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3D Dynamic Walking on Stepping Stones with Control Barrier ...

3D Dynamic Walking on Stepping Stones with Control BarrierFunctionsQuan Nguyen1, Ayonga Hereid2, Jessy W. Grizzle3, Aaron D. Ames2, Koushil Sreenath1 Abstract 3D dynamical Walking subject to precise footstepplacements is crucial for navigating real world terrain with dis-crete footholds. We present a novel methodology that combinescontrol Lyapunov functions to achieve periodic Walking andcontrol Barrier functions to enforce strict constraints on steplength and step width unified in a single optimization-basedcontroller. We numerically validate our proposed method bydemonstrating Dynamic 3D Walking at m/s on DURUS, a23 degree-of-freedom underactuated humanoid INTRODUCTIONThe primary advantage of legged locomotion on roboticsystems is the ability to traverse terrain not acc

3D Dynamic Walking on Stepping Stones with Control Barrier Functions Quan Nguyen1, Ayonga Hereid 2, Jessy W. Grizzle3, Aaron D. Ames , Koushil Sreenath1 Abstract—3D dynamical walking subject to precise footstep placements is crucial for navigating real world terrain with dis-

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Transcription of 3D Dynamic Walking on Stepping Stones with Control Barrier ...

1 3D Dynamic Walking on Stepping Stones with Control BarrierFunctionsQuan Nguyen1, Ayonga Hereid2, Jessy W. Grizzle3, Aaron D. Ames2, Koushil Sreenath1 Abstract 3D dynamical Walking subject to precise footstepplacements is crucial for navigating real world terrain with dis-crete footholds. We present a novel methodology that combinescontrol Lyapunov functions to achieve periodic Walking andcontrol Barrier functions to enforce strict constraints on steplength and step width unified in a single optimization-basedcontroller. We numerically validate our proposed method bydemonstrating Dynamic 3D Walking at m/s on DURUS, a23 degree-of-freedom underactuated humanoid INTRODUCTIONThe primary advantage of legged locomotion on roboticsystems is the ability to traverse terrain not accessible bywheeled devices this can be canonically represented byterrain with discrete footholds such as Stepping Stones .

2 Yetcurrent approaches to handling this terrain type use simplisticmethods both at the level of models and Control of walkingrobots to plan footstep locations and quasi-statically drivea fully-actuated humanoid robot to achieve the desired footplacements. The overarching goal of this work is to create aformal framework that will enable bipedal humanoid robotsto achieve Dynamic and rapid locomotion over a randomlyplaced set of Stepping placement for fully-actuated legged robots essen-tially rely on quasi-static Walking using the ZMP criterionthat requires slow Walking speeds and small steps [13],[7].

3 Impressive results in footstep planning and placementsin obstacle filled environments with vision-based sensingwas carried out in [6]. However, these ZMP-based methodsimpose strict restrictions on the Walking gaits as they relyon kinematics of quasi-static motions or simple dynamicalmodels such as the linear inverted pendulum with masslesslegs, see [8], [18]. Moreover, these methods typically requirefull-actuation, cannot handle compliance well, and are notapplicable for Dynamic Walking with faster Walking paper presents initial results on precise footstepplacement for 3D Dynamic Walking .

4 The proposed methodis based on feedback Control of the full nonlinear andunderactuated hybrid Dynamic model of bipedal robots toachieve periodic Dynamic Walking gaits with formal stabilityguarantees that enforce the safety-critical foot placementconstraints. The main contribution of this paper is a novel1 Dept. of Mechanical Engineering, Carnegie Mellon University, Pitts-burgh, PA of Mechanical Engineering, Georgia Institute of Technology,Atlanta, GA of Electrical Engineering and Computer Science, Univ. of Michi-gan, Ann Arbor, MI work is supported by NSF grants IIS-1526515, IIS-1526519, 1: The problem of dynamically Walking in 3D overa randomly generated set of discrete footholds.

5 Simulationvideo: Control strategy that can guarantee precisefootstep placement for 3D Dynamic Walking of a high-dimensional bipedal robot. We achieve this by combining (a) Control Lyapunov function based quadratic programs (CLF-QPs) [9] for enforcing virtual constraints, that represent anominal gait, while simultaneously respecting the saturationlimits of the actuators, and (b) Control Barrier functionbased quadratic programs (CBF-QPs) [4] to guarantee statedependent safety goal of this paper is to relax the tracking behaviorof the nominal gait by enforcing a set of state-dependentsafety constraints, governed by Control Barrier functions , thatguide the swing foot trajectory to the discrete method enables dealing with a large range of desiredfoothold separations with precise placement of footsteps onsmall footholds.

6 This requires simultaneously guaranteeingprecise step length and step width constraints at foot work builds off our recent work on precise footstepplacement for planar (2D) Walking [14]. In comparison toour prior work, this paper makes the following additionalcontributions: We consider 3D Dynamic bipedal Walking in contrast toplanar Walking . We extend the applicability of our method to a 23degree-of-freedom bipedal system (up from 7 degree-of-freedom model in planar Walking ) establishing scal-ability of the proposed method.

7 We consider additional degrees of underactuation in theform of compliant 2: The Humanoid DURUS is a23degree-of-freedomsystem, which break down into15actuated joints,2passivesprings at the feet, and a6degree-of-freedom floating coordinates of DURUS are illustrated with the redarrows representing the positive rotation (or translation) axisof the robot joints. In addition to step length constraints in planar Walking ,we simultaneously address both step length and stepwidth rest of the paper is organized as follows. Section IIpresents the hybrid dynamical model of DURUS, a 3D hu-manoid robot.

8 Section III revisits Control Lyapunov function-based quadratic programs (CLF-QPs). Section IV presentsexponential Control Barrier functions (ECBFs) for enforcingsafety constraints. Section V presents the proposed ECBF-CLF-QP based feedback controller for enforcing precisefootstep placement for 3D Dynamic Walking . Section VIpresents numerical validation of the controller on , Section VII provides concluding HYBRIDSYSTEMMODELIn this section, we briefly discuss the two-domain hybridsystem model for the flat foot Walking of underactuated 3 Drobot, DURUS.

9 Hybrid zero dynamics Control framework isalso concisely introduced as a way to synthesize Walking gaitfor the DURUS ModelDURUS is an underactuated humanoid robot with15actuated joints and two passive springs (see Fig. 2), designedand built by SRI International for the study of high efficiencymulti-domain bipedal locomotion [12], [17]. The two passivelinear springs are attached to the end of each ankle joint,such that they are rigidly perpendicular to the foot, and aredesigned for reducing energy loss and mitigating mechanicalshocks at this paper, the generalized floating-base coodinates,q=[pb, b,qr]T Q=R3 SO(3) Qr, of the robot is usedto model this high-dimensional humanoid, wherepb R3isthe Cartesian position and b SO(3)is the orientation ofFig.

10 3: The directed cycle structure of the multi-domainhybrid system model for flat-foot humanoid body base frameRb which is attached to the center ofthe pelvis link with respect to the world frame, andqr Qris the 17-dimensional joint coordinates of DURUS, as shownin Fig. 2 (see [17] for more details.)B. Hybrid System Model for Bipedal WalkingDue to the existence of both continuous and discretedynamics, bipedal locomotion is naturally modeled as ahybrid Control system [11]. The existence of passive springsleads to two continuous domain behaviors in the case offlat-foot Walking : adouble supportdomain, where both feetare on the ground, and asingle supportdomain, where thenon-stance foot is above the ground while only the stancefoot stays on the ground, as shown in Fig.


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