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Research Papers

# Performance Analysis and Feedback Control of ATRIAS, A Three-Dimensional Bipedal Robot

[+] Author and Article Information
Alireza Ramezani

Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: aramez@umich.edu

Jonathan W. Hurst

Professor
Department of Mechanical Engineering,
Oregon State University,
Corvallis, OR 97331
e-mail: jonathan.hurst@oregonstate.edu

Kaveh Akbari Hamed

Department of Electrical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: kavehah@umich.edu

J. W. Grizzle

Professor
Department of Electrical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: grizzle@umich.edu

Because of the way coordinates have been assigned, Bs is a constant matrix. Moreover, because the actuators are independent, Bs has (full) rank equal to the number of actuators, 6.

We anticipate that the assumption of yaw torque in the form of viscous friction will better approximate the behavior of the robot in the laboratory, where a passive foot significantly will reduce yaw, while still allowing the foot to pitch and roll.

Virtual leg is defined as a virtual line connecting the pivot point of the stance leg to the hip joint.

Essentially means the second derivatives of the six outputs in Eq. (13) depend on six inputs in a full rank or independent manner.

Frontal plane is attached to the torso and the normal axis is aligned with y axis of the torso frame.

When the decoupling matrix is invertible, Dzero is guaranteed to be invertible as well.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received December 3, 2012; final manuscript received October 7, 2013; published online December 9, 2013. Assoc. Editor: Luis Alvarez.

J. Dyn. Sys., Meas., Control 136(2), 021012 (Dec 09, 2013) (12 pages) Paper No: DS-12-1421; doi: 10.1115/1.4025693 History: Received December 03, 2012; Revised October 07, 2013

## Abstract

This paper develops feedback controllers for walking in 3D, on level ground, with energy efficiency as the performance objective. Assume The Robot Is A Sphere (ATRIAS) 2.1 is a new robot that has been designed for the study of 3D bipedal locomotion, with the aim of combining energy efficiency, speed, and robustness with respect to natural terrain variations in a single platform. The robot is highly underactuated, having 6 actuators and, in single support, 13 degrees of freedom. Its sagittal plane dynamics are designed to embody the spring loaded inverted pendulum (SLIP), which has been shown to provide a dynamic model of the body center of mass during steady running gaits of a wide diversity of terrestrial animals. A detailed dynamic model is used to optimize walking gaits with respect to the cost of mechanical transport (CMT), a dimensionless measure of energetic efficiency, for walking speeds ranging from 0.5 $(m/s)$ to 1.4 $(m/s)$. A feedback controller is designed that stabilizes the 3D walking gaits, despite the high degree of underactuation of the robot. The 3D results are illustrated in simulation. In experiments on a planarized (2D) version of the robot, the controller yielded stable walking.

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## Figures

Fig. 2

The right leg of the robot. (a) The 4-bar parallelogram that forms each leg and a conceptual representation of the series-elastic actuators. (b) The configuration variable q3R is the rotation of the right hip in the frontal plane; q3L is similar. (c) q1R and q2R are the angles between the right upper links and the torso; q1L and q2L are defined similarly.

Fig. 1

Untethered 3D bipedal robot ATRIAS 2.1

Fig. 8

Control effort during one step for (a) u1R, u1L, (b) u2R, u2L, and (c) u3R, u3L corresponding to a fix point with nominal walking speed of 1.0(m/s) and CMT 0.096

Fig. 9

Actuation power during one step for (a) p1R, p1L, (b) p2R, p2L, and (c) p3R, p3L corresponding to a fix point with nominal walking speed of 1.0(m/s) and CMT 0.096

Fig. 4

(a) Desired trajectory for right (stance) leg knee angle qgrRKnee, (b) first derivative of the desired trajectory, and (c) the second derivative of the desired trajectory when order of the NURB basis function is 3

Fig. 5

Leg shortly after impact, showing the springs absorbing the impact energy

Fig. 7

Evolution during one step of (a) the right leg angle qLAR and the left leg angle qLAL, (b) the right knee angle qKneeR and the left knee angle qKneeL, (c) the right hip joint angle qHipR and the left hip joint angle qHipL for an optimal walking motion with the nominal velocity of 1.0(m/s) and CMT 0.096

Fig. 3

(a) Basis functions, (b) first derivative of the basis functions, and (c) second derivative of the basis functions, all with respect to the timing variable s for each control point Bi when the order k is six (solid line) and three (dashed line)

Fig. 6

Circles represent computed CMT using Bézier curves for the desired trajectories versus walking speed from 0.5 to 1.4(m/s). The thick solid line is a cubic interpolation of these data. Squares represent computed CMT using NURB curves versus walking speeds 0.5, 0.7, 0.9, 1.1, and 1.4(m/s). The dashed line is a cubic interpolation of these data. Losses at the harmonic drives are ignored

Fig. 11

Top curve is the (xy)-plane projection of the center of mass and foot positions to achieve steering along a desired path with a 30 deg deflection with respect to the y-axis. The lower curve is the commanded path, shifted by 2.5 m. Convergence to the desired path is clear.

Fig. 12

Convergence of the trajectories to the fixed point after perturbing the initial condition

Fig. 10

NURB curve of order 5 with 15 control point is fitted to the trajectory of the horizontal distance between the end of the left leg and the COM in the frontal plane, solid curve is the distance trajectory, dashed curve is the fitted data and the squares are the control points of the NURB curve.

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