6. System Evaluation
6.1. System Mechanical Responses
6.1.1. Step Response
The response of the system was first measured for step inputs. The following graphs show the system’s responses when the (x,y) components of the desired position were switched from (900,0) to (400,0) and vice versa. Since no change in the y position was required, these step inputs isolated the linear subsystem from the rotational subsystem and was able to evaluate linear motion alone.
The graphs show that the linear subsystem was able to reach a steady-state error of less than 10 mm, and that the response was nearly identical with and without the 5-pound load. Even though the carriage was able to travel the entire range of linear motion in roughly one second, the position vs. time graphs make the response look somewhat slow, particularly immediately
following the step input. The graphs below show velocity vs. time data for the same responses.
0 200 400 600 800 1000
0.00 0.50 1.00 1.50
X-Position (mm)
Time (s)
Position Response to Step Input in X-Direction
Step Input 0 lbs 5 lbs
0 200 400 600 800 1000
0.00 0.50 1.00 1.50
X-Position (mm)
Time (s)
Position Response to Step Input in X-Direction
Step Input 0 lbs 5 lbs
Figure 6-1: Position vs. Time Responses of the Linear System to Step Inputs in X-Direction 880
890 900 910
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These velocity vs. time graphs help show that the slow start to the step response is due to the acceleration limits placed on the motion by the controller. The true velocity follows the ideal velocity very closely, this ideal velocity is just limited by comfort constraints rather than
mechanical constraints.
The below graphs show the system’s responses when the (x,y) components of the desired position were switched from (700,300) to (700,0) and vice versa. Again, these points were selected as an attempt to isolate rotational motion from linear motion. While this motion does in fact require some movement along the linear subsystem, since the controller operates in the x-y coordinate system this step input was used as an approximation of rotational movement alone.
-1,000 -800 -600 -400 -200 0 200
0.00 0.50 1.00 1.50
X-Velocity (mm/s)
Time (s)
Velocity Response to Step Input in X-Direction
Ideal 0 lbs 5 lbs
-200 0 200 400 600 800 1,000
0.00 0.50 1.00 1.50
X-Velocity (mm/s)
Time (s)
Velocity Response to Step Input in X-Direction
Ideal 0 lbs 5 lbs
-100 0 100 200 300 400
0.00 1.00 2.00
Y-Position (mm)
Time (s)
Position Response to Step Input in Y-Direction
Step Input 0 lbs 5 lbs
-100 0 100 200 300 400
0.00 0.50 1.00 1.50 2.00
Y-Position (mm)
Time (s)
Position Response to Step Input in Y-Direction
Step Input 0 lbs 5 lbs Figure 6-2: Velocity vs. Time Responses of Linear System to Step Inputs in X-Direction
Figure 6-3: Position vs. Time Responses of System to Step Inputs in Y-Direction
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It is clear from the response data that additional load has a larger impact on rotational motion than on linear motion. With a load of 5 lbs, the system takes roughly two seconds to travel the range of rotational motion. Although the response to a step input in the y-direction is somewhat slower, the steady-state error is still less than 15 mm, which is essential for the ultimate goal of assisting the user in reaching the desired location. The velocity data of the responses in the y-direction are shown below.
The disorganized data is a result of the relatively low resolution of the Arduino ADC combined with the small change in the rotational potentiometer’s voltage. The velocity
responses in the y-direction show that the true velocity in the y-direction slightly lags the desired velocity, and with an additional load isn’t able to accelerate as quickly as desired. This may be addressed more directly in the future with better motor equations and in-the-loop moment of inertia calculations, but the y-direction step responses show the system is able to reach minimal steady-state error, which is the main concern for the device’s purposes.
-400 -200 0 200 400 600 800
0.00 1.00 2.00
Y-Velocity (mm/s)
Time (s)
Velocity Response to Step Input in Y-Direction
Ideal 0 lbs 5 lbs
-800 -600 -400 -200 0 200 400
0.00 0.50 1.00 1.50 2.00
Y-Velocity (mm/s)
Time (s)
Velocity Response to Step Input in Y-Direction
Ideal 0 lbs 5 lbs
Figure 6-4: Velocity vs. Time Response of System to Step Inputs in Y-Direction
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The following graphs show the system’s responses when the (x,y) components of the desired position were switched from (900,0) to (600,250) and vice versa. This step input was used to evaluate the combination of linear and rotational motion.
The graphs show that the system as a whole reaches a steady-state error of less than 10mm, and that the velocity is fairly well driven to its desired value. The graph below shows the trajectory of the end-effector during the entire system step response.
0 100 200 300 400 500
0.00 0.50 1.00 1.50
Position (mm)
Time (s)
Position Response to Step Input in Both Directions
Step Input 0 lbs 5 lbs
0 100 200 300 400 500
0.00 0.50 1.00 1.50 2.00
Position (mm)
Time (s)
Position Response to Step Input in Both Directions
Step Input 0 lbs 5 lbs
0 200 400 600 800 1,000
0.00 0.50 1.00 1.50
Velocity (mm/s)
Time (s)
Velocity Response to Step Input in Both Directions
Ideal 0 lbs 5 lbs 0
200 400 600 800 1,000
0.00 0.50 1.00 1.50
Velocity (mm/s)
Time (s)
Velocity Response to Step Input in Both Directions
Ideal 0 lbs 5 lbs
Figure 6-5: Position vs. Time Responses of System to Step Inputs in Both Directions
Figure 6-6: Velocity vs. Time Response of System to Step Inputs in Both Directions
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The graphs show a slight delay in rotational motion to linear motion. Although the trajectory does seem to be more disrupted with an additional load, the trajectories on the whole appear to be fairly direct.