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Concept# Trajectory

Summary

A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete trajectory is defined by position and momentum, simultaneously.
The mass might be a projectile or a satellite. For example, it can be an orbit — the path of a planet, asteroid, or comet as it travels around a central mass.
In control theory, a trajectory is a time-ordered set of states of a dynamical system (see e.g. Poincaré map). In discrete mathematics, a trajectory is a sequence (f^k(x))_{k \in \mathbb{N}} of values calculated by the iterated application of a mapping f to an element x of its source.
Physics of trajectories
A familiar example of a trajectory is the path of a projectile, such as a thrown ball or rock. In a significantly simplified model, the object moves only under the influenc

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In this thesis we describe a strategy to control robotic knees and ankles. A dynamical system is used to generate a position trajectory to control a servo motor replacing the missing joint. The dynamical system consists in a pool of coupled oscillators modeling a central pattern generator (CPG). As a first step, anthropometric trajectories of the knee and ankle are learned by the system through the convergence of the oscillators to the specific frequencies, corresponding amplitudes and phase relations. The same system is then used to play back these trajectories. As a sensory feedback to trigger the playback we use one adaptive frequency oscillator to synchronized with the acceleration from the thigh. We use a bipedal model in a physics-based robot simulation environment to test the proposed system. Finally we present a simple hardware implementation of this system on the Agonist-Antagonist Active Knee prototype.

2010As robots start pervading human environments, the need for new interfaces that would simplify human-robot interaction has become more pressing. Robot Programming by Demonstration (RbD) develops intuitive ways of programming robots, taking inspiration in strategies used by humans to transmit knowledge to apprentices. The user-friendliness of RbD is meant to allow lay users with no prior knowledge in computer science, electronics or mechanics to train robots to accomplish tasks the same way as they would with a co-worker. When a trainer teaches a task to a robot, he/she shows a particular way of fulfilling the task. For a robot to be able to learn from observing the trainer, it must be able to learn what the task entails (i.e. answer the so-called "What-to-imitate?" question), by inferring the user's intentions. But most importantly, the robot must be able to adapt its own controller to fit at best the demonstration (the so-called "How-to-imitate?" question) despite different setups and embodiments. The latter is the question that interested us in this thesis. It relates to the problem of optimizing the reproduction of the task under environmental constraints. The "How-to-imitate?" question is subdivided into two problems. The first problem, also known as the "correspondence problem", relates to resolving the discrepancy between the human demonstrator and robot's body that prevent the robot from doing an identical reproduction of the task. Even though we helped ourselves by considering solely humanoid platforms, that is platforms that have a joint configuration similar to that of the human, discrepancies in the number of degrees of freedom and range of motion remained. We resolved these by exploiting the redundant information conveyed through the demonstrations by collecting data through different frames of reference. By exploiting these redundancies in an algorithm comparable to the damped least square algorithm, we are able to reproduce a trajectory that minimizes the error between the desired trajectory and the reproduced trajectory across each frame of reference. The second problem consists in reproducing a trajectory in an unknown setup while respecting the task constraints learned during training. When the information learned from the demonstration no longer suffice to generalize the task constraints to a new set-up, the robot must re-learn the task; this time through trial-and-error. Here we considered the combination of trial-and-error learning to complement RbD. By adding a trial-and-error module to the original Imitation Learning algorithm, the robot can find a solution that is more adapted to the context and to its embodiment than the solution found using RbD. Specifically, we compared Reinforcement Learning (RL) – to other classical optimization techniques. We show that the system is advantageous in that: a) learning is more robust to unexpected events that have not been encountered during the demonstrations and b) the robot is able to optimize its own model of the task according to its own embodiment.

Sample return capsules, as the Apollo Command Module have been widely used to ad- vance the knowledge and planning of manned lunar and planetary return missions. Such reentry vehicles undergo extreme thermal conditions, caused by shock-heated air during their super-orbital atmospheric re-entry. Such extreme conditions can result in failure of the aeroshell structure and loss of important payload. This technological challenge is ad- dressed by the use of ablative thermal protection systems (TPS), which dissipate the heat away from the vehicles front wall via ablative products release into the boundary layer. Additionally, such velocity and temperature magnitudes during reentry conditions intro- duce significant radiative heat loads, filling the shock layer with radiators that react with the ablative species injected by the capsule wall. Therefore, accurate numerical modeling techniques are required, so that the thermophys- ical, thermochemical environment of a reentry capsule can be successfully reproduced and predicted. The present work aims to numerically rebuild certain significant trajectory points, containing the peak heating points of the Apollo 4 terrestrial re-entry. This re- quires the coupling of the resolved flow-field with radiative and ablative effects in order to accurately predict the convective and radiative heat flux for each trajectory point. The results will be compared to previous calculations and existing flight data. The numerical simulations are performed in 2D thermal non-equilibrium with a compress- ible explicit Navier-Stokes solver, coupled to a radiation database and a thermal material response code to implement the ablative effects. The calculations are performed also in 3D, using a commercial implicit Navier-Stokes solver. The results will be used to reproduce the capsules trajectory and verify the accuracy of the associated Modeling Tools.

2014