How can robots navigate an environment without clear boundaries using their motors and sensors? You will learn how to create robot bodies and behaviors that use limbs and other appendages to exert physical forces that grant dependable motion in a dynamic and complex environment.

We create a method for creating straightforward dynamical abstractions that somewhat automate the creation of intricate sensory programmes. Mobility in animals and robots, the kinematics and dynamics of devices with legs, and the design of dynamical behavior using energy landscapes are some of the specific subjects that will be explored.

## Robotics: Mobility Coursera Quiz Answers

### Robotics: Mobility Week 1 Quiz Answers

#### Quiz 1: 1.1 Why and how do animals move

Q1. In Newton’s F=ma*F*=*ma*, what does F*F* denote? (single word lower case answer expected)

**Force**

Q2. Why is bioinspiration encouraged over biomimicry

- Animals have behaviors that are not of interest to roboticists
- Evolution can never start from a blank slate
- Natural and physical materials behave differently
**All of the above**

Q3.

<image: https://d3c33hcgiwev3.cloudfront.net/imageAssetProxy.v1/it-4jOlJEeWIdgqHxZs34w_d9b77146dd3a4616aed0f32f81dbb635_Screen-Shot-2016-03-13-at-2.29.13-PM.png?expiry=1638748800000&hmac=3srRHVEyZ7LBtuggXbb7QAdyiVr7oduiVOnQVdECIcA>

If an animal or robot wants to turn left, it should swing its tail

- Up
- Down
**Left**- Right

Q4. Which is an example of intelligence “baked into” the body?

- A baby learning to crawl
- A cat swinging its tail to balance
**Kangaroo tendons acting as springs**- A dog shedding its winter coat

#### Quiz 2: 1.2 Bioinspiration

Q1. The speed of airplanes is often described in units of Mach (the speed of sound at standard sea level conditions). What is the speed of Helios 2 in Mach?

**207**

Q2. Acceleration is sometimes measured in g (acceleration due to gravity at the Earth’s surface). What is the cheetah’s acceleration in units of g?

**1.3**

Q3. Turning left “at the house with the red roof” is an example of navigation using which part of the hippocampus?

- Grid cells
**Place cells**

Q4. Turning left “after 25 meters” is an example of navigation using which part of the hippocampus?

**Grid cells**- Place cells ;

#### Quiz 3: 1.3 Legged Mobility: dynamic motion and the management of energy

Q1.

<image: https://d3c33hcgiwev3.cloudfront.net/imageAssetProxy.v1/YXbddeTtEeWufRJaRfO1AQ_bf02ac0b6685f3b14f421d38e79d6020_Screen-Shot-2016-03-08-at-12.19.10-AM.png?expiry=1638748800000&hmac=8gNRhWpGX7hetBZKjzRqR5lXyyZQhG0piQ_TRPXVPjI>

In the above image, we see three robots with the leg forces on the ground drawn in blue. Match the robots with their resulting jumps.

- Robot “a” will jump the highest up, robot “b” will jump the furthest sideways.
- Robot “c” will jump the highest up, robot “b” will jump the furthest sideways.
**Robot “b” will jump the highest up, robot “a” will jump the furthest sideways.**- Robot “a” will jump the highest up, robot “a” will jump the furthest sideways.

Q2. Hamiltonian systems have their total energy (sum of potential and kinetic energies) conserved. Which of these can best be modeled as a Hamiltonian system.

- A block sliding down a ramp covered in honey
- A car decelerating at a red;
- An air conditioner unit
**A pendulum in a clock swinging back and forth**

Q3.

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Assuming these two motors are about the same size, which one will likely produce more torque and which one will have higher inertia?

**The motor on the right will produce more torque and have higher inertia.**- The motor on the left will produce more torque and have higher inertia.
- The motor on the right will produce more torque, the motor on the left will have higher inertia.
- The motor on the left will produce more torque, the motor on the right will have higher inertia.
- Both motors will produce the same torque and have the same inertia.

Q4. Which are examples of programming work?

**A robot running over rough terrain**- A robot using a sensor to scan the environment
**A robot opening a door**- A robot making a decision about which way to go ;

#### Practice Quiz 1: 1.4 Nonlinear mechanical systems

Q1. Which of the following are ** nonlinear** mechanical systems with degree of freedom \theta

*θ*?

- \ddot\theta = -\frac{g}{l}\theta- \frac{b}{m}\dot\theta
*θ*¨=−*lg**θ*−*mb**θ*˙ - \ddot\theta = -\theta^2
*θ*¨=−*θ*2 - \ddot\theta = -\frac{g \sin\theta}{l} – \frac{b}{m}\dot\theta
*θ*¨=−*lg*sin*θ*−*mb**θ*˙ - \ddot\theta = – \dot\theta
*θ*¨=−*θ*˙

Q2. Assume a nonlinear system has total energy \eta = \frac{m l^2}{2}\dot\theta^2 + mgl (\theta – 1)^2 *η*=2*ml*2*θ*˙2+*mgl*(*θ*−1)2 and that the system is dissipative such that the total energy decreases to zero. What do you expect the final angle \theta*θ* to be when the system comes to a rest? You can assume that there are perturbations that move the system off any unstable fixed points. (Note: your answer should be numeric)

Preview will appear here…

Q3. Assume that the nonlinear system in 2 with total energy \eta = \frac{m l^2}{2}\dot\theta^2 + mgl (\theta – 1)^2 *η*=2*ml*2*θ*˙2+*mgl*(*θ*−1)2 is actually the model of some robot arm you are building and wish to control so that \theta*θ* goes to the answer of 2. If the mass is doubled and the length is quadrupled, how will the answer to 2 change?

- It will go up
- It will stay the same
- It will go down

Q4. Assume you are designing a robot controller to make a robot walk to the top of a hill. You’ve figured out the low-level robot control to prevent your robot from falling over and to give it basic commands like move forwards and turn, but you still have to control the robot to navigate safely to the hilltop without hitting any obstacles.

Your available sensor inputs are an IMU that gives you the robot orientation (so you can tell if the robot is moving up the hill or down the hill) and a LiDAR device that tells you if there are obstacles such as trees in the way. Assume that you have figured out a way to encode the task of safely getting to the top of the hill as an energy-like basin using these sensor inputs and a energy-like function. Which one of the following could be true about this energy-like function? Remember that the controller will seek to decrease this function.

- The top of the hill represents the lowest energy state, while the obstacles represent high energy states.
- The top of the hill represents the highest energy state, while the obstacles represent low energy states.
- The top of the hill represents the highest energy state, while the obstacles represent high energy states.
- The top of the hill represents the lowest energy state, while the obstacles represent low energy states. ;

### Robotics: Mobility Week 2 Quiz Answers

#### Quiz 1: 2.1.1 Walking like a rimless wheel

Q1. Energetic cost of transport is a measure commonly used to evaluate the energetic efficiency of robotic locomotion, and is given by the energy needed to travel a given distance with a robot, divided by the energy needed to lift the robot the same distance vertically against gravity. Which of the below trigonometric relations gives the energetic cost of transport (COT) for the rimless wheel powered by gravity as a function of ramp angle? In this context, ramp angle is the angle that the ramp makes with the ground. Use the horizontal distance traveled in your calculations to represent distance traveled.

- COT = Sin(rampAngle)
- COT = Cos(rampAngle)
**COT = Tan(rampAngle)**- COT = Csc(rampAngle)
- COT = Sec(rampAngle)
- COT = Cot(rampAngle)

Q2. The Cornell Ranger has a mass of 9.91 kg and walked a total distance of 65,243 m using 1,774,800 J of battery energy. What is the cost of transport for the Cornell Ranger? Assume a gravitational acceleration of 9.81 m/s^2 in calculating your answer. ;

**0.28**

Q3. What ramp angle is required to give a cost of transport of 0.2 for the rimless wheel, which is approximately equal to the efficiency of human walking? Note that ramp angle is the angle that the ramp makes with the ground.

- 6 degrees
**11 degrees**- 31 degrees
- 45 degrees

Q4. We’ve seen that energy is often used very efficiently for movement in biology as compared to in robots. The cost of storing energy (in terms of weight) is another factor that comes into play in determining locomotion ability — the more weight is stored in fuel, the more mass the actuators need to move! Let’s compare how much it weights to store a given amount of energy in biological fuel — i.e. food — as compared with electricity to power a robot.

How many kg of battery weight would it be necessary for a human-like robot to walk a marathon? Assume that a marathon takes 5960 KJ to walk (this is an extremely roughly estimate assuming a 72 kg person walking it with a energetic cost of transport of 0.2), and that the lithium-polymer batteries have a mass of 0.002kg per KJ of energy storage. Round to the nearest kilogram.

Just to put this in perspective, this same energy can be contained in less than 7 candy bars, which have a total weight of less than 0.4kg!

**12** ;

#### Quiz 2: 2.1.2 Running like a spring-loaded pendulum

Q1. Given a robot with a leg length of 0.4m, what is the estimated maximum possible walking speed in meters per second according to the Froude number? Assume an acceleration due to gravity of 10m/s^2.

**2**

Q2. Assume that the energetically optimal walk-to-run transition speed for a robotic biped occurs at a Froude number of 0.5. What is the expression for the optimal speed of transitioning from walking to running in terms of leg length d? Assume a gravitational acceleration of 10m/s^2. Enter you answer in simplest form, using sqrt() to represent any square roots. (Ex: The square root of 2*d is written as sqrt(2*d)).

**sqrt(0.5 10d)**

Q3. Which of the following energy plots best represents the energy during stance of the center-of-mass of a robot running with SLIP-like dynamics? The blue curve shows kinetic energy and the red curve shows gravitational potential energy. Assume the horizontal axis is time and the vertical axis is energy. **1 point**

<image: https://d3c33hcgiwev3.cloudfront.net/imageAssetProxy.v1/hKUpnuSTEeWufRJaRfO1AQ_78af4438f83632f6c473464bb029da23_run.jpg?expiry=1638748800000&hmac=pMRE5ya2CJHq0RP3H4TRGZ-LGFt21AOzOXDMuYVUQI8>

<image: https://d3c33hcgiwev3.cloudfront.net/imageAssetProxy.v1/lu0IgOSTEeWufRJaRfO1AQ_dfca02c4347a7ea143ed9ee37088708c_walk.jpg?expiry=1638748800000&hmac=QP8G5Ee4FXX8bT1klEfbDgs_R3i5CP29kRb0JS1FMOI>

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Q4. Here is actual data for the center-of-mass trajectory of a robot running primarily in the directions labeled y and z. From this data, which template best explains the motion observed?

<image: https://d3c33hcgiwev3.cloudfront.net/imageAssetProxy.v1/1JQZceSPEeW–hK2gi_BIw_8d85c1bc59747a3917eae4c4260afba2_dyno_climber.png?expiry=1638748800000&hmac=V3D56m8CYZ6OdLYIqSNLQZCcNKwsBkM30KqicC4Yebc>

**Lateral leg spring (LLS)**- Rimless wheel
- Ackermann vehicle ;

#### Quiz 3: 2.1.3 Controlling the spring-loaded inverted pendulum

Q1. Today’s quiz will tune the Raibert hopper in Matlab. As a first step, this question covers tuning the vertical hopping behavior.

If you haven’t yet set up the Matlab environment, we have a course resource titled “Setting up MATLAB” here. If you have not worked with Matlab before, there are also 2 tutorials provided in Week 1 to that should provide a sufficient introduction to Matlab to allow you to complete the quiz questions.

Download the MATLAB file called Quiz_2_1_1_Raibert_Hopper. You run the simulation by running Quiz_2_1_1_Question_1 with no arguments.

Your first task is to tune the thrust duration using the slider so that the hopper reaches a steady state hopping height between 1m and 1.1m (denoted by the horizontal green and red lines in the generated plot) at the end of the simulation. Give the resulting thrust duration in seconds that meets these goals as your answer.

**0.00071429**

Q2. Now we will tune the forward speed controller. You run the simulation by running Quiz_2_1_1_Question_2 with no arguments.

Recall that the neutral point is estimated at each step, and that the forward speed controller consists of a proportional controller that puts the toe in front of or behind the neutral point to obtain the desired speed. What proportional gain to the forward speed controller causes the robot to accelerate from rest to a speed between 0.9 m/s and 1.1 m/s without ever going over 1.1 m/s?

**0.087439**

Q3. Now we will tune the pitch controller to achieve a desired pitch of \frac{\pi}{12}12*π* radians. You run the simulation by running Quiz_2_1_1_Question_3 with no arguments.

Your task is to finish writing the PD controller contained in pitchController.m. Keep a derivative term of kd_phi = 5. What value of kp_phi bounds the pitch within the green and red lines in the graph generated by the simulation?

**5**

Q4. Now we will put all 3 controller parts together. You run the simulation by running Quiz_2_1_1_Question_4 with no arguments.

In this simulation you combine the 3 previous controllers. Can you make the robot move how you want it to? The intention of this question is just for you to gain some intuition regarding how these controllers interact, and to have some fun seeing that very simple controllers can get the robot to move robustly. When you are done, simply enter 42 as the answer below.

**42**

#### Quiz 4: 2.2.1 Metrics and Scaling: mass, length, strength

Q1. If a cube’s side length is doubled, what happens to its perimeter, surface area, and volume?

**Perimeter: 2x****Area: 4x****Volume: 8x**- Perimeter: 2x
- Area: 2x
- Volume: 2x
- Perimeter: 2x
- Area: 2x
- Volume: 4x
- Perimeter: 2x
- Area: 2x
- Volume: 8x ;

Q2. A small ice cube with side length L and a larger ice cube of length 2L are left sitting out in the sun. If the small ice cube takes 10 minutes to melt, how long will the large one take, in minutes?

Hint: You don’t need to know anything about thermodynamics to solve this; heat transfer (the rate at which the ice cube heats up) is proportional to the surface area of the ice cube, and thermal mass (the amount of heat required to melt the ice cube) is proportional to volume. We assume that the melting time is proportional to this ratio.

**20
Solution: 20 -> 10 minutes * Volume / Area**

Q3.

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If a larger bone in a larger animal is 3x longer than a smaller bone in a smaller animal, and compressive stress in both bones is the same, **how many times thicker must the larger bone be?**

Hint: stress is mass divided by area. Assuming constant density of the animals, mass is proportional to volume. Also assume that the volume scales with length^3. Remember that the primary compression occurs axially along the length of the bone. How much larger must the thickness be to maintain the same (mass)/(cross sectional area) ratio?

Correct answers should be submitted with one decimal place, (e.g, 23.5) ;

**0.11
V = 3 33 = 27 -> sqrt(V) = 5.19 ~ 5.2**

Q4. Pair the following scenarios with the dominant mode of loading:

- 1) a rubber band being stretched
- 2) a diving board being jumped on
- 3) a can being stood on
- 4) a bolt being tightened
**1) tension****2) bending****3) compression****4) torsion**- 1) compression
- 2) torsion
- 3) tension
- 4) bending
- 1) torsion
- 2) shear
- 3) bending
- 4) tension
- 1) bending
- 2) tension
- 3) torsion
- 4) compression
- 1) shear
- 2) compression
- 3) bending
- 4) tension ;

#### Quiz 5: 2.2.2 Materials, manufacturing, and assembly

Q1. With everything else being equal (stiffness, strength, dimensions, etc.) which type of material will heat up more if stressed in tension until failure?

**Ductile**- Brittle

Q2. If a 10cm diameter, 0.5m long cylinder is loaded with 20kg mass in tension, what is the stress (force divided by cross-sectional area) in the material in Pascals (the SI unit for stress: N/m^2)? (Remember, we are concerned with axial tension loads)

**24972.4**

Q3. If the above cylinder then stretches by 8cm, what is the strain (extension normalized by initial length) in %?

**16**

Q4. Necking is a phenomena where plastic deformation induces significant strain resulting in decreased cross sectional area. If the above cylinder experiences “necking” will the stress in the neck be larger or smaller than the stress in the rest of the sample?

- Smaller
**Larger**;

#### Quiz 6: 2.2.3 Design: figures of merit, robustness

Q1. Given the following material properties:

Aluminum 6061-T6 has a density of 2.7 g/cm^3, a Young’s modulus of 69 GPa, and an ultimate tensile strength of 310 MPa.

ABS Plastic has a density of 1.3 g/cm^3, a Young’s modulus of 2.2 GPa, and an ultimate tensile strength of 70 MPa.

If a “spring” were to be made by pulling a 10cm long cylinder of material in tension with a desired stiffness of 5000kN/m, what should the diameter (in millimeters) be for an Aluminum “spring”?

This wikipedia page may be helpful, specifically the section : “Force exerted by stretched or contracted material”.

**3.04**

Q2. If the “spring” were to be made of ABS plastic instead, what should the diameter be (in millimeters)?

**17.01**

Q3. What would be the mass of the aluminum “spring” (in grams)?

**1.96**

Q4. What would be the mass of the plastic “spring” (in grams)?

**31.97**

Q5. If the aluminum “spring” were pulled 3mm, would it break?

**Yes**- No

Q6.If the plastic “spring” were pulled 3mm, would it break?

- Yes
**No**;

#### Quiz 7: 2.3.1 Actuator technologies

Question 1

Which of these tasks is significantly dependent on high actuator bandwidth (frequency at which actuator can work well)?

**High speed pick and place**- A self-driving car making a passing maneuver
- A robot arm picking up a fragile object
**A legged robot catching itself after a stumble**

Q2. The speed torque curve establishes a relationship between these two physical properties in electric motors. Since power is the product of torque and angular speed, where does maximum power occur?

- At stall (no speed)
- At 1/4 no-load speed
**At 1/2 no-load speed**- At 3/4 no-load speed
- At no-load speed

Q3. Torque in an electromagnetic actuator is proportional to current by the constant K_t*Kt* ( in Nm/A, often specified by motor manufacturers). The dominant mode of wasted energy in these actuators is due to Joule heating which scales with current squared. Where does maximum wasted energy occur?

**At stall (no speed)**- At 1/4 no-load speed
- At 1/2 no-load speed
- At 3/4 no-load speed
- At no-load speed

Q4. In the above diagram, if both arrows are held fixed and the position of the central node is x*x* and its velocity is v*v*, what is the differential equation representing this system? Assume the spring has rest length x_0*x*0.

- F = bx – k(x_0 – v)
*F*=*bx*−*k*(*x*0−*v*) - Fx = v(b + k) + x_0
*Fx*=*v*(*b*+*k*)+*x*0 **F + bv = k(x – x_0)**;*F*+*bv*=*k*(*x*−*x*0)

### Robotics: Mobility Week 3 Quiz Answers

#### Quiz 1: 3.1.1 Review of kinematics (MATLAB)

Q1. For all of this quiz, download and unzip the attached file.

Using MATLAB, open the Q1 folder and edit “computeForwardKinematics.m” and compute forward kinematics for a single revolute link, replacing the trivial solution (x=0, y=0). Then run “singlelinkforward.m”To check that this is correct, drag the slider to \theta_1 = 2*θ*1=2 what is the x coordinate of the end effector?

Assume L_1=1*L*1=1.

–**0.41**

Q2. Using MATLAB, open the Q2 folder and edit “computeRRForwardKinematics.m” and compute forward kinematics for a revolute-revolute robot, replacing the trivial solution for both the elbow (joint connecting L_1*L*1 and L_2*L*2) and end effector (orange dot). To check that this is correct, run “rrForward.m” and drag the sliders to \theta_1 = 1*θ*1=1 and \theta_2 = 0.75*θ*2=0.75 what is the y coordinate of the end effector?

Assume L_1=1*L*1=1, and L_2=1*L*2=1. ;

**1.83**

Q3. Using MATLAB, open the Q3 folder and edit “computeRRInverseKinematics.m” and compute inverse kinematics for a revolute-revolute robot. To check that this is correct, run “rrInverse.m” and drag the sliders to x = 0*x*=0 and y = 1.5*y*=1.5. What is the value of \theta_2*θ*2, assuming that 0<\theta_1*θ*1<\frac{\pi}{2}2*π* and 0<\theta_2*θ*2<2\pi2*π*?

Assume L_1=1*L*1=1, and L_2=1*L*2=1.

Note: There are technically two correct configurations, yet we are limiting the possible angles of \theta_1*θ*1 in such a way that only one of the solutions is valid.

**1.44**

Q4.

<image: https://d3c33hcgiwev3.cloudfront.net/imageAssetProxy.v1/l2vAGelOEeWIdgqHxZs34w_149a15c94fef76d665cc7af309116a19_Screen-Shot-2016-03-13-at-3.05.33-PM.png?expiry=1638748800000&hmac=F7Vo7tDgOzkRUl4LG70h1PMzkM4_bJpKUS2yu5uivWM>

Using MATLAB, open the Q4 folder and edit “computeMiniForwardKinematics.m” and compute the forward kinematics for a parallel robot. To check that this is correct, run “miniForwardNoToe.m” and drag the sliders to \theta_1 = 3.5*θ*1=3.5 and \theta_2 = 1.5*θ*2=1.5 what is the value of x?

Assume the shorter link is of length 1 and the longer link is of length 2.

**1.02**

#### Quiz 2: 3.1.2 Introduction to dynamics and control

Q1. One possible simplification to the dynamics of the spring loaded inverted pendulum (SLIP) model is to neglect gravity altogether. Using your knowledge of Lagrangian mechanics, which of the below equations of motion are generated from this simplification?

<image: https://d3c33hcgiwev3.cloudfront.net/imageAssetProxy.v1/SZxGXOiJEeWufRJaRfO1AQ_e6b72205dcef11d0d99616ce3f69666d_Screen-Shot-2016-03-12-at-2.32.50-PM.png?expiry=1638748800000&hmac=vp4LF7-zbCR_O-AGgSthJj4iav-rc_lEeZEs3ARzT2o>

Note: The kinetic and potential energies are given using polar coordinates. The correct answer below is in simplest form and involves a cancellation. You can assume that the radial coordinate never goes to zero.

<image: https://d3c33hcgiwev3.cloudfront.net/imageAssetProxy.v1/ic8JR-T3EeWBDQ73-3lhaw_65770baad3641c5c29f772859dada910_incorrect1.png?expiry=1638748800000&hmac=ucUecI3B0W_rw_nGBfntTjan55F5gQY5k5AjSYDUG9Q>

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<image: https://d3c33hcgiwev3.cloudfront.net/imageAssetProxy.v1/lvk-ueT3EeWBDQ73-3lhaw_db5b42b7e67077ad034dda445380d635_incorrect2.png?expiry=1638748800000&hmac=LoIdLu36I4-Aen_V2dT7_MFL–Nq1Id0UBBE0ghtGn8>

Q2. Which of the following PD controllers — used to directly give motor commands u — is implemented correctly for the goal of having the motors to track the desired state? Let the desired state is given by the subscript “des”, the proportional term be given by kp, and the derivative term be given by kd. **1 point** ;

<image: https://d3c33hcgiwev3.cloudfront.net/imageAssetProxy.v1/PjgToeT5EeW–hK2gi_BIw_1ef5c2cd1920fd96dfb989f97b609d3a_pd_incorrect_3.png?expiry=1638748800000&hmac=Ww3zogJyMocidU64sQOu9xkAdjA-Jj1ICfRf4tuIscg>

<image: https://d3c33hcgiwev3.cloudfront.net/imageAssetProxy.v1/AFH6FuT5EeWufRJaRfO1AQ_915f09032e2a9e2f28dc591640b5abc0_pd_correct.png?expiry=1638748800000&hmac=MOvGyI1l56ykGXVvZBZoWUcAGAjN_GR8o02kqO5IbbU>

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Q3. PD control allows the motors to add stiffness (via the P term) and damping (via the D term) in different directions. Often when controlling robotic legs it proves useful to have these directions align with some coordinate system at the foot. For example, you might want the motors to make the leg seem very stiff when the foot pushes against the ground in the vertical direction but not as much in the horizontal direction.

Assume a 2 degree of freedom leg actuated by 2 motors. Which of the following equations is a PD controller that gives motor commands to track a desired x and y foot position such that the leg stiffness and damping can be independently controlled in the x and y directions? Hint: it may be helpful to review the end of the kinematics section showing how forces at the end effector are mapped to motor torques.

Let J*J* be the kinematic Jacobian, mapping motor velocities into toe velocities. Let k_{px}, k_{py}*kpx*,*kpy* be the x*x* and y*y* proportional gains, and k_{dx}, k_{dy}*kdx*,*kdy* be the x*x* and y*y* derivative gains. ;

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#### Quiz 3: 3.2.1 Sprawled posture runners

Q1. Assume the center of mass (CoM) is at (0,0)(0,0), the left corner is at (-1,0)(−1,0), the bottom is at (1,-1)(1,−1), and the top is at (1,1)(1,1). What is d1*d*1? Enter 3 decimal places.

Hint: https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line

**0.447**

Q2. The lecture introduced the idea of a support polygon (see the figure below and [Ting, 1994] for more information). The cockroach is “quasistatically stable” when the projection of the CoM is within the support polygon at all times. In this question and the next, we will think about the limits this model imposes on locomotion speed.

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Assume that the CoM moves along a vertical line through the origin as the cockroach runs forward, and that the toe locations are fixed at (-1,0)(−1,0), (1,-1)(1,−1), and (1,1)(1,1), in units of cm (about right for a small cockroach).

What is the maximum CoM displacement that keeps a positive quasistatic stability margin (in cm)?

**1**

Q3. (building off the previous question)

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Assume that the CoM moves along a vertical line through the origin as the cockroach runs forward, and that the toe locations are fixed at (-1,0)(−1,0), (1,-1)(1,−1), and (1,1)(1,1), in units of cm (about right for a small cockroach).

Assuming the stride frequency and duty cycle are such that each tripod is maintained for 0.5 seconds, what is the maximum speed such that quasistatic stability is guaranteed (in cm/s)?

**2**

Q4. In this question you will think about if RHex’s “C” legs are better or worse than stick legs for a vertical leap. Imagine the two leg designs on a vertical rail and frictionless floor (as shown above); the legs rotate in the direction of the red arrow and the body is propelled upwards.

Recall Gavin’s lecture on effective mechanical advantage (EMA) from 3.1.1. Which of these curves roughly captures the EMA versus height **trend** for each leg?

*Hint: you don’t need to calculate the kinematics. Think about how much the hip joint rotates per infinitesimal displacement in height of the “robot” as a function of the height. Start with the stick leg.***1 point**

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- blue – half circle leg, gray – stick leg
- red – half circle leg, green – stick leg
- gray – half circle leg, red – stick leg
**green – half circle leg, blue – stick leg**

#### Quiz 4: 3.2.2 Quadrupeds

- Walk to gallop
- Walk to trot
- Gallop to walk
**Trot to gallop**

Q2. This ostrich is a _________________.

**Digitigrade**- Unguligrade
- Plantigrade
- None of the above

Q3. From the evidence of this cost of transport (CoT) plot of animals and robots from MIT, which of these is a more efficient mode of locomotion?

- Running
- Flying
**Swimming**- There is no correlation between these modalities and CoT

Q4. From the evidence of this CoT plot of animals and robots from MIT, how much power is a human running at 3m/s consuming (in Watts)?

Assume the mass of the human is m=80*m*=80 kg, and the CoT is 10^{-0.5}10−0.5.

**743**

#### Quiz 5: 3.2.3 Bipeds

You’ve seen the SLIP model a few times now. You may have noticed that this model does not allow for a biped to stand still (recall inverted pendulums from 1.2.3). What additional actuation/appendage is required for a monoped (sagittal projection of a biped) to stably stand? (There may be more than one correct answer, and assume that multiple answers are independent of each other, i.e., A OR B and not A and B)

**Ankle torque****An actuated tail attached at the hip (but not allowed to touch the ground)****A tail allowed to touch the ground**

Q2. How many degrees of freedom (DOFs) does sagittal Jerboa (model above) have in stance (toe is pinned to the ground)?

*Image from De and Koditschek [2015b]*

*Hint: *First, count the total unconstrained DOFs for the Jerboa, including (x, z)(*x*,*z*), the leg DOFs (\theta_1, \theta_2)(*θ*1,*θ*2), and the orientation DOFs (\phi_1, \phi_2)(*ϕ*1,*ϕ*2). In stance, you have to consider the TWO constraints imposed by the fact that the toe is pinned to the ground.

**4**

Q3. How many non-degenerate* degrees of freedom (DOFs) does sagittal Jerboa (model above) have in **flight** (massless leg)?

*We call a DOF with zero apparent inertia (such as a massless end effector on an actuated joint) degenerate, since theoretically any actuator torque would cause infinite acceleration.

*Image from De and Koditschek [2015b]*

*Hint: *start from the total DOFs in the unconstrained Jerboa (as in the previous question), but now eliminate the DOFs that have zero mass associated with them.

**4**

#### Quiz 6: Simply stabilized SLIP (MATLAB)

Q1. In the following cartoon robots, an encoder measures leg angle relative to body, and an IMU measures absolute body angle.

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In this cartoon robot, assuming the control input is desired leg angle, which sensors need to be used to implement the simply stabilized controller? (Keep in mind the ‘disk body’ has no degree of freedom)

- IMU
- Encoder
**None**

Q2. In this cartoon robot, which sensors are needed to implement the Raibert stepping controller (from week 2)?

- IMU
**Encoder**- None

Q3. In this cartoon robot, assuming the control input is desired leg angle (relative to the body), which sensors are needed to implement the simply stabilized controller?

**IMU**- Encoder
- None

Q4. In this cartoon robot, assuming the control input is desired leg angle (relative to the body), which sensors are needed to implement the Raibert stepping controller?

**IMU****Encoder**- None

Q5. In the attached zip file, you will find a MATLAB GUI (run “slipHW.m”). In the GUI that appears, you must set the parameters through the input boxes (as described below), and click “run” to run the simulation and present results.**slip**ZIP FileDownload file

You will find a simplistic implementation of SLIP with a tunable fixed stepping angle. You will change the value of the stepping angle (**in radians**) to get a sense of the basin of this simple controller.

What is a valid stepping angle (in units of radians) for speed 0.50.5 m/s?

*Hint: change the stepping angle till there are sustained bounces, i.e., the stride-to-stride apex zz and speed \dot xx˙ should not change much. A “close enough” answer is sufficient.*

*Hint 2: the correct answer lies between 0.120.12 and 0.140.14 radians, and you may need to go down to \pm 0.0001±0.0001 radians of precision to get it to work. This question is meant to convey how small the basin of the fixed stepping angle controller really is!*

**0.1294**

Q6.

(building off the previous question)

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Experiment with a couple of more values of speed, and finding a good stepping angle. Which of the following plot trends represents the stepping angle – speed relation well? NOTE: Enter “gray”, “green”, “blue”, or “red” (not including quotes).

**blue** ;

### Robotics: Mobility Week 4 Quiz Answers

#### Quiz 1: 4.1.1 Sequential and Parallel Composition

Q1. In the execution of this sequential composition, how many controllers need to be sequenced for initial point x_1*x*1? (enter a numeric answer)

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Q2. In this scenario, \Phi_1Φ1 makes the goal an attracting fixed point. Which of these is a correct reason that controllers \Phi_2Φ2 and \Phi_3Φ3 are needed in addition?

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- To control the rate of contraction
**The basin of \Phi_1Φ1 is not large enough**- A sequence of controllers is better than a single controller
- The goal of \Phi_1Φ1 is not the correct goal location

Q3. Why is \Phi_3 \rightarrow \Phi_1Φ3→Φ1 not a valid sequential composition?**1 point**

**The basin of \Phi_1Φ1 does not include the goal of \Phi_3Φ3**- Without \Phi_2Φ2, the overall basin of attraction is too small

#### Quiz 2: 4.2.1 Why is parallel hard?

Q1. Which of these is a template for a trotting horse?

*Hint: this is a review of earlier material, specifically around 1:30–2:30 of lecture 2.1.1 where Jeff introduces templates*

**SLIP**- Compass walker
- LLS
- Rimless wheel

Q2. Recall from lecture the **de-**coupled RL-RC circuit (R_{21} = \infty*R*21=∞), where \dot x = Ax*x*˙=*Ax*, and A*A* is a 2×2 diagonal matrix with values A_{11}*A*11 and A_{22}*A*22. What is the solution x_1(t)*x*1(*t*) if starting at t=0*t*=0? Use “x10” for x_1(0)*x*1(0), “x20” for x_2(0)*x*2(0), and “A11” for A_{11}*A*11 etc. To denote e^x*ex*, use “exp(x)”.

Hint: for those in need of a refresher on ODEs, you might find this helpful.

Preview will appear here…

**x10 exp(A11t)**

Q3. Which of these is the reason the ”heavy toe” simulation of the Raibert hopper fails?

- The stepping controller affects the hopping height
- The thrust controller affects the body pitch
**The stepping controller affects the pitch**- The pitch controller affects the hopping height

#### Quiz 3: 4.2.3a RHex

Q1. Why was RHex designed to have one actuator per hip?

- To have adequate specific power per individual motor
**To have adequate robot specific power (power/mass)**- Because cockroaches have one muscle per leg
- Two actuators per hip would be redundant

Q2. Which are some of the steps that had to be taken to anchor a pogo stick on RHex (referred to as “tuning” in the lecture)?

- A physical pogo stick had to be attached to RHex
**The compliance (stiffness magnitudes) had to be tuned****The clock had to be tuned properly**- The running speed had to be limited to exactly 1 bodylength/sec

#### Quiz 4: 4.3.1 Compositions of vertical hoppers

Q1. How many virtual legs are used in a quadrupedal walk? (refer: 3.2.2)

Q2. How many virtual legs are used by a hopping kangaroo?

Q3. Suppose the cartoon robot below is in stance (toe is pinned to the ground), and a hip torque is applied in the direction of the arrow. Then

- the leg will rotate in the clockwise direction, the body will not experience any moments
- the body will experience a counterclockwise moment
- the leg will rotate in the counterclockwise direction, the body will not experience any moments
- the body will experience a clockwise moment

Q4. Assume all the virtual or physical toes are in stance (i.e. pinned to the ground). The arrows in the left figure are physical leg extension forces, and the arrow in the right figure is a virtual hip torque. Note that this *counterclockwise* virtual hip torque would result in a *clockwise* acceleration of the body.

To apply the shown virtual hip torque on the right, in the physical model on the left,

- both legs should apply the same extension force
- this is not possible without using physical hip torques
- the left leg should apply a higher extension force
- the right leg should apply a higher extension force

Q5. Assume the physical robot on the left is in stance. The equivalent “virtual leg” is shown in the image on the right. In this we think about how physical torques can move the virtual hip. Assume that a torque in the direction of the arrows is positive.

Imagine the right physical hip applies 00 Nm. Then, a torque of 11 Nm applied by the left physical hip moves the virtual hip to the **left**. Similarly, if the left hip applies -1−1 Nm and the right hip applies 00 Nm, the virtual hip moves to the **right**.

Which of the following would translate the pivot point of the virtual hip to the **left**?

- Left hip applies 11 Nm, right hip applies -1−1 Nm.
- The left toe is positioned further left in flight
- Left hip applies 22 Nm, right hip applies -1−1 Nm.
- Left hip applies -2−2 Nm, right hip applies 11 Nm.

#### Quiz 5: MATLAB: composition of vertical hoppers

Q1. We have made a **very simplistic **MATLAB simulation with a Minitaur-like body and a **composition of identical, independent vertical hoppers**. This means that in the simulation, each leg is controlled as if it were a vertical hopper, and the controllers are truly decoupled.

The following questions should help you see the power of this kind of composition, and also how the body design might affect behavio

Q2. Select d=0.2*d*=0.2 and leave it for the following three s.

Increasing the gain from 0, with d=0.2*d*=0.2, find the smallest value of vertical hopper gain such that both of the vertical hoppers have sustained bounces.

Hint: look at the hybrid mode plot; “3” means that both legs are in stance. Make sure that the system doesn’t settle in to mode 3 after a few bounces; i.e., modes *other than* 3 (when either vertical hopper has some aerial phase) are visible till the end of the simulation.

**12**

Q3. With d=0.2*d*=0.2, find a value of VH gain such that “bounding” occurs, i.e. the vertical hoppers get out of sync, and there is no double stance (i.e. the hybrid mode does not reach 3 after a few initial bounces)

**3x**

Q4. With d=0.2*d*=0.2, what is the value of VH gain such that the equilibrium z*z* (of the CoM) is \approx 0.2≈0.2.

**36**

Q5. Now we shall see how the body dynamics (specifically, as parameterized by the length of the body) affects our composition. Change to d=0.1*d*=0.1 and leave it for the following three s.

With d=0.1*d*=0.1, find a value of vertical hopper gain such that ”bounding” occurs, i.e. the vertical hoppers get out of sync, and there is no double stance (i.e. the hybrid mode does not reach 3 after a few initial bounces).

*Note: You will observe that the shorter body does not like to passively desynchronize the vertical hoppers as well as the longer body…telling us that more active desynchronization is needed (not implemented in this simulation).*

**15x**

Q6. With d=0.1*d*=0.1, what is the value of vertical hopper gain such that the equilibrium z*z* (of the CoM) is \approx 0.2≈0.2?

*Note: it might oscillate about this value unless you run the simulation for a longer time, but try to guess the steady-state value it is converging to.*

**40**

#### Quiz 6: 4.3.3 Same body, different compositions

Q1. The inertial reorientation template plant (pictured below) dynamically only has 1 DOF, because

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- of linear momentum conservation
**of angular momentum conservation**- the tail angle is not a DOF
- the body angle is not a DOF

Q2. Suppose we modify the template plant for inertial reorientation and pin the body away from the CoM. Gravity is present. How many DOFs does this system have?

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**2**

#### Quiz 7: 4.3.4 Transitions

Q1.To get some idea of the ground reaction complex (GRC), take the example of an nn-legged stool being dropped to the ground, and assume that as it falls, contacts can be made but never broken (i.e. if a leg touches the ground it doesn’t lift off again), and also that there are no “simultaneous” contacts. Define the “sitting” mode as the one where all legs are in contact.

Define the term “*word*” as a sequence of contact modes in a transitional behavior. For example, 000 \to 001 \to 011 \to 111000→001→011→111 is a single *word *comprised of 4 *letters *(contact modes). For further reference into the vocabulary analogy take a look at

How many possible “*words*” are there from aerial to sitting for n=2*n*=2?

**2**

Q2. (Same setup as previous question) How many words are there from aerial to sitting for n=3*n*=3?

**6**

Q3.

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The three digits inside the robot are a binary string representing current contact mode, with 1 signifying an active contact, and 0 and inactive one. If “LB = left body, RB = right body, T = toe”, the correct sequence of labels is

**LB,T,RB**- T,LB,RB
- LB,RB,T
- RB,LB,T

**Review: **

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