Quality of Computer animation relies on many things but good rigging is vital.  Without rigging animation would be, at the very best, very difficult to execute.   Imagine trying to manipulate a marionette (stringed puppet) without the strings.
Almost all animated entities within the 3D computer environment can benefit from some sort easy interface or harness that makes the job of animation easier.  The most obvious rig is a set of joints for a human character. Imagine trying to animate one without a jointed skeleton, it can be done but with great difficulty.   IK (Inverse Kinematics) is another typical tool that greatly enhances the work flow of animation.
Rigging a character like a human or animal figure or any entity that moves requires a lot more sophistication that just those of joint sets and IK handles.   An easy to use, flexible system that takes into account all the actions required for the animation and leaves nothing to chance is what a character rig should do.   The process of animation deals, in essence, with timing and action and this is difficult enough to deal with, without having to figure out the mechanics of the
character while coping with the choreography.
The role of the rig is to enable animation to be executed easily and be intuitive to use.
To give an extreme example of rigging in the visual effects world of feature film/tv, let us take the case of a car parked in a deserted street that suddenly explodes. It rises up into the air in a ball of flame with pieces of metal finally crashing down on all sides.   If that was done for real then the car might disintegrate but Iím afraid, the results would be very uninteresting. Stunts like these are carefully choreographed and carefully rigged.   There is a rig responsible for catapulting the car in the air, another to produce the ball of fire generated for visual impact and another for the pieces of metal carefully designed to break off with their own individual characteristics.   The mighty sound of the explosion is added afterwards of course, as the real sound of this event would hardly raise an eyebrow.   Although we may not be faking a visual effect, like the one above, the character rig should incorporate every detail of what is required from the action and the sequences that are to be embarked upon.

What is required of a good character rig?

It should include:-
Intuitive and easy to find control handles.
Simplicity in use but can incorporate complexity in outcome.
Robustness and reliability.
Ability to Reset. It needs to be capable of going back to it's default position easily.
Good documentation so that if a team is responsible for the rig then everyone is clear as to what and where everything is.

What should be rigged?

Human or animal character that has to animate or act.
Prop or normally inert object which interacts with or is used by a character..
Mechanical objects like a childís swing or a steam engine.
Any geometry which requires more than just transformations..
Banks or arrays of items like an array of lights or large groups of props which canít be individually choreographed.

What is required to rig in Maya?

A comprehensive brief of what the rigged character or entity has to do within the animation sequences.
Storyboards are good for this.

Clean and error free geometry:-
    with no construction history,
    with no transform values on the geometry node or its hierarchy,
    which should be aligned and positioned precisely on the origin,
    which is properly named
    with meaningfull pivot points
    which has a well planned hierarchical structure.

Blend shapes nodes.
Shape changing nodes like the lattice, wire, cluster and non linear deformers
Joint sets.
IK handles and spline IKs
Soft Skinning.
Curves or other entities for the construction of handles.
Constraint nodes
Additional attributes for specialist channels.
Utility nodes
Feedback interfaces if necessary.
Other support nodes like locators and distance nodes

Here is an example of a simple rig brief:-
Character = Floor brush:
What is it used for? = brushing the floor.
Who uses it? = character A.
How does character A hold the brush? = in both hands
Does character A ever put the brush down? = yes
Is the brush used for anything else? = no

Example of work flow, to build the floor brush rig:-
Rig brush bristles to flex when in contact with the floor.
Arrange for character A's hands to be selectively constrained to the brush.

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