Abstract
Multi-legged robots are analyzed and controlled based on a reduced-order single-legged planar model. However, the rolling motion is ignored in the discussion of the planar model, making a quantitative discussion of comprehensive stability and energy cost impossible to conduct. In this paper, a progressive process for analyzing the rolling motion of an RHex-style robot is presented. A planar bounding-in-place model whose stiffnesses of the left leg and right leg are different is analyzed. Two new orbits that represent the rolling motion more appropriately than the pronking orbit are found. To further investigate the rolling motion with the consideration of a half-circular compliant leg, a three-dimensional model, called the roll two-leg, clock-torqued, spring-loaded inverted pendulum model with a rolling contact (RTL-CTR-SLIP), is developed. It is proved via the simulation that the pronking orbit fails to serve as a passively stable rolling orbit in the three-dimensional model. The dynamics of rolling motion and the power consumption are also investigated via the simulation. Finally, the empirical data show a regular periodic rolling motion exists as an RHex-style robot runs, and the simulation of the RTL-CTR-SLIP model partially predicts the rolling dynamic and the energy cost of a real-world RHex-style robot.
Similar content being viewed by others
Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
References
Bruzzone, L., Quaglia, G.: Review article: locomotion systems for ground mobile robots in unstructured environments. Mech. Sci. 3(2), 49–62 (2012)
Rubio, F., Valero, F., Llopis-Albert, C.: A review of mobile robots: concepts, methods, theoretical framework, and applications. Int. J. Adv. Rob. Syst. 16(2), 1729881419839596 (2019)
Full, R.J., Koditschek, D.E.: Templates and anchors: neuromechanical hypotheses of legged locomotion on land. J. Exp. Biol. 202(23), 3325–3332 (1999)
Alexander, R.M.: Elastic Mechanisms in Animal Movement. Cambridge University Press, Cambridge (1988)
Holmes, P., et al.: The dynamics of legged locomotion: models, analyses, and challenges. SIAM Rev. 48(2), 207–304 (2006)
Seipel, J., Holmes, P.: A simple model for clock-actuated legged locomotion. Regular Chaotic Dyn. 12(5), 502–520 (2007)
Shen, Z.H., Seipel, J.E.: A fundamental mechanism of legged locomotion with hip torque and leg damping. Bioinspir. Biomim. 7(4), 046010 (2012)
Ankarali, M.M., Saranli, U.: Stride-to-stride energy regulation for robust self-stability of a torque-actuated dissipative spring-mass hopper. Chaos 20, 033121 (2010)
Hamzaçebi, H., Morgül, Ö.: On the periodic gait stability of a multi-actuated spring-mass hopper model via partial feedback linearization. Nonlinear Dyn. 88(2), 1237–1256 (2017)
Raibert, M.H.: Legged Robots that Balance. MIT press, USA (1986)
Poulakakis, I., Papadopoulos, E., Buehler, M.: On the stability of the passive dynamics of quadrupedal running with a bounding gait. Int. J. Robotics Res. 25(7), 669–687 (2006)
Poulakakis, I., Smith, J.A., Buehler, M.: Modeling and experiments of untethered quadrupedal running with a bounding gait: the Scout II robot. Int. J. Robotics Res. 24(4), 239–256 (2005)
Kimura, H., Fukuoka, Y., Cohen, A.H.: Adaptive dynamic walking of a quadruped robot on natural ground based on biological concepts. Int. J. Robotics Res. 26(5), 475–490 (2007)
Fukuoka, Y., Kimura, H., Cohen, A.H.: Adaptive dynamic walking of a quadruped robot on irregular terrain based on biological concepts. Int. J. Robotics Res. 22(3–4), 187–202 (2003)
Cham, J.G., et al.: Fast and robust: Hexapedal robots via shape deposition manufacturing. Int. J. Robotics Res. 21(10–11), 869–882 (2002)
Cham, J.G., Karpick, J.K., Cutkosky, M.R.: Stride period adaptation of a biomimetic running hexapod. Int. J. Robotics Res. 23(2), 141–153 (2004)
Kim, S., Clark, J.E., Cutkosky, M.R.: iSprawl: design and tuning for high-speed autonomous open-loop running. Int. J. Robotics Res. 25(9), 903–912 (2006)
Saranli, U., Buehler, M., Koditschek, D.E.: RHex: A simple and highly mobile hexapod robot. Int. J. Robotics Res. 20(7), 616–631 (2001)
Lin, P.-C., Komsuoglu, H., Koditschek, D.E.: A leg configuration measurement system for full-body pose estimates in a hexapod robot. IEEE Trans. Rob. 21(3), 411–422 (2005)
Lin, P.-C., Komsuoglu, H., Koditschek, D.E.: Sensor data fusion for body state estimation in a hexapod robot with dynamical gaits. IEEE Trans. Rob. 22(5), 932–943 (2006)
Chou, Y.C., et al.: Bio-inspired step-climbing in a hexapod robot. Bioinspiration Biomim 7: 036008 (2012)
Song, X., et al. An adaptive step-climbing method for a RHex-style robot. in 2021 40th Chinese Control Conference (CCC) (2021)
Razzaghi, P., Khatib, E.A., Hurmuzlu, Y.: Nonlinear dynamics and control of an inertially actuated jumper robot. Nonlinear Dyn. 97(1), 161–176 (2019)
Huang, K.-J., Huang, C.-K., Lin, P.-C.: A simple running model with rolling contact and its role as a template for dynamic locomotion on a hexapod robot. Bioinspir. Biomim. 9(4), 046004 (2014)
Lu, W.-C., Yu, M.-Y., Lin, P.-C.: Clock-torqued rolling SLIP model and its application to variable-speed running in a hexapod robot. IEEE Trans. Rob. 34(6), 1643–1650 (2018)
Hu, C.-J., et al.: A torque-actuated dissipative spring loaded inverted pendulum model with rolling contact and its application to hexapod running. Bioinspir. Biomim. 14(2), 026005 (2019)
Yang, W.-S., Lu, W.-C., Lin, P.-C.: Legged robot running using a physics-data hybrid motion template IEEE Transactions Robotics (2021)
Chou, Y.C., et al.: Model-based development of leaping in a hexapod robot. IEEE Trans. Rob. 31(1), 40–54 (2015)
McGeer, T.: Passive dynamic walking. Int. J. Robotics Res. 9(2), 62–82 (1990)
Kajita, S., et al.: The 3D linear inverted pendulum mode: a simple modeling for a biped walking pattern generation. in Proceedings 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems. Expanding the Societal Role of Robotics in the Next Millennium (Cat. No.01CH37180). (2001)
Seipel, J.E., Holmes, P.: Running in three dimensions: analysis of a point-mass sprung-leg model. Int. J. Robotics Res. 24(8), 657–674 (2005)
Wensing, P.M., Orin, D.E.: 3D-SLIP steering for high-speed humanoid turns. in 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems (2014)
Wensing, P.M., Orin, D.E.: High-speed humanoid running through control with a 3D-SLIP model. in 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems (2013)
Faraji, S., Ijspeert, A.J.: 3LP: a linear 3D-walking model including torso and swing dynamics. Int. J. Robotics Res. 36(4), 436–455 (2017)
Sharbafi, M.A., et al.: A 3D Template Model for Healthy and Impaired Walking. in 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). (2018)
Xiong, X., Ames, A.D.: Motion decoupling and composition via reduced order model optimization for dynamic humanoid walking with CLF-QP based Active Force Control. in 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (2019)
Xiong, X. Ames, A.D.: Coupling Reduced Order Models via Feedback Control for 3D Underactuated Bipedal Robotic Walking. in 2018 IEEE-RAS 18th International Conference on Humanoid Robots (Humanoids). (2018)
Huang, C.-K., et al.: Model-based bounding on a quadruped robot. in 2016 IEEE International Conference on Robotics and Automation (ICRA) IEEE (2016)
Burden, S., et al.: Heterogeneous leg stiffness and roll in dynamic running. in Proceedings 2007 IEEE International Conference on Robotics and Automation IEEE (2007)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42. Springer, New York (2013)
Haldane, D.W., Fearing, R.S.: Roll oscillation modulated turning in dynamic millirobots in 2014 IEEE International Conference on Robotics and Automation (ICRA). 2014. IEEE
Berkemeier, M.D.: Modeling the dynamics of quadrupedal running. Int. J. Robotics Res. 17(9), 971–985 (1998)
Chatzakos, P., Papadopoulos, E.: A parametric study on the passive dynamics of straight-ahead level-ground quadrupedal running in 2009 International Conference on Advanced Robotics 2009 IEEE
De, A., Koditschek, D.E.: Vertical hopper compositions for preflexive and feedback-stabilized quadrupedal bounding, pacing, pronking, and trotting. Int. J. Robotics Res. 37(7), 743–778 (2018)
Acknowledgements
This work was supported by Ministry of Science and Technology (MoST), Taiwan, under contract: MOST 109-2813-C-002-161-E, MOST 109-2634-F-002-039-, and MOST 110-2634-F-002-038-.
Funding
This work was supported by Ministry of Science and Technology (MoST), Taiwan, under contract: MOST 109–2813-C-002–161-E, MOST 109–2634-F-002–039-, and MOST 110–2634-F-002–038-.
Author information
Authors and Affiliations
Contributions
All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by I-Chia Chang, Chih-Hsiang Hsu, Hong-Sheng Wu and Pei-Chun Lin. The first draft of the manuscript was written by I-Chia Chang, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors have no relevant financial or non-financial interests to disclose.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Chang, IC., Hsu, CH., Wu, HS. et al. An analysis of the rolling dynamics of a hexapod robot using a three-dimensional rolling template. Nonlinear Dyn 109, 631–655 (2022). https://doi.org/10.1007/s11071-022-07481-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-022-07481-9