assignments

Massachusetts Institute of Technology Department of Mechanical Engineering

2.12 Introduction to Robotics

Problem Set No.1 Out: September 14, 2005 Due: September 21, 2005

Problem 1 The torque-speed characteristics of a DC motor have been determined, as shown in the figure below. Note that u is the voltage applied to the motor armature, m and

m are, respectively, the output torque and angular velocity of the motor shaft. From this plot,

(a) Obtain the motor constant, torque constant, and armature resistance of the motor. (b) When the motor is rotating at 100 radians per second, what is the back emf

voltage induced at the armature? (c) When the motor is producing 2x10-3 Nm of torque, how much power is dissipated

as heat at the motor windings?

m

m0

4 x 10-3 Nm

160 r/s

u = 20 volts

200 r/s

u = 24 volts

Figure 1 Torque-speed characteristics of a DC motor

Problem 2 The DC motor in Problem 1 is now used for driving a single axis robot arm with a gear reducer. See the figure below. The inertia of the motor rotor including the shaft and the pinion gear is Im = 1.0 x 10-4 kgm2 , while the arm inertia about the joint axis including the joint axis and the large gear is Ia= 1.6 x 10-1 kgm2. Obtain the gear ratio that maximizes the angular acceleration of the joint axis, when the angular velocity is almost zero. Ignore gravity and viscous damping. Also obtain the maximum angular acceleration when the maximum armature voltage is 30 volts.

1

Ia= 1.6 x 10-1 kgm2 Im = 1.0 x 10-4 kgm2

axis

Joint Axis

Gearing

Robotic Arm

DC Motor

Figure 2 Single axis robot arm with DC motor and gearing Problem 3 Consider a single axis robot control system with an optical shaft encoder measuring both position and velocity of the joint axis. The transfer function from input armature voltage u to the joint angle is given by

)(1

)()()(

21 asassussG

+==

where parameters are 53.0,16.0 21 == aa . Answer the following questions.

(a) Consider the position and velocity feedback control, as shown in the first figure below. Determine the position feedback gain kp and the velocity feedback gain kv so that the 0-100% rise time is 0.4 sec and that the maximum overshoot is 5 %.

(b) Consider the integral control along with the position and velocity feedbacks, as shown in the second figure below. Sketch a root locus for appropriate values of the velocity feedback gain kv and the integral gain kI , and discuss stability, settling time, and steady-state error. [I believe that you have learned these in 2.004. If not, please let me know.]

)(1

21 asas +

skv

pk _ +

_ +

)(1

21 asas +

skv

)1(skk Ip + _

+ _

+

Figure 3 Block diagrams

2

Problem 4 A DC motor connected to a switching power transistor is shown in Figure 4-a. Answer the following questions. a). The torque constant of the motor is 5.0 x 10-2 Nm/A. What is the voltage across the motor armature as the motor rotates at 100 rad/s with a zero torque load? b). Figure 1-b shows profiles of the transistor voltage and current, Vce and Ic, when the transistor turns on and off. The maximum voltage is 10 volts, and the maximum current is 0.5 A. It takes 5 s for the transistor to turn on, and 10 s to turn-off. During the turn-on and turn-off transition periods both voltage and current vary linearly, as shown in the figure. How much heat (cal) is generated at the transistor in one second when this transistor is used for a uni-polar PWM amplifier of 10 kHz PWM frequency? Note that the mechanical equivalent of heat quantity is 4.2 joule/cal.

Ic

R

Vce

V

Figure 4-a DC motor connected

to a switching transistor

V

Transistor Voltage

& Current

Turn-OFF Turn-ON 0

Ic Vce

5 s 10 s

100 s

Time

Figure 4-b Switching characteristics of the transistor

3

Issued: September 26, 2005 Due: Wednesday, October 5, 2005

2.12 Introduction to Robotics Problem Set 2: Robot Programming

The goal of this problem set and the associated lab sessions on Thursday, September 28th and Friday, September 29th, is for you to develop a mobile robot motion planning and control algorithm for a simulated demining robot.

1. First, using matlab, write an algorithm to generate a series of waypoints that will cover a 5 by 5 meter area, (the box.txt environment in simple sim, our demining robot simulator) using each of three different strategies: (a) backandforth (mowing the lawn) motions, (b) spiraling, and (c) random motions. Your program should write the waypoints out to a file, in the form of (x, y) coordinates in two columns. Read in the data from the file and generate a plot of the waypoints. (Dont specify waypoints too close to one another; a separation for example of at least 0.5 meters is desirable.)

2. Next, write a C program to generate the waypoint files that you generated using matlab in part 1.

3. Next, run simple sim on an athena linux workstation (well show you how in Lab). First, teach yourself to manually steer the robot around a simple environment with four mines. Perform two or three runs where you create a data log file for manual control to activate all the mines, and load and plot the data to reconstruct the trajectory that you manually executed. Details for the data logging format will be provided in lab. Plot both the true (x, y) trajectory based on the simulated robot state (columns 2 and 3 of the data file), and the deadreckoned robot trajectory computed for you by the simulator by integrating the encoders on the robot wheels. Compare the two trajectories. Try to do this for two different motion control strategies (e.g., conservative vs. agressive velocity control). How quickly can you reach all the mines?

4. As discussed in class, simple sim in with a prebuilt trajectory controller, that can read in a set of waypoints and then systematically perform trajectory control to try to reach each of the waypoints in turn (assuming no obstacles!). Run simple sim in this mode providing the waypoint files that you automatically generated above in part 1 to see how the waypoint controller of the simulator performs on your waypoint lists. Does it do a good job? Can you think of ways to do better?

5. Now, add your own C code to the file user_code.cpp to implement your own trajectory control algorithm in simple sim. To start, try to simply to integrate your waypoint generator with a simple waypoint controller, and see how it performs. Log data and plot the results in matlab. How does your controller compare to the builtin simple sim controller tested in part 4, and to the results that you obtained under manual control?

6. (Optional): Develop a more complex motion controller that would be capable of running in a more complex environment with numerous obstacles (such as maze.txt) and/or could achieve good coverage despite large amounts of deadreckoning error. For example, can you implement a finite state machine that switches between the three modes of long transits, bouncing off walls, and spiraling motions, that seems to be the mode of operation of the roomba robot vacuum cleaner? Does your code outperform a simple controller in a more cluttered environment? How fast can you find all the mines? (If you do not feel that you have the prior C programming experience to attempt this, then feel free to sketch out a potential solution strategy on paper.)



Problem Set No.3 Out: October 3, 2005 Due: October 12, 2005

Problem 1 Shown below is a construction robot having two revolute joints and one prismatic joint. Notice that the axis of the prismatic joint has an offset of from the first revolute joint at the origin O. Namely, the distance between point O and point A is , a constant, and the angle between OA and AB is 90 degrees, a constant as well. Joint 2 is a prismatic joint, whose displacement is given by distance d, a variable. Using the geometric parameters and joint displacements shown in the figure, answer the following questions.

1

1

Joint 1

Link 3

Link 2

Link 1

Joint 3

Joint 2

Link 0

3

1

e

y

3

x

e

e

yx

d B

A

O Link 0

1

Figure 1 Construction robot with two revolute joints and one prismatic joint

(a) Obtain the kinematic equations relating the end-effecter position and orientation to the joint displacements.

(b) Joint 1 can rotate between 45 degrees and 135 degrees, and joint 3 can rotate from 90 degrees to +90 degrees, while joint 2 can move from to . Sketch the workspace of the end-effecter E within the xy plane.

1 12

(c) Solve the inverse kinematics problem to find joint displacements leading the end-effecter to a desired position and orientation: eee yx ,, .

Problem 2 Shown below is the schematic of a three dof articulated robot arm. Although this arm looks three-dimensional, its kinematic equations can be obtained in the same way as that of planar robots. For joints 2 and 3 alone, consider a vertical plane containing links 2 and 3. As for joint 1, consider the projection of the endpoint onto the xy plane. Answer the following questions, using the notation shown in the figure.

(a) Obtain the kinematic equations relating the endpoint coordinates, , to joint angles

eee zyx ,,

321 ,, . (b) Solve the inverse kinematics problem, i.e. obtain the joint coordinates, given the endpoint

coordinates. Obtain all of the multiple solutions, assuming that each joint is allowed to rotate 360 degrees.

(c) Sketch the arm configuration for each of the multiple solutions.

x Joint 1

Link 1

1

Joint 3

Joint 2

2

y

3

1

2

z

Link 2

Link 3

e

e

e

zyx

Endpoint

Figure 2 Schematic of 3 dof articulated robot arm

Problem 3 Shown below is a robot arm with three revolute joints. Coordinate system is

fixed to Link 0. Axis is fixed to Link 1. Joint angle 000 zyxO

1x 1 is measured about the joint axis OA (z0 axis) from to . The second joint axis BC is horizontal, and joint angle 0x 1x 2 is measured from axis to axis , which is fixed to Link 2, as shown in the figure. Joint angle 1x 2x 3 is measured about the joint axis CD from axis to Link 3,i.e. line DE. Link dimensions are OA=1, AB=1, BC=1, CD=0, and DE=1. (For the purpose of explaining the kinematic structure, points C and D are shown to be different points, but they are the same, i.e. the length CD is zero.) Note also that

. Answer the following questions.

2x

oCDEBCDABCOAB 90====

Link 3 E

0x

D

1

1

3

2

2

30z

2x 1x

Link 2

Link 0

B

CA

O

0y

Figure 3 Kinematic structure of 3 DOF robot a). Obtain the coordinates of point C viewed from the base coordinate system . 000 zyxO b). Assuming that all the joints are allowed to rotate 360 degrees, determine the workspace of the robot. Sketch the workspace envelope, and show the size and dimensions of the envelope in your sketch.




Problem 1 An astronaut is operating a shuttle manipulator with an inspection end effecter attached to the tip of the arm. For the sake of simplicity we consider only the three revolute joints, 321 ,, , and the three links, as shown in the figure. A Cartesian coordinate system, O-xy, is attached to the object to be inspected. The distance of the coordinate origin O is L from the location of the first joint. Answer the following questions using the notation shown in the figure.

a) Obtain the forward kinematic equations relating the end effecter position and orientation, eee yx ,, , to the three joint angles, 321 ,, . Note that the end effecter position and

orientation are viewed from the Cartesian coordinate system attached to the object, O-xy. b) Obtain the Jacobian matrix associated with the kinematic equations of Part a). Sketch a

block diagram of Resolved Motion Rate Control where the astronaut uses a joystick for generating velocity commands, , with reference to the Cartesian coordinate system O-xy.

eeyex yvxv === ,,

c) The inspection end effecter must be moved along the object surface, i.e. the x-axis, at a constant speed, . The gap between the inspection sensor and the object surface must be constant,

sec/20 cmvxd =cmyed 10= , and the orientation must be kept horizontal,

. Compute the time trajectories of the three joint angles as well as the joint velocities when the end effecter moves from Point A at

oed 90=

cmyx 10,0 == , to Point B at . The link lengths are cmymx 10,15 == cmmm 40,10,10 321 === , and the

distance to the object is . Plot position and velocity profiles using MATLAB. mL 5.10=

Figure 1 Shuttle manipulator inspecting an object surface

All the angles are measured in the right hand sense. 2 in the figure is therefore negative.

Joint 3 3

x Inspection

e

e

yx

e

y

End Effecter

O

Object

1

1

y

x2

2 3

B

O A

L

Problem 2 Shown below is the same articulated robot as the one in the previous problem set. The robot has three revolute joints that allow the endpoint to move in three-dimensional space. However, this robot has some singular points inside the workspace. Answer the following questions.

a) Obtain each column vector of the Jacobian matrix based on its geometric interpretation, as discussed in class. (Consider the endpoint velocity created by each of the joints while immobilizing the other joints.)

b) Obtain the Jacobian via direct differentiation of the kinematic equations relating the end-effecter coordinates to joint displacements. Compare the result with the geometric approach in part a).

c) For this question and the next question only, assume 121 == for brevity. Compute the Jacobian matrix for the arm configuration:

32,

6,

2 321 ===

and obtain the determinant of the Jacobian. d) Obtain the joint velocities that move the endpoint with desired velocities 0,2,1 === zyx vvv

at the instant of the arm configuration in part c). e) Obtain the joint angles of singular configurations by solving the singularity condition:

. 0det =Jf) Based on the results of part e), sketch the arm posture for each of the singular

configurations. Show where in the workspace it becomes singular and in which direction the endpoint cannot be moved at a non-zero velocity.

Figure 2 Schematic of a three dof articulated robot

x Joint 1

Link 1

1

Joint 3

Joint 2

2

y

3

1

z

2

Link 3

Link 2

e

e

e

zyx

Endpoint

Problem 3 Shown below is a planar 3 d.o.f. robotic leg standing on the ground. Three joint angles, 321 ,, , all measured from the ground, are used as an independent set of generalized

coordinates uniquely locating the system. The second figure below shows the front view of the robot including actuators and transmission mechanisms. Actuator 1 generates torque 1 between link 0 and link 1. Note that the body of Actuator 1 is fixed to link 0, while its output shaft is connected to link 1. Actuator 2 is fixed to Link 3, and its output torque 2 is transmitted to Joint 2, i.e. the knee joint, through the mass-less belt-pulley system with a gear ratio of 1:1. Actuator 3 is fixed to Link 3, while its output shaft is connected to Link 2. All actuator torques 321 ,, are measured in a right hand sense, as shown by the arrows in the figure. Displacements of the individual actuators are denoted 321 ,, , and are measured in the same direction of the torque. The location of the hip, i.e. Link 3, is represented by the coordinates of its center of mass, , and angle measured from the base coordinate system fixed to Joint 1, as shown in the figure.

hh yx ,

a). Obtain the Jacobian matrix relating infinitesimal joint angles 321 ,, to infinitesimal changes to the hip position and orientation, , . hh yx ,b). Obtain the Jacobian matrix relating joint velocities to actuator angular velocities

. 321 ,,

321 ,, c). Obtain actuator angular velocities when the hip is moving horizontally at a constant speed,

321 ,, 0,0, === hh yVx .

Figure 3 Leg robot

1 1

2

y

x

2

O

Link 3

3

Joint 1

Joint 3 (hip)

Joint 2

Link 2

Link 1

Link 0

Actuator 1

Belt-Pulley Mass-less

Transmission

Actuator 2

Actuator 3

Joint 1

Joint 3

3

Ground

h

h

yx

2

1

(a) Side view (b) Front view

Rear Front

1

2

3




Problem 1 Consider a mass-less rod of length l constrained by two sliding joints at both ends A and B, as shown in the figure below. The rod is connected to a spring of spring constant k at A and is pulled down by mass m at B. Friction is negligible. Let be the angle between the horizontal line and the rod. Using the Principle of Virtual Work, show that the rod is in equilibrium at the angle that satisfies the following relationship:

kmg

= tan)cos1(

where g is acceleration of gravity. Assume that at 0= the spring force is zero.

A

k

B

mg Figure 1

Problem 2 A planar robot with three revolute joints is shown below. Let i and be the angle of joint i and the length of link i , respectively, and

i

eee yx ,, be the end-effecter position and orientation viewed from the base coordinate frame, as shown in the figure. In performing a class of tasks, the end-effecter orientation doesnt have to be specified. Namely, the number of controlled variables is two, while the number of degrees of freedom is three. Therefore the robot has a redundant degree of freedom.

At an arm configuration of obtain the 2x3 Jacobian matrix relating the end-effecter position to joint displacements. We want to generate an endpoint force of

ooo 225,45,135 321 ===

NFNF yx 2,10 == . Obtain the equivalent joint torques needed for generating the endpoint force.

Endpoint

Figure 2 Three degree-of-freedom redundant robot arm

Note: In the following problem, numerical values of link lengths and other geometric parameters are not given, but you can solve the problem using given functions

alone. ),(),( 21211 sshsh

3

e

e

yx

3

2

y mmm 1,2,3 321 === 2

11

x

2

Problem 3 'Text and diagram removed for copyright reasons. See Problem 4.2, description and figure, in Asada and Slotine, 1986.' (1) At a given configuration of 1 and 2 we want to move the endpoint at a specified velocity, v = [vx, vy] T with reference to the base coordinate system OO - xy. Obtain the cylinder speeds, s1 and s2, that produce the desired endpoint velocity. Hint: Use derivatives of functions h1 (s1) and h2 (s1,s2). (2) Let f1 and f2 be the forces exerted by the cylinders, HC1 and HC2, respectively. Each force acts in the longitudinal direction of the cylinder, and is defined to be positive in the direction of expanding the cylinder. We want to push an object at the arm's endpoint. Obtain the cylinder forces, f1 and f2, required for exerting an endpoint force of Fx = 0 and Fy = F, assuming that all the joints are frictionless. Also ignore gravity.

3


2.12 Introduction to Robotics Problem Set No.6

Out: October 31, 2005 Due: November 9, 2005 Problem 1 A robot arm is drawing a line with a ruler, as shown below. The ruler is held by another robot arm, which is not shown. Assume no friction and a quasi-static process. a). Obtain natural and artificial constraints, using the C-frame attached to the ruler. b). Sketch the block diagram of the hybrid position/force control system in accordance with the natural and artificial constraints obtained above. Also obtain the projection matrices Pa and Pc associated with the natural and artificial constraints.

Figure 1 Ruler and C-frame

Problem 2

Shown below is an office robot drying ink with blotting paper attached to a semicircular roller of radius R. The roller should not slide but roll on the paper in order to avoid smearing the wet signature. Assuming that the process is quasi-static and friction-less, we want to perform the task using hybrid position/force control. Obtain natural and artificial constraints in terms of velocities and forces at the robot endpoint E. Describe the constraints with respect to the coordinate system fixed to the desk, O-xyz. Note that Point E is in the middle of the top surface of the semicircular roller. Is the rolling-contact requirement a natural constraint or an artificial constraint? [The key is to differentiate constraints that physics dictates, i.e. natural constraints, from the type of trajectories that you want the robot to follow in order to accomplish a given task, i.e. artificial constraints.]

z

zv

yv yxv

x

Problem 3

The planar 3 d.o.f. robotic leg considered in a previous problem set is shown below. Three joint angles, 321 ,, , all measured from the ground, are used as an independent set of generalized coordinates uniquely locating the system. The second figure below shows the front view of the robot including actuators and transmission mechanisms. As before, Actuator 1 generates torque 1 between link 0 and link 1. Actuator 2 is fixed to Link 3, and its output torque 2 is transmitted to Joint 2, i.e. the knee joint, through the mass-less belt-pulley system with a gear ratio of 1:1. Actuator 3 is fixed to Link 3, while its output shaft is connected to Link 2. All actuator torques

321 ,, are measured in a right hand sense, as shown by the arrows in the figure. Displacements of the individual actuators are denoted 321 ,, , and are measured in the same direction of the torque. The location of the hip, i.e. Link 3, is represented by the coordinates of its center of mass, , and angle measured from the base coordinate system fixed to Joint 1, as shown in the figure.

hh yx ,

In order to walk in rough terrain, the robot wants to make its knee, ankle, and hip

joints compliant so that disturbances acting on the body may be alleviated. All the disturbance forces acting on the foot can be represented collectively with an equivalent linear force and moment acting at the hip: . See the figure below. Using compliance (stiffness) control, we want to support the hip with a desired stiffness defined as:

Tyx MFF ],,[=F

=

h

h

y

x

y

x

yx

kk

k

MFF

0

0

z

O Figure 2 Office robot drying ink with blotting paper

x

SusanHockfield

Blotting Paper

RR

y

EE

R

2

where is linear and angular displacements of the hip, and the elements of the stiffness matrix, , are appropriate positive values. For the leg configuration and the joint angles shown in the figure, obtain the joint feedback gain matrix K that provides the desired stiffness given above. Also obtain the feedback gain matrix in the actuator space.

Thh yx ],,[ =p

kkk yx ,,

Link 3

Joint 2

Link 2

Link 1

Link 0

Actuator 1

Belt-Pulley Mass-less

Transmission

Actuator 2

Actuator 3

Joint 1

Joint 3 2

3

1 1

2

3

Front

1 1

2

y

x

2

O

3

h

h

yx

Joint 3 (hip)

Rear

Joint 1

(b) Front view Ground (a) Side view

112

434

2

1

3

2

1

==

=

=

=

MFF

y

x

3

2

(c) Hip compliance

1

Figure 3 Leg robot revisited

3



Out: November 9, 2005 Due: November 16, 2005

Problem 1

A two degree-of-freedom robot arm with one prismatic joint is shown below. The direction of the prismatic joint is perpendicular to the centerline of the first link. As shown in the figure, joint angle and distance z between the tip of the first link and the mass centroid of the second link are used as generalized coordinates. The first actuator fixed to the base link produces torque about the first joint, while the second actuator located at the tip of the first link generates linear force f acting on the second link. Using the parameters shown in the figure, answer the following questions.

a). Obtain the moment of inertia reflected to Joint 1 when the second joint is fixed at 0zz = . At which arm configuration does the moment of inertia become minimal? b). Obtain the centrifugal force acting on Link 2 when the first joint is rotating at a constant angular velocity & ? Also obtain the torque induced by the centrifugal force upon Joint 1, i.e. the joint torque needed for canceling out the centrifugal effect. c). Obtain the Coriolis force acting on Link 2 when the first joint is rotating at a constant angular velocity & and the second joint is moving at a constant linear velocity z& . Also obtain the torque induced by the Coriolis force upon Joint 1. d). Obtain the linear velocity vector of each mass centroid, Vci, as functions of generalized coordinates and their time derivatives. e). Obtain the linear acceleration vector of each mass centroid, aci. f). Obtain Newton-Eulers equations of motion by drawing Free-body-diagrams of the individual links. g). Eliminate constraint forces involved in the Newton-Euler equations, and obtain closed-form dynamic equations relating actuator torques, and f, to &&&,, and zzz &&&,, .

Vc2

11, mI

,

f1l

z

O

1cl

22 , mIVc1

Joint 1 Revolute joint

Joint 2 Prismatic joint

Link 2

Link 1

Figure 1 Mass properties and link parameters of a two d.o.f. arm

Problem 2 Figure 2 shows the schematic of a three degree-of-freedom rehabilitation bed/chair

system. The seat is tilted with Actuator 1 fixed to the base frame. The back leaf and the footrest are driven together by Actuator 2 fixed to the seat. Note that the motor shaft of Actuator 2 is connected to a belt-pulley mechanism to move the footrest together with the back leaf. The headrest is moved with Actuator 3 fixed to the seat through another belt-pulley mechanism as shown in the figure. Figure 3 shows the kinematic structure and joint variables along with geometric and mass parameters of the individual links. Note that joint angles 32 and are measured from the seat, while angle 1 is from the base frame.

Figure 3 Mass parameters of the bed-chair system

Figure 2 Powered rehabilitation bed-chair

2

Actuator 1

Actuator 2

Back Leaf

Head Rest

Base Frame

Belt

Foot Rest

Seat

Actuator 3

Belt

m0, I0

m2, I2

m3, I3

Link 1

Link 0

Link 2

m1, I1

O1

O0 O2

O3

C0

C1

l1

y l2

C2

C3

l3

lc0lc1

lc2

Link 3

lc3

l0

x

Endpoint E

12

3

2

Figures by MIT OCW.

The closed-form equations of motion are in form:

+++=

+++=

+++=

3332231133

3232221212

3132121111

HHH

HHH

HHH

(1)

where Hij is the i-j element of the 3x3 inertia matrix }{ ijH=H associated with the joint coordinates. Answer the following questions.

a) Explain the physical meaning of the inertia matrix elements , respectively. Show which part of the link inertia is associated with each of . Be sure which type of motion, translation and/or rotation, is involved in .

2211 and HH

2211 and HH

2211 and HHb) Based on the physical interpretation in part a), obtain , respectively. Use the

mass parameters shown in the figure: m2211 and HH

i is mass, Ii the moment of inertia at the centroid Ci ; the distance between i-th joint axis and the mass centroid of the link. ci

3



Out: November 21, 2005 Due: November 30, 2005 Problem 1 Shown below is a vehicle similar to the 2.12 mobile robot having a pair of powered wheels and a frictionless caster. The radius of the wheels is r=3 cm, while the distance between the two wheels is 2b=20 cm. The angular velocity of the right wheel is , and that of the left

wheel is r

l . Each powered wheel is equipped with a shaft encoder to measure the angular velocity. Answer the following questions.

r

l

0

0

YX

)()(

f

f

tYtX

)( ft

0

X

Y

2b

r

Figure 1 Vehicle trajectory

Time sec

Angular Velocities

rad/sec

10

8

6

4

2

0 3 5 8 tf = 10

r

l

Figure 2 Time profiles of the wheel velocities

a). At time t = 0, the vehicle was at position cmYcmX 20,20 00 == with reference to the inertial reference frame O-XY and at orientation 00 = measured from

oved. The time profiles of the wheel angular velocitgure 2. Compute the posit

the positive X axis. See Figure 1. Then the vehicle m ies during the movement were recorded, as shown in Fi ion and orientation of the vehicle at time based on the time s shown in Figure 2. Assume no slip. To go back to the initial position and orientation,

sec10=ft profile

00 , YX , 0 , a feedback control law is now employed. Let us consider the following control method. As illustrated in Figure 3, let be the angle between the direction of the vehicle, i.e. line AB, and the direction of the destination from the current position of the vehicle, line AC.

[ ] )()(),(2arctan 00 ttXXtYY = The primary goal is to reduce the distance between the current position and the destination ,

)(),( tYtX

00 , YX2

0 ))( Yt hould move in the direction giv

20 ())(( YXtXD +=

To reduce this distance D the vehicle s en by angle . At the same time the vehicle should be oriented in the direction of 00 = at the destination. Therefore, the vehicle should reduce the difference in orientation:

)(0 t during the movement towards the destination.

b ll these, let us consider the following feedback law:

kkDkD+

where v is the vehicle forward velocity and

=

To com ine av = =

is the angular velocity of the vehicle rotation. Answer the following questions.

Destination

Figure 3 Feedback law

0

0 00 , YX

)(),( tYtX )(t B

C

DA

2

b). Obtain the Jacobian relating the vehicle forward velocity v and rotati on velocity to the

angular velo f

cities o the right and left wheels, and r l . c). Sketch an approximate trajectory of the vehicle from the final position obtained in Part a), i.e.

)(),(),( fff ttYtX , back to the original position and orientation, 00 , YX , 0 . Find approprvalues for the feedback gains, kkkD ,, .

iate

For Extra Credit:

edback gains are changed. What will happen if

e). If

d). Discuss whether the vehicle can reach the exact destination when the fe

? Dkk

Probl The objective of this assignment is to build the dynamic model of the 2.12 arm being used for the final project, and obtain feedforward torques for manipulating an end-effecter in a vertical plane. As you already know, both actuators of the 2.12 arm are fixe

em 2

d to the base link, and e actuator torque of the second motor,th 2 , is transmitted from joint 1 to joint 2 through a belt-

pulley mechanism. Actuator displacements 21 and , which are absolute angles measured from the base axis, are used as generalized coordinates, and actuator torques 1 and 2 correspond to the actuator displacements, forming virtual work: 2211 +=Work .

a) Obtain mass properties of each link, i.e. , as defined in Figure 4. Figure 6 illustrates the disjointed ar puting the mass properties). Each arm link consists of an aluminum x 50 mm x 20 mm and two masses at both ends of the link. For simplicity s at both ends are treated as mass particles having no moment of inertia. Obtain ass, the location of the center of mass, and the moment of inertia about the center of mass, , for each link.

b) Obtain feedforward actuator torques for compensating for the gravity load of the arm when displacements

icii Im ,,m links (Details are ignored for com

bar of 275 mm, the masse

the micii Im ,,

21 and are measured. For extra credit:

c) Obtain equations of motion in terms of generalized coordinates 21 and and actuator torques 1 and 2 . Discuss why no Coriolis term is involved in the equations of motion.

d) Consider the cosine curves shown in Figure 7 for the trajectories of 21 and . Compute

the angular velocities and accelerations, , along the trajectories, and then

obtain the feedforward torques for tracking the trajectories from time to

2121 ,and, 0=t ftt = .

y

1 x

2

O 1

2

1

1c

22

2

, mI

2c

11

1

, mI

Figure 4 Variables and parameters of the 2.12 arm

Figure 5 Mechanism of the arm

4

Link 1 Link 2: The same dimension

as Link 1

0.3 kg

0.5 kg

0.1 kg

230 mm

20 mm

50 mm

1 kg 275 mm

Figure 6 Link dimensions and masses attached to the links

90

1 deg.

45

45

-45

2 deg.

0 1 2 3 tf = 4 Time sec

0 1 2 3 tf = 4 Time sec

Figure 7 Trajectories

5

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