From Bipedal Walking to Quadrupedal Locomotion: Full-Body Dynamics Decomposition for Rapid Gait Generation

This paper systematically decomposes a quadrupedal robot into bipeds to rapidly generate walking gaits and then recomposes these gaits to obtain quadrupedal locomotion. We begin by decomposing the full-order, nonlinear and hybrid dynamics of a three-dimensional quadrupedal robot, including its continuous and discrete dynamics, into two bipedal systems that are subject to external forces. Using the hybrid zero dynamics (HZD) framework, gaits for these bipedal robots can be rapidly generated (on the order of seconds) along with corresponding controllers. The decomposition is achieved in such a way that the bipedal walking gaits and controllers can be composed to yield dynamic walking gaits for the original quadrupedal robot — the result is the rapid generation of dynamic quadruped gaits utilizing the full-order dynamics. This methodology is demonstrated through the rapid generation (3.96 seconds on average) of four stepping-in-place gaits and one diagonally symmetric ambling gait at 0.35 m/s on a quadrupedal robot — the Vision 60, with 36 state variables and 12 control inputs — both in simulation and through outdoor experiments. This suggested a new approach for fast quadrupedal trajectory planning using full-body dynamics, without the need for empirical model simplification, wherein methods from dynamic bipedal walking can be directly applied to quadrupeds

From Bipedal Walking to Quadrupedal Locomotion: Full-Body Dynamics Decomposition for Rapid Gait Generation

This paper systematically decomposes a quadrupedal robot into bipeds to rapidly generate walking gaits and then recomposes these gaits to obtain quadrupedal locomotion. We begin by decomposing the full-order, nonlinear and hybrid dynamics of a three-dimensional quadrupedal robot, including its continuous and discrete dynamics, into two bipedal systems that are subject to external forces. Using the hybrid zero dynamics (HZD) framework, gaits for these bipedal robots can be rapidly generated (on the order of seconds) along with corresponding controllers. The decomposition is achieved in such a way that the bipedal walking gaits and controllers can be composed to yield dynamic walking gaits for the original quadrupedal robot — the result is the rapid generation of dynamic quadruped gaits utilizing the full-order dynamics. This methodology is demonstrated through the rapid generation (3.96 seconds on average) of four stepping-in-place gaits and one diagonally symmetric ambling gait at 0.35 m/s on a quadrupedal robot — the Vision 60, with 36 state variables and 12 control inputs — both in simulation and through outdoor experiments. This suggested a new approach for fast quadrupedal trajectory planning using full-body dynamics, without the need for empirical model simplification, wherein methods from dynamic bipedal walking can be directly applied to quadrupeds

From Bipedal to Quadrupedal Locomotion, Experimental Realization of Lyapunov Approaches

Possibly one of the most significant innovations of the past decade is the hybrid zero dynamics (HZD) framework, which formally and rigorously designs a control algorithm for robotic walking. In this methodology, Lyapunov stability, which is often used to certificate a dynamical system's stability, was introduced to the control law design for a hybrid control system. However, the prerequisites of precise modeling to apply the HZD methodology can often be too restrictive to design controllers for uncertain and complex real-world hardware experiments. This thesis addresses the problem raised by noisy measurements and the intricate hybrid structure of locomotion dynamics. First, the HZD methodology's construction is based on the full-order, hybrid dynamics of legged locomotion, which can be intractable for control synthesis for high-dimensional systems. This thesis studies the general structure of hybrid control systems for walking systems, ranging from 1D hopping, 2D walking, 2D running, and 3D quadrupedal locomotion on rough terrains. Further, we characterize a walking behavior--gait--as a solution (execution) to a hybrid control system. To find these solutions, which represent a "gait," we employed advanced numerical methods such as collocation methods to parse the solution-finding problem into the open- and closed-loop trajectory optimization problems. The result is that we can find versatile gaits for ten different robotic platforms efficiently. This includes bipedal running, bipedal walking on slippery surfaces, and quadrupedal robots walking on sloped terrains. The numerous solution-finding examples expand the applicability of the HZD framework towards more complex dynamical systems. Further, for the uncertain and noisy real-world implementation, the exponential stability of the continuous dynamics is an ideal but restrictive condition for hybrid stability. This condition is especially challenging to satisfy for highly dynamical behaviors such as bipedal running, which loses ground support for a short period. This thesis observes the destabilizing effect of the noisy measurements of the phasing variable. By reformulating the traditional input-to-state stability (ISS) concept into phase-uncertainty to state stability, we are able to synthesize a robust controller for bipedal running on DURUS-2D. This time+state-based controller formally guarantees stability under noisy measurements and stabilizes the 1.75 m/s running experiments. Lastly, robotic dynamics have long been characterized as the interconnection of rigid-body dynamics. We take this perspective one step further and incorporate controller design into the formulation of coupled control systems (CCS). We first view a quadrupedal robot as two bipedal robots connected via some holonomic constraints. In a dimensional reduction manner, we develop a novel optimization framework, and the computational performance is reduced to a few seconds for gait generation. Furthermore, we can design local controllers for each bipedal subsystem and still guarantee the overall system's stability. This is done by combining the HZD framework and the ISS properties to contain the disturbance induced by the other subsystems' inputs. Utilizing the proposed CCS methods, we will experimentally realize quadrupedal walking on various outdoor rough terrains.</p

First Steps Towards Full Model Based Motion Planning and Control of Quadrupeds: A Hybrid Zero Dynamics Approach

The hybrid zero dynamics (HZD) approach has become a powerful tool for the gait planning and control of bipedal robots. This paper aims to extend the HZD methods to address walking, ambling and trotting behaviors on a quadrupedal robot. We present a framework that systematically generates a wide range of optimal trajectories and then provably stabilizes them for the full-order, nonlinear and hybrid dynamical models of quadrupedal locomotion. The gait planning is addressed through a scalable nonlinear programming using direct collocation and HZD. The controller synthesis for the exponential stability is then achieved through the Poincaré sections analysis. In particular, we employ an iterative optimization algorithm involving linear and bilinear matrix inequalities (LMIs and BMIs) to design HZD-based controllers that guarantee the exponential stability of the fixed points for the Poincaré return map. The power of the framework is demonstrated through gait generation and HZD-based controller synthesis for an advanced quadruped robot, —Vision 60, with 36 state variables and 12 control inputs. The numerical simulations as well as real world experiments confirm the validity of the proposed framework

Coupled Control Systems: Periodic Orbit Generation with Application to Quadrupedal Locomotion

A robotic system can be viewed as a collection of lower-dimensional systems that are coupled via reaction forces (Lagrange multipliers) enforcing holonomic constraints. Inspired by this viewpoint, this letter presents a novel formulation for nonlinear control systems that are subject to coupling constraints via virtual “coupling” inputs that abstractly play the role of Lagrange multipliers. The main contribution of this letter is a process—mirroring solving for Lagrange multipliers in robotic systems—wherein we isolate subsystems free of coupling constraints that provably encode the full-order dynamics of the coupled control system from which it was derived. This dimension reduction is leveraged in the formulation of a nonlinear optimization problem for the isolated subsystem that yields periodic orbits for the full-order coupled system. We consider the application of these ideas to robotic systems, which can be decomposed into subsystems. Specifically, we view a quadruped as a coupled control system consisting of two bipedal robots, wherein applying the framework developed allows for gaits (periodic orbits) to be generated for the individual biped yielding a gait for the full-order quadrupedal dynamics. This is demonstrated on a quadrupedal robot through simulation and walking experiments on rough terrains

Coupled Control Systems: Periodic Orbit Generation with Application to Quadrupedal Locomotion

A robotic system can be viewed as a collection of lower-dimensional systems that are coupled via reaction forces (Lagrange multipliers) enforcing holonomic constraints. Inspired by this viewpoint, this letter presents a novel formulation for nonlinear control systems that are subject to coupling constraints via virtual “coupling” inputs that abstractly play the role of Lagrange multipliers. The main contribution of this letter is a process—mirroring solving for Lagrange multipliers in robotic systems—wherein we isolate subsystems free of coupling constraints that provably encode the full-order dynamics of the coupled control system from which it was derived. This dimension reduction is leveraged in the formulation of a nonlinear optimization problem for the isolated subsystem that yields periodic orbits for the full-order coupled system. We consider the application of these ideas to robotic systems, which can be decomposed into subsystems. Specifically, we view a quadruped as a coupled control system consisting of two bipedal robots, wherein applying the framework developed allows for gaits (periodic orbits) to be generated for the individual biped yielding a gait for the full-order quadrupedal dynamics. This is demonstrated on a quadrupedal robot through simulation and walking experiments on rough terrains