Robot motion can be described in several alternative representations, including joint configuration or end-effector spaces. These representations are often used for manipulation or navigation tasks but they are not suitable for tasks that involve close interaction with the environment. In these scenarios, collisions and relative poses of the robot and its surroundings create a complex planning space. To deal with this complexity, we exploit several representations that capture the state of the interaction, rather than the state of the robot. Borrowing notions of topology invariances and homotopy classes, we design task spaces based on winding numbers and writhe for synthesizing winding motion, and electro-static fields for planning reaching and grasping motion. Our experiments show that these representations capture the motion, preserving its qualitative properties, while generalising over finer geometrical detail. Based on the same motivation, we utilise a scale and rotation invariant representation for locally preserving distances, called interaction mesh. The interaction mesh allows for transferring motion between robots of different scales (motion re-targeting), between humans and robots (teleoperation) and between different environments (motion adaptation). To estimate the state of the environment we employ real-time sensing techniques utilizing dense stereo tracking, magnetic tracking sensors and inertia measurements units. We combine and exploit these representations for synthesis and generalization of motion in dynamic environments. The benefit of this method is on problems where direct planning in joint space is extremely hard whereas local optimal control exploiting topology and metric of these novel representations can efficiently compute optimal trajectories. We formulate this approach in the framework of optimal control as an approximate inference problem. This allows for consistent combination of multiple task spaces (e.g. end-effector, joint space and the abstract task spaces we investigate in this thesis). Motion generalization to novel situations and kinematics is similarly performed by projecting motion from abstract representations to joint configuration space. This technique, based on operational space control, allows us to adapt the motion in real time. This process of real-time re-mapping generates robust motion, thus reducing the amount of re-planning. We have implemented our approach as a part of an open source project called the Extensible Optimisation library (EXOTica). This software allows for defining motion synthesis problems by combining task representations and presenting this problem to various motion planners using a common interface. Using EXOTica, we perform comparisons between different representations and different planners to validate that these representations truly improve the motion planning.

Complex realistic human motion sequences satisfying environmental constraints can be created by motion capture, which is a reliable way to reproduce human motions. However, motion capture data is difficult to modify in order to obtain variant motion sequences for multiple tasks. In this thesis, a system for synthesizing complex realistic human motion sequences based on a small group of sample motions to satisfy constraints is proposed. Methods are proposed for the system to preprocesses raw motion capture data to create sample motions that can be easily modified for the purpose of meeting specific requirements, while maintaining the subtleties of the origin motion capture data. Methods for the system to scan user-input constraints, to choose the best sample motion and synthesize the motion sequence based on route affected by the constraint are also proposed. Each generated motion piece is blended with the default motion, and thus a motion sequence composed of several pieces of motion based on constraints will be generated. Artifacts that arise during motion generation are identified and handled properly. Experimental results will show that the system can create cyclical sample motions from motion capture data, generate motion pieces based on environmental constraints, and synthesize complex realistic human motion sequences.