Many engineering problems are naturally posed as optimal control problems. It may involve moving between two points in the fastest possible way, or to put a satellite into orbit with minimum energy consumption. Many optimal control problems are too difficult to be solved analytically and therefore require the use of numerical methods. The numerical methods that are the most widespread are the so-called direct methods. However, there is one major drawback with these. If the problem is non-convex, the solution is not guaranteed globally optimal, that is, the absolute best, instead it is guaranteed locally optimal, that is the best in its vicinity. To compensate for this, the problem should be solved several times, under different conditions, in order to investigate whether the solution is a good candidate for the global optimum. CasADi is a software specifically designed for dynamic optimization. It has gained wide spread in recent years because it provides all the necessary building blocks for dynamic optimization. This has given individual engineers and scientists the ability to independently formulate and solve all sorts of optimal control problems. However, this requires good theoretical knowledge of the necessary numerical methods. The advantage of a toolbox, which solves general optimal control problems, is that the underlying numerical methods have been tested and shown to function on optimal control problems with known solutions. This means that the user does not need exhaustive knowledge of the numerical methods involved, but can focus on formulating and solving optimal control problems. The main contribution of this thesis is an optimal control toolbox for MATLAB based on CasADi. The toolbox does not require expert knowledge of the numerical methods, but provides an alternative lower level abstraction that allows for more complex problem formulations. The toolbox implements two direct methods, direct multiple shooting and direct collocation. This allows a problem formulation with many degrees of freedom. The most important property of the toolbox is that the discretization can be changed, without the problem formulation needing to be altered. This way the user can easily change the conditions for his/her problem. The thesis describes how the two implemented direct methods work, and the design choices made. It also describes what remains to test and evaluate, and the problems that have been used as a reference during the development process.

In pursuit of producing robots capable of achieving the dexterity exhibited by animals in nature, roboticists have begun to explore the application of robotic tails. This thesis will explore the design, optimization, construction, and implementation of an articulated serpentine robotic tail. Numerous serpentine tail prototypes have been designed and tested; however, they have not yet been integrated with a mobile base. The main challenges preventing the incorporation of serpentine tails with mobile bases include: (1) the large size and inflexible packaging associated with the actuation unit for the tail, (2) the relatively low power to weight ratios of the existing serpentine tail systems, and (3) the complexity of optimizing the tails physical parameters. Therefore, to address these issues, a novel layout for a serpentine robotic tail actuation unit along with a design optimization methodology for the tail are proposed. The actuation unit will feature a power dense and modular design which allows for flexibility in packaging. Simulation results along with experimental data gathered using a prototype of the design will be reviewed in order to quantify the performance of the actuation unit. Following, a design optimization methodology which uses a modified direct collocation technique will be presented. The optimization allows for the simultaneous optimization of both a trajectory and the physical structure of a tail. Representative results of this technique will be presented and compared against more traditional methods for design optimization. To conclude the on-going and future work for both the actuation unit and optimization methodology will be stated. / Master of Science / Robotic tails largely fall into two categories based on their construction. These two categories are pendulum and serpentine structure. Pendulum structure tails consist of a long rigid rod with a weight attached to the end of it which can be swung to assist in controlling the orientation of the base which it is attached to. Serpentine tails are characterized by their ability to articulate and move in three dimensions similar to cat or monkey tails. The non-rigid structure of the tail opens up many new possibilities for their use. However, these possibilities come at the cost of design complexity. To date this complexity has led to designs for serpentine tails which are too heavy or unwieldy to be easily added to a mobile base. Additionally, the complexity of the tail structure itself make it difficult to optimize the design as has been done previously with pendulum designs. In an effort to overcome these challenges this thesis presents a novel design for a tail actuation unit and design optimization methodology. The actuation unit design is more power dense and provides greater flexibility in its layout than previous designs. This makes it much easier to adapt to and integrate with a mobile base. This will be demonstrated through the creation of a prototype tailed quadruped featuring the new actuation unit. The optimization methodology will use a technique known as direct collocation which has previously been developed for optimal path planning. This technique accommodates the complexities of serpentine tail designs and allows for the parameters such as length and weight of the tail to be optimized. The conclusion of the thesis will present the on-going and future work for both the actuation unit and optimization technique.

The turbocharger is an important component of competitive environmentally friendly vehicles. Mathematical models are needed for controlling turbochargers in modern vehicles. The models are parameterized using data, gathered from turbocharger testing ingas stands (a flow bench for turbocharger, where the engine is replaced with a combustion chamber, so that the exhaust gases going to the turbocharger can be controlled with high accuracy). Collecting the necessary time averaged data is a time-consuming process. It can take more than 24 hours per turbocharger. To achieve a sufficient level of accuracy in the measurements, it is required to let the turbocharger system reach steady state after a change of operating point. The turbocharger material temperatures are especially slow to reach steady state. A hypothesis is that modern methods in control theory, such as numeric optimal control, can drastically reduce the wait time when changing operating point. The purpose of this thesis is to provide a method of time optimal testing of turbo chargers.  Models for the turbine, bearing house and compressor are parameterized. Well known models for heat transfer is used to describe the heat flows to and from exhaust gas and charge air, and turbocharger material, as well as internal energy flows between the turbocharger components. The models, mechanical and thermodynamic, are joined to form a complete turbocharger model, which is validated against measured step responses. Numeric optimal control is used to calculate optimal trajectories for the turbo charger input signals, so that steady state is reached as quickly as possible, fora given operating point. Direct collocation is a method where the optimal control problem is discretized, and a non-linear program solver is used. The results show that the wait time between operating points can be reduced by a factor of 23. When optimal trajectories between operating points can be found, the possibility of further gains, if finding an optimal sequence of trajectories, are investigated. The problem is equivalent to the open traveling salesman, a well studied problem, where no optimal solution can be guaranteed. A near optimal solution is found using a genetic algorithm. The developed method requires a turbocharger model to calculate input trajectories. The testing is done to acquire data, so that a model can be created, which is a catch-22 situation. It can be avoided by using system identification techniques. When the gas stand is warming up, the necessary model parameters are estimated, using no prior knowledge of the turbocharger.

While the interest in the Martian moons increases, the low-thrust propulsion technology is expected to enable novel mission scenarios but is associated with unique trajectory design challenges. Accordingly, the current investigation introduces a multi-phase low-thrust design framework. The trajectory of a potential spacecraft that departs from the Earth vicinity to reach both of the Martian moons, is divided into four phases. To describe the motion of the spacecraft under the influence of gravitational bodies, the two-body problem (2BP) and the Circular-Restricted Three Body Problem (CR3BP) are employed as lower-fidelity models, from which the results are validated in a higher-fidelity ephemeris model. For the computation and optimization of low-thrust trajectories, direct collocation algorithm is introduced. Utilizing the dynamical models and the numerical scheme, the low-thrust trajectory design challenge associated each phase is located and tackled separately. For the heliocentric leg, multiple optimal control problems are formulated between the planets in heliocentric space over different departure and arrival epochs. A contour plot is then generated to illustrate the trade-off between the propellant consumption and the time of flight. For the tour of the Martian moons, the science orbits for both moons are defined. Then, a new algorithm that interfaces the Q-law guidance scheme and direct collocation algorithm is introduced to generate low-thrust transfer trajectories between the science orbits. Finally, an end-to-end trajectory is produced by merging the piece-wise solutions from each phase. The validity of the introduced multi-phase formulation is confirmed by converging the trajectories in a higher-fidelity ephemeris model.<br>

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The ever increasing demands on today's engine performance and emissions control is forcing the automotive industry to make use of innovative solutions. One of these is to apply the technology of VNT turbos on commercial petrol vehicles. When using a VNT turbo, the aspect ratio of the turbine can be altered while driving to suit the current operating window. In order to actually gain performance while using a VNT, the vanes have to be properly controlled using a suitable control strategy. In this project, direct collocation have been utilized through the usage of YOP which is an adaptation of CasADi in MATLAB to solve non-linear optimization problems. Comprehensive models of the turbocharger and the cylinders have been built and validated to properly represent a VEP4 LP engine from AUROBAY. The models are implemented in YOP to create and simulate different OCPs using the turbo speed as state and position of the vanes as control signal. With this model in YOP together with the air mass flow per second as reference, a good reference following together with decent values for relevant parameters can be accomplished. Other objective functions such as minimum time and maximal volumetric efficiency are also investigated in the project which yield likewise results. From the results it can be concluded that this type of model and control strategy can be used with success when studying optimal control of a VNT turbo.

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