% FK/IK using homogenous matrices %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% This script demonstrates how homogenous matrices can be used to create
% both a forward kinematics (FK)and an inverse kinematics (IK) model of 
% a serial kinematics manipulator, with 5 % motors (q1,q2,q3,q4,q5) 
% and 4 user coordinates (X,Y,Z,theta).
%
% The script uses a chain of Homogenous Transformation Matrices to create 
% the FK model. And the Moore-Penrose Pseudo-Inverse of the Jacobian 
% to iteratively solve the IK model (Newton-Raphson).
%
% 5.10.2017 Wayne Glettig
  
clear all;
syms offsetX offsetY offsetZ ...
     alpha beta gamma...
     q1 q2 q3 q4 q5;

%% Here the Transformation Matrix HBA is constructed from a translation 
% matrix and 3 rotation matrices, using symbolic variables.
%Translation:
HXYZ = [1,0,0,offsetX;...
        0,1,0,offsetY;...
        0,0,1,offsetZ;...
        0,0,0,1];
%Rot around X:
HRX = [1,0        ,0           ,0;...
       0,cos(alpha),-sin(alpha),0;...
       0,sin(alpha), cos(alpha),0;...
       0,0        ,0           ,1];
%Rot around Y:
HRY = [ cos(beta),0,sin(beta),0;...
       0         ,1,0        ,0;...
       -sin(beta),0,cos(beta),0;...
       0         ,0,0        ,1];
%Rot around Z:
HRZ = [cos(gamma),-sin(gamma),0,0;...
       sin(gamma), cos(gamma),0,0;...
       0         ,0          ,1,0;...
       0         ,0          ,0,1];
%Create HBA (first rotation then translation)
HBA=HXYZ*HRZ*HRY*HRX;
%Create HAB (inverse)
HAB=inv(HBA);


%% Here the Kinematic Chain is created (q1,q2,q3,q4,q5 are motor coordinates)
%Essentially the HBA matrix above is filled with the offsetXYZ and angle
%values to assemble the chain. The chain consists here of the reference
%frames 0-1-2-3-4-5-6-7:
%Frame 0: MATLAB darstellungsframe (for plot3)
%Frame 1: Beamline UCS
%Frame 2: on Z Stage
%Frame 3: on X Stage
%Frame 4: on RX Stage
%Frame 5: on Fast X Stage
%Frame 6: on Fast Y Stage
%Frame 7: Sample pos

H10 = subs(HBA,[offsetX,offsetY,offsetZ,alpha,beta,gamma],...
               [0,0,0,90/180*pi,0/180*pi,90/180*pi]);
H21 = subs(HBA,[offsetX,offsetY,offsetZ,alpha,beta,gamma],...
               [0,0,q1,0,0,0]); %Z Stage
H32 = subs(HBA,[offsetX,offsetY,offsetZ,alpha,beta,gamma],...
               [q2,0,0,0,0,0]); %X Stage
H43 = subs(HBA,[offsetX,offsetY,offsetZ,alpha,beta,gamma],...
               [0,0,0,0,q3/180*pi,0]); %Rotation RY
H54 = subs(HBA,[offsetX,offsetY,offsetZ,alpha,beta,gamma],...
               [q4,0,0,0,0,0]); %Fast Stage X
H65 = subs(HBA,[offsetX,offsetY,offsetZ,alpha,beta,gamma],...
               [0,q5,0,0,0,0]); %Fast Stage Y
H76 = subs(HBA,[offsetX,offsetY,offsetZ,alpha,beta,gamma],...
               [0,0,0,0,0,0]); %sample Offset
H70 = H10*H21*H32*H43*H54*H65*H76;
H07 = inv(H70);

%% The FK model (forward kinematics) can be defined as 
% [X;Y;Z;theta] = f(q1,q2,q3,q4,q5)
H71 = H21*H32*H43*H54*H65*H76; %UCS: reference frame 1
f= H71*[0;0;0;1]; %XYZ position of orgin of reference frame 7
f(4)=q3;          %theta angle directly connected to q3

%% The IK model is more difficult to find. Different IK solution methods 
% exist, this is one of the basic ones, using the Pseudo-inverse of the
% Jacobian matrix to converge a solution towards a solution
% (Newton-Raphson)

% Jacobian of FK f function derived over motor coordinates
J= jacobian(f,[q1,q2,q3,q4,q5]);
Jinv = pinv(J);

tic
% Algorithm for IK (q are motor coords, x are user coords)
xt=[2;1;0;32]; %SET TARGET USER COORDS 
q0=[0;0;0;0;0]; %motor start values
qc = q0; %set current q values for loop
loopcond = 1;
loopcounter=0;
while loopcond
    xc = vpa(subs(f, [q1,q2,q3,q4,q5], qc')); %get current x values based on q 
    deltax=xt-xc; %get x error (target - current)
    if abs(deltax)<1e-9 %if abs error small enough, get out of loop
        loopcond=0;
    end
    Jinvc=vpa(subs(Jinv, [q1,q2,q3,q4,q5], qc')); %inv Jacobian with current q
    deltaq=Jinvc*deltax; %By multiplying the x error with Jinv, a q correction can be deduced
    qc = qc+deltaq;%update current motor values
    loopcounter=loopcounter+1;
end
q = qc %output q as the motor coordinates
toc

%% Plot the results (construct sample position using motors)
Oin1 = [0;0;0;1];
e1in1 = [1;0;0;1];
e2in1 = [0;1;0;1];
e3in1 = [0;0;1;1];

clf;
hold on;
grid on;
axis vis3d;
axis([-10 10 -10 10 -10 10]);
title('Sample Position, constructed with (q1,q2,q3,q4,q5)');
xlabel('Beamline Z-Axis');
ylabel('Beamline X-Axis');
zlabel('Beamline Y-Axis');


for i= 0:1
H60c = subs(H70, [q1 q2 q3, q4, q5], q');

Oin0  = H60c*Oin1;
e1in0 = H60c*e1in1;
e2in0 = H60c*e2in1;
e3in0 = H60c*e3in1;

plot3([Oin0(1),e1in0(1)],[Oin0(2),e1in0(2)],[Oin0(3),e1in0(3)],'r-');
plot3([Oin0(1),e2in0(1)],[Oin0(2),e2in0(2)],[Oin0(3),e2in0(3)],'g-');
plot3([Oin0(1),e3in0(1)],[Oin0(2),e3in0(2)],[Oin0(3),e3in0(3)],'b-');
plot3([-10,0],[0,0],[0,0],'r-')
end