4 Degrees of Freedom Robot Arm With a Twist

4 Degrees of Freedom Robot Arm With a Twist

STM32F4 - Processor in the Loop
  • Inverse Kinematics of a Two-Link Planar Elbow Manipulator (Symbolic Math)
  • Solving the 4 Degree of Freedom (Matlab Script)
  • Simulink Matlab Function Block Implementation
  • Simulation with Simscape Multibody
  • Processor in the loop with STM32F4 Discovery Board
    • What we need
    • Results
  • Next Steps
This is a documentation of steps being done to be able to solve the 6 Degree of Freedom Robot Arm bought online. This goes through the process of solving the inverse kinematics of a Two-Link Planar Elbow Manipulator which is the basis of our case.

 Wrist and Clamps was not solve here &_&

Inverse Kinematics of a Two-Link Planar Elbow Manipulator


 xd, yd = location of the tip of the manipulator    
q1d, q2d = angles of the links (counter-clockwise is positive) 
q3d = dummy angle
q4d = dummy angle
L1, L2 = lengths of the links (constant)
th1d := Symbol::subScript(Symbol("theta"),"1d"):
th2d := Symbol::subScript(Symbol("theta"),"2d"):
th3d := Symbol::subScript(Symbol("theta"),"3d"):
th4d := Symbol::subScript(Symbol("theta"),"4d"):
L1 := Symbol::subScript(Symbol("rho"),1):
L2 := Symbol::subScript(Symbol("rho"),2):
th1d, th2d, th3d, th4d, L1, L2;

Using Pythagoras theorem on the right triangle (law of cosines)
We can be able to add the L1 and L2cos(q2d) as X and L2sin(q2d) as Y
Since we have an assumption target location Xd and Yd we can be able to
solve q2d via Pythaoras theorem on the right triangle
eq1 := xd^2 + yd^2;
eq2 := L1 + (L2*(cos(th2d)));
eq3 := L2*sin(th2d);
eq4 := eq1 = eq2^2 + eq3^2
eq5 := simplify(eq4)
solving for cos(q2d)

assume(L1 > 0 and L2 > 0 ):
eq6 := op(solve(eq5,cos(th2d)),1);
q1D = q4D - q3D
eq7 := arccos(xd/(sqrt(xd^2 + yd^2)))
eq8 := L2*eq6
eq9 := cos(th3d) = (L1 + eq8)/sqrt(eq1)
eq10 := op(solve(eq9,cos(th3d)),1);
eq11 := arccos(eq10)
eq12 := eq7 - eq11
eq13 := th2d = arccos(eq6)

TH1des := subs(eq12),L1 = L_1,L2 = L_2;
TH2des := subs(eq13),L1 = L_1,L2 = L_2;



Solving from the top:

newth1des = th1des+th2des;
newth2des = th1des;

Solving the 4 Degree of freedom:

This is to check that the solution that we have from the Symbolic Math Toolbox is correct or not
%% Solving for a 4 Degree of Freedom Robot ARM
% this is just an example of solving 4 DoF Robot ARM via making one link
% (Stationary) from 0 to 360

%% Define the Target Location
xt = 0.5;
yt = 0.5;
zt = 0.5;
%% Define your robot arm body parameters
l1 = 1;
l2 = 1;
l3 = 1;
h0 = 0;

%% Top view angle : th0 (base angle)
% solve for th0
th0 = atand(yt/xt);
if th0 < 0
    th0 = th0+180;
elseif isnan(th0)
    th0 = 90;
end
display(th0)
%% For loop for Visualization
% we need to imagine that we have moved into a different Dimention hence
% you can si that we have change the angle to line lenghts to ut and wt
wt = sqrt(xt^2 + yt^2);
% Find the distance from the base of the bicep to the target
dist2target = sqrt(wt^2 + (zt-h0)^2);
ut = sqrt(dist2target^2 - wt^2);

%solve for the angle
angle = atand(ut/wt);
for counter = 0:360
th1 = counter;
xb = l1*cosd(th1);
yb = l1*sind(th1);
% solve the th1des and th2des with the ik_doublelink that we have solved
% earlier from our MuPad
[th1des, th2des] = ik_doublelink_mathworks(wt-xb,ut-yb,l2,l3);

% This is just for Visualization, you only need to get the th1des and
% th2des (Solving X and Y)
xl2 = l2*cosd(th1des);
yl2 = l2*sind(th1des);
xl3 = l3*cosd(th1des+th2des);
yl3 = l3*sind(th1des+th2des);
%display the x,y and r value
display([wt ut dist2target])
display([th1 th1des th2des])

% translate to solve from the top
% basically we made a PARALLELOGRAM from the solution with th1des and th2des
% and the angle origin 0 is on the +X axis
newth1des = th1des+th2des;
newth2des = th1des;

newxl2 = l2*cosd(newth1des);
newyl2 = l2*sind(newth1des);

newxl3 = l3*cosd(newth2des);
newyl3 = l3*sind(newth2des);

subplot(1,2,1)
plot(wt,ut,'rx')
xlim([-1 2]);
ylim([-1 2]);
hold on
plot([0,xb],[0,yb])
plot([xb,xb+xl2],[yb,yb+yl2])
plot([xb+xl2,xb+xl2+xl3],[yb+yl2,yb+yl2+yl3])

subplot(1,2,2)
plot(wt,ut,'rx')
xlim([-1 2]);
ylim([-1 2]);
hold on
plot([0,xb],[0,yb])
plot([xb,xb+newxl2],[yb,yb+newyl2])
plot([xb+newxl2,xb+newxl2+newxl3],[yb+newyl2,yb+newyl2+newyl3])
pause(0.1)
end

This is for a fixed 45 Degree angle at the bicep

This is for a fixed 0-360 Degree angle at the bicep

Simulink Matlab Function Block Implementation:

function [tAngle,bAngle,fAngle,wAngle] = IK(xt,yt,zt)
%#codegen
persistent th0 th1 th2 th3
% Initialize joint angles on the first pass
if isempty(th0)
    th0 = 0;
    th1 = 60;
    th2 = 0;
    th3 = 0;
end

l1 = 1;
l2 = 1;
l3 = 1;
h0 = 0;

% solve for th0
th0 = 90 - atand(xt/yt);

if isnan(th0)
    th0 = 90;  
end

wt = sqrt(xt^2 + yt^2);
% Find the distance from the base of the bicep to the target
dist2target = sqrt(wt^2 + (zt-h0)^2);
ut = sqrt(dist2target^2 - wt^2);

%solve for the angle
%angle = atand(ut/wt);

xb = l1*cosd(th1);
yb = l1*sind(th1);

[th1des, th2des] = ik_doublelink_mathworks(wt-xb,ut-yb,l2,l3);

% translate to solve from the top
% basically we made a PARALLELOGRAM from the solution with th1des and th2des
% and the angle origin 0 is on the +X axis
th2 = th1des+th2des;
th3 = th1des;


tAngle = th0;
bAngle = th1;
fAngle = th2;
wAngle = th3;

end

function [th1des,th2des] = ik_doublelink_mathworks(xd,yd,l1,l2)
if sqrt(xd^2 + yd^2) < l1+l2
    % Calculate the forearm link angle
    c2 = (xd.^2 + yd.^2 - l1^2 - l2^2)/(2*l1*l2);
    s2 = sqrt(1 - c2.^2);
    th2des = atan2(s2, c2);
else % protect for relative floating point accuracy at th2 = 0
    c2 = 1;
    s2 = 0;
    th2des = 0;
end   

% Calculate the bicep link angle
k1 = l1 + l2*c2;
k2 = l2*s2;
% Mantain the sign of th1 positive
if xd < 0 && yd < 0 % if point is in the third quadrant add 2*pi
    th1des = 2*pi + atan2(yd, xd) - atan2(k2, k1);
else
    th1des = atan2(yd, xd) - atan2(k2, k1);
end

th1des = rad2deg(th1des);
th2des = rad2deg(th2des);
end

Simulation with Simscape Multi Body


Processor in the loop with STM32F4 Board

What we need:
  1. STM32F4Discovery Board
  2. USART Serial Converter (FTDI Basic)
  3. Two jumper wires for TX and RX
  4. STM32-Mat Support Package for Matlab/Simulink
  5. Matlab/Simulink License
    • Matlab
    • Simulink
    • Matlab Coder
    • Simulink Coder
    • Embedded Coder

Results: 
There is no difference between the simulation model and the code being generated and deployed to the STM32f4Discovery Board (Check Scope 2 Output)

The function IK is being called every 0.001 secs (fundamental timestep), according to the Code Execution profile of the IK function, the Maximum Execution time is around 339905 which we need to multiply by 1e-09 (Default time setting for the measurement) hence the IK Algorithm to solve the 4 DoF angles would take about 339.91 micro seconds hence calling the function every 1 milli seconds is enough to finish the function in solving the IK of the 4 DoF link arm.

Next Steps:

  • Conversion of the Degrees to Servo Motor Control Signal
  • Need to add in a 3.3v to 5.0v converter since the STM32Discovery Boards are operating at 3.3 volts
Credits to Mathworks and my Techsource AEG

How a Differential Equation Becomes a Robot

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