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ME5286 Robotics

Spring 2014

Quiz 1

Total Points: 30

You are responsible for following these instructions. Please take a minute and read them completely.

1. Put your name on this page, any other page you write on, and your blue book.

2. This quiz has 11 pages (including this cover page) and contains 2 problems. There are 7 parts to

problem # 1 and 3 parts to problem # 2.

3. This quiz is open book and open notes. You may use a calculator. You may not use any device

that is capable of wireless communication.

4. To get full credit, your response must have a single, correct solution reported with appropriate

units. Partial credit is awarded, so be sure to show your work.

5. If you believe a problem statement is missing a necessary parameter, make an assumption and

carry on. Be sure to specify the exact nature of your assumption.

6. If you get stuck and cannot derive the solution to one part that you will need for a subsequent

part, assume an answer and carry on.

7. If you can’t get an answer, or you believe your answer is incorrect, and cannot find the problem

in the time available, write a brief explanation of what you think is wrong, why you don’t believe

your answer is correct, and how you would continue to find the correct solution.

Name:

Student ID:

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PROBLEM 1 (25 points)

The S-one robot by SCHAFT Inc. is a bipedal humanoid robot which won the DARPA robotics chal-

lenge for 2013. The robot weights 95 kg and stands 1.48 meters tall. It is comprised of 2 arms and 2 legs

along with an array of sensors on the upper torso. The walking motion of the robot is counterintuitive as

its knees bend behind the torso while walking. Figure 1.1 shows the S-one robot from front and backside

views.

Figure 1.1: The S-one robot as seen from behind (Left Figure) and looking at its front (Right Figure).

A. (2 points) You are given the rotational axis for each joint in one leg of a robot similar to the S-one

robot (Figures 1.2 and 1.3).The world coordinate frame [ ] is inertial with the robot’s for-

ward motion in the + direction. Determine the number of degrees of freedom. Then decide on a

base coordinate frame for the leg and state why. Draw the base coordinate frame [ ] for the

leg on Figure 1.6.

B. (3 points) Write the unit vectors

for each of the joint axes for the shown figures (Figures 1.2 and

1.3), as expressed in the base coordinate frame [ ]. The direction of these vectors should be

determined from the joints’ rotational direction as shown (Figure 1.5), based on the right-hand rule

convention. Assume that joint axes B and F are parallel and have no offset in the Y axis.

C. (4 points) Given the base coordinate frame you chose in Part A, clearly draw and label the remaining

link coordinate frames on Figure 1.6. Note that the order in which you label the remaining coordinate

frames will depend on your selection of the base coordinate frame.

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Figure 1.2. The joint axis are labeled for joints A-G of the left leg of the S-one. Note that joint A is not

necessarily the ‘first’ joint Note also that the axes A, B and C intersect at a single point and the axes E, F,

and G intersect at a single point.

Figure 1.3. The joint axes (A-G) of the left leg on the S-one (holding a fire hose). Notice that joints C, D,

and E are coming out of the page. (Side view)

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Figure 1.4. The dimensions of a robot similar to the S-one leg are given. Lines are drawn through the hip

and foot joint axes. (Ignore the blue color, this is the same robot as Fig. 1.1-1.3)

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Figure 1.5. The dimensions of a robot similar to the S-one leg are given. Notice this sketch is not to scale.

The rotational directions are also given for each joint. Note the rotational axes of joints C, D, and E are

coming out of the page and is positive along the walking direction.

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Figure 1.6. A sketch of a robot similar to the S-one is provided for sketching joint coordinate frames.

The joint locations are the same as those indicated in Figures 1.2 – 1.5.

D. (6 points) The elements of the matrices from the ( ) to the joint can be calculated using

the Denavit-Hartenburg convention. Fill in the table below with the Denavit-Hartenburg variables

for each of the matrices. Note that the joints will start from your choice of base coordinate

frame and work up (or down) the leg. Use the appropriate geometric dimensions from Figures 1.4 and

1.5.

Joint i θ (degrees) d (mm) a (mm) α (degrees)

1

2

3

4

5

6

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E. (5 points) At one instant in time, the joints move through their zero positions (as shown in Figures 1.4

and 1.5). Compute the matrices ,

, , and

using the Denavit-Hartenburg variables from part

C for this instant.

F. (4 points) Find the homogeneous transformation matrix from your chosen base coordinate frame

[ ] to the coordinate frame at the knee joint (Joint D).

G. (1 point) Explain in words what you think the advantage is of using the rear bending knee in the S-

one robot.

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PROBLEM 2 (5 points)

Google utilizes hundreds of vehicles each year to drive along roads throughout the world in order to

develop their street view maps. One such vehicle is pictured in figure 2.1

Figure 2.1 A Google maps street view vehicle.

The vehicle is equipped with two primary sensors. The first is a camera that can determine the position

and angle of an object on the road relative to the camera lens. This object is determined by this camera to

be at a position (J,K,L) relative to the camera. The variables J, K and L represent distances in the cameras

coordinate frame ( )

The second sensor is a GPS receiver located on top of the vehicle but behind the camera unit. This GPS

unit receives global X,Y,Z position updates and computes a heading relative to state plane coordinate

system which can then be used to find the car position as well as object positions.

The coordinate frames for the camera, the GPS unit and an object on the road are given in Figures 2.2 –

2.4 as ( ), ( ), and ( ), respectively. Notice the Y axes for the coordinate

frames are parallel and are aligned in the direction of travel of the vehicle. Each coordinate frame follows

the right hand rule convention.

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Figure 2.2 The Google maps vehicle with camera and GPS coordinate frames labeled.

Figure 2.3: A rear view sketch of the Google maps vehicle with camera, GPS and object coordinate

frames labeled. Note: the Y axis for each coordinate frame is into the page.

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Figure 2.4: A side view sketch of the Google maps vehicle with camera and GPS coordinate frames la-

beled. Note that this sketch is not to scale.

Figure 2.5: A top down view of the vehicle in the state plane coordinate system.

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The vehicle is located on a road such that the world frame is stationary with respect to the ground. In this

setup, the GPS unit is assumed to be located at [0.0m, -1.5m, -1.0m] relative to the camera unit (Figure

2.4). Assume that the camera and GPS unit never move relative to each other.

A. (1 point) Determine the homogenous transformation matrix to go from the camera coordinate frame

to the GPS coordinate frame. In other words, find .

B. (1 point) At a given instant in time a general offset from the object to the camera unit is given by

(J,K,L) (Figures 2.3 and 2.4). Determine the homogenous transformation matrix to go from the cam-

era coordinate frame to the object coordinate frame. In other words, find .

C. (3 points) At a given moment in time the GPS receiver is found to be at [778190 m, 480350 m] rela-

tive to a state plane coordinate system with a heading angle (Figure 2.5). At the same mo-

ment in time, the object on the road is found by the camera to be at [5.0 m, 0.0 m, -2.5 m]. Deter-

mine the position and orientation of the object on the road relative to the state plane coordi-

nate system.