Kinematics Of Robots (Position Analysis)

Kinematics Of Robots(Position Analysis)By R. C. Saini

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IntroductionTo determine where the robots hand is?Forward Kinematic Equation if all joint variable are know, we can determine where the robots hand (End-effector).

To calculate what each joint variable is?Inverse Kinematic Equation Calculate what each joint variable must be in order to locate the hand at a particular point and a particular orientation.www.rcsaini.com2ROBOTS AS MECHANISMMultiple type robot have multiple DOF.

3 Dimensional, open loop and chain mechanisms

Fig. A one-degree-of-freedom closed-loop four-bar mechanism

Fig. (a) Closed-loop versus (b) open-loop mechanism www.rcsaini.com33www.rcsaini.comwww.rcsaini.comMATRIX REPRESENTATIONRepresentation of a Point in SpaceA point P in space : 3 coordinates relative to a reference frame

Fig. Representation of a point in space

www.rcsaini.com4MATRIX REPRESENTATIONRepresentation of a Vector in SpaceA Vector P in space : 3 coordinates of its tail and of its head

Fig. Representation of a vector in space

W = scale factorwww.rcsaini.com5MATRIX REPRESENTATIONRepresentation of a Frame at the Origin of a Fixed-Reference FrameEach Unit Vector is mutually perpendicular : normal, orientation, approach vector

Fig. Representation of a frame at the origin of the reference frame

www.rcsaini.com6MATRIX REPRESENTATIONRepresentation of a Frame in a Fixed Reference FrameEach Unit Vector is mutually perpendicular : normal, orientation, approach vector

Fig. Representation of a frame in a frame

www.rcsaini.com7MATRIX REPRESENTATIONRepresentation of a Rigid BodyAn object can be represented in space by attaching a frame to it and representing the frame in space.

Fig. Representation of an object in space

www.rcsaini.com8HOMOGENEOUS TRANSFORMATION MATRICESA transformation matrices must be in square form.

It is much easier to calculate the inverse of square matrices. To multiply two matrices, their dimensions must match.

www.rcsaini.com9REPRESENTATION OF TRANSFORMATINSA transformation is defined as making a movement in space. A pure translation. A pure rotation about an axis. A combination of translation or rotations.

Representation of a Pure Translation

Fig. Representation of an pure translation in space

www.rcsaini.com10REPRESENTATION OF TRANSFORMATINSRepresentation of a Pure Translation

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REPRESENTATION OF TRANSFORMATINSRepresentation of a Pure Rotation about an Axis Assumption : The frame is at the origin of the reference frame and parallel to it.

Fig. Coordinates of a point in a rotating frame before and after rotation.

Fig. Coordinates of a point relative to the reference frame and rotating frame as viewed from the x-axis. www.rcsaini.com12REPRESENTATION OF TRANSFORMATINSRepresentation of a Pure Rotation about an Axis

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REPRESENTATION OF TRANSFORMATINSRepresentation of a Pure Rotation about an Axis Example--www.rcsaini.com14

REPRESENTATION OF TRANSFORMATINSRepresentation of Combined TransformationsA number of successive translations and rotations.

Fig. Effects of three successive transformations

Fig. Changing the order of transformations will change the final result www.rcsaini.com15REPRESENTATION OF TRANSFORMATINSRepresentation of Combined Transformations

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REPRESENTATION OF TRANSFORMATINSRepresentation of Combined Transformationswww.rcsaini.com17

REPRESENTATION OF TRANSFORMATINSRepresentation of Combined TransformationsRelative to current framewww.rcsaini.com18

Point P [7, 3,1]INVERSE OF TRANSFORMATION MATIRICESInverse of a matrix calculation steps :

Fig. The Universe, robot, hand, part, and end effecter frames. www.rcsaini.com19

INVERSE OF TRANSFORMATION MATIRICESwww.rcsaini.com20

FORWARD AND INVERSE KINEMATICS OF ROBOTSForward Kinematics Analysis: Calculating the position and orientation of the hand of the robot. If all robot joint variables are known, one can calculate where the robot is at any instant.Fig. The hand frame of the robot relative to the reference frame.

www.rcsaini.com21FORWARD AND INVERSE KINEMATICS OF ROBOTSForward and Inverse Kinematics Equations for PositionForward Kinematics and Inverse Kinematics equation for position analysis :

Cartesian (gantry, rectangular) coordinates.

Cylindrical coordinates.

Spherical coordinates.

Articulated (anthropomorphic, or all-revolute) coordinates.

www.rcsaini.com22Forward and Inverse Kinematics Equations for PositionCartesian (Gantry, Rectangular) CoordinatesAll actuator is linear.A gantry robot is a Cartesian robot.

Fig. Cartesian Coordinates.

www.rcsaini.com23Forward and Inverse Kinematics Equations for PositionCylindrical Coordinates2 Linear translations and 1 rotation translation of r along the x-axis rotation of about the z-axis translation of l along the z-axis

Fig. 2.19 Cylindrical Coordinates.

www.rcsaini.com24Forward and Inverse Kinematics Equations for PositionSpherical Coordinates2 Linear translations and 1 rotation translation of r along the z-axis rotation of about the y-axis rotation of along the z-axis

Fig. Spherical Coordinates.

www.rcsaini.com25Forward and Inverse Kinematics Equations for PositionArticulated Coordinates3 rotations: Denavit - Hartenberg representation

Fig. Articulated Coordinates.

www.rcsaini.com26FORWARD AND INVERSE KINEMATICS OF ROBOTSForward and Inverse Kinematics Equations for Orientation

Roll, Pitch, Yaw (RPY) angles

Euler angles

Articulated joints

www.rcsaini.com27Forward and Inverse Kinematics Equations for OrientationRoll, Pitch, Yaw(RPY) AnglesRoll: Rotation of about -axis (z-axis of the moving frame)Pitch: Rotation of about -axis (y-axis of the moving frame)Yaw: Rotation of about -axis (x-axis of the moving frame)

Fig. RPY rotations about the current axes.

www.rcsaini.com28Forward and Inverse Kinematics Equations for OrientationEuler AnglesRotation of about -axis (z-axis of the moving frame) followed byRotation of about -axis (y-axis of the moving frame) followed byRotation of about -axis (z-axis of the moving frame).

Fig. Euler rotations about the current axes.

www.rcsaini.com29Forward and Inverse Kinematics Equations for OrientationArticulated Joints

Consult again slide no. 19.

www.rcsaini.com30FORWARD AND INVERSE KINEMATICS OF ROBOTS

Assumption : Robot is made of a Cartesian and an RPY set of joints. Assumption : Robot is made of a Spherical Coordinate and an Euler angle.Another Combination can be possibleDenavit-Hartenberg Representationwww.rcsaini.com31DENAVIT-HARTENBERG REPRESENTATION OF FORWARD KINEMATIC EQUATIONS OF ROBOTDenavit-Hartenberg Representation :Simple way of modeling robot links and joints for any robot configuration, regardless of its sequence or complexity.Transformations in any coordinates is possible.Any possible combinations of joints and links and all-revolute articulated robots can be represented.

Fig. A D-H representation of a general-purpose joint-link combinationwww.rcsaini.com32DENAVIT-HARTENBERG REPRESENTATION OF FORWARD KINEMATIC EQUATIONS OF ROBOTDenavit - Hartenberg Representation procedures: Start point: Assign joint number n to the first shown joint.Assign a local reference frame for each and every joint before or after these joints. Y-axis does not used in D-H representation.

www.rcsaini.com33DENAVIT-HARTENBERG REPRESENTATION OF FORWARD KINEMATIC EQUATIONS OF ROBOTProcedures for assigning a local reference frame to each joint:All joints are represented by a z-axis. (right-hand rule for rotational joint, linear movement for prismatic joint)The common normal is one line mutually perpendicular to any two skew lines. Parallel z-axes joints make a infinite number of common normal. Intersecting z-axes of two successive joints make no common normal between them(Length is 0).

www.rcsaini.com34DENAVIT-HARTENBERG REPRESENTATION OF FORWARD KINEMATIC EQUATIONS OF ROBOTSymbol Terminologies : : A rotation about the z-axis. d : The distance on the z-axis. a : The length of each common normal (Joint offset). : The angle between two successive z-axes (Joint twist) Only and d are joint variables.

www.rcsaini.com35DENAVIT-HARTENBERG REPRESENTATION OF FORWARD KINEMATIC EQUATIONS OF ROBOTThe necessary motions to transform from one reference frame to the next.

Rotate about the zn-axis an able of n+1 (Coplanar).Translate along zn-axis a distance of dn+1 to make xn and xn+1 colinear.Translate along the xn-axis a distance of an+1 to bring the origins of xn+1 together. Rotate zn-axis about xn+1 axis an angle of n+1 to align zn-axis with zn+1-axis.

www.rcsaini.com36THE INVERSE KINEMATIC SOLUTION OF ROBOT Determine the value of each joint to place the arm at a desired position and orientation.

www.rcsaini.com37THE INVERSE KINEMATIC SOLUTION OF ROBOT

www.rcsaini.com38INVERSE KINEMATIC PROGRAM OF ROBOTS A robot has a predictable path on a straight line, Or an unpredictable path on a straight line.A predictable path is necessary to recalculate joint variables.(Between 50 to 200 times a second)To make the robot follow a straight line, it is necessary to break the line into many small sections. All unnecessary computations should be eliminated.

Fig. Small sections of movement for straight-line motionswww.rcsaini.com39DEGENERACY AND DEXTERITYDegeneracy : The robot looses a degree of freedomand thus cannot perform as desired. When the robots joints reach their physical limits, and as a result, cannot move any further.

In the middle point of its workspace if the z-axes of two similar joints becomes colinear. Fig. An example of a robot in a degenerate position.

Dexterity : The volume of points where one can position the robot as desired, but not orientate it.

www.rcsaini.com40THE FUNDAMENTAL PROBLEM WITH D-H REPRESENTATIONDefect of D-H presentation : D-H cannot represent any motion about the y-axis, because all motions are about the x- and z-axis.

Fig. The frames of the Stanford Arm.

#da1100-9022d109030d1004400-9055009066000TABLE THE PARAMETERS TABLE FOR THE STANFORD ARMwww.rcsaini.com41Thank You

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