A 2-link planar robot arm has link lengths L₁ = 1m and L₂ = 0.8m, with joint angles θ₁ = 45° and θ₂ = 30°. The forward kinematics equations are: x = L₁·cos(θ₁) + L₂·cos(θ₁ + θ₂), y = L₁·sin(θ₁) + L₂·sin(θ₁ + θ₂). The end-effector position is approximately:
Ax ≈ 1.41m, y ≈ 1.41m
Bx ≈ 1.56m, y ≈ 1.27m
Cx ≈ 0.71m, y ≈ 0.71m
Dx ≈ 1.80m, y ≈ 0.95m
Forward kinematics is computed by summing the contributions of each link's position. The first link extends L₁ in direction θ₁; the second extends L₂ in direction θ₁ + θ₂ (absolute frame). Substituting the values into the equations and computing the cosines and sines at the given angles yields the position.
Question 2 Multiple Choice
A 3-DOF robot arm has a reachable workspace that forms a sphere of radius 1.5m (sum of all link lengths). A task requires the end-effector to reach a point 2.0m from the base. What is true about inverse kinematics for this task?
AInverse kinematics has exactly one solution in the reachable configuration space
BInverse kinematics has no solution; the point is outside the robot's workspace
CInverse kinematics has infinitely many solutions because the robot is redundant
DInverse kinematics has two solutions: one with elbow-up and one with elbow-down orientation
This geometric picture generalizes to higher dimensions: the number of inverse kinematics solutions depends on how many times the solution manifold intersects the constraint surface. For a 6-DOF arm with a 6-DOF pose specification (3 position + 3 orientation), solutions typically range from zero (outside workspace) to a discrete set of isolated configurations, often including multiple self-consistent solutions reflecting different 'elbow' configurations or arm bends.
Question 3 Short Answer
A SCARA robot (selective compliance arm for robotic assembly) is a 4-DOF arm with four revolute joints: two horizontal planar joints, a vertical lift joint, and a wrist rotation. How many solutions does inverse kinematics typically have for a given target position and orientation?
Think about your answer, then reveal below.
Model answer: A SCARA arm typically has two solutions: elbow-up and elbow-down configurations for the planar motion, while the vertical and rotation DOFs are determined uniquely by the target height and orientation. The redundancy in the planar 2-DOF arm reaching a 2-DOF target (x, y) creates the two-solution family.
The SCARA's structure makes inverse kinematics partially decoupled: the planar 2-DOF subsystem solves for (x, y) position (two solutions), the vertical joint directly sets z position, and the wrist orientation joint directly sets rotation. This design choice makes the arm easier to control than a fully general 6-DOF arm.