Figure P41 illustrates a robot arm that has two “links” connected by two “joints”—a should...

Figure P41 illustrates a robot arm that has two “links” connected by two “joints”—a shoulder or base joint and an elbow joint. There is a motor at each joint. The joint angles are θ1 and θ2. The (x, y) coordinates of the hand at the end of the arm are given by

where L1 and L2 are the lengths of the links.

Figure P41

Polynomials are used for controlling the motion of robots. If we start the arm from rest with zero velocity and acceleration, the following polynomials are used to generate commands to be sent to the joint motor controllers.

where θ1(0) and θ2(0) are the starting values at time t = 0. The angles θ1(tf) and θ2(tf) are the joint angles corresponding to the desired destination of the arm at time tf. The values of θ1(0), θ2(0), θ1(tf), and θ2(tf) can be found from trigonometry, if the starting and ending (x, y) coordinates of the hand are specified.

a. Set up a matrix equation to be solved for the coefficients a1, a2, and a3, given values for θ1(0), θ1(tf), and tf. Obtain a similar equation for the coefficients b1, b2, and b3.

b. Use MATLAB to solve for the polynomial coefficients given the values tf = 2 secs, θ1(0) = -19°, θ2(0) = 44°, θ1(tf) = 43°, and θ2(tf) = 151°. (These values correspond to a starting hand location of x = 6.5, y = 0 ft, and a destination location of x = 0, y = 2 ft, for L1 = 4 and L2 = 3 ft.)

c. Use the results of part b to plot the path of the hand.