Joint Types

As shown in the above figure, joints can be grouped into lower order pairs having one DoF and higher order pairs offering two or three DoF. For serial manipulators (a.k.a., robot arm or kinematic chain), the most common joint types are revolute and prismatic allowing one rotational and one translational DoF, respectively.

As seen in the previous exercise, two rigid bodies in free space connected by a one DoF revolute joint would only have a total of 2*6 - 5 = 7 DoF. For modeling purposes, higher order pairs can always be replaced by a collection of lower order pairs, so without loss of generality, we can restrict our attention to only the revolute and prismatic 1 DoF joint types.

Let’s look at another example to make this concept more explicit. Let's define the number of 1-DoF joints to be equal to n.

The serial manipulator shown here has n=3 joints: 2 revolute (cylinders) and 1 prismatic (square). Each joint connects two links so the total number of links is n+1 = 4. Notice that the revolute described by \theta_1 connects the ground (i.e., fixed “base link”) to the link between joints 1 and 2. Thus the total number of DoF for any serial manipulator with three 1-DoF joints is:

The takeaway is this: For serial manipulators with only revolute and/or prismatic joints, the number of degrees of freedom is always equal to the number of joints. The exception to this rule is when both ends of the manipulator are fixed (closed chain linkage), as shown in the image below.

If a manipulator has more DoF than is required for its given task, it is said to be kinematically redundant or Overconstrained. For example, if a three DoF arm is used to locate a rigid body on a plane, it would have one degree of redundancy since only two generalized coordinates are needed to locate a point on a plane. However, to control both the position and orientation of a rigid body on a plane, a manipulator with at least three DoF is required.

Kinematically redundant manipulators have a number of advantages. The extra DoF means that they are more dexterous (End effector can reach more points with an arbitrary orientation) and better at avoiding obstacles. Because they have more flexibility in terms of path planning they can also be more energetically efficient. However, the redundancy does come at a cost, in that they are more difficult to control.

SCARA

The selective compliance assembly robot arm known by SCARA is one of the famous serial robotic manipulators designed and built in 1981. The SCARA is composed of two revolute joints and a prismatic joint at its tip.