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What's the difference between an underactuated system, and a nonholonomic system? I have read that "the car is a good example of a nonholonomic vehicle: it has only two controls, but its configuration space has dimension 3.". But I thought that an underactuated system was one where the number of actuators is less than the number of degrees of freedom. So are they the same?

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They are different things. An underactuated system does mean that the number of independent control inputs is fewer than the number of degrees of freedom you are trying to command. This can happen for holonomic systems when they encounter a singularity, or for nonholonomic systems when they are commanded to move in a direction they cannot achieve. The state space method of looking at this is the controllability matrix, and in kinematics it is the rank of the Jacobian matrix, but you probably don't need to go into that level of detail to understand an underactuated system.

Nonholonomic means quite a different thing. A holonomic system can have its control kinematics expressed as geometric (including trigonometric) equations. A nonholonomic system, on the other hand, cannot be expressed as closed-form geometric equations. Instead, the best you can do is to write differential equations to express the kinematics.

Here's an example: think of a mobile robot with wheels. If you ask it to ascend to 10 meters in the air, it is underactuated. If you ask it to go to position (100 meters, 20 meters) from its current location, then the robot needs to turn its two drive wheels to get to the new position. Since it is a nonholonomic system, you cannot just count the number of rotations of each wheel to determine where it ended up, because the sequence of rotations matters, not just the total number of rotations (for example, it would end up in a different place if it actuated only the left wheel for so many rotations, then the right. as compared to if it actuated both wheels together for that many rotations). For a robot manipulator that is holonomic, you can locate it's end effector position by knowing the total number of degrees of rotation of each motor, and you don't need to care in what order the robot moved its axes.

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    $\begingroup$ Great answer SteveO, I wonder if it would be clearer if we could generate a table of examples with Holonomic/Nonholonomic on one axis, and Underactuated/Trivially underactuated/Fully actuated on the other. $\endgroup$ – Mark Booth Sep 13 '16 at 11:25
  • $\begingroup$ Comment from Matthew T Watson (not enough rep to comment): I have issue with the following comment Since it is a nonholonomic system, you cannot just count the number of rotations of each wheel to determine where it ended up, because the sequence of rotations matters, not just the total number of rotations (for example, it would end up in a different place if it actuated only the left wheel for so many rotations, then the right. as compared to if it actuated both wheels together for that many rotations). $\endgroup$ – Chuck Jan 3 '19 at 15:07
  • $\begingroup$ If this is the case, then surely that would mean any omnidirectional wheeled robot is nonholonomic, as a different number of wheel rotations would occur if the robot moved 1m forward versus performing a 90 degree rotation, moving 1m sideways, and performing a -90 degree rotation. Yet, these robots are usually referred to as holonomic, even see the wikipedia article on holonomy in robotics. I feel like either this post or this commonly used statement are erroneous, but I am not personally sure what the correct answer should be. $\endgroup$ – Chuck Jan 3 '19 at 15:08
  • $\begingroup$ Can you comment on this SteveO? $\endgroup$ – Chuck Jan 3 '19 at 15:08
  • $\begingroup$ Wiki is correct. My answer is for typical wheeled robots (one DOF wheels) each. With two such drive wheels, the robot cannot instantaneously move in a direction other than the direction it is facing. Yet, by coordinating the sequence of wheel rotations, as in parallel parking, it can end in a position perpendicular from its starting point (using an infinite number of paths). The Omni wheels do not require this motion - they can directly move in two independent directions instantaneously, and therefore can be described kinematically, without needing the differential motion equations. $\endgroup$ – SteveO Jan 3 '19 at 22:17
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Underactuation leads to nonholonomic constraints (non-integrable kinematic or dynamic equations) in mechanical systems.

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  • $\begingroup$ On Stack Exchange we are are looking for comprehensive answers that provide some explanation and context. Very short answers cannot do this, so please edit your answer to explain why it is right, ideally with citations. Answers that don't include explanations may be removed. What does your answer add that wasn't already explained by SteveO? $\endgroup$ – Mark Booth Sep 13 '16 at 11:23

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