Real-Time Planning with Primitives for Dynamic Walking over Uneven Terrain

We present an algorithm for receding-horizon motion planning using a finite family of motion primitives for underactuated dynamic walking over uneven terrain. The motion primitives are defined as virtual holonomic constraints, and the special structure of underactuated mechanical systems operating subject to virtual constraints is used to construct closed-form solutions and a special binary search tree that dramatically speed up motion planning. We propose a greedy depth-first search and discuss improvement using energy-based heuristics. The resulting algorithm can plan several footsteps ahead in a fraction of a second for both the compass-gait walker and a planar 7-Degree-of-freedom/five-link walker.Comment: Conference submissio

Trajectory generation for multi-contact momentum-control

Simplified models of the dynamics such as the linear inverted pendulum model (LIPM) have proven to perform well for biped walking on flat ground. However, for more complex tasks the assumptions of these models can become limiting. For example, the LIPM does not allow for the control of contact forces independently, is limited to co-planar contacts and assumes that the angular momentum is zero. In this paper, we propose to use the full momentum equations of a humanoid robot in a trajectory optimization framework to plan its center of mass, linear and angular momentum trajectories. The model also allows for planning desired contact forces for each end-effector in arbitrary contact locations. We extend our previous results on LQR design for momentum control by computing the (linearized) optimal momentum feedback law in a receding horizon fashion. The resulting desired momentum and the associated feedback law are then used in a hierarchical whole body control approach. Simulation experiments show that the approach is computationally fast and is able to generate plans for locomotion on complex terrains while demonstrating good tracking performance for the full humanoid control

Dynamically Stable 3D Quadrupedal Walking with Multi-Domain Hybrid System Models and Virtual Constraint Controllers

Hybrid systems theory has become a powerful approach for designing feedback controllers that achieve dynamically stable bipedal locomotion, both formally and in practice. This paper presents an analytical framework 1) to address multi-domain hybrid models of quadruped robots with high degrees of freedom, and 2) to systematically design nonlinear controllers that asymptotically stabilize periodic orbits of these sophisticated models. A family of parameterized virtual constraint controllers is proposed for continuous-time domains of quadruped locomotion to regulate holonomic and nonholonomic outputs. The properties of the Poincare return map for the full-order and closed-loop hybrid system are studied to investigate the asymptotic stabilization problem of dynamic gaits. An iterative optimization algorithm involving linear and bilinear matrix inequalities is then employed to choose stabilizing virtual constraint parameters. The paper numerically evaluates the analytical results on a simulation model of an advanced 3D quadruped robot, called GR Vision 60, with 36 state variables and 12 control inputs. An optimal amble gait of the robot is designed utilizing the FROST toolkit. The power of the analytical framework is finally illustrated through designing a set of stabilizing virtual constraint controllers with 180 controller parameters.Comment: American Control Conference 201

Development of a Hybrid Powered 2D Biped Walking Machine Designed for Rough Terrain Locomotion

Biped robots hold promise as terrestrial explorers because they require a single discrete foothold to place their next step. However, biped robots are multi-input multi-output dynamically unstable machines. This makes walking on rough terrain difficult at best. Progress has been made with non-periodic rough terrain like stairs or inclines with fully active walking machines. Terrain that requires the walker to change its gait pattern from a standard walk is still problematic. Most walking machines have difficulty detecting or responding to the small perturbations induced by this type of terrain. These small perturbations can lead to unstable gait cycles and possibly a fall. The Intelligent Systems and Automation Lab at the University of Kansas has built a three legged 2D biped walking machine to be used as a test stand for studying rough terrain walking. The specific aim of this research is to investigate how biped walkers can best maintain walking stability when acted upon by small perturbations caused by periodic rough terrain. The first walking machine prototype, referred to as Jaywalker has two main custom actuation systems. The first is the hip ratchet system. It allows the walker to have either a passive or active hip swing. The second is the hybrid parallel ankle actuator. This new actuator uses a pneumatic ram and stepper motor in parallel to produce an easily controlled high torque output. In open loop control it has less than a 1° tracking error and 0.065 RPM velocity error compared to a standard stepper motor. Step testing was conducted using the Jaywalker, with a passive hip, to determine if a walker with significant leg mass could walk without full body actuation. The results of testing show the Jaywalker is ultimately not capable of walking with a passive hip. However, the walking motion is fine until the terminal stance phase. At this point the legs fall quickly towards the ground as the knee extends the shank. This quick step phenomenon is caused by increased speeds and forces about the leg and hip caused by the extension of the shank. This issue can be overcome by fully actuating the hip, or by adding counterbalances to the legs about the hip

A Comparative Study on the L-1 Optimal Event-Based Method for Biped Walking on Rough Terrains

This paper is concerned with a comparative study of biped walking on rough terrains. Given a bipedal robot capable of walking on a flat ground with periodic behavior, whose motion can be described by a limit cycle with the Poincare map, we consider whether the robot remains stable on rough terrain, in which geometrical uncertainties of the terrain are assumed to be persistent and bounded. More precisely, the l(infinity)-induced norm is defined on the Poincare map and taken as a performance measure evaluating a robot walking with the bounded persistent uncertainties. To minimize the performance measure and achieve an optimal walking performance, we further provide a systematic controller design scheme consisting of a inner-loop continuous-time controller and a outer-loop event-based controller, in which the latter is described as a sort of the l(1) optimal controller. Finally, the validity as well as the effectiveness of our proposed methods in biped walking on a rough terrain are demonstrated through simulation studies.11Yscopu

Push recovery with stepping strategy based on time-projection control

In this paper, we present a simple control framework for on-line push recovery with dynamic stepping properties. Due to relatively heavy legs in our robot, we need to take swing dynamics into account and thus use a linear model called 3LP which is composed of three pendulums to simulate swing and torso dynamics. Based on 3LP equations, we formulate discrete LQR controllers and use a particular time-projection method to adjust the next footstep location on-line during the motion continuously. This adjustment, which is found based on both pelvis and swing foot tracking errors, naturally takes the swing dynamics into account. Suggested adjustments are added to the Cartesian 3LP gaits and converted to joint-space trajectories through inverse kinematics. Fixed and adaptive foot lift strategies also ensure enough ground clearance in perturbed walking conditions. The proposed structure is robust, yet uses very simple state estimation and basic position tracking. We rely on the physical series elastic actuators to absorb impacts while introducing simple laws to compensate their tracking bias. Extensive experiments demonstrate the functionality of different control blocks and prove the effectiveness of time-projection in extreme push recovery scenarios. We also show self-produced and emergent walking gaits when the robot is subject to continuous dragging forces. These gaits feature dynamic walking robustness due to relatively soft springs in the ankles and avoiding any Zero Moment Point (ZMP) control in our proposed architecture.Comment: 20 pages journal pape

3LP: a linear 3D-walking model including torso and swing dynamics

In this paper, we present a new model of biped locomotion which is composed of three linear pendulums (one per leg and one for the whole upper body) to describe stance, swing and torso dynamics. In addition to double support, this model has different actuation possibilities in the swing hip and stance ankle which could be widely used to produce different walking gaits. Without the need for numerical time-integration, closed-form solutions help finding periodic gaits which could be simply scaled in certain dimensions to modulate the motion online. Thanks to linearity properties, the proposed model can provide a computationally fast platform for model predictive controllers to predict the future and consider meaningful inequality constraints to ensure feasibility of the motion. Such property is coming from describing dynamics with joint torques directly and therefore, reflecting hardware limitations more precisely, even in the very abstract high level template space. The proposed model produces human-like torque and ground reaction force profiles and thus, compared to point-mass models, it is more promising for precise control of humanoid robots. Despite being linear and lacking many other features of human walking like CoM excursion, knee flexion and ground clearance, we show that the proposed model can predict one of the main optimality trends in human walking, i.e. nonlinear speed-frequency relationship. In this paper, we mainly focus on describing the model and its capabilities, comparing it with human data and calculating optimal human gait variables. Setting up control problems and advanced biomechanical analysis still remain for future works.Comment: Journal paper under revie