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1. Discrete Mechanics and Optimal Control for Passive Walking with Foot Slippage

   Simoes, A; Bloch, A; Colomobo, L (June 2023, Proc. American Control Conference 2023)

   Forced variational integrators are given by the discretization of the Lagrange-d’Alembert principle for systems subject to external forces, and have proved useful for numerical simulation studies of complex dynamical systems. In this paper we model a passive walker with foot slip by using techniques of geometric mechanics, and we construct forced variational integrators for the system. Moreover, we present a methodology for generating (locally) optimal control policies for simple hybrid holonomically constrained forced Lagrangian systems, based on discrete mechanics, applied to a controlled walker with foot slip in a trajectory tracking problem. 

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2. Discrete Mechanics and Optimal Control for Passive Walking with Foot Slippage

   Simoes, A; L´opez-Gordon, A; Bloch, A; Colombo, L (June 2023, Proceedings ACC)

   Forced variational integrators are given by the discretization of the Lagrange-d’Alembert principle for systems subject to external forces, and have proved useful for numerical simulation studies of complex dynamical systems. In this paper we model a passive walker with foot slip by using techniques of geometric mechanics, and we construct forced variational integrators for the system. Moreover, we present a methodology for generating (locally) optimal control policies for simple hybrid holonomically constrained forced Lagrangian systems, based on discrete mechanics, applied to a controlled walker with foot slip in a trajectory tracking problem. 

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3. Balance Recoverability and Control of Bipedal Walkers With Foot Slip

   https://doi.org/10.1115/1.4053098  

   Mihalec, Marko; Trkov, Mitja; Yi, Jingang (May 2022, Journal of Biomechanical Engineering)

   Abstract Low-friction foot/ground contacts present a particular challenge for stable bipedal walkers. The slippage of the stance foot introduces complexity in robot dynamics and the general locomotion stability results cannot be applied directly. We relax the commonly used assumption of nonslip contact between the walker foot and the ground and examine bipedal dynamics under foot slip. Using a two-mass linear inverted pendulum model, we introduce the concept of balance recoverability and use it to quantify the balanced or fall-prone walking gaits. Balance recoverability also serves as the basis for the design of the balance recovery controller. We design the within- or multi-step recovery controller to assist the walker to avoid fall. The controller performance is validated through simulation results and robustness is demonstrated in the presence of measurement noises as well as variations of foot/ground friction conditions. In addition, the proposed methods and models are used to analyze the data from human walking experiments. The multiple subject experiments validate and illustrate the balance recoverability concept and analyses. 

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4. Convex geometric motion planning of multi-body systems on lie groups via variational integrators and sparse moment relaxation

   https://doi.org/10.1177/02783649241296160  

   Teng, Sangli; Jasour, Ashkan; Vasudevan, Ram; Ghaffari, Maani (November 2024, The International Journal of Robotics Research)

   This paper reports a novel result: with proper robot models based on geometric mechanics, one can formulate the kinodynamic motion planning problems for rigid body systems as exact polynomial optimization problems. Due to the nonlinear rigid body dynamics, the motion planning problem for rigid body systems is nonconvex. Existing global optimization-based methods do not parameterize 3D rigid body motion efficiently; thus, they do not scale well to long-horizon planning problems. We use Lie groups as the configuration space and apply the variational integrator to formulate the forced rigid body dynamics as quadratic polynomials. Then, we leverage Lasserre’s hierarchy of moment relaxation to obtain the globally optimal solution via semidefinite programming. By leveraging the sparsity of the motion planning problem, the proposed algorithm has linear complexity with respect to the planning horizon. This paper demonstrates that the proposed method can provide globally optimal solutions or certificates of infeasibility at the second-order relaxation for 3D drone landing using full dynamics and inverse kinematics for serial manipulators. Moreover, we extend the algorithms to multi-body systems via the constrained variational integrators. The testing cases on cart-pole and drone with cable-suspended load suggest that the proposed algorithms can provide rank-one optimal solutions or nontrivial initial guesses. Finally, we propose strategies to speed up the computation, including an alternative formulation using quaternion, which provides empirically tight relaxations for the drone landing problem at the first-order relaxation. 

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5. Efficient Computation of Higher-Order Variational Integrators in Robotic Simulation and Trajectory Optimization

   https://doi.org/10.1007/978-3-030-44051-0_40  

   Fan, T.; Schultz, J.; Murphey, T. (May 2020, Workshop on the Algorithmic Foundations of Robotics)

   This paper addresses the problem of efficiently computing higher- order variational integrators in simulation and trajectory optimization of mechanical systems as those often found in robotic applications. We develop O(n) algorithms to evaluate the discrete Euler-Lagrange (DEL) equations and compute the Newton direction for solving the DEL equations, which results in linear-time variational integrators of arbitrarily high order. To our knowledge, no linear-time higher-order variational or even implicit integrators have been developed before. Moreover, an O(n2) algorithm to linearize the DEL equations is presented, which is useful for trajectory optimization. These proposed algorithms eliminate the bottleneck of implementing higher-order variational integrators in simulation and trajectory optimization of complex robotic systems. The efficacy of this paper is validated through comparison with existing methods, and implementation on various robotic systems—including trajectory optimization of the Spring Flamingo robot, the LittleDog robot and the Atlas robot. The results illustrate that the same integrator can be used for simulation and trajectory optimization in robotics, preserving mechanical properties while achieving good scalability and accuracy. 

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