Analytical motion planning on 3D-Lie groups

In this thesis optimal analytical controls derived using the framework of geometric control theory are used to plan motions for real engineering systems, and their performance assessed. Beginning with the nonholonomic wheeled robot on the 3-D Special Euclidean group, the co-ordinate free Maximum Principle of optimal control is used to create a kinematic motion planner via parametric optimisation. The reachable sets of the planner are investigated, and an obstacle avoidance framework is created and demonstrated. The practical applicability of this motion planner is assessed in both the unit speed and arbitrary speed cases. Subsequently, the natural motions of an axisymmetric and asymmetric rigid body are derived in convenient quaternion form, which is particularly suitable for practical implementation. The initial angular velocities of the rigid body are parametrically optimised to produce attitude reference tracks for a small spacecraft. It is shown through comparison with a standard proportional-derivative controller that the natural motions require lower accumulated torque to track. Additional testing of the axisymmetric references, which are comprised of simple trigonometric functions, found that the references can be utilised in a "bang-off-bang" manner in low disturbance environments to save on computation and control effort. Next, the general solution to the optimal control problem of a rigid body constrained to spin at a constant rate is derived. This takes the form of a Weierstrass elliptic function which is impractical for motion planning. However, a particular case is identified that is comprised of trigonometric functions and utilised to plan efficient motions for a spinning solar sail and compared in simulation to a pure spin benchmark. The geometric references are found to limit the spacecraft body rates, while tracking requires lower accumulated torque in most cases. Finally, an actuator study is carried out to identify the technology requirements for reference tracking for a spinning solar sail.In this thesis optimal analytical controls derived using the framework of geometric control theory are used to plan motions for real engineering systems, and their performance assessed. Beginning with the nonholonomic wheeled robot on the 3-D Special Euclidean group, the co-ordinate free Maximum Principle of optimal control is used to create a kinematic motion planner via parametric optimisation. The reachable sets of the planner are investigated, and an obstacle avoidance framework is created and demonstrated. The practical applicability of this motion planner is assessed in both the unit speed and arbitrary speed cases. Subsequently, the natural motions of an axisymmetric and asymmetric rigid body are derived in convenient quaternion form, which is particularly suitable for practical implementation. The initial angular velocities of the rigid body are parametrically optimised to produce attitude reference tracks for a small spacecraft. It is shown through comparison with a standard proportional-derivative controller that the natural motions require lower accumulated torque to track. Additional testing of the axisymmetric references, which are comprised of simple trigonometric functions, found that the references can be utilised in a "bang-off-bang" manner in low disturbance environments to save on computation and control effort. Next, the general solution to the optimal control problem of a rigid body constrained to spin at a constant rate is derived. This takes the form of a Weierstrass elliptic function which is impractical for motion planning. However, a particular case is identified that is comprised of trigonometric functions and utilised to plan efficient motions for a spinning solar sail and compared in simulation to a pure spin benchmark. The geometric references are found to limit the spacecraft body rates, while tracking requires lower accumulated torque in most cases. Finally, an actuator study is carried out to identify the technology requirements for reference tracking for a spinning solar sail