## Question about units when calculating arm end-effector velocity

## Question about units when calculating arm end-effector velocity

(OP)

I have a data set for a simulated arm moving in a single plane (as if sliding along a tabletop). The arm has 2 joints (shoulder and elbow). My data set includes the shoulder and elbow joint angles (in radians) at each timestep.

My goal is to calculate the end-effector velocity of the hand/wrist of this modeled arm. So far, I have been using the Jacobian matrix for this system, using the following:

velocity_x = dx/dt = [-L1*sin(theta_sh) - L2*sin(theta_sh + theta_elb)]d(theta_sh)/dt - L2*sin(theta_sh + theta_elb)*d(theta_elb)/dt

velocity_y = dy/dt = [ L1*cos(theta_sh) + L2*cos(theta_sh + theta_elb)]d(theta_sh)/dt + L2*cos(theta_sh + theta_elb)*d(theta_elb)/dt

where theta_sh is shoulder angle (in radians), theta_elb is elbow angle (in radians), L1 is the length (in meters) of the upper arm segment, and L2 is the length (in meters) of the forearm segment.

After I have calculated the individual x and y velocities, I take the square root of the sum of squares to calculate the overall curvilinear velocity of the hand/wrist.

From my literature reviews, I would expect a human-like arm to have a curvilinear peak velocity in the range of ~0.1 to 3 meters/second (or ~10 - 300 cm/sec). Currently, when I use the units mentioned above, my resulting peak velocity values are around 0.04, which is clearly not consistent with either m/s or cm/s units.

If I multiply the resulting curvilinear velocity values by (180/3.14159), this brings them into a reasonable range, although I'm not sure if this adjustment is justified.

If I convert the joint angles to degrees and perform the calculations, the resulting velocity values are in the correct range (for meters/second units), but the plots look jagged, whereas the velocity plots calculated using joint angles in radians are smooth, as expected.

Does anyone have advice about how I should convert my units in order to yield an end-effector velocity in the range of ~0.1 to 3 meters/second? My data have been vetted for validity, so this is not a problem with the data set itself.

Thanks in advance for your guidance!

My goal is to calculate the end-effector velocity of the hand/wrist of this modeled arm. So far, I have been using the Jacobian matrix for this system, using the following:

velocity_x = dx/dt = [-L1*sin(theta_sh) - L2*sin(theta_sh + theta_elb)]d(theta_sh)/dt - L2*sin(theta_sh + theta_elb)*d(theta_elb)/dt

velocity_y = dy/dt = [ L1*cos(theta_sh) + L2*cos(theta_sh + theta_elb)]d(theta_sh)/dt + L2*cos(theta_sh + theta_elb)*d(theta_elb)/dt

where theta_sh is shoulder angle (in radians), theta_elb is elbow angle (in radians), L1 is the length (in meters) of the upper arm segment, and L2 is the length (in meters) of the forearm segment.

After I have calculated the individual x and y velocities, I take the square root of the sum of squares to calculate the overall curvilinear velocity of the hand/wrist.

From my literature reviews, I would expect a human-like arm to have a curvilinear peak velocity in the range of ~0.1 to 3 meters/second (or ~10 - 300 cm/sec). Currently, when I use the units mentioned above, my resulting peak velocity values are around 0.04, which is clearly not consistent with either m/s or cm/s units.

If I multiply the resulting curvilinear velocity values by (180/3.14159), this brings them into a reasonable range, although I'm not sure if this adjustment is justified.

If I convert the joint angles to degrees and perform the calculations, the resulting velocity values are in the correct range (for meters/second units), but the plots look jagged, whereas the velocity plots calculated using joint angles in radians are smooth, as expected.

Does anyone have advice about how I should convert my units in order to yield an end-effector velocity in the range of ~0.1 to 3 meters/second? My data have been vetted for validity, so this is not a problem with the data set itself.

Thanks in advance for your guidance!

## RE: Question about units when calculating arm end-effector velocity

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## RE: Question about units when calculating arm end-effector velocity

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## RE: Question about units when calculating arm end-effector velocity

This should not be the case. Since you are taking derivatives, the result should be choppy, since any data errors are magnified.

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## RE: Question about units when calculating arm end-effector velocity

In response to IRstuff:

>> "but the plots look jagged, whereas the velocity plots calculated using joint angles in radians are smooth, as expected."

>> This should not be the case. Since you are taking derivatives, the result should be choppy, since any data errors are magnified.

I have attached a PDF that shows 3 versions of the plot, each created using a different strategy. I'm not sure if it's accurate to categorize the third plot (the one I described as "jagged") as "choppy"; I suspect that the qualitative differences between plot 3 vs. plots 1 and 2 is the derivative being taken on units of radians vs. degrees.

In response to IceBreakerSours:

>> What sort of analysis are you carrying out?

I'm comparing the performance of different controllers for this simple arm system. Right now, I am just working with a single, 100-timestep data set so that I can determine the proper processing method for these velocity plots, but eventually, I'll be processing large sets of numerous tasks. My ultimate goal here is to take my joint angle data and to determine, with correct units, the end-effector velocity of the hand/wrist, so that I can contrast different controllers' behavior on this performance characteristic.

The PDF I attached also includes my Matlab code. It seems that the first plot in this attachment is reasonable in its appearance (using units of m/s), but I am just wondering if I am justified in multiplying the tiny resulting velocities I get from my angles-in-radians calculations by (180/pi) to convert to degrees, *after* the velocity processing has been complete... Thanks for any additional insights anyone has.

## RE: Question about units when calculating arm end-effector velocity

q = jointAngles(:,3:6) assumes RADIANS in the table

q = jointAngles(:,3:6)*(3.14159/180.0) assumes degrees in the table, since you get deg * rad/deg = rad

Likewise, had you read the manual, you'd see that sin and cos require arguments in RADIANS:

http://www.mathworks.com/help/symbolic/mupad_ref/s...

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## RE: Question about units when calculating arm end-effector velocity

My fundamental question remains: when I do use the joint angles in radians to calculate the endpoint-effector velocity, the resulting velocities have tiny values (e.g. peak velocity value of a typical movement is 0.037), which small magnitudes do not correspond to any typical units used to report velocities (e.g. meters/second or cm/second).

Is it legitimate to take the small velocity values that result from my calculations, and multiply them by (180/pi) to yield velocities in units of meters/second? If that is not legitimate, how should I go about 'translating' my small velocity values into physiologically-meaningful units?

Thanks for your input!

## RE: Question about units when calculating arm end-effector velocity

But, I see your problem, now. This is one of the reasons I use Mathcad, which understands and tracks units. You did a difference function and called it a "derivative." As you probably tecall from calculus, since you did the symbolic math correctly, is that a derivative is the ratio of differentials. So, your derivatives need to include the difference in time. In Mathcad, the result would have been glaringly obvious, because the answer would only have meters instead of meters/second.

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## RE: Question about units when calculating arm end-effector velocity

## RE: Question about units when calculating arm end-effector velocity

Thanks again for your responses, IRstuff and IceBreakerSours; your posts really helped me to work through this issue to the correct solution.