3. Manipulator Kinematics. Division of Electronic Engineering Prof. Jaebyung Park

Transcription

1 3. Manipulator Kinematics Division of Electronic Engineering Prof. Jaebyung Park

2 Introduction Kinematics Kinematics is the science of motion which treats motion without regard to the forces that cause it. Position, velocity, acceleration, and all higher order derivatives of the position variables are studied w.r.t time or any other variables. Position and orientation of the manipulator linkages in static situations. The position and orientation of the end-effector relative to the base are computed as a function of the joint variables.

3 Link Description Lower pair Lower pair means the connection between a pair of bodies when the relative motion is characterized by two surfaces sliding over one another. Most manipulators have revolute joints or have sliding joint called prismatic joints.

4 Link Description For the purposes of obtaining the kinematic equations, A link is considered only as a rigid body which defines the relationship between two neighboring joint axes of a manipulator. Two link parameters for link i-1 Link length is measured along the mutually perpendicular line as a i-1. Link twist is the relative location of two axes and denoted as α i-1.

5 Link Connection Description Intermediate links in the chain Link offset is the distance along the common joint axis between two neighboring links. (d i at joint axis i) Joint angle is the amount of rotation about this common joint axis. (θ i ) Link parameters For a revolute joint, Joint variable: θ i Fixed link parameters: a i, α i, d i For a prismatic joint, Joint variable: d i Denavit-Hartenberg notation

6 Convention for Affixing Frames to Links Link frame attachment procedure 1. Identify the joint axes and imagine (or draw) infinite lines along them. For steps 2 through 5 below, consider two of these neighboring lines (at axes i and i+1). 2. Identify the common perpendicular between them, or point of intersection. At the point of intersection, or at the point where the common perpendicular meets the ith axis, assign the link frame origin. 3. Assign the Z axis pointing along the ˆi ith joint axis. 4. Assign the Xˆ i axis pointing along the common perpendicular, or if the axes intersect, assign Xˆ i to be normal to the plane containing the two axes. 5. Assign the Yˆi axis to complete a righthand coordinate system. 6. Assign {0} to match {1} when the first joint variable is zero. For {N} choose an origin location and Xˆ N direction freely, but generally so as to cause as many linkage parameters as possible to become zero.

7 Convention for Affixing Frames to Links Link parameters in terms of the link frames a i i d i i Zˆi Zˆi 1 = the distance from to measured along Z Z ˆi ˆi 1 Xˆ ˆ i 1 i Xˆ ˆ i 1 i = the angle between and measured about = the distance from to X measured along = the angle between and X measured about Xˆ i Zˆi Xˆ i Zˆi

8 Convention for Affixing Frames to Links Example 1 of 3DOF manipulator Frame Assignment

9 Convention for Affixing Frames to Links Example 1 of 3DOF manipulator (Cont ) Link Parameters

10 Convention for Affixing Frames to Links Example 2 of 3DOF and one prismatic joint Frame Assignment

11 Convention for Affixing Frames to Links Example 2 of 3DOF and one prismatic joint (Cont ) Link Parameters

12 Convention for Affixing Frames to Links Example 3 of 3DOF nonplanar manipulator Two Possible Frame Assignment

13 Convention for Affixing Frames to Links Example 3 of 3DOF nonplanar manipulator (Cont ) Two More Possible Frame Assignment

14 Manipulator Kinematics Derivation of link transformations Transform vectors defined in {i} to their description in {i-1} or where or

15 Manipulator Kinematics Concatenating link transformations Single transformation that relates frame {N} to frame {0}

16 Frames with Standard Names Thebaseframe,{B} The station frame, {S} The wrist frame, {W} The tool frame, {T} The goal frame, {G}