在Matlab中求解非线性方程

Question

I have a homework question that I cannot answer. Here is the question prompt:

1. Define Eq. 8.3 and Eq. 8.4 in a function. This function should take a vector of joint angles ( and ) as input and should return a column vector containing the two functions ( and ) evaluated at those angles. The function should contain the link lengths and values of and for calculation. Include your code in your solution.

Equations 8.3 and 8.4 are:

(8.3)

(8.4)

where

Here is the function I wrote:

function F = rob_arm (alphag, betag)
F = (1).*cos(alphag)+ (1).*(cos(alphag+betag))-(1) ;
  (1).*sin(alphag) + (1).*(sin(alphag+betag))-(1.1) ;
end

Because alpha and beta are matrices of different sizes, I used meshgrid to create alphag and betag , and used those matrices to calculate the values of rob_arm . After four hours of messing with this, I'm not even sure what the question is asking anymore, and the TA's are not currently answering emails. I wrote the following code, to try and force rob_arm into a single column:

alpha = 0:pi/100:(pi/2); %define angle alpha
beta = 0:pi/100:pi; %define angle beta

[alphag, betag] = meshgrid (alpha, beta); %mesh grid alpha and beta b/c different matrix dimensions
arm_pos = rob_arm (alphag, betag);

for ii = 1:1:101
    for k = 1:1:51
 col_vec (1,1:1:5151) = arm_pos(ii,k);
    end
end

Ignoring the query to create a column vector, the resulting output, arm_pos is good output. I can graph it and I get a very pretty picture of all the possible points that this robot arm can 'reach.'

But because I am dumb and have been trying this for many hours, it's not saving successive values of rob_arm into col_vec , it just replaces it each time and I end up with a 1x1 matrix. Ultimately, the goal will be to use the Newton-Raphson method to determine the zeroes of this function, but that's a long ways off. I am thinking if I can get all of the values calculated by rob_arm into a single column, then I can answer this question.

The next question is:

2. Create a separate function that accepts input of a single row vector containing the pair of angles and . The output of the function should be the Jacobian (a 2 by 2 matrix). First, calculate the derivatives of Eqs. 8.3 – 8.4 by hand, then put them into your function. Include your function code in your solution

Which I will need to ask for clarification on, because I don't understand how a 1 x 51 matrix ( alpha ) and a 1 x 102 matrix ( beta ) could be accepted into a single row vector, that would then output a 2x2 matrix. I know what the Jacobian is, and it is the partial derivitives of my two functions, not a matrix of values.

If anyone wants to give me a hand, that would be super awesome.

Answer 1

You function definition is not right, you should use something like

function F = rob_arm (alphag, betag) 
 F = [(1).*cos(alphag)+ (1).*(cos(alphag+betag))-(1) ; ...   
      (1).*sin(alphag) + (1).*(sin(alphag+betag))-(1.1)]; 
end

Without the three dots " ... " MATLAB performs this calculation

 (1).*sin(alphag) + (1).*(sin(alphag+betag))-(1.1);

but does nothing with the result, and returns

  F = (1).*cos(alphag)+ (1).*(cos(alphag+betag))-(1) ; 

which is not what you want, half the F values are missing.