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To browse Academia.edu and the wider internet faster and more securely, please take a few seconds to upgrade your browser. You can download the paper by clicking the button above. The student who has worked on the Assume the dimensions of the base and other parts of the structure of the robot are as shown. Assume the dimensions of the base and other parts of the structure of the robot are as shown. Assume the dimensions of the base and other parts of the structure of the robot are as shown. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 7 Problem 2.3 Will the three vectors p, q, and r in Problem 2.2 form a traditional frame. If not, find the necessary unit vector s to form a frame between p, q, and s. Estimated student time to complete: 15-20 minutes Prerequisite knowledge required: Text Section(s) 2.4 Solution: As we saw in Problem 2.2, since q ? r is not a unit vector, it means that q and r and not perpendicular to each other, and therefore, they cannot form a frame. However, p and q are perpendicular to each other, and we can select s to be perpendicular to those two. Find the new location of the point relative to the reference frame. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 13 Problem 2.9 Derive the matrix that represents a pure rotation about the z-axis of the reference frame. Estimated student time to complete: 10 minutes. Apply the following transformations to frame B and find AP. Using the 3-D grid, plot the transformations and the result and verify it. Apply the following transformations to frame B and find AP.
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This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 1? 6 ?? 4? ? 1? 21 Problem 2.17 The frame B of problem 2.16 is rotated 90D about the a-axis, 90D about the y-axis, then translated 2 and 4 units relative to the x- and y-axes respectively, then rotated another 90D about the n-axis. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 0 3? 1 5 ?? 0 ?1? ? 0 1? 22 Problem 2.18 Show that rotation matrices about the y- and the z-axes are unitary. About what axes are these rotations supposed to be. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 27 Problem 2.23 Suppose that a robot is made of a Cartesian and RPY combination of joints. Therefore, the robot moves 4, 6, and 9 units along the x-, y-, and z-axes. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 28 Problem 2.24 Suppose that a robot is made of a Cartesian and Euler combination of joints. Therefore, the robot moves 4, 6, and 9 units along the x-, y-, and z-axes. Determine what angles should be used to achieve the same result if RPY is used instead. Estimated student time to complete: 30 minutes Prerequisite knowledge required: Text Section(s) 2.12 Solution: In this solution, the locations of the origins of some of the frames are arbitrary. Therefore, intermediate matrices might be different for each case. However, the final answer should be the same. The length of each link l1 and l2 is 1 ft.
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Estimated student time to complete: 20-30 minutes Prerequisite knowledge required: Text Section(s) 2.12 Solution: In this solution, the locations of the origins of some of the frames are arbitrary. Therefore, intermediate matrices might be different for each case. However, the final answer should be the same. Therefore, intermediate matrices might be different for each case. However, the final answer should be the same. To express the height from the table, the 27-in base height must be added to the pz value. Please also note that no particular reset position is specified. Therefore, the results relate to a position and orientation that would correspond to the given transformations from a reset position. We assume at reset, there is 90 degrees between x0 and x1. Estimated student time to complete: 20-30 minutes Prerequisite knowledge required: Text Section(s) 2.12 Solution: In this solution, the locations of the origins of some of the frames are arbitrary. Therefore, intermediate matrices might be different for each case. However, the final answer should be the same. Fill out the parameters table. Write an equation in terms of A-matrices that shows how U TH can be calculated. Estimated student time to complete: 20-30 minutes Prerequisite knowledge required: Text Section(s) 2.12 Solution: In this solution, the locations of the origins of some of the frames are arbitrary. Therefore, intermediate matrices might be different for each case. However, the final answer should be the same. Estimated student time to complete: 20-30 minutes Prerequisite knowledge required: Text Section(s) 2.12 Solution: In this solution, the locations of the origins of some of the frames are arbitrary. Therefore, intermediate matrices might be different for each case. However, the final answer should be the same. Estimated student time to complete: 20-30 minutes Prerequisite knowledge required: Text Section(s) 2.
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12 Solution: In this solution, the locations of the origins of some of the frames are arbitrary. Therefore, intermediate matrices might be different for each case. However, the final answer should be the same. Please also note that no particular reset position is specified. Estimated student time to complete: 20-30 minutes Prerequisite knowledge required: Text Section(s) 2.12 Solution: In this solution, the locations of the origins of some of the frames are arbitrary. Therefore, intermediate matrices might be different for each case. However, the final answer should be the same. C1o y 0 C2 S 3 S 2 S3 C3 0. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 46 Problem 2.40 Derive the inverse kinematic equations for the robot of Problem 2.37. Estimated student time to complete: 20-30 minutes if Problem 2.37 is already solved Prerequisite knowledge required: Text Section(s) 2.13 Solution: We assume all positions are made relative to the base of the robot, and therefore, U T0 is not included in the solution. Using the Denavit-Hartenberg representation and the joint parameters shown, we get. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 48 CHAPTER THREE Problem 3.1 Suppose the location and orientation of a hand frame is expressed by the following matrix. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 49 Problem 3.2 As a result of applying a set of differential motions to frame T shown, it has changed an amount dT as shown. Find the magnitude of the differential changes made ( dx, dy, dz.
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This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 51 Problem 3.4 The initial location and orientation of a robot’s hand are given by T1, and its new location and orientation after a change are given by T2. a. Find a transformation matrix Q that will accomplish this transform (in the Universe frame). b. Assuming the change is small, find a differential operator. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 54 Problem 3.6 Two consecutive frames describe the old (T1) and new (T2) positions and orientations of the end of a 3-DOF robot. The corresponding Jacobian relative to T1, relating to T1 dz, T1. The corresponding Jacobian, relating to dz. The corresponding inverse Jacobian of the robot at this location is also given. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 58 Problem 3.9 A camera is attached to the hand frame T of a robot as given. The corresponding inverse Jacobian of the robot relative to the frame at this location is also given. The robot makes a differential motion, as a result of which, the change dT in the frame is recorded as given. a. Find the new location of the camera after the differential motion. b. Find the differential operator. The corresponding inverse Jacobian of the robot at this location relative to this frame is also shown. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 62 Problem 3.
11 The hand frame T of a robot is given. The corresponding inverse Jacobian of the robot at this location is also shown. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 64 Problem 3.12 Calculate the T6 J 21 element of the Jacobian for the revolute robot of Example 2.25. Estimated student time to complete: 20 minutes Prerequisite knowledge required: Text Section(s) 3.10 Solution: The robot has 6 revolute joints. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 65 Problem 3.13 Calculate the T6 J16 element of the Jacobian for the revolute robot of Example 2.25. Estimated student time to complete: 20 minutes Prerequisite knowledge required: Text Section(s) 3.10 Solution: The robot has 6 revolute joints. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 66 Problem 3.14 Using Equation (2.34), differentiate proper elements of the matrix to develop a set of symbolic equations for joint differential motions of a cylindrical robot and write the corresponding Jacobian. Estimated student time to complete: 10 minutes Prerequisite knowledge required: Text Section(s) 3.10-3.11 Solution: For a cylindrical coordinate (without additional rotation) there can only be 3 variables. Estimated student time to complete: 10 minutes Prerequisite knowledge required: Text Section(s) 3.10-3.11 Solution: For a spherical coordinate (without additional rotation) there can only be 3 variables.
This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 68 Problem 3.16 For a cylindrical robot, the three joint velocities are given for a corresponding location. Find the three components of the velocity of the hand frame. Find the three components of the velocity of the hand frame. Find the required three joint velocities that will generate the given hand frame velocity. Find the required three joint velocities that will generate the given hand frame velocity. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 74 Problem 4.2 Calculate the total kinetic energy of the link AB, attached to a roller with negligible mass, as shown. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 75 Problem 4.3 Derive the equations of motion for the 2-link mechanism with distributed mass, as shown.This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 79 Problem 4.5 Using Equations 4.49 to 4.54, write the equations of motion for a 3-DOF revolute robot and describe each term. Attached to the object is a frame, which describes the orientation and the location of the object.The object’s location relative to the base frame of a robot is described by RT0. Calculate the coefficients for a third-order polynomial joint-space trajectory. Determine the joint angles, velocities, and accelerations at 1, 2, and 3 seconds.
It is assumed that the robot starts from rest, and stops at its destination. Calculate the coefficients for a third-order polynomial joint-space trajectory and plot the joint angles, velocities, and accelerations. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 86 Problem 5.3 The second joint of a 6-axis robot is to go from initial angle of 20D to an intermediate angle of 80D in 5 seconds and continue to its destination of 25D in another 5 seconds. Calculate the coefficients for third-order polynomials in joint-space. Plot the joint angles, velocities, and accelerations. Assume the joint stops at intermediate points. Estimated student time to complete: 30-40 minutes (with plotting) Prerequisite knowledge required: Text Section(s) 5.5.1 Solution: There are two segments to this motion. The eighth equation can be generated by making assumptions such as a maximum allowable acceleration or an intermediate velocity. For this problem we will assume that the joint will come to a stop at the intermediate point. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 88 Problem 5.4 A fifth-order polynomial is to be used to control the motions of the joints of a robot in joint-space. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 91 Problem 5.6 A robot is to be driven from an initial position through two via points before it reaches its final destination using a 4-3-4 trajectory. The positions, velocities, and time duration for the three segments for one of the joints are given below.
Find the joint variables for the robot if the path is divided into 10 sections. Each link is 9 inches long. Estimated student time to complete: 60-75 minutes (with plotting) Prerequisite knowledge required: Text Section(s) 5.6 Solution: The inverse kinematic equations can be found in different ways. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 95 Problem 5.8 The 3-DOF robot of Example 5.7, as shown in Figure 5.18, is to move from point (3, 5, 5) to point (3, -5, 5) along a straight line, divided into 10 sections. Find the angles of the three joints for each intermediate point and plot the results. Estimated student time to complete: 30 minutes, depending on programming expertise Prerequisite knowledge required: Text Section(s) 5.6 Solution: Using the inverse kinematic equations of the robot from Example 5.7, the joint angles can be found as shown. Show that the angle criterion is met. Can you determine from the root locus whether or not the system is stable. Show that the angle criterion is met. Can you determine whether or not the system may become unstable as the gain changes. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 114 Problem 6.16 For the system of Problem 6.11, find the roots, the gain, and the steady-state error for a settling time of less that 1 second and overshoot of 4 or less. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 115 Problem 6.
17 For the following system, find the roots, the gain, and steady-state error for the fastest response and a settling time of less than 2 seconds and an overshoot of less than 4. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 116 Problem 6.18 For the system of problem 6.17, select the locations and the proportional and integral gains to change it to a proportional-plus-integral system with zero steady-state error. Find a proper location for the zero and the loop gain. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 118 Problem 6.20 For the system of problem 6.19, add an integrator to the system to make it into a PID system in order to achieve a zero steady-state error. Find the location of an additional zero and proportional, derivative, and integral gains. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 119 CHAPTER SEVEN Problem 7.1 A motor with a rotor inertia of 0.030 Kgm2 and maximum torque of 12 Nm is connected to a uniformly distributed arm with a concentrated mass at its end, as shown in Figure P.7.1. Ignoring the inertia of a pair of reduction gears and viscous friction in the system, calculate the total inertia felt by the motor and the maximum angular acceleration it can develop if the gear ratio is a) 5, b) 50, c) 100. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 120 Problem 7.
2 Repeat Problem 1, but assume that the two gears have 0.002 Kgm2 and 0.005 Kgm2 inertias respectively. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner. 121 Problem 7.3 The three-axis robot shown in Figure P.7.3. is powered by geared servomotors attached to the joints by worm gears. Each link is 22 cm long, made of hollow aluminum bars, each weighing 0.5 Kg. The center of mass of the second motor is 20 cm from the center of rotation. The worst case scenario for the elbow joint is when the arm is fully extended, as shown.