Machine Kinematics
solved, then drawn
Concept

Trajectory — where it appears

A path through a mechanism's configurations with a schedule on it, as opposed to the path alone, which carries no time. Two mechanisms following the same path with different schedules have the same geometry and different accelerations, which is why cam design is about the schedule.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

How far a joint-space move bows off the line. Every joint runs from its start value to its end value at a constant rate — the simplest possible move, and the one that can never be refused, because every point along it is a set of joint values and every set of joint values is a pose. The tool does not travel in a straight line while it happens. It bows away by 402 mm over a move of 1223 mm, which is 32.9% of the distance travelled and enough to hit something that the straight line would have missed.

A straight line at constant speed

Run every joint from its start value to its end value and the tool bows 402 mm off the line between them. Insist on the line instead and the arm will follow it — until the path passes near a singularity, where the joint rates a metre a second demands grow as one over the distance, measured at an exponent of −1.010.

serial · Serial Three shortest paths. The tool's route between the same two poses, under three interpolations. The straight one is position interpolated along a line with the rotation carried separately, which is what most controllers do. The screw path is the single turn about a single axis that Chasles's theorem says takes any pose to any other — the only one of the three that mentions no coordinate system — and it is 18.0% longer. The joint-space path is what the arm does when nobody asks for anything in particular, and it is 26.8% longer again. The screw axis itself is drawn: pitch -0.096 m per radian.

The distance between two poses

Ask an arm to take the shortest route between two poses and three reasonable definitions give three different paths, of lengths 1.223, 1.443 and 1.550 metres. The disagreement is not numerical. There is no distance between two rigid poses until somebody chooses a length to measure a radian in, and on this arm the choice changes which of two poses is nearer at 74.5 millimetres.

serial · Serial

Named alongside it

The objects these essays reach for when they reach for this one.

ConditioningInterpolationSerial manipulatorChasles's theoremInverse kinematicsJacobianJoint spaceMetricPoseScrewSingularityTask space

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