<inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula>- and <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_\infty$</tex-math></inline-formula>-Optimal Model Predictive Controllers for Robust Legged Locomotion
This paper formally develops robust optimal predictive control solutions that can accommodate disturbances and stabilize periodic legged locomotion. To this end, we build upon existing optimization-based control paradigms, particularly quadratic programming (QP)-based model predictive controllers (M...
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2024-01-01
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| Series: | IEEE Open Journal of Control Systems |
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| Online Access: | https://ieeexplore.ieee.org/document/10543084/ |
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| author | Abhishek Pandala Aaron D. Ames Kaveh Akbari Hamed |
| author_facet | Abhishek Pandala Aaron D. Ames Kaveh Akbari Hamed |
| author_sort | Abhishek Pandala |
| collection | DOAJ |
| description | This paper formally develops robust optimal predictive control solutions that can accommodate disturbances and stabilize periodic legged locomotion. To this end, we build upon existing optimization-based control paradigms, particularly quadratic programming (QP)-based model predictive controllers (MPCs). We present conditions under which the closed-loop reduced-order systems (i.e., template models) with MPC have the continuous differentiability property on an open neighborhood of gaits. We then linearize the resulting discrete-time, closed-loop nonlinear template system around the gait to obtain a linear time-varying (LTV) system. This periodic LTV system is further transformed into a linear system with a constant state-transition matrix using discrete-time Floquet transform. The system is then analyzed to accommodate parametric uncertainties and to synthesize robust optimal <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_\infty$</tex-math></inline-formula> feedback controllers via linear matrix inequalities (LMIs). The paper then extends the theoretical results to the single rigid body (SRB) template dynamics and numerically verifies them. The proposed robust optimal predictive controllers are used in a layered control structure, where the optimal reduced-order trajectories are provided to a full-order nonlinear whole-body controller (WBC) for tracking at the low level. The developed layered controllers are numerically and experimentally validated for the robust locomotion of the A1 quadrupedal robot subject to various disturbances and uneven terrains. Our numerical results suggest that the <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula>- and <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_\infty$</tex-math></inline-formula>-optimal MPC controllers significantly improve the robust stability of the gaits compared to the normal MPC. |
| format | Article |
| id | doaj-art-b6c0b9bc9246411c9e69621d8fa9ad80 |
| institution | Kabale University |
| issn | 2694-085X |
| language | English |
| publishDate | 2024-01-01 |
| publisher | IEEE |
| record_format | Article |
| series | IEEE Open Journal of Control Systems |
| spelling | doaj-art-b6c0b9bc9246411c9e69621d8fa9ad802025-01-09T00:03:03ZengIEEEIEEE Open Journal of Control Systems2694-085X2024-01-01322523810.1109/OJCSYS.2024.340799910543084<inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula>- and <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_\infty$</tex-math></inline-formula>-Optimal Model Predictive Controllers for Robust Legged LocomotionAbhishek Pandala0https://orcid.org/0000-0002-7424-5508Aaron D. Ames1https://orcid.org/0000-0003-0848-3177Kaveh Akbari Hamed2https://orcid.org/0000-0001-9597-1691Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA, USADepartment of Mechanical and Civil Engineering, California Institute of Technology, Pasadena, CA, USADepartment of Mechanical Engineering, Virginia Tech, Blacksburg, VA, USAThis paper formally develops robust optimal predictive control solutions that can accommodate disturbances and stabilize periodic legged locomotion. To this end, we build upon existing optimization-based control paradigms, particularly quadratic programming (QP)-based model predictive controllers (MPCs). We present conditions under which the closed-loop reduced-order systems (i.e., template models) with MPC have the continuous differentiability property on an open neighborhood of gaits. We then linearize the resulting discrete-time, closed-loop nonlinear template system around the gait to obtain a linear time-varying (LTV) system. This periodic LTV system is further transformed into a linear system with a constant state-transition matrix using discrete-time Floquet transform. The system is then analyzed to accommodate parametric uncertainties and to synthesize robust optimal <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_\infty$</tex-math></inline-formula> feedback controllers via linear matrix inequalities (LMIs). The paper then extends the theoretical results to the single rigid body (SRB) template dynamics and numerically verifies them. The proposed robust optimal predictive controllers are used in a layered control structure, where the optimal reduced-order trajectories are provided to a full-order nonlinear whole-body controller (WBC) for tracking at the low level. The developed layered controllers are numerically and experimentally validated for the robust locomotion of the A1 quadrupedal robot subject to various disturbances and uneven terrains. Our numerical results suggest that the <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula>- and <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_\infty$</tex-math></inline-formula>-optimal MPC controllers significantly improve the robust stability of the gaits compared to the normal MPC.https://ieeexplore.ieee.org/document/10543084/Nonlinear systems and controloptimal controlroboticsrobust controlstability of nonlinear systems |
| spellingShingle | Abhishek Pandala Aaron D. Ames Kaveh Akbari Hamed <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula>- and <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_\infty$</tex-math></inline-formula>-Optimal Model Predictive Controllers for Robust Legged Locomotion IEEE Open Journal of Control Systems Nonlinear systems and control optimal control robotics robust control stability of nonlinear systems |
| title | <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula>- and <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_\infty$</tex-math></inline-formula>-Optimal Model Predictive Controllers for Robust Legged Locomotion |
| title_full | <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula>- and <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_\infty$</tex-math></inline-formula>-Optimal Model Predictive Controllers for Robust Legged Locomotion |
| title_fullStr | <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula>- and <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_\infty$</tex-math></inline-formula>-Optimal Model Predictive Controllers for Robust Legged Locomotion |
| title_full_unstemmed | <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula>- and <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_\infty$</tex-math></inline-formula>-Optimal Model Predictive Controllers for Robust Legged Locomotion |
| title_short | <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula>- and <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_\infty$</tex-math></inline-formula>-Optimal Model Predictive Controllers for Robust Legged Locomotion |
| title_sort | inline formula tex math notation latex mathcal h 2 tex math inline formula and inline formula tex math notation latex mathcal h infty tex math inline formula optimal model predictive controllers for robust legged locomotion |
| topic | Nonlinear systems and control optimal control robotics robust control stability of nonlinear systems |
| url | https://ieeexplore.ieee.org/document/10543084/ |
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