A common robot has three parallel rotary joints and a single linear joint in the vertical direction, for four degrees of freedom at the tool tip.
Select figure to enlarge.

For years, most motion system designers wanting to create motion in a 2D plane or in 3D space would use a Cartesian mechanism. The perpendicular action of Cartesian actuators makes several things easy for the system designer. First, Cartesian coordinates are commonly used and familiar. Second, having Cartesian actuators provides for a direct mapping of tool coordinates to actuator positions. Third, the same direct mapping applies to velocities and accelerations, making these easily calculable (and limitable) for each actuator. In addition, perpendicular Cartesian actuators create no significant dynamic interaction between the axes, so simple single-feedback control loops can be used on each axis. Finally, there is virtually no change in system dynamics over the travel of the actuators because the moment of inertia experienced by each actuator essentially stays constant.

However, modern computing power and controller technology are making it so non-Cartesian actuators can be used where they weren’t feasible or cost-effective before. This in turn permits a great deal more choice in the physical configuration of mechanisms, often yielding startling increases in performance. How are all these benefits of Cartesian systems outweighed by alternative configurations? Well, Cartesian systems do have their limitations. In almost all Cartesian mechanisms one axis carries another perpendicular axis; this requires much greater power and can yield lower performance for the carrying axis than other methods. Too, Cartesian mechanisms don’t produce the extended working area that serial-link mechanisms can provide (the area possible with a standard robot arm, for example.) Nor can Cartesian systems increase mechanical stiffness and reduce measurement errors as hexapods and other parallel-link mechanisms do.

Kinematics

To maximize non- Cartesian system performance, full exploration and definition of the kinematic situation is required.

The subject of applied kinematics deals with the geometric relationship between a mechanism’s tool tip (or end effector) and the underlying linear and rotary joints that cause its movement. A forward-kinematic transformation computes the tool-tip coordinates from the joint positions. Conversely, the inversekinematic transformation computes the joint position from the tool-tip coordinates. Both of these transformations are needed if the end-user wants to program motion in tool-tip coordinates. The forward-kinematic calculations must be done at the outset when starting from an arbitrary configuration to establish the starting tool-tip coordinates for the initial move. The inverse-kinematic calculations must be done at least for the end point of every move.

If a controlled path is also desired, these calculations must also be done at closely spaced intervals along that path.

Kinematic mechanism analysis

Kinematics equations
Limiting the elbow angle θ_E to positive values and selecting positive arc-cosine solutions gives the inverse kinematic equations as shown.
Select figure to enlarge.

As an example, let’s consider the common SCARA robot. This mechanism has three parallel rotary joints turning about vertical axes — shoulder S, elbow E, and wrist W — and a single linear joint V in the vertical direction. The tool tip therefore has four degrees of freedom: three translational (X, Y, Z) and one rotational (C rotating about Z).

Limiting ourselves to positive values of the elbow angle θ_E that produce the right-armed case (done by selecting the positive arc-cosine solutions) we can write the inverse-kinematic equations as shown in the figure above.

Even this simple mechanism brings to light a couple of tricky issues commonly encountered with non-Cartesian mechanisms. The first problem is that there can be multiple solutions to the inverse-kinematic transformation: in this case, the right-armed and left-armed configurations. (Here, “right-armed” refers to the counterclockwise bend of the elbow joint, which looks like a human’s right arm from the top view.) Some method must be used to select which is used.

Continue on page 2