# Inverse Kinematics Computation -- why are alpha angle values not included?

Robotics Asked by hey on January 6, 2022

Given a desired transform matrix of the end effector relevant to the base frame of the P560: John J. Craig, in his book, Introduction to Robotics
Mechanics and Control, computes the inverse kinematic solutions of a Puma 560, with (correct me if wrong) Modified DH parameters and gets the following set of equations for theta angles: and I noticed that there are no alpha angles in these calculations.

So, my question is why aren’t the alpha angle values not used in the calculation for the desired pose with the given end effector transform? Why is it independent of the axes twist angles of the robot?

Check out Equation 3.6:

$$^{i-1}_iT = left[ begin{array}{c} ctheta_i & -stheta_i & 0 & r_{i-1} \ stheta_icalpha_{i-1} & ctheta_icalpha_{i-1} & -salpha_{i-1} & -salpha_{i-1}d_i \ stheta_isalpha_{i-1} & ctheta_isalpha_{i-1} & calpha_{i-1} & calpha_{i-1}d_i \ 0 & 0 & 0 & 1 end{array} right]$$

and Figure 3.21, "Link parameters of the PUMA 560":

$$begin{array}{c} i & alpha_{i-1} & r_{i-1} & d_i & theta_i \ \ 1 & 0 & 0 & 0 & theta_1 \ 2 & -90^{circ} & 0 & 0 & theta_2 \ 3 & 0 & a_2 & d_3 & theta_3 \ 4 & -90^{circ} & a_3 & d_4 & theta_4 \ 5 & 90^{circ} & 0 & 0 & theta_5 \ 6 & -90^{circ} & 0 & 0 & theta_6 \ end{array}$$

Here I've used $r$ instead of $a$ to prevent any confusion between $alpha$ and $a$. In Equation 3.6, $s$ means $sin$ and $c$ means $cos$.

To your question, $alpha$ is more related to the orientation of two joints relative to each other as opposed to the motion of the joints relative to each other. That is, $alpha$ is generally fixed by physical construction and thus, as a parameter, isn't time-dependent.

So hopefully you can see that, from Figure 3.21, all of the $alpha$ values are either +/- 90 degrees or zero. Equation 3.6 then uses it only in sine and cosine terms, meaning that they can all be reduced to +/-1 or zero.

I really, really hate when authors jump through simplifications like this without a heads up, so I totally understand the confusion.

Answered by Chuck on January 6, 2022

## Related Questions

### How to design a robot chassis so as to avoid bending issues?

1  Asked on October 3, 2021 by jash

### Can I use a breadboard to power multiple SG90 servos?

0  Asked on October 3, 2021

### Moment of Inertia of a Dumbbell

2  Asked on October 3, 2021 by kucar

### Are there any tutorials for OMPL in Python?

1  Asked on October 3, 2021 by batini

### Arduino and stepper motor

2  Asked on February 19, 2021

### Is it OK to apply too many volts to Actobotics HD motors?

1  Asked on February 9, 2021

### Is the covariance matrix in the extended Kalman filter guaranteed to be positive definite (ignoring numerical errors)?

1  Asked on January 20, 2021

### Pictures coordinates to robot coordinates

1  Asked on January 14, 2021 by dairon

### Not able to communicate with Roomba with serial port

1  Asked on January 9, 2021 by aalian-khan

### Directly tapping Create2 battery through vacuum brush

2  Asked on January 6, 2021 by xabigarde

### How to obtain the state space representation of a computed torque control system?

3  Asked on December 27, 2020 by james_erikson

### Is it compulsary to learn Arduino programming in order to use Udoo Neo as an IOT device?

1  Asked on December 24, 2020 by farhan-ahmed

### Path planning algorithm better than RRT* and Artificial Potential Field

2  Asked on December 21, 2020 by nikthebrick

### How can I allow for this code to switch from remote control to autonomous?

1  Asked on December 15, 2020 by stackcode12

### Not getting expected performance from kalman filter+mahalanobis distance

1  Asked on December 14, 2020

### A Delivery Drone

1  Asked on December 6, 2020 by will

### How to convert vertical motion to horizontal?

5  Asked on December 4, 2020 by boardbite

### What should be the required operating voltage for Servo motors in my 5 DOF robotic arm, with 50 kg payload

0  Asked on November 18, 2020 by muzammil-ibrahim

### Monobrick Communication Library-reading nxt sensors

2  Asked on October 31, 2020 by jpcreeper13

### Res-Scaling method beats PID controller. How?

1  Asked on October 26, 2020 by pe-dro