I must determine the resolution of a control system when using an 8-bit encoder for a single-rotation joint

The length of the link is 50 cm and the arm moves 180 degrees. Can anyone help me?

Out of the cosine theorem, $$d = \ell \sqrt{2\left(\ell-cos\right)}$$, I calculated the distance traveled by a linear axis and the rotation arc through the rotation connection.

The circumfence is $$2\pi R$$. And $$d$$, for my drawing is $$d=50\sqrt{2\left(50+1\right)}=50\sqrt{101}$$

The distance is 502 cm and the rotation is 9000 cm. From here, I don't know what to do. I can't find any equation for the resolution.

I only have the values that I said, and I need to determine the resolution for the control system, when an encoder of 8-bit is used. I don't have anything else..This is the whole exercise. I understand if it's not clear enough. It isn't for me, either.

• Comments are not for extended discussion; this conversation has been moved to chat. I'll edit the question to include any relevant details from the comments. Feb 7 '19 at 16:07

I think I understand what you need. You have an arm (length L) rotating (a maximum of Θmax radians) around an axis (similar to the hand of a watch), and the motion is generated by some motor. The motor's position is read by an encoder with a resolution of 8 bits.

We assume that the 8 bits cover the angles between 0 and 2*PI radians. In this case, the resolution of the encoder is:

resEnc=2*PI/(2^8)

The resolution of the angle of the arm would be:

res(Θ)=resEnc

In the same way, the resolution of movement of the tip of the arm is:

resTip = l * res(Θ)

Note: we do not know how the motor is connected to the arm and we do not know how the encoder is connected tot he motor.

Note2: I may have understood why d is needed :D

• "the arm moves 180 degrees" -> 8 Bit for 0-PI Feb 7 '19 at 14:47
• That is true only if you assume that there is a gear adaptation between motor and encoder. But it is usual (because it is easy and cheap) to connect encoder to motor coaxially. Without proper attention, one may not even notice that there is an encoder. Let alone the case when the motor has embedded encoded - the encoder is not even visible. Feb 7 '19 at 15:39
• Disclaimer: I have only very limited experience with encoders and mechanical design. Feb 7 '19 at 15:40
• @FooBar - This is a very sparse question, and there's no information on where the encoder is located. Absent a specific statement otherwise, though, I would assume the encoder is attached to the thing that is to be measured. That is, I would assume the encoder is attached at the joint where the arm is located. You could get better resolution attaching it to the motor but, as you mentioned in the comments on the question, this looks like a homework question and, more importantly, there's no information given regarding a gear ratio. Feb 7 '19 at 16:05

2*pi/(2^8) is correct..............

• Welcome to Robotics robotno. Thanks for your answer but we are looking for comprehensive answers that provide some explanation and context. Very short answers cannot do this, so please edit your answer to explain why it is right, ideally with citations. Answers that don't include explanations may be removed. Feb 17 '19 at 11:27
• I've left longer feedback on your other answer over here. I'm removing the "delete my answer" section here. I think your answer is correct, but you don't prove it. You just say "___ is correct." Please expand on this answer to teach the person asking the question. This is not a forum, it's intended to be a learning tool. Try to write answers that teach someone why the correct answer is what it is. Feb 17 '19 at 11:47