Direct Kinematics - By Remotely Triggering Stationary Base Robot (Theory) _ Virtual Robotics Lab _...

8/12/2019 Direct Kinematics - By Remotely Triggering Stationary Base Robot (Theory) _ Virtual Robotics Lab _ Mechanical Engineering _ IIT GUWAHATI Virtual Lab

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8/11/13 Dir ect Kinematics - By Remotely Trig gering Stationary Base Robot (Theory) : Virtual Robotics Lab : Mechanical Engi neering : IIT GUWAHATI Virtual

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Introduction Theory Setup Demonstration Quiz Reference Feedbac k

DIRECT KINEMATICS:The primary objective of the robotic manipulator is to control both the position and orientation of the tool in threedimensional space. This requires the formulation of a relationship between the joint variables and the position andorientation of the tool which is termed as the direct kinematics problem.

COORDINATE FRAMES:Fundamental Rotation:

Fig: Fundamental rotations in R 3

Let F={f 1, f 2 , f 3 } represent a fixed coordinate frame and M={m 1, m 2 , m 3 } represent a mobile coordinate frame in the spaceR 3 , with both the frames having the same origin. The mobile coordinate frame is obtained from the fixed coordinate frame

by rotating the former about one of the u nit v ectors of the latter. T he result ing coordinate transformation matrix is called afundamental rotation matrix.Initially, M is coincident with F. For an arbitrary vector P in the space R 3 , we have,

Now,

In matrix form,

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Where,

is the resulting coordinate transformation matrix or fundamental rotation matrix which maps M coordinates into fixed Fcoordinate.

Fig: Rotation o f M about f 1 by angle

If M is rotated about the f 1 axis of the fixed coordinate frame F by an angle , then, f 1=m 1 and since, m 1 is orthonormal to both m 2 and m 3 , we have

Since the dot product is equal to the cosine of the angle between the unit vectors, we have

Similarly, if M rotates about the f 2 axis by an angle , then

and if M rotates about the f 3 ax is by an angle , t hen


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Composite Rotations:

Composite Rotations:The yaw, pitch and roll are the three fundamental rotations, each of which is represented by a matrix. When a number of fundamental rotations are applied to a robotic tool, it corresponds to a number of rotation matrices multiplied together

with the product matrix representing a sequence of rotations about the unit vectors. Th is form of mul tiple rotations istermed as composite rotations.Since the operation of matrix multiplication is not commutative, the order in which the fundamental rotations areperformed makes a difference in the resulting composite rotation. After one rotation has been performed, the axes of thetwo coordinate frames are no longer coincident and the subsequent rotations of the tool could be performed about the unit

vectors of either th e fixed coordinate frame, F or the rotated coordinate frame, M.The algorithm for constructing a composite rotation matrix is given below:1. The rotation matrix is initialized to R=I, which implies that the orthonormal coordinate frames, F and M, arecoincident.

2. If mobile coordinate frame M is to be rotated by an amount about the k th unit vector of the fixed coordinate frame

F, R is premultiplied by R k ( ).

3. If the mobile coordinate frame M is to be rotated by an amount about its own k th unit vector, R is postmultiplied

by R k ( ).4. If there are more fundamental rotations to be performed, step 2 is repeated otherwise stopped. The resulting compositerotation matrix R maps mobile M coordinates into fixed F coordinates.

The identity matrix I corresponds to no rotation at all.Rotation of frame M about the unit vectors of frame F are represented by premultiplication by the appropriate fundamentalrotation matrix.Rotation of frame M about one of its own unit vectors are represented by post multiplication by the appropriatefundamental rotation matrix. Homogeneous Coordinates:The orientation of a robotic tool is characterized by pure rotations where the origin of the transformed coordinate frame issame as the origin of the original coordinate frame. However, position of the tool relative to a coordinate frame attached tothe robot base can be characterized by one type of transformation known as translation where the origin of the translatedcoordinate frame is not same as the origin of the original coordinate frame.

Let q be a point in space R 3 and F be an orthonormal coordinate frame for R 3 .

is any non-zero scale factor, thus the homogeneous coordinates of q w.r.t. F are denoted as,

i.e., the homogeneous coordinate of q in R 3 are represented by a vector in four dimensional space R 4 . The originalphysical three dimensional vector from its four dimensional homogeneous coordinates can be recovered using the relation,

where is a 3x4 h omogeneous coordinate conversion matrix defined by ,

To change a physical point in three dimensional space expressed in terms of it homogeneous coordinates from onecoordinate frame to another, a 4x4 homogenous transformation matrix, T is used. T can be partition into four separatesubmatrices as given below,

Where, R=rotation matrix which represents the orientation of the mobile coordinate f rame relativ e t o the fixed referenceframe. It specifies the orientation of the tool.p=translation vector which represents the position of the origin of the mobile coordinate frame relative to the fixedreference frame. It specifies the position of the tool tip.

=perspective vector and specifies a point of perspective.

= scaling factor set to unity.

Translations and Rotations:


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For two orthonormal coordinate frames, F and M, that are initially coincident, if M is rotated by an amount about thek th unit vector of F, then in terms of homogeneous coordinates this operation can be represented by a 4x4 matrix Rot(

), where

R k ( )= k th fundamental rotation matrix

Rot( )=fundamental homogeneous ro ta tion matrixThe translation vector p has been set to zero. the homogeneous coordinate can also be used to represent translations. Fortwo initially coincident orthonormal coordinate frames, F and M, if origin of mobile coordinate frame M is to be translated

by an amount p k along the k th unit vector of F for , then in terms of homogeneous coordinates this operation can be represented by a 4x4 matrix Tran(p), where

Tran(p) is the fundamental homogeneous translation matrix with translation vector p and rotation matrix R set to identity matrix I.Homogeneous transformation matrices represent both rotation and translation of mobile frames w. r. t. fixed frame.

Composite Homogeneous Transformation:Since matrix multiplication is not a commutative operation, the order in which rotations and translations are performed isdetermined by the following algorithm:

1. The transformation matrix is initialized to T=I which corresponds to the orthonormal coordinate frames F and M being coincident.

2. Rotation and translations are represented using separate homogeneous transformation matrices.3. Composite rotations are represented as separate fundamental homogeneous rotation matrices.4. If M is to be rotated about or translated along a unit vector of F, premultiply the homogeneous transformation

matrix T by the appropriate fundamental homogeneous rotation or translation matrix.5. If M is to be rotated about or translated along one of its own unit vectors, postmultiply T by the appropriate

fundamental homogeneous rotation or translation matrix.6. For more fundamental rotations or translation to be performed, step 4 is repeated else stopped.

Resulting composite homogeneous transformation matrix T maps mobile M coordinates into fixed F coordinates.

Screw Transformation: A linear displacement along an ax is combined with an angular displacement about the same axis is referred to as a screw transformation.For F and M being initially coincident fixed and mobile orthonormal coordinate frames respectively, if M is translated along

the kth

unit vector of F by a displacement of and rotated about the kth

unit vector of F by an angle of , theresulting composite homogeneous coordinate transformation matrix is called a k th fundamental screw transformationmatrix.

Kinematic Parameters:There are two types of kinematic parameters, viz. joint parameters and link parameters.Joint Parameters: every adjacent pair of links which connected by either a revolute or prismatic joint. The joint parametersspecify the relative position and orientation of two successive links.


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Fig Joint angle and joint distance d

In the above diagram joint k connects link k- 1 to link k and axis z k-1 is aligned with the axis of joint k. The two jointparameters associated with joint k are

Joint angle ( ): it is the rotation about z k-1 needed to make axis x k-1 parallel with axis x k .

Joint distance ( ): it is the translation along z k-1

needed to make axis x k-1

intersect with axis x k

.For a revolute joint, the joint angle is variable and joint distance is fixed while for a prismatic joint the joint angle is fixedand the joint distance is variable. Link Parameters: since there is a link between two successive joints, link parameters specify the relative position andorientation of the axes of two successive joints.

Fig: Link length and link twist angle

In the above diagram link k connects joint k to joint k+1 and an axis x k is the common normal between the axes of joint kand joint k+1. the two link parameters are

Link length( ): it is the translation along x k needed to make axis z k-1 intersect with the axis z k .

Link twist angle( ): it is the rotation about x k needed to make axis z k-1 parallel with axis z k .The two link parameters are always constant for both revolute and prismatic joints.For an n -axis robot, there are n+1 links interconnected by n joints, with joint k connecting link k-1 to link k .The joints and links of a robotic manipulator are numbered outwards starting from the fixed base, which is termed as link 0to the tool which is termed as link n


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1 0 0

TWO-AXIS PLANAR ARTICULATED ROBOT:Two axis planar articulated robot is shown below:

Fig: A two-axis planar articulated robotThe two physical links are thin cylinders or rods. There are two joints, therefore two axes and three links. The fixed base islink 0 and there will be three coordinate frames for each link.The kinematic parameters, using D-H algorithm are given by,

Axis d

1 0 0

2 0 0

FOUR-AXIS SCARA ROBOT:


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Source: ww w.adept.com A SCARA (Select ive Com pliance Assembly Robot Arm) robot is an important example of robotic m anipulator.The SCARA robot shown above has three revolute joints and one prismatic joint. Using D-H algorithm, we can construct thelink-coordinate diagram as shown below,

Fig: A four-axis SCARA Robot (Adept One )

The kinematics parameters of the four axis SCARA robot is given by,Axis d

1

2 0 0

3 0 0 0

4 0 0

PUMA ROBOT:The PUMA robot is one of the most famous industrial robots with 6 rotary joints.The robot is shown in the figure below:


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Source: www .berglas.org The kinematic parameters of the P UMA robot a re as follows :

Joints d

1 90 0 0 -90

2 0 149.5 432 0

3 90 0 0 90

4 0 432 0 -90

5 0 0 0 90

6 0 55.5 0 0

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