IRB1_HW006 תרגיל בית 6

Question 1

We are given the following:

with the following D–H Parameters:

The transformation matrices are given as:

Part a

Generalized Coordinates
For this RP manipulator: where:

• is the revolute joint angle
• is the prismatic joint extension

Position Analysis
From the given transformation matrices, the key positions are:

End-effector position:

Center of mass of link (at distance from base):

Velocity Analysis
Taking time derivatives:

End-effector velocity:

Center of mass velocity:

Kinetic Energy Calculation:
Translational kinetic energy of the distributed link:

Rotational kinetic energy of the link:

Kinetic energy of point mass:

Total kinetic energy:

Potential Energy
Since gravity acts along the axis and all motion occurs in the - plane (as evident from the transformation matrices), there is no change in gravitational potential energy:

Lagrangian and Equations of Motion
The Lagrangian is:

Applying the Euler-Lagrange equations from Equation :

For :

For :

Final Matrix Form
Following the standard form from Equation :

We get:

Part b

We are given:

where , , are the starting and ending points respectively, is the dimension of the Euclidean space, and is a polynomial of degree :

a. Determine and
For this RP manipulator, from the transformation matrices we see that the gripper moves in the - plane:

For the gripper’s motion to begin and end with zero velocity and acceleration, we have the following constraints:

Position constraints:

Velocity constraints:

• (zero initial velocity)
• (zero final velocity)

Acceleration constraints:

• (zero initial acceleration)
• (zero final acceleration)

With total constraints, we need a polynomial of degree:

b. Compute coefficients
The polynomial is:

where . The derivatives are:

Applying constraints:
From , , and , we get:

From , , and :

Solving:

Final polynomial:

c. Expressions for , , and
Position:

Velocity:

Acceleration:

where:

d. Joint values, velocities, and accelerations
From the position analysis, the gripper position is:

Joint values (Inverse Kinematics):

Joint velocities:
Using the chain rule and the Jacobian relationship:

Joint accelerations:

e. Generalized forces
Using our solution from Part a:

where:

Explicitly: