Accurate and efficient treatment of electrostatics is a crucial step in computational analyses of biomolecular structures and dynamics. In this study, we have explored a second-order finite-difference numerical method to solve the widely used Poisson-Boltzmann equation for electrostatic analyses of realistic biomolecules. The so-called immersed interface method was first validated and found to be consistent with the classical weighted harmonic averaging method for a diversified set of test biomolecules. The numerical accuracy and convergence behaviors of the new method were next analyzed in its computation of numerical reaction field grid potentials, energies, and atomic solvation forces. Overall similar convergence behaviors were observed as those by the classical method. Interestingly, the new method was found to deliver more accurate and better-converged grid potentials than the classical method on or nearby the molecular surface, though the numerical advantage of the new method is reduced when grid potentials are extrapolated to the molecular surface. Our exploratory study indicates the need for further improving interpolation/extrapolation schemes in addition to the developments of higher-order numerical methods that have attracted most attention in the field.

The Poisson-Boltzmann Surface Area (PBSA) method, when coupled with molecular dynamics (MM, thus the term MMPBSA) simulations has been widely adopted as an efficient and reliable free energy simulation method to model molecular recognition, such as protein-ligand binding interactions. In this dissertation, we first reviewed the recent developments and applications of the MMPBSA method. This is followed with our analysis of different computational alternatives on their impacts to the agreement of MMPBSA calculations with experiment. Most importantly, we studied the performance of nonpolar solvation models and found that the modern approach that separately models hydrophobic and dispersion interactions dramatically reduces RMSD’s of computed relative binding affinities with respect to experiment. Next, we focus on the quality of the PBSA method in modeling ionic interactions by comparing with both explicit solvent simulations and experiment. Specifically, the molality-dependent chemical potentials for sodium chloride (NaCl) electrolyte were simulated in both SPC/E explicit solvent and PBSA solvent models. Our comparative analysis shows that the molality-dependent chemical potentials of NaCl are reproduced well with both linear PB and nonlinear PB, with nonlinear PB agrees better with SPC/E and experiment. However, for non-electrostatics/van der Waals solvation free energies, the absence of molality dependence in the PBSA method dramatically reduces its quality in modeling of the salt-dependent ionic interactions. Given the above analyses of PBSA method in various applications, we explored a new numerical method for more accurate and robust electrostatic modeling of biomolecular systems. The new method was found to deliver more accurate and better-converged grid potentials than the classical method on or nearby the molecular surface, important for highly demanding applications in direct application to molecular dynamics simulations. Finally, we focus on another important issue to utilize the PBSA method in molecular dynamics simulations. Specifically, we implemented and analyzed a range of standard surface fitting schemes to calculate molecular surface electric fields and dielectric boundary forces, crucial for porting PBSA methods to molecular dynamics engines. Given the developments presented here, we are in a better position to apply PBSA method for biomolecular simulations for more robust free energy simulations with a wider range of applications.

Robots are designed to perform tasks with precision and efficiency, often surpassing human capabilities. They not only enhance productivity in manufacturing but also work in areas inaccessible or hazardous to humans. In homes, robotic vacuum cleaners and assistants make daily life more convenient. In healthcare, they assist in complex surgeries and patient care. Moreover, the integration of machine learning and large models further equips robots with intelligence, enabling them to learn from their environments and make informed decisions without additional human commands. However, as those models or policies are learned offline, it is still an open question of whether the robots can reliably interact in the new scenarios online. To address this question, this dissertation delves into the development of efficient and robust methods that empower robots to learn and plan manipulation trajectories in real-time, and to self-adjust their skills based on live sensor feedback.

The core of this dissertation is anchored in three principal challenges:

Efficient Trajectory Optimization: Model predictive control (MPC) plays an important role in online robotic planning. In these scenarios, a robot will optimize its future actions based on the dynamics model. Therefore, the development of efficient formulations and algorithms for MPC and trajectory optimization is crucial for real-time robot learning and skill adaptation. In Part I of the dissertation, we introduce algorithms and formulations tailored for real-time robotic trajectory optimization. Our focus is on enhancing the computational speed of the optimization process while ensuring the resulting trajectory maintains high quality. These advancements can equip robots with the ability to swiftly plan and adjust their movements in novel environmental conditions.

Adaptive Model Learning for Deformable Object Manipulation: Building on the efficient MPC formulation presented in Part I, the second part of the dissertation shifts its focus to the real-time enhancement of robotic behaviors. This enhancement involves dynamically adjusting the robot's dynamics model based on real-world sensor data. Part II specifically addresses online model learning for the manipulation of deformable objects, such as ropes and fabrics. These tasks, characterized by their complex and non-linear state changes, present substantial challenges for model-based methods, yet are critically important for everyday applications. In this part of the dissertation, we introduce algorithms capable of learning the dynamics of these objects and effectively manipulating them. Our research emphasizes the integration of real-time sensory feedback with machine learning algorithms. This synergy is crucial for the continuous refinement and updating of the dynamics model. We illustrate how such real-time adjustments are vital for enabling robotic systems to modify their manipulation strategies effectively.

Real-time Control Policy Adaptation for Contact-Rich Manipulation Tasks: While the MPC and adaptive model learning can provide robots with more reliable trajectories, another crucial aspect for real-time execution is low-level force control, which calculates the motor torque for reliable environmental interaction. Part III of the dissertation concentrates on optimizing control policies for robotic systems engaged in contact-rich manipulation tasks such as assembly, pivoting, and screwing. The main emphasis is on the ability of robotic systems to dynamically adjust their control policies in response to force and torque feedback. This level of adaptation is critical for the successful completion of tasks requiring delicate and precise physical interactions. By doing so, we demonstrate the improvement of the robot's motion under massive contacts.