As hard as we try, we can never make two axes exactly parallel, we cannot machine parts to exact dimensions, and we can never be totally sure where a zeroed configuration is located.  Those are the supposed to be easy parts of kinematic analysis.  Still, in a more practical setting, we must think about backlash (which might be gravity-dependent) and sensor resolutions and nonlinearities.  Building a kinematic model for real robot can be quite tedious, and we need these models for the inverse kinematics problem.  This is the field of robot calibration.  My current research is examining methods for carrying out this process both with an without machine learning using the 3 degree-of-freedom manipulator above.

Angles are not the only way to measure orientation.  Constrained points, i.e. points on a circle, also provide equivalent orientation information.  In addition, the mathematics can typically be performed without trigonometric functions.  This leads to faster computation, particularly on less powerful processors.  Read more about kinematic modeling and control with using a point on a circle in the following paper: Angle-Free Mobile Robot Position Control!  Preliminary work with unit-quaternions is also described in: 

• Application and benefits of using unit-quaternions for mobile robot kinematics and control

• Quaternion-Based Kinematic Modeling and Control for Differential Drive Mobile Robots!

The transposed Jacobian control technique is a technique that originated in robotic manipulator control.  The idea is to determine the force and torque profile that will drive an end-effector to a desired pose and then transform this force and torque profile in to joint torques.  The technique can also be applied to mobile robots where, perhaps, one wishes to drive a tool or instrument on the robot to a certain location in space.  Several of my papers that address and solve this problem are:

•  A Transposed Jacobian Position Control for Differential Drive Mobile Robots 

• Transposed Jacobian Control of a Mobile Robot

Another paper that addresses this topic in a more advanced manner, End-Effector Trajectory Tracking Control on Differentially Driven Mobile Robot, is currently under review.  

The phrase "space robot" is used to refer to space satellites possessing a robotic manipulator.  Unlike terrestrial manipulation where a manipulator's based is fixed to the Earth which moves very little in response to manipulator motion, a satellite can move a significant amount in response to manipulator motion.  This creates an interesting problem.  My Ph.D. work in this area focused on modeling the dynamics of these system with unit-quaternions and also the control of them with quaternion elements.  The high-degree of freedom nature of the systems, and the fact that simulation is the primary tool for studying them, led me to develop methods for extremely fast simulation of the equations of motion.  The above video is demonstrates the result from integrating the equations of motion for a 18 degree of freedom system with a 4th order Runge-Kutta in near-real time!  Although much of my recent research has made use of this technique and those results have been publish, the specific technique has not yet been published.  It will be present in a forthcoming book.