### Week 01: Modern Robotics, Course 4: Robot Motion Planning and Control Quiz Answers

#### Quiz 01: Lecture Comprehension, Overview of Motion Planning (Chapter 10.1)

Q1. Which of these refers to the ability of a motion planner to find a solution when one exists?

- Computational complexity
**Completeness**- Optimality

Q2. If the robot is a dynamic mechanical system, and the inputs u*u* are forces, what is the state of the system?

- The configuration q
*q*. **The configuration q***q*and the velocity v*v*.

#### Quiz 02: Lecture Comprehension, C-Space Obstacles (Chapter 10.2.1)

Q1. Consider a two-joint RP robot with joint limits on both the revolute joint and the prismatic joint. In the absence of obstacles, how many connected components does the free C-space have?

**1**- 2
- 3

#### Quiz 03: Lecture Comprehension, Graph Search (Chapter 10.2.4)

Q1. 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.

**Zero**.- Infinity.
**The exact minimum cost to get to the goal node.**

Q2. Can the estimated total cost assigned to a node ever increase?

- Yes.
**No.**

Q3. 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?

**The solution found by A* may not be optimal.**- A* may fail to find a solution when one exists.

#### Quiz 04: Lecture Comprehension, Complete Path Planners (Chapter 10.3)

Q1. What properties does a C-space roadmap satisfy? Select all that apply.

**For each connected component of the free C-space, there is a connected component of the roadmap.**- Every free configuration is represented by a point on the roadmap.
**There is a simple planner that can find a free path from any free configuration to a point on the roadmap.**- From every free configuration, there is a straight-line path to a point on the roadmap.

#### Quiz 05: Chapter 10 through 10.3, Motion Planning

Q1. The lecture on complete path planners describes a planner that finds plans for a polygon translating among polygonal obstacles. Which of the following properties is satisfied by the planner? Select all that apply.

- The planner finds solutions that go exactly to the goal configuration, as opposed to those that go approximately to the goal configuration.
- The planner finds optimal (shortest) solutions, as opposed to simply satisficing solutions.
- The planner is complete.

Q2. The figure below shows the C-space of a 2R robot. Remembering how the C-space wraps around, how many connected components does the free C-space have? C-space obstacles are represented as gray.

Q3. Two graph nodes, A and B, are in the OPEN list of an A* search. Node A has a heuristic_cost_to_go of 20. Node B has a heuristic_cost_to_go of 10. Which node will be explored from next?

- A
- B
- Not enough information to say.

Q4. For the graph below, which of the following is a correct characterization? Select all that apply.

- Weighted
- Unweighted
- Directed
- Undirected
- Tree

### Modern Robotics, Course 4: Robot Motion Planning and Control Week 02 Quiz Answers

#### Quiz 01: Lecture Comprehension, Grid Methods for Motion Planning (Chapter 10.4)

Q1. If the C-space is represented by three coordinates, and each coordinate range is subdivided into 10 intervals, how many cells are in the C-space grid representation?

- 10 x 3 = 30
**10^3 = 1,000**- 3^10 = 59,049

Q2. Which of the following statements about a motion planner is correct?

- Resolution completeness is a stronger notion than completeness.
**Completeness is a stronger notion than resolution completeness.**

Q3. A C-space cell is marked as “collision” if any configuration in the cell is in collision. Otherwise the cell is marked as “free.” The configuration that represents a cell is the point at the middle of the cell. Consider two adjacent 4-connected cells in a 2-dimensional grid. True or false: the straight-line path between two adjacent free cells is necessarily collision free

**True**- False

#### Quiz 02: Lecture Comprehension, Sampling Methods for Motion Planning (Chapter 10.5, Part 1 of 2)

Q1. A probabilistic roadmap (PRM) representing the free C-space is generated by sampling configurations in C-space (the nodes of the graph) and then connecting nearby nodes by edges that correspond to collision-free paths found by a simple local planner. What methods could be used to generate the sampled configurations? Select all that apply.

**Random sampling from a uniform distribution over the C-space.****Random sampling from a non-uniform distribution over the C-space.****Sampling by a deterministic algorithm.**

#### Quiz 03: Lecture Comprehension, Sampling Methods for Motion Planning (Chapter 10.5, Part 2 of 2)

Q1. The RRT algorithm generates new edges using a local planner, which takes as input a sampled state x_{\text{samp}}*x*samp and a state x_{\text{nearest}}*x*nearest (already in the search tree). What properties should the local planner satisfy? Select all that apply. **(This is consistently one of the most incorrectly answered questions in this course, so think about it carefully!)**

- The local planner should return a path from x_{\text{nearest}}
*x*nearest to x_{\text{samp}}*x*samp. **The local planner should execute quickly.**- The local planner should only return collision-free paths.

Q2. Is an RRT better described as a general “graph” or more specifically as a “tree?”

**Tree**- Graph

#### Quiz 04: Lecture Comprehension, Virtual Potential Fields (Chapter 10.6)

Q1. First we construct a potential field representing the goal configuration and the obstacles: the potential is minimized at the goal configuration and is high near the obstacles. Which of the following might be a good choice for a robot controller based on the potential field P?

- Command robot velocities equal to the gradient of P.
- Command robot forces equal to the negative of the gradient of P.
**Command robot forces equal to the negative of the gradient of P, plus some damping forces.****Command robot velocities equal to the negative of the gradient of P.**

Q2. The point of using virtual potential fields is to drive the robot to the global minimum of the potential, which should be at the goal configuration. The global minimum is also a local minimum, which is one example of a critical point of the potential function, where the gradient goes to zero (i.e., there is no “downhill” or “uphill” direction, when considering only the derivatives of the potential function; there may be a downhill or uphill direction considering second or higher derivatives). Other critical points include local minima that are not at the goal configuration, local maxima, and saddles (which are local minima in some directions and local maxima in other directions).

Which of the following critical points represent significant problems for virtual potential field controllers? Select all that apply.

**Local minima not at the goal configuration.**- Saddles.
- Local maxima.

#### Quiz 05: Lecture Comprehension, Nonlinear Optimization (Chapter 10.7)

Q1. In the “shooting” approach to nonlinear optimization, you guess the controls (represented by some finite parametrization of the control history) and use simulation to calculate the resulting trajectory of the robot. You iterate this process to find a set of controls that satisfies the constraints and (locally) minimizes the cost. To assist this process, which gradients might you have to calculate? Select all that apply.

**The gradient of a representation of the trajectory with respect to the controls.****The gradient of the constraints with respect to the trajectory.**- The gradient of the controls with respect to the cost.
**The gradient of the cost with respect to the controls.**

Q2. Which formulation of a nonlinear optimization problem would you expect to have more design variables and constraints, shooting or collocation?

- Shooting.
**Collocation.**

#### Quiz 06: Chapter 10.4 through 10.7, Motion Planning

Q1. Which properties does a basic RRT planner satisfy? Select all that apply.

- Complete.
- Optimal.
- Single-query planning.

Q2 If you represent the 6-dimensional C-space of a rigid body as a grid, with 20 discrete values along each of the 6 dimensions, how many cells does the C-space grid have

Q3. If a potential field is quadratic in the configuration, which of the following describes its gradient?

- Linear in the configuration.
- Cubic in the configuration.
- Neither of the above.

Q4. A probabilistic roadmap (PRM) approach is likely to be preferred to an RRT in which situation? (Choose only one.)

- When many motion plans will be generated in the same static obstacle environment.
- When the obstacle environment changes between motion plans.

### Modern Robotics, Course 4: Robot Motion Planning and Control Week 03 Quiz Answers

#### Quiz 01: Lecture Comprehension, Control System Overview (Chapter 11.1)

Q1. What is hybrid motion-force control?

- Switching between controlling motions and forces.
**Controlling motions in some directions and forces in other directions.**- Controlling the mass-spring-damper properties of a robot.

Q2. If the controller acts based on sensor feedback, this is:

**Closed-loop control.**- Open-loop control

#### Quiz 02: Lecture Comprehension, Error Response (Chapter 11.2.1)

Q1. Which of the following are components of the transient error response? Select all that apply.

- The steady-state error.
**The 2% settling time.****The overshoot.**

#### Quiz 03: Lecture Comprehension, First-Order Error Dynamics (Chapter 11.2.2.1)

Q1. For the decaying response \theta_e(t) = \theta_e(0) e^{-at}*θe*(*t*)=*θe*(0)*e*−*at*, a>0*a*>0, what is the time constant \mathfrak{t}t of the decay?

**\mathfrak{t}=at=***a*- \mathfrak{t} = 1/at=1/
*a*

Q2. For the first-order error dynamics

\dot{\theta}_e(t) +(k/b) \theta_e(t) = 0*θ*˙*e*(*t*)+(*k*/*b*)*θ**e*(*t*)=0,

k*k* is a (virtual) spring and b*b* is a (virtual) damper. The time constant of the dynamics is \mathfrak{t} = b/kt=*b*/*k*.

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

- If \mathfrak{t}<0t<0, the step error response has overshoot.
**If \mathfrak{t}<0t<0, the error dynamics is unstable.****For two different first-order error dynamics, one with a time constant \mathfrak{t}_1 > 0t1>0 and the other with a time constant \mathfrak{t}_2 > 0t2>0, where \mathfrak{t}_1 > \mathfrak{t}_2t1>t2, the 2% settling time for the system with time constant \mathfrak{t}_2t2 is shorter.**

Q3. For the first-order error dynamics \dot{\theta}_e + (1/\mathfrak{t}) \theta_e = 0*θ*˙*e*+(1/t)*θe*=0, which of the following is closest to the 2% settling time for the step error response?

- \mathfrak{t}t
- 2\mathfrak{t}2t
**4\mathfrak{t}4t**

### Quiz 04: Lecture Comprehension, Second-Order Error Dynamics (Chapter 11.2.2.2)

Q1. The second-order error dynamics

\ddot{\theta}_e(t) + 2 \zeta \omega_n \dot{\theta}_e(t) + \omega_n^2 \theta_e(t) = 0*θ*¨*e*(*t*)+2*ζ**ω**n**θ*˙*e*(*t*)+*ω**n*2*θ**e*(*t*)=0

has a characteristic equation

s^2 + 2 \zeta \omega_n s + \omega_n^2=0*s*2+2*ζ**ω**n**s*+*ω**n*2=0,

with roots s_{1,2} = -\zeta \omega_n \pm \omega_n \sqrt{\zeta^2-1}*s*1,2=−*ζωn*±*ωn**ζ*2−1. Assume that \zeta*ζ* and \omega_n*ωn* are both positive. Which of the following could possibly describe the step error response? Select all that apply.

- The error grows without bound.
**A decaying sinusoid.****A monotonically decaying signal with no overshoot.**

Q2. Refer back to the equations at the beginning of Question 1. Which of the following statements is true? Select all that apply.

**The roots are complex conjugates if the system is underdamped.****The roots are real if the system is critically damped or overdamped.**- A critically damped system with roots at s = -2
*s*=−2 has a faster settling time than a critically damped system with roots at s=-4*s*=−4.

#### Quiz 05: Lecture Comprehension, Motion Control with Velocity Inputs (Chapter 11.3, Part 1 of 3)

Q1. The joint controller is \dot{\theta} = K_p \theta_e*θ*˙=*Kp**θe*. If the reference has \dot{\theta}_d = 0*θ*˙*d*=0, which of the following describes the error dynamics?

**First-order homogeneous differential equation.**- First-order nonhomogeneous differential equation.
- Second-order homogeneous differential equation.
**Second-order nonhomogeneous differential equation.**

Q2. The joint controller is \dot{\theta} = K_p \theta_e*θ*˙=*Kp**θe*. If the reference has \dot{\theta}_d = k \neq 0*θ*˙*d*=*k*=0, which of the following describes the error dynamics?

- First-order homogeneous differential equation.
- First-order nonhomogeneous differential equation.
- Second-order homogeneous differential equation.
- Second-order nonhomogeneous differential equation.

Q3. A controller that commands a joint velocity proportional to a position error is called a proportional (P) controller, \dot{\theta} = K_p \theta_e*θ*˙=*Kp**θe*. This can reduce the error \theta_e*θe* to zero if the reference trajectory is \theta_d = k*θd*=*k* for some constant k*k* (i.e., \dot{\theta}_d =0*θ*˙*d*=0), but it cannot reduce error to zero if the reference trajectory is \theta_d(t) = kt*θd*(*t*)=*kt* (a constant velocity k \neq 0*k*=0). Which of the following is a correct explanation why?

- To reduce error to zero, K_p
*Kp* must be larger than k*k*. **To maintain zero error, the controller must command a constant velocity k>0***k*>0. The P controller requires constant nonzero error to command a constant nonzero velocity.

#### Quiz 06: Lecture Comprehension, Motion Control with Velocity Inputs (Chapter 11.3, Part 2 of 3)

Q1. The PI control law for a velocity-controlled joint is

\dot{\theta}(t) = K_p \theta_e(t) + K_i \int_0^t \theta_e(\text{t}) d \text{t}*θ*˙(*t*)=*K**p**θ**e*(*t*)+*K**i*∫0*t**θ**e*(t)*d*t.

If the reference trajectory satisfies \theta_d(t) = kt*θ**d*(*t*)=*k**t* (i.e., \dot{\theta}_d(t) = kt*θ*˙*d*(*t*)=*k**t*) for some k\neq 0*k*=0, the error dynamics satisfies

\ddot{\theta}_e(t) + K_p \dot{\theta}_e(t) + K_i \theta_e(t) = 0.*θ*¨*e*(*t*)+*K**p**θ*˙*e*(*t*)+*K**i**θ**e*(*t*)=0.

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

**If the error dynamics are stable, then increasing K_p***Kp* will increase the damping ratio.- If the error dynamics are stable, then increasing K_i
*Ki* will increase the damping ratio. - If the error dynamics are stable, then increasing K_p
*Kp* will increase the natural frequency \omega_n*ωn* of the error dynamics.

#### Quiz 07: Lecture Comprehension, Motion Control with Velocity Inputs (Chapter 11.3, Part 3 of 3)

Q1. If X_d(t)*Xd*(*t*) is the SE(3)*SE*(3) representation of the desired end-effector frame in the space frame, and \dot{X}_d*X*˙*d* is its time derivative at a given time, then X_d^{-1} \dot{X}_d*Xd*−1*X*˙*d* is the se(3)*se*(3) representation of the twist at that time in the end-effector frame, and \dot{X}_d X_d^{-1}*X*˙*d**Xd*−1 is the se(3)*se*(3) representation of the twist in the space frame {s} (as we learned in Chapter 3). In the feedforward plus PI feedback control law described in the video, what is \mathcal{V}_dV*d*?

- X_d^{-1} \dot{X}_d \in se(3)
*Xd*−1*X*˙*d*∈*se*(3). - \dot{X}_d X_d^{-1} \in se(3)
*X*˙*d**Xd*−1∈*se*(3). **The 6-vector \mathcal{V}_dV***d* satisfying [\mathcal{V}_d] = X_d^{-1} \dot{X}_d[V*d*]=*Xd*−1*X*˙*d*.- The 6-vector \mathcal{V}_dV
*d* satisfying [\mathcal{V}_d] = \dot{X}_d X_d^{-1}[V*d*]=*X*˙*d**Xd*−1.

Q2. \mathcal{V}_dV*d* is the twist for the desired end-effector trajectory expressed in the desired end-effector frame {d}. [\text{Ad}_{X_{bd}}] \mathcal{V}_d[Ad*X**b**d*]V*d* is this same twist, but expressed in the current end-effector frame {b}. The feedforward plus feedback control law is

\mathcal{V}_b = [\text{Ad}_{X_{bd}}] \mathcal{V}_d + K_p X_e + K_i \int X_e d\text{t}V*b*=[Ad*X**b**d*]V*d*+*K**p**X**e*+*K**i*∫*X**e**d*t.

Select all the statements below that are true.

**When the feedback gains are zero, the error X_e***Xe* remains constant.- If we used \mathcal{V}_dV
*d* instead of [\text{Ad}_{X_{bd}}] \mathcal{V}_d[Ad*Xbd*]V*d* as the feedforward term, and the feedback gains are zero, the error X_e*Xe* remains constant. **If we used \mathcal{V}_dV***d* instead of [\text{Ad}_{X_{bd}}] \mathcal{V}_d[Ad*Xbd*]V*d* as the feedforward term, the new control law would command the same twist as the original control law when the error X_e*Xe* is zero.

Q3. What is the error twist X_e*Xe*?

- X_{sb}^{-1} X_{sd}
*Xsb*−1*Xsd*. **\log (X_{sb}^{-1} X_{sd})log(***Xsb*−1*Xsd*).

Q4. Once a controller twist \mathcal{V}_bV*b* is calculated, the commanded joint velocities can be calculated as \dot{\theta} = J_b^\dagger(\theta) \mathcal{V}_b*θ*˙=*Jb*†(*θ*)V*b*. Under what circumstances can the inverse Jacobian J_b^{-1}(\theta)*Jb*−1(*θ*) be used instead of the pseudoinverse?

**The Jacobian is 6 \times 66×6 and full rank.**- The Jacobian is square.

#### Quiz 08: Chapter 11 through 11.3, Robot Control

Q1. Which of the following is the characteristic equation of

4\ddot{x} + 8 \dot{x} + 12 x = 04*x*¨+8*x*˙+12*x*=0?

- 4 \ddot{s} + 8 \dot{s} + 12 s = 04
*s*¨+8*s*˙+12*s*=0. - s^2 + 2 s + 3 = 0
*s*2+2*s*+3=0.

Q2. What is the characteristic equation corresponding to

\dot{x} = \left[

*x*˙1*x*˙2*x*˙3

\right] = \left[

00−310−201−1

\right] \left[

*x*1*x*2*x*3

\right]*x*˙=⎣⎢⎡*x*˙1*x*˙2*x*˙3⎦⎥⎤=⎣⎢⎡00−310−201−1⎦⎥⎤⎣⎢⎡*x*1*x*2*x*3⎦⎥⎤?

- s^3+s^2+2s+3=0
*s*3+*s*2+2*s*+3=0. - s^3-s^2-2s-3=0
*s*3−*s*2−2*s*−3=0.

Q3. Referring to the previous question: is the differential equation stable or unstable? (You can use a test in the book, or use software to calculate roots or eigenvalues.)

- Stable.
- Unstable.

Q4. Which of the following are hallmarks of the response of an underdamped system? Select all that apply.

- Overshoot
- Oscillation
- Steady-state error

Q5. What is the damping ratio \zeta*ζ* for the second-order linear dynamics described by the characteristic equation s^2+2s + 2 =0*s*2+2*s*+2=0? Enter a numeric value, accurate to within 0.0

Q6. What is the damped natural frequency \omega_d*ωd* for the dynamics \ddot{x} + 3\dot{x} + 9x =0*x*¨+3*x*˙+9*x*=0? Enter a numeric value, accurate to within 0.01

Q7. The roots of the characteristic equations of the second-order error dynamics for two different controlled systems, A and B, are shown below. The two roots for B are collocated. Which of the following statements are true? Select all that apply.

- The error response of system A is less damped.
- The error response of system B has more oscillation.
- The error response of system B has a faster settling time.
- The error dynamics of system B are underdamped.
- The error dynamics of system A are overdamped.

Q8. Consider the robot joint error dynamics described by the characteristic equation

s^2 + K_p s + K_i =0*s*2+*K**p**s*+*K**i*=0,

where the reference motion of the joint has a constant velocity and K_p*Kp* and K_i*Ki* are the P and I control gains, respectively, in your PI joint velocity controller \dot{\theta} = K_p \theta_e + K_i \int \theta_e d\text{t}*θ*˙=*Kp**θe*+*Ki*∫*θe**d*t. If you want critical damping and you have committed to choosing K_i = 10*Ki*=10, what value of K_p*Kp* should you use? Enter a numeric value, accurate to within 0.1

Q9. What are the units for the K_p*Kp* value you entered in Question 8?

- 1/s
- radian/s
- s/radian

Q10. For Question 8, what is the 2% settling time of your error dynamics for your solution to K_p*Kp*? Enter a numeric value, accurate to within 0.1 s

Q11. Consider again the robot joint error dynamics described by the characteristic equation

s^2 + K_p s + K_i =0*s*2+*K**p**s*+*K**i*=0.

If we fix K_p=20*Kp*=20, what is the 2% settling time if we choose K_i*Ki* for critical damping? Enter a numeric value, accurate to within 0.01 s.

### Modern Robotics, Course 4: Robot Motion Planning and Control Week 04 Quiz Answers

#### Quiz 01: Lecture Comprehension, Motion Control with Torque or Force Inputs (Chapter 11.4, Part 1 of 3)

Q1. For PD control of a single joint to a constant setpoint in zero gravity, as discussed in the video, the error dynamics can be written as a second-order linear homogeneous differential equation. The characteristic equation corresponding to this differential equation is

s^2 + ((b+K_d)/M) s + K_p/M = 0*s*2+((*b*+*K**d*)/*M*)*s*+*K**p*/*M*=0,

and we can control the locations of the two roots in the complex plane by choosing the control gains K_p*Kp* and K_d*Kd*. Which of the following statements is true? Select all that apply.

**To increase the damping ratio, we could decrease K_p***Kp*.- The response is better if the roots are further to the right in the complex plane.
**The response is better if the roots are closer to the real axis.****The steady-state error is zero if the error dynamics are stable.**- It is not possible to have critical damping if K_d=0
*Kd*=0.

#### Quiz 02: Lecture Comprehension, Motion Control with Torque or Force Inputs (Chapter 11.4, Part 2 of 3)

Q1. What is the order of the differential equation describing the error dynamics for a torque-controlled robot using PID control?

- First order.

- Second order.
**Third order.**

Q2. Which of the following is a reason to add an integral control term (i.e., to use PID control instead of PD control)?

- To improve the transient error response.
**To improve the steady-state error response.**

Q3. Why do many real robot controllers avoid the use of integral control (i.e., PD control is favored over PID control)?

**Stability is more important than steady-state error.**- Integral control is not useful for practical systems.

#### Quiz 02: Lecture Comprehension, Motion Control with Torque or Force Inputs (Chapter 11.4, Part 3 of 3)

Q1. The joint-space computed-torque control law calculates the commanded joint acceleration as the sum of the feedforward acceleration \ddot{\theta}_d*θ*¨*d* and a PID or PD feedback correction term. How are commanded joint accelerations converted to commanded joint torques and forces?

- Using the inertia matrix model \tilde{M}(\theta)
*M*~(*θ*). **Using the inertia matrix model \tilde{M}(\theta)***M*~(*θ*) and added torques due to the model of gravitational and velocity-product terms \tilde{h}(\theta,\dot{\theta})*h*~(*θ*,*θ*˙).

Q2. The task-space computed torque control law uses a model of the end-effector mass matrix \tilde{\Lambda}Λ~ and the end-effector gravitational and velocity-product terms \tilde{\eta}*η*~ to compute a commanded end-effector wrench \mathcal{F}_bF*b*. How is this wrench converted to a set of joint torques and forces?

**\tau = J_b^{{\rm T}}(\theta) \mathcal{F}_b***τ*=*Jb*T(*θ*)F*b*.- \tau = J_b^{-1}(\theta) \mathcal{F}_b
*τ*=*Jb*−1(*θ*)F*b*. - \tau = J_b(\theta) \mathcal{F}_b
*τ*=*Jb*(*θ*)F*b*. - \tau = J_b^\dagger(\theta) \mathcal{F}_b
*τ*=*Jb*†(*θ*)F*b*.

#### Quiz 03: Lecture Comprehension, Force Control (Chapter 11.5)

Q1. Usually a force controller does not use a model of the robot’s mass matrix and velocity-product terms. What is one reason?

- It is difficult to get a good model of the mass matrix and velocity-product terms.
**Typically the velocity and acceleration of the robot is negligible during a force control task.**

#### Quiz 04: Lecture Comprehension, Hybrid Motion-Force Control (Chapter 11.6)

Q1. The robot end-effector firmly grips the handle of a drawer, that can only slide in and out with one degree of freedom. The robot is controlled with a hybrid motion-force controller. The controlled motion is the opening and closing of the drawer. What is the rank of the matrix P(\theta)*P*(*θ*) that projects arbitrary end-effector wrenches to the components of the wrench that cause motion without creating forces against the constraints?

**1**- 5
- 6

#### Quiz 05: Chapter 11.4 through 11.6, Robot Control

Q1. Consider a single-joint robot driven by a PID controller that commands a joint torque. Assume the characteristic equation of the error dynamics can be written in the form

s^3 + \frac{b+K_d}{M}s^2 + \frac{K_p}{M} s + \frac{K_i}{M} = 0*s*3+*M**b*+*K**d**s*2+*M**K**p**s*+*M**K**i*=0,

and let M = 1*M*=1, b = 2*b*=2, K_d = 3*Kd*=3, and K_p = 4*Kp*=4. What is the maximum value of K_i*Ki* for which the system is stable? Enter a numeric value, accurate to within 0.1

Q2. For the previous problem, what is the minimum value of K_i*Ki* for which the system is stable? Enter a numeric value, accurate to within 0.1

Q3. For the error characteristic equation of Question 1, we would like to choose the control gains K_d*Kd*, K_p*Kp*, and K_i*Ki* so that all three roots are at s = -4*s*=−4. If M=1*M*=1 and b=2*b*=2, what gain K_d*Kd* must we choose? Enter a numeric value, accurate to within 0.1

Q3. In the computed-torque control law, the control law calculates the commanded acceleration of the joints

\ddot{\theta} = \ddot{\theta}_d + K_d \dot{\theta}_e + K_p \theta_e + K_i \int \theta_e(t) d\text{t}*θ*¨=*θ*¨*d*+*K**d**θ*˙*e*+*K**p**θ**e*+*K**i*∫*θ**e*(*t*)*d*t.

To convert this to commanded joint torques, what else is needed? Select all that apply.

- A model of the inertia matrix as a function of the actual robot configuration.
- A model of the inertia matrix as a function of the desired robot configuration at this instant.
- A model of the gravitational and velocity-product terms as a function of the current configuration, velocity, and commanded acceleration.
- A model of the gravitational and velocity-product terms as a function of the actual robot configuration and velocity.
- A model of the gravitational and velocity-product terms as a function of the desired robot configuration and velocity.

Q5. Why are integral gains K_i*Ki* often chosen as zero in robot controllers?

- Most robots move slowly, so the integral gain is unnecessary.
- The danger of inducing instability due to the extra dynamics (extra state).
- Proportional gains serve the same role as integral gains.

Q6. Consider an 8R robot arm at a configuration away from a singularity, i.e., its Jacobian has rank 6 and it can move its end-effector in any twist direction in the 6-dimensional twist space. Now, we rigidly attach its end-effector to the end-effector of a 3R robot arm (also away from a singularity), which allows the end-effector of the 8R to move instantaneously in only three degrees of freedom.

Now consider the eight-dimensional space of joint torques of the 8R. There is a subspace A of joint torques that causes internal motion of the 8R robot (with no motions or forces at its end-effector) and a subspace B of joint torques that causes forces against the constraints provided by the 3R but no motion of the 8R (internal or at the end-effector).

What is the dimension of the subspace

Q7. From the previous question: what is the dimension of the subspace B?

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