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I am the moment trying to read and understand this paper Task Constrained Motion Planning in Robot Joint Space but seem to have a hard time understanding the math.

The paper describes how to perform task constrained motion planning in cases where a frame is constrained to a specific task.
the problem the paper tackles is when sampling in joint space, randomized planners typically produce samples that lie outside the constraint manifold. the method they proposed methods use a specified motion constrain vector to formulate a distance metric in task space and project samples within a tolerance distance of the constrain.

Given the this I am seem to a bit confused on some simple terms they define in this paper.

Examples. How is a task space coordinate defined ? what information does it have?

they compute the $$\Delta x = T_e^t(q_s)$$ which is transformation matrix of the end effector with respect to the task frame.

What I don't get is why the end effector? and why the end effector with respect to the task frame?

Secondly. Later in the paper they write down an expression that relates the task space to the joint space motion. They do it using the Jacobian, but seem to miss explaining (in my opinion) what $E(q_s)$ actually do.

$$J(q_s) = E(q_s)J^t(q_s)$$

What is said about it in the paper is that

Given the configuration $q_s$, instantaneous velocities have a linear relationship $E(q_s)$

why the need of instaneous? what is the definition of an instantaneous component? how does it differ from the information given by the jacobian?

Basically i don't understand how and why the mapping is as it is?..

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"Why the end effector?" Because a robot working on a task uses its end effector to interact with the objects that make up the task. For example, it is common to align the jaws ("fingers") of a parallel-jaw gripper with the sides of an object to be picked up. The simplest way to describe that object is by using a fixed coordinate system in which the task objectives are easy to visualize. Maybe, for example, $\hat x$ is pointed in the direction of travel of a conveyor belt. Since a robot can be placed with its base frame in an arbitrary relationship to the task space, it is usually easier to use the task coordinate system and make the transposes described.

Regarding $E$, look at the paper's appendix. The author uses this matrix to relate the Euler angles of the robot to the task system.

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