Robot w/ 2 Links: Finding Max. Endpoint Error & Bit Resolution

[engineer23]

Homework Statement

I have a device with two links (see attached diagram). The device uses the joint angles to determine the position of the end effector (x,y).

a) Potentiometers are 2% accurate. Find the maximum endpoint error for the device.

b) If you use optical encoders, how many bits are needed to make the endpoint error less than 2 mm?

Homework Equations

Based on the diagram,
I have x = .5 cos (theta1) - .5 cos (theta1 + theta 2) and
y = .5 sin (theta1) - .5 sin(theta1+theta2)

Is my trig right?

2% pot. means that accuracy is +/- 7.2 degrees (since 2% of 360 is 7.2)

The Attempt at a Solution

For part a, do I just choose two arbitrary angles, add and find the difference between the actual x,y position and the x,y position at max error? (ex. choose 30 and 120 as joint angles, calculate position, then recalculate with 37.2 and 127.2, and find difference between position values)

I am a little confused on b. I want resolution to be 2 mm. How do I find max/min (x,y) position? Range/resolution = 2^n...solve for n?

 

[KataKoniK]

it is important to approach problems with a systematic and logical approach. In this case, we have a device with two links and we want to determine the maximum endpoint error and the number of bits needed for a certain resolution.

First, let's address part a. The trigonometric equations you have written for x and y appear to be correct. To find the maximum endpoint error, we can use the given accuracy of the potentiometers (2%) and the fact that it corresponds to an angle of +/- 7.2 degrees. This means that for any given joint angle, the actual angle could be off by +/- 7.2 degrees. Using this information, we can find the maximum endpoint error by calculating the difference between the actual position and the position at the maximum error angle for each joint angle. As you mentioned, we can choose two arbitrary joint angles and calculate the difference to get an idea of the maximum endpoint error. However, to get a more accurate result, it would be better to calculate this for all possible joint angles and take the maximum value.

Moving on to part b, we want to determine the number of bits needed for the endpoint error to be less than 2 mm. To do this, we can use the formula you mentioned: Range/resolution = 2^n. In this case, the range would be the maximum endpoint error we calculated in part a (in meters), and the resolution would be 2 mm (0.002 meters). Solving for n will give us the number of bits needed for the desired resolution. However, keep in mind that this is just an estimation and the actual number of bits needed may vary depending on the specific system and its accuracy.

In conclusion, it is important to approach problems step by step and using the given information. In this case, we used the given accuracy of the potentiometers and trigonometric equations to determine the maximum endpoint error, and then used this value to calculate the number of bits needed for a certain resolution. It is always good to double check your calculations and make sure they make sense in the context of the problem.

 

[piercas]

I would approach this problem by first clarifying the context and assumptions of the device and its use. From the diagram, it appears that the device is a robotic arm with two links that can move in two dimensions to position the end effector (x,y). The joint angles are used to determine the position of the end effector, and the accuracy of this determination is dependent on the type of sensors used.

In part a, the homework statement mentions the use of potentiometers with 2% accuracy. This means that the measured joint angles can vary by +/- 2% of the actual angle. Using trigonometry, we can calculate the maximum possible error in the position of the end effector by choosing two arbitrary joint angles and finding the difference between the actual position and the position at maximum error. However, it may be more useful to determine the maximum error across all possible joint angles, as this will give a more comprehensive understanding of the accuracy of the device.

In part b, the homework statement asks for the number of bits needed for the endpoint error to be less than 2 mm when using optical encoders. This would require calculating the maximum and minimum possible positions of the end effector, and then using the given resolution of 2 mm to determine the number of bits needed for the encoder. This calculation will also depend on the range of motion of the device and the precision of the encoder.

Overall, the trigonometry presented in the homework statement appears to be correct, but it is important to consider the context and assumptions of the device in order to fully understand and solve the problem.