1. Position Analysis

For this section, the input gear reduction ratio will be ignored so the input is assumed to be the 'L' link 2 position.

Figure 3, animation of the mechanism in motion.

Derivation

The system as mentioned earlier consists of a driving crank slider that is coupled with a driven clank slider that manipulates the rest of the structure.

Figure 4, Initial linkages of the clank sliders.

In order to find the angles of links 4 and 5, I need to find the position of point C, the sliding linkage.

Driver Vector Loop

$\vec{R}_2 + \vec{R}_3 - \vec{R}_1 = \vec{0}$

$a e^{j\theta_2} + b e^{j\theta_3} - c_x - c_y = 0$

(Eq. 1)

(Eq. 2)

Where C_yis a constant value and C_x is a function of θ₂. This loop was put into a math solver to find C_x .

Driven Vector Loop

$\vec{R}_4 - \vec{R}_5 - \vec{R}_1 = \vec{0}$

$de^{j\theta_4} - fe^{j\theta_5} - c_x - c_y = 0$

(Eq. 3)

(Eq. 4)

Which can again be put into a solver to find the proximal angle, θ₄, and the distal angle, θ₅

These angles are important because the rest of the mechanism is parallel to links 4 and 5. This means no other vector loops are necessary.

End Effector Position

Figure 5, all linkages of the mechanism (not to scale)

Solving for the end effector is done by using links 4 and 5 as building block vectors, defining point A as origin, and defining point C as a function of θ₂.

$\vec{E} = ge^{j\theta_4}$