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

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

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

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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Grimes, J. A., and Hurst, J. W., 2012, “The Design of ATRIAS 1.0 a Unique Monoped, Hopping Robot,” Proceedings of the 2012 International Conference on Climbing and Walking Robots and the Support Technologies for Mobile Machines, pp. 548–554.
Collins, S. H., Ruina, A., Tedrake, R., and Wisse, M., 2005, “Efficient Bipedal Robots Based on Passive-Dynamic Walkers,” Science, 307, pp. 1082–1085. [CrossRef] [PubMed]
Blickhan, R., 1989, “The Spring Mass Model for Running and Hopping,” J. Biomech., 22(11–12), pp. 1217–1227. [CrossRef] [PubMed]
McMahon, T. A., and Cheng, G. C., 1990, “The Mechanics of Running: How Does Stiffness Couple With Speed?,” J. Biomech., 23, pp. 65–78. [CrossRef] [PubMed]
Farley, C. T., Glasheen, J., and McMahon, T. A., 1993, “Running Springs: Speed and Animal Size,” J. Exp. Biol., 185,, pp. 71–86. [PubMed]
Full, R. J., and Farley, C. T., 2000, “Musculoskeletal Dynamics in Rhythmic Systems—A Comparative Approach to Legged Locomotion,” Biomechanics and Neural Control of Posture and Movement, J. M.Winters and P. E.Crago, eds. Springer-Verlag, New York.
Seyfarth, A., Geyer, H., Gunther, M., and Blickhan, R., 2001, “A Movement Criterion for Running,” J. Biomech., 35, pp. 649–655. [CrossRef]
Raibert, M., 1986, Legged Robots that Balance, MIT Press, Cambridge, MA.
Zeglin, G., and Brown, H. B., 1998, “Control of a Bow Leg Hopping Robot,” IEEE International Conference on Robotics and Automation.
Ahmadi, M., and Buehler, M., 2006, “Controlled Passive Dynamic Running Experiment With the ARL Monopod II,” IEEE Trans. Robotics, 22, pp. 974–986. [CrossRef]
Sreenath, K., Park, H.-W., Poulakakis, I., and Grizzle, J. W., 2011, “Compliant Hybrid Zero Dynamics Controller for Achieving Stable, Efficient and Fast Bipedal Walking on MABEL,” Int. J. Robot. Res., 30(9), pp. 1170–1193. [CrossRef]
Sreenath, K., Park, H.-W., and Grizzle, J., 2012, “Design and Experimental Implementation of a Compliant Hybrid Zero Dynamics Controller With Active Force Control for Running on MABEL,” International Conference on Robotics and Automation (ICRA), pp. 51–56.
Park, H.-W., Sreenath, K., Ramezani, A., and Grizzle, J., 2012, “Switching Control Design for Accommodating Large Step-Down Disturbances in Bipedal Robot Walking,” International Conference on Robotics and Automation (ICRA), pp. 3331–3450.
Park, H.-W., Ramezani, A., and Grizzle, J., 2013, “A Finite-State Machine for Accommodating Unexpected Large Ground-Height Variations in Bipedal Robot Walking,” IEEE Trans. Rob. Autom., 29, pp. 45–50. [CrossRef]
Geyer, H., Seyfarth, A., and Blickhan, R., 2006, “Compliant Leg Behaviour Explains the Basic Dynamics of Walking and Running,” Proc. R. Soc., London, Ser. B, 273, pp. 2861–2867. [CrossRef]
Laboratory, D. R., 2012, “SolidWorks—Model Parameters for ATRIAS Based on Solidworks,” http://code.google.com/p/atrias/
Hodgins, J. K., and Raibert, M. H., 1991, “Adjusting Step Length for Rough Terrain Locomotion,” IEEE Trans. Rob. Autom., 7(3), pp. 289–298. [CrossRef]
Ahmadi, M., and Buehler, M., 1999, “The ARL Monopod II Running Robot: Control and Energetics,” IEEE International Conference on Robotics and Automation, pp. 1689–1694.
Chevallereau, C., Abba, G., Aoustin, Y., Plestan, F., Westervelt, E. R., Canudas-de-Wit, C., and Grizzle, J. W., 2003, “RABBIT: A Testbed for Advanced Control Theory,” IEEE Control Syst. Mag., 23(5), pp. 57–79. [CrossRef]
Corp., H., “Asimo Humanoid Robot,” http://world.honda.com/asimo/
McGeer, T., 1988, “Stability and Control of Two-Dimensional Biped Walking,” Center for Systems Science, Simon Fraser University, Burnaby, B.C., Canada, Technical Report 1.
McGeer, T., 1990, “Passive Dynamic Walking,” Int. J. Robot. Res., 9(2), pp. 62–82. [CrossRef]
Collins, S. H., Wisse, M., and Ruina, A., 2001, “A 3-D Passive Dynamic Walking Robot With Two Legs and Knees,” Int. J. Robot. Res., 20, pp. 607–615. [CrossRef]
Pratt, J., and Pratt, G., 1998, “Exploiting Natural Dynamics in the Control of a Planar Bipedal Walking Robot,” Proceedings of the Thirty-Sixth Annual Allerton Conference on Communication, Control, and Computing.
Playter, R., Buehler, M., and Raibert, M., 2006, “Bigdog,” Proceedings of SPIE International Society for Optical Engineering, G. R.Gerhart, C. M.Shoemaker, and D. W.Gage, eds., SPIE, Vol. 6230.
Robinson, D. W., Pratt, J. E., Paluska, D. J., and Pratt, G. A., 1999, “Series Elastic Actuator Development for a Biomimetic Walking Robot,” IEEE/ASME International Conference on Advanced Intelligent Mechatronics, pp. 561–568.
Boaventura, T., Semini, C., Buchli, J., Frigerio, M., Focchi, M., and Caldwell, D. G., 2012, “Dynamic Torque Control of a Hydraulic Quadruped Robot,” IEEE Conference on Robotics and Automation, pp. 1189–1894.
Kim, S., 2012, “Biomimetics Robotics Lab,” http://sangbae.scripts.mit.edu/biomimetics/videos/
Chevallereau, C., Grizzle, J., and Shih, C., 2009, “Asymptotically Stable Walking of a Five-Link Underactuated 3D Bipedal Robot,” IEEE Trans. Rob. Autom., 25(1), pp. 37–50. [CrossRef]
Westervelt, E. R., Grizzle, J. W., Chevallereau, C., Choi, J., and Morris, B., 2007, Feedback Control of Dynamic Bipedal Robot Locomotion. Control and Automation, CRC Press, Boca Raton, FL.
Tuttle, T., and Seering, W., 1996, “A Nonlinear Model of a Harmonic Drive Gear Transmission,” IEEE Trans. Rob. Autom., 12(3), pp. 368–374. [CrossRef]
Kennedy, C., and Desai, J., 2005, “Modeling and Control of the Mitsubishi pa-10 Robot Arm Harmonic Drive System,” IEEE/ASME Trans. Mechatron., 10(3), pp. 263–274. [CrossRef]
Hurmuzlu, Y., and Chang, T., 1992, “Rigid Body Collisions of a Special Class of Planar Kinematic Chains,” IEEE Trans. Syst. Man Cybern., 22(5), pp. 964–971. [CrossRef]
Grizzle, J. W., Abba, G., and Plestan, F., 2001, “Asymptotically Stable Walking for Biped Robots: Analysis Via Systems With Impulse Effects,” IEEE Trans. Autom. Control, 46, pp. 51–64. [CrossRef]
Westervelt, E., Grizzle, J., and Koditschek, D., 2003, “Hybrid Zero Dynamics of Planar Biped Walkers,” IEEE Trans. Autom. Control, 48(1), pp. 42–56. [CrossRef]
Canudas-de-Wit, C., 2004, “On the Concept of Virtual Constraints as a Tool for Walking Robot Control and Balancing,” Annu. Rev. Control, 28, pp. 157–166. [CrossRef]
Isidori, A., 1995, Nonlinear Control Systems: An Introduction, 3rd ed., Springer-Verlag, Berlin, Germany.
Sreenath, K., Park, H., Poulakakis, I., and Grizzle, J., 2011, “A Compliant Hybrid Zero Dynamics Controller for Stable, Efficient, and Fast Bipedal Walking on MABEL,” Int. J. Robot. Res., 30, pp. 1170–1193. [CrossRef]
Morris, B., and Grizzle, J. W., 2009, “Hybrid Invariant Manifolds in Systems With Impulse Effects With Application to Periodic Locomotion in Bipedal Robots,” IEEE Trans. Autom. Control, 54(8), pp. 1751–1764. [CrossRef]
Piegl, L., and Tiller, W., 1997, The NURBS Book, Springer, New York.
Ruina, A., 2012, “Cornell Ranger, 2011 4-Legged Bipedal Robot,” Nov. http://ruina.tam.cornell.edu/research/topics/locomotion_and_robotics/ranger/Ranger 2011/index.html
Shih, C., Grizzle, J., and Chevallereau, C., 2012, “From Stable Walking to Steering of a 3D Bipedal Robot With Passive Point Feet,” Robotica, 30, pp. 1119–1130. [CrossRef]


Grahic Jump Location
Fig. 1

Untethered 3D bipedal robot ATRIAS 2.1

Grahic Jump Location
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.

Grahic Jump Location
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)

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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

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Fig. 5

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

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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

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Fig. 12

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

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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

Grahic Jump Location
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

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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

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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.

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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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