# Guidance: Convex hull, null space and convex basis etc

• Inner_Peace
In summary, the conversation discusses the closure properties of a robotic grasp and the confusion surrounding mathematical terms such as convex hull, convex basis, convex combination, free space, and null space. The suggestion is made to google these terms and look for tutorials or books on the subject, with Amazon being a recommended source. A brief explanation of each term is also provided.
Inner_Peace
Hi friends!
I am getting started with a research paper that discusses the closure properties of a robotic grasp. There are of lot of mathematical terms that confuse me like 'convex hull' , convex basis, convex combination of vectors, a free subset, nullspace etc. I might have studied some of them in University mathematics but that seems a long ago. Could you please suggest me places where should I start looking for concept building ? Any good tutorials or books?

Thank you ! :)

Have you tried to google these terms?

I don't see this as having much to do with vectors. A "convex set" in $R^n$ is a set such that for any two points p and q in the set the line segment between p and q is in the set also.

The "convex hull" of a set, A, is the smallest convex set that has A as a subset. One way to construct the convex hull of a set is to add all line segments between any two points in the set.

I don't recognize the terms "convex basis", "convex combination", or "free space".

The "null space" of a linear Transformation, T, from one vector space to another, is the set of all vectors, v, such that T(v)= 0. One can show that the null space is a subspace of the domain vector space.

1 person
Amazon.com has several books on the subject. You will need to look at the reviews to see what fits your needs.
A convex basis is a set of vectors that can be added together with positive weights (all weights 0<=w<=1 that sum to 1). Those weighted sums are the convex combinations.

1 person

Hello! It's great that you are diving into a research paper on robotic grasping. The terms you mentioned, such as convex hull, null space, and convex basis, are all important mathematical concepts in the field of robotics and specifically in grasp planning.

To start building your understanding of these concepts, I recommend looking into linear algebra and optimization theory. These subjects will provide the foundation for understanding convex hulls, null spaces, and convex bases. In particular, understanding convex optimization will be crucial for grasping research as it deals with finding optimal solutions for problems with convex constraints.

Some good resources for learning about these topics include textbooks such as "Convex Optimization" by Stephen Boyd and Lieven Vandenberghe, "Linear Algebra and Its Applications" by David C. Lay, and "Robotics: Modelling, Planning and Control" by Bruno Siciliano and Oussama Khatib.

Additionally, there are many online tutorials and lectures available on these topics, such as those on YouTube or through online learning platforms like Coursera or edX.

I also recommend reaching out to experts in the field of robotics and grasp planning for guidance and further resources. They may also be able to provide valuable insights and explanations on these concepts.

Overall, building a strong understanding of these mathematical concepts will greatly benefit your research and allow you to effectively analyze and develop solutions for robotic grasping problems. Best of luck with your research paper!

## 1. What is a convex hull?

A convex hull is the smallest convex set that contains all points in a given set. It can also be defined as the intersection of all convex sets that contain the given set.

## 2. How is a convex hull computed?

A convex hull can be computed using various algorithms, such as the Gift wrapping algorithm, Graham scan algorithm, or Quickhull algorithm. These algorithms use the given set of points to determine the outermost points that form the convex hull.

## 3. What is the null space of a matrix?

The null space of a matrix is the set of all vectors that are mapped to the zero vector when multiplied by the matrix. In other words, it is the set of vectors that produce a zero result when multiplied by the given matrix.

## 4. How is the null space of a matrix determined?

The null space of a matrix can be determined by finding the inverse of the matrix and multiplying it by the zero vector. Alternatively, it can also be determined by solving the system of linear equations represented by the matrix.

## 5. What is a convex basis?

A convex basis is a set of linearly independent vectors that span the same space as the convex hull of a given set of points. It is used in convex optimization to represent the feasible region of a problem in terms of a smaller number of vectors.

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