Kinematic Synthesis of Robots


Our current focus is on the computer aided synthesis of constrained serial chains. There are 15 spatial serial chain primitives that have 35 special cases, as opposed to planar constrained serial chians where there are two basic primitives. This provides a rich opportunity for the invention of devices to provide leveraged spatial movement.

The planar constrained serial chain primitives are the RR and RP chains show in this figure.

The following table is the list of spatial constrained serial chain primitives, without considering permutations of the joints.

Our Synthetica 1.0 software provides a structure for the synthesis, animation and evaluation of constrained serial chains.

Synthesis algorithms for the RR, TS and special cases of the CS and RPS chains have been integrated so far.

Select here to load an animation of Synthetica showing the construction and analysis of an RRR spatial robot.

Select here to load an animation showing the synthesis of an RRTS spatial linkage.

Here are other linkage systems constructed and analysed using Sythetica 1.0:

This is a parallel constrained robot formed by two TPR serial chains. The blue trajectory is a goal path that interpolates a set of specified frames. The green path is the trajectory generated by Synthetica that lies in the workspace of this chain.

This is a parallel constrained robot formed by three RPS serial chains. The grey trajectory is a goal path and the blue path is the trajectory generated by Synthetica.

Select here to download a copy of our Synthetica 1.0 software.

Bennett Linkage Animations

The kinematic synthesis of robots begins with the spatial RR chain which is the simplest spatial robot. An interesting result discovered in the late 1960's is that for any synthesis problem there are two real solutions that can be assembled to form Bennett's linkage.

Select here for an animation (2.1MB) of a Bennett Linkage (Tsai and Roth 1973).
Select here for a more detailed animation (5.2MB).

Synthesis of Spatial RR Chains

This Mathematical notebook derives the design equations for three position synthesis of a spatial RR chain. Cut and paste the text into a new Mathematical notebook.

The centerpiece is the symbolic elimination of special design parameters in four non-linear equations in four unknowns to obtain a single cubic equation. Along the way, we develop and use operations for line geometry, including determining the principal axes of a cylindroid.


If the axes of the joints in an RR chain intersect in a point, then it is a spherical RR chain. Two of these chains are assembed to form a spherical 4R closed chain, or spherical linkage.

Our computer aided linkage design software SphinxPC was developed to explore the computer aided synthesis of spherical linkages. The task is specified by four spatial orientations and the software generates an evaluates a map of solutions for the designer to examine.

This software also provides for the synthesis of planar 4R linkages, whick is based on an almost identical mathematical formulation. In this case, four planar positions are specified.

The most recent paper describing SphinxPC is:

D. Alan Ruth, and J. Michael McCarthy: The Design of Spherical 4R Linkages for Four Specified Orientations.

Here is our SphinxPC Manual. Select here to download our SphinxPC software

Spherical Linkage Deployment Devices

SphinxPC was used to design innovative spherical linkages for deploying and stowing accessories for the wheel-chair bound. These devices provide singly-actuated spatial movement that meet the needs of the disabled, expanding the opportunity for independent activity.

Here are photos of a wheelchair with prototype devices installed.

Select this link for more spherical linkages and animations which show student work from my class MAE 183 Computer-Aided-Design of Mechanisms.