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Kinematics & Grasping. Goal : understand ideal robot mechanisms Robot positions and shapes as a function of control parameters – kinematics. Need to know: Representing mechanism geometry Standard configurations Degrees of freedom Grippers and graspability conditions. - PowerPoint PPT Presentation

Kinematics & GraspingNeed to know: Representing mechanism geometryStandard configurationsDegrees of freedomGrippers and graspability conditions

Goal : understand ideal robot mechanisms

Robot positions and shapes as a function of control parameters kinematics

3D Coordinate SystemsLeft handed (Right handed reverses +Z direction)

Vectors & Points in 3D

Local Reference FramesLocal (translated) coordinates Global (untranslated) coordinates Local could also have further sub-local frames

TranslationsMove to

RotationsRotate about Z axis

Rotation Matrix Representation I Much more compact and clearer

Rotation Matrix Representation II

Other Rotation RepresentationsAll equivalent but different parameters: Yaw, pitch, roll Azimuth, elevation, twist Axis + angle Slant, tilt, twist Quaternion

Full Rotation SpecificationNeed 3 angles for arbitrary 3D rotationLock & key exampleRotation angles :

Warning: rotation order by convention but must be used consistently:

Full Transformation SpecificationEach connection has a new local coordinate system

Need to specify 6 degrees of freedom3 rotation + 3 translation

Kinematic Chainsis at:In C2:

In C1:

In C0:

Homogeneous Coordinates IMessy when more than 2 links, as in robotSo: pack rotation and translation into

Homogeneous coordinate matrix

Extend points with a 1 from 3-vector to 4-vectorExtend vectors with a 0 from 3-vector to 4-vectorPack rotation and translation into 4x4 matrix:

3 rotation parameters:3 translation parameters:

Homogeneous coordinates IIIn C2:

In C1:

In C0:

Longer chains for robot arms (e.g. 6 links):

Adding JointsPrismatic joint: sliding structures parameterize one translation direction per joint E.g.:

slides in the X directionRevolute joint: rotates parameterize one rotation angle per joint

Degrees of Freedom Controllable DoF: number of joints Effective DoF: number of DoF you can get after multiple motions

car has 2 controllable (move, turn) but can adjust XY position and orientation so 3 effective DoF

Task DoF: configurations in space dimensionality: 2D : 3 (x,y,angle) 3D : 6 (x,y,z,3 angles)

Forward and Inverse KinematicsForward: Given joint angles, find gripper position Easy for sequential joints in robot arm: just multiply matrices

Inverse: Given desired gripper position, find joint angles Hard for sequential joints geometric reasoning

Configuration Space IAlternative representation to scene coordinates

Number of joints=JJ-dimensional space Binary encoding: 0:invalid pose, 1: free space Real encoding: distance from goal configuration

Point in C.S = configuration in real spaceCurve in C.S. = motion path in real space

Configuration Space II

Sequential & Parallel MechanismsSimplified into 2DSerial manipulator Parallel manipulatorvary: vary:

Computing Positions & Parameters

SPECIFYING ROBOT POSITIONSActuator level: specify voltages that generate required joint angles.2. Joint level: specify joint angles and let system calculate voltages.3. Effector level: specify tool position and let system compute joint angles.4. Task level: specify the required task and let the system compute the sequence of tool positionsMost robot programming is at level 2 or 3.

Grippers and Grasping Gripper: special tool for general part manipulation Fingers/gripper: 2, 3, 5 Joints/finger: 1, 2, 3Your hand: 5 fingers * 3 DoF + wrist position (6) = 21 DoF. Whew!

Barret Hand2 parallel fingers (spread uniformly)1 opposable fingerDoF: 4 fingers (2 finger joints bend uniformly)

Barret Hand

Finger Contact GeometryCoefficient of friction at fingertip: Surface normal: direction perpendicular to surface:Friction cone: angles within about surface normalForce direction: direction finger pushedNo-slip condition:

Force Closure

Other Grasping CriteriaSome heuristics for a good grasp: Contact points form nearly equilateral triangle Contact points make a big triangle Force focus point near centre-of-mass

Grasp AlgorithmIsolate boundaryLocate large enough smooth graspable sectionsCompute surface normalsPick triples of grasp pointsEvaluate for closure & select by heuristicsEvaluate for reachability and collisionsCompute force directions and amountPlan approach and finger closing strategyContact surface & apply grasping force Lift (& hope)

Kinematics Summary Need vector & matrix form for robot geometry Geometry of joints & joint parameters Forward & inverse kinematics Degrees of freedom Grippers & grasping conditions