Precision-point synthesis problems for design of four-bar linkages have typically been formulated using two approaches. The exclusive use of path-points is known as “path synthesis,” whereas the use of poses, i.e., path-points with orientation, is called “rigid-body guidance” or the “Burmester problem.” We consider the family of “Alt–Burmester” synthesis problems, in which some combination of path-points and poses is specified, with the extreme cases corresponding to the classical problems. The Alt–Burmester problems that have, in general, a finite number of solutions include Burmester's original five-pose problem and also Alt's problem for nine path-points. The elimination of one path-point increases the dimension of the solution set by one, while the elimination of a pose increases it by two. Using techniques from numerical algebraic geometry, we tabulate the dimension and degree of all problems in this Alt–Burmester family, and provide more details concerning all the zero- and one-dimensional cases.

This letter investigates the mathematical relationships between the positions and orientations at the segment tips of a piecewise constant curvature (PCC) continuum robot with up to three segments. For one-segment, a reachability criterion is proposed, simplifying the calculation of the neighboring orientation. For two-segments, a reachability criterion is proposed and the redundancy of its inverse kinematics solution is found, establishing a circle of tip locations. For three-segments, the redundancy of the inverse kinematics includes tips that lie on a sphere providing a closed-form solution to the inverse kinematics problem. These relationships are derived from the unique characteristics of the bisecting plane of a single segment. The degenerate cases for the solutions are also addressed. These outcomes stem from a specific PCC parametrization, with implications extending to the general PCC model. Note that this study is grounded solely in simulation.

This paper considers single degree-of-freedom (DOF), closed-loop linkages with a designated input angle and one design parameter. For a fixed value of the design parameter, a linkage has input singularities, that is, turning points with respect to the input angle, which break the motion curve into branches. Motion of the linkage along each branch can be driven monotonically from the input. As the design parameter changes, the number of branches and their connections, in short the topology of the motion curve, may change at certain critical points. Allowing the design parameter to vary, the singularities form a curve called the critical curve, whose projection is the singularity trace. Many critical points are the singularities of the critical curve with respect to the design parameter. The critical points have succinct geometric interpretations as transition linkages. This paper presents a general method to compute the singularity trace and its critical points. As an example, the method is used on a Stephenson III linkage, and a range of the design parameter is found where the input angle is able to rotate more than one revolution between singularities. This characteristic is associated with critical points that appear as cusps on the singularity trace.

This work seeks to systematically model and solve the equations associated with the kinematics of spherical mechanisms. The group of special unitary matrices, SU(2), is utilized throughout. Elements of SU(2) are employed here to analyze the three-roll wrist and the spherical Watt I linkage. Additionally, the five orientation synthesis of a spherical four-bar mechanism is solved, and solutions are found for the eight orientation synthesis of the Watt I linkage. Using SU(2) readily allows for the use of a homotopy-continuation-based solver, in this case Bertini. The use of Bertini is motivated by its capacity to calculate every solution to a design problem.

This paper presents a planar four-bar approximate motion synthesis technique that uses only pole locations. Synthesis for rigid-body guidance determines the linkage dimensions that guide a body in a desired manner. The desired motion is specified with task positions including a location and orientation angle. Approximation motion synthesis is necessary when an exact match to the task positions cannot be obtained. A linkage that achieves the task positions as closely as possible becomes desired. Structural error refers to the deviations between the task positions and the linkage’s generated positions. A challenge in approximate motion synthesis is that structural error involves metrics that include location and orientation. A best-fit solution is not evident because the structural error is based on an objective function that combines the location and orientation. Such solutions lack bi-invariance because a change in reference for the motion changes the values of the metric. This work uses only displacement poles, described solely by their coordinates, as they sufficiently characterize the relative task positions. The optimization seeks to minimize the distance between the poles of the task positions and the poles of the generated positions. The use of poles results in a bi-invariant statement of the problem.

This paper considers single-degree-of-freedom, closed-loop linkages with a designated input angle and one design parameter. For a fixed value of the design parameter, a linkage has turning points (dead-input singularities), which break the motion curve into branches such that the motion along each branch can be driven monotonically from the input. As the design parameter changes, the number of branches and their connections, in short the topology of the motion curve, may change at certain critical points. As the design parameter changes, the turning points sweep out a curve we call the “turning curve,” and the critical points are the singularities in this curve with respect to the design parameter. The critical points have succinct geometric interpretations as transition linkages. We present a general method to compute the turning curve and its critical points. As an example, the method is used on a Stephenson II linkage. Additionally, the Stephenson III linkage is revisited where the input angle is able to rotate more than one revolution between singularities. This characteristic is associated with cusps on the turning point curve.