebodied human walking. These results suggest that body mechanics, independent of neurophysiological mechanisms such as leg dominance, may contribute to ablebodied gait asymmetry.
 Control and Planning of 3D Dynamic Walking with Asymptotically Stable Gait Primitives.
R. Gregg, A. Tilton, S. Candido, T. Bretl, and M. Spong. IEEE Transactions on Robotics, vol. 28, no. 6, pp. 14151423, 2012.
(Abstract, PDF, Movie 1, Movie 2, Movie 3)
In this paper we present a hierarchical framework that enables motion planning for asymptotically stable 3D bipedal walking in the same way that planning is already possible for ZMP 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 a priori rules for sequential composition with other primitives, reducing a highdimensional feedback motion planning problem into a lowdimensional discrete tree search. As a constructive example, we develop this planning framework for the 3D compassgait 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 biped stably from start to goal in threedimensional workspaces with obstacles. We finally show how this framework generalizes to more complex models by planning walking paths for an underactuated fivelink biped.
 ReductionBased Control of ThreeDimensional Bipedal Walking Robots.
R. Gregg and M. Spong. International Journal of Robotics Research, vol. 26, no. 6, pp. 680702, 2010. Reprint.
(Abstract, PDF, Full Text, Supplemental Equations, Straight Walking Movie, Turning Movie)
In this paper we develop the concept of reductionbased control,
which is founded on a controlled form of geometric reduction known
as functional Routhian reduction. We prove a geometric property of
general serialchain robots termed recursive cyclicity, identifying the
inherent robot symmetries that we exploit with the Subrobot Theorem.
This shows that any serialchain robot can be decomposed for arbitrarily
lowerdimensional analysis and control. We apply this method
to construct stable directional threedimensional walking gaits for a
fourdegreeoffreedom hipped bipedal robot. The controlled reduction
decouples the biped’s sagittalplane motion from the yaw and
lean modes, and on the sagittal subsystem we use passivitybased control
to produce known planar limit cycles on flat ground. The unstable
yaw and lean modes are separately controlled to 2periodic orbits
through their shaped momenta. We numerically verify the existence of
stable 2periodic straightwalking limit cycles and demonstrate turning
capabilities for the controlled biped.
 Experimental Effective Shape Control of a Powered Transfemoral Prosthesis.
R. Gregg, T. Lenzi, N. Fey, L. Hargrove, and J. Sensinger. In the 2013 IEEE Int. Conf. Rehabilitation Robotics, Seattle, WA.
(Abstract, PDF, Experiment Movie)
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 conformsthrough movement of ground reaction forces and leg jointsduring the stance period of gait. Ablebodied 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 ablebodied subject used a bypass 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.
 Biomimetic Virtual Constraint Control of a Transfemoral Powered Prosthetic Leg.
R. Gregg and J. Sensinger. In the 2013 American Control Conference, Washington, DC.
(Abstract, PDF, Movie)
This paper presents a novel control strategy for a powered kneeankle prosthesis based on biomimetic virtual constraints. We begin by deriving kinematic constraints for the "effective shape" of the human leg during locomotion. This shape characterizes ankle and knee motion as a function of the Center of Pressure (COP)the point on the foot sole where the ground reaction force is imparted. Since the COP moves monotonically from heel to toe during steady walking, we adopt the COP as the phase variable of an autonomous feedback controller. We show that our kinematic constraints can be enforced virtually by an output linearizing controller that uses only feedback available to sensors onboard a prosthetic leg. This controller produces walking gaits with humanlike knee flexion in simulations of a 6link biped with feet. Hence, both knee and ankle control can be coordinated by one simple control objective: maintaining a constantcurvature effective shape.
 Towards Formal Verification Methods for Robotic LowerLimb Prostheses and Orthoses.
R. Gregg and U. Topcu. In the 2013 Medical Cyber Physical Systems Workshop, CPS Week, Philadelphia, PA.
(Abstract, PDF)
Populations including lowerlimb amputees, stroke survivors, and the elderly have an increased risk for falls, which significantly limits mobility and quality of life. Recently developed robotic prostheses and orthoses are capable of actively assisting mobility in these populations. New risk management and design control tools are needed to meet the evolving regulatory demands for these increasingly complex systems, which can impart large torques and forces that could potentially harm the user. This paper attempts to introduce formal verification methods to this rapidly developing field by discussing some topical problems, the motivating example of a prosthetic leg, and potential verification metrics that could be used to certify the safety of these robotic medical devices in a pragmatic and costeffective manner.
 Stable OpenLoop Brachiation on a Vertical Wall.
N. Rosa, A. Barber, R. Gregg, K. Lynch. In the 2012 IEEE International Conference on Robotics and Automation, St. Paul, MN.
(Abstract)
This paper presents a hybrid mechanical model for the Gibbot, a robot that dynamically climbs along a flat vertical wall in a manner analogous to gibbons swinging between branches in the forest canopy. We focus on one particular style of gait, continuouscontact brachiation, which always has one handhold in contact with the wall. We use zero cost, unstable solutions corresponding to horizontal brachiation, originally found by Gomes and Ruina, as templates to generate openloop stable gaits in arbitrary directions. The first case considered is passive brachiation down a shallow slope, roughly corresponding to upsidedown locomotion of the wellstudied compassgait biped. We then consider underactuated brachiation with a constant forcing term at the elbow to produce openloop stable descending and ascending gaits.
 Functional Asymmetry in a FiveLink 3D Bipedal Walker.
R. Gregg, Y. Dhaher, and K. Lynch. In the 2011 International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), Boston, MA.
(Abstract, PDF)
This paper uses a symmetrical fivelink 3D biped model to computationally investigate the cause, function, and benefit of gait asymmetry. We show that for a range of mass distributions, this model has asymmetric walking patterns between the left and right legs, which is due to a phenomenon known as perioddoubling bifurcation. 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 human walking. These results suggest that natural mechanics could be responsible for asymmetry in ablebodied walking, rather than neurophysiological mechanisms such as leg dominance.
 Controlled Reduction of a FiveLink 3D Biped with Unactuated Yaw. R. Gregg. In the 2011 IEEE Conference on Decision and Control, Orlando, FL.
(Abstract, PDF, Movie)
This paper presents a formulation of controlled geometric reduction with one degree of underactuation for mechanical systems with an unactuated cyclic variable subject to passive damping. We show that the first control term in the fully actuated case reduces to passive jointvelocity feedback, which can be equivalently provided by viscous friction. The underactuated control strategy is applied to a fivelink 3D biped with a hip, torso, knees, and unactuated yaw at the foot contact point. We show asymptotically stable walking in the presence of passive yawing for realistic friction coefficients.
 The Simplest Parkour Model: Experimental Validation and Stability Analysis.
A. Long, R. Gregg, and K. Lynch. In the 2011 International Conference on Climbing and Walking Robots, Paris, France.
(Abstract, Experiment Video)
We describe and experimentally validate the Simplest Parkour Model (SPM) for the ParkourBot, a planar dynamic climbing robot equipped with two springy BowLegs. By controlling the leg angles and injected energy at impact, the ParkourBot is capable of climbing up and down in a rigid chute on an inclined air table. The SPM consists of a point mass and two massless legs. The legs are assumed to be infinitely stiff, resulting in an instantaneous stance phase and a closedform solution of the hybrid dynamics. In this paper, we show that the SPM is a good predictor of the actual experimental behavior. Using the SPM we compute the fixed points, stability and basins of attraction of period1 limit cycles.
 The Basic Mechanics of Bipedal Walking Lead to Asymmetric Behavior.
R. Gregg, A. Degani, Y. Dhaher, and K. Lynch. In the 2011 IEEE International Conference on Rehabilitation Robotics, Zurich, Switzerland.
(Abstract, PDF, Poster, Fast Forward Presentation)
This paper computationally investigates whether gait asymmetries can be attributed in part to basic bipedal mechanics independent of motor control. Using a symmetrical rigidbody model known as the compassgait biped, we show that changes in environmental or physiological parameters can facilitate asymmetry in gait kinetics at fast walking speeds. In the environmental case, the asymmetric family of highspeed gaits is in fact more stable than the symmetric family of lowspeed gaits. These simulations suggest that lower extremity mechanics might play a direct role in functional and pathological asymmetries reported in human walking, where velocity may be a common variable in the emergence and growth of asymmetry.
 A Control Theoretic Approach to RobotAssisted Locomotor Therapy.
R. Gregg, T. Bretl, and M. Spong. In the 2010 IEEE Conference on Decision and Control, Atlanta, GA. Reprint.
(Abstract, PDF, Experiment 1 Video, Experiment 2 Video)
This paper proposes a control theoretic strategy
for human walking gait assistance based on underactuated
potential energy shaping. We design a simple control law that
lessens the perceived weight of the patient’s center of mass
through a robotic anklefoot orthosis (AFO) with one actuated
degreeoffreedom. We then adopt a passive “compassgait”
bipedal walker as an implicit model of human locomotor
behavior, which we simulate to draw beneficial implications
for rehabilitation such as energy regulation, improved stability,
and progressive training by Lyapunov funneling. Given current
challenges in developing effective robotassisted locomotor therapies,
this paper offers a novel systematic approach to control
strategy design for gait training and athome assistance.
 Constrained Accelerations for Controlled Geometric Reduction: SagittalPlane Decoupling in Bipedal Locomotion.
R. Gregg, L. Righetti, J. Buchli, S. Schaal. In the 2010 IEEE International Conference on Humanoid Robots, Nashville, TN.
(Abstract, PDF, Movie)
Energyshaping control methods have produced
strong theoretical results for asymptotically stable 3D bipedal
dynamic walking in the literature. In particular, geometric
controlled reduction exploits robot symmetries to control momentum
conservation laws that decouple the sagittalplane
dynamics, which are easier to stabilize. However, the associated
control laws require highdimensional matrix inverses multiplied
with complicated energyshaping terms, often making
these control theories difficult to apply to highlyredundant
humanoid robots. This paper presents a first step towards
the application of energyshaping methods on real robots by
casting controlled reduction into a framework of constrained
accelerations for inverse dynamics control. By representing
momentum conservation laws as constraints in acceleration
space, we construct a general expression for desired joint
accelerations that render the constraint surface invariant. By
appropriately choosing an orthogonal projection, we show that
the unconstrained (reduced) dynamics are decoupled from the
constrained dynamics. Any accelerationbased controller can
then be used to stabilize this planar subsystem, including
passivitybased methods. The resulting control law is surprisingly
simple and represents a practical way to employ control
theoretic stability results in robotic platforms. Simulated walking
of a 3D compassgait biped show correspondence between
the new and original controllers, and simulated motions of a
16DOF humanoid demonstrate the applicability of this method.
 Asymptotically Stable Gait Primitives for Planning Dynamic Bipedal Locomotion in Three Dimensions.
R. Gregg, T. Bretl, and M. Spong. In the 2010 IEEE International Conference on Robotics and Automation, Anchorage, AK.
(Abstract, PDF, Slides, Random Walk Movie, Planned Walking Movie)
This paper applies geometric reductionbased control
to derive a set of asymptotically stable dynamic walking
gaits for a 3D bipedal robot, each corresponding to walking
along a nominal arc of constant curvature for a fixed number
of steps. We show that any such set of asymptotically stable gait
primitives may be composed in arbitrary order without causing
the robot to fall, so any walking path that is a sequence of
these gaits may be followed by the robot. This result enables
motion planning for bipedal dynamic walkers, which are fast
and energetically efficient, in a similar manner to what is already
possible for biped locomotion based on Zero Moment Point
(ZMP) equilibrium constraints.
 ReductionBased Control of Branched Chains: Application to ThreeDimensional Bipedal Torso Robots.
R. Gregg and M. Spong. In the 2009 IEEE Conference on Decision and Control, Shanghai, China.
(Abstract, PDF, Slides, Supplemental Equations, Straight Walking Movie, Turning Movie)
This paper revisits the concept of controlled reduction,
a symmetrybased method of decomposing the control of
mechanical systems into lowerdimensional problems. Founded
on a geometric property of serial kinematic chains termed
recursive cyclicity, the present work extends this property
to branched chains by defining the minimal irreducible tree
structure. We then generalize the Subrobot Theorem to branched
chains, showing that we can use control to reduce any robot’s
dimensionality to that of its irreducible tree structure. This
method is applied to a 5DOF bipedal robot with a hip and
torso – we construct stable straightahead and turning gaits in
3D space based on known sagittalplane limit cycles.
 Bringing the CompassGait Bipedal Walker to Three Dimensions.
R. Gregg and M. Spong. In the 2009 IEEE International Conference on Intelligent Robots and Systems, St. Louis, MO.
(Abstract, PDF, Slides, 4DOF Eqns, 5DOF Eqns, 4DOF Movie, 5DOF Movie)
The planar compassgait biped has been extensively
studied in the dynamic walking community, motivated by
the gravitybased pendular efficiencies of human walking. These
results can be extended to three dimensions using controlled
geometric reduction for openchain robots, by which stable 3
D walking gaits are built from known sagittalplane limit cycles.
We apply this method to the standard and withtorso compassgait
(hipless) bipeds, showing straightahead walking gaits (i.e.,
stable 1step periodic limit cycles) as well as hstep turning in
full circles (i.e., stable hperiodic limit cycles). These constantcurvature
maneuvers are composed of stable 1periodic turning
gaits modulo heading change, demonstrating two types of gaits
for directional dynamic walking in three dimensions
 Reductionbased Control with Application to ThreeDimensional Bipedal Walking Robots.
R. Gregg and M. Spong. In the 2008 American Control Conference, Seattle, WA. Reprint.
(Abstract, PDF, Supplemental Equations, Straight Walking Movie, Turning Movie)
This paper develops the concept of reductionbased
control, which is founded on a controlled form of
geometric reduction known as functional Routhian reduction. We
introduce a geometric property of general serialchain robots
termed recursive cyclicity, leading to our presentation of the
Subrobot Theorem. This shows that reductionbased control can
arbitrarily reduce the dimensionality of any serialchain robot,
so that it may be controlled as a simpler “subrobot” while
separately controlling the divided coordinates through their
conserved momenta. This method is applied to construct stable
directional 3D walking gaits for a 4d.o.f. hipped bipedal robot.
The walker’s sagittalplane subsystem can be decoupled from
its yaw and lean modes, and on this planar subsystem we
use passivitybased control to construct limit cycles on flat
ground. Due to the controlled reduction, the unstable yaw and
lean modes are separately controlled to 2periodic orbits. We
numerically verify the existence of stable 2periodic limit cycles
and demonstrate turning capabilities for the controlled biped.
 A Geometric Approach to ThreeDimensional Hipped Bipedal Robotic Walking.
A. Ames, R. Gregg, and M. Spong. In the 2007 IEEE Conference on Decision and Control, New Orleans, LA.
(Abstract, PDF)
This paper presents a control law that results in
stable walking for a threedimensional 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 threedimensional 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.
 Stably Extending TwoDimensional Bipedal Robotic Walking to Three Dimensions.
A. Ames and R. Gregg. In the 2007 American Control Conference, New York City, New York.
(Abstract, PDF)
In this paper we develop a feedback control law
that results in stable walking gaits on flat ground for a threedimensional
bipedal robotic walker given stable walking gaits
for a twodimensional 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 twodimensional
bipedal robot walking on flat ground. These are related to
walking gaits for a threedimensional (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.
 Towards the Geometric Reduction of Controlled 3D Bipedal Walking Robots.
A. Ames, R. Gregg, E. Wendel, S. Sastry. In the IFAC 3rd Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Nagoya, Japan.
(Abstract)
The purpose of this paper is to apply methods from geometric mechanics
to the analysis and control of bipedal robotic walkers. We begin by introducing a
generalization of Routhian reduction, functional Routhian Reduction, which allows
for the conserved quantities to be functions of the cyclic variables rather than
constants. Since bipedal robotic walkers are naturally modeled as hybrid systems,
which are inherently nonsmooth, in order to apply this framework to these systems
it is necessary to first extend functional Routhian reduction to a hybrid setting.We
apply this extension, along with potential shaping and controlled symmetries, to
derive a feedback control law that provably results in walking gaits on flat ground
for a threedimensional bipedal walker given walking gaits in twodimensions.
 Is there Life after Zeno? Taking Executions past the Breaking (Zeno) Point.
A. Ames, H. Zheng, R. Gregg, S. Sastry. In the 2006 American Control Conference, Minneapolis, MN.
(Abstract)
Understanding Zeno phenomena plays an important
role in understanding hybrid systems. A natural—and
intriguing—question to ask is: what happens after a Zeno
point? Inspired by the construction of (Filippov, 1999), we propose a method
for extending Zeno executions past a Zeno point for a class
of hybrid systems: Lagrangian hybrid systems. We argue that
after the Zeno point is reached, the hybrid system should switch
to a holonomically constrained dynamical system, where the
holonomic constraints are based on the unilateral constraints
on the configuration space that originally defined the hybrid
system. These principles are substantiated with a series of
examples.
 Challenges for Control Research: Control of Powered Prosthetic Legs.
R. Gregg, L. Hargrove, and J. Sensinger. The Impact of Control Technology, 2nd ed., T. Samad and A.M. Annaswamy (eds.), IEEE Control Systems Society, 2014, available at www.ieeecss.org. (Abstract, PDF)
Amputee locomotion is slower, less stable,
and requires more metabolic energy than
ablebodied locomotion. Lowerlimb amputees
fall more frequently than ablebodied individuals
and often struggle to navigate inclines such
as ramps, hills, and especially stairs. These
challenges can be attributed largely to the use
of mechanically passive prosthetic legs, which
do not contribute positively to the energetics
of gait, as do the muscles of the biological
leg. Powered (or robotic) prosthetic legs could
significantly improve mobility and quality of
life for lowerlimb amputees (including nearly
one million Americans), but control challenges
limit the performance and clinical feasibility
of today's devices
 On the Geometric Reduction of Controlled ThreeDimensional Bipedal Robotic Walkers.
A. Ames, R. Gregg, E. Wendel, and S. Sastry. Lagrangian and Hamiltonian Methods for Nonlinear Control, ser. Lecture Notes in Control and Information Sciences, vol. 366. Nagoya, Japan: Springer, 2007, pp. 183196.
(Abstract, Full Text)
The purpose of this paper is to apply methods from geometric mechanics to the analysis and control of bipedal robotic walkers. We begin by introducing a generalization of Routhian reduction, functional Routhian Reduction, which allows for the conserved quantities to be functions of the cyclic variables rather than constants. Since bipedal robotic walkers are naturally modeled as hybrid systems, which are inherently nonsmooth, in order to apply this framework to these systems it is necessary to first extend functional Routhian reduction to a hybrid setting. We apply this extension, along with potential shaping and controlled symmetries, to derive a feedback control law that provably results in walking gaits on flat ground for a threedimensional bipedal walker given walking gaits in two dimensions.
© 2009 Robert D. Gregg, IV
