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is an example of a trivially underactuated system: it has two degrees of freedom (one for its support's motion in the horizontal plane, and one for the angular motion of the pendulum), but only one of them (the cart position) is actuated, and the other is only indirectly controlled. Although
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does not explicitly represent the number of actuators present in the system. Indeed, there may be more actuators than degrees of freedom and the system may still be underactuated. Also worth noting is the dependence of
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imposed by the wheels. That is, a car cannot accelerate in a direction perpendicular to the direction the wheels are facing. A similar argument can be made for boats, planes and most other vehicles.
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dictate that the dynamics of mechanical systems are inherently second order. In general, these dynamics can be described by a second order
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To understand the mathematical conditions which lead to underactuation, one must examine the dynamics that govern the systems in question.
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When this condition is met, there are acceleration directions that can not be produced no matter what the control vector is.
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31:. This condition can occur for a number of reasons, the simplest of which is when the system has a lower number of
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Furthermore, in many cases the dynamics for these systems can be rewritten to be affine in the control inputs:
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N. Dechev, W.L. Cleghorn, and S. Naumann, “Multiple Finger, Passive
Adaptive Grasp Prosthetic Hand,”
626:. That is, there may exist states in which an otherwise fully actuated system becomes underactuated.
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M. Saliba, and C.W. de Silva, "An
Innovative Robotic Gripper for Grasping and Handling Research,"
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The class of underactuated mechanical systems is very rich and includes such diverse members as
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to describe mechanical systems that cannot be commanded to follow arbitrary trajectories in
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Underactuated
Robotics: Learning, Planning, and Control for Efficient and Agile Machines
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On the concept of virtual constraints as a tool for walking robot control and balancing
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naturally extremely unstable, this underactuated system is still controllable.
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When expressed in this form, the system is said to be underactuated if:
361:{\displaystyle {\ddot {q}}=f_{1}(q,{\dot {q}},t)+f_{2}(q,{\dot {q}},t)u}
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College of
Mechanical and Nuclear Engineering, Kansas State University
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Annual
Reviews in Control, 28 (2004), pp. 157–166. (Elsevier)
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642:A standard automobile is underactuated due to the
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141:{\displaystyle {\ddot {q}}=f(q,{\dot {q}},u,t)}
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584:{\displaystyle f_{2}(q,{\dot {q}},t)}
526:{\displaystyle f_{2}(q,{\dot {q}},t)}
216:{\displaystyle u\in \mathbb {R} ^{m}}
180:{\displaystyle q\in \mathbb {R} ^{n}}
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19:is a technical term used in
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702:, pp. 975–979, 1991.
644:nonholonomic constraints
68:Newton's laws of motion
41:trivially underactuated
675:Tedrake, Russ (2008).
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54:, and even
48:automobiles
737:Categories
662:References
476:Note that
62:Definition
611:˙
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341:˙
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244:is time.
199:∈
163:∈
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52:airplanes
33:actuators
650:See also
630:Examples
21:robotics
150:Where:
56:animals
682:(PDF)
35:than
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23:and
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.