Dear Arash Marashian

It seems, there is a glitch in the mathematical logic of your so-called "complex equation". Since, you have differentiated the variables; x_r(tau), y_r(tau), psi_r(tau), hence they should vary with regard to tau. In other words tau is a variable similar to time, and therefore x_r, y_r, and psi_r, might be time signals.

Hence, t and tau are both variables as you have already designated them by syms command at MATLAB prompt line.

Let's get an equation similar in format to your so-called complex equation. An equation like:

eq= 2*(1-lambda)^2*(phrase3),

where we assumed other expressions in your equation are zero. We already know that:

phrase3 = 0.5*(psi - psi_r)*psi_r_dot;

hence the aforementioned equation would cast into:

eq= 2*(1-lambda)^2*(0.5*(psi - psi_r)*psi_r_dot),

which has to variables; t and tau.

At the last line of your MATLAB code, you wrote:

sol = solve(eq == 0 , tau , 'IgnoreAnalyticConstraints',1) ,

But as discussed, the equation has two variables t and tau. So, I ask you, how it would be possible to analytically or numerically solve a single equation with two variables; t and tau. It is something like finding tau, in the equation sin(t+tau)+tan(sqrt(t+tau))+1=0,

which seems to have no easy explicit solution, but even if an explicit analytic solution exists, then it is in the form of: tau=f(t), where f is simply a function.

If t is known as a numerical value, then you can insert it in the function f, to get the numerical value of tau as: tau=f(t).

So to get a numerical solution, a relation between t and tau should be known, as for example, t=tau+1.

For a numerical solution you should assign a value of t, for example t=1, and then numerically calculate the value of tau.

Pass x, instead of tau, as rightly pointed out by Saeb AmirAhmadi Chomachar I don't really know if this is what you want, but please check the solution below it.

syms x y psi tau t

c1 = 1;c2 = 1.5;lambda = 0.1;

x_r(tau) = 0.8486*tau - 0.6949;

y_r(tau) = 5.866*sin(0.1257*tau + pi);

psi_r(tau) = 0.7958*sin(0.1257*tau - pi/2);

x_r_dot = 0.8486;

y_r_dot(tau) = 0.7374*cos(0.1257*tau + pi);

psi_r_dot(tau) = 0.1*cos(0.1257*tau - pi/2);

phrase1 = c1/2*(cos(psi)*(x - x_r) + sin(psi)*(y - y_r))*(cos(psi)*x_r_dot + sin(psi)*y_r_dot);

phrase2 = c1/2*(-sin(psi)*(x - x_r) + cos(psi)*(y - y_r))*(-sin(psi)*x_r_dot+cos(psi)*y_r_dot);

phrase3 = 0.5*(psi - psi_r)*psi_r_dot;

eq = -2*(1-lambda)^2*(phrase1 + phrase2 + phrase3) - 2*lambda^2*(t - tau);

eqn = rewrite(eq,'log');

sol = solve(eqn == 0 , x , 'IgnoreAnalyticConstraints',1);

pretty(sol)

Dear Hassan Nazeer Chaudhry

Thanks for the comment. I do not understand why you added the command; eqn= rewrite(eq,'log')?. It seems you meant to transform the equation into logarithmic format, hence, I interpret you have missed phrase for phase (?).

But my other guess is that, you wanted to get the equation simplified by taking logarithm of the coupled terms, as for example: log (psi*psi_dot) =log(psi)+log(psi_dot), hence the analytic solution would be more convenient to be obtained. If this is your goal, which I think it is, then I agree with you, and that's O.K.

Thank you again.

Saeb AmirAhmadi Chomachar , thanks for your answer,

as I mentioned tau is PI_0.1(X,t) in the photo. so tau is stated in terms of X = [x(t),y(t),psi(t)]^T and t. And yes tau is time-like but it is not equal to t.

I actually approximate the sin(.) and cos(.) with Taylor's series and Padè algorithm, but the answer got too complex, like root(...).

• Taylor approximation:

app_sin(tau) = taylor(sin(tau) , tau , 'order' , 8);

app_cos(tau) = taylor(cos(tau) , tau , 'order' , 8);

• Padè approximation:

app_sin(tau) = pade(sin(tau) , tau);

app_cos(tau) = pade(cos(tau) , tau);

Hassan Nazeer Chaudhry , thanks for your time, but this equation is supposed to be solved in terms of tau, not x. Did you have a specific reason for solving it in terms of X?

Best regards,

Arash

Dear Arash Marashian

It appears that, your so-called "complex equation" should be explained more through a more clear mathematical statement.

Let us solve, the last part of your equation marked up in the image attached.

Assume:

psi_r(tau) = 0.7958*sin(0.1257*tau - pi/2);

psi_r_dot(tau) = 0.1*cos(0.1257*tau - pi/2);

Hence, the red-marked equation takes the form of:

1/2*(psi(t)- 0.7958*sin(0.1257*tau - pi/2))* 0.1*cos(0.1257*tau - pi/2)+2*(0.1)^2*(t - tau)=0.

This equation has a typical solution in form of tau=f(t,psi(t)).

But if you add the other phrases, then the solution will be in shape of:

tau=f(t, X(t),y(t),psi(t)).

Warning: You have designated at the prompt line:

syms x y psi tau t

Hence, MATLAB interprets all these parameters as unknown analytic values in LeftHandSide of the equation, while you have only one single algebraic equation to glean these 5 variables.

My suggestion:

At the prompt line of MATLAB code only designate tau as the symbolic variable:

syms tau,

But at the bad luck, there is a format problem in MATLAB, if you do not designate other variables as syms the solution will not be available ! And you cannot do anything to resolve this default MATLAB format challenge.

Outwit:

I think Maple can adroitly do this task. I have done it for too many equations before. I think Wolfram Mathematica is also a good match for this task.

Write your equation, and ask Maple, or Mathematica to solve it, while you only designate tau as the variable to be analytically found. If the equation is not toughly nonlinear or transcendental, an analytical solution will be fortunately rendered by Mathematica or Maple, as tau=f(t, X(t),y(t),psi(t)).

Also beware to use the mathematical trick pointed out by Mr. Chaudhry, I mean rewrite the equation in logarithm format to decouple the hard nonlinearities and get the original equation simpler, before you feed it to Maple or Mathematica for the final solution.

That's it.

Saeb AmirAhmadi Chomachar , Thanks for your detailed answer, but I'm not familiar with Maple and I don't understand where you said: only designate tau as the symbolic variable. How is it possible to designate only tau, however, we have other variables that are unknown?

Dear Arash Marashian

You should only designate tau as a variable, because you have only one single equation, and you can only glean one variable from that equation. You can designate only one variable, to abide by the mathematical logic. You should freeze all other variables as they are analogous to a number. If you want to solve ax+b=0, then although through a mathematical viewpoint, the parameters a , x, and b, are variables, but computer cannot think like a human, and through the eye of computer, only one of the three parameters (a, x, b) should be designated as a variable, to give the solution in terms of the other variables as: x=-b/a, or b=-ax, or a=-b/x.

I mention an example here:

syms a b c x

eqn = a*x^2 + b*x + c == 0;

solx = solve(eqn, x)

solution would be:

solx = -(b + (b^2 - 4*a*c)^(1/2))/(2*a) -(b - (b^2 - 4*a*c)^(1/2))/(2*a)

MatLab solves this equation through a predefined setup in its memory, and does not think about this equation. It is senseless on this equation. If you get the equation a bit larger, MatLab usually leaves you with the message:

Warning: Unable to find explicit solution. For options, see help.

This message is sent, because MATLAB manipulates the equation as an implicit equation (see the image attached), and an implicit equation has certainly no simple and easy explicit solution in one variable, if the variables are more than one, or the equation is a bit complicated, containing cross-coupled terms.

https://en.wikipedia.org/wiki/Implicit_function

https://en.wikipedia.org/wiki/Diophantine_equation

MatLab only boasts its fake capacity to solve this quadratic equation

through already knowing the famous solution. For too many other

algebraic equations, specifically nonlinear, MatLab certainly gets incapacitated.

But some software as Maple or Mathematica, solve these types of single algebraic equations

by designating only one variable to be gleaned in their syntax. Even a system of

two or three independent nonlinear algebraic equations are usually solved by Maple easily.

I have solved some toughly nonlinear algebraic equations with Maple,

only with a click.

Maple is very powerful to solve such equations.

Mathematica is also powerful to solve such nonlinear algebraic equations.

You can try Mathematica (I think online version is also available).

Mathematica is a simple and fast software for too many math problems.

MatLab abbreviates Matrix Laboratory. It would be interesting for you to know that

MatLab relies on Maple for analytical solutions, while I had read

somewhere in an online source that, Mfile of MatLab, does actually means Maple file.

As MatLab is the matrix laboratory, it is definitely capable to do

numerical math operations with a great efficiency. However, in analytical

math operations Maple is a super-duper, as I have experienced it myself for

too many problems.

Try Maple or Mathematica to solve your equation, and designate only tau

as a variable to be gleaned by the solver syntax.

Feel free to share with me, any other comment or query you might have on.

Good Luck.

Dear Arash Marashian , it seems unlikely that an explicit solution exists for this equation. If it does, it would be a lot to ask from Matlab's symbolic toolbox to find it. Note that, because of the involved trigonometric functions, the number of solutions may depend on x, y, psi, t. For some combinations, there may even be infinitely many solutions.

Since you also asked for numerical solutions: If you prescribe the values of x, y, psi, t, you can find a corresponding solution for tau. I tried this using both 'fzero' and 'vpasolve' without difficulties (for some random value of x, y, psi, t as I have no idea what these values mean). Once again, keep in mind there may be many solutions. You may plot 'eq(tau)' for some chosen values of x, y, psi, t to see that it can oscillate around zero and, hence, produce many different solutions.

Hauke Gravenkamp , Thanks for your answer, actually [x,y,psi] will be derived from a differential equation, which is the dynamical system equation that is coupled with tau. Hence I think I don't have an explicit value for them (?)

Dear Arash Marashian

Your recent statement is again a logical dilemma. If you are running the system of coupled ODEs for the [x,y,psi], then t is the time variable in the simulation which naturally varies from zero to infinity. Since, tau is a a time-like variable; e.g. tau=t+5, therefore, tau similarly varies from 5 to infinity. So, are you looking to find the simulation time (?!).

Moreover, you have the xr(tau), and x_dot_r(tau) and similarly other variables in terms of explicit tau. Hence, you can place them in your so-called "complex equation" , and find a variable as psi(t)+k y(t)=f(tau). Then reverse the notation by only changing the place-holder of t and tau, by hand, to get; psi(tau)+k y(tau)=f(t). An example would be: sin(tau)+tau^2+sqrt(tau)=t^2+t+1. Afterward, you insert a nominal value for t, e.g. t=1.5, and calculate numerical value of the variable tau for that nominal time, based on this recent equation which relates tau to t.

Another statement of your so-called "complex equation" with a clear description, would be helpful, to get us known, what are you actually looking for?