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In robotics, holonomicity refers to the relationship between the controllable degrees of freedom and total degrees of freedom of a given robot (or part thereof).

  1. Holonomic system:
    If the controllable degrees of freedom is equal to the total degrees of freedom then the robot is said to be holonomic.
  2. Non-holonomic system:
    If the controllable degrees of freedom are less than the total degrees of freedom it is non-holonomic.
  3. Redundant system:
    A robot is considered to be redundant if it has more controllable degrees of freedom than degrees of freedom in its task space.

A redundant system is also called holonomic. But a holonomic system may not be redundant.

Non-holonomic example:

An automobile (a car) is an example of a non-holonomic vehicle. The vehicle has three degrees of freedom—its position in two axes, and its orientation relative to a fixed heading. Yet it has only two controllable degrees of freedom—acceleration/braking and the angle of the steering wheel—with which to control its position and orientation.

The resulting phenomenon is: the velocity of the car is always the same or inverse as the orientation of the car, if there is no skidding or sliding. Thus, not every path in phase space is achievable. The non-holonomicity of a car makes parallel parking and turning in the road difficult.

Remark: we just compare the number of DOF. A controllable state may not be one of the total state. For example, the state of a care is position and orientation angle. The controllable state is acceleration and orientation (input). The holonomic property is only determined by the number.

Holonomic & Redundant example:

A human arm, by contrast, is a holonomic, redundant system because it has seven (controllable) degrees of freedom (three in the shoulder – rotations about each axis, two in the elbow – bending and rotation about the lower arm axis, and two in the wrist, bending up and down (i.e. pitch), and left and right (i.e. yaw)) and there are only six physical degrees of freedom in the task of placing the hand (x, y, z, roll, pitch and yaw), while fixing the seven degrees of freedom fixes the hand. See also sub-Riemannian geometry for a discussion of holonomic constraints in robotics.


Edit: The holonomic, non-holonomic and redundant  look the same as fully-actuated, under-actuated and over-actuated.


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