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I am a researcher for a robotic soccer project, and im developing a guidance algorithm in which I need to be able to predict the position of our robots over time, given only that they have a constant acceleration. These are omnidirectional wheeled robots, which means they can accelerate instantly in any given direction under normal conditions. This is not true, however, when the robot reaches its maximum speed, and I haven't been able to define the equations of motion for the robots under this particular case.

To clarify: suppose that at t=0 you have a point-like robot moving in an X-Y plane with a certain, known positionp(0)=[p_{x}(0) p_{y}(0)]^{T}and velocityv(0) = [v_{x}(0) v_{y}(0)]^{T}, and you try to apply a constant accelerationa= [a_{x}a_{y}]^{T}to it. If the speed of the robot is below a certain limit v_{max}, then the equations describing the position of the robot over time are trivial (p(t) =p(0)+v(0)*t+(1/2)*a*t^{2}). However, when the robot reaches its maximum speed (i.e. ||v(t)||=v_{max}) then, obviously, it will not be able to accelerate in the direction of its velocity vector, and so only the component ofawhich is orthogonal tovwill have any effect, which will be to rotate the velocity vector until bothaandvare aligned. My goal is then to obtainp(t) in these conditions. My main problem is that in this case, the useful component of the acceleration (let's call ita_{c}(t)) depends instantaneously onv(t), and I need to describe it in a simple manner so that it's possible to integrate that twice without falling into a mathematical analysis nightmare (my mathematical skills are sadly limited for an engineer). So far my only idea was to projectaontovvia an inner product and obtaina_{c}(t) =a-(a.v(t))*(v(t)/v_{max}), but this is probably silly.

Any help is appreciated.

PS: This is my first post here, so excuse me if this isn't in the right section.

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# Determining the trajectory of an uniformly accelerated object with speed restrictions

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