Robotic's lab 5: inverse kinematics

Inverse kinematics

We use the strategy to define a wirst and use the geometric method. In order to do that, we use the next isometric view:

As we can see, the first joint can be calculated as:

$q_1=atan2(y,x)$

Then, we proceed with the thrid joint. As the method say we proyect the vector to the wirst on the x2-y2 plane. We get:

First, we note that $L=\sqrt((z_c-L_1)^2+x_c^2+y_c^2)$

We construct a triangule with the sides $L,L_2$ and $L_3$. From this, we can use the cosine theorem and get:

$L^2 = L_2^2 + L_3^2 - 2L_2L_3cos(180-q3)$

We solve the equation for $cos(q_3)$ and finally get the next equation system:

$c_{q3}=\frac{L^2-(L_2^2+L_3^2)}{2L_2L_3}$

$s_{q_3}=\sqrt{1-c_{q3}}$

$q_3=atan2(s_{q3},c_{q3})$

Then we examine the second and fourth joint. For this purpouse we use the next proyection:

From the image:

$90=q_2+\alpha+\gamma \implies q_2=\alpha+\gamma-90$

For $\alpha$ we construct a right triangule and use $atan2$:

$\alpha = atan2(z_c-L_1,\sqrt(x^2+y^2))$

As above, we use another right triangule for $\gamma$:

$\gamma = atan2(L_3s_{q3},L_2+L_3c_{q3})$

Finally for the last joint we know that the sum of every joint angle is the desired orientation:

$\theta = 90 + q_2 + q_3 + q_4$

We solve for $q_4$:

$q_4 = \theta -(90 + q_2 + q_3)$

Video:

Conclusions

Contributors

• Juan Sebastian Duenas ([email protected])
• German Andres Urbina Gutierrez ([email protected])
• Brayan Daniel Barrera Galvis ([email protected])

Reference