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DONG QiuhuangCollege of Mechanical and Electronic Engineering, FAFU.Mathematical BasisManipulator-Mechanism Design 2Mathematical basisIntroduction4and tools will be some sort of mechanism.How to define the manipulate mathematical quantities （数学量）that represent location of the body?IntroductionRigid Body Motion （刚体运动）Position andOrientation Mathematical Quantities (Coordinates)Velocities, Forces （速度和力）must define coordinate systems representation.Mathematical basisDescription （描述）83×1 position vector.DescriptionCatesian coordinate system (笛卡尔坐标系):9Description10Description of Orientations （姿态描述）position , but also need to describe its orientation in the space.DescriptionHow to describe the orientation of a body ?Description of Orientationsreference system.Description12Description of OrientationsDescription1314Description of OrientationsDescriptionDescription of OrientationsDescriptionObtain the projection of that vector onto the unit directions of its reference coordinate.Example:Compute the Rotation MatrixRotation about axisExample:Rotation about axis:Rotation about axis:Description of OrientationsDescription19Description20manipulator hand is a position DescriptionWe define such a entity which contain the pair of position and Mathematical basiscoordinate system .工具坐标系目标坐标系固定坐标系基座坐标系Mapping23Mathematics of changing descriptions of the same quantity from frame to frame.1. Translation(平移）已知S 点在坐标系{B}中的表达，那么在坐标系{A}中如何表达？242. Rotation （旋转）求矢量在坐标系{A}三个主轴上的投影。
•ReviewRobotRoboticsManipulatorMobile RobotKinmatics¾forward knematics¾inverse kinematics¾velocity kinematicsDynamicsCartesian SpacePositionOrientationJoints¾Prismatic Joint¾Revolute Joint¾Ball JointCoordinate System (frame)¾Global Reference Coordinate System ¾Joint Reference Coordinate System ¾Tool Reference Coordinate SystemChapter 2 Spatial Description and TransformationCh2.1Introduction •Robotic manipulation: the parts and tools will be moved around in space by some sort of mechanism.•How to define and manipulate mathematical quantities that represent location of the boy?We must define coordinate systems and develop representation.•There should be a universe coordinate system.Ch2.2Description •Description: used to specify attributes of various objects with which an manipulation system deals.•These objects include parts, tools, and the manipulator itself.•Description: position, orientation, and frame• A frame can be used as a description of one coordinate system relative to another.{, }U AABORG R P2.Rotation•We have introduced the notation of describing anorientation by three unit vectors denoting the principal axes of a body-fixed frame.ˆˆˆˆˆˆB AA A AA B BBBB A B A X R X Y Z Y Z ⎡⎤⎢⎥⎡⎤==⎢⎥⎣⎦⎢⎥⎢⎥⎣⎦•Rotation matrix which contains three unit vector is utilized to describe an orientation.1()()A B B TBAAR R R −==Example 2.1 in textbook3.Mapping involving General Frame•Problem: we know the description of a vector with respect to frame {B}, and we would like to know it’s description with respect to frame {A}.•General case:(1) frame {B} might not has the same orientation withrespect to frame {A}.(2) frame {B} and {A} might not have coincident origins.Example 2.2 in textbookCh2.4OperatorsMain objectives: translate/rotate points/vectors, or both 1.Translation operatormove a point in space a finite distance along a give vector direction.Example: translate a vector along the direction vector ofA P1A Q2. Rotational operatorOperate on a vector and changes that vector to a new vector by means of a rotation 1A P 2A P21()AAP R P =•The mathematics describe of rotational operator is same asthe one of mapping rotation, only the interpretation is different.•The rotation matrix that rotates vectors through somerotation, R , is the same as the rotation matrix that describes a frame rotated by R relative to the reference frame.•Example 2.3 in the text book3. Transformation operators:As a entity consists vector and rotation matrix, a frame has another interpretation as a translation operator.21AAP T P =3101R Q T ×⎡⎤=⎢⎥⎣⎦The transform that rotates by R and translates by Q is the same as the transform that describes a frame rotated by R and translated by Q relative to the reference frame.•Example 2.4 in the text bookCh2.5Summary•Homogeneous:0001RP T ⎡⎤=⎢⎥⎣⎦(1)Description of a framedescribes frame {B} relative frame {A}0001A AA BBORG BR P T ⎡⎤=⎢⎥⎣⎦(2) Transform mapping:maps to A BT B P A P (3) Transform operator (translate and rotate) on to create 2AP 1APThank You!。
