6. robot kinematics and dynamics (1)

6. Basic CoordinatesRotation and Translation in basic of Robot Coordinates, Homogeneous Transformg

Euler Angles, Denavit-Hartenberg Notation

Affiliated Professor : Jaeyoung LeeE-mail : [email protected]

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한국 IT직업전문학교

Topicsp

Basic Coordinates

Coordinate Frames & Denavit Hartenberg NotationCoordinate Frames & Denavit-Hartenberg Notation

Forward and Inverse Kinematics of Robot Manipulators

Manipulator Jacobianp

Review of Lagrangian Dynamics

Lagrangian Dynamics of Robot Manipulators

Newton-Euler Equations of Robot Manipuators

Forward and Inverse Dynamics

Robot StaticsRobot Statics

Robot Trajectory Planning

Robot Control

로봇공학개론 2


Basic Coordinates

Operators for Translation and Rotation

- Translation - RotationTranslation Rotation



Basic Coordinate

Coordinate Frames




Basic Coordinate

Different View Based Upon Different Coordinates




Basic Coordinate

Rotation and Translation Combined

- A single rotation after a single translation:A single rotation after a single translation:

- A single rotation and a single translation after another pair of translation and rotation:

- Scheme shown above to represent transform operations is not a good way of simplifying notation and computations.

- We need consistent transforms for translations and rotations:- We need consistent transforms for translations and rotations: homogeneous transform



Basic Coordinate

Rotation Matrix R

- Vector p can be represented with respect to the two different coordinates, i.e., coordinate A and B

- We can to find a rotation matrix such that

- Noting that x_A, y_A and z_A represent the components of p along the x, y, and z axes of coordinate A.coordinate A.



Basic Coordinate

Rotation Matrix R

- Noting that x A, y A and z A represent the components of p along the x, y, and zNoting that x_A, y_A and z_A represent the components of p along the x, y, and z axes of coordinate A.

- Therefore,



Basic Coordinate

Rotation Matrix R

- Therefore,Therefore,

- Similarly,

- Thus, Therefore,

- Rotational matrix R is orthonormal (orthogonal).



Basic Coordinate

Homogeneous Transform

- In homogeneous transforms, translation and orientations have identicalIn homogeneous transforms, translation and orientations have identical characteristics.

- Position vector [x y z]T is represented by an augmented vector [x y z 1]T .

- Translation is represented by a matrix:Translation is represented by a matrix:



Basic Coordinate

- Rotation is represented by a matrix:



Basic Coordinate

- If translation and rotation are combined



Basic Coordinate

- If translation and rotation are combined

Therefore,

- Inverse of homogeneous transform



Basic Coordinate

More On Rotation : Euler Angles

- Many different sets of Euler angles existMany different sets of Euler angles exist

- Z-Y-Z, Z-Y-X, and etc,

- Example : Z-Y-X : Rotate around z-axis, and then around y-axis, and finally around x-axisaround x axis.

- Rotational Transformation (Post-multiplication)



Basic Coordinate




Basic Coordinate


- X-Y-Z Fixed Angles (Pitch-Yaw-Roll)X Y Z Fixed Angles (Pitch Yaw Roll)

- Note

Rotational Transformation (Pre multiplicatin)- Rotational Transformation (Pre-multiplicatin)



Basic Coordinate

Equivalent Angle-Axis

- Represented by a unit vector that indicates the axis of rotation and an angle toRepresented by a unit vector that indicates the axis of rotation and an angle to rotate.

- If the axis of rotation is k = [K_x K_y K_z]^T,



Basic Coordinate

Denavit-Hartenberg Notation

- Conventions

Joint i moves link i

Frame i is fixed at (attached to) link i and located at the distal joint, which means that frame i is located at joint i+1.


Z-axes are aligned with joint axes.


Basic Coordinate

4 Link Parameters

- Link Length(L n): the shortest distance between two joint axes at bothLink Length(L_n): the shortest distance between two joint axes at both ends of Link n.

- Link Twist(alpha_n): "twist" angle of two z-axes at both ends of Link n in the direction of x n -axisin the direction of x_n axis.

- Link Offset (d_n): distance from the origins of Frame n-1 to Frame n in the direction of Z_n-1-axis.

i l h l f 1 i i i h di i- Joint Angle(Thetat_n): angle from x_n-1-axis to x_n -axis in the directionof z_n-1-axis.



Basic Coordinate

Miscellaneous Conventions

- If z-axes meet each other x n – axis in general is defined to be z n-If z axes meet each other, x_n axis, in general, is defined to be z_n1Ⅹz_n.

- Positive direction of z-axis can be either.



Basic Coordinate

Miscellaneous Conventions

- The origin of frame 0 is usually coincide with that of frame 1The origin of frame 0 is usually coincide with that of frame 1.

- Z-axis of frame 0 is aligned with z-axis of joint 1.

- For a prismatic joint, z-axis is aligned with the sliding direction of the j i I hi li k ff d i h j i i bljoint. In this case, link offset d is the joint variable.

- Homogeneous transform :

- Homogeneous transform can be obtained directly by observing the dcoordinates in most cases.



Basic Coordinate

Example :



Basic Coordinate

Example :

h b d l d dNote that T^0_1 can be directly determined



Kinematics & Inverse Kinematics

Forward Kinematics

- For a given set of joint displacements, the end-effector position and orientation can be calculated. (Forward Kinematics)

Inverse Kinematics

- For a given set of end-effector position

and orientation, joint displacements are

computed.

- We have to solve nonlinear equations;We have to solve nonlinear equations;

however, we may not able to solve for

solutions analytically.




Kinematics & Inverse Kinematics for a 2-D Revolute-Jointed Manipulator

Forward Kinematics





F d Ki tiForward Kinematics





E d Eff t P itiEnd-Effector Position End-Effector Orientation





Inverse KinematicsInverse Kinematics- For a given tip position [x y]^T, the joint displacement [theta_1 theta_2]^Tis to be determined.

Geometric Method (Using cosine rule)





Inverse KinematicsInverse Kinematics






Sign: "+", if -p < theta_2 < 0, i.e., at "elbow up“ configuration "-", otherwise.





Algebraic MethodAlgebraic Method

left-hand-side: a constant matrixleft hand side: a constant matrixright-hand-side: a matrix whose elements are functions of joint angles






Let

Then





Algebraic MethodAlgebraic MethodSimilarly,

Usingg





Algebraic MethodAlgebraic Methodwhere

Therefore,




Kinematic Decoupling

If the final 3 consecutive axes meet at a point, the process of solving inverseIf the final 3 consecutive axes meet at a point, the process of solving inverse kinematics is simplified.

Three axes z4 , z5 , and z6 meet at a wrist centermeet at a wrist center.

The position of the wrist center, w, is determined by q1, q2 , and q3 only.




Kinematic Decoupling

- From the target tool position d and orientation R, wrist centerFrom the target tool position d and orientation R, wrist centerposition can be computed:

hwhere

- Once the wrist center is known, the first 3 joint variables, q1, q2, and q3 , canOnce the wrist center is known, the first 3 joint variables, q1, q2, and q3 , can be computed.

- The orientation matrix R^0_3 can be computed after q1, q2 , and q3 are d t i ddetermined.

- The rest of joint variables, q 4, q5 , and q6 , can be computed from R_6^3, which in turn can be computed bywhich in turn can be computed by




SCARA Robot




SCARA Robot

- Inverse Kinematics of SCARA Robot




SCARA Robot

- Rotation matrix R can be in the form ofRotation matrix R can be in the form of




SCARA Robot

- Using a geometry formed by projecting the manipulator configurationUsing a geometry formed by projecting the manipulator configuration upon a horizontal plane, which forms a 2-link manipulator,

- Again, similarly to a 2-link manipulator,

- and




SCARA Robot

- Solvability : A manipulator is defined to be solvable, if all the sets of joint variablesSolvability : A manipulator is defined to be solvable, if all the sets of joint variables for a given position and orientation can be determined by an algorithm. Many manipulators have multiple solutions for a single configuration. For example, PUMA 560 has 8 solutions.

W k Th l f h h d ff f i l h- Workspace : The volume of space that the end-effector of a manipulator can reach. Within the workspace, inverse kinematics solutions exist.

- Closed Form Solution : Inverse kinematic solutions can be grouped into closed form solutions and numerical solutions Closed form solutions are noniterative such assolutions and numerical solutions. Closed form solutions are noniterative such as algebraic expressions; whereas numerical solutions are iterative. Numerical solutions are computationally more expensive and slower.

- Pieper's work : A general 6 d.o.f. manipulator does not have a closed form solution. p g pIf the three consecutive axies intersect at a point, a closed form solution exists. For most commercial manipulators, the last consecutive axes intersect at a point.

- Notes: Even if inverse kinematic solutions are found, they may not be physically realizable due to joint angle limitationsrealizable due to joint angle limitations.



HW 6

139p, 연습문제 1



REFERENCE

기초로봇 공학

제 2편 : 로봇 제어의 시작- 제 2편 : 로봇 제어의 시작

Introduction to Robotics, by J, J. Craig , y , g

- Ch 2. Spatial descriptions and transformations

- Ch 3. Manipulator kinematics

- Ch 4. Inverse manipulator kinematics


6. robot kinematics and dynamics (1)

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