Roboterhand im Raum — sechs Achsen, acht Lösungen (3D)
About this app
Ein Sechs-Achs-Roboter mit Kugelhandgelenk in vier Stationen: Ort und Richtung des Greifers sind sechs Zahlen, zwei Drehungen ergeben je nach Reihenfolge verschiedene Würfel (R₂·R₁ ≠ R₁·R₂), acht Geisterroboter halten das Werkzeug exakt gleich und manche rammen den Tisch, und an einer Naht verklemmt das Handgelenk bei q₅ = 0.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
What it is about
A hand pose in space consists of six numbers: three for the position of the tool and three for its orientation. That is why an industrial robot has six axes. The orientation is described by a rotation matrix whose columns are the axes of the tool. Two rotations in a row give a matrix product, and the order matters. For a robot with a spherical wrist, inverse kinematics from the hand pose to the joint angles splits into position and orientation. It yields up to eight solutions.
What the app shows
A six-axis robot with a table and a slanted screw in 3D, in four stations. Six fields show which of the six quantities the gripper already matches. Two dice turn through the same two rotations in different orders, next to the rotation matrices and their products. Up to eight translucent robots hold the tool exactly the same way; the app checks each solution with the forward calculation and colours collisions with the table, workpiece or floor red. Finally the gripper draws a seam just past the wrist singularity, with an angle chart, gimbal rings and the determinant.
Try it yourself
In station 1, move to the screw first with three axes, then with six, and compare how many quantities match. In station 2, choose your own axes and angles and see which face ends up on top. In station 3, drag the target around and count how many poses stay free; clicking a ghost moves the robot there. In station 4, move the seam and watch how fast axis 4 has to turn.
Frequently asked questions
- Why does an industrial robot have six axes?
- Because a tool pose in space has six degrees of freedom: three for the position (x, y, z) and three for the orientation. With fewer axes a robot can reach points but cannot set every direction. On articulated robots, axes 1 to 3 bring the wrist to the position and axes 4 to 6 turn the tool.
- Why does the order of rotations matter?
- Rotations in space are described by rotation matrices, and two rotations in a row correspond to the matrix product. Matrix multiplication is not commutative: in general A·B ≠ B·A. If you turn a die first about x and then about y, a different face ends up on top than in the reverse order. Translations, by contrast, can be swapped.
- How many inverse kinematics solutions does a six-axis robot have?
- For a robot with a spherical wrist, whose last three axes intersect in one point, there are up to eight solutions: shoulder front or back, elbow up or down, wrist normal or flipped. General six-axis robots without this design can have up to 16 solutions. Joint limits and obstacles decide which solution is usable.
- What can I try in the “Robot hand in space” app?
- You set six joints or let the robot move to the screw, compare two rotation orders on dice with the matching matrices, show up to eight solutions as ghost robots and let the gripper draw a seam close to the wrist singularity. Joint limits are left out for simplicity, collisions are checked roughly.
Subject: Engineering mechanics
More from Engineering mechanics
- Bending stress & moment of inertia
- Düsenjet bis v_max (3D)
- Energy & oscillation
- Truss analysis (method of joints)
- Equilibrium & support reactions
- Kinematics
- Kinetics & Newton's second law
- Forces & vector addition
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