Dear Yaser Nabavi,

the distribution of error values for joint angles and joint angle velocities depend heavily on the underlying robot system, in detail:  the kinematics, the tolerances from manufacturing, the way the joints are controlled, the method of measuring the values for joint angle and joint angle velocity.

In addition, i do not think, that modelling the error by white noise is appropriate.

Do you really need "white noise", saying, a random value with an equally distributed (flat) spectrum?

Or, do you want to have the value distribution around the presumed values following a normal distribution (Gaussian distribution) ?

Can you provide us with the presumed reason for assuming noisy, thus incorrect measurements? Is the positioning of the joints noisy, or the measurement of the joint angles noise, incorrect? How is the joint angle velocity obtained? by physical measurement or by a numerically deriving the joint angles over time? 

If you have an industrial robot arm, the tolerances, and thus the residual errors,  are at least one order of magnitude smaller than using a self-made robot arm.

Without this information it will be hard to estimate any values.

Feel free to ask, if anything is unclear.

regards

Nils Goerke

Depends on the sensor system. Try to analyze yours from the sensor data-sheets. Or you can process the actual measurements if they're available.

The position angle noise is generally a function of the resolution of the encoder used.  When using an encoder, where the digital signal is kept clean, the "noise" is just the quantization.  The quantization is a function of the pulses-per-revolution of the encoder (ppr), and is uniformly distributed between +/- one pulse divided by 4 for quadrature decoding.  Thus, if you have a 500ppr encoder with quadrature decoding, you can expect the quantization to be less than 360/500/4 = 0.18 degrees. 

The joint velocity is measured in a number of ways.  It may involve the use of a tachometer, rate gyro, or may be based on the encoder readings.  These each have distinct noise patterns.  The encoder, for example can give fairly small errors ("noise"), but the value is based on the velocity.  In general, most velocity measurements suffer at low speeds (i.e., near zero).  The data-sheets are the way to go on this if they have the data.

Thanks all for valuable comments. I will present more information about my problem.

What which I'm working on is about developing a mathematical formulation for estimation of mass properties of a space-robot. (With applications like Active Space Debris Removal)

Up until now I developed the formulation and used numerical simulation to analyze its performance in estimation of mass properties of space robot elements assuming ideal measurements.

My new challenge is that how noise affects the convergence and accuracy of the results.

Considering the comments, accurate modeling of joints parameters measurement is professional and complicated task which is case dependent. I want to avoid such these complexities in this step of my research.

Considering complexity and cost of current space robots like Canadarms it seems trivial that they should be well equipped, but how?  I found no technical details about their measurements.

So what which I'm looking for is the simplest model for measurement of joint parameters of a space-robot with reasonable range of parameters to analyze the consistency of developed estimation algorithm with current technologies. As I’m not sure that these models are accessible in the case of space-robots, I think that the best at hand solution is to use appropriate models for ground-based robots instead.

I really appreciate it if someone provide suggestion or information with this problem.

Thanks again and excuse me for my lengthy text

Hi Yaser,

I'm quite familiar with research on space manipulator, having just completed a PhD thesis on active space debris removal. I have never heard of anyone caring about the measurement noise on the joint angles / velocities on space manipulators, except for the people (e.g., at MDA) involved in the low-level joint control systems. If you look at the system specifications overview for the SSRMS (Canadarm2):

http://adsabs.harvard.edu/full/1991NASCP3113...15K

You can see that the joint positions are measured by a redundant set of brushless joint angle resolvers (which give analog, absolute angle measurements), I don't know what the resolution of the ADC (analog-digital converter) is for those angle measurements, but the standard practice with resolvers is to have an ADC resolution well above the noise of the original analog signal. In other words, the error model for those measurements is simply down to +/- the resolution of the ADC used. I believe that the control frequency of the SSRMS is 3 kHz, but don't quote me on that, I'm not sure.

For the joint velocities, the SSRMS uses brushless motors (which are the only thing used in space manipulators, because brushed motors create way too much electro-magnetic interferences that could jeopardize the operation of other systems on-board the ISS or other spacecraft). The motor rate of turning can be measured directly from the commutations on the motor, which is what they use, and since the motor motion is highly gear-reduced (1845:1), this means that the resolution on the angular velocity measurement of the actual joint is ridiculously high.

All these facts combined, I would say that "noise" on the joint angle / velocity measurements is completely negligible. As I said, I have never seen anyone consider it in their analysis. Also, you have to consider that joints on space manipulators typically have reaction torque sensors as well, which, if you want to do estimation of mass properties, is going to be of great interest, needless to say.

So, in summary, it is pretty safe to assume that space manipulators, like the SSRMS and others (ERA, or JEMRMS), are extremely well-equipped when it comes to sensor systems.

However, you will notice that they all have accuracy specifications at the end-effector around 10 mm / 1 degree, which, if you are used to working with ground-based industrial manipulators, looks absolutely terrible. The reason for that end-effector accuracy is the flexibility of the links, which is a major source of problems when controlling or estimating anything related to a space manipulator. They are all very light-weight flexible manipulators and therefore, any motion you do on them is going to induce significant vibrations on the links, and thus, on the joint angles too. Therefore, joint angles / velocities might display a lot of "noise", but it's not measurement noise, it's physical vibrations of the links. And, it's obviously not white noise, as each link have their own discrete sets of vibrational modes with associated frequencies. So, in the overall system, the measurement noise on the joints is negligible compared to the vibrations of the links, so that should be your main concern.

If you really need specific details on the design specifications or noise levels at the joints, I suggest that you simply email people who work directly with those systems (e.g., people at MDA), that can point you to the best resources. Otherwise, I know that most industrial manipulators, especially Kuka robots, have the same type of setup at the joints to measure angle / velocity (angle from redundant resolvers, and velocity from pre-gear-reduced commutations on brushless motors), and usually controlled around 2 to 4 kHz and 16bit ADC resolution (if I remember correctly). So, these are extremely accurate and high sample-rate measurements, which is why I have never heard of anyone who was worried about that "noise".

Also, if you are going to be working on estimation of mass properties in relation to space manipulators and active debris removal, I highly suggest that you look up the PhD thesis of my ex-colleague Thai Chau Nguyen Huynh, who has done a lot of work on reactionless control and online estimation of mass properties of the manipulator's payload (space debris):

https://www.researchgate.net/profile/Thai_Chau_Nguyen_Huynh

http://digitool.library.mcgill.ca/thesisfile117066.pdf

And you might want to get into contact with Prof. Sharf, you has been supervising many students over the years and done a lot of work on various aspects on the active space debris removal problem with a robotic manipulator.

Good luck to you! I hope this was helpful!

Dear Mikael

Thanks really for your useful comments, suggestions and information. Although I didn't find direct detailed information about the measurement accuracies in Canadarm reference but its information are really helpful in my research. It seems that it needs more analyzes (like what which you did) and so I should work more on it.

I used the works which are supervised by Proff. Sharf as one of my most important references although I am interested in more contacts with her. 

Considering the type of formulations which I used in my research I hope that I would be able to extend it to elastic cases . But considering the complexities of elastic problem I preferred to start with simpler assumptions like ideal measurement. 

As you mentioned most references assumed that noise is negligible but In my research I think that I should be more careful about this assumption because that  I am using Force-Based approach which needs more measurements than most other researches which used Momentum-Based approach.

For example, in this approach, one of the parameters which should be measured or estimated (in parallel) is joint acceleration which as I know it can not be measured directly. May be using linear accelerometer one can estimate this parameter but its measurement is also susceptible to noise.

I will use the information which you provided in my researches and thanks again for your time  and useful comments and suggestions.