$29.$ In the labeling scheme for robot dynamics, frame $\{0\}$ is fixed in space and frame $\{1\}$ is at the center of mass of link $1.$ Where is frame $\{$n$+1\}?$

a$.\_\_\_$There is no frame $\{$n$+1\}.$

b$.\_\_\_$It is at the center of mass of link n$+1.$

c$.\_\_\_$It is at the end$-$effector, at a location fixed relative to link n$.$

$30.$ In the forward iterations of the recursive Newton$-$Euler dynamics, what terms are summed to create the twist Vi of link $\{$i$\}($expressed in $\{$i$\}$ coordinates$)?$ Select all that apply.

a$.\_\_\_\_$A term representing the twist of link $\{$i$1\},$ expressed in $\{$i$\}.$

b$.\_\_\_\_$A term representing the twist of link $\{$i$+1\},$ expressed in $\{$i$\}.$

c$.\_\_\_\_$A term representing the twist of link $\{$ii$\}$ due to the joint velocity $\backslash$theta $$i$.$

d$.\_\_\_\_$A term representing the twist of link $\{$ii$\}$ due to the joint acceleration $\backslash$theta $$i$.$

$31.$ In the backward iterations, the wrench Fi at link i is calculated as the sum of the wrench that must be transmitted to link i$+1($but expressed in frame $\{$i$\})$ and the wrench needed for link i by the rigid$-$body dynamics. How is the joint force or torque $\backslash$tau i$$ related to Fi$?$

a$.\_\_\_$They are equivalent.

b$.\_\_\_\backslash$tau i$$ is the dot product of Fi and the joint axis Ai$.$

$32.$ Which of these refers to the ability of a motion planner to find a solution when one exists?

a$.\_\_\_$Computational complexity

b$.\_\_\_$Completeness

c$.\_\_\_$Optimality

$33.$ If the robot is a dynamic mechanical system, and the inputs u are forces, what is the state of the system?

a$.\_\_\_$The configuration q

b$.\_\_\_$The configuration q and velocity v

$35.$ Which of the following is a valid optimistic cost$-$to$-$go from a graph node, assuming that there is a path from the node to the goal? Select all that apply.

a$.\_\_\_$Zero

b$.\_\_\_$Infinity

c$.\_\_\_$The exact minimum cost to get to the goal

$37.$ If the optimistic cost$-$to$-$go heuristic returns an estimated cost$-$to$-$go from a node that is higher than the actual minimum cost to get to the goal node, what is the effect on A$*$ search?

a$.\_\_\_$The solution found by A$*$ may not be optimal

b$.\_\_\_$A$*$ may fail to find a solution when one exists.

$38.$ What is hybrid motion$-$force control?

a$.\_\_\_$Switching between controlling motions and forces.

b$.\_\_\_$Controlling motions in some directions and forces in other directions.

c$.\_\_\_$Controlling the mass$-$spring$-$damper properties of a robot.

$39.$ If the controller acts based on sensor feedback, this is:

a$.\_\_\_$Closed$-$loop control.

b$.\_\_\_$Open$-$loop control.

$40.$ If the robot is at a state $($s$,$s$)$ where no feasible acceleration s$$ exists that satisfies the actuator force and torque bounds, what happens?

a$.\_\_\_$One or more of the actuators is damaged.

b$.\_\_\_$The robot leaves the path.

c$.\_\_\_$The robot must begin to decelerate, s$<0.$

$41.$ For the first$-$order error dynamics

$\backslash$theta $$e$($t$)+($k$/$b$)\backslash$theta e$($t$)=0,$

k is a $($virtual$)$ spring and b is a $($virtual$)$ damper. The time constant of the dynamics is t$=$b$/$k$.$

Which of the following statements is true? Select all that apply.

a$.\_\_\_$If t$<0,$ the step error response has overshoot.

b$.\_\_\_$If t$<0,$ the error dynamics is unstable.

c$.\_\_\_\_$For two different first$-$order error dynamics, one with a time constant t$1>0$ and the other with a time constant t$2>0,$ where t$1>$t$2,$ the $2\%$ settling time for the system with time constant t$2$ is shorter.

$42.$ For the first$-$order error dynamics $\backslash$theta $$e$+(1/$t$)\backslash$theta e$=0,$ which of the following is closest to the $2\%$ settling time for the step error response?

a$.\_\_\_$t

b$.\_\_\_2$t

c$.\_\_\_4$t

$43.$ What is the order of the differential equation describing the error dynamics for a torque$-$controlled robot using PID control?

a$.\_\_\_$First order.

b$.\_\_\_$Second order.

c$.\_\_\_$Third order.

$44.$ Which of the following is a reason to add an integral control term $($i$.$e$.,$ to use PID control instead of PD control$)?$

a$.\_\_\_$To improve the transient error response.

b$.\_\_\_$To improve the steady$-$state error response.

$45.$ Why do many real robot controllers avoid the use of integral control $($i$.$e$.,$ PD control is favored over PID control$)?$

a$.\_\_\_$Stability is more important than steady$-$state error

b$.\_\_\_$Integral control is not useful for practical systems.

$46.$ What does "contact kinematics" refer to$?$

a$.\_\_\_$The feasible motions between rigid bodies in contact.

b$.\_\_\_$A description of the forces that can be applied through a contact.

$47.$ The dynamic equations of motion of a robot include a mass matrix times joint accelerations, a quadratic velocity$-$product term, and a gravity term. Which of these could be part of a quasistatic analysis?

a$.\_\_\_$Mass matrix times acceleration.

b$.\_\_\_$Velocity$-$product term.

c$.\_\_\_$Gravity term.