robotics final notes

1. A coordinate frame is defined by ( B )
A. A single point in space
B. Three orthogonal unit vectors
C. A set of Euler angles
D. A single rotation matrix

2. A vector’s value changes when ( D )
A. The coordinate frame changes
B. The vector’s magnitude changes
C. The vector’s direction changes
D. All of the above

3. A transformation between two frames expresses: ( B )
A. The shape of the object
B. The relationship between the frames
C. The object’s mass distribution
D. The time evolution of motion

4. An orthonormal basis satisfies: ( C )
A. 𝐢 ⋅ 𝐣 = 1
B. ∣ 𝐢 ∣= 2
C. 𝐢 ⋅ 𝐣 = 0, ∣ 𝐢 ∣= 1
D. None of the above

5. A rotation matrix is: ( B )
A. Any 3×3 matrix
B. A 3×3 matrix with orthogonal columns of unit length
C. A symmetric matrix
D. A diagonal matrix

6. For a valid rotation matrix 𝑅: ( A )
A. 𝑅் 𝑅 = 𝐼 and det (𝑅) = 1
B. 𝑅் 𝑅 = 0
C. 𝑅ି ଵ = −𝑅்
D. det (𝑅) = −1

7. The inverse of a rotation matrix equals: ( B )
A. Its determinant
B. Its transpose
C. Its negative
D. None

8. The standard rotation matrix about the z-axis by angle θ is: ( B )
A. ൥
1 0 0
0 cos 𝜃 −𝑠𝑖𝑛𝜃
0 𝑠𝑖𝑛𝜃 cos 𝜃
൩
B. ൥
cos 𝜃 −𝑠𝑖𝑛𝜃 0
𝑠𝑖𝑛𝜃 cos 𝜃 0
0 0 1
൩
C. ൥
cos 𝜃 𝑠𝑖𝑛𝜃 0
−𝑠𝑖𝑛𝜃 cos 𝜃 0
0 0 1
൩
D. ൥
0 −1 0
1 0 0
0 0 1
൩
9. The result of successive rotations is obtained by: ( B )
A. Adding the rotation angles
B. Multiplying rotation matrices in order
C. Subtracting the rotation matrices
D. Averaging the rotation matrices

10. Rotating a coordinate frame vs rotating a vector leads to: ( B )
A. The same mathematical operation
B. The transpose operation

C. No diƯerence
D. A translation

11. A position vector represents: ( C )
A. Orientation only
B. Distance between two points
C. The location of a point relative to a reference frame
D. A direction only

12. Rigid body motion preserves: （ C ）
A. Distances between points
B. Angles between lines
C. Both A and B
D. Neither

13. Rigid body motion is described by: （ A）
A. A rotation and a translation
B. A scaling factor
C. A shear transformation
D. None

14. The composition of two rigid transformations is: （C ）
A. Not a rigid transformation
B. A scaling
C. Another rigid transformation
D. Undefined

15. The homogeneous transformation matrix has size: ( B )
A. 3×3
B. 4×4

C. 2×2
D. 5×5

16. A 3D point in homogeneous coordinates is represented as: （B）
A. [ 𝑥 𝑦 𝑧 ]୘
B. [ 𝑥 𝑦 𝑧 1 ]୘
C. [ 𝑥 𝑦 𝑧 0 ]୘
D. [ 𝑥 𝑦 ]୘

17. The point transformation in homogeneous form is: （A）
A. 𝑝ᇱ = 𝐓𝑝
B. 𝑝ᇱ = 𝐑𝑝
C. 𝑝ᇱ = 𝑝 + 𝑡
D. 𝑝ᇱ = 𝐓 + 𝑝

18. If 𝑻଴
ଵ and 𝑻ଵ
ଶ are known, then 𝑻଴
ଶis: ( B )
A. 𝑻ଵ
ଶ + 𝑻଴
ଵ
B. 𝑻଴
ଵ𝑻ଵ
ଶ
C. 𝑻ଵ
ଶ𝑻଴
ଵ
D. None

19. The homogeneous transformation 𝐓 represents: ( C )
A. Only rotation
B. Only translation
C. Both orientation and position
D. Scaling

20. The Denavit–Hartenberg (DH) method is primarily used to: ( B )

A. Describe the dynamic behavior of a robot
B. Represent the geometric relationship between links and joints
C. Analyze the control system of a manipulator
D. Compute the mass properties of links

21. Each link in the DH convention is characterized by four parameters: ( B )
A. α, β, γ, θ
B. a, α, d, θ
C. x, y, z, φ
D. p, R, T, ω

22. In the DH convention, the z-axis of each frame is aligned with: ( B )
A. The link length direction
B. The joint axis
C. The x-axis of the previous link
D. The gravity direction

23. For a revolute joint, which DH parameter is variable? ( D )
A. a
B. α
C. d
D. θ

24. A typical 3-link cylindrical robot has which joint type sequence? ( C )
A. RRR
B. PPR

C. RPP
D. PRR

25. The workspace of a cylindrical robot is best described as: ( C )
A. A cube
B. A rectangular box
C. A cylindrical volume
D. A spherical shell

26. A spherical wrist allows: ( C)
A. Linear motion in three directions
B. Pure translational movement
C. Orientation control independent of position
D. Force amplification

27. In a spherical wrist, the three wrist joint axes: ( B )
A. Are parallel
B. Intersect at a single point
C. Are skewed
D. Are orthogonal but not intersecting

28. Forward kinematics determines: ( B )
A. Joint variables from a desired end-effector pose
B. End-effector pose from given joint variables
C. Forces from torques
D. Velocities from accelerations

29. Forward kinematics is computed by: ( B )
A. Numerical iteration
B. Successive multiplication of link transformation matrices
C. Differentiation of joint angles
D. Integration of velocities

30. Inverse kinematics involves: ( A )
A. Finding joint variables for a desired end-effector pose
B. Determining link masses from torques
C. Computing inertia tensors
D. Calculating dynamic forces

31. Inverse kinematics generally has: ( C )
A. A single unique solution
B. No solution
C. Multiple possible solutions
D. Random solutions

32. A skew-symmetric (anti-symmetric) matrix 𝑆 satisfies: ( B )
A. 𝑺 = 𝑺்
B. 𝑺் = −𝑺
C. 𝑺் 𝑺 = 𝐼
D. det (𝑺) = 1

33. The diagonal elements of any skew-symmetric matrix are always: ( C )
A. 1
B. -1
C. 0
D. Equal to the determinant

34. For a vector 𝜔 = [𝜔௫, 𝜔௬, 𝜔௭]் , its skew-symmetric matrix is: ( B )
A. ቎
0 𝜔௭ −𝜔௬
−𝜔௭ 0 𝜔௫
𝜔௬ −𝜔௫ 0
቏
B. ቎
0 −𝜔௭ 𝜔௬
𝜔௭ 0 −𝜔௫
−𝜔௬ 𝜔௫ 0
቏
C. ቎
0 𝜔௬ 𝜔௭
𝜔௬ 0 𝜔௭
𝜔௭ 𝜔௬ 0
቏
D. None of the above

35. The matrix form 𝑺(𝝎)×𝒗 = 𝝎 × 𝒗 expresses: ( C )
A. The projection of v
B. The dot product
C. The cross product using a matrix operator
D. The inverse of rotation

36. Angular velocity describes: ( B )
A. Change in position over time
B. Rate of change of orientation over time
C. The curvature of a path
D. The speed of linear translation

37. Angular velocity can be represented as: ( B )
A. A scalar
B. A vector
C. A tensor
D. A matrix

38. The matrix 𝑆(𝜔) represents: ( C )
A. Linear velocity
B. Rotational acceleration
C. The instantaneous rate of rotation (angular velocity)
D. A scaling transformation

39. For a point 𝑃 at position vector 𝒓, its linear velocity is: ( B )
A. 𝑟̇ = 𝑹𝒗
B. 𝒗௉ = 𝒗ை + 𝝎 × 𝒓ை௉
C. 𝒗௉ = 𝝎 ⋅ 𝒓ை௉
D. 𝒗௉ = 𝝎𝒓ை௉

40. A rigid body’s instantaneous motion is described by: ( C )
A. Linear velocity only
B. Angular velocity only
C. Both linear and angular velocities
D. Acceleration and velocity

41. The Jacobian matrix relates: ( B )
A. Forces and torques
B. Joint velocities and end-effector velocities
C. Joint positions and accelerations
D. End-effector forces and displacements

42. For a 6-DOF manipulator, the Jacobian matrix has dimensions: ( C )
A. 3×3
B. 3×6
C. 6×6
D. 6×3

43. The Jacobian is often divided into: ( A )
A. Translational and rotational parts
B. Forward and inverse parts
C. Active and passive parts
D. Static and dynamic parts

44. Each column of the Jacobian corresponds to: ( B )
A. A Cartesian axis
B. The effect of one joint’s motion on the end-effector velocity
C. A transformation matrix

D. A torque vector

45. The Jacobian represents: ( B )
A. The gradient of energy
B. The instantaneous velocity mapping between joint space and task space
C. The potential field
D. The control gain matrix