Galileo Sphere Robot (GSR) Kinematics

ROBOT KINEMATICS FOR GALILEO SPHERE ROBOT

Robot definition as a group of axes with a predefined point type

The robot kinematics will be assigned to a new model type (15) with a XYZPR (x,y,z, pitch, roll) point type. The following line defines the GALILEO robot as a new system variable:

Common Shared GALILEO As Group Axnm = A1 Axnm = A2 Axnm = A3 Axnm = A4  Axnm = A5 model = 15 of xyzpr

Kinematics of the robot

Joint Coordinates:

Axis name Joint Type Range Units Theta1 J1 Rotary axis [-180,180] deg Theta2 J2 Rotary axis [-10,90] deg Z1 J3 Linear [Z1min,Z1max] mm Z2 J4 Linear [Z2min,Z2max] mm Theta3 J5 Rotary axis Multi turn deg

There are three rotary axes:

• theta1 – big horizontal rotary (J1 or joint #1) • theta2 – pitch axis (J2 or joint #2) • theta3 – tool rotation (J5 or joint #5)

And two linear axes:

• Z1 (J3 or joint #3) • Z2 (J4 or joint #4) The Z1 is the linear axis without connecting rod. General presentation of the robot joint point in MC-Basic language is (list of numeric expressions inside curly brackets):

{J1, J2, J3, J4, J5}

Cartesian coordinates:

Cartesian coordinates (world) of the robot TCP (Tool Center Point) are (Xw, Yw, Zw). Orientation of the robot tool’s Z axis is always expressed as a rotation needed to rotate the Z world (Zw) axis into Z tool (Zt) axis. It can be expressed with 3 Euler angles (Yaw, Pitch, Roll) of the ZYZ order of rotations.

Yaw angle:

Rotating (Xw, Yw, Zw) system around Zw axis into (X1, Y1, Z1)

Pitch angle:

Rotating (X1, Y1, Z1) system around Y1 axis into (X2, Y2, Z2)

Roll angle:

Rotating (X2, Y2, Z2) system around Z2 axis into (X3, Y3, Z3) which coincides to (Xt, Yt, Zt)

As the given robot has only 5 DOF (Degrees Of Freedom) we are reducing the orientation representation to Pitch & Roll angles only. That corresponds to rotations around Y2 (axis obtained after rotating Yw with θ1) and Z3 tools Z axis (in that order).

General presentation of the robot Cartesian point in MC-Basic language is (list of numeric expressions inside curly brackets preceded with hash sign):

1. {X, Y, Z, Pitch, Roll}