Modern Robotics, Course 2: Robot Kinematics

Question 1

True or false? The PoE formula in the space frame only correctly calculates the end-effector configuration if you first put the robot at its zero configuration, then move joint n to θn​, then move joint n−1 to θn−1​, etc., until you move joint 1 to θ1​.

• True

• False

Consider the screw axis Si​ used in the PoE formula. Which of the following is true?

• Si​ represents the screw axis of joint i, expressed in the end-effector frame {b}, when the robot is at its zero configuration.

  
  

• Si​ represents the screw axis of joint i, expressed in the end-effector frame {b}, when the robot is at an arbitrary configuration θ.

• Si​ represents the screw axis of joint i, expressed in the space frame {b}, when the robot is at an arbitrary configuration θ.

• Si​ represents the screw axis of joint Si, expressed in the space frame {b}, when the robot is at its zero configuration.

When the robot is at an arbitrary configuration θ, does the screw axis corresponding to motion along joint i, represented in {s}, depend on θi−1​?

• No

• Yes

When the robot is at an arbitrary configuration θ, does the screw axis corresponding to motion along joint i, represented in {b}, depend on θi−1​?

• Yes

• No

When the robot arm is at its home (zero) configuration, the axis of joint 3, a revolute joint, passes through the point (3,0,0) in the {b} frame. The axis of rotation is aligned with the z^b​-axis of the {b} frame. What is the screw axis B3​?

• 
  

  (0,0,1,−3,0,0)

• 
  

  (0,0,1,0,−3,0)

• (0,0,1,0,0,−3)

In the image below, imagine a frame {c} on the axis of joint 2 and aligned with the {s} frame. What is the screw axis of joint 1 expressed in the frame {c}?

• (0,0,1,0,10,0)

• 
  

  (0,0,1,0,0,10)

True or false? The Jacobian matrix depends on the joint variables.

• True

• False

True or false? The Jacobian matrix depends on the joint velocities.

• True

• False

True or false? Row i of the Jacobian corresponds to the end-effector velocity when joint i moves at unit speed and all other joints are stationary.

• True

• False

Consider a square Jacobian matrix that is usually full rank. At a configuration where one row of the Jacobian becomes a scalar multiple of another row, is the robot at a singularity?

• Yes

• No