**Physical Address**

304 North Cardinal St.

Dorchester Center, MA 02124

Q1. What does “contact kinematics” refer to?

**The feasible motions between rigid bodies in contact.**- A description of the forces that can be applied through a contact.

Q2. 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? Select all that apply.

- Mass matrix times acceleration.
**Velocity-product term.****Gravity term.**

Q3. Consider unit vectors along the three coordinate axes (x, y, and z) of three-dimensional space. These unit vectors are represented by the coordinates (1,0,0)(1,0,0), (0,1,0)(0,1,0), and (0,0,1)(0,0,1). Which of the following statements is correct? Select all that apply.

**The linear span of the three vectors is the full three-dimensional space.**- The positive span of the three vectors is the full three-dimensional space.
**The convex span of the three vectors is a two-dimesional triangle (and its interior) in the three-dimensional space.**- The positive span of the three vectors, along with a fourth unit vector in the -x−
*x*direction, is the entire three-dimensional space.

Q4. What is the minimum number of vectors needed to positively span a six-dimensional space?

Q1. Which of these statements is correct? Select all that apply.

- In a first-order contact analysis, the local contact normal is either unknown or ignored.
**In a first-order contact analysis, the local contact curvature is either unknown or ignored.**

Q1. Any planar twist can be visualized as an instantaneous motion about a center of rotation (CoR), represented by two real numbers, the coordinates (x_{{\rm c}},y_{{\rm c}})(*x*c,*y*c), and a ++ or -− label. A twist, on the other hand, uses three real numbers. What information can a twist represent that a CoR cannot? Select all true statements below.

**A translational motion cannot be represented by a CoR with finite (x_{{\rm c}},y_{{\rm c}})(***x*c,*y*c) coordinates.- A CoR cannot represent the rate of motion.
- A CoR cannot represent twists that have rotation

Q2. Select all true statements below.

**The positive linear span of two ++ CoRs is the line segment between the two CoRs, labeled ++.**- The positive linear span of two -− CoRs is the line segment between the two CoRs, labeled ++.
- The positive linear span of a ++ CoR and a -− CoR is the line segment between the two CoRs, labeled ++ and -−.
**The positive linear span of a ++ CoR and a -− CoR is the line containing the two points, excluding the segment between the two points, where the ray connected to the ++ CoR is labeled ++ and the ray connected to the -− CoR is labeled -−.**

Q1. A planar body is in point contact with a stationary restraint. The contact normal, into the body, is on the line L*L*. Select all of the following statements that are true.

**Feasible CoRs on L***L*are labeled \pm±.**Feasible CoRs to the left of the contact normal are labeled ++ (counterclockwise).**- Feasible CoRs to the right of the contact normal are labeled ++ (counterclockwise).
**Feasible CoRs not on L***L*are labeled B, for breaking.**Feasible CoRs at the contact point are labeled R, for rolling.****Feasible CoRs on L***L*, but not at the contact point, are labeled S, for sliding.

Q2. If a single rigid body is in contact with multiple stationary constraints, the set of feasible CoRs for the rigid body (a representation of the feasible twist cone) is

- the union of the feasible CoRs for the individual contacts.
**the intersection of the feasible CoRs for the individual contacts.**

Q1. Consider a body in point contact with multiple stationary restraints. Select all true statements.

**A planar body is in first-order form closure if there are no feasible CoRs by a planar graphical analysis.**- If a body is in form closure by a second-order analysis (considering local curvature at the contacts), then it is in form closure by a first-order analysis.
- If a first-order analysis shows that there exist twists that cause breaking contact (B) at all of the contacts, the body may still be in form closure by a higher-order analysis.
**If a body is in first-order form closure, then it is in form closure by any higher-order analysis.**

Q1. In the figure above, three stationary contacts, labeled 1, 2, and 3, are acting on a planar rigid body (not shown). Each contact is drawn as a large dot (where the contact occurs) and an arrow in the normal direction pointing into the body. Also shown is the line on which the normal vector lies, drawn as a dashed line. The figure also shows seven points in the plane, labeled A through G.

The cone of twists for the body that respect the contact constraints can be represented using centers of rotation. In that representation, each point in the plane either gets a ++ label, a -− label, or no label.

Which of the points A through G gets a ++ label? Select all that apply.

- A
- B
- C
- D
- E
- F
- G

Q2. Referring again to the figure in Question 1, which of the points A through G gets a -− label? Select all that apply.

- A
- B
- C
- D
- E
- F
- G

Q3. Refer again to the figure in Question 1. The planar body is not in form closure. You will add **one more contact** to put the body in form closure. Assume you are able to add a contact at any of the points labeled A through G, and that you can choose the contact normal to be in any direction. Then which points would allow you to achieve form closure by adding a contact at that point? Select all that apply.

- A
- B
- C
- D
- E
- F

- G

Q4. The figure above shows two stationary contacts, 1 and 2, acting on a planar rigid body (not drawn). Each contact is drawn as a dot at the contact location and the inward-pointing contact normal. Also shown are four points, labeled A through D (D is at contact 2). Each of these four points has a label ++, -−, or \pm±, indicating the directions of rotation the stationary contacts allow at this point. Each of the points A through D also has a contact mode associated with it, indicating the contact situation at points 1 and 2. Each contact can be rolling (R), breaking (B), or sliding (Sl if the body slides to the left relative to the stationary contact and Sr if it slides to the right). So if the body is sliding to the left relative to the stationary contact at 1 and breaking free at contact 2, the contact mode is SlB (the contact label at contact 1 is given first).

What is the contact mode for the center of rotation at A? Your answer should be a string with only the characters R, B, S, l, or r, with no whitespaces

Q5. Refer again to Question 4. What is the contact mode for the center of rotation at B? Your answer should be a string with only the characters R, B, S, l, or r, with no whitespace

Q6. Refer again to Question 4. What is the contact mode for the center of rotation at C? Your answer should be a string with only the characters R, B, S, l, or r, with no whitespace

Q7. Refer again to Question 4. What is the contact mode for the center of rotation at D? Your answer should be a string with only the characters R, B, S, l, or r, with no whitespace

Q8. Consider a spatial rigid body being contacted by two stationary point contacts. What is the dimension of the twist cone for the rigid body that maintains a rolling (R) contact at one of the contacts and a sliding (S) contact at the other contact? Assume the general case, where the equality constraints corresponding to each of the two labels are independent. Your answer should be a nonnegative integer (i.e., 0, 1, 2, 3, 4, 5, or 6)

Q9. A spatial rigid body is contacted by two stationary point contacts, one at (x_1,y_1,z_1) = (2,0,0)(*x*1,*y*1,*z*1)=(2,0,0) with a contact normal into the body in the z*z*-direction \hat{n}_1 = (0,0,1)*n*^1=(0,0,1), and one at (x_2,y_2,z_2) = (0,-3,0)(*x*2,*y*2,*z*2)=(0,−3,0), also with a contact normal into the body in the z*z*-direction \hat{n}_2 = (0,0,1)*n*^2=(0,0,1). Now consider the body moving with the twist \mathcal{V} = (2,0,0,0,1,0)V=(2,0,0,0,1,0). What is the contact mode? The label at each contact is B (breaking), S (sliding), R (rolling), or P (penetrating; the twist is disallowed by the contact). Your answer should be a two-character string. For example, if the contact is penetrating at contact 1 and sliding at contact 2, your answer should be PS (the contact label at contact 1 is written first)

Q10. Consider a spatial body contacted by two stationary point contacts as described in the previous question. Now consider the body moving with a different twist \mathcal{V} = (2,0,0,0,1,6)V=(2,0,0,0,1,6). What is the contact mode? The label at each contact is B (breaking), S (sliding), R (rolling), or P (penetrating; the twist is disallowed by the contact). Your answer should be a two-character string. For example, if the contact is penetrating at contact 1 and sliding at contact 2, your answer should be PS (the contact label at contact 1 is written first).

Q11. The figure above shows four centers of rotation, two labeled ++ and two labeled -−. We want to construct the positive span of these rotation centers (i.e., the rotation-center representation of the planar twist cone generated by the positive span of the twists represented by the rotation centers). In this rotation-center representation of the twist cone, every point in the plane receives the label ++, -−, b (meaning ‘both’ for \pm±), or 0 (meaning there is no label), indicating which direction(s) of rotations about that point are included in the twist cone, if any.

What labels are assigned to points in the regions labeled 1, 2, 3, and 4 (which are separated by dashed lines between the rotation centers)? Your answer should be a four-character string consisting only of the characters ++, -−, b, or 0. For example, if points in region 1 have the ++ label, points in 2 have the 0 label, points in 3 have the b label, and points in 4 have the -− label, you would answer ++0b-− (give the labels in order).

Q1. You are pushing a 50 kg box over the floor at a constant speed. The floor applies 500 N of force to the box in the upward direction (counteracting the downward pull of gravity). The friction coefficient \mu*μ* between the floor and the box is 0.5. How much force (in Newtons) are you applying tangent to the floor

Q2. A point contact on a spatial body with a friction coefficient \mu > 0*μ*>0 creates a friction cone with a 3-dimensional interior. True or false: The corresponding wrench cone also has a 3-dimensional interior in the 6-dimensional wrench space.

**True.**- False.

Q1. A planar wrench (with a nonzero linear component) can be represented by a line of force in the plane. All nonnegative scalings of this wrench can be represented by labeling points left of the line with a ++, points to the right of this line with a -−, and points on the line with a \pm±. How is the positive span of two planar wrenches represented?

- Each point labeled ++ for both wrenches gets the label ++. Each point labeled -− for both wrenches gets the label -−. Each point labeled \pm± for both wrenches gets the label \pm±. Each point that gets the label \pm± for one wrench and the label ++ for the other gets the label ++. Each point that gets the label \pm± for one wrench and the label -− for the other gets the label -−. All other points get no label.
- Each point labeled ++ for both wrenches gets the label ++. Each point labeled -− for both wrenches gets the label -−. Each point labeled \pm± for both wrenches gets the label \pm±. Each point that gets the label ++ for one wrench and the label -− for the other gets the label \pm±.

Q3. The planar wrench cone A is represented by a single region of points with the moment label ++. The planar wrench cone B is represented by a smaller region of points labeled ++, completely contained inside the region for wrench cone A. Which wrench cone contains more of the wrench space in its interior?

- A
- B

Q1. Consider the moment-labeling representation of the wrenches that can be provided by a set of frictional point contacts with a planar body. What kind of moment-labeling representation corresponds to force closure?

- All points in the plane have the same moment label.
- No point in the plane has a label.

Q2. If the contacts have friction \mu>0*μ*>0, what is the minimum number of point contacts that can yield force closure of a 3D spatial body

Q3. If the contacts are frictionless (\mu = 0*μ*=0), what is the minimum number of point contacts that can yield force closure of a 3D spatial body?

Q1. When we solve a problem in rigid-body mechanics with point contacts with friction, we solve for the contact forces and the object motions (e.g., velocities in the case of quasistatic mechanics). Each contact has one of three labels (breaking B, sliding S, or rolling R). Given a known label, how many total force and motion equality constraints are provided by a contact in the plane (2D)

Q2. When we solve a problem in rigid-body mechanics with point contacts with friction, we solve for the contact forces and the object motions (e.g., velocities in the case of quasistatic mechanics). Each contact has one of three labels (breaking B, sliding S, or rolling R). Given a known label, how many total force and motion equality constraints are provided by a contact in space (3D)?

Q1. In (a), (b), (c), and (d) below, you see a meter stick supported by two fingers below. The contact label for each finger could be R (rolling), B (breaking), Sl (the meter stick slips to the left relative to the finger), or Sr (the meter stick slips to the right relative to the finger). Which of the following shows the correct moment labeling for the set of wrenches that can be applied to the meter stick by the fingers for the contact mode SrSr?

- (a)
- (b)
- (c)
**(d)**

Q1. Assume there are 5 rigid bodies in an assembly on the table. There are 15 frictional point contacts total, between the rigid bodies and between the rigid bodies and the table. You want to determine if the assembly can stand (i.e., be at static equilibrium) in gravity. How many nonnegative coefficients k_i*ki* are needed to represent all the possible contact forces

Q2. Again assume there are 5 rigid bodies in an assembly on the table, and you want to determine if the assembly can be in static equilibrium. How many equality constraints, provided by the static equilibrium equations, must be satisfied by the contact force coefficients k_i*ki*?

Q1. A thin rod is supported by two stationary circular fingers, one below the rod at a distance of 2 units from the rod’s center of mass and one above the rod at a distance of 3 units from the rod’s center of mass. Gravity pulls down the rod with a force of 10 Newtons. To balance the forces and moments on the rod, what force magnitudes f_1*f*1 and f_2*f*2 must the two fingers apply? Write your answer as two integers separated by a comma, f_1*f*1,f_2*f*2, without units. In other words, if you think f_1*f*1 should be 20 Newtons and f_2*f*2 should be 5 Newtons, then you should write

20,5

(**Note:** The two values will be positive integers, so don’t write any negative signs or decimal points, as that will confuse the automatic grading!)

Q2. A planar box sits on a table and is pushed to the right by a robot finger. The contact between the finger and the box is frictionless, but there is friction between the box and the table, as indicated by the friction cones. Gravity acts downward at the center of mass (not shown) and pushes the box against the table. The left image shows the finger and the table; the right image shows the lines of action of the finger’s pushing force and the edges of the friction cones with the table.

As the finger begins to push the box quasistatically, the box either slides to the right at both contacts with the table (contact mode SrSr), tips over the rightmost contact with the table (contact mode BR), or tips over the rightmost contact while also sliding (contact mode BSr).

If the contact mode is SrSr, the contact wrench the table applies to the box can be any positive linear combination of which friction cone edges? The friction cone edges are indicated as a, b, c, and d, so if you think the contact force can be any positive linear combination of a, b, and d, but not c, you should write abd. Don’t put any spaces in your answer, write the letters in increasing order (e.g., write abd, don’t write adb), and leave out any friction cone edges that cannot contribute to the table contact wrench

Q3. Referring again to Question 2, if the contact mode is BR, the contact wrench the table applies to the box can be any positive linear combination of which friction cone edges? The friction cone edges are indicated as a, b, c, and d, so if you think the contact force can be any positive linear combination of a, b, and d, but not c, you should write abd. Don’t put any spaces in your answer, write the letters in increasing order (e.g., write abd, don’t write adb), and leave out any friction cone edges that cannot contribute to the table contact wrench

Q4. Referring again to Question 2, if the contact mode is BSr, the contact wrench the table applies to the box can be any positive linear combination of which friction cone edges? The friction cone edges are indicated as a, b, c, and d, so if you think the contact force can be any positive linear combination of a, b, and d, but not c, you should write abd. Don’t put any spaces in your answer, write the letters in increasing order (e.g., write abd, don’t write adb), and leave out any friction cone edges that cannot contribute to the table contact wrench.

Q5. (Continuing from the previous questions.) Depending on the mass distribution of the box, the center of mass (CM) could be inside the light gray area labeled A, the red area labeled B, the blue area labeled C, the pink area labeled D, the green area labeled E, the yellow area labeled F, or the dark gray area labeled G. These areas come close to, but do not touch, the dashed force lines that bound them.

During quasistatic pushing, the contact forces acting on the box, from the pushing finger and the table, must balance the downward gravity force. If the contact mode is SrSr (the box slides to the right without tipping) as the finger moves to the right, in which areas might the center of mass be? Select all possible areas for the location of the CM so that quasistatic force balance is satisfied. (An option is correct if a small region of correct CM locations lies in that area; not all CM locations in that area have to be consistent with SrSr.)

- A
- B
- C
- D
- E
- F
- G

Q6. Following up on the previous question: If the contact mode is BR (the box tips over the right contact with the table) as the finger moves to the right, in which areas might the center of mass be? Select all possible areas for the location of the CM so that quasistatic force balance is satisfied. (An option is correct if a small region of possible CM locations lies in that area; not all CM locations in that area have to be consistent with BR.)

- A
- B
- C
- D
- E
- F
- G

Q7. The figure above shows a C-shaped object and four potential finger contact locations, labeled A, B, C, and D. The friction cone at each of these contacts are shown. If your robot hand has only two fingers, so you can only choose two of these contact locations, which two contact locations give a force-closure grasp? There may be more than one answer; select all that apply.

- A and B.
- A and C.
- A and D.
- B and C.
- B and D.
- C and D.

Q8. A rigid spatial body is contacted by four frictional point contacts. Each friction cone is approximated as a four-sided polyhedral convex cone, as in Figure 12.18 in the book. In the linear programming test for force closure (Chapter 12.2.3), how many rows and columns does the F*F* matrix have? Write your answer as r,c where r is the number of rows and c is the number of columns. For example, if your answer is 5 rows and 3 columns, you should simply write

Q1. In this chapter, what kinds of mobile robot models do we use?

- Second-order dynamic models, mapping wheel torques to chassis accelerations.
**First-order kinematic models, mapping wheel velocities to chassis velocities.**

Q1. Consider a wheel with rollers that allow free sliding, like a mecanum wheel or an omniwheel. In the wheel frame, the unit linear velocity corresponding to the driving direction is v_d = (1,0)*v**d*=(1,0). The unit linear velocity corresponding to the free sliding direction is v_s = (0.71, 0.71)*v**s*=(0.71,0.71). If the total wheel velocity is (0,1) = k_s v_s + k_d v_d(0,1)=*k**s**v**s*+*k**d**v**d*, where k_s*k**s* and k_d*k**d* are the scalar sliding and driving speeds, respectively, what is the sliding speed k_s*k**s*?

Quiz 03:

**More Peer-graded Assignment Solutions >>**

Boosting Productivity through the Tech Stack Peer-graded Assignment Solutions

Essential Design Principles for Tableau Peer-graded Assignment Solutions

Creating Dashboards and Storytelling with Tableau Peer-graded Assignment Solutions

Intro to AR/VR/MR/XR: Technologies, Applications & Issues Graded Assignments

Conversational Selling Playbook for SDRs Peer-graded Assignment Solution