Reachability and Real-Time Actuation Strategies for the Active SLIP Model

Running and hopping follow similar patterns for different animals, independent of the number of legs employed. An aerial phase alternates with a ground contact phase, during which the center of mass moves as if a spring were compressed and then extended to recover stored elastic energy. Hence, consisting of a point mass mounted on a massless spring leg, the Spring Loaded Inverted Pendulum (SLIP) is a prevalent model for analyzing running and hopping. In this work we consider an actuated version of the SLIP model, with a series elastic actuator added to the leg, serving the purposes of adding/removing energy to/from the system and of modifying dynamics during stance, toward achieving non-steady locomotion on varying terrain. While the SLIP model has been a topic of research in legged locomotion for several decades, studies on the effect of actuation on the system's behavior are still not complete.The goal of this thesis is to explore how a series elastic actuator applied to the SLIP model's leg can change the system's dynamics. This, in turn, enables a variety of long-term planning strategies for using limited footholds and design non-steady gaits while simultaneously recovering from unexpected perturbations, both sensorial and due to a limited knowledge of the terrain profile.We principally investigate how, through actuation, we can solve partially or completely the system's equations of motion, to enforce a desired trajectory and reach a desired state. We also determine the reachable state space of the model using several different actuation strategies, investigating the variation of the reachable set with respect to particular actuator motions and providing relationships between local actuator displacements throughout stance and location of the reached apex state. We then propose a control strategy based on graphical and numerical studies of the reachability space to drive the system to a desired state, with the ability to reduce the effects of sensing errors and disturbances happening at landing as well as during ground contact

Accurate Step Length Control Strategies for Underactuated and Realistic Series Elastic Actuated Hoppers via High Order PFL

Among the different types of legged robots, hopping robots, aka hoppers, can be classified as one of the simplest sufficient models that capture the important features encompassed in dynamic locomotion: underactuation, compliance, and hybrid features. There is an abundance of work regarding the implementation of highly simplified hopper models, the prevalent example being the spring loaded inverted pendulum (SLIP) model, with the hopes of extracting fundamental control ideas for running and hopping robots. However, real world systems cannot be fully described by such simple models, as real actuators have their own dynamics including additional inertia and non-linear frictional losses. Additionally, implementing feedback control for hopping systems with significant amounts of compliance is difficult as the input variable does not instantaneously change the leg length acceleration. The current state-of-the-art of step length control in the presence of non-steady state motions required for foothold placement is not precise enough for operation in the real world. Therefore, an important step towards demonstrating high controllability and robustness to real-world elements is in providing accurate higher order models of real-world hopper dynamics, along with compatible control strategies. Our modeling work is based on a series-elastic actuated (SEA) hopping robot prototype constructed by our lab group, and we provide verifying hardware results that high order partial feedback linearization (HOPFL) can be implemented directly on the leg state of the robot. Using HOPFL, we investigate two paths of compatible trajectory generation that can accomplish desirable tasks such as precise foothold planning. We investigate the practicality of using SLIP-based trajectory generation techniques on more realistic hopping robots, and show that by implementing HOPFL directly on the robot's leg, we can make use of computationally fast SLIP-based approximations, account for non-trivial pitch dynamics, and improve the state-of-the-art of precision step length control for SEA hoppers. We also consider control strategies towards hoppers for which SLIP-based trajectories may not be compatible, by planning all ground reaction force vector (GRF) components during the stance phase concurrently, using a lower order and very general model to construct trajectories for the system's center of mass (CoM), and maintain body stability by controlling the orientation of the GRF directly. While not purely analytical as our SLIP-based approaches, this method is general enough to work on a variety of hopping robots that are not necessarily kinematically structured resembling the classical SLIP model

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

On the periodic gait stability of a multi-actuated spring-mass hopper model via partial feedback linearization

Spring-loaded inverted pendulum (SLIP) template (and its various derivatives) could be considered as the mostly used and widely accepted models for describing legged locomotion. Despite their simple nature, as being a simple spring-mass model in dynamics perspective, the SLIP model and its derivatives are formulated as restricted three-body problem, whose non-integrability has been proved long before. Thus, researchers proceed with approximate analytical solutions or use partial feedback linearization when numerical integration is not preferred in their analysis. The key contributions of this paper can be divided into two parts. First, we propose a dissipative SLIP model, which we call as multi-actuated dissipative SLIP (MD-SLIP), with two extended actuators: one linear actuator attached serially to the leg spring and one rotary actuator attached to hip. The second contribution of this paper is a partial feedback linearization strategy by which we can cancel some nonlinear dynamics of the proposed model and obtain exact analytical solution for the equations of motion. This allows us to investigate stability characteristics of the hopping gait obtained from the MD-SLIP model. We illustrate the applicability of our solutions with open-loop and closed-loop hopping performances on rough terrain simulations. © 2017, Springer Science+Business Media Dordrecht

Control Implementation of Dynamic Locomotion on Compliant, Underactuated, Force-Controlled Legged Robots with Non-Anthropomorphic Design

The control of locomotion on legged robots traditionally involves a robot that takes a standard legged form, such as the anthropomorphic humanoid, the dog-like quadruped, or the bird-like biped. Additionally, these systems will often be actuated with position-controlled servos or series-elastic actuators that are connected through rigid links. This work investigates the control implementation of dynamic, force-controlled locomotion on a family of legged systems that significantly deviate from these classic paradigms by incorporating modern, state-of-the-art proprioceptive actuators on uniquely configured compliant legs that do not closely resemble those found in nature. The results of this work can be used to better inform how to implement controllers on legged systems without stiff, position-controlled actuators, and also provide insight on how intelligently designed mechanical features can potentially simplify the control of complex, nonlinear dynamical systems like legged robots. To this end, this work presents the approach to control for a family of non-anthropomorphic bipedal robotic systems which are developed both in simulation and with physical hardware. The first is the Non-Anthropomorphic Biped, Version 1 (NABi-1) that features position-controlled joints along with a compliant foot element on a minimally actuated leg, and is controlled using simple open-loop trajectories based on the Zero Moment Point. The second system is the second version of the non-anthropomorphic biped (NABi-2) which utilizes the proprioceptive Back-drivable Electromagnetic Actuator for Robotics (BEAR) modules for actuation and fully realizes feedback-based force controlled locomotion. These systems are used to highlight both the strengths and weaknesses of utilizing proprioceptive actuation in systems, and suggest the tradeoffs that are made when using force control for dynamic locomotion. These systems also present case studies for different approaches to system design when it comes to bipedal legged robots

Hopping, Landing, and Balancing with Springs

This work investigates the interaction of a planar double pendulum robot and springs, where the lower body (the leg) has been modified to include a spring-loaded passive prismatic joint. The thesis explores the mechanical advantage of adding a spring to the robot in hopping, landing, and balancing activities by formulating the motion problem as a boundary value problem; and also provides a control strategy for such scenarios. It also analyses the robustness of the developed controller to uncertain spring parameters, and an observer solution is provided to estimate these parameters while the robot is performing a tracking task. Finally, it shows a study of how well IMUs perform in bouncing conditions, which is critical for the proper operation of a hopping robot or a running-legged one