Using belts and pulleys is far less expensive than using precision gearheads, and still has minimal backlash. High ratios get messy but not really needed with stronger motors and balanced or spring-loaded arm sections.

My goals for this arm:
Since the size of each arm section depends on the size of its "child" section, I needed to start with the wrist and work my way back to the shoulder and base.

The general design of the arm is below: Click thumbnails for full size

 The Wrist & EE-Disc Box 
The wrist box serves two purposes:
  - Provide a mount for the gripper with pass-thru cable (EE-Disc).
  - Able to spin the EE-Disc (and gripper) any number of times.

The Forearm holds the wrist box at its end. It also holds the motor that tilts the wrist box (and gripper) up and down 180° or more.

Some drawings: Click thumbnails for full size

Some pics: Click thumbnails for full size

I have switched back to using Arduino boards rather than the Beaglebone Black to run the stepper motors. Why? ...
Keeping with the "KISS" philosophy, I now use seven(7) separate Arduino Nano boards - one for each motor joint plus the gripper motor. Why? ...
So here's some pics of the revised motor control box ...

Some videos ...