UFMF4X-15-M

Q1

a) Linear segments with parabolic blends can be used to generate smooth trajectories. Which statement is correct?

i) The velocity is never continuous but has transients at many points along the trajectory.

ii) The acceleration is a sequence of constant values at certain intervals.

iii) The second derivative of the position is quadratic during the blends.

iv) Integrating the position three times results in the desired acceleration profile for the entire trajectory.   (2 marks)

b) Which equation is used as part of the computation of linear segments with parabolic blends for trajectories along via points?

(2 marks)

c) The joint velocity is a function of ...

i ... time squared and can be described by a cubic equation when using cubicd polynomials for a joint trajectory.

ii ... time and can be described by a quadratic equation when using cubicd polynomials for a joint trajectory.

iv ... angular acceleration which in turn is a function of the joint diameter, in case of translational joints.           (2 marks)

d) How can the homogeneous coordinate transformation matrix be used to specify the position and orientation of the end effector of a robot arm.

i) Attach a frame. Ft to the end effector. Let F0 be the fixed frame. Then  where  gives the position of the end effector and  gives the orientation of the end effector. Rt0 gives the direction cosines of (xt, yt, zt) relative to (x0, y0, z0).

ii) This is not possible.

iii) Attach a frame. Ft to the end effector. Let F0 be the fixed frame. Then  where  gives the end effector singularity and  gives the Null position of the manipulator.

iv) Attach a frame. to the end-effector and the base of the manipulator and then apply a cubic spline to connect the two. This is to be repeated for all joints and results in the forward kinematic solution for the robot.   (2 marks)

e) The Jacobian Matrix can be used to relate forces applied to the manipulator end effector to forces at the joints as follows:

                           (2 marks)

f) Why is the inverse kinematic solution important in controlling a robot manipulator.

i) Robot manipulators are usually programmed at world coordinates (tool space). However, control is performed at the joint level. Hence, world coordinates have to be transformed to joint coordinates to control the robot manipulator.

ii) Robot manipulators are usually programmed at joint level (joint coordinates). However, control is performed at the tool (or end-effector) level. Hence, joint coordinates have to be transformed to world coordinates to control the robot manipulator.

iii) Robot manipulators are usual programmed by hand, hence, inverse kinematics are not needed and never used to control the robot.

iv) This is a trick question. Since inverse kinematics is mathematically proven to lead to singularities in any situation or configuration, it is better to use reverse dynamics for the control of a robot.      (2 marks)

g) What does it mean for a robot to be in a singular configuration?

i) It creates a road map connecting vertices of manipulator frames that can be used for control and path planning.

ii) A singularity describes mathematically what happens inside a black hole. Hence, if a manipulator approaches a singular configuration it will turn into a black hole and effectively turn inside out.

iii) At singularities, the solution of the inverse Jacobian tends to infinity, because the determinant of the Jacobian becomes zero. Hence, the closer the configuration of the manipulator comes to the singularity the larger are the velocities required to move the manipulator. This is not desired.

iv) It is a mapping technique that creates paths between robot and visible obstacles only.         (2 marks)

h) How can the manipulator Jacobian matrix be used to find a solution to the inverse kinematic problem.

i) The Inverse Jacobian can only be used, if the robot manipulator moves through a well-defined zero position (called Null-Space).

ii) If the Inverse Jacobian can be determined, based on the start configuration and ΔX (pose change), then Δθ (the change in joint configuration) can be found, and [Θ] as a function of X + ΔX can be derived.

iii) The Inverse Jacobian can only be employed to compute the Forward Kinematics of a manipulator. iv) The Jacobian matrix provides an analytical IK solution.   (2 marks)

j) What are the advantages and disadvantages of using the Jacobian matrix to solve the inverse kinematics problem, compared to the analytical solution of the inverse kinematic problem?

i) The Inverse Jacobian method has the following advantages: it is fast to compute (always real-time capable) and always diverges to infinity.

ii) The main advantage is that the inverse Jacobian method is generally applicable to all manipulators; disadvantages: it is a numerical method whose accuracy depends on the step size; it is plagued by singularities.

iii) An advantage is that the Jacobian matrix does not need to be inverted when using it for computing the inverse kinematics of a manipulator.

iv) The Jacobian can only be used to compute the inverse kinematics, if the robot is made up of a combination of rotary and prismatic links.          (2 marks)

k) What are issues that make it difficult to find analytical solutions to inverse kinematics for manipulators?

i) Computing the inverse kinematics is as straightforward as computing forward kinematics.

ii) The required equations are often non-linear, transcendental (in terms of cos & sin functions) and coupled. Hence, no general solution method exists; a solution may not exist at all; there may be multiple solutions.

iii) The required equations are linear, non-transcendental (in terms of cos and sin) and decoupled; hence, only one exact solution can be found.

iv) The main difficulty is that the solutions suffer from the curse of dimensionality.       (2 marks)

l) Mark the following statements with TRUE or FALSE:

i) A quaternion is a type of higher complex number that can be written as: w + ix + jy + kz, where w, x, y, z are real numbers.

ii) In the previous equation, i, j, k are imaginary numbers for which the following is valid:

i2 = j2 = k2 = -1 and ij = k = ji.

iii) Using quaternions we can avoid Gimbal lock, which is what can occur when representing orientation with rotation matrices.

iv) Using quaternion, four parameters are needed to express a rotation, whereas rotation matrices need 9 entries. However, quaternion multiplication is not as plain as matrix multiplication.        (2 marks)

m) The dynamic equation of a robot is  If q is the joint vector of the robot, explain what

i) matrix M(q) corresponds to

ii) what forces corresponds to the term 

iii) what forces correspond to the term g(q).                                (2marks)

n) What are the ZYZ Euler angles that describe the rotation of frame. {b} with respect to frame. {A} below:

                                     (1 mark)

Q2

A smooth trajectory for a manipulator joint can be calculated employing linear functions with parabolic blends. Use this type of function to solve part (a) of this question.

a) The trajectory of one particular joint of a robotic manipulator is specified as follows: Path points: θ1 = 50°, θ2 = 100°, θ3= 10°. The magnitude of the acceleration to use is 20°/sec2 at each path point. The maximum speed at which the joint can move is limited to 10°/sec. The manipulator is motionless at the start of the trajectory and at the end of the trajectory.

i) Make three sketches showing the angular position, velocity and acceleration versus time for this trajectory. Show in your sketch positions ( 1, 2, 3), duration times (td12, td23), blend times (t1, t2, t3), and linear times (t12, t23).   (6 marks)

ii) Now, determine all blend times (t1, t2, t3), and all linear times (t12, t23), the signs of all accelerations, and the durations of the two segments, td12 and td23, respectively. Assume that the robot joint moves at maximum speed during the linear segments.  (8 marks)

b) An alternative approach is to use cubic splines to generate smooth trajectories. Draw a graph showing two cubic splines connecting start, intermediate and end points of the single manipulator joint. Also, provide sketches for the velocity and acceleration profiles. There are no calculations required for this part of the question. For your sketch use the angular values provided and the segment times calculated under (a), where possible.   (6 marks)

c) Discuss advantages and disadvantages of using cubic splines and linear segments with parabolic blends for trajectory generation. Relate your answer to the results above.   (5 marks)

Q3

The first three links of an RPR robot manipulator are shown in Figure 3a. The three parameters are 1, d2, and 3, respectively.

Figure 3a

a) State the Denavit-Hartenberg (D-H) convention for attaching frames on a manipulator. Add the labels where missing from the frames shown in Figure 3 and verify then that all frames attached to the manipulator conform. to the D-H convention.   (5 marks)

b) Find the Denavit-Hartenberg parameters for the shown manipulator.  (5 marks)

c) Using the Denavit-Hartenberg method, find the forward kinematic equations for the shown manipulator.  (8 marks)

d) Assume the planar 2RPR parallel robot in Figure 3b has two legs. Calculate the length of joints p1 and p2 when the end-effector is at position [2 4]T .

Figure 3b

(4 marks)

e) Find the homogenous transformations T01, T02 between the frames shown in Figure 3c.

Figure 3c

(3 marks)

Q4

A three degree of freedom planar manipulator (RPR) with two rotary joints (O1 and O2) and a prismatic joint (J) is shown in Figure 4. Link R2 has a constant length of 5 units and link r1 has a variable length. The base of the robot (at origin O1) is fixed.

Figure 4

a) Using an analytical approach, find the joint coordinates ( 1, 2, r1) when the manipulator is in the following two configurations, config1: (Px1, Py1 α1) = (8.5cm, 8.5cm, 55 o), and config2: (Px2, Py2 α2) = (9.0cm, 9.0cm, 60 o), where {Pxi, Pyi} describe a coordinate pair in the planar workspace of the manipulator with respect to origin O1 and αi describes the tool angle measured from the positive x-axis.   (7 marks)

b) It is now required to move the tool from config1 to config2. Find the corresponding joint coordinates using the inverse Jacobian method.   (7 marks)

c) Compare the results found under (a) and (b) and calculate the error that is made when the Jacobian Method is used. Briefly discuss.   (4 marks)

d) When the robot is in configuration config1, the tool is required to exert the following forces on a work piece in contact with the end effector: fx = 1.0 unit along the x-axis and fy = 1.0 units along the y-axis. Find the actuator forces required in order for the tool to exert the above forces. Make use of the Jacobian that you have calculated under (c).         (7 marks)