A waterjet system in accordance with at least some embodiments includes a carriage, a motion assembly configured to move the carriage horizontally relative to a workpiece, and a cutting head carried by the carriage. The waterjet system can also include a kinematic chain through which the cutting head is operably connected to the carriage. The kinematic chain can include first, second, and third joints rotatably adjustable about different first, second, and third axes, respectively. The carriage and the first and second joints can be configured to move the cutting head along a path relative to the workpiece while the cutting head directs a jet toward the workpiece to form a product. The third joint can be configured to shift a kinematic singularity away from the path to reduce or eliminate delay and corresponding reduced cutting accuracy associated with approaching the kinematic singularity.