Active balance of humanoids with foot positioning compensation and non-parametric adaptation☆
Introduction
Robust locomotion is fundamental for a humanoid robot to maneuver in an unstructured and undetermined human environment. Currently, humanoid walking is commonly realized by planning the Center of Mass (CoM) trajectories so that the resultant Zero Moment Point (ZMP) [1] trajectory follows a desired ZMP trajectory, which is normally determined by predefined foot positioning [2], [3], [4]. For real time implementation, a humanoid robot was represented by a mass-concentrated model and a simplified dynamics model is generally used [5].
However, these control methods can realize humanoids walking under ideal environment instead of complex real world, where balanced walking of a humanoid robot requires dynamic stability. To achieve this objective, strategies have been proposed to compensate for nonzero variations in momentum [6] or regulate CoM directly through sensory feedback control [7]. These approaches actually imitate the instantaneous reactive balance strategies of human beings, but their performances are significantly limited. The supporting feet of a humanoid robot forms a supporting polygon on the ground. Because the humanoid robot’s foot can only push the ground, the forces available to control the CoM is limited by the area of this supporting polygon once the foot steps are determined. And this supporting area constraint makes the robot unable to adjust its controllability freely to large disturbances.
Besides the instantaneous reactive balance strategies, dynamics adjustable foot positioning is also an effective strategy for active balance. Experience such as taking several steps to recover from a stumble is a good illustration. Biomechanically motivated studies in [8] indicate that dynamic balance is realized more often than not on adjustable subsequent steps to ensure complete recovery and this foot positioning strategy is the key to balance recovery. On the other hand, the effectiveness of dynamic adjustable foot positioning can be explained as follows: the modification of foot positioning changes the available force region to control the CoM, hence appropriate foot positioning may provide corresponding forces to regulate the CoM trajectory as desired.
In this paper, a Foot Positioning Compensator (FPC) is proposed to dynamically adjust the foot positioning of a humanoid robot according to the real-time state of this robot’s CoM. To take into account the constraint of limited supporting area of a real humanoid robot, a Constrained Inverted Pendulum Model (CIPM) is used to apply the available acceleration of CoM to determine the future state of CoM over control command. A Constraint Nonlinear Optimization Programming (CNOP) scheme [9] is used to find optimal FPC output from CoM state based on CIPM dynamics. Due to its computational complexity, this searching-based CNOP scheme is only applied offline to generate training samples for online FPC model.
Although the CIPM takes into account supporting area constraints, substantial model errors still exist for a real full-body humanoid robot. Factors such as the multi-body dynamics and other mechanical effects (links elasticity, gear backlash, uncertain ground friction etc.) cannot be sufficiently modeled. It therefore makes sense for a humanoid robot to autonomously learn its real dynamics and adapt its compensator accordingly in a real environment. On this point, an adjustable non-parametric regression model is applied online to represent the mapping from CoM state to FPC output. To adapt the online FPC model fast and efficiently, a sample-efficient online local adaptation algorithm is also proposed for the non-parametric model.
The rest of this paper is organized as follows. Section 2 reviews works related to humanoid disturbance rejection using foot positioning adjustment. Section 3 presents a constrained inverted pendulum model for a humanoid robot. Section 4 introduces a FPC strategy. Section 5 develops a non-parametric FPC model and Section 6 an adaptation method. Section 7 presents both simulation and experimental results to show the performance of the proposed FPC strategy. Section 8 provides the conclusions.
Section snippets
Related work
Recently, increasing attention is directed to applying adjustable foot positioning for disturbance rejection in humanoid robot walking. For a standing robot to resist unexpected impact, a Maximal Constraint Positively Invariant (CPI) Sets method [10] is proposed to determine whether a robot needs to take a step to recover from disturbance. The prediction is based on the Linear Inverted Pendulum Model (LIPM) [5] and the consideration of the supporting area constraint, but the stride of the
Coordinate systems
A typical full-body humanoid robot is illustrated in Fig. 1. Two coordinate systems are used: the first one is the world coordinate system , with its origin on the ground and its , axis formed a plane parallel to the ground; the second one is called the supporting coordinate system , which is attached on the supporting foot. Its origin is the origin point of the supporting foot with its axis pointing in the forward direction of the supporting foot, and axis to the outside.
Linear inverted pendulum model
The
Foot positioning strategies
When a humanoid robot is walking, large disturbances may cause its VZMP move outside the supporting polygon and lead the system out of balance. To mimic human’s reaction to large disturbances during walking, it is natural to change next foot positioning.
Humanoid walking is generally considered as a cyclic process. This process can be divided into four phases based on the state of supporting feet: two single support (SS) phases and two double support (DS) phases. The duration of the single
Non-parametric FPC model
The prediction of FPC based on the CNOP scheme faces two major drawbacks: computation complexity and lack of model flexibility. They make FPC prediction difficult to implement in online humanoid robot walking control. On the other hand, considering the uncertainty in humanoid walking control, mapping from the CoM state to FPC output can also be treated as a Bayesian regression problem. The CoM state is the input vector and the FPC output are some noisy realization of an underlying functional
Local adaptation for online FPC
To adapt the online FPC models for a humanoid system, a sample-efficient and computationally fast local adaptation algorithm for the HSpGP model is proposed in this section.
Humanoid robot
To validate the proposed method, robot NAO produced by Aldebaran robotics is used for the experiments. The robot has 21 degrees of freedom (DOF), with 11 DOFs for the two legs. The height of this robot is about 50 cm during walking and its weight is 4.35 kg. The sensors data and the walking control module update at a rate of 50 frames per second (FPS). Besides, the robot is equipped with an AMD GEODE 500 MHz CPU. The difficulty to develop walking approaches on this robot is that each leg
Conclusion
A foot positioning compensator is developed in this paper for a full-body humanoid to retrieve its balance during continuous walking. The policy of the foot positioning is first pre-computed offline based on the constrained dynamics model using constrained nonlinear optimization programming. The policy is then interpreted using a heteroscedastic sparse Gaussian process model for the online computation efficiency. To make the FPC policy coincide with the full-body humanoid dynamics, a MAP-like
Acknowledgments
The authors would like to express thanks to technicians in the Laboratory of Robotics and Intelligent Systems of Tongji University for their assistance during experiments.
Chengju Liu received the B.E. degree from Qingdao University of Science and Technology, China, in 2007, and the Ph.D. degree in control theory and control engineering from Tongji University in 2011. After graduation, from March 2011 to June 2013, she was a postdoctoral researcher in Tongji University. From October 2011 to July 2012, she is a research associate in the BEACON Center for the study of evolution in action in Michigan State University. She is currently an associate professor in the
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Chengju Liu received the B.E. degree from Qingdao University of Science and Technology, China, in 2007, and the Ph.D. degree in control theory and control engineering from Tongji University in 2011. After graduation, from March 2011 to June 2013, she was a postdoctoral researcher in Tongji University. From October 2011 to July 2012, she is a research associate in the BEACON Center for the study of evolution in action in Michigan State University. She is currently an associate professor in the School of Electronics and Information Engineering, Tongji University. Her research interests include motion control of legged robots and evolutionary computation.
Tao Xu born in 1985, received his B.E. degree from Tongji University, China, in 2006, and the Ph.D. degree in control theory and control engineering from Tongji University in 2013. His research interests include legged robots locomotion, machine learning and probabilistic robotics.
Danwei Wang received his Ph.D. and MSE degrees from the University of Michigan, Ann Arbor in 1989 and 1984, respectively. He received his B.E. degree from the South China University of Technology, China in 1982. Since 1989, he has been with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. Currently, he is professor, Division of Control and Instrumentation. He has served as general chairman, technical chairman and various positions in international conferences. He is an associate editor for the International Journal of Humanoid Robotics and served as an associate editor of Conference Editorial Board, IEEE Control Systems Society from 1998 to 2005. He was a recipient of Alexander von Humboldt fellowship, Germany. His research interests include robotics, control theory and applications. He has published in the areas of iterative learning control, repetitive control, robust control and adaptive control systems, manipulator/mobile robot dynamics, path planning, and control.
Qijun Chen graduated from the Department of Automatic Control Engineering at Huazhong University of Science and Technology in 1987. In 1990, he got his master’s degree from the Department of information and control engineering, Xi’an Jiaotong University, and was granted Ph.D. from the Department of Electrical Engineering of Tongji University in 1999. He is now the full professor in the College of Electronic and Information Engineering, Tongji University. His research interests include robotics, environmental perception and understanding of mobile robot, networked system.
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This work was supported in part by the National Natural Science Foundation of China (Grant No. 61203344, 61573260, 91120308), and the Fundamental Research Funds for the Central Universities (0800219309), the Basic Research Project of Shanghai Science and Technology Commission (Grant No. 12JC1408800).