Dynamic motion of quadrupedal robots on challenging terrain: a kinodynamic optimization approach

Abstract

The dynamic motion of quadrupedal robots on challenging terrain generally requires elaborate spatial–temporal kinodynamic motion planning and accurate control at higher refresh rate in comparison with regular terrain. However, conventional quadrupedal robots usually generate relatively coarse planning and employ motion replanning or reactive strategies to handle terrain irregularities. The resultant complex and computation-intensive controller may lead to nonoptimal motions or the breaking of locomotion rhythm. In this paper, a kinodynamic optimization approach is presented. To generate long-horizon optimal predictions of the kinematic and dynamic behavior of the quadruped robot on challenging terrain, we formulate motion planning as an optimization problem; jointly treat the foot’s locations, contact forces, and torso motions as decision variables; combine smooth motion and minimal energy consumption as the objective function; and explicitly represent feasible foothold region and friction constraints based on terrain information. To track the generated motions accurately and stably, we employ a whole-body controller to compute reference position and velocity commands, which are fed forward to joint controllers of the robot’s legs. We verify the effectiveness of the developed approach through simulation and on a physical quadruped robot testbed. Results show that the quadruped robot can successfully traverse a 30° slope and 43% of nominal leg length high step while maintaining the rhythm of dynamic trot gait.

Key words： quadrupedal robot; kinodynamic planning; nonlinear optimization; challenging terrain; whole-body control

Acknowledgements

This work was supported by the Foundation of Engineering Research Center of Hubei Province for Clothing Information, China (Grant No. 2023HBCI05), the Hubei Provincial Natural Science Foundation General Program, China (Grant No. 2022CFB563), and the Hubei Key Laboratory for New Textile Materials and Applications, Wuhan Textile University, China (Grant No. FZXCL202311).

Conflict of Interest

The authors declare that they have no conflict of interest.

References

1 Wilshin S, Reeve M A, Spence A J. Dog galloping on rough terrain exhibits similar limb coordination patterns and gait variability to that on flat terrain. Bioinspiration & Biomimetics, 2021, 16(1): 015001

2 Liu Q Y, Chen X D, Han B, Luo Z W, Luo X. Learning control of quadruped robot galloping. Journal of Bionic Engineering, 2018, 15(2): 329–340

3 Zhang G T, Li Y B, Ma S G. Trotting and pacing locomotion of a position-controlled quadruped robot. In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems. Prague: IEEE, 2021, 7456–7463

4 Spröwitz A, Tuleu A, Vespignani M, Ajallooeian M, Badri E, Ijspeert A J. Towards dynamic trot gait locomotion: design, control, and experiments with Cheetah-cub, a compliant quadruped robot. The International Journal of Robotics Research, 2013, 32(8): 932–950

5 Koo I M, Trong T D, Lee Y H, Moon H, Koo J, Park S, Choi H R. Biologically inspired gait transition control for a quadruped walking robot. Autonomous Robots, 2015, 39(2): 169–182

6 Mercy T, Van Parys R, Pipeleers G. Spline-based motion planning for autonomous guided vehicles in a dynamic environment. IEEE Transactions on Control Systems Technology, 2018, 26(6): 2182–2189

7 Fukui T, Fujisawa H, Otaka K, Fukuoka Y. Autonomous gait transition and galloping over unperceived obstacles of a quadruped robot with CPG modulated by vestibular feedback. Robotics and Autonomous Systems, 2019, 111: 1–19

8 Roennau A, Heppner G, Nowicki M, Zoellner J M, Dillmann R. Reactive posture behaviors for stable legged locomotion over steep inclines and large obstacles. In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems. Chicago: IEEE, 2014, 4888–4894

9 Barasuol V, Buchli J, Semini C, Frigerio M, De Pieri E R, Caldwell D G. A reactive controller framework for quadrupedal locomotion on challenging terrain. In: Proceedings of the IEEE International Conference on Robotics and Automation. Karlsruhe: IEEE, 2013, 2554–2561

10 Xin G Y, Lin H C, Smith J, Cebe O, Mistry M. A model-based hierarchical controller for legged systems subject to external disturbances. In: Proceedings of the IEEE International Conference on Robotics and Automation. Brisbane: IEEE, 2018, 4375–4382

11 Li Q, Sun P, Zhao C L, Luo X. Adaptive strategies for quadruped robot to climb high-slope terrain without priori information. In: Proceedings of the IEEE International Conference on Robotics and Biomimetics. Jinghong: IEEE, 2022, 1396–1401

12 Zico Kolter J, Ng A Y. The Stanford littledog: a learning and rapid replanning approach to quadruped locomotion. The International Journal of Robotics Research, 2011, 30(2): 150–174

13 Zucker M, Ratliff N, Stolle M, Chestnutt J, Bagnell J A, Atkeson C G, Kuffner J. Optimization and learning for rough terrain legged locomotion. The International Journal of Robotics Research, 2011, 30(2): 175–191

14 Kalakrishnan M, Buchli J, Pastor P, Mistry M, Schaal S. Learning, planning, and control for quadruped locomotion over challenging terrain. The International Journal of Robotics Research, 2011, 30(2): 236–258

15 Jenelten F, Miki T, Vijayan A E, Bjelonic M, Hutter M. Perceptive locomotion in rough terrain––online foothold optimization. IEEE Robotics and Automation Letters, 2020, 5(4): 5370–5376

16 Chen L, Ye S S, Sun C M, Zhang A D, Deng G Y, Liao T J. Optimized foothold planning and posture searching for energy-efficient quadruped locomotion over challenging terrains. In: Proceedings of the IEEE International Conference on Robotics and Automation. Paris: IEEE, 2020, 399–405

17 Bellicoso C D, Jenelten F, Gehring C, Hutter M. Dynamic locomotion through online nonlinear motion optimization for quadrupedal robots. IEEE Robotics and Automation Letters, 2018, 3(3): 2261–2268

18 Fankhauser P, Bjelonic M, Bellicoso C D, Miki T, Hutter M. Robust rough-terrain locomotion with a quadrupedal robot. In: Proceedings of the IEEE International Conference on Robotics and Automation. Brisbane: IEEE, 2018, 5761–5768

19 Aceituno-Cabezas B, Mastalli C, Dai H K, Focchi M, Radulescu A, Caldwell D G, Cappelletto J, Grieco J C, Fernández-López G, Semini C. Simultaneous contact, gait, and motion planning for robust multilegged locomotion via mixed-integer convex optimization. IEEE Robotics and Automation Letters, 2018, 3(3): 2531–2538

20 Bjelonic M, Sankar P K, Bellicoso C D, Vallery H, Hutter M. Rolling in the deep––hybrid locomotion for wheeled-legged robots using online trajectory optimization. IEEE Robotics and Automation Letters, 2020, 5(2): 3626–3633

21 Di Carlo J, Wensing P M, Katz B, Bledt G, Kim S. Dynamic locomotion in the MIT cheetah 3 through convex model-predictive control. In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems. Madrid: IEEE, 2018, 1–9

22 Bledt G, Kim S. Implementing regularized predictive control for simultaneous real-time footstep and ground reaction force optimization. In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems. Macau: IEEE, 2019, 6316–6323

23 Grandia R, Farshidian F, Ranftl R, Hutter M. Feedback MPC for torque-controlled legged robots. In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems. Macau: IEEE, 2019, 4730–4737

24 Neunert M, Stäuble M, Giftthaler M, Bellicoso C D, Carius J, Gehring C, Hutter M, Buchli J. Whole-body nonlinear model predictive control through contacts for quadrupeds. IEEE Robotics and Automation Letters, 2018, 3(3): 1458–1465

25 Jenelten F, Grandia R, Farshidian F, Hutter M. TAMOLS: terrain-aware motion optimization for legged systems. IEEE Transactions on Robotics, 2022, 38(6): 3395–3413

26 Winkler A W, Bellicoso C D, Hutter M, Buchli J. Gait and trajectory optimization for legged systems through phase-based end-effector parameterization. IEEE Robotics and Automation Letters, 2018, 3(3): 1560–1567

27 Bjelonic M, Grandia R, Geilinger M, Harley O, Medeiros V S, Pajovic V, Jelavic E, Coros S, Hutter M. Offline motion libraries and online MPC for advanced mobility skills. The International Journal of Robotics Research, 2022, 41(9–10): 903–924

28 Li Q, Qian L T, Sun P, Luo X. Energy-efficient dynamic motion planning of quadruped robots via whole-body nonlinear trajectory optimization. In: Proceedings of the IEEE International Conference on Mechatronics and Automation. Guilin: IEEE, 2022, 1610–1615

29 Li Q, Qian L T, Wang S H, Sun P, Luo X. Towards generation and transition of diverse gaits for quadrupedal robots based on trajectory optimization and whole-body impedance control. IEEE Robotics and Automation Letters, 2023, 8(4): 2389–2396

30 Siciliano B, Khatib O. Springer Handbook of Robotics. 2nd ed. Cham: Springer, 2016, 11–36

31 Andersson J A E, Gillis J, Horn G, Rawlings J B, Diehl M. CasADi: a software framework for nonlinear optimization and optimal control. Mathematical Programming Computation, 2019, 11(1): 1–36

32 Wächter A, Biegler L T. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical Programming, 2006, 106(1): 25–57

33 Felis M L. RBDL: an efficient rigid-body dynamics library using recursive algorithms. Autonomous Robots, 2017, 41(2): 495–511

34 DiehlMBock H GDiedamHWieberP B. Fast direct multiple shooting algorithms for optimal robot control. In: Diehl M, Mombaur K, eds. Fast Motions in Biomechanics and Robotics: Optimization and Feedback Control. Heidelberg: Springer, 2006, 65–93

35 Norby J, Yang Y, Tajbakhsh A, Ren J, Yim J K, Stutt A, Yu Q, Flowers N, Johnson A M. Quad-SDK: full stack software framework for agile quadrupedal locomotion. In ICRA Workshop on Legged Robots, 2022