11/10/2015Handout 41 Robotics kinematics: D-H Approach

11/10/2015Handout 42 General idea for robot kinematics: revisit

11/10/2015Handout 43 End-effector World frame

11/10/2015Handout 44 Robotics kinematics: Definition, Motor and End-effector  Each component has a coordinate system or frame.  Kinematics reduces to the relationship between the frames.  Further, if one frame is set up on the ground called world frame, the “absolute” position and orientation of the end-effector is known. The relationship between different frames = kinematics

11/10/2015Handout 45 How to set up or assign a local frame to each component of the robot? What is called a component? What is called a joint?

11/10/2015Handout 46 Robot Kinematics: Logics of presentation: Kinematics: what Coordinate system: way to describe motion Relation between two coordinate systems Definition of component and joint: robot structure

11/10/2015Handout 47 World frame

11/10/2015Handout 48 Link: Component with only considering its joint line but neglecting its detailed shape. Next slide (Fig. 2-21) shows various types of joints

11/10/2015Handout 49 Fig Joint types Kinematic pair types Neglecting the details of the joint but relative motions or relative constraints between two connected links Degrees of freedom of joint: the number of relative motions between two links that are in connection

11/10/2015Handout 410 Fig.2-22 General configuration of link

11/10/2015Handout 411 The geometrical parameters of the general link are: - The mutual perpendicular distance: a i-1 - The link twist: i-1 Fig shows two links that are connected, which leads to the following geometrical parameters: - d i link offset - joint angle From axis i-1 to axis i From axis a(i-1) to axis a(i) along axis i From axis a(i-1) to axis a(i)

11/10/2015Handout 412 Fig. 2-23

11/10/2015Handout 413  Denavit-Hartenberg (D-H) notation for describing robot kinematic geometry.  It has the benefit that only four parameters describe completely robot kinematic geometry. The above four parameters define the geometry of Link (i-1).  The shortcoming is that the four parameters defined across two links, e.g., for Link (i-1) in the above, the four parameters are defined based on Link (1-1) and Link (i).

11/10/2015Handout 414 Alternative way to define D-H parameters Definition of DH parameters for Link (i) will cross two links as well, that is, Link (i) and Link (i-1). In this class, we take the previous one.

11/10/2015Handout 415 Labeling of links: towards a unified representation  The base link or ground 0.  The last link n.  For other links, 1, 2, …., n-1.

11/10/2015Handout 416 Robot Kinematics: Logics of presentation: Kinematics: what Coordinate system: way to describe motion Relation between two coordinate systems Definition of component and joint: robot structure Assign a local frame to each link (D-H notation)

11/10/2015Handout 417 Rule to assign a frame to each link (intermediate links)  The Z-axis of frame (i), Zi is coincident with the joint axis i. The origin of frame (i) is located at the intersection point on axis i of the common perpendicular line between axis i and axis i+1.  Xi points along the common perpendicular line between axis i and axis I+1, particularly directed from axis i to axis i+1. In the case that the common perpendicular distance is zero, Xi is normal to the plane which is spanned by axis Zi and Zi+1.  Yi is formed by the right-hand rule based on Xi and Zi.

11/10/2015Handout 418 An example for link i-1 and link i Fig. 2-24

11/10/2015Handout 419 The rule for the D-H coordinates of frame (0) and frame (n): For frame (0), the rule is as follows: 1.Define Z0 coincident with Z1 such that a o = Define X0 such that αo = Additionally, define the origin of frame (0) such that d 1 = 0.0 if joint 1 is revolute, or θ 1 = 0.0 if joint 1 is prismatic. For frame (n), the rule is as follows: 1.Define Xn such that αn = 0.0.  Z axes are all normal to the paper plane, including Z0.  Z0 is coincident with Z0 and Z1, so a0=0.0, α0 = 0.0.  X0 is set such that d0=0.0.  X3 is set such that d3=0.

11/10/2015Handout 420 Summary of the D-H parameters If the link frames have been attached to the links following the foregoing convention, the definitions of the link parameters are (for link i):  ai : the distance from Zi to Zi+1, measured along Xi.  di : the distance from Xi to Xi-1 measured along Zi.  αi : the angle between Zi, and Z i+1 measured about Xi.  Θi: the angle between X i-1 and Xi, measured about Zi. Remark: Choose a i > 0 since it corresponds to the distance; however, other three parameters could be a number with signs (plus, minus).

11/10/2015Handout 421 A note about non-uniqueness in assignment of D-H frames: The convention outlined above may not result in a unique assignment of the frame to the link. 1.There are two choices of the direction of Zi when defining Zi axis with joint axis i. 2.When axes i and i+1 are parallel, there are multiple choices of the location of the origin for frame (i). 3.When axes i and i+1 are in intersection, there are two choices of the direction of Xi. 4.When Zi and Zi+1 are coincident, there are multiple choices for Xi as well as for the location of the origin of frame (0).

11/10/2015Handout 422 Fig.2-25 Example 1

11/10/2015Handout 423 Fig.2-26 Link 0 Link 1 Link 2

11/10/2015Handout 424 Example 2

11/10/2015Handout 425

11/10/2015Handout 426 Parameter table to be given in the classroom

11/10/2015Handout 427 Summary 1.Link and joint concept. 2.D-H notation for link. 3.Assign frames to links based on D-H. 4.Benefit of D-H: a minimum number of parameters to describe links and joints. 5.Shortcoming of D-H: parameters must cross two consecutively connected links.

11/10/2015Handout 428 Robot Kinematics: Logics of presentation: Kinematics: what Coordinate system: way to describe motion Relation between two coordinate systems Definition of component and joint: robot structure Assign a local frame to each link (D-H notation) Kinematic equation

11/10/2015Handout 429 Robot kinematics: The relationship among the D-H frames In the previous discussion, D-H frames are established, that is, we have 0, 1, 2, …, n frames established based on the D-H notation and rule. In this slide, we discuss the mathematical representation for this relationship.

11/10/2015Handout 430 The goal is to find the relationship matrix for frame i-1 and frame i

11/10/2015Handout 431 The idea is to put a series of frames between them, denoting them as FR, FQ, FP. As such, frame i-1  FR  FQ  FP  frame i.

11/10/2015Handout 432 The idea is to put a series of frames between them, denoting them as FR, FQ, FP. As such, frame i-1  FR  FQ  FP  frame i. T from i-1 to FR T from FR to FQ T from FQ to FP T from FP to i

11/10/2015Handout 433 Transformation matrix between two DH frames

11/10/2015Handout 434 Forward kinematics  General idea: suppose that we have n moving likes.  Solution: forward kinematics, given the motor’s motion, to find the position and orientation of the end-effector.  The position of the end-effector and the orientation of the end-effector completely describe the end-effector.  The position of the end-effector can be the position of the origin of the frame (on the end-effector).  The orientation of the end-effector is represented in the R matrix between the frame (on the end-effector) and the world frame or frame to the ground, i.e., {0}.

11/10/2015Handout 435 The problem is: known the right side variable to find the left side variable.

11/10/2015Handout 436 Inverse kinematics

11/10/2015Handout 437 The problem is: known the left side variable to find the right side variable.

11/10/2015Handout 438 Example 1

11/10/2015Handout 439

11/10/2015Handout 440 T matrix here

11/10/2015Handout 441 Forward kinematics

11/10/2015Handout 442 Inverse kinematics

11/10/2015Handout 443 Summary 1.Transformation matrix between two DH frames. 2.General equations for forward kinematics. 3.General equations for inverse kinematics.