With the onset of the age of space travel, asteroid missions have been steadily gaining interest. The pristine nature of asteroids due to their preserved state since the formation of the Solar System is an opportunity to unravel many mysteries about the Solar System. Also with the ever-growing need for resources, asteroids prove to be a plentiful source. With the discovery of asteroids in the close vicinity of our planet and a probable threat to the preservation of life, a need for defence missions has also arisen. The ever-growing need for better computational speed and accuracy has led to the development of new representations for attitude and position of the spacecraft in the past. The usual methods of representations of position and attitude (pose) are the Cartesian coordinates and quaternions. A recent development is the simultaneous representation of the pose of the spacecraft using dual quaternions which are eight-dimensional vectors. In this thesis, an attempt to incorporate these state-of-the-art guidance technologies to asteroid missions has been made. The novelty in this thesis is, using dual quaternions for SC pose and attitude representation to autonomously guide the spacecraft for mapping an asteroid and perform a touch and go descent using sampling-based motion predictive optimisation and successive convexification respectively. After the development and verification of these algorithms, different scenarios have been designed to find out the robustness of the dynamic successive convexification method. The scenarios have been run by both the Cartesian quaternion and dual quaternion based algorithms. They are found to behave similarly in their results, besides the latter being computationally more expensive as proved by different theses so far. The algorithm performs better than the one developed by Szmuk et al. (2017), but faces difficulties with badly scaled problems as is the nature of missions to asteroids. The software used for the thesis is ECOS, which faces numerical problems in these mission scenarios even when the problem is feasible and has an optimal result. The availability of a solution to the optimal control problem depends on the scaling of the penalty weights used in the cost function to penalise virtual controls and trust region, which are used to prevent infeasibility and bound the problem, respectively. It also depends on finding an appropriate final time for the descent along with the number of nodes for discretisation. This research proves, that successive convexification indeed provides a speedy solution for an autonomous precision descent but needs further work to make it robust and stable. The outcome of this thesis is to carry out further research to understand the complex relations, between the scale of the problem, simulation parameters for the optimal control problem and final time to make the algorithm robust and safe for an autonomous mission. Another important future research prospect is to incorporate the sampling based model predictive optimisation, for the SC to autonomously map the target body and help in selecting a landmark for a descent.