Modeling and Optimized Gait Planning of Biped Robots with Different Leg Mechanisms

This research focuses on modeling and gait generation optimization of four different real biped models that include practical extended models of the theoretical SLIP and compass gait as a novelty of the work. The first model is kneed Biped model without spring that is a 5-rigid-link robot with four actuators in its hip and knees. The second model, kneed biped model with springs in shins is very similar to the first model, but its shins have linear springs. The 3rd model is a semi-telescopic springy biped model and the 4th model is a semi-compass gait with kneed swing leg. Optimization parameters of their walking gait, objective functions and constraints are presented and successive stages of optimization are completed to find the optimal gaits. The efficiency of the gaits and required motor torques for the optimal gait of each model are illustrated

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

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 Poincaré 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 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

Robust and Economical Bipedal Locomotion

For bipedal robots to gain widespread use, significant improvements must be made in their energetic economy and robustness against falling. An increase in economy can increase their functional range, while a reduction in the rate of falling can reduce the need for human intervention. This dissertation explores novel concepts that improve these two goals in a fundamental manner. By centering on core ideas instead of direct application, these concepts are aimed at influencing a wide range of current and future legged robots. The presented work can be broken into five major contributions. The first extends our understanding of the energetic economy of series elastic walking robots. This investigation uses trajectory optimization to find energy-miminizing periodic motions for a realistic model of the walking robot RAMone. The energetically optimal motions for this model are shown to closely resemble human walking at low speeds, and as the speed increases, the motions switch abruptly to those resembling human running. The second contribution explores the energetic economy of the real robot RAMone. Here the model used in the previous investigation is shown to closely match reality. In addition, this investigation demonstrates a concrete example of a trade-off between energetic economy and robustness. The third contribution takes a step towards addressing this trade-off by deriving a robot constraint that guarantees safety against falling. Such a constraint can be used to remove considerations of robustness while conducting future investigations into economical robot motions. The approach is demonstrated using a simple compass-gait style walking model. The fourth contribution extends this safety constraint towards higher-dimensional walking models, using a combination of hybrid zero dynamics and sums-of-squares analysis. This is demonstrated by safely modifying the pitch of a 10 dimensional Rabbit model walking over flat terrain. The final contribution pushes the safety guarantee towards a broader set of walking behaviours, including rough terrain walking. Throughout this work, a range of models are used to reason about the economy and robustness of walking robots. These model-based methods allow control designers to move away from heuristics and tuning, and towards generalizable and reliable controllers. This is vital for walking robots to push further into the wild.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/153459/1/nilssmit_1.pd

Feedback Control of a Bipedal Walker and Runner with Compliance.

This dissertation contributes to the theoretical foundations of robotic bipedal locomotion and advances the experimental state of the art as well. On the theoretical side, a mathematical formalism for designing provably stable, walking and running gaits in bipedal robots with compliance is presented. A key contribution is a novel method of force control in robots with compliance. The theoretical work is validated experimentally on MABEL, a planar bipedal testbed that 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. The methods of virtual constraints and hybrid zero dynamics, originally developed for rigid robots with a single degree of underactuation, are extended and applied to MABEL, which has 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. A second time-invariant feedback controller is designed such that the closed-loop system not only respects the natural compliance of the open-loop system, but also enables active force control within the compliant hybrid zero dynamics and results in exponentially stable running gaits. Several experiments are presented that highlight different aspects of MABEL and the feedback design method, ranging from basic elements such as stable walking, robustness under perturbations, energy efficient walking to a bipedal robot walking speed record of 1.5 m/s (3.4 mph), stable running with passive feet and with point feet. On MABEL, the full hybrid zero dynamics controller is implemented and was instrumental in achieving rapid walking and running, leading upto a kneed bipedal running speed record of 3.06 m/s (6.8 mph).Ph.D.Electrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/89801/1/koushils_1.pd

Passive dynamic walking with knees : a point foot model

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2007.Includes bibliographical references (p. 57-59).In this thesis, a hybrid model for a passive 2D walker with knees and point feet is presented. The step cycle of the model has two phases of continuous dynamics: one with an unlocked knee configuration and a second one with a locked knee configuration. These stages are modeled as three-link and two-link pendulums correspondingly. The model switches between the two stages at knee-strike and heel-strike, which are both discrete events modeled as instantaneous inelastic collisions. The dynamics of this model were fully derived analytically. Furthermore, a stable gait was found given a set of physical parameters and initial conditions. A basic stability analysis of this stable limit cycle was performed around the fixed point of the Poincar6 return map examined right after heel-strike. This thesis also presents the design and construction of a planar robot based on this kneed walker model, as well as a careful examination of its correspondence to the motion predicted by the model simulation. The goal is to be able to study the nonlinear dynamics of simplified dynamic models which are also physically realizable, in order to build robots based on them in a more rigorous and reproducible manner. The work presented here aims to bridge the gap between existing theoretical models and successful experiments in passive dynamic walking.by Vanessa F. Hsu Chen.M.Eng

Feedback Control Design for MARLO, a 3D-Bipedal Robot.

This work develops feedback controllers for bipedal walking in 3D on level ground, both in simulation and experimentally. MARLO is a new robot that has been designed for the study of 3D-bipedal locomotion, with the aim of combining energy efficiency, speed, and robustness with respect to natural terrain variations in a single platform. The robot is highly underactuated, having six actuators and, in single support, 13 degrees of freedom. Its sagittal plane dynamics are designed to embody the spring loaded inverted pendulum (SLIP), which has been shown to provide a dynamic model of the body center of mass during steady running gaits in a wide diversity of terrestrial animals. A detailed dynamic model is used to optimize walking gaits with respect to the cost of mechanical transport (cmt), a dimensionless measure of energetic efficiency. A feedback controller is designed that balances the robot during the quiet standing mode, and another feedback policy is developed such that the robot can take a transition step from quiet standing to walking. A feedback controller is designed that stabilizes steady-state 3D walking gaits, despite the high degree of underactuation of the robot. These controllers are combined through a state machine that handles switching among the three controllers controllers. In experiments on planarized (2D) and untethered (3D) versions of the robot with point feet and passive feet (prosthetic feet) walking over flat ground or on a ramp with a shallow slope, the adaptability of the designed controller to the environment (planar or untethered, flat ground or ramp), and to the morphology of the robot (point feet or passive feet), is demonstrated. In experiments on a planarized version of the robot with passive feet, the controller yielded stable walking after starting from quiet standing, autonomously and without any intervention from the operator. In experiments on an untethered (3D) version of the robot, the controller was able to balance the robot over flat ground or on a shallow ramp during the quiet standing mode. In addition, the controller yielded six-untethered ``human-like'' steps after starting from quiet standing, autonomously without any intervention from the operator.PhDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/102339/1/aramez_1.pd

Energy Shaping of Mechanical Systems via Control Lyapunov Functions with Applications to Bipedal Locomotion

This dissertation presents a method which attempts to improve the stability properties of periodic orbits in hybrid dynamical systems by shaping the energy. By taking advantage of conservation of energy and the existence of invariant level sets of a conserved quantity of energy corresponding to periodic orbits, energy shaping drives a system to a desired level set. This energy shaping method is similar to existing methods but improves upon them by utilizing control Lyapunov functions, allowing for formal results on stability. The main theoretical result, Theorem 1, states that, given an exponentially-stable limit cycle in a hybrid dynamical system, application of the presented energy shaping controller results in a closed-loop system which is exponentially stable. The method can be applied to a wide class of problems including bipedal locomotion; because the optimization problem can be formulated as a quadratic program operating on a convex set, existing methods can be used to rapidly obtain the optimal solution. As illustrated through numerical simulations, this method turns out to be useful in practice, taking an existing behavior which corresponds to a periodic orbit of a hybrid system, such as steady state locomotion, and providing an improvement in convergence properties and robustness with respect to perturbations in initial conditions without destabilizing the behavior. The method is even shown to work on complex multi-domain hybrid systems; an example is provided of bipedal locomotion for a robot with non-trivial foot contact which results in a multi-phase gait