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This work aims to enhance the linear inverted pendulum model (LIPM) for bipedal robot control. While the LIPM simplifies the dynamics by assuming homogeneity, it fails to capture critical nonlinear dynamics observed in real-world scenarios. To address this limitation, we propose the non-homogeneous LIPM (NH-LIPM), which incorporates a non-homogeneous term in the traditional LIPM dynamics. The NH-LIPM is augmented with controllable inputs, allowing for greater parameter control compared to the LIPM. Through regression analysis and the use of the Recursive Least Squares algorithm with forgetting, we extract and adaptively tune the NH-LIPM parameters. Evaluation through high-fidelity simulation and experimentation on a 30-degree-of-freedom humanoid demonstrates that the NH-LIPM offers improved velocity tracking control, particularly when ankle torque with damping control is added. This model provides a flexible framework for simultaneously controlling the center of mass velocity and position, enabling precise reference tracking and enhanced bipedal locomotion. A video is in this shortened link: http://tiny.cc/NHLIPM 

The ability of quadrupedal robots to follow commanded velocities is important for navigating in constrained environments such as homes and warehouses. This paper presents a simple, scalable approach to realize high fidelity speed regulation and demonstrates its efficacy on a quadrupedal robot. Using analytical inverse kinematics and gravity compensation, a task-level controller calculates joint torques based on the prescribed motion of the torso. Due to filtering and feedback gains in this controller, there is an error in tracking the velocity. To ensure scalability, these errors are corrected at the time scale of a step using a Poincar´e map (a mapping of states and control between consecutive steps). A data-driven approach is used to identify a decoupled Poincar´e map, and to correct for the tracking error in simulation. However, due to model imperfections, the simulation-derived Poincar´e map-based controller leads to tracking errors on hardware. Three modeling approaches – a polynomial, a Gaussian process, and a neural network – are used to identify a correction to the simulation-based Poincar´e map and to reduce the tracking error on hardware. The advantages of our approach are the computational simplicity of the task-level controller (uses analytical computations and avoids numerical searches) and scalability of the sim-to-real transfer (use of low-dimensional Poincar´e map for sim-to-real transfer). A video is in this shortened link: http://tiny.cc/humanoids23