Automated driving is where automobiles meet robotics. With the recent advances in intelligence, sensor technology, wireless technology, and computation power, we are inching ever closer to realising full autonomy in a vehicle. We are nowhere near the end of the line, however. Automated vehicles will have to interact with static obstacles like pavements, dividers, and poles and dynamic elements like pedestrians and other vehicles. This opens up a range of sub-topics on privacy, safety, and decision-making. While driving on the road in the presence of other agents(human or autonomous), safety constraints and collision avoidance are of paramount importance. Even with this achieved, we need a good performance in the latency of processing each iteration; we need constraints to be followed and, of course, ensure minimal errors in our control. Through this thesis, we introduce automated driving, its general pipeline stack, and industry standards for autonomy. We then get ourselves up-to-date with the latest trends and advances in motion planning, decision-making, and control of automated vehicles. Once that is covered, we narrow our focus towards using a scenario-based approach to safe trajectory planning. We delve in-depth into safety constraints guaranteed by this approach and discuss previous results obtained by using this method with pedestrians in an urban setting. This thesis aims to extend this previously used scenario-based planning method to a multi-vehicle implementation. Two methods of modelling scenario distributions(assumed to be Gaussian) are proposed, implemented, and compared using evaluation metrics like computation times, safety, and time taken to reach the goal. The first method models each obstacle vehicle as a series of obstacles linked together by the vehicle constraints. Thus, each vehicle is represented by multiple collision regions corresponding to each obstacle. The second method models the vehicle as having a single collision region with multiple scenario distributions. This extension's safety/risk guarantee is shown both theoretically and experimentally. Experiments are conducted in simulation on an urban straight road and at a T-Junction with multiple obstacle vehicles, and performances are compared, not only between these two methods but also with each obstacle modelled as a single disc, which is the baseline implementation. Conclusions are then made based on the performance metrics, and further improvements are proposed. It is shown that modelling the vehicle as a series of linked scenarios improves over the baseline method and the multiple obstacle discs implementation in terms of safety and computation times respectively.