Prerequisite knowledge required: Text Section s 2. Apply the following transformations to frame B and find AP. Using the 3-D grid, plot the transformations and the result and verify it. Find the new location and orientation of the frame. Estimated student time to complete: Prerequisite knowledge required: Text Section s 2. Solution: Since transformation matrices are unitary, we calculate the inverses simply by transposing the rotation part and calculating the position part by: 0.
Estimated student time to complete: 5 minutes Prerequisite knowledge required: Text Section s 2. About what axes are these rotations supposed to be?
These rotations must be relative to the current moving frame in order to prevent changing the position of the frame. Estimated student time to complete: 20 minutes Prerequisite knowledge required: Text Section s 2. Solution: a The equations representing position in a spherical coordinates may be set equal to the given values as: i.
Since orientation is not specified, no more information is available to check the results. 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. We want to place the hand on the part. The 3,2 element of spherical coordinate transformation matrix should be zero. This is not. Estimated student time to complete: 30 minutes Prerequisite knowledge required: Text Section s 2.
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. The length of each link l1 and l2 is 1 ft. Therefore, there is really no need to use the traditional method of inverse kinematic mentioned in Section 2. To express the height from the table, the in base height must be added to the pz value. 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. Please note how frames 2, 3, and 4 relate to each other by referring to the side view of the arm. Reprinted with permission from Staubli Robotics. Please also note that there can be an additional frame between frames 1 and H. In that case, the two joint variables will be represented by two separate matrices. Estimated student time to complete: minutes if Problem 2. What is the effect of a differential rotation of 0.
Find the new location of the hand. What is the differential operator relative to the reference frame? What is the differential operator relative to the frame A? Find a transformation matrix Q that will accomplish this transform in the Universe frame. By inspection, find a differential translation and a differential rotation that constitute this operator. The corresponding inverse Jacobian of the robot at this location is also given.
The robot makes a differential motion, as a result of which, the change in the frame dT is recorded as given. Find the new location of the camera after the differential motion. Find the differential operator. Find the joint differential motion values associated with this move.
Find how much the differential motions of the hand-frame T D should have been instead, if measured relative to frame T, to move the robot to the same new location as in part a. 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.
The corresponding inverse Jacobian of the robot at this location relative to this frame is also shown. Find which joints must make a differential motion, and by how much, in order to create the indicated differential motions.
Find the change in the frame. Find the new location of the frame after the differential motion. Find how much the differential motions given above should have been if measured relative to the Universe, to move the robot to the same new location as in Part c. The corresponding inverse Jacobian of the robot at this location is also shown. Find how much the differential motions given above should have been, if measured relative to Frame T, to move the robot to the same new location as in Part c.
Estimated student time to complete: 20 minutes Prerequisite knowledge required: Text Section s 3. From Equation 3. Estimated student time to complete: 10 minutes Prerequisite knowledge required: Text Section s 3. Find the three components of the velocity of the hand frame. Estimated student time to complete: 10 or 20 minutes Prerequisite knowledge required: Text Section s 3.
Find the required three joint velocities that will generate the given hand frame velocity. Kinematic: First we write equations describing the kinematic relationships. Estimated student time to complete: 30 minutes Prerequisite knowledge required: Text Section s 4. Estimated student time to complete: minutes Prerequisite knowledge required: Text Section s 4. Di terms are gravity terms. Estimated student time to complete: 1 hour Prerequisite knowledge required: Text Section s 4.
From Equation 4. Attached to the object is a frame, which describes the orientation and the location of the object. Find the equivalent forces and torques acting on the object relative to the current frame.
Assuming that the two parts must be aligned together for this purpose, find the necessary forces and moments that the robot must apply to the part relative to its hand frame.
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. Estimated student time to complete: minutes Prerequisite knowledge required: Text Section s 5. Calculate the coefficients for a third-order polynomial joint-space trajectory and plot the joint angles, velocities, and accelerations.
Estimated student time to complete: minutes with plotting Prerequisite knowledge required: Text Section s 5. Calculate the coefficients for third-order polynomials in joint-space. Plot the joint angles, velocities, and accelerations.
Assume the joint stops at intermediate points. This results in seven equations. 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. Find the necessary blending time for a trajectory with linear segments and parabolic blends and plot the joint positions, velocities, and accelerations.
The positions, velocities, and time duration for the three segments for one of the joints are given below. Determine the trajectory equations and plot the position, velocity, and acceleration curves for the joint.
The same can be found using the matrix equation of Equation 5. Find the joint variables for the robot if the path is divided into 10 sections.
Each link is 9 inches long. 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. Show that the angle criterion is met. Can you determine from the root locus whether or not the system is stable? Estimated student time to complete: 30 minutes Prerequisite knowledge required: Text Section s 6. Can you determine whether or not the system may become unstable as the gain changes?
The roots are always on the left side, and therefore, the system is always stable. Estimated student time to complete: minutes Prerequisite knowledge required: Text Section s 6. Find a proper location for the zero and the loop gain. Angle deficiency: 3. Find the location of an additional zero and proportional, derivative, and integral gains.
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 Compare the results. Estimated student time to complete: minutes Prerequisite knowledge required: Text Section s 7. Each link is 22 cm long, made of hollow aluminum bars, each weighing 0. 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. Assume the inertias of the worm gears are negligible. Therefore, a new motor must be picked. Two other motors are available, one with the inertia of 0. Skip to main content. By using our site, you agree to our collection of information through the use of cookies.
To learn more, view our Privacy Policy. Log In Sign Up. Download Free PDF. William Haxworth. Download PDF. A short summary of this paper. Download full file from buklibry. This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system and the Internet without prior expressed consent of the copyright owner.
Assume the dimensions of the base and other parts of the structure of the robot are as shown. Estimated student time to complete: minutes Prerequisite knowledge required: Text Section s 1. 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.
Estimated student time to complete: minutes Prerequisite knowledge required: Text Section s 2. Express the vector in matrix form. If not, find the necessary unit vector s to form a frame between p, q, and s.
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