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Research Papers

Contact-Dependent Balance Stability of Biped Robots

[+] Author and Article Information
Carlotta Mummolo, William Z. Peng

Department of Mechanical and
Aerospace Engineering,
New York University,
Brooklyn, NY 11201

Carlos Gonzalez

Department of Mechanical
and Aerospace Engineering,
New York University,
Brooklyn, NY 11201

Joo H. Kim

Department of Mechanical and
Aerospace Engineering,
New York University,
Brooklyn, NY 11201
e-mail: joo.h.kim@nyu.edu

1Corresponding author.

Manuscript received September 22, 2017; final manuscript received December 20, 2017; published online February 27, 2018. Assoc. Editor: Andreas Mueller.

J. Mechanisms Robotics 10(2), 021009 (Feb 27, 2018) (13 pages) Paper No: JMR-17-1319; doi: 10.1115/1.4038978 History: Received September 22, 2017; Revised December 20, 2017

A theoretical–algorithmic framework for the construction of balance stability boundaries of biped robots with multiple contacts with the environment is proposed and implemented on a robotic platform. Comprehensive and univocal definitions of the states of balance of a generic legged system are introduced with respect to the system's contact configuration. Theoretical models of joint-space and center of mass (COM)-space dynamics under multiple contacts, distribution of contact wrenches, and robotic system parameters are established for their integration into a nonlinear programing (NLP) problem. In the proposed approach, the balance stability capabilities of a biped robot are quantified by a partition of the state space of COM position and velocity. The boundary of such a partition provides a threshold between balanced and falling states of the biped robot with respect to a specified contact configuration. For a COM state to be outside of the stability boundary represents the sufficient condition for falling, from which a change in the system's contact is inevitable. Through the calculated stability boundaries, the effects of different contact configurations (single support (SS) and double support (DS) with different step lengths) on the robot's balance stability capabilities can be quantitatively evaluated. In addition, the balance characteristics of the experimental walking trajectories of the robot at various speeds are analyzed in relation to their respective stability boundaries. The proposed framework provides a contact-dependent balance stability criterion for a given system, which can be used to improve the design and control of walking robots.

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Figures

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Fig. 1

The distinction between the states of balance of a biped robot is relative to a specified contact configuration. While the SS and DS contact configurations are shown as examples, the proposed definitions of balanced and unbalanced (or falling) states apply to any multi-contact configuration between a generic legged system and the environment.

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Fig. 2

A robotic system in multicontact configuration with the environment. Frame {X, Y, Z} represents the global coordinate system, frame {b} the floating base reference frame, frame {p} (p = 1, 2, 3 in this figure) the contact reference frames, and CSp the convex polygon approximating the pth contact surface.

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Fig. 3

The contact surfaces and wrenches in two-dimensional SS and DS contact configurations for a biped robot

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Fig. 4

Planar model of the biped robot DARwIn-OP in SS (left) and DS (right) contact configurations. The base frame {b} is coincident with the global frame {X, Y}, and its unactuated translation and rotation are represented by a system of fictitious joints. The joint angle variables q2 and q7 represent the ankle, q3 and q6 the knee, and q4 and q5 the hip revolute joints. The biped system's COM positions corresponding to the given configurations are also shown.

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Fig. 5

Examples of contact-specific COM workspace areas. Biped system is shown in home configurations chosen for SS and DS, respectively.

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Fig. 7

COM workspace discretization strategy to evaluate the balance stability boundary (i.e., velocity extrema) at selected grid points in a given direction of interest û

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Fig. 6

Joint torque and velocity limits for the actuator model of the given biped robot

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Fig. 9

Contact-specific COM workspaces with the corresponding discretized grid points for balance stability boundary construction. The COM paths corresponding to one step of the robot's five walking motions are also shown, where the markers indicate the beginning and the end of each SS (black) and DS (gray) path.

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Fig. 8

Biped robot DARwIn-OP: link parameters (adapted from Ref. [36]) corresponding to the planar model (Fig. 4) and the experimental walking parameters

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Fig. 10

Balance stability boundary in the COM X-state space for the biped robot in SS contact configuration, for which BOS dimension corresponds to the foot length fl. The upper and lower curves of the stability boundary correspond to the directions of interest û=[1  0]T and û=[−1  0]T, respectively. The vertical lines indicate the COM workspace limits for SS, estimated for the selected y¯=0.208 m.

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Fig. 11

Balance stability boundary in the COM X-state space for the biped robot in various DS contact configurations, with step length sli and BOS dimensions equal to sli + fl. The upper and lower curves of the stability boundaries correspond to the directions of interest û=[1  0]T and û=[−1  0]T, respectively. The vertical lines indicate the COM workspace limits for DS, estimated for the selected y¯=0.208 m and the specified sli. All the SS and DS stability boundaries are compared on the COM X-state space (bottom-right).

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Fig. 12

Balance stability characteristics of the biped robot's various walking trajectories. The forward walking progression is in the positive X direction. The BOS dimension is equal to fl for SS (markers indicate the rear and front ends of the stance foot) and sli + fl for DS for i = 1–5 (markers indicate the rear end of the rear foot and the front end of the front foot).

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