Unit 6: I’m organizing the notes around the planning method, Cartesian straight-line generation, the two-link planar kinematics, and the Orangewood arm implementation. The equations will distinguish Cartesian path generation from joint-space execution, since that distinction is central to obtaining a genuinely straight end-effector path.# Unit 6: Trajectory Planning - Practice Quiz

1 What is the main objective of straight-line trajectory planning for a two-axis planar robot?

2 In which space is a straight-line end-effector path normally defined?

3 Which information is essential for generating a straight-line path?

4 Which equation represents linear interpolation between a start point and an end point ?

5 What does the interpolation parameter represent in ?

6 What does the interpolation parameter represent in ?

7 What is the purpose of inverse kinematics in straight-line trajectory implementation?

8 What is the purpose of forward kinematics for a two-axis planar robot?

9 How many primary joint variables are controlled in a two-axis planar articulated robot?

10 What is a waypoint in a robot trajectory?

11 Why are multiple intermediate points generated along a straight-line path?

12 What does sampling time specify during trajectory execution?

13 Which method best helps the end effector follow a Cartesian straight line?

14 For uniform linear interpolation with equal sampling intervals, how are the Cartesian waypoints normally spaced?

15 Which quantities are commonly sent as commands to the joints of a two-axis articulated robot?

16 What should the algorithm do if a generated waypoint lies outside the robot's reachable workspace?

17 Which robot parameters are needed in the kinematic model of a two-link planar arm?

18 Why is smooth time scaling used in trajectory planning?

19 Which plot is most useful for checking whether the end effector followed a straight planar path?

20 What is a basic safety practice when first testing a trajectory on the Orangewood Robotic Arm?

21 For a straight-line Cartesian trajectory from to , which equation correctly generates an intermediate point using the path parameter ?

22 A trajectory planner divides a straight-line motion into 10 equal intervals and includes both the initial and final positions. How many Cartesian waypoints must be generated?

23 A two-link planar arm has and must reach . Using the inverse-kinematics branch with positive , which joint configuration is obtained?

24 Why is direct linear interpolation of and generally unsuitable when the Orangewood arm's end effector must follow a straight Cartesian line?

25 The end effector moves from to in 5 equal intervals. What Cartesian increment is applied per interval?

26 Which velocity profile is most appropriate when the arm should accelerate smoothly, move at a bounded speed, and decelerate before reaching the endpoint?

27 For a two-link planar robot, the determinant of the position Jacobian is . At which configuration is the Jacobian singular?

28 What is the main practical risk of commanding a Cartesian trajectory very close to a kinematic singularity?

29 Each Cartesian waypoint has elbow-up and elbow-down inverse-kinematics solutions. Which selection rule best avoids sudden joint jumps along the trajectory?

30 A planned trajectory contains joint-angle waypoints separated by . What command update frequency is required to send one waypoint per control period?

31 Due to floating-point accumulation, the final interpolated waypoint is slightly different from the requested endpoint. What is the most reliable implementation practice?

32 The end effector must travel along a straight line in at constant Cartesian speed. What speed should be commanded?

33 For a planar arm with link lengths and , which target is outside the arm's reachable workspace?

34 A Cartesian waypoint is geometrically reachable, but both inverse-kinematics solutions violate an Orangewood arm joint limit. How should the planner handle that waypoint?

35 Trajectory points are defined in a table coordinate frame, but the inverse-kinematics model uses the robot base frame. What must be done before solving inverse kinematics?

36 Which method best verifies in software that the generated joint trajectory follows the intended straight Cartesian path before execution?

37 Which normalized time-scaling function gives zero path velocity at both and while moving from to ?

38 A trajectory is sampled every , including samples at both and . How many joint commands are generated?

39 The trajectory algorithm calculates joint angles in radians, but the Orangewood control interface expects degrees. Which conversion should be applied before sending commands?

40 During testing, the measured end-effector path shows noticeable deviation between correctly computed Cartesian waypoints because the commands are widely spaced in time. Which modification is most likely to improve line tracking?

41 A two-link planar arm has . For the Cartesian waypoint , which joint solution corresponds to the branch with ?

42 A trajectory planner linearly interpolates the joint angles between and for an arm with . What is the perpendicular Cartesian deviation at the joint-space midpoint from the straight line joining the two endpoint positions?

43 For a two-link planar robot with and at , what joint velocity produces the Cartesian velocity ?

44 Which condition identifies a kinematic singularity of a two-axis planar articulated arm when both link lengths are nonzero?

45 During Cartesian interpolation, two inverse-kinematics solutions exist at every waypoint. Which strategy best prevents a discontinuous elbow-branch change while still respecting joint limits?

46 A straight Cartesian path has length and must be completed in using a symmetric trapezoidal speed profile. Each acceleration or deceleration phase lasts . What is the required cruise speed?

47 Along a Cartesian line parameterized by distance , inverse kinematics gives . If the joint-speed limits are , what is the maximum allowable path speed at that point?

48 For a desired Cartesian acceleration at a nonsingular configuration, which expression correctly computes the joint acceleration needed to track the path?

49 Cartesian waypoints lie exactly on a straight segment, but the Orangewood arm controller interpolates independently in joint space between successive commands. Which modification most directly reduces geometric deviation between commands?

50 A planar arm has and . A requested straight path runs from to . Why must the planner reject the complete segment even though both endpoints are reachable?

51 A straight line is specified in the laboratory frame, but the inverse-kinematics implementation treats those coordinates as if they were already expressed in the robot base frame. Which outcome is most likely when the base has an unmodeled planar rotation and translation?

52 Because of floating-point error, inverse kinematics computes for a waypoint nominally on the workspace boundary. What is the most robust implementation?

53 Which normalized time-scaling polynomial produces zero velocity and zero acceleration at both ends of a straight-line trajectory?

54 Near a singular configuration, which resolved-rate formula provides damped least-squares joint velocities for a desired Cartesian velocity ?

55 Successive inverse-kinematics outputs for one joint are and . If the joint can rotate through the boundary, what should the trajectory implementation do before computing velocity?

56 During execution, the measured joint following error repeatedly exceeds the safety threshold while command packets continue to be transmitted. What is the safest planner-controller response?

57 Let the desired line run from to . Which expression gives the perpendicular distance of a measured point from the infinite desired line?

58 A tool-tip offset is attached to the second link, but trajectory planning uses the wrist-joint position as the end-effector point. What systematic validation result should be expected?

59 A computed joint trajectory exceeds a velocity limit by a factor of and an acceleration limit by a factor of . If the same geometric path is uniformly stretched in time by a factor , what minimum satisfies both limits?

60 A trajectory is sampled with nominal period . The implementation uses times and explicitly appends if it is not already present. How many command samples are generated?