Robotics, Fall 2006 Lecture 3: Homogenous Transformations (Translation & Rotation) Copyright © 2005, 2006 Jennifer Kay

Review: Transforming Points Between Coordinate Frames Simple start: translation along one axis

Review We have talked about two concepts. It is extremely important that you do not confuse the two 1. How do we take a point that is in frame j coordinates and convert it to be in frame k coordinates? 2. How do we compute the transformation between frame j and frame k (i.e. how would we move frame j to line it up with frame k)?

Review: General Matrix for Arbitrary Translation Along the X,Y,Z axes Suppose that to get from frame p to frame q: – Move a units along p’s x axis – Move b units along p’s y axis – Move c units along p’s z axis To take a point from q coordinates to p coordinates, premultiply by:

Review The transformation that takes a point in j coordinates and computes its location in k coordinates. Easiest way to come up with the matrix: first figure out how to move frame k to frame j.

What about Rotation? How do we transform the k coordinate frame into the j coordinate frame? (i.e., what is )

First: Visually compute the following Consider the origin of the j axis, (in j coordinates). What is its location in k coordinates? Consider the point (in j coordinates). What is its location in k coordinates? x j y j z j z k x k y k

Transformation Matrices for Rotations about the Z axis = Rot z(-90) Rot z(θ) = =? x j y j z j z k x k y k

Sanity check: Let’s test our matrix on the examples we did by hand (1) We said that x j y j z j z k x k y k = = j k= ?

Sanity check: Let’s test our matrix on the examples we did by hand (2) j = = ? We said that k x j y j z j z k x k y k =

So Where Are We? Now: We can compute the F (and T) matrices when: – The two frames only differ by translation along some combination of translations along the x, y, and z axes. – The two frames only differ by a rotation about one of the z axes. NEXT – The general case: any combination of rotations and translations.

Two ways to compute the relationship between two frames “Moving Axes” Approach “Fixed Axes” Approach Both approaches give the same result You should learn to do both, because some problems are easier to solve with moving axes, and others are easier with fixed axes.

Introduction to Solving the General Problem What sequence of moves do you need to make to compute ? Try it with frame models!

: Moving Axes (Many ways to do it) One approach: Start in world coordinates xwxw ywyw zwzw FwFw g

One approach: xwxw ywyw zwzw 1. Rotate about x w by –90 degrees. Call the resulting frame Frame 1, and its axes x1, y1, z1 x1x1 z1z1 y1y1 : Moving Axes FwFw g

2. Rotate about z1 by –90, call the resulting frame Frame 2, with axes x2, y2, z2 One approach: x1x1 z1z1 y1y1 1. Rotate about x w by –90 degrees. Call the resulting frame Frame 1, and its axes x1, y1, z1 x2x2 y2y2 z2z2 : Moving Axes FwFw g

2. Rotate about z1 by –90, call the resulting frame Frame 2, with axes x2, y2, z2 One approach: 3. Translate by (0,0,5) relative to coord frame 2 x2x2 y2y2 z2z2 1. Rotate about x w by –90 degrees. Call the resulting frame Frame 1, and its axes x1, y1, z1 xgxg ygyg zgzg : Moving Axes FwFw g

Moving Axes Notes Every step along the way is based on the resulting frame from the previous step. – E.g. Rotation 2 is based on frame1’s axes, NOT based on the world axes. Summary of our moves: 1. Rot x (-90) 2. Rot z (-90) 3. Trans (0,0,5)

Every move will be relative to the world coordinate frame’s axes. : Using Fixed Axes FwFw g

One approach: xwxw ywyw zwzw 1. Rotate about x w by –90 degrees (now we’re yellow). : Fixed Axes FwFw g

xwxw ywyw zwzw 2. Rotate about y w by –90 degrees (now we’re blue). One approach: 1. Rotate about x w by –90 degrees (now we’re yellow). : Fixed Axes FwFw g

One approach: zwzw xwxw ywyw xgxg ygyg zgzg 2. Rotate about y w by –90 degrees (now we’re blue). 1. Rotate about x w by –90 degrees (now we’re yellow). 3. Translate by (0,5,0) relative to world coords (now we’re green) : Fixed Axes FwFw g

Fixed Axes Notes Every move is relative to your initial frame Summary of our moves: 1. Rot x (-90) 2. Rot y (-90) 3. Trans (0,5,0)

Moving vs. Fixed Moving – Each move is relative to the frame resulting from the previous one – Summary of our moves: 1. Rot x (-90) 2. Rot z (-90) 3. Trans (0,0,5) Fixed – Each move is done relative to the original frame – Summary of our moves: 1. Rot x (-90) 2. Rot y (-90) 3. Trans (0,5,0)

So How Do We Get to the Matrix? A whole pile of equations You do not need to memorize these – in a test situation, I will give you a copy You can find them on page 16 of your reading You should become comfortable using them

Step 1: The Transformation Equations Trans (a,b,c) = Rot x (θ) = Rot y (θ) = Rot z (θ) =

Using Moving Axes Rot x (-90) Rot z (-90) Trans (0,0,5) – Summary of our moves (moving axes): 1. Rot x (-90) 2. Rot z (-90) 3. Trans (0,0,5) 1. List the moves from left to right 2. Replace each move with the appropriate matrix 3. Multiply the matrices together

Using Fixed Axes Trans (0,5,0) Rot y (-90) Rot x (-90) – Summary of our moves (fixed axes): 1. Rot x (-90) 2. Rot y (-90) 3. Trans (0,5,0) 1. List the moves from right to left 2. Replace each move with the appropriate matrix 3. Multiply the matrices together

Fixed Axes: Rotation Warning! E.g.: Rotate w axis –90 o about y g Think about rigidly locking the frames together as you rotate!

Remember! Moving Axes – List the equations from left to right Fixed Axes – List the equations from right to left

Let’s do some examples …… Trans (a,b,c) = Rot x (θ) = Rot y (θ) = Rot z (θ) =