[SOLVED] Cs4610/cs5335: homework 1

HW Q1: (Spong, Problem 2-15) If the coordinate frame A is obtained from
the coordinate frame B by a rotation of π/2 about the x-axis followed by a
rotation of π/2 about the fixed y-axis, find the rotation matrix R representing the composite transformation. Sketch the initial and final frames.
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Figure 1:
HW Q2: (Spong, Problem 2-37) Consider the diagram in Figure 1. A robot
is set up 1 meter from a table. The table top is 1 meter high and 1 meter
square. A frame o1 x1, y1, z1 is fixed to the edge of the table as shown. A
cube measuring 20 cm on a side is placed on top of the table and at the
center of the table with frame o2 x2, y2, z2 established at the center of the
cube as shown. A camera is situated directly above the center of the block
2m above the table top with frame o3 x3, y3, z3 attached as shown. Find
the homogeneous transformations relating each of these frames to the base
frame o0 x0, y0, z0. Find the homogeneous transformation relating the frame
o2 x2, y2, z2 to the camera frame o3 x3, y3, z3.
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Figure 2:
HW Q3: For each of the three degree of freedom manipulators shown in
Figure 2, find the forward kinematics map. Assume that the link lengths are
given to you in as l1, l2, l3, etc. Specifically, in (i), l1 is the vertical height
of the first joint and l2 is the length of the arm segment after θ1,2. In (ii)
and (iii), l1 is the vertical height of the first joint, l2 is the length of the arm
segment after θ1,2 and before θ3, and l3 is the length of the arm after θ3. In
(iv), the first two link lengths are as defined above. The last link length is
determined by the translational joint, θ3, which determines the additional
amount by which the hand extends beyond the l2 part of the arm.
HW Q4: Let J (θ) : R
n →R
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be the Jacobian of a manipulator. Show
that the manipulability measure µ =
√
det(JJT
) is given by the product of
the singular values J(θ); that is,
µ =
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∏
i=1
σi(θ).
Thus, µ3(θ) is zero if and only if the Jacobian is singular.
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HW Q5: A point in a manipulator’s workspace is said to be isotropic if the
condition number (the ratio of the largest to smallest singular values) of the
Jacobian is 1.
(a) Calculate conditions under which a two-link planar manipulator has
isotropic points and sketch their location in the plane.
(b) Discuss why isotropic points are useful for tasks which involve applying
forces against the environment.
(a) (b)
Figure 3: Illustration of Q1. (a) is before moving the arm. (b) is after
moving the arm to the configuration calculated in your function.
PA Q1: Implement the function in Q1.m. This function will use the built-in
inverse kinematics function in the RTB to calculate a joint configuration that
corresponds to a desired end effector position (just position, not orientation).
The function will take as input a robot (encoded as a SerialLink class) and
a desired position (encoded as a 3×1 vector). It will calculate a target joint
configuration that will cause the end of the robot arm to reach a point at the
center of the sphere (see Figure 3). This function should work for arbitrary
desired positions. Note that in order to use the inverse kinematics functions
in the toolbox, you need to apply the appropriate mask vector.
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PA Q2: Achieve the same result as in PA Q1, but this time using Jacobian
pseudoinverse or Jaboian transpose iterations. The exact solution found by
your function will probably be different from what you found in PA Q1.
However, the end effector should reach the same position.
PA Q3: In Q2, you used Jacobian pseudoinverse or transpose iterations to
solve for the IK. However, notice that you function returns only a single vector of joint angles corresponding to a final configuration. In this question,
I want you to use Jacobian Pseudoinverse control to find a trajectory that
moves the end effector to the goal position in precisely a straight line and
with a specific velocity. The input to the function is a 3 × 1 goal position, a
parameter epsilon that specifies the maximum allowed distance of the manipulator from the goal position at termination, and a parameter velocity
that specifies exactly how far the end effector should move on each time
step. The output of this function should be an n × 9 trajectory that moves
the end effector exactly velocity distance per row of the matrix.
(a)
Figure 4: Tracing out a circle in the workspace
PA Q4: Using the result of Q3, write a function that finds a trajectory (i.e.
an n × 9 matrix of joint angles) that moves the end effector in a circle as
specified in the hw1.m code (Figure 4) at constant velocity. Each row of the
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output trajectory matrix should cause the end effector to move its position
by exactly the velocity specified.
(a) (b)
Figure 5: Joint configurations found by my code before (a) and after (b)
running the code in Q3.
(a)
PA Q5: Now, imagine that there is a two-fingered hand on the end of the
arm. Use Jacobian pseudoinverse control to move the arm/hand so that
the two fingers capture the sphere by moving each finger to one side of the
sphere as shown in hw1.m. There are now TWO objectives in this problem
(to move each finger to the desired spot), not just one. I want you to solve
this problem by formulating a new Jacobian matrix that reflects the twopart objective. Notice that the configuration of the arm and two fingers is
now encoded as an 11-dof (degree of freedom) configuration rather than a
9-dof configuration. The first seven joints are the arm joints. The next two
joints are for finger f1. The final two joints are for finger f2.