QUESTION
We will solve the problem in 3-dimensions.We will take spherical coordinates.As final point coordinates are known which is range R and elevation q,i.e.(R,q),we have to solve the problem in coordinates of two points (x1,y1,z1) and (x2,y2,z2).Because only these points will vary while moving the arms of robot,which is of length L1,L2and L3.
SOLUTION
x1=L1sin(theta1)cos(phi1);
y1=L1sin(theta1)sin(phi1);
z1=L1cos(theta1);
x2-x1=L2 sin(theta2)cos(phi2);
y2-y1=L2 sin(theta2)sin(phi2);
z2-z1=L2cos(theta2);
x3-x2-x1=L3 sin(theta3)cos(phi3);
y3-y2-y1=L3 sin(theta3)sin(phi3);
z3-z2-z1=L3cos(theta3);
solving for x3,y3,z3,
x3= L1sin(theta1)cos(phi1)+ L2 sin(theta2)cos(phi2)+ L3 sin(theta3)cos(phi3);
y3= L1sin(theta1)sin(phi1)+ L2 sin(theta2)sin(phi2)+ L3 sin(theta3)sin(phi3);
z3= L1cos(theta1)+ L2cos(theta2)+ L3cos(theta3);
as ‘phi’ is angle from vertical,and ‘theta’ is angle from y axis.
We have,
Sin(theta1)=x1/L1
Cos(phi1)=y1/L1
Similarly converting for all,
We have,
Also
Solving,
CODE AND PLOT OF VARIATION OF Y2 WITH Y1—
R=90;q=60;L1=20;L2=30;L3=40;
y1=1:1:20;
y2=L2.*((R.*(1-q/L3)-(((L1.^2)-y1.^2).^(1/2)).*y1/L1)./((((q-(R.^2)./L3).^2)-(((L1.^2)-y1.^2).^(1/2))).*L2).^(1/2));
plot(y1,y2);
xlabel(‘y1’);
ylabel(‘y2’);
CODE AND PLOT OF VARIATION OF X2 WITH X1—
L1=20;L2=30;L3=40;R=90;q=60;
x1=1:1:20;
x2=((L2.*((q-(R.^2)./L3))-((x1.^2)./L1)).^2);
plot(x1,x2);
xlabel(‘x1’);
ylabel(‘x2’);
CODE FOR VARIATION OF Z2—
R=90;q=60;L1=20;L2=30;L3=40;
y1=1:1:20;
y2=L2.*((R.*(1-q/L3)-(((L1.^2)-y1.^2).^(1/2)).*y1/L1)./((((q-(R.^2)./L3).^2)-(((L1.^2)-y1.^2).^(1/2))).*L2).^(1/2));
z2=y1+y2;
plot(y1,z2);
xlabel(‘y1’);
ylabel(‘z2’);
KI41
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