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Position Estimation 1

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Position Estimation 1:Relevant Aspects of the Physics of Measurement1 Context of Measurement1.1 Five Elements of Measurement Context. a unit system (e.g. meters, seconds). a number system (e.g. base 10 weighted positional). a coordinate system (e.g. directions north, east). a reference frame to which the measurement is ascribed (e.g. your car).. a reference frame with respect to which the measurement is made (e.g. the earth).. Lets agree to use the metric system that is based sensibly on the wavelengths of photon emissions of certain atomic transition.... Let us further agree to use a base 10 (not binary or hex or octal,) weighted positional number system (not like MCMLXXIII).. Let us further agree to erect an orthogonal right-handed Cartesian coordinate system where:. the direction called north is aligned with the projection of the earth’s instantaneous spin axis onto the local surface tangent plane.. the direction called up is aligned with the direction of gravitation. (north and up are distinct everywhere but the poles).. the direction called east is determined from the other two such that north cross up equals east.. I observed the motion of your car relative to my radar gun (which was fixed with respect to my police car, which was fixed with respect to the earth).. The observed value of my measurement within this context was 30 units south.1.2 Three Elements of Measurement Complexity1.2.1 Measurement Order. scalars. possessing magnitude only. represented by a single number. e.g. mass, fluid pressure. vectors. possessing magnitude in a particular direction. represented by a column of numbers. e.g. position, velocity, acceleration. tensors. possessing magnitude in all (or n principle) directions. represented by matrices and up. e.g. stress, inertia1.3 Measurement Number. discrete - one only. systems - finite enumerable number. continuum - infinite (we call this a “field” or “body”)1.4 Measurement Dimension. one-dimensional (distance along the road). two-dimensional (position on the earth’s surface). three-dimensional (position in the atmosphere)2 Coordinate Systems and Reference Frames (or and all that)2.1 Coordinate Systems2.2 Reference Frames2.3 Mutually Stationary Frames2.4 The Galilean Transformation2.5 Accelerating Frames - The Coriolis Law3 Inertial Frames. All of Newton’s laws hold only in inertial frames. For the second law (F=ma):. The forces are contact (electrostatic), gravitation, and electromagnetic. Except for nuclear forces, there are no other forces in nature.. The acceleration is the inertial acceleration.3.1 Apparent Forces3.2 Equivalence of Gravitation and Acceleration3.3 Inertial Space. An inertial frame is one which is:. Far removed from any massive object (no gravitation), or.... Under no net gravitational force, or.... Free to move under the influence of gravitation (freefall). Third case subsumes the others.. Newton thought that acceleration was the issue but a free falling body can accelerate.. Examples of inertial frames are:. Intergalactic space (far removed). Center of the earth (no net g’s). Spacecraft in orbit (why are you “weightless” if you are being held in orbit by the earth’s gravity?). Aircraft in freefall (NASA trains astronauts in these). Anything that rotates wrt the stars is not inertial.3.4 Earth’s Sidereal Rotation. Earth rotates wrt the stars about once a mean solar day. The surface of the earth is not an inertial frame. Tangential velocity at the equator is 1000 mph wrt center of earth. Velocity of earth wrt sun is 70,000 mph !!. sidereal day is 364/365 of solar day (4 minutes shorter). earth rate of rotation is 15 degrees per hour3.5 Geocentric Inertial Frame. Origin at center of the earth.. Nonrotating wrt stars.. Therefore, rotates once a year wrt sun.. Imagine observer sitting there:. Such an observer can write Newton’s Laws in that frame for a vehicle moving on the surface of the earth.. This is where you start to write the equations of inertial navigation.3.6 Effect of Earth’s Rotation. The surface of the earth is not an inertial frame. Every you used in high school was actually wrong.. Instruments fixed wrt surface of the earth will measure the earth’s rotation (wrt stars) when stationary.. Faucault proved that the earth rotates in 1850 with a gyroscope.. Examples of “failure” of Newton’s laws on the earth:. Gyroscope at equator pointing “up” rotates once per day.. Faucault pendulum in UN building in New York.. Objects don’t fall in straight lines.. Gyrocompass oscillates due to earth’s rotation.4 Example: A Paradox?4.1 Earth-Fixed Observer. This observer is fixed to the surface of the earth just like we are now.. Weight of brick cancelled by reaction of table.. Newtons Laws are satisfied:. Since sum of forces is zero, no acceleration takes place.4.2 Stars-Fixed Observer. A net deficit of the reaction relative to the weight giving rise to the circular motion.. A extremely slight moment caused by the force distribution of left side relative to right.5 Summary. Robot rollover requires the motion of the cg. You can’t put sensors there and it often moves anyway.. Inertial navigation is based fundamently on all this stuff.6 NotesMobile Robot SystemsAlonzo Kelly February 9, 2005Position Estimation 1: 11.1Five Elements of Measurement ContextPosition Estimation 1:Relevant Aspects of the Physics of Measurement 1 Context of Measurement• Everything they never told you in physicsthat you will need for inertial navigation.• This will start out trivial and then getsomewhat subtle pretty quickly.1.1 Five Elements of Measurement Context• Suppose a state policeman use his radar gunto clock you at 30 meters/second (70 mph) onyour way to work. He pulls you over and says“you were going 30 m/s southbound on I-279”. What does he REALLY mean.• The expression of any measurement inphysics generally requires:• a unit system (e.g. meters, seconds)• a number system (e.g. base 10 weightedpositional)• a coordinate system (e.g. directionsnorth, east)• a reference frame to which themeasurement is ascribed (e.g. your car).• a reference frame with respect to whichthe measurement is made (e.g. the earth).Mobile Robot SystemsAlonzo Kelly February 9, 2005Position Estimation 1: 21.2Three Elements of Measurement Complexity• So he REALLY means to say• Lets agree to use the metric system thatis based sensibly on the wavelengths ofphoton emissions of certain atomictransitions. Not the British


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