# Lecture 10

# Trajectory planning

* for this moment we are planning each DOF individually and combining every joint to get final trajectory.
* we want smoother because it uses less energy and easy to control.


* What is joint space ?
* If we want to perticular trajectory it is quite hard to do it in joint space because there is no way to know that it had followed that trajectory. whereas it is easy in cartesian space.
* we have discontinuity in cartesian space i.e, we might plan striaght line in C-plane but in joint space it might be impossible to achieve.
* in reality we use hybrid approch to reduce computaion but to also avoid obstacles..


* Didn't understood how velocity will reach infinty in first slide?
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* here u is generic variable i.e, when doing actual calculations we have to replace it with x,y,z or alpha,beta,gamma.... etc.
### Single cubic polynomial:
* i am not the graph what it is representing like there are three things given position, degree, time. what is degree ?: Basically POsition,velocity are name of that perticular graph.


* we are basically solving here some equations.
* This graphs are just examples, therefore equations and graphs shown are distinct of each other.


* so with cubic polynomial we can't control accelaration as we have already have found out in terms of U0, u0_dot,uf,uf_dot.
* acceraltion is not fixed i.e, it is changing with time but we can't control it with goal configuration.
* heuristics: some constrains
* what is the difference btw u1 and u2 ? so he is refering to u1 as velocity of 1st segment and u2 as velocity of 2nd segment, but what is segment?[28:53](https://youtu.be/7wIqGavQjTQ?t=1733): Basically a segment is btw two points thorugh which robot should move.
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* now here we are finding blending time from below formula in which acceleration, intial and final position are known.
### Linear interpolation:


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* proff of below equations is not given,so we need to find it.


* Going through via points is not that important because we are actually using them to avoid obstacles but if need to do it we can do it in following way:
* didn't understood 3rd point how it will stop after repeating point?


### Higher order polynomials:

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