Automation sequences, robotics and coordinated motion

Make state transitions depend on evidence

A finite-state model names task conditions such as IDLE, MOVE, VERIFY and COMPLETE. Each transition has a guard: a condition that must be satisfied before the change. Actions and timeouts belong to specified states. A timer expiring is evidence of elapsed time, not evidence that a clamp closed or an axis arrived. A separate observation must support that physical claim.

Define priority when several conditions occur together. A fault transition may need to take precedence over the normal completion transition. Commands should be distinguishable from acknowledgements and physical confirmation. Repeating a command after a lost acknowledgement can repeat an action unless its meaning and permitted retries are explicit. These are requirements to analyse, not generic permission to restart machinery.

Relate a path to the mechanism

Forward kinematics relates joint coordinates to a tool pose; inverse kinematics asks which joint coordinates produce a desired pose. There can be multiple solutions or none within limits. Near a singular configuration, a small tool change can demand a large joint change. A path in tool coordinates therefore needs checking against joint, collision and actuator constraints.

An axis reaching its endpoint before another can make an unintended tool path. A shared progress variable can coordinate a simple Cartesian move: x=x₀+sΔx and y=y₀+sΔy with 0≤s≤1. The path geometry comes from the coordinate changes; the function s(t) supplies the timing. Velocity and acceleration limits concern the derivatives of that timing law and the mechanism mapping.

Locate a point from two rotating links

Consider a fictional planar arm with two rigid links, each 0.20 m long, and ideal rotating joints. Put the base at (0,0), with x right and y up. Let q₁ be the first link’s counterclockwise angle from x, and q₂ the second link’s angle relative to the first, so its absolute direction is q₁+q₂. For a link of length L at angle θ, its horizontal and vertical projections are Lcosθ and Lsinθ: cosine and sine give the corresponding signed fractions of its length. Adding the two link projections gives x=0.20cosq₁+0.20cos(q₁+q₂) and y=0.20sinq₁+0.20sin(q₁+q₂). All positions are in metres; the examples below use degrees.

Choose q₁=60° and q₂=−120°. The elbow is (0.100,0.1732) m, and the second link points at −60°, so its vertical projection cancels the first: the tool point is (0.200,0) m. Choosing q₁=−60°, q₂=120° puts the elbow below the axis and reaches the same point. This is two inverse solutions for point position, not the same full pose: the final link directions differ. Joint limits, collisions or an orientation requirement could remove one or both. A target (0.45,0) m has no solution for this fixed-base arm, because even its total link length is only 0.40 m.

Near the straight configuration, a small tool step can require substantial joint change. For the symmetric family q₁=α, q₂=−2α, the vertical components cancel and x=0.40cosα. Straightening from α=10° to 5° moves x from 0.393923 to 0.398478 m, about 4.555 mm. Straightening the next equal 5° reaches 0.400000 m, only another 1.522 mm. The same change in joint angles produces less outward motion as the arm approaches full extension. Maintaining a chosen outward tool speed can therefore demand increasing joint rates, before actual limits are considered.

At the exactly straight configuration, rotation about either joint gives the tip an instantaneous velocity perpendicular to the straight links. Combining the joint velocities can give vertical tip velocity, but no horizontal instantaneous velocity. This loss of a local motion direction is a kinematic singularity. It does not mean the arm mechanically locks: it can rotate, and beginning to bend can subsequently bring the point inward. Reaching a position, following a feasible path and confirming the task’s physical result remain separate questions.

Separate planned motion, servo response and task confirmation

A planner supplies a trajectory; a feedback loop attempts to follow it; task logic decides whether observed conditions permit the next action. None guarantees the other automatically. A good tracking trace does not establish that the tool held the right object. A valid sequence diagram does not establish that the mechanism can produce the requested acceleration.

Record the active state, target, actual position, applied actuator signal and relevant confirmations on a common timeline. Frame definitions matter: a point expressed relative to the robot base differs from one relative to a moved fixture. Position repeatability under fixed conditions is distinct from absolute accuracy in an external frame.

Two axes share one progress variable

Move a fictional Cartesian point from (0, 0)m to (.3, .4)m along a straight path. Its length is √(.3²+.4²)=.5 m. Using s=t/2 for the interior of a 2 s move gives dx/dt=.15 m/s and dy/dt=.20 m/s, so tool speed is .25 m/s. The figure uses equal spatial drawing scales.

This linear interior law jumps velocity at the endpoints. A real trajectory needs a suitable acceleration/deceleration profile and constraint checks, which can increase duration. A task transition to VERIFY should depend on a defined measured settling condition, not simply on the nominal 2 s having elapsed.

The Cartesian shared-progress calculation commands point coordinates directly. A rotating-link arm would first need joint configurations consistent with those points in its declared base frame, and a continuous feasible choice along the path. Neither the 0.50 m path length nor a successful endpoint inverse solution supplies joint-rate, collision or task-confirmation evidence. The two-link illustration deliberately ignores loads, flexibility and joint limits so that this geometric distinction is visible.

Try the reasoning

1. If y finishes first while x is still moving, is the path necessarily straight?

   Show the answer and reasoning

   No. Endpoint agreement alone does not fix the intermediate relationship y=(.4/.3)x. The axes need coordinated progress.

2. Does an acknowledgement prove the commanded physical action occurred?

   Show the answer and reasoning

   No. It may confirm receipt or acceptance only. Its contract and independent physical confirmation must establish what happened.

3. A joint solution reaches the required pose. Is the planned move complete?

   Show the answer and reasoning

   No. Intermediate collisions, joint rates, acceleration, loads and task conditions still need checking; an endpoint solution alone is insufficient.

4. Change both planar link lengths to 0.15 m. Find the tip for q₁=90°, q₂=−90°, then compare q₁=0°, q₂=90°. Can either arrangement reach (0.32,0) m?

   Show the answer and reasoning

   The first arrangement has an elbow at (0,0.15) m and a horizontal second link, giving tip (0.15,0.15) m. The second has elbow (0.15,0) m and a vertical second link, reaching the same point with a different final-link direction. These are position alternatives, subject to any real limits. Neither can reach (0.32,0) m: the maximum distance from the fixed base is 0.15+0.15=0.30 m. Changing the joint angles cannot extend the rigid links or establish a feasible path through obstacles.

What to carry forward

Combine guarded task transitions with feasible trajectories and observed physical completion.