$1.$ The number of degrees of freedom of a robot is:

a$.\_\_\_\_$ the dimension of its configuration space.

b$.\_\_\_\_$ the number of real numbers needed to specify its configuration.

c$.\_\_\_\_$ the number of points on the robot.

d$.\_\_\_\_$ the number of joints of the robot.

e$.\_\_\_\_$ the number of bodies comprising the robot.

f$.\_\_\_\_$ the number of freedoms of the bodies minus the number of independent constraints

between the bodies.

$2.$ To deform one n$-$dimensional space into another topologically equivalent space, which operations are you allowed to use? Select all that apply.

a$.\_\_\_\_$Stretching

b$.\_\_\_\_$Cutting.

c$.\_\_\_\_$Gluing.

$3.$ True or false? An n$-$dimensional space can be topologically equivalent to an m$-$dimensional space, where m$!=$n$.$

a$.\_\_\_\_$True.

b$.\_\_\_\_$False.

$4.$ True or false? An explicit parametrization uses fewer numbers to represent a configuration than an implicit representation.

a$.\_\_\_$True.

b$.\_\_\_$False.

$5.$ True or false? A nonholonomic constraint implies a configuration constraint.

a$.\_\_\_$True.

b$.\_\_\_$False.

$6.$ True or false? A Pfaffian velocity constraint is necessarily nonholonomic.

a$.\_\_\_$True.

b$.\_\_\_$False.

$7.$ True or false? The workspace depends on the robot's joint limits but the task space does not.

a$.\_\_\_$True.

b$.\_\_\_$False.

$8.$ When your thumb points along an axis of rotation, positive rotation about the axis is defined by the direction your fingers curl if you use which thumb?

a$.\_\_\_$Right thumb

b$.\_\_\_$Left thumb

$9.$ When we refer to a frame attached to a moving body, we always consider a stationary frame $\{$b$\},$ because

a$.\_\_\_\_$the motion of all other frames is expressed relative to $\{$b$\}.$

b$.\_\_\_\_\{$b$\}$ is the stationary frame that is coincident $($at a particular instant$)$ with the frame attached to the moving body.

$12.$ True or false? The Jacobian matrix depends on the joint variables

a$.\_\_\_$True.

b$.\_\_\_$False.

$13.$Which column of the space Jacobian does not depend on the joint configuration $\backslash$theta $?$

a$.\_\_\_\_$The first column.

b$.\_\_\_\_$The last column.

c$.\_\_\_\_$All columns depend on $\backslash$theta $.$

$14.$ The body Jacobian is not affected by the choice of

a$.\_\_\_$the space frame $\{$s$\}.$

b$.\_\_\_$the body frame $\{$b$\}.$

$15.$ Consider a robot with $7$ joints and a space Jacobian with a maximum rank of $6$ over all configurations of the robot. At the current configuration, the rank of the space Jacobian is $5.$ Which of the following statements is true?

a$.\_\_\_$The robot is redundant with respect to the task of generating arbitrary end$-$effector twists.

b$.\_\_\_$The robot is kinematically deficient with respect to the task of generating arbitrary end$-$effector twists.

c$.\_\_\_$The robot is at a singularity$.$

$16.$ Consider a robot with $7$ joints and a space Jacobian with a maximum rank of $3$ over all configurations of the robot. At the current configuration, the rank of the space Jacobian is $3.$ Which of the following statements is true? Select all that apply.

a$.\_\_\_$The robot is redundant with respect to the task of generating arbitrary end$-$effector twists.

b$.\_\_\_$The robot is at a singularity$.$

c$.\_\_\_$The space Jacobian is "fat."

$17.$ Consider a robot with $8$ joints and a body Jacobian with rank $6$ at a given configuration. For a given desired end$-$effector twist Vb$,$ what is the dimension of the subspace of joint velocities $($in the $8-$dimensional joint velocity space$)$ that create the desired twist?

a$.\_\_\_2$

b$.\_\_\_0$

c$.\_\_\_$The desired twist cannot be generated.

$18.$ Consider a robot with a full rank Jacobian as it approaches a singular configuration. As it approaches a singular configuration, what happens to the manipulability ellipsoid?

a$.\_\_\_$The length of one principal axis approaches zero.

b$.\_\_\_$The length of one principal axis approaches infinity.

c$.\_\_\_$The interior "volume" of the ellipsoid approaches zero.

d$.\_\_\_$The interior "volume" of the ellipsoid approaches infinity.

$20.$ What are advantages of numerical inverse kinematics over analytic inverse kinematics?

a$.\_\_\_$It can be applied to open$-$chain robots with arbitrary kinematics.

b$.\_\_\_$It requires an initial guess at the solution.

c$.\_\_\_$It returns all possible inverse kinematics solutions.

$22.$ True or false? If you grab the end$-$effector of a robot and try to move it around by hand, the apparent mass $($to you$)$ depends on the configuration of the robot.

a$.\_\_\_\_$True

b$.\_\_\_\_$False.

$23.$ True or false? If you apply $($by hand$)$ a linear force to the end$-$effector of a robot, the end$-$effector will accelerate in the same direction as the applied force.

a$.\_\_\_\_$Always true.

b$.\_\_\_\_$Always false.

c$.\_\_\_\_$Sometimes true, sometimes false.

$24.$ How is the center of mass of a rigid body defined?

a$.\_\_\_\_$The point at the geometric centroid of the body.

b$.\_\_\_\_$The point at the mass$-$weighted $($or density$-$weighted$)$ centroid of the body.

$26.$ How is the center of mass of a rigid body defined?

a$.\_\_\_$The point at the geometric centroid of the body.

b$.\_\_\_$The point at the mass$-$weighted $($or density$-$weighted$)$ centroid of the body.

$29.$ In the labeling scheme for robot dynamics, frame $\{0\}$ is fixed in space and