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

Treadmill Platform for Quadrupedal Robots

Cal Poly Legged Robots, led by Professor Refvem and Professor Xing, has been leading Cal Poly’s attempts to simulate, produce, and test their legged robots. The initial testing of the locomotion of these robots can be dangerous to the robot since any bugs in the code could cause the robot to fall over and harm itself. Our responsibility as a team was to deliver a portable platform for testing the locomotion capabilities containing a fall prevention mechanism. In short, we have designed a platform for this purpose that consists of a treadmill surrounded by a wheeled chassis with a system of ropes, pulleys, and a winch for a fall prevention mechanism. Our method of lifting the robot is a success in two ways. First, our method only requires the addition of four eyebolts to the robot, a rather minor modification. Second, our method does not impede the motion of the robot when it is running normally. However, it was found that our method requires 1-1.7 seconds to lift the robot (depending on where the robot is located on the treadmill) – a rather crippling amount of time to lift a robot, seeing as how it is desired for the robot to run at 8 mph on the treadmill. Regardless, our design provides a great starting point for future Senior Project teams to improve upon it. It is our hope that our design will allow Cal Poly Legged Robots to further the development of legged robots and to generate interest, both at Cal Poly and hopefully around the world, in this area of study

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

Whole-Body MPC and Online Gait Sequence Generation for Wheeled-Legged Robots

Our paper proposes a model predictive controller as a single-task formulation that simultaneously optimizes wheel and torso motions. This online joint velocity and ground reaction force optimization integrates a kinodynamic model of a wheeled quadrupedal robot. It defines the single rigid body dynamics along with the robot's kinematics while treating the wheels as moving ground contacts. With this approach, we can accurately capture the robot's rolling constraint and dynamics, enabling automatic discovery of hybrid maneuvers without needless motion heuristics. The formulation's generality through the simultaneous optimization over the robot's whole-body variables allows for a single set of parameters and makes online gait sequence adaptation possible. Aperiodic gait sequences are automatically found through kinematic leg utilities without the need for predefined contact and lift-off timings, reducing the cost of transport by up to 85%. Our experiments demonstrate dynamic motions on a quadrupedal robot with non-steerable wheels in challenging indoor and outdoor environments. The paper's findings contribute to evaluating a decomposed, i.e., sequential optimization of wheel and torso motion, and single-task motion planner with a novel quantity, the prediction error, which describes how well a receding horizon planner can predict the robot's future state. To this end, we report an improvement of up to 71% using our proposed single-task approach, making fast locomotion feasible and revealing wheeled-legged robots' full potential.Comment: 8 pages, 6 figures, 1 table, 52 references, 9 equation

Online Gait Transitions and Disturbance Recovery for Legged Robots via the Feasible Impulse Set

Gaits in legged robots are often hand tuned and time based, either explicitly or through an internal clock, for instance, in the form of central pattern generators. This strategy requires trial and error to identify leg timings, which may not be suitable in challenging terrains. In this letter, we introduce new concepts to quantify leg capabilities for online gait emergence and adaptation, without fixed timings or predefined foothold sequences. Specifically, we introduce the Feasible Impulse Set, a notion that extends aspects of the classical wrench cone to include a prediction horizon into the future. By considering the impulses that can be delivered by the legs, quantified notions of leg utility are proposed for coordinating adaptive lift-off and touch-down of stance legs. The proposed methods provide push recovery and emergent gait transitions with speed. These advances are validated in experiments with the MIT Cheetah 3 robot, where the framework is shown to automatically coordinate aperiodic behaviors on a partially moving walkway