Robotics Assignment

Series 1

1) Suppose that three coordinate frames o1x1y1z1, o2x2y2z2, o3x3y3z3 are given, and suppose

;

Find the matrix

2) Consider the following figure; Find the homogenous transformations

,

representing the transformation between the tree frames. Show that

.

3) Consider the following figure. 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 2m 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.

4) In Problem 3, suppose that, after the camera is calibrated, it is rotated 90 about z3.

Recompute the above coordinate transformations.

5) In Problem 3, 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.

Good luck

Dr. Abbas Chatraei

Robotics Assignment

Series 2

3. Derive the forward kinematics equations for the following KUKA robot using DH-

convention ;

3) Solve the inverse kinematics problem of the above robots separately.

رباتیک سوم تکلیف سری

Robotic Assignment

Series 4

1) Consider a rigid body undergoing a pure rotation with no external forces acting on it. The kinetic

energy is then given as

with respect to a coordinate located at the center of mass and whose coordinate axes are principal axes.

Take as generalized coordinates the Euler angles , , and show that the Euler-Lagrange equations

of motion of the rotating body are

2) The two-link robot arm shown in the figure below is attached to the ceiling and under the influence of

the gravitational acceleration g = 9.8062 m/sec2; (x0, y0, z0) is the reference frame; 1, 2 are the

generalized coordinates; d1, d2 are the lengths of the links; and m1 , m2 are the respective masses.

Under the assumption of lumped equivalent masses, the mass of each link is lumped at the end of the

link.

(a) Find the link transformation matrices , i = 1, 2.

(b) (b) Find the inertia matrix Ji for each link.

(c) (c) Derive the Lagrange-Euler equations of motion by first finding the elements in the M(q), h(q,

), and c(q) matrices.

3) Use the Lagrange-Euler formulation to derive the equations of motion for the two-link

RP robot arm shown below, where (x0, y0, z0) is the reference frame, and d are the

generalized coordinates, and m1, m2, are the link masses. Mass m1 of link 1 is assumed to

be located at a constant distance r1 from the axis of rotation of joint 1, and mass m2 of link

2 is assumed to be located at the end point of link 2.

Good Luck

Dr. A. Chatraei

Robotic Assignment

Series 5

1) A single-link rotary robot is required to move from q(0) = 30° to q(2) = 100° in 2 s. The joint

velocity and acceleration are both zero at the initial and final positions. Plan a path for this

robot.

2) In trajectory planning by cubic polynomial functions show that the determinant of the

coefficient matrix is (tf − t0)/4.

3) Suppose we wish a manipulator to start from an initial configuration at time t0 and track a

conveyor. Discuss the steps needed in planning a suitable trajectory for this problem.

4) Write three Matlab m-files cubic.m, quintic.m, lspb.m, to generate Cubic, Quintic and LSPB

trajectories, respectively , given appropriate initial data.

5) Consider the following two-link robot arm and assume that each link is 1 m long. The robot

arm is required to move from an initial position (x0, y0) = (1.96, 0.50) to a final position

(xf, yf) = (1.00, 0.75). The initial and final velocity and acceleration are zero. Plan the path for

this robot.

Good Luck

Dr. Abbas Chatraei