Links, pairs, kinematic chains and degrees of freedom

Why it matters

Every machine, from an engine crank train to a pick-and-place robot, is built from rigid links joined by joints. Before you calculate a single velocity or force you must know how many independent inputs (motors or actuators) the system needs, and whether it will move at all. Counting links, pairs and degrees of freedom answers that in a minute and catches design errors such as locked linkages before anything is built.

Key ideas

Link (element). A resistant body that transmits motion and force to other bodies. Usually rigid (crank, connecting rod, frame), but a belt in tension or oil in a hydraulic line also counts, because it transmits force in one direction. Links are classified by how many joints (nodes) they carry:

• binary link: 2 nodes; ternary link: 3 nodes; quaternary link: 4 nodes.
• The fixed link (frame) is still a link and must be counted.

Kinematic pair. Two links in contact so that a definite relative motion is possible.

• By contact: lower pair = surface contact (revolute/turning, prismatic/sliding, screw, cylindrical, spherical, planar); higher pair = point or line contact (cam and follower, gear teeth, ball in a race, wheel on rail).
• By constraint: each planar lower pair (revolute or prismatic) removes 2 of the 3 planar freedoms and leaves 1. A planar higher pair (e.g. cam and flat follower with slip) removes 1 and leaves 2.
• By closure: self-closed (form closure, e.g. pin in a hole) or force-closed (spring or gravity keeps a follower on a cam).

Kinematic chain. Links joined by pairs so that the relative motion of every link is completely constrained, i.e. the chain has one definite motion for a given input. In classical theory of machines a chain is a closed loop. Fixing one link of a kinematic chain gives a mechanism; a mechanism used to transmit useful work is a machine. If the assembly has no relative motion it is a structure (or a frame).

Degrees of freedom (mobility). The number of independent coordinates (inputs) needed to define the configuration of the whole mechanism. A free rigid body in a plane has 3 DOF (x, y, θ); in space it has 6.

• DOF = 1: constrained mechanism, one actuator drives it (four-bar, slider-crank).
• DOF = 0: a structure, no motion (e.g. a triangle of three links).
• DOF < 0: a statically indeterminate (redundant, pre-loaded) structure.
• DOF ≥ 2: needs that many independent inputs, e.g. a 5-bar linkage with two motors, or a planar robot arm.

Counting rules that decide the answer.

• A pin joining m links counts as m − 1 revolute pairs (a compound or multiple joint). Three links on one pin = 2 joints.
• A rolling contact with no slip behaves like a 1-DOF pair (count as a lower pair); a contact with rolling and sliding is a higher pair (h = 1).
• Redundant (idle) freedom: a roller follower's spin about its own pin adds 1 to the count but does not change the output motion. Subtract it to get the effective DOF.
• Gruebler/Kutzbach counts only numbers of links and joints. Special geometry (parallel links, as in a parallelogram added to a parallelogram) can make an apparently zero-DOF linkage move. Such over-constrained exceptions are real, so treat the formula as a check, not a law.

Open chains in mechatronics. A serial robot arm is an open chain; its DOF is simply the sum of its joint freedoms (a 6-axis arm with six revolute joints has 6 DOF). The closed-chain formulas below are for loops, and they also give the right answer for serial chains with every link counted.

Formulas

F = 3(n − 1) − 2j − h (Kutzbach criterion, planar)

• F = degrees of freedom (mobility), dimensionless
• n = number of links including the frame
• j = number of lower pairs (1-DOF pairs: revolute, prismatic, pure rolling), with compound joints counted as m − 1
• h = number of higher pairs (2-DOF pairs)
• Applies to planar mechanisms; ignores special geometry.

3n − 2j − 4 = 0 (Gruebler's criterion, for F = 1 with only lower pairs)

• Gives j = 3n/2 − 2, so n must be even: 4 links need 4 pairs, 6 links need 7 pairs, 8 links need 10 pairs.

F = 6(n − 1) − 5j₁ − 4j₂ − 3j₃ − 2j₄ − j₅ (spatial Kutzbach)

• jₖ = number of pairs that allow k relative freedoms (revolute, prismatic, screw: k = 1; cylindrical: 2; spherical: 3).

F_effective = F − (number of redundant freedoms)

Worked examples

Example 1 (standard): slider-crank and four-bar. Given: a slider-crank mechanism (frame, crank, connecting rod, slider).

1. Count links: n = 4.
2. Count lower pairs: crank–frame (R), crank–rod (R), rod–slider (R), slider–frame (P) → j = 4. Higher pairs h = 0.
3. Formula: F = 3(n − 1) − 2j − h
4. F = 3(4 − 1) − 2(4) − 0 = 9 − 8 = 1 Answer: F = 1, so one input (crank rotation) fixes everything. The same count holds for a four-bar linkage (four R pairs).

Example 2 (GATE level): cam with a roller follower, plus a compound joint check. Given: a disc cam rotating on a fixed pin drives a roller on a translating follower; the roller rolls and slips on the cam.

1. Links: frame, cam, follower, roller → n = 4.
2. Lower pairs: cam–frame (R), follower–frame (P), roller–follower (R) → j = 3.
3. Higher pair: roller–cam contact with slip → h = 1.
4. F = 3(4 − 1) − 2(3) − 1 = 9 − 6 − 1 = 2
5. One of these is the free spin of the roller about its pin, which does not affect the follower: redundant freedom = 1.
6. F_effective = 2 − 1 = 1 Answer: F = 2 by the formula, effective DOF = 1.

Now a linkage with a compound joint: 6 links, of which three meet at one pin; besides that pin there are 5 simple revolute pairs.

1. The triple pin counts as 3 − 1 = 2 pairs, so j = 5 + 2 = 7.
2. F = 3(6 − 1) − 2(7) = 15 − 14 = 1 Answer: F = 1 (a constrained six-bar). Counting the triple pin as 1 joint would wrongly give F = 3.

Common mistakes

• Forgetting to count the fixed link (frame) in n.
• Counting a pin that joins three links as one joint instead of two.
• Treating a cam or gear contact as a lower pair; counting a pure-rolling wheel as a higher pair (pure rolling is a 1-DOF constraint).
• Reporting F = 2 for a roller follower without removing the roller's redundant spin.
• Saying a negative answer "means the mechanism has 0 or 1 DOF". A negative F means a redundant, pre-stressed structure; F = 0 means a just-rigid structure.
• Using the planar formula for a spatial linkage (e.g. a universal joint or a Stewart platform); use the spatial form.
• Trusting the count blindly for special geometry such as a double parallelogram, which moves even though the formula gives 0.

For GATE ME

Expect one-mark questions that give a sketch of a linkage and ask for the DOF, often hiding a compound joint, a slider, a rolling wheel or a roller follower. Also common: identify the type of a pair (sliding, turning, rolling, screw, spherical; lower or higher; closed or open) and classify chain / mechanism / structure. Practise counting links and pairs on six- and eight-link sketches until you do it in under a minute, and always state whether a contact rolls, slides or both.

Quick check

1. How many lower pairs does a pin joining four links count as?
2. A linkage has n = 5 links and j = 6 revolute pairs. What is F, and what is the assembly called?
3. Is a ball-and-socket joint a lower or a higher pair, and how many freedoms does it allow?
4. For a constrained planar chain with only lower pairs, how many pairs do 8 links need?
5. Why does a roller follower give F = 2 by Kutzbach?

Answers: 1. Three. 2. F = 3(4) − 12 = 0, a structure. 3. Lower pair (surface contact), 3 rotational freedoms. 4. j = 3(8)/2 − 2 = 10. 5. The roller's free spin about its own pin is counted; it is a redundant freedom, so the effective DOF is 1.