Background: I have access to a UR-10, 6DOF robotic arm through my school (I'm very new to robotics). I know the desired set of linear speeds that I want in the x,y, z-direction in terms of the end effector ([x,y,z,rx =0, ry,=0, rz =0]). Using an analog controller I receive linear speeds in the x,y, z-direction ranging from -.1 -.1m/s.

I found the forward kinematics for the UR-10 online and begin to derive the Jacobian Matrix. (If anyone has the Jacobian matrix for a UR-10 that would be awesome.) Since I'm only interested in the linear motion, where rx,ry,rz =0 I thought I could simplify my Jacobian to a 3x3 matrix.

I realized that by doing so I would be unable able to solve for all the joints speeds 1-6.

[Jacobian^-1][Velcoity Vector] = [Joint Speeds]

[3x3][3x1] = [3x1] Joint Speed vector

However, I need a 6x1, so I have the speed for each joint.

What am I doing wrong?

What are the other 3 equations I would need to define a full 6x6 Jacobian and solve for the appropriate joint speeds?

EDIT: I foresee a problem that since my linear speeds change incrementally there may be singularities when calculating my Inverse Jacobian how could I work around that?

TLDR; Basically what I want to do is create linear motion using the speedJ command 

O a 6dof robot, linear motion can be created just by using the first 3 joints. Think of the first 3 joints as the positioning system and the last 3 joints as the orientation system.