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I400/I590/B659: Intelligent Robotics

I400/I590/B659: Intelligent Robotics. Preliminaries: Vectors. Lab 1. Install Klamp’t python API In Klampt /Python/demos folder: run python gltemplate.py ../../data/athlete_fractal_1.xml Navigate with the mouse (hold Shift and Ctrl to pan and zoom) to get a good screenshot of the robot

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I400/I590/B659: Intelligent Robotics

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  1. I400/I590/B659: Intelligent Robotics Preliminaries: Vectors

  2. Lab 1 • Install Klamp’t python API • In Klampt/Python/demos folder: run python gltemplate.py ../../data/athlete_fractal_1.xml • Navigate with the mouse (hold Shift and Ctrl to pan and zoom) to get a good screenshot of the robot • Press ‘s’ to start simulating • Take a screenshot, submit it in .jpg or .png format • Due next Wednesday • Ask your peers / myself for help installing • Document installation problems, email to me (hauserk@indiana.edu)

  3. Agenda • Vector algebra: Representing and manipulating points and directions in 2D, 3D, and higher dimensions

  4. Vectors • Represent physical quantities that exist in some “space” • Both direction and magnitude • Represent ordered collections of related numbers • In this class: • 2D and 3D: Positions, velocities, accelerations, forces, pixel positions… • Higher D: RGBA colors, configurations, robot-environment system states, torques…

  5. Example #1 • Bob is standing at point A. He moves north one meter. He then moves east 2 meters. How far is he now from point A? A

  6. Example #1 • Bob is standing at point A. He moves north one meter. He then moves east 2 meters. How far is he now from point A? A

  7. Example #1 • Bob is standing at point A. He moves north one meter. He then moves east 2 meters. How far is he now from point A? B A

  8. Example #1 • Bob is standing at point A. He moves north one meter. He then moves east 2 meters. How far is he now from point A? (2,1) B (0,0) A

  9. Example #1 • Bob is standing at point A. He moves north one meter. He then moves east 2 meters. How far is he now from point A? (2,1) B A The displacement vector from A to B is (it is oriented vertically for reasons to be discussed later)

  10. Notation • A displacement vector starting at point A and ending at pint B is denoted • Capitals • Not used much in this class • A plain vector will be denoted • Point of origin is either unspecified, assumed, or does not need to be made explicit to be meaningful (e.g., velocity) • Lowercase • Eventually we’ll drop the arrow

  11. Example #1 • Bob is standing at point A. He moves north one meter. He then moves east 2 meters. How far is he now from point A? ? (2,1) B A The length of the vector is . This is also known as the norm of the vector and is denoted .

  12. Norms • Definition: If is a vector, then • (Pythagorean theorem) • [Subtle point: distance is not the same as distance traveled! Bob traveled 3m, but ended up about 2.23m away from the start]

  13. Comment • What about if Bob repeats the same procedure, arriving at point C? C (2,1) B A

  14. Comment • What about if Bob repeats the same procedure, arriving at point C? (4,2) C (2,1) B A

  15. Comment • What about if Bob repeats the same procedure, arriving at point C? (4,2) C (2,1) B (0,0) A The displacement vector from A to B is? The displacement vector from B to C is? The displacement vector from A to C is?

  16. Example #2: Multiplication • Dan is standing at point A. He moves in the same direction that Bob originally did, but goes 50% farther. Where does he stand? D (2,1) B A

  17. Example #2: Multiplication • Dan is standing at point A. He moves in the same direction that Bob originally did, but goes 50% farther. Where does he stand? D (2,1) B A Displacement vector from A to D is

  18. Scalar multiplication • To differentiate it from a vector, a “regular number” is known as a scalar • Scalar-vector multiplication scales both elements of the vector by the same amount. If a, b, and c are scalars, then: • It commutes: • Scalar division is the same as multiplying by 1/c • [As usual, don’t divide by zero]

  19. Scalar multiplication • Scalar multiplication “passes through” the norm operation: • (Why?)

  20. Points vs. Displacements • A point X in space can be represented for the purposes of calculations (out of the realm of pure thought) as a displacement vector from some special reference point O, called the origin • The representation of a point P changes depending on the choice of O • When comparing or manipulating two points, their representations as vectors must use the same origin! • A displacement vector from point X to Y is the same regardless of the choice of origin • [Note: what about the orientation of the reference axes? More later]

  21. Example #3: Vector addition • Suppose Bob is at B=(2,1). Bob then moves South 2 meters and east 3 more meters, arriving at E. What are E’s coordinates? B (2,1) (3,-2) A E

  22. Example #3: Vector addition • Suppose Bob is at B=(2,1). Bob then moves South 2 meters and east 3 more meters, arriving at E. What are E’s coordinates? B (2,1) (3,-2) A E Vector addition : produces a new vector in which ‘s first element is equal to the sum of the first elements of and , and its second element is equal to the sum of the second elements.

  23. Example #4: Vector subtraction • Dan is at back at D=(3,1.5). Along which vector would he have to move in order to reach E=(5,-1)? D (3,1.5) A (5,-1) E Vector subtraction : produces a new vector in which ‘s first element is equal to the first element of minus the first element of , and its second element is equal to the second element of minus the first element of .

  24. Vector subtraction: another view • With normal numbers, • With vectors, • (Try it)

  25. Distances D • How far is D from E? • Step 1: calculate • Step 2: calculate the norm • In other words, the distance between D and E is ? (3,1.5) A (5,-1) E

  26. Interpolation • To go from D to E gradually, you can use linear interpolation D (3,1.5) A E (5,-1) for u[0,1] a scalar parameter Or… Or…

  27. Higher dimensions • Everything is exactly the same, but the vectors contain more elements • 3D: • 4D: • Etc… • Scalar-vector multiplication, vector addition and subtraction remain essentially the same

  28. Norms in higher dimensions • 3D: if , then (why?) • 4D: if • Etc…

  29. Standard vector spaces • Cartesian space • 1D space: ℝ • 2D space: ℝ2 • Etc… ℝn • It can be shown that any space of objects that transform like vectors do, with scalar-vector multiplication and vector-vector addition, is isomorphic to ℝn for some n**as long as it has finite dimension

  30. Cheat sheet • Notation for n-dimensional vectors: • , , etc • Space of n-dimensional vectors: ℝn(Cartesian space) • Norm: • Scalar-vector multiplication: • Norms and scalar multiplications: • Vector-vector addition: • Vector-vector subtraction: • No such thing as vector-vector multiplication, vector-scalar addition.

  31. Implementation in Python • Lists: • E.g., [2,1], or [0.5,-0.8] • Operators +, -, *, / do not work in the same way • Use klampt.vectorops.{add,sub,mul,div} • Norm: klampt.vectorops.norm • Distance: klampt.vectorops.distance • Numpy arrays: • E.g., numpy.array([2,1]), or numpy.array([0.5,-0.8]), • Operators +, -, *, / work as desired • Norm: numpy.linalg.norm • Distance: numpy.linalg.norm(x-y)

  32. Next time • Matrix algebra, linear transformations (Principles A.E) • No class on Monday

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