How to include variable offsets in polynomial equations

I am attempting to display a series of equations, of which some need offsets to the variable. The equations are (at maximum) cubic polynomials, however some require an offset while others do not.

No offset: a*t^3 + d

Offset: a*(t-5)^3 + b(t-5)^2 + d

The offset will be the constant in each polynomial.

Is there an easy way to apply this? Code is given below, the offset of each polynomial is equal to t_0 of the function. So for segment 2, the offset is t_1 should be shown as (t-2)^n in the polynomial:

clear all;
clc;

syms t real;

% Position in [deg deg m]
q_0 = [0 0 0]';
q_1 = [-8 45 0.2]';
q_2 = [-90 90 0.4]';

% Speeds in [deg/s deg/s m/s]
q_dot_0 = [0 0 0]';
q_dot_1 = [10 40 0.2]';
q_dot_2 = [0 0 0]';

% Times in s
t_0 = 0;
t_1 = 2;
t_2 = 4;

coefficients = zeros(2,4,3);

% Segment 1: 0 < t < 2
coefficients(1,:,1) = cpsCoefficients(q_0(1), q_1(1), q_dot_0(1), q_dot_1(1), t_0, t_1);
coefficients(1,:,2) = cpsCoefficients(q_0(2), q_1(2), q_dot_0(2), q_dot_1(2), t_0, t_1);
coefficients(1,:,3) = cpsCoefficients(q_0(3), q_1(3), q_dot_0(3), q_dot_1(3), t_0, t_1);

% Segment 2: 2 < t < 4
coefficients(2,:,1) = cpsCoefficients(q_1(1), q_2(1), q_dot_1(1), q_dot_2(1), t_1, t_2);
coefficients(2,:,2) = cpsCoefficients(q_1(2), q_2(2), q_dot_1(2), q_dot_2(2), t_1, t_2);
coefficients(2,:,3) = cpsCoefficients(q_1(3), q_2(3), q_dot_1(3), q_dot_2(3), t_1, t_2);

eqn(1,1) = vpa(poly2sym(fliplr(coefficients(1,:,1)),t));
eqn(2,1) = vpa(poly2sym(fliplr(coefficients(2,:,1)),t));
eqn(3,1) = vpa(poly2sym(fliplr(coefficients(1,:,2)),t));
eqn(4,1) = vpa(poly2sym(fliplr(coefficients(2,:,2)),t));
eqn(5,1) = vpa(poly2sym(fliplr(coefficients(1,:,3)),t));
eqn(6,1) = vpa(poly2sym(fliplr(coefficients(2,:,3)),t));

eqn = string(eqn); %For table display purposes

table_segment_1 = table(eqn(1:2), eqn(3:4), eqn(5:6),'RowNames',{'Segment 1 (0 < t < 2)','Segment 2 (2 < t < 4)'},'VariableNames',{'Joint 1','Joint 2','Joint 3'});
disp(table_segment_1);

function [coeffs] = cpsCoefficients(theta_0, theta_f, theta_dot_0, theta_dot_f, t_0, t_f)
%Calculates and returns the CPS coefficients
a0 = theta_0;
a1 = theta_dot_0;
a2 = (3*(theta_f-theta_0)-(2*theta_dot_0+theta_dot_f)*t_f)/t_f;
a3 = (2*(theta_0-theta_f)+(theta_dot_0+theta_dot_f)*(t_f-t_0))/((t_f-t_0)^3);
coeffs = [a0 a1 a2 a3];
end

Answer 1

[Bjorn Gustavsson]

Have a look at the help and documentation of taylor. That should help you a good part of the way - especially if you use the 'expansionpoint' variable you will get rather close to what you want:

>> syms X
>> A = sym('A',[1,4]);
>> f = A(1) + A(2)*X + A(3)*X^2 + A(4)*X^3;
>> asd = taylor(f,X,'expansionpoint',5);
% Returns 
%
% asd =
% A1 + 5*A2 + 25*A3 + 125*A4 + (A3 + 15*A4)*(X - 5)^2 + A4*(X - 5)^3 + (X - 5)*(A2 + 10*A3 + 75*A4)
a0 = 12;
a1 = 7;
a2 = 5;
a3 = -3;
zxc = subs(asd,A,[a0 a1 a2 a3]);
% Returns: 
% zxc =
% 
% 637 - 40*(X - 5)^2 - 3*(X - 5)^3 - 168*X

HTH

[Connor LeClaire]

It appears that MATLAB automatically expands and simplifies the first order term which changes the 0th order term, is there a way to keep the offset instead of automatically simplifying?

[Bjorn Gustavsson]

So it seems. I don't know and lack interest to find out...

[Connor LeClaire]

right, ok thanks!

[Bjorn Gustavsson]

My pleasure, glad it helped.