engineering

Hand–eye calibration problem

Hand–eye calibration problem is a engineering topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Hand–eye calibration problem rather than just read about it. In short: In robotics and mathematics, the hand–eye calibration problem (also called the robot–sensor or robot–world calibration problem) is the problem of determining the transformation between a robot end-effector and a sensor or sensors (camera or laser scanner) or between a robot base and the world coordinate system. It is conceptually analogous to biological hand–eye coordination (hence the name).

Key takeaways

Hand–eye calibration problem belongs to engineering; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hand–eye calibration problem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hand–eye calibration problem from memory before moving on to harder problems.

Reference excerpt

In robotics and mathematics, the hand–eye calibration problem (also called the robot–sensor or robot–world calibration problem) is the problem of determining the transformation between a robot end-effector and a sensor or sensors (camera or laser scanner) or between a robot base and the world coordinate system. It is conceptually analogous to biological hand–eye coordination (hence the name). It takes the form of AX=ZB, where A and B are two systems, usually a robot base and a camera, and X and Z are unknown transformation matrices. A highly studied special case of the problem occurs where X=Z, taking the form of the problem AX=XB. Solutions to the problem take the forms of several types of methods, including separable closed-form solutions, simultaneous closed-form solutions, and iterative solutions. The covariance of X in the equation can be calculated for any randomly perturbed matrices A and B. The problem is an important part of robot calibration, with efficiency and accuracy of the solutions determining the speed accuracy of the calibrations of robots.

Methods Many different methods and solutions developed to solve the problem, broadly defined as either separable, simultaneous solutions. Each type of solution has specific advantages and disadvantages as well as formulations and applications to the problem. A common theme throughout all of the methods is the common use of quaternions to represent rotations.

Separable solutions Given the equation AX=ZB, it is possible to decompose the equation into a purely rotational and translational part; methods utilizing this are referred to as separable methods. Where RA represents a 3×3 rotation matrix and tA a 3×1 translation vector, the equation can be broken into two parts:

RARX=RZRB RAtX+tA=RZtB+tZ The second equation becomes linear if RZ is known. As such, the most frequent approach is to solve for Rx and Rz using the first equation, then using Rz to solve for the variables in the second equation. Rotation is represented using quaternions, allowing for a linear solution to be found. While separable methods are useful, any error in the estimation for the rotation matrices is compounded when being applied to the translation vector. Other solutions avoid this problem.

Simultaneous solutions Simultaneous solutions are based on solving for both X and Z at the same time (rather than basing the solution of one part off of the other as in separable solutions), propagation of error is significantly reduced. By formulating the matrices as dual quaternions, it is possible to get a linear equation by which X is solvable in a linear format. An alternative way applies the least-squares method to the Kronecker product of the matrices A⊗B. As confirmed by experimental results, simultaneous solutions have less error than separable quaternion solutions.

Iterative solutions Iterative solutions are another method used to solve the problem of error propagation. One example of an iterative solution is a program based on minimizing ||AX−XB||. As the program iterates, it will converge on a solution to X independent to the initial robot orientation of RB. Solutions can also be two-step iterative processes, and like simultaneous solutions can also decompose the equations into dual quaternions. However, while iterative solutions to the problem are generally simultaneous and accurate, they can be computationally taxing to carry out and may not always converge on the optimal solution.

The AX=XB case The matrix equation AX=XB, where X is unknown, has an infinite number of solutions that can be easily studied by a geometrical approach. To find X it is necessary to consider a simultaneous set of 2 equations A1X=XB1 and A2X=XB2; the matrices A1, A2, B1, B2 have to be dermined by experiments to be performed in an optimized way.

The 2D laser profile scanner case

[ p b 1 ] = [ R b T b 0 1 ] ⋅ [ R s T s 0 1 ] ⋅ [ p s 1 ] {\displaystyle {\begin{bmatrix}p_{b}\\1\end{bmatrix}}={\begin{bmatrix}R_{b}&T_{b}\\0&1\end{bmatrix}}\centerdot {\begin{bmatrix}R_{s}&T_{s}\\0&1\end{bmatrix}}\centerdot {\begin{bmatrix}p_{s}\\1\end{bmatrix}}}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Hand–eye calibration problem

Start with the simplest possible case. Write down what Hand–eye calibration problem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In engineering, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hand–eye calibration problem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hand–eye calibration problem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hand–eye calibration problem

In research: Hand–eye calibration problem appears in engineering research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Hand–eye calibration problem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Hand–eye calibration problem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Robot control, so understanding it makes those chapters shorter.
In everyday life: Look for Hand–eye calibration problem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Hand–eye calibration problem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hand–eye calibration problem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Hand–eye calibration problem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hand–eye calibration problem in simple terms?

Why does Hand–eye calibration problem matter?

Because it connects several engineering ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hand–eye calibration problem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hand–eye calibration problem.