- source: [Slides](http://www.matthewpeterkelly.com/tutorials/trajectoryOptimization/cartPoleCollocation.svg#frame2704) and [video of talk](https://www.youtube.com/watch?v=wlkRYMVUZTs&t=1354s) ## Assumptions in this talk - Note that his talk has assumed the following assumptions: - continuous systems but not discrete; the latter is shown in blue) - problems are single-phase and not multiphase problems (also shown in bigger blue with the dashed lines between curves) - Kelly does point the listener to use cases of such multiphase problems in his paper. He also gives clear examples of where multiphase problems apply, such as in: - walking robots, when the foot hits the ground; and - multi-stage rockets, where stages of rockets break-off. ![[Pasted image 20221029172902.png]] - his talk also assumes smooth functions with integrals. The functions: - must also have [[11a Consistent functions|consistency]] and - non-linearity is permissible. For example, see the nonlinear term in the equation alongside text saying "smooth" (sine and also cosine with a product of the state in the picture in [[11a Consistent functions]]) ## Trajectory Optimisation Problems can be transcribed into a Non-Linear Program - The paper and talk go over the idea of "[[11b transcription, which is an approach to transform the trajectory optimisation into a constrained parameter optimisation problem]]". - This Constrained Parameter Optimisation Problem falls under a class of Nonlinear Optimization Problem (NLP). - The key differences between TO and NLP are in bullet points of the image in [[11b transcription, which is an approach to transform the trajectory optimisation into a constrained parameter optimisation problem| the page on transcription]]. ## There are various aspects to transcription ### Indirect vs direct methods ![[Pasted image 20221031152951.png]] - Indirect methods of trajectory optimisation were first developed and used by the aerospace community to optimise multi-stage rockets. - However, the robotics community frequently uses direct techniques. - The main difference is in the sequence of steps to solve the optimal control problem. - Indirect methods optmize then discretize - direct methods discretize the optimal control problem and then solve these equations. - Indirect methods are more accurate. - Direct methods are easier to pose and solve. - Direct methods are more recenty and used in teh robotics and manipulaton community but direct methods were made first. ### Shooting vs Collocation Methods - In both direct and indirect methods, the actual discretisation is done in one of two ways. The key differences are shown below: ![[Pasted image 20221031153021.png]] - Collocation methods use implicit integrators to a discretise (or approximate) the dynamics whereas shooting methods use explicit integrators to discretise (or approximate) the dynamics. ### H-Methods v/s P-methods ![[Pasted image 20221031153915.png]] - These basically tell you how to increase or improve the accuracy of a method. - h-method increases number of trajectory segments. That is, it asks the simulator to take more time-steps on the same order method. Typically used in lower order transcription methods. - p-,methods are used in high-order methods; instead of taking more steps, it tells the simulator to use a higher-order method. - The most sophisticated systems use a combination of two methods by intelligently deciding when to use which method.