Question

Problem 1: Consider the two-link planar elbow manipulator shown below with link information: a1-1, a2 2 92 |Link l a, lai la, 1 6, 1 lai | 0 | 0|0; variable re 0.7 (a) Suppose that the manipulator needs to reach Po- 1.5.Compute the all possible sets of solutions of Computetheallpossib aanipulatorneedstoreachpo the joint variables. (b) For each set of solutions you computed in (a), sketch the robot's configuration to verify that the end-effector does reach the specified location.

Answer 1

Location of joint 2 with respect to origin:

$x_1 = l_1cos\theta_1;y_1 = l_1sin\theta_1$

Location of end point with respect to joint 2:

$x_2 = l_2cos(\theta_1+\theta_2);y_2 = l_2sin(\theta_1+\theta_2)$

Location of endpoint with respect to origin:

$x = l_1cos\theta_1+l_2cos(\theta_1+\theta_2);y =l_1sin\theta_1 +l_2sin(\theta_1+\theta_2)$

Given:

11 = 1: 12 = 2

= 0.7: y = 1.5

COS

squaring the equations;

$\Rightarrow 0.7^2 = (cos\theta_1+2cos(\theta_1+\theta_2))^2;1.5^2 = (sin\theta_1 +2sin(\theta_1+\theta_2))^2$

add the 2 equations:

$\Rightarrow 0.7^2 +1.5^2 = (cos\theta_1+2cos(\theta_1+\theta_2))^2+ (sin\theta_1 +2sin(\theta_1+\theta_2))^2$

$\Rightarrow 2.74 = cos^2\theta_1+4cos^2(\theta_1+\theta_2)+ sin^2\theta_1 +4sin^2(\theta_1+\theta_2)+4cos\theta_1cos(\theta_1+\theta_2)+4sin\theta_1sin(\theta_1+\theta_2)$

$\Rightarrow 2.74 = 5+4(cos\theta_1cos(\theta_1+\theta_2)+sin\theta_1sin(\theta_1+\theta_2))$

$\Rightarrow 2.74 = 5+4cos\theta_2$

$\Rightarrow cos\theta_2 = -0.5650$

$\Rightarrow \theta_2 = 180^0\pm55.6^0$

124.4, 235.6

substitute these in:

COS

$\Rightarrow 0.7 = cos\theta_1+2cos(\theta_1+124.4);1.5 = sin\theta_1 +2sin(\theta_1+124.4)$

$\Rightarrow 0.7 = cos\theta_1+2(cos\theta_1cos124.4-sin\theta_1sin124.4);1.5 = sin\theta_1 +2(sin\theta_1cos124.4+cos\theta_1sin124.4)$

$\Rightarrow 0.7 = cos\theta_1-1.13cos\theta_1-1.65sin\theta_1;1.5 = sin\theta_1 -1.13sin\theta_1+1.65cos\theta_1$

$\Rightarrow 0.7 =-0.13cos\theta_1-1.65sin\theta_1;1.5 = -0.13sin\theta_1+1.65cos\theta_1$

solving the 2 equation:

$\Rightarrow sin\theta_1 = -0.493;cos\theta_1 = 0.870$

$\Rightarrow \theta_1 = -29.54^0$

and

COS

$\Rightarrow 0.7 = cos\theta_1+2cos(\theta_1+235.6);1.5 = sin\theta_1 +2sin(\theta_1+235.6)$

$\Rightarrow 0.7 = cos\theta_1+2(cos\theta_1cos235.6-sin\theta_1sin235.6);1.5 = sin\theta_1 +2(sin\theta_1cos235.6+cos\theta_1sin235.6)$

$\Rightarrow 0.7 = cos\theta_1-1.13cos\theta_1+1.65sin\theta_1;1.5 = sin\theta_1 -1.13sin\theta_1-1.65cos\theta_1$

$\Rightarrow 0.7 =-0.13cos\theta_1+1.65sin\theta_1;1.5 = -0.13sin\theta_1-1.65cos\theta_1$

$\Rightarrow \theta_1 = 159.55$

124.4, 235.6
$\theta_1 = -29.54,159.55$

a simple check gives the pairs of solutions: