1、Chapter 2 Spatial descriptions and transformations2.1 Introduction2.2 Descriptions:Positions,orientation,and frames2.3 Mappings:Changing descriptions from frame to frame2.4 Operators:Translations,rotations,and transformations2.5 Summary of interpretations2.6 Transformation arithmetic2.7 Transform eq
2、uations2.8 More on representation of orientation12.1 Introduction Robotic manipulation-move around in space Representing position,orientation Coordinate systems Universe coordinate system22.2 Descriptions:Positions,orientation,and frames2.2.1 Description of a position32.2.2 Description of an orienta
3、tion It is necessary not only to represent a point in space,but also to describe the orientation 4 Coordinate system B has been attached to the body to give orientation of the body.We denote the unit vectors giving the principal directions of coordinate system B asWritten in terms of A,they are call
4、ed 56 Stack these three unit vectors together as a 33 matrix,namely rotation matrix7 rij is simply the projection of that vector onto the unit directions of its reference frameDot productDirection cosineFurther inspect rows of the matrix8Hence,the description of frame A to relative B,is given by the
5、 transpose of This suggestion that the inverse of a rotation matrix is equal to its transpose.92.2.3 Description of a frame For convenience,the point of a manipulator whose position we will describe is chosen as the origin of the body-attached frame.10112.2.4 Mappings:Changing descriptions from fram
6、e to frame121314Mappings involving general framesThe origin of frame B is not coincident with that of frame A but has a general vector offset15 The form of(2.17)is not as appealing as the conceptual form We define a 44 matrix operator and use 41 position vectors,so that(2.17)has the structure16Homog
7、eneous transform17BP=3 7 0T2.4 Operators2.4.1 Translational operators is translated by a vector AQ18Write this operator as a matrix operatorq is the magnitude of the translation along the vector Q192.4.2 Rational operators20The correct rotational operator isGiven 2.4.3 Transformation operators2122No
8、te:This example is numerically exactly the same as Example 2.2,but the interpretation is quite different.2.5 Summary of interpretations232.6 Transformation arithmetic242.6.1 Compound transformationsHave CP and wish to find AP25Frame C is know relative to frame B,which is know relative to frame A.We
9、can transform CP to BP as262.6.2 Inverting a transformWe know the value of ,sometime wish to get 272.7 Transform equations A frame D can be expressed as products of transformation in two different ways2829Consider that all transforms are known except ,we easily find its solution to be 2.8 More on representation of orientation2.8.1 X-Y-Z fixed angles One method of describing the orientation of a frame B is as follows:30where c is shorthand for cos,s for sin.31 Inverse problem:extracting angles from a rotation matrix322.8.2 Z-Y-X Euler angles33342.8.2 Z-Y-Z Euler angles35
