Robot arms, tool position and inverse kinematics: sources & model

Tilt a real robot model while keeping its gripper center on a target. Compare two routes between the same poses, discover what stops a reach, and build a paper arm with two ways to touch one point.

What supports the explanation?

What this model assumes

1. Rigid geometric teaching model of the pinned ROBOTIS description. No physical robot is connected or controlled.
2. Base yaw is fixed at zero for the target task. A planar point and tilt do not represent arbitrary 3D tool orientation.
3. Within published joint intervals is not a safety label. Self-collision, obstacles, contact and routes between arbitrary requested poses are not checked.
4. Authored playback and calculated residuals are not measurements of motor speed, repeatability, absolute accuracy, force or payload.
5. The gripper-center point is the published end-effector frame. Finger-frame spacing is not presented as a measured fingertip opening.
6. The real study uses a different sensing attachment. Three photographs do not establish its exact joint angles, trajectory, contact forces or our model’s accuracy.
7. Paper strips demonstrate unrestricted two-link centerlines with permitted overlap. Their 40–200 mm reach interval is not the workspace of the source robot.
8. The exact model being investigated: The manufacturer describes four positioning joints plus a gripper. This experiment fixes the base yaw q1 at zero and works in the vertical X/Z plane: X is forward, Z is up, Y is lateral. Camera orbit changes your view, not the coordinates. The fingers stay at a fixed display setting. All numerical dimensions and joint intervals come from the pinned URDF, not from a mixture of current product-page examples or other robot revisions.
9. A pose is more than a point: A configuration is the list of joint values. A tool pose includes position and orientation. Here the positive-up tool tilt is φ = −(q2 + q3 + q4). This arm does not provide arbitrary independent roll, pitch and yaw. The model starts from the source frame 12 mm forward of world zero, with its shoulder 59.5 mm above that frame. The shoulder-to-elbow offset is 24 mm forward and 128 mm up: its length is about 130.23 mm, not 128 mm.
10. Solve the wrist first: For a requested tool center (x,z) and tilt φ, subtract the 126 mm tool offset: wx = x − 12 − 126 cos φ; wz = z − 59.5 − 126 sin φ, using millimeters and radians. The remaining two link lengths are L1 = √(24² + 128²) mm and L2 = 124 mm. The cosine rule gives cos β = (wx² + wz² − L1² − L2²)/(2L1L2). Values genuinely outside [−1,1] mean no geometric solution for that direction; they must not be clamped into reach.
11. Candidate does not mean permitted: The two signs of β can give two geometric arrangements. They must be converted into the actual source joint angles and checked against q2 ∈ [−1.5,1.5], q3 ∈ [−1.5,1.4], q4 ∈ [−1.7,1.97] radians. At the reference point, a +1 radian tilt has geometric candidates but none within every interval. Near extension, this arm does have a narrow example with two interval-valid arrangements; their elbow centers differ by about 12.7 mm. A broad textbook elbow flip would misrepresent this particular mechanism.
12. A precise midpoint result, not a precision claim: Our authored path goes from tilt 0 to −0.6 rad at target (240,0,160) mm. Halfway joint interpolation puts the tool at approximately (250.3967,0,159.2647) mm: a 10.4227 mm geometric miss. This is the midpoint miss, not a proven global maximum and not measured hardware error. Solving the fixed-center constraint at each progress value makes the equations agree within numerical rounding; tiny residuals do not establish extraordinary physical accuracy.
13. What a singularity means here: For geometric link directions α and β, the planar wrist Jacobian has determinant L1L2 sin β. At the investigated straight configuration β = 0, its rank drops from two to one: the available instantaneous wrist velocities lose a direction. At fixed tool tilt, an additional 1 mm radially outward is beyond the two links’ total length. A finite 1 mm inward request can be reached by bending. Losing an instantaneous direction does not mean the robot can never move again. These geometric angle rates are not the source joint encoder rates without a coordinate conversion.
14. The useful point changes with the tool: A tool-center transform tells the calculation which point matters. In an authored transfer example, add a 50 mm probe along the tool axis while keeping the old point definition. At tilt −0.6 rad its tip shifts about 41.3 mm forward and 28.2 mm downward, totaling 50 mm. This hypothetical probe is not part of the source mesh or the photographed research sensor. A different tool can also change physical loads, which this lesson does not calculate.
15. Geometry, a route, then control: A geometric path lists arrangements; a timed trajectory additionally says when to visit them. A real controller uses sensors and actuator commands to carry out its task. Contact tasks may require force or impedance objectives as well as positioning. This lesson’s comparison timing is authored for inspection, not a prediction of motor speed. Modern Robotics’ planning and control chapters explain these additional layers. No collision, stiffness, backlash, loading or sensor-error model is executed here.

What has been checked

Analytical reference cases, conservation or transition invariants, finite drawing commands, bounded setup parsing, discovery and route integrity are checked automatically. These checks do not establish anatomical fidelity, learner outcomes or browser/device compatibility. Independent subject review, learner trials, comprehensive accessibility review and browser video encoding checks remain pending.

Each source supports the associated claim. Sources do not certify this implementation or its visuals.

About the cover illustration

Actual rendered ROBOTIS OpenManipulator-X meshes from pinned commit 9187eca0920458be04d2399906388f55242f81f1, Apache 2.0. Original source topology and rigid transforms; Brytalearn materials, camera, stage and target markers. A geometric teaching model, not measured hardware motion. Separate Marzani et al. research photographs are credited inside the lesson.