Robot Modeling and Control. Mark W. Spong

coordinates (x, y, z). If Q specifies the orientation of the end effector frame with respect to the base frame, and T is the vector from the base frame to the origin of the end effector frame, show that the coordinates of p with respect to the base frame are given by (2.77)in which (0, x, y, z) is a quaternion with zero as its real component.

      35 Verify Equation (2.67).

      36 Compute the homogeneous transformation representing a translation of 3 units along the x-axis followed by a rotation of about the current z-axis followed by a translation of 1 unit along the fixed y-axis. Sketch the frame. What are the coordinates of the origin o1 with respect to the original frame in each case?

      37 Consider the diagram of Figure 2.13. Find the homogeneous transformations representing the transformations among the three frames shown. Show that .

      38 Consider the diagram of Figure 2.14. A robot is set up 1 meter from a table. The table top is 1 meter high and 1 meter square. A frame o1x1y1z1 is fixed to the edge of the table as shown. A cube measuring 20 cm on a side is placed in the center of the table with frame o2x2y2z2 established at the center of the cube as shown. A camera is situated directly above the center of the block 2 meters above the table top with frame o3x3y3z3 attached as shown. Find the homogeneous transformations relating each of these frames to the base frame o0x0y0z0. Find the homogeneous transformation relating the frame o2x2y2z2 to the camera frame o3x3y3z3.

      39 In Problem 2–38, suppose that, after the camera is calibrated, it is rotated 90° about z3. Recompute the above coordinate transformations.

      40 If the block on the table is rotated 90° about z2 and moved so that its center has coordinates [0, .8, .1]T relative to the frame o1x1y1z1, compute the homogeneous transformation relating the block frame to the camera frame; the block frame to the base frame.

      41 Consult an astronomy book to learn the basic details of the Earth’s rotation about the sun and about its own axis. Define for the Earth a local coordinate frame whose z-axis is the Earth’s axis of rotation. Define t = 0 to be the exact moment of the summer solstice, and the global reference frame to be coincident with the Earth’s frame at time t = 0. Give an expression R(t) for the rotation matrix that represents the instantaneous orientation of the earth at time t. Determine as a function of time the homogeneous transformation that specifies the Earth’s frame with respect to the global reference frame.

      42 In general, multiplication of homogeneous transformation matrices is not commutative. Consider the matrix product Determine which pairs of the four matrices on the right-hand side commute. Explain why these pairs commute. Find all permutations of these four matrices that yield the same homogeneous transformation matrix, .

Figure 2.13 Diagram for Problem 2–37.

Figure 2.14 Diagram for Problem 2–38.

      Notes and References

Rigid body motions and the groups SO(n) and SE(n) are often addressed in mathematics books on the topic of linear algebra. Standard texts for this material include [9], [30], and [49]. These topics are also often covered in applied mathematics texts for physics and engineering, such as [143], [155], and [182]. In addition to these, a detailed treatment of rigid body motion developed with the aid of exponential coordinates and Lie groups is given in [118].

      Notes

      1 1 We will use , to denote both coordinate axes and unit vectors along the coordinate axes depending on the context.

      2 2 It should be noted that other conventions exist for naming the roll, pitch, and yaw angles.

      3 3 The definition of rigid motion is sometimes broadened to include reflections, which correspond to detR = −1. We will always assume in this text that detR = +1 so that R ∈ SO(3).