Principles of Robotics and Automation is the signature subject of the GATE Robotics and Automation (RA) 2027 paper — the Part A section that defines the discipline. Its questions revolve around a few powerful ideas: counting degrees of freedom with the Grubler criterion, describing orientation with rotation matrices, and combining rotation and translation into homogeneous transformations. This guide reviews each with its working formulas and solves five GATE-style problems in full.
TABLE OF CONTENTS
Manipulators & joints
A robot manipulator is an open kinematic chain of rigid links joined by joints. The two lower-pair joints that dominate the syllabus are the revolute (R) joint, which rotates, and the prismatic (P) joint, which slides — each contributes one degree of freedom. A serial arm is named by its joint sequence, so a typical industrial arm is an RRR (articulated) manipulator, and a full 6-axis robot has six joints.
Degrees of freedom & the Grubler criterion
The degrees of freedom (DOF) of a mechanism is the number of independent inputs needed to fix its configuration. For a planar mechanism, Grubler’s (Kutzbach) criterion counts them from the number of links n and the number of one- and two-DOF joints. A free rigid body has 3 DOF in the plane and 6 DOF in space.
Figure 1. A planar four-bar linkage: n = 4, four revolute joints, giving one degree of freedom.
Rotation matrices
Orientation in space is described by a 3×3 rotation matrix, whose columns are the axes of the rotated frame expressed in the reference frame. Rotation matrices are orthonormal: their determinant is +1 and their inverse equals their transpose. The elementary rotation about z by an angle θ is the most-used building block.
Homogeneous transformations
A homogeneous transformation matrix combines a rotation R and a translation P into a single 4×4 matrix, so that one multiplication carries a point from one frame to another: AP = R·BP + Porg. Chaining these transforms link by link is exactly how a manipulator’s forward kinematics is built.
Forward kinematics & end-effectors
Forward kinematics maps the joint variables to the pose of the end-effector (the gripper or tool). For a serial arm this is the ordered product of the link transforms from base to tool. Because the map is unique, forward-kinematics questions always have a single deterministic answer — see the dedicated forward-kinematics guide for fully worked manipulator examples.
Worked examples (GATE-style)
Example 1 — DOF of a four-bar linkage
Find the degrees of freedom of a planar four-bar mechanism (4 links, 4 revolute joints).
Solution. DOF = 3(n−1) − 2j1 − j2 = 3(4−1) − 2(4) − 0 = 9 − 8.
DOF = 1 — a single input crank fully drives the mechanism.
Example 2 — DOF of a slider-crank
Find the DOF of a slider-crank mechanism (4 links; 3 revolute + 1 prismatic joint, all one-DOF pairs).
Solution. There are 4 one-DOF joints (j1 = 4, j2 = 0). DOF = 3(4−1) − 2(4) = 9 − 8.
DOF = 1 (the prismatic slider counts as a lower pair, just like a revolute).
Example 3 — rotating a point
A point at (2, 0, 0) is rotated 90° about the z-axis. Find its new coordinates.
Solution. Rz(90°) = [0 −1 0; 1 0 0; 0 0 1]. Applying it: x′ = 0×2 − 1×0 = 0; y′ = 1×2 + 0×0 = 2; z′ = 0.
New point = (0, 2, 0) — the point swings from the x-axis onto the y-axis.
Example 4 — degrees of freedom in space
How many degrees of freedom does a single unconstrained rigid body have (a) in a plane and (b) in space?
Solution. In a plane: two translations (x, y) plus one rotation = 3 DOF.
In space: three translations (x, y, z) plus three rotations (roll, pitch, yaw) = 6 DOF. This is why a general-purpose arm needs six joints.
Example 5 — DOF of a five-bar linkage
Find the DOF of a planar five-bar linkage (5 links, 5 revolute joints).
Solution. DOF = 3(5−1) − 2(5) = 12 − 10.
DOF = 2 — a five-bar needs two actuated inputs, which is why it is used for planar parallel positioning.
Key formulas
FORMULA SHEET
Planar DOF: 3(n−1) − 2j1 − j2
Spatial DOF: 6(n−1) − 5j1 − 4j2 − …
Rigid body: 3 DOF (planar), 6 DOF (spatial)
Rotation: Rz(θ) = [cθ −sθ 0; sθ cθ 0; 0 0 1]
Orthonormal: R−1 = RT, det(R) = +1
Transform: AP = R·BP + Porg
Common mistakes to avoid
- Miscounting joints in Grubler: every lower pair (revolute or prismatic) is a one-DOF joint j1; only special pairs like a cam or gear contact are j2.
- Forgetting the ground link — n includes the fixed frame, and the criterion uses (n−1).
- Transposing a rotation the wrong way: the inverse rotation is the transpose, so R(−θ) = R(θ)T.
- Adding translation before rotation: a homogeneous transform rotates the point first, then adds the origin offset.
- Assuming more joints means more DOF — closed loops remove DOF, which is exactly what Grubler captures.
GATE ROBOTICS & AUTOMATION 2027
Master Principles of Robotics for GATE RA
Get in-depth lectures on DOF, transformations, kinematics and automation, with PYQs and full-length tests, taught by Piyush Wairale (IIT Madras).
Frequently asked questions
What is the Grubler criterion used for?
Grubler’s (Kutzbach) criterion computes the degrees of freedom of a mechanism from its link and joint counts. It tells you how many independent actuators are needed to drive the mechanism — one for a four-bar, two for a five-bar, and so on.
Why does a general robot arm need six degrees of freedom?
A rigid body in 3-D space has six degrees of freedom — three position and three orientation. To place an end-effector at any position and any orientation within its workspace, a manipulator therefore needs at least six independent joints.
What is the difference between forward and inverse kinematics?
Forward kinematics computes the end-effector pose from known joint values and always has a unique answer. Inverse kinematics does the reverse and may have multiple solutions or none. See the dedicated forward-kinematics guide for worked manipulator problems.
This solved-problems guide is part of the complete GATE RA 2027 Syllabus overview. For the full topic breakdown, see the Principles of Robotics & Automation syllabus guide, and work through the Forward Kinematics solved problems.
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