This means you need to pick an arbitrary axis for when the rotation is close to the identity and this produces a weird edge case: converting from angle-axis and back again can give you a different result to what you started with. Consider the identity rotation: the angle is clearly zero, but in angle-axis the axis could be any axis. Maybe things will be clearer with a bit of code.īecause all of these concepts are so interlinked, when it comes down to actually implementing them, it seems natural to implement them in terms of each other, and probably the first approach most people take is to define everything in terms of angle-axis conversion.īut it can actually be beneficial to think more carefully about the differences between all of them. If you have a set of angular velocities you want to integrate, you simply multiply them by the "dt", convert them all back from the scaled-angle-axis format to whatever rotation format you are using, and multiply all those rotations together.If we have two rotations, and we multiply one by the inverse of the other, convert the result to scaled-angle-axis, and divide by the "dt" we have exactly the "angular velocity" between these rotations - just as if we were computing the velocity via finite difference.The angular velocity is related in a slightly different way: in essence it's the difference between two rotations, stored in the "scaled-angle-axis" representation, and scaled by some "dt": Or putting it another way, if we want to compute the angle and axis of a rotation we can take its "log", and the direction of the result will represent the "axis", while the magnitude will represent half the "angle" of rotation around that axis.In other words, the "log" computes the "axis" of a rotation, scaled by the "half angle" of a rotation.Taking the "log" of a rotation to get the 3D exponentially mapped version is exactly the same as computing the scaled-angle-axis representation and then dividing it by two.The confusion around all these three concepts comes down to the fact that the exponential map, the angle-axis representation, and the angular velocity all do more or less the same thing. Usually this is simply called the log of a rotation. a quaternion, or rotation matrix), and converts it into the corresponding exponentially mapped 3D vector. The second is simply the inverse of the exponential map - a function which takes a rotation in some form (e.g. The result is a new 3D vector which I like to call the scaled-angle-axis representation. The first, which does not seem to have a standard name (sometimes I've heard it called the "helical" representation of a rotation), is to convert a rotation into angle-axis, take the axis, and scale it by the angle of the rotation. The angular velocity: The angular rate of change of a rotation with respect to time.īut before we go into more detail there are a few more concepts which we need to define:. The angle-axis representation: A representation of rotation consisting of an axis, and an angle of rotation around that axis.The exponential map: A specific function which takes a 3D vector and produces a rotation from it.These three concepts are all closely related, so to make sure we're all on the same page, this is exactly how I'm defining them: If you've used rotations when programming graphics, physics, or video games, there are three concepts you've probably seen mentioned, but may not have understood exactly what the difference was between them all: Finally, a new sort of continuous dependence is proved, that of the fixed time local flow on the parameter which plays an important role in the establishment of the homeomorphism of the exponential map.Exponential Map, Angle Axis, and Angular Velocity Created on July 10, 2021, 5:16 p.m. Using geometric descriptions of suitable topologies for vector fields and for local diffeomorphisms, the homeomorphism of the exponential map is derived by a uniform treatment for all cases of regularities. Moreover, all manners of regularity in state are considered, from the minimal locally Lipschitz dependence to holomorphic and real analytic dependence. That is to say, we give a presheaf construction of the exponential map for vector fields with measurable time-dependence and continuous parameter-dependence in the category of general topological spaces. An appropriate substitute is devised for the exponential map, not by trying to force flows to be globally defined by any compact assumptions on the manifold, but by categorical development of spaces of vector fields and flows, thus allowing for systematic localisation of such spaces. However, this map does not exists due to the lack of completeness of flows for general vector fields. The exponential map that characterises the flows of vector fields is the key in understanding the basic structural attributes of control systems in geometric control theory.
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