Less time we have seen what is inverse kinematics and how to solve it.
And we have also seen two methods of solving inverse kinematics, namely analytical way and numerical
way in terms of analytical way.
We have said that while it is faster and more initiative, it becomes extremely complex in terms of
redundant robot manipulators because they have higher degree of freedoms.
While we have say that we can utilize kinematic decoupling to make the process easier.
This method belongs to specific robots, namely six degree of freedom and spherical wrist.
So what we will do if we have redundant robots, which may not have closed loop solution or if there
is but very complex, what will we do if our robot doesn't have spherical wrist?
How we can solve inverse kinematics problem in such kind of situations?
We will use a numerical approach.
Let's explain this method more clearly.
Assume that we have a function of GQ, which is equal to X the minus f q here X is desirable and the
vector pose that we want to obtain.
While F Q is nothing but forward, kinematics function.
So we give Q joint configurations to function f it returns US end effector pose at the joint configuration.
Our goal is to make that function, namely GQ zero.
Because then we will have our design joint configuration of Q D, which will make our robot reach the
desired and the vector pose of XD.
So we will eventually lasso our inverse kinematics problem.
No, the problem is how we can make G.Q becomes zero.
We have two methods for doing that.
The first is Jacobean inverse method.
The second is Jacobean transpose method.
Both of these solutions are iterative because numerical inverse climatiques iterative solution.
Let's start with Jacob in inverse method.
First, we expand forward kinematics function of F with Taylor series about a configuration of Kudi,
which is desired configuration.
Higher order terms, which is denoted by h o t because, as you know, we can go without stopping the
Taylor expansion.
So if we hold, we don't need higher order terms.
So if we neglect higher order terms, then we will have this equation, which contains inverse of Jacobean.
Be careful here.
We assume that all the Jacobean is square and invert.
So in order to find true, we start with initial guess of Q zero and then find Q1, Q2 and so on up
to when we converge q d through an iterative way in order to that code.
In order to do that calculation, this equation is needed, which is iterative.
Way of converging to the here over is nothing but step size, and it's greater than zero.
It can be chosen as constant or K dependent, so it changes at each iteration.
When we neglect the higher order terms, we can expect local convergence, namely Q, has to be near
solution or kudi.
Additionally, as inverse kinematics have multiple solutions, we can have different kudi depending
on initial guess of Q zero.
What if Jacobean is singular and non-convertible?
Then we will use less.
So the inverse Jacobean instead?
Now let's switch to the Giacobbe and transpose method.
So we will first define quadratic optimization problem, which is given like this.
So if you have read about optimization, one of the widely used ways of solving them is through gradient
descent, which requires us to know the direction of minimization and steps that we take at each iteration
through that direction.
We can find direction of minimization through finding gradient of the cost function, and it is given
with that equation, which contains transpose of Jacobean.
Then we can find optimal solution, which is kudi using gradient descent algorithm, which is also iterative.
As you can see, this method includes transpose of Jacobean, which have advantages of being much more
easier to calculate.
Also, we don't have issues of Jacobean singularity.
However, while we have these advantages, we have this advantage of slower convergence.
See you on the next lesson.