Forward Kinematics Cedric Fischer and Michael Mattmann Institute of - PowerPoint PPT Presentation

Forward Kinematics Cedric Fischer and Michael Mattmann Institute of Robotics and Intelligent Systems Department of Mechanical and Process Engineering (DMAVT) ETH Zurich 1

Forward/Inverse Kinematics § Kinematics: To describe the motion of the manipulator without consideration of the forces and torques causing the motion : A Geometric Description. Forward Kinematics To determine the position and orientation of the end effector with the given values for the joint variables. θ 1 , θ 2 , θ 3 → R 0 3 Inverse Kinematics To determine the joint variables with the given the end effector � s position and orientation. R 0 3 → θ 1 , θ 2 , θ 3 2

Rigid body motion q : a point attached to the rigid body  R ab ( t ) � p ab ( t ) the rigid body motion of the frame B attached to the body, g ab ( t ) = 0 1 y, relative to a fixed or inertial frame A 3

Forward Kinematics with Screw Theory: POE § Forward Kinematics defines a transformation between the joint space and the task space § Joint Space: § Defined by the independent angles theta § Configuration of robot joints § Task Space: § Defined by position and orientation of end-effector § Cartesian space 4

Forward Kinematics with Screw Theory: POE § General forward kinematics map § Written using the product of exponentials formula: ˆ ˆ ˆ ξ 1 θ 1 e ξ 2 θ 2 ...e ξ n θ n g st (0) g st ( θ 1 , θ 2 , ..., θ n ) = e ˆ ˆ ˆ ξ 1 θ 1 e ξ 2 θ 2 ...e ξ n θ n g st (0) g st ( θ ) = e § Product of exponentials uses only two frames! Base frame S and tool frame T • 5

Forward Kinematics with Screw Theory: Example § Start from the general formula: ˆ ˆ ˆ ξ 1 θ 1 e ξ 2 θ 2 ...e ξ n θ n g st (0) g st ( θ ) = e § Find and Screw parameter and calculate exponentials: § Compute forward kinematics: 6

The Denavit-Hartenberg Convention § In general, we would need 6 independent parameters to define the transformation between two � neighboring � coordinate frames § The D-H convention reduces the problem to 4 parameters by a clever choice of the origin and orientation for the coordinate frames § Cancellations occur! A i = Rot z, θ i Trans z,d i Trans x,a i Rot x, α i         c θ i − s θ i a i 0 0 1 0 0 0 1 0 0 1 0 0 0 s θ i c θ i c α i − s α i 0 0 0 1 0 0 0 1 0 0 0 0         =         0 0 1 0 0 0 1 d i 0 0 1 0 0 s α i c α i 0         0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1   c θ i − s θ i c α i s θ i s α i a i c θ i s θ i c θ i c α i − c θ i s α i a i s θ i   =   s α i c α i d i 0   0 0 0 1 7

The Denavit-Hartenberg Convention § Assume two features! DH1: The axis X i is perpendicular to Z i-1 DH2: The axis X i intersects the axis Z i-1 • angle from x i-1 to x i measured in a plane normal to z i-1 θ i : joint angle d i : link o ff set • distance from o i-1 to intersection of x i and z i-1 measured along z i-1 a i : link length • distance between z i-1 and z i measured along x i α i : link twist • angle between z i-1 and z i measured in a plane normal to x i 8

The Denavit-Hartenberg Convention (Example) § Forward Kinematics with Denavit-Hartenberg convention d, the robot is better conditioned. DH1: The axis X i is perpendicular to Z i-1 θ 2 θ 1 DH2: The axis X i intersects the axis Z i-1 y 1 y 2 z t z 3 a α d θ Link y 3 1 0 90 5 90 x 1 x 2 y t x 3 2 5 0 0 θ 1 z 1 z 2 x t 3 3 -90 0 θ 2 t 0 0 0 -90 z b y b x b 9

Assignment 4 d, the robot is better conditioned. g 0 t (0) a) (by inspection) θ 2 θ 1 b) Screw parameters frame 0 h, l, M ˆ c) ξ i ξ i frame 0 g b 0 (0) d) g bt ( θ ) = g b 0 g 0 t ( θ ) e) • Figure, chart, video… ˆ ˆ ξ 1 θ 1 e ξ 2 θ 2 g 0 t (0) = g b 0 e 10