Using the same set of parameters from last article, we have the plant transfer function
(1)with poles at . Let’s observe the Bode plot for this plant in Figure 1. We see the noticeable peak at frequency around 2.8 Hz, corresponding to the flexible mode of harmonic drive. The other peak at 16.7 Hz is more damped, and it happens at higher frequency so has less impact on the closed-loop performance.
- steady-state error less than 0.01 below 0.01 Hz
- high-frequency magnitude of close-loop is attenuated to 0.1 above 1 Hz
- phase margin at least 40 degrees
- below 0.01 Hz
- above 1 Hz
- has at least 40 degrees phase margin, or
(2)which does not quite meet the specs no 1 and 2, but is good enough for illustration purpose. The script hmd_lshape.sce gives you the plots for open and closed loop transfer functions in Figure 2 and 3, respectively. The low-frequency bound is violated slightly at the corner, and high-frequency bound at the peak of lower flexible mode.
ConclusionIn this article we discuss preliminary control design for the robot joint actuated by DC motor with harmonic drive, using the linear model developed from our previous work. Using the well-studied classical loopshaping design approach, we get a quite simple controller, with only one zero and two poles, that yields superior performance than a standard PID controller. This result is reasonable, since the PID is only a second-order transfer function with not much degree-of-freedom to compensate the undesirable plant dynamics. Indeed, the integral term of PID in this case is somehow useless. That’s why in  they often use only PD control.
Exercises1. Tweak the controller (2) to see if you can achieve one that does not violate any bound. 2. For the plant transfer function (1), one can get quite decent response using a controller with only one pole in the form
(3)Using the script file hmd_lshape.sce, find your choice of a and b that gives better response than from the PID controller. Hint: the magnitude response of already descends at slope -20 dB/decade below the first resonance peak. Where would you place the controller pole then?
Scilab Script and Model Files
The files can be downloaded from links embedded in the text. Both are also contained in this zip file: hmd_lshape.zip
- M.W.Spong, S. Hutchinson and M. Vidyasagar, Robot Modeling and Control. John Wiley & Sons. 2006.
- V. Toochinda, Robot Analysis and Control with Scilab and RTSX, Mushin Dynamics, 2014.