Homework #1 Due 1/23/08

Read: Craig, Ch. 1.

1. This exercise uses the simple two-degree-of-

freedom holonomic-robot model shown to illustrate a kinematics

problem.

(a) Write an expression for the

forward kinematics associated with point P in the y-direction,

i.e. y_P = y_P(q₁,q₂).

Note that the counterpart to this for the x-direction is

x_P = x_P(q₁,q₂) = L₁cos(q₁)+L₂cos(q₁+q₂).

(b) Referring to the Jacobian as defined below, apply

the expressions above for x_P and y_P as functions of q₁ and q₂

to determine the four elements of the Jacobian [J], also

as functions of q₁ and q₂.

2. This exercise uses the same system to

illustrate a kinetics or dynamics problem.

Assume that a motor torque t₁ is applied from

the base onto the blue member 1 as indicated.

At the same time a torque t₂ is applied from

the blue member 1 onto the green member 2.

Note that both torques as described act counterclockwise in

the positive directions.

We seek to put the governing kinetics (or dynamics)

equations of motion into the standard form:

d²q₁/dt²=f₁(q₁,q₂,dq₁/dt,dq₂/dt,t₁,t ₂)

d²q₂/dt²=f₂(q₁,q₂,dq₁/dt,dq₂/dt,t₁,t ₂)

using the free-body diagrams. Assume that

members 1 and 2 have the masses and lengths

indicated, that they are narrow with uniform

cross sections, and that the masses of the

actuators can be neglected.