The stability and safety of robotic systems are heavily impacted by delays and parametric uncertainties due to external disturbances, modeling inaccuracies, reaction forces, and variations in dynamics. This work addresses the effects of parametric uncertainties in the application of payload transportation by robotic systems that involve time delays and state constraints. The problem is split into two research questions: control of a quadrotor UAV in the presence of delays and control of robotic systems with state constraints. The first two papers explore the approaches for remotely operated quadrotors in the presence of delays and uncertainties. Specifically, the first paper surveys the existing methods for controlling a payload-carrying UAV and further presents a class of control techniques in theory that focus on time-delayed systems. The second paper proposes an adaptive control solution for the tracking control of a quadrotor UAV to transport various unknown payloads in the presence of unknown time-varying delays. The proposed controller is robust to modeling uncertainties and does not require knowledge of the uncertainties' bounds. The performance of the controller is verified on a MATLAB-SIMULINK simulated environment. The final three papers deal with enforcing state constraints on tracking control to ensure the safety of the robots in the presence of parametric uncertainties. The third paper exploits state constraints in the post-grasping scenario of the space debris disposal application. This work proposes a robust control for a space robot to follow the desired trajectory without any violation to safely grasp, carry, and release unknown payloads in their respective regions. The controller is tested in a MATLAB-SIMULINK environment with the dynamics of a planar space robot. The fourth paper introduces an adaptive control technique without any a priori knowledge of the system dynamics or the bounds of uncertainties to impose state constraints in control. The proposed controller is designed for a generic Euler-Lagrangian system in the presence of parametric uncertainties, where the state-dependent nature of the uncertainties introduces unboundedness in the overall uncertainty. The controller is validated in simulation using a robotic manipulator in a pick-and-place operation. The final paper proposes an adaptive controller for the tracking control of an experimental planar space robot. The proposed controller enforces constraints on the robot's states and their derivatives on the tracking control for transporting different payloads without any knowledge of the dynamics of the robot or the bounds of the uncertainties. The controller is validated on the experimental space robot. The stability of the proposed controllers is studied analytically using the Lyapunov theory. The results are presented with various plots and numerically analyzed on the metrics of root mean squared errors and peak errors.