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.
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Watch the joints, then watch the dot. The shoulder, elbow and wrist can turn together while the gripper center stays on the target. The pieces keep their lengths. Their arrangement changes. Finding that arrangement from a tool goal is called inverse kinematics.
Both poses put the gripper center on the target. Will halfway joint angles keep it there?
At the reference midpoint, joint blending misses by about 10.4 mm. Fixed-center coordination uses a different joint arrangement. The links keep their lengths.
Start with the gold gripper-center dot inside the orange target ring. Tilt the gripper a little. Several joints change together to keep the center still. The point is a chosen reference in the tool; it is not a promise that a finger is gripping or touching anything.
Imagine placing a key beside a keyhole. Getting the key to the right place is only part of the task: its direction matters too. Here, you choose a point and a tilt. Changing the tilt changes where the wrist must be, even when the gripper center stays put.
Forward kinematics starts with joint angles and calculates where the tool goes. Inverse kinematics starts with the requested tool position and direction and finds candidate joint angles. There may be no solution, one solution or several. The pinned ROBOTIS geometry and the Modern Robotics inverse-kinematics explanation are the sources for this calculation.
In Compare two paths, both rules start and finish on the same target. At halfway, simply blending the joint angles moves the gripper center about 10.4 mm away. Coordinating the joints to keep the center fixed gives a different middle arrangement. Neither rule is universally better: the right path depends on the task.
A request can need more link length than the arm has. Or a mathematical arrangement can reach the point but require a joint to turn beyond its published interval. Those are different failures. Keep the last allowed pose visible while inspecting the requested target and the reason it failed.
In the real research photographs, an OpenManipulatorX moves a tactile sensing tool along a curved textured surface. That attachment is different from our standard gripper. The photographs connect positioning to a real task; they do not measure our virtual arm’s accuracy or prove a collision-free route.
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.
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.
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.
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.
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.
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.
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.
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.
Marzani and colleagues used an OpenManipulatorX with a BioIn-Tacto sensing module to gather texture data on curved surfaces. Their Figure 1 shows the beginning, midpoint and end of an exploration. That real application needs coordinated positioning and a suitable tool; the published image is not a video or a calibration measurement.
A tool that must stay at a point while changing direction needs a different path constraint from a tool that only needs to arrive at the same endpoints. The two computed paths demonstrate this distinction without claiming that joint interpolation is inherently wrong.
The point of interest can move when an end tool changes. Tool-center configuration is a standard robotics concept, described in manufacturer documentation such as Universal Robots’ TCP manual. That different manufacturer’s manual supplies the concept, not this arm’s dimensions.
Mark two centers 120 mm apart on one folded strip and two centers 80 mm apart on the other. Measure between the dots, not the strip ends. You can draw the whole construction instead. If using the learning-pack template, print at actual size and check its 100 mm line with a ruler.
On the larger sheet mark base O. Draw X to the right and Z upward. Put target P 120 mm right and 80 mm up from O. Predict whether two fixed-length strips can join O to P, and whether there is more than one arrangement.
Put the first strip’s base mark on O. Overlap its other mark with the first mark of the second strip: this is the elbow. Hold the marks together lightly. Put the last mark on P. One arrangement has its elbow 120 mm right of O, at the same height. Trace it.
Keep O and P fixed. Find another elbow location without changing the center distances. Trace it in a different color or dashed line. The ideal second elbow is about 46.15 mm right and 110.77 mm up from O. Hand measurements need not match those decimals perfectly.
Try points 210 mm and 20 mm from O. The freely turning ideal links reach at most 120 + 80 = 200 mm and no closer than 120 − 80 = 40 mm. Both test points are unreachable. Keep the base and lengths fixed; paper overlap is allowed in this simple construction.
Draw a short arrow at P to add a requested facing direction. Explain why “touch P” and “touch P facing this way” give different information. Your arrow is a request, not an extra working joint; these paper hinges do not copy the real robot’s joint stops.
Write your prediction, sketch both arrangements and record any measured gap. Explain one difference between this model and a physical robot: restricted joints, a real tool, sensing, control or possible collisions. A small drawing error is worth recording.
How can two different elbow positions place the same endpoint on P—and why might a real robot reject one of them?
Original paper geometry activity with unrestricted ideal centerlines and permitted strip overlap. It is not a robot performance or collision test. No sharp fastener, cutting tool, purchase or powered mechanism is required; an adult may help prepare folded strips.
No; several joints can change together while preserving that point. Correct. Coordinated joint changes can preserve the tool-center position while changing its direction, within the model's limits.
Inverse kinematics. Correct. This works backward from a requested tool pose to candidate joint values.
Mark that candidate as outside the joint limit. Correct. It can be inspected as a mathematical candidate, but should not be executed as an allowed robot pose.
No; in the checked example the midpoint misses by about 10.4 mm. Correct. Coordinating the joints using a fixed-center constraint produces a different intermediate configuration.
The two wrist links would need more combined length. Correct. Their distance from shoulder to wrist is already at its geometric maximum in this direction.
The new tip is displaced; the tool-center transform must represent the new point. Correct. In the checked tilted case the error components are about +41.3 mm forward and −28.2 mm vertically, totaling 50 mm.
No; this educational model is not a hardware controller or safety assessment. Correct. It teaches geometry. Real operation needs the manufacturer's setup and a suitable, validated control system.
They show a research robot using a tactile tool at several stages along a curved surface. Correct. The photograph and original paper support that specific research context.
Commit 9187eca0920458be04d2399906388f55242f81f1. Numerical geometry uses this revision consistently; current web examples may differ.
ROBOTIS · pinned source geometrySource XYZ-up, meters and radians. STL millimeters are scaled once. The separate 19 mm link2 visual offset is not an extra kinematic link.
ROBOTIS · complete URDFManufacturer specifications are not this lesson’s measured performance. Current illustrative URDF snippets are not mixed into our pinned chain.
ROBOTIS · specificationLynch and Park, author-owned teaching resource. The paper exercise is an original construction; no source figures are copied.
Modern Robotics · inverse kinematicsOur planar task and finite inward example must be distinguished from general six-dimensional motion.
Modern Robotics · singularitiesOur endpoint solver and path comparison do not execute a collision planner.
Modern Robotics · motion planningThese are additional layers beyond the geometric model, not simulated performance claims.
Modern Robotics · control overviewDifferent manufacturer: conceptual reference only, not our arm’s dimensions or executable hardware instructions.
Universal Robots · tool-center configurationData in Brief 59 (2025), 111312, Figure 1; Marzani, Khatibi, Masinjila, Prado da Fonseca and Alves de Oliveira. Real sensing attachment differs from our standard gripper.
Marzani et al. · original tactile experimentROBOTIS package authors Hye-Jong KIM, Sungho Woo, Wonho Yun and Woojin Wie. License and credits accompany the source download. Brytalearn presentation is not manufacturer endorsement.
ROBOTIS · Apache 2.0 geometry license© 2025 The Author(s). Credit Marzani et al., article and Figure 1, retain the license link and identify any responsive panel crop. Full original remains available.
CC BY 4.0 · research photograph licenseIndependent subject review is pending.
Read the sources and model assumptions