Pole arrangements that introduce prismatic joints into the design space of four- and five-position rigid-body synthesis

Although a general five-position, rigid-body guidance problem admits a discrete number of revolute–revolute (RR) dyads, this paper identifies arrangements of five task positions that result in a center-point curve. For these special arrangements, a one-dimensional set of revolute-prismatic (RP) dyads exist to achieve the task positions. Other five-position arrangements are identified where a one-dimensional set of prismatic-revolute (PR) dyads exist to achieve the task positions. For a general case of five task positions, neither PR nor RP dyads are possible. In a general case of four-position rigid-body guidance problems, a unique PR dyad and RP dyad exist. Four-position arrangements are identified where the associated center-point curve includes the line at infinity and admits a PR dyad with a line of slide in any direction. Likewise, arrangements of the four positions are identified where the circle-point curve includes the line at infinity, permitting a one-dimensional set of RP dyads. These special four and five positions lead to dyads that can be coupled to solve a rigid-body guidance synthesis problem with a PRRP or RPPR device which is generally not possible. These solutions are particularly useful in design situations where actuation through a prismatic joint is desired.