Robotics Fundamentals 1, Chapter 1: Challenge 1 solution (count the degrees of freedom)

Formulas: planar mechanism    M = 3 (n - 1) - 2 j1 - j2        (n counts the ground link; j1 = hinges and sliders; j2 = two-motion joints)
          spatial mechanism   M = 6 (n - 1) - 5 j1             (all joints one-motion)

(a) Planar five-bar linkage: ground + 4 moving links, so n = 5; five hinges, so j1 = 5.
        M = 3 x (5 - 1) - 2 x 5 = 12 - 10 = 2
    Two DOF: two motors (usually both on the ground link) are needed to drive it fully. This is why five-bar linkages are used as fast two-motor "pick and place" mechanisms.

(b) Slider-crank: ground, crank, connecting rod, slider, so n = 4.
    Joints: crank-ground (hinge), crank-rod (hinge), rod-slider (hinge), slider-ground (prismatic): four joints that each allow one motion, so j1 = 4.
        M = 3 x (4 - 1) - 2 x 4 = 9 - 8 = 1
    One DOF: one motor turning the crank (or pushing the slider) drives the whole motion. It is the mechanism of a piston engine.

(c) Planar arm: ground + 3 links, so n = 4; three hinges, so j1 = 3.
        M = 3 x 3 - 2 x 3 = 9 - 6 = 3
    Three motors. The hand can reach a point (x, y) and also point in a chosen direction (the hand angle).

(d) Spatial arm: ground + 4 links, so n = 5; four hinges, so j1 = 4.
        M = 6 x (5 - 1) - 5 x 4 = 24 - 20 = 4
    Four motors. The hand cannot reach every position and orientation (that needs 6), but a 4-DOF arm is common for simple desktop arms and sorting jobs.

(e) Triangle: ground + 2 links, so n = 3; three hinges, so j1 = 3.
        M = 3 x (3 - 1) - 2 x 3 = 6 - 6 = 0     -> a rigid structure: it cannot move (this is why bridges and robot frames use triangles).
    Square frame: ground + 3 links, so n = 4; four hinges, so j1 = 4.
        M = 3 x 3 - 2 x 4 = 9 - 8 = 1           -> it can move (it folds into a parallelogram or a rhombus), so a square frame needs a diagonal brace to be a structure.

(f) Quadruped with 3 joints per leg:
    Actuators:  4 legs x 3 joints = 12 motors.
    Body DOF in the world:  the trunk is a free body (6 DOF) plus 12 joint angles = 18.
    The trunk's own 6 DOF have no motor of their own, so the robot is underactuated: 12 actuators for 18 DOF. The trunk's position and orientation are controlled indirectly, by pushing on the ground through the feet that are in contact.

(g) Differential-drive robot on a flat floor:
    Body DOF: 3 (x, y, heading).    Actuators: 2 (one per wheel).
    It is not underactuated in the usual sense because the robot can reach any pose, but it has a non-holonomic constraint: the wheels do not let it move sideways, so it cannot move directly to a point beside it. It must turn, drive and turn again (Robotics Fundamentals 4 covers the motion).

Check on method: always count the ground as a link (n includes it), and count only joints that allow one motion in j1. A common slip is to use the number of moving links for n, which gives an answer that is too high by 3.