ZMP-Based Trajectory Generation for Bipedal Robots Using Quadratic Programming

ZMP-Based Trajectory Generation for Bipedal Robots Using Quadratic Programming

Sergei Savin
DOI: 10.4018/978-1-5225-9924-1.ch007
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In this chapter, the problem of trajectory generation for bipedal walking robots is considered. A number of modern techniques are discussed, and their limitations are shown. The chapter focuses on zero-moment point methods for trajectory generation, where the desired trajectory of that point can be used to allow the robot to keep vertical stability if followed, and presents an instrument to calculate the desired trajectory for the center of mass for the robot. The chapter presents an algorithm based on quadratic programming, with an introduction of a slack variable to make the problem feasible and a change of variables to improve the numeric properties of the resulting optimization problem. Modern optimization tools allow one to solve such problems in real time, making it a viable solution for trajectory planning for the walking robots. The chapter shows a few results from the numerical simulation made for the algorithm, demonstrating its properties.
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Walking robots are one of the focuses of modern robotics, and had remained so for the last decades. Even though there had been a significant amount of research and significant achievements in the field, it still remains one of the main challenges for robotics research.

Stable walking on horizontal surfaces had been demonstrated a variety of times. One of the best known examples are ASIMO robots produced by Honda (Chestnutt et al., 2005; Chestnutt et al., 2007; Sakagami et al., 2002). That robot was able to walk stably, climb stairs and was one of the first anthropomorphic bipedal robots. Early ASIMO models used energy inefficient gaits and were not demonstrated being able to walk on rough terrain.

It should be noted that although at this stage of humanoid robot development, the best implemented models were slow and energy-inefficient, there were parallel research such as Raibert’s hoppers and runners (Thompson & Raibert, 1990, Pratt 2000) and passive dynamic walkers (McGeer, 1990a, 1990b). Those robots exhibited not only stable but energy efficient gaits and provided ideas for the following generations of walking robots of different types.

A number of highly dynamic quadruped robots had been developed, including BigDog by Boston Dynamics (Raibert et al., 2008; Playter et al., 2006). Those robots provided a demonstration of capabilities of full-scale controllable walking robots, including demonstrations of walking on ice, grassland, forest undergrowth, climbing stairs and withstanding kicks from humans. Those early quadrupeds inspired and were followed by newer models such as ANYmal by ETH Zurich (Hutter et al., 2016Hutter et al., 2017), Spot and Spot Mini by Boston Dynamics and others. Unlike bipedal robots, quadrupeds have lesser constraints on their admissible regimes of walking and have better possibilities to remain vertically stable. This allows them to demonstrate more dynamics motion and serve as pioneers in solving many of the tasks that are common for walking robots.

The progress in the walking robotics was highly improved by DRC (DARPA Robotics Challenge) and its co-events (such as DARPA virtual challenge). That event assembled a number of different walking robots and allowed the teams to test them against a set of challenges, including egress from a car, operating a valve, walking over rough terrain and others. This allowed not only to assess the current state of the walking robotics (Atkeson et al., 2015, 2016), but also to point out the best control strategies (Long, 2017; Feng et al., 2015; Karumanchi et al., 2017; Tsagarakis et al., 2017) and to discuss the future research directions needed for the further improvements in the field as became evident during the event. Most of the teams that took part in the project provided extensive discussions of their results (Yi et al., 2015; Johnson et al., 2015, 2017; Radford at al., 2015), which allowed to easily assess different methodologies and compare further projects to those approaches using them as state of the art indicators.

Since then, a lot of attention has been given to the motion of bipedal robots through uneven terrain. This is a challenging task with a number of independent components. For example, it can be divided into the tasks of 1) moving over unknown terrain where the robot cannot plan its footsteps in advance, 2) moving over partial footholds, 3) recovering balance after making a step with supporting surface properties different from what was expected, 4) moving in regimes other than walking, such as running or jumping and others. There are some theoretical and applied works that provide solutions for this problem (Zheng et al., 2010; Kanoulas et al., 2018; Deits & Tedrake, 2014; Focchi et al., 2018; Short & Bandyopadhyay, 2018; Mastalli et al., 2018).

Key Terms in this Chapter

Bipedal Robot: A robot with two legs.

Vertical Stability: A potential ability of a walking robot to remain from making unplanned contacts with the supporting surface or uncontrollably losing contacts with the supporting surface (falling).

Robot Feet: Elements of the robot’s structure designed to remain in periodic contact with the supporting surface during walking, in a similar manner as do feet of humans.

Anthropomorphic Robot: A robot with a shape and morphology generally similar to that of a human body.

Zero-Moment Point: A point on the supporting surface relative to which the reaction forces’ torques are balanced. Used to find stable trajectories for the center of mass of the robot.

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