I want to simulate the behaviour of a 2-DOF robotic manipulator, which is described by the following

Sonia Gay 2022-06-23 Answered
I want to simulate the behaviour of a 2-DOF robotic manipulator, which is described by the following model:
M ( q ) q ¨ = C ( q , q ˙ ) q ˙ G ( q ) + τ           ( 1 )
Considering the fact that the 2x2 mass matrix M is positive definite, I could use the inverse matrix and break down the problem into the 4 first order ordinary differential equations and simulate it:
x 1 = q 1 x 1 ˙ = x 2
x 2 = q 1 ˙ x 2 ˙ = M 1 ( 1 , : ) C [ x 2 x 4 ] T M 1 ( 1 , : ) G + M 1 ( 1 , : ) τ x 3 = q 2 x 3 ˙ = x 4
x 4 = q 2 ˙ x 4 ˙ = M 1 ( 2 , : ) C [ x 2 x 4 ] T M 1 ( 2 , : ) G + M 1 ( 2 , : ) τ
Suppose I would like to use a solver that takes as an argument the mass matrix (a MATLAB ODE solver in particular) and don't use its inverse because this will also simplify the computation of the jacobian (I intend to simulate a 7-DOF robotic manipulator after that so providing the mass matrix would be great). How can I write the initial equation (1) as a series of first order ordinary differential equations and be able to simulate its response by using some software solvers ?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

britspears523jp
Answered 2022-06-24 Author has 28 answers
Let us choose η = [ q q ˙ ] . Then
[ I 0 0 M ( q ) ] η ˙ = [ 0 I 0 C ( q , q ˙ ) ] η + [ 0 G ( q ) + τ ] .
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-11-20
Show that y=x2sin(x) and y=0 are both solutions of
x2y4xy+(x2+6)y=0
and that both satisfy the conditions y(0)=0 and y(0)=0. Does this theorem contradict Theorem A? If not, why not?
asked 2021-03-07
Solve differential equationxu(x)=u24
asked 2021-03-07

Solve the correct answer linear equations by considering y as a function of x, that is, y=y(x).dydx+y=cosx

asked 2021-09-06
Using only the Laplace transform table, obtain the Laplace transforms of the following function:
cosh(4t)4t
where ”cosh” stands for hyperbolic cosine and cosh x=exex2
asked 2020-11-29
Part II
29.[Poles] (a) For each of the pole diagrams below:
(i) Describe common features of all functions f(t) whose Laplace transforms have the given pole diagram.
(ii) Write down two examples of such f(t) and F(s).
The diagrams are: (1)1,i,i.(2)1+4i,14i.(3)1.(4) The empty diagram.
(b) A mechanical system is discovered during an archaeological dig in Ethiopia. Rather than break it open, the investigators subjected it to a unit impulse. It was found that the motion of the system in response to the unit impulse is given by w(t)=u(t)et2sin(3t2)
(i) What is the characteristic polynomial of the system? What is the transfer function W(s)?
(ii) Sketch the pole diagram of the system.
(ii) The team wants to transport this artifact to a museum. They know that vibrations from the truck that moves it result in vibrations of the system. They hope to avoid circular frequencies to which the system response has the greatest amplitude. What frequency should they avoid?
asked 2021-11-23
Solve th second order linear equations:
y+4y=cos2x
asked 2022-01-22
Solving an ordinary differential equation:
Prove if y(t)+3y(t)=6t+5,y(0)=3, then y(t)=2e3t+2t+1.