Free Essay

Parallel Kinematic Machine

In: Science

Submitted By arjunam16
Words 6225
Pages 25



A parallel mechanism is a closed-loop mechanism of which the end-effector is connected to the base by a multitude of independent kinematic chains. Generally it comprises two platforms which are connected by joints or legs acting in parallel.
In recent years, parallel kinematic mechanisms have attracted a lot of attention from the academic and industrial communities due to their potential applications not only as robot manipulators but also as machine tools. The dream of all developers in Machine Tools has always been to combine the flexibility and envelope of the robots with the accuracy and stiffness of traditional Machine Tools. In the last 20 years the focus of this development has been Parallel Kinematics Machines so called PKM. This technology means that the motions in X, Y and Z are performed by three or more parallel axis that gives an outstanding stiffness and accuracy with a maintained flexibility and envelope.
Generally, the criteria used to compare the performance of traditional serial robots and parallel robots are the workspace, the ratio between the payload and the robot mass, accuracy, and dynamic behavior. In addition to the reduced coupling effect between joints, parallel robots bring the benefits of much higher payload-robot mass ratios, superior accuracy and greater stiffness; qualities which lead to better dynamic performance. The main drawback with parallel robots is the relatively small workspace.


Objective of our major project is to model, design and fabricate a 2-axis Parallel Kinematic Machine and to test the same for a Point to Point contour movement. Report has been divided into four main modules namely Modelling, Design, Fabrication and Testing of the PKM.
Report starts with considerable amount of literature survey carried out to better define our problem statement. Report then moves on to the modelling of the PKM which includes inverse kinematic, dynamic and workspace analysis of the PKM. Once modelling is completed, next module is the Design which includes dimensional and structural design. Details of the fabrication of the PKM are next discussed followed by design and development of electronic interface. Finally the fully fabricated PKM s tested for Point to point movement using a graphical interface which consisted of a Personal computer driving the motor through an IC based Micro controller. Next part of the report contains the experimentation and testing results for the output accuracy and resolution.Report concludes with discussing the scope for further improvements and the works which could be taken up as future development.


Design and Analysis of a Three Degrees of freedom Parallel Kinematic Machine by Xiaolin Hu(2002) discusses about the historical sequence in the development of PKM. A great deal of research on parallel robots has been carried out worldwide, and a large number of parallel mechanism systems have been built for various applications, such as remote handling, machine tools, medical robots, simulators, micro-robots, and humanoid robots. The first design for industrial purposes was completed by Gough in the United Kingdom, and implemented as a tire testing machine in 1955. Some years later, Gough’s compatriot Stewart published a design for a flight simulator.
The design illustration of the Stewart-Gough platform is shown in Fig. 1.

Fig. 1 The Stewart-Gough platform
Thereafter, many applications of parallel robots can be found in various industries and fields, such as manufacturing production configurations, Micro parallel robot for medical applications, and assembly robot arms in automotive applications deep sea exploration, etc. More recently, they have been used in the development of high precision machine tools. Machine tools used by industries are the conventional Serial Kinematics Mechanisms(SKMs).
Fundamental Comparison of the Use of Serial and Parallel Kinematics for Machines Tools by Jiri Tlusty(1999) talks about the requirement of a PKM structure. This is discussed briefly here.
There are two requirements for these machine structures:
• Robust design for higher power utilization, and
• Lightweight construction for high feeds and accelerations
These two requirements are rather contradictory in SKMs.
The conventional SKMs appears to have reached their limits both physically (considering the feed rates and accelerations) and also in terms of purchase cost. This has necessitated a totally new concept in machine tool construction - the Parallel Kinematics Machine (PKM). There are several types of PKMs: the triglides, tripods and the chaste hexapods and hex glides.
The visible advantages of PKMs over SKMs are their simplicity and light weight construction. The parallel, closed frame in PKMs results in the improved dynamic characteristics combined with lower maintenance costs. This is in contrast to SKMs where sturdy slides are needed in each direction leading to cost and weight escalations .Optimal Design of Parallel Kinematics Machines with 2 Degrees of Freedom by Radu Bălan et al. (2006) discusses the optimization of design of 2 axis PKM.
The optimization of machine tools with parallel kinematics can be based on the following objectives functions:
• workspace,
• the overall size of the machine tool,
• kinematic transmission of forces and velocities, • stiffness,
• acceleration capabilities,
• dexterity,
• accuracy,
• the singular configurations,
• isotropy.
In the design process we want to determine the design parameters so that the parallel Kinematics machine fulfills a set of constraints. These constraints may be extremely different but we can mention:
• workspace requirement,
• maximum accuracy over the workspace for a given accuracy of the sensors,
• maximal stiffness of the Parallel Kinematics Machines in some direction,
• minimum particular forces for a given load,
• maximum velocities or accelerations for given actuator velocities and accelerations.
Parallel Kinematic Machine Tools – Current State and Future Potentials by M. Weck et. al (2009) discussed about the current state of development of PKM across the world and also talks about the future developments in design and scope


As discussed in chapter 2, a lot of innovative PKM designs have come up in the recent years differentiated mainly by the degrees of freedom. However, as the degrees of freedom increases, the level of complexity involved shoots up. By thoroughly studying the analysis procedure involved for various degrees of freedom, it was decided to go with a 2-degress of freedom PKM, which could be successfully modelled, designed and fabricated within the specified time. The modelling and analysis procedure adopted for the same is as follows.


Two most suitable mechanisms for a two-axis PKM are 1. Constant Length Strut PKM

Fig.2- Constant Length Strut PKM

2. Variable Length Strut PKM Fig.3- Variable Length Strut PKM

Out of the two, based on economic considerations (the high cost of telescopic links required for a variable length strut PKM), the first one that is Constant length Strut PKM is selected. Nevertheless, Position analysis is conducted for the Variable Length Strut PKM as shown below:-

Fig.4- Variable Length Strut PKM Analysis
Let q1 and q2 be the respective movements of the telescopic links.
(p,q) being the coordinates of the endpoint.
Inverse Kinematic equations are:- q1=xp2+yp2 (3.1) q2=(b-xp)2+yp2 (3.2)

Forward Kinematics:-
From eqn.(1), we have yp=q12-xp2 (3.3)
Also, from eqn.(2), xp=q12+b2-q222b (3.4)

Thus, values of xp and yp can be determined.

As mentioned above, based on cost constraint, a constant length PKM is chosen for design and fabrication.

3.2.1 Position Analysis of a Constant Length Strut PKM:-

Fig.5-Constant Length Strut PKM Analysis
Let (p,q) be the x and y coordinates of the end point to be reached respectively.
From the Figure, We have cosα=aa2+b2 sinα=ba2+b2

cosβ=a2+b22a sinβ=3a2-b22a


=a2-acosαcosβ-sinαsinβ = a2-aaa2+b2*a2+b22a-ba2+b2*3a2-b22a =b3a2-b22a2+b2

Putting ba=m, p=a2*3-m2m21+m2 (3.5)

Similarly, q=acosγ-3a2+y =acos90-(α+β)-3a2+y =asinα+β-3a2+y =asinαcosβ+cosαsinβ-3a2+y =aba2+b2*a2+b22a-aa2+b2*3a2-b22a-3a2+y =y+b2+a23a2-b2a2+b2 (3.6)
3.3.1 Assumptions 1. Maximum workspace to be covered is assumed to be the typical size of a paper roughly = 32×16 cm 2. For stability, maximum angle θ is constrained to be equal to 90.
Which takes the following configuration:-

Fig.6-Constraint on Stability
From this configuration, the link length can be suitably assumed as a= 16 cm.


A matlab programme to simulate the real machine movement to reach the required position was prepared ( appendix ). Inverse Kinematics was carried out with the help of the programme. This was validated by making the programmes to plot some typical shapes including line, square, circle etc. The plot was compared with the actual graphical plot obtained on graph paper using the same data.

Fig.7 Square Validation Fig.8 Straight Line Validation

Fig9. Circle Validation

Also, we obtained the total working envelope or the workspace by providing the required constraints to the MATLAB Programme.

Fig.10 Workspace

3.5 OBSERVATIONS REGARDING WORKSPACE * Workspace of the PKM was plotted using MATLAB . * Plot showed that Points at the peripheries were too close to distinct as compared to the central region. * This Indicates that in order to plot distinct points in the periphery resolution of the actuator should be as high as 0.01mm. * Thus resolution is a major limiting factor in case of the determination of usable workspace. * This also explains the error observed in the plot used for validation as the drawings were based on 1mm resolution of actuator which is comparable to practical cases. * Three configuration of singularity were found. These points are critical in case of end effector movement during working. The singularities are as shown:- (1) (2) (3) Fig 11- Points of Singularities Problems associated with singularity are:- 1. Loss of freedom of movement 2. Limiting the workspace 3. Loss of control Thus, singularity brings in a major constraint on the usable workspace. So, as discussed in section 3.2, we had to limit the maximum angle between the links to 90(Fig.6) and this constraint was also added to the Matlab Programmes. Thus, singularity was inherently avoided for the further procedure. 3.6 VELOCITY AND ACCELERATION ANALYSIS From Fig.5, We have p=a2*3-m2m21+m2 q=y+b2+a23a2-b2a2+b2-3a2
Velocity along x-direction, Vp=dpdt=dpdm*dmdx*dxdt=1a*Vr*dpdm (3.7)
Acceleration along x-direction, ap=d2pdt2=d2pdm2*dmdx*dxdt*1a*Vr+dpdm*1a*ur =1a2*Vr2*dpdm+1a*dpdm*ar

Jerk, d3pdt3=1a2*Vr2*d3pdm3*1a*Vr+1a2*d2pdm2*2Vr*ar+ d2pdm2*1a*Vr*1a*ar+pa*Jr

Assuming jerk of the slider=0, and jerk of end effector=0,
We have,

1 a2*Vr2*d3pdm3*1a*Vr+1a2*d2pdm2*2Vr*ar+ d2pdm2*1a2*Vr*ar=0 or Vr*1aVr2*d2pdm2+3*ar*dpdm=0 ar=-Vr2* d3pdm33 * d2pdm2 (3.8)
Similarly for q,
Velocity, Vq = dqdt=dqdb*dbdx*dxdt=Vr*dqdb (3.9)

Acceleration, d2qdt2=d2qdb2*Vr2+dqdb*ar Jerk, d3qdt3=Vr3*d3qdb3*1+d2qdb2*2Vr*ar+grdqdb=0 Vr3*d3qdb3+d2qdb2*2Vr*ar+Vr*ar*d2qdb2=0 ar=-Vr2d3qdb33d2qdb2 Using (7) and (8),
Knowing Vp =200-500 mm/min, a=16 cm,
We obtain Vr=4 mm/s to 9 mm/s

The Variations of the slider velocity and acceleration are plotted using MATLAB as shown.

Fig 12 Plot between Velocity and X axis of Workspace Fig 13:Plot between Acceleration and X axis of Workspace



The link bar can be designed as a cantilever beam.

Figure 14: Link Bar

Deflection, ∆x=Ft×L33EI (4.1)

* Weight of the link is negligible * Material used is aluminum; E=70GPa

Taking a permissible deflection of 2mm, d is calculated as d≅5mm

Figure 15: Cross Section of Link

Therefore cross section of the link will be 5mm×10 mm

Specification of the link is given as * Material : Aluminum * Dimensions(l×b×h) : 160mm×10mm×5 mm


Link joint is proposed to be used at the slider

Figure 16: Cross section to be considered

4.2.1 Moments acting on the pin

Figure 17: Moments acting on the pin

Force F=m×a= .5×.17

Tensile stress ST=MTyI= 10×160×r3.14×r44 (4.2)

Axial Stress Sa=MTyI= .85×16×r3.14×r44 (4.3)
As width of the link is 10mm, the pin diameter has to be less than that and therefore we choose an M6×1 pin made of steel.

ST+ Sa=10×160×43.14×33+ .085×160×43.14×33=76.09 N/mm2

Yield limit for steel is Y = 350Mpa
Therefore factor of safety is 4.6

4.2.2 Check for joint strength

Figure 18: Joint strength of pin

Joint strength SJ=McI= 10.085×160×2.510×5312=38.72 MPa (4.4)

Factor of safety =8038.72=2.06


The linear actuator is selected as electromechanical (servo/stepper motor with ball/roller screw) with sliding rails and possible provision to incorporate a feedback system.

Fig19. CAD model of the two axis PKM


For getting velocity of slider = 5mm/s, and pitch of screw = 3mm
Motor should have an rpm given by N = v×60p (5.1) N= 5×603=100rpm

The total load that acts on one motor (on single actuautor) should not be more than 1.5kgcm

The motor should be able to provide the required accuracy and speed. Based on economic considerations a DC motor was selected. A 6A DC motor was used initially. But the electronic circuit could only provide a maximum of 1A. Moreover the high torque resulted in higher vibrations in the machine. So that option was ruled out. Next option was to use a stepper motor.
But the stepper motor produced discontinuous rotations. It also led to jamming of the lead screw and resulted in poor accuracy.

After repeated testing a lower torque DC motor was finalized. It worked on a 12 volt supply and its no load rotational speed was 100 rpm and gave a torque of more than 1.5kgcm.


The link bar material had to be selected keeping in mind the following aspects * Loads acting * Structural rigidity * Minimum Weight * Low inertia
Taking into account the following factors aluminium was selected for the link bar


The fixture material was to be designed keeping in mind the rigidity of the setup. It should also account for the vibrations due to imperfections of the motor setup. The fixture was designed as semi fixed type to account for the vibrations. It was decided to use wood for making the fixture



6.1. PROCUREMENT STAGE: 1. Lead Screw: We had designed for a lead screw of outer diameter as 16 mm. But it was not readily available in the market. So we purchased an M20 lead screw for our mechanism. 2. Mild Steel: Mild steel was purchased to use it as basic supporting structure or skeleton of the mechanism. 3. M6 Screw: 6 M6 screw was purchased to use as pin joints and for fixing coupler to the shaft. 4. M20 Nut: It is used to make the linear actuator on lead screw. 5. Aluminium: Aluminum is used to fabricate the arms of the mechanism. It was preferred in the design since it was light weighted. It is also used as a connector element between coupler and motor shaft. 6. Motor: Initially a high torque low rpm motor was purchased. Later a stepper motor with 1.8 degree step is also purchased. And finally a 12V, 100 rpm DC motor is finalized for the design. 7. Ply Wood: Ply wood is used as the base for the mechanism. 8. Motor Mount: Motor mount is made from wood. 9. Ball Bearings: 4 ball bearings (SKF 6304) were purchased to make the smooth rotation of lead screw possible.

6.2. REDESIGN STAGE: Motor: The motor initially used was of high torque and slow speed. It was chosen to rotate the lead screw rotate smoothly even if there is some imperfection in design. But it requires a high ampere current more than 3 A to work smoothly. Since the electronic circuit was not able to provide current more than 1 A, it was abandoned.
Second choice was to use a stepper motor of 1.8 degree step per pulse. That means it rotates 1.8 degrees per pulse given by the circuit. An additional H bridge was attached to main circuit to accommodate this. But the stepper motor with required torque was not available in the market. More over the stepper motor work in steps and it doesn’t give continuous rotation, which may result in jerks in the end effecter/tool.
Finally a DC motor with low ampere requirement with 12 V and 100 rpm was chosen. It was integrated to the circuit and it gave required torque as output to rotate the lead screw. So the DC motor was finalized for the design. Lead screw: Lead screw in the design was not available in market. So the available lead screw M20 is purchased. The initial pitch was taken as 2mm. now it increased to 2.5 mm pitch. Since the outer diameter increased, new bearings had to be purchased.

Slider velocity: Since the designed lead screw was not available readily in the market, we had to purchase M20 lead screw. So, slight changes were made in the mechanism.
Pitch of the lead screw =2.5mm
For getting a slider velocity of 5mm,
Motor rpm = 60 vp (6.1) = 60 x 52.5 = 120 rpm. The motor available was only 100 rpm or 150 rpm. 100 rpm was fixed considering the stability of the mechanism. Now the modified slider velocity (theoretical) = 100 x 2.560 = 4.166 mm/s Work Space: Since the lead screw was changed, the clearances must be increased for the smooth functioning of mechanism. This further restricted the available workspace in the previous design. It was then decided to update the workspace. The arm length is increased from 150 mm to 200 mm and the slider length again increased to 400 mm. Required changes were made to the Mat Lab program to accommodate this. Ball Bearing & Nut: Ball bearing purchased initially was having 16 mm internal diameter. New ball bearing was again purchased with an internal diameter of 20 mm. M20 nut is purchased to make the linear actuator. Aluminium Arm/link: The pre designed aluminium arm/link was too small for the new configuration. So in order to ensure the smooth functioning an aluminium square rod with 25x5 dimensions is chosen even though that much dimension wasn’t necessary from point of view of strength of mechanism.


Skeleton: Skeleton for the mechanism is made of MS flat. The flat strip with two 500 mm length and two 300mm length pieces were cut and welded to form a rectangle type structure. Four legs were provided from the corners with a height of 120mm. the top part of the structure is then cleaned and leveled. At most care was taken to make the top platform in same level.


Fig.20 Dimensions of the structure

Bearings: bearings are then welded at a distance of 400 mm end to end and 300mm side to side on both sides. Bearings are first spot welded. Final welding was only done after all components are assembled.

Guide Way: A guide way is made of a circular mild steel rod and a bush having an inner diameter a little more than that of outer diameter of the rod. This will work as a smooth guide way and prevent the nut from rotating along with lead screw and thereby converting the nut into a linear actuator. The guide way is spot welded in between the bearings in the 400 mm side.

Lead Screw & Nut: Lead screw is then inserted through one of the bearings. Nut is put in it and then the other end of lead screw is taken out through the bearing at opposite side as given in the design. The nut is then positioned just above the bush which is now on the guide way. Now the nut is spot welded to the bush and thereby preventing any relative motion between nut and the bush. Now when the lead screw freely rotates, the nut won’t rotate along, instead it moves linearly through the lead screw in between the bearings.
The other side of linear actuator is also fabricated in this way. Now in order to prevent the axial movement of the lead screw, a restrictor is welded on its one end. Care was taken to give a clearance on both ends in order to ensure smooth functioning.

Arm/Link: Arm is made from aluminium. The arm is first made by attaching two aluminium rods of 200x25x5 dimension. A pin joint is made at the joint of the arm and a 8mm hole is provided with the help of a bush to hold the tool.
Arm is then connected to the nut with a pin joint made by M6 nut. The whole mechanism is made such a way that it can be de-assembled and reassembled quickly by removing the nuts and bolts. The pin joint allows the relative rotation between linear actuator and the arm.

Coupler: Two couplers were made to attach the lead screw with the shaft of the motor. Coupler is designed in such a way than it can be removed easily if we need to remove or re insert the lead screw without any obstruction. The coupler is made from mild steel and two threaded holes were drilled radially from outside to tighten the coupler and aluminum connector

Aluminium Connector: Aluminium connector was chosen to connect the shaft with the coupler. Aluminum was chosen in order to protect the shaft in cases when high torque comes and the lead screw get jammed, then instead of the shaft get ruined, aluminium will undergo wear and saves the shaft.

Shrink Fit: The shaft was connected to aluminium connector using shrink fit. i.e., the inner diameter of hole of aluminium connector is slightly less than that of shaft and shaft get perfectly tight fit when shrink fit is used

Platform: Platform for the workspace was made by fixing plywood to the lower side of the skeleton platform. Graph paper is fixed to its upper surface to mark the axis.

Motor Mount: Motor mount is designed and fabricated with wood. Two vertical wood pieces were attached to a base wood and the motor is attached to the wooden mount by using a metal clamp which is screwed to the wood. Clearances were provided to the motor in order to accommodate the eccentricities arises during the attachment of motor shaft to the lead screw.


Once the whole mechanical setup was ready, now the next objective was to design and fabricate an electronic interface to control our mechanism. Idea was a interface where the operator inputs the end coordinates where the end effector has to reach and rest is analysed and controlled by a micro controller and gives the necessary instructions to the motors to make the end effectors reach the required co-ordinates.

First step in the process was to have an outlay of our hardware and how it will function. It was decided that the input can be entered via a graphical interface of a programme which could run in any personal computer. This input will be analysed and then relayed to the micro controller using a HyperTerminal USB connector. Micro controller would then send the appropriate instructions to the motor to rotate for the required time frame so that the end effector would reach the points as needed. We would also have a LCD display which will additionally display the status o f the process – ‘running’ and ‘complete’ etc.
Since we were dealing with DC motor, the movement of lead screw were to be controlled with the duration of time the motor would rotate. This is the parameter to be controlled electronically. It was decided that the quantization value will be half a second.
So the next step was to make a programme for the micro controller which would take in the duration of seconds a motor would run and then relay it to the Logic circuit for the proper movement and rotation of the motor. This programme was done in the assembly language to obtain maximum efficiency.
First a pseudo code was generated for the ease of coding and later it was developed into the final code. Algorithm of this code is such that it would first take the input as from the user which is the time period of rotation of each motor, it would then, first implement the smaller of the two time periods for both motor, it would then make the appropriate motor of the two rotate for additional time period equivalent to the difference of the two time period.
As an example if the motor A has to rotate for 10 seconds and B for 20 seconds for reaching end points (1,2), the microcontroller would first make both the motors rotate for 10 seconds together and then the next motor B for additional 10 seconds.
Microcontroller would also remember the last position reached so that when the next input comes in form of the duration of seconds to reach a new end point, microcontroller already remembers the number of seconds it has already moved and only the remainder would be implemented now.
As a continuation of the above example, now if the input is to rotate 15 seconds of motor A and 30 seconds of motor B, as an increment to the already reached previous position, Motor A would now need to rotate only 5 seconds and motor B for 10 seconds.
It is self explanatory that as the next input as 0 and 0 seconds for each motor, micro- controller would make motor rotate in 15 seconds for motor A and 30 seconds for motor B in reverse direction.

The working of this code was then simulated using assembly language programme simulation packages.
A MATLAB interface for inputting the co-ordinates of the end effectors was also made. This code would also display the output, the Input provided (duration of rotation) to each motor via the hyper terminal to the micro controller.
Once this was done, the next big step was the selection of Micro controller to use and the circuit for the same. This was first designed on paper and components decided.
Some of the major components required are –
8051 – Microcontroller
IC- 7414D
LM329 IC
LCD display,
Capacitors and resistors, wires, bread board, PCB etc.

Fig.21: schematic diagram of the 8051 input and output signal flow.

Fig.22: schematic diagram of the total circuit of microcontroller
Once these where procured the circuit was replicated on a Bread board and tested. Programme was also burned on to the 8051 micro controller.
The data sheet of the 8051 used for their design process is provided in the appendix.
Next this verified circuit was soldered permanently on a PCB for more efficient and lasting setup. The final testing was done on this setup and it was successfully validated.

Now it was integrated with the mechanical setup to control of the motor. Once this was done, circuit was ready for practical validity. Using the MATLAB interface, input was provided as the consequent display of status was successfully displayed on the LCD display but the IC would blow off during the starting of the implementation itself. It was later understood that it was the result of overflow of current via the IC which lead to the burning. To rectify this, a buffer circuit was added to the existing circuit which would absorb the excess current. The 5 V power for the circuit and 12V supply for the motor were now provided via a regulated power supply unit to prevent further damage.
Once these problems were rectified, the model started to function as planned and needed. Many points were inputted and the PKM machine was able to reach the point with maximum precision
Once the PKM machine was successfully tested and errors rectified, the setup was ready to be passed on for further experimentation and testing for various parameters like speed, accuracy, resolution etc.


After the completion of the fabrication and the integration of electronics of the model we then went for testing of the mechanism. Testing the mechanism is the final phase of the project. The main aim of the testing was to find out the parameters like speed, accuracy, error and resolution of the mechanism and to make more improvements in the mechanism if possible, hence improving the performance of the mechanism. The parameters like accuracy, resolution etc are important parameters that show the working efficiency of the mechanism. Hence these were our prime focus.
The main parameters measured were:

8.1. SPEED
The first step in the testing phase was the measurement of the speed. The speed refers to the velocity of the two sliders along the slide rail. Speed is a very important parameter when it comes to PKM because it most operations that are done in a PKM require an optimum speed of the sliders. It is this slider velocity that decides the final speed of the end effector.
The speed of the slider was calculated by measuring the time taken to cover a distance of about 50 mm using a stop watch and then dividing it by the distance covered. This process was repeated a few times and the average of the values was found out so as to minimize the error in the measurement of the speed. The same method was used for measuring both the slider velocities. Thus we measured the speed (i.e the slider velocity) which was found to be around 3.33 mm/s .The measured velocity of both the sliders were found to be approximately equal. The slider velocity mainly depended on the motor rpm, which in our case was 100 rpm. A higher slider velocity would have been obtained if a higher rpm motor was used.
Resolution is the ability to 'resolve' differences; that is, to draw a distinction between two things. High resolution means being able to resolve small differences. Basically, resolution quantifies how close lines can be to each other and still visibly be resolved. The present resolution ranges for most mechanisms are constantly shifting because of the technology changes.
Since 0.5 seconds is the quantization for the time for the microcontroller i.e. since the minimum incremental value that has been programmed is 0.5 seconds, so corresponding to it we get a measurable resolution 1.4 mm, i.e.
Velocity of the sliders = 3.33 mm/s
Distance covered in 3.4 seconds = 10 mm
Distance covered in 1 second = 10 ÷ 3.4 = 2.94 mm
Quantization time for the microcontroller = 0.5 seconds
Hence, Resolution of the mechanism = 0.5 × 2.94 = 1.4 mm As seen in section 3.5 where the workspace of the PKM was plotted using MATLAB. The plot showed that points at the peripheries were too close to distinct as compared to the central region indicating that in order to plot distinct points in the periphery, resolution of the actuator should be as high a 0.01mm. Thus resolution is a major limiting factor in case of the determination of the usable workspace.

The accuracy of a system refers to how much the system, whether in measurement or control, deviates from the truth. It is the degree of closeness of measurements of a quantity to that quantity's actual value or true value. Accuracy is usually established by repeatedly measuring some traceable reference standard .To be meaningful, accuracy must really refer to 'worst case accuracy'. The accuracy of a system can never exceed its resolution.
The accuracy of the two axis PKM was measured by plotting twenty random points in a graph sheet using the mechanism and then comparing the actual and the obtained positions of these twenty random points. We also plotted the variation in tracing few patterns. The errors were tabulated for each point and then finally the cumulative error or average error was calculated. The average error calculated using the twenty random points was about 0.389 mm.
The sample images of the two points that were traced are shown below:

Fig.23: sample images of some traced points.


As per the objective, we were successfully able to design and fabricate a 2 axis PKM which could carry put point to point movement with remarkable precision and accuracy. The electronic interface with automated control really enhanced the functionality and the utility of the device. However, we faced a lot of challenges which is worth mentioning.
Selection of right motor which is the heart of the PKM is a task to be carried out with care as it has to provide the exact amount of torque and linear velocity, at the same time has to be under the permissible amperage and voltage to be compatible with the electronic circuit.
The fabrication process has to be reviewed with care especially about the linear actuator. Even a small error in the linearity of the linear actuator caused quite a considerable difference in accuracy and torque required.
In this regard, we would suggest incorporating a feedback mechanism to the existing machine to further improve the accuracy. Selection of linear actuator can also be done more judiciously for better efficiency but of course cost is a great concern in this regard.
So, these matters need to be kept in mind when designing and future fabrication of similar setup. We would like to mention a few of significant challenges and directions for the future work.

· Work volume optimization (without losing stiffness)
· Alternative kinematic configurations
· Improved modularity/configurability
· Low friction, high stiffness joints
· Improved machine dynamics
· Application/market development
· Improved control system capabilities
· Improved speed and convenience of calibration

This attempt at PKM is a humble step which beholds a lot of scope of improvement. The next step to the existing design would be to add a Machine tool to the end effectors thereby making it a fully functional machining apparatus. Also, the number of degrees of freedom could be enhanced from the present 2 degrees of freedom

Appendix I MATLAB code for Position Analysis clear all close all clc syms m; %create a symbolic variable n % a=16; % p=0:0.01:8; sol=solve('(3-m^2)*m^2/(m^2+1)*a^2/(4*p^2)=1',m); %solve for n in terms of N clear a p; a=16; p=3; syms q; q= 6; sol=a*sol; disp(eval(sol(3))); b= eval(sol(3)); syms q1; sol1=solve('(b/2)+ (a/2)*sqrt((3*a^2-b^2)/(a^2+b^2))- (sqrt(3)*a/2)-q1=0',q1 ); syms q2; q2=eval(sol1); disp(eval(sol1)); syms y; sol2=solve('q-q2-y=0',y); y= eval(sol2); disp(eval(sol2)); syms x; sol4=solve('b+y-x=0',x); disp(eval(sol4)); Appendix II MATLAB code for Workspace Analysis Code to generate Workspace. X=0; Y=0; syms x m p q r a=16; p=sqrt((3-m^2)*m^2/(m^2+1)*a^2/(4)); q=(x/2)+(a/2)*sqrt((3*a^2-x^2)/(a^2+x^2))-(sqrt(3)*a/2); i=0; j=0; s=1; for Y=0:0.25:32 if Y+16<32 r=16; else r=32-Y; end for X=Y:0.25:Y+r x=X-Y; m=(X-Y)/16; T(s,1)=eval(p); T(s,2)=Y+eval(q); s=s+1; end end for X=0:0.25:32 if X+16<32 r=16; else r=32-X; end for Y=X:0.25:X+r x=Y-X; m=(X-Y)/16; T(s,1)=-1*eval(p); T(s,2)=X+eval(q); s=s+1; end end i=1; while i<s+1; scatter(T(i,1),T(i,2),'.') i=i+1; hold on end


[1] Vinodh, S., “Axiomatic modelling of agile production system design,” International Journal of Production Research, Vol. 49:11, pp. 3251-3269, 2011 [2] Gunasekaran, A., “Agile manufacturing: Enablers and An implementation framework,” International Journal of Production Research, Vol.36:5, pp. 1223-1247, 1998 [3] Aitken, J., Christopher, M., and Towill, D., “Understanding, Implementing and Exploiting Agility and Leanness,” International Journal Of Logistics Research and Applications, Vol. 5:1, pp. 59-74, 2002 [4] Chan, T.S., and Zhang, C., “Modelling for agile Manufacturing systems,” International Journal of Production Research, Vol.39:11, pp. 2313-2332, 2001…...

Similar Documents

Free Essay

Parallel Computing and Algorithms

...i SASTRA UNIVERSITY Shanmugha Arts, Science, Technology and Research Academy Thirumalaisamudram Thanjavur – 613 402 BCSCCS705 PARALLEL COMPUTING AND ALGORITHMS LAB B.TECH (COMPUTER SCIENCE AND ENGINEERING) 7TH SEMESTER ii List of Exercises: 1. Basic arithmetic operations in parallel 2. Find out factorial of a number 3. Generation of Fibonacci series, finding prime numbers in an interval. 4. Evaluate the integral of a function 5. Merging of two sorted lists 6. Parallel tree traversals 7. Matrix multiplication 8. Enumeration sort 9. Odd-Even transposition sort 10. Bitonic merge 11. Quick sort 12. Single source shortest path iii Exercise 1 Aim: Basic Arithmetic Operations in parallel To perform a set of arithmetic operations in parallel using a cluster of computers Procedure: Step 1: Identify a set of numbers over which arithmetic operations are to be done Step 2: Identify the rank in the cluster Step 3: Split the set of numbers into domains and assign each domain to their corresponding processor Step 4: Get the result from each computer in the cluster and assimilate them at the master Step 5: Verify the time taken for the completion of each task. Algorithm:function sum(+,identity,a) = if #a == 1 then [identity] else let e = even_elts(a); o = odd_elts(a); s = scan_op(op,identity,{op(e,o): e in e; o in o}) in interleave(s,{op(s,e): s in s; e in e}); iv Input:- An array of integers Output: - The sum 3,2,7,6 P1 18 0,5,4,8 P2 17 62 2,0,1,5 P3 8...

Words: 1116 - Pages: 5

Free Essay

Electric Machines

...Science 6.685 Electric Machines Class Notes 1: Electromagnetic Forces c 2003 James L. Kirtley Jr. September 5, 2005 1 Introduction Bearings Stator Stator Conductors Rotor Air Gap Rotor Conductors Shaft End Windings Figure 1: Form of Electric Machine This section of notes discusses some of the fundamental processes involved in electric machinery. In the section on energy conversion processes we examine the two major ways of estimating electromagnetic forces: those involving thermodynamic arguments (conservation of energy) and field methods (Maxwell’s Stress Tensor). But first it is appropriate to introduce the topic by describing a notional rotating electric machine. Electric machinery comes in many different types and a strikingly broad range of sizes, from those little machines that cause cell ’phones and pagers to vibrate (yes, those are rotating electric machines) to turbine generators with ratings upwards of a Gigawatt. Most of the machines with which we are familiar are rotating, but linear electric motors are widely used, from shuttle drives in weaving machines to equipment handling and amusement park rides. Currently under development are large linear induction machines to be used to launch aircraft. It is our purpose in this subject to develop an analytical basis for understanding how all of these different machines work. We start, however, with a picture of perhaps the most common of electric machines. 2 Electric Machine......

Words: 5431 - Pages: 22

Premium Essay


...Worksheet Kinematics 1. A body starts from rest and reaches a speed of 5 m/s after travelling with uniform acceleration in a straight line for 2 s. Calculate the acceleration of the body. 2. A body starts from rest and moves with uniform acceleration of 2m/s2 in a straight line. a. Calculate the velocity after 5s. b. Calculate the distance travelled in 5s. c. Find the time taken for the body to reach 100m from its starting point. 3. An airplane accelerates down a runway at 3.20 m/s2 for 32.8 s until is finally lifts off the ground. Determine the distance traveled before takeoff. 4. A car starts from rest and accelerates uniformly over a time of 5.21 seconds for a distance of 110 m. Determine the acceleration of the car. 5. Upton Chuck is riding the Giant Drop at Great America. If Upton free falls for 2.6 seconds, what will be his final velocity and how far will he fall?   6. A race car accelerates uniformly from 18.5 m/s to 46.1 m/s in 2.47 seconds. Determine the acceleration of the car and the distance traveled.   7. A feather is dropped on the moon from a height of 1.40 meters. The acceleration of gravity on the moon is 1.67 m/s2. Determine the time for the feather to fall to the surface of the moon.   8. Rocket-powered sleds are used to test the human response to acceleration. If a......

Words: 781 - Pages: 4

Free Essay

Serial Communication over Parallel

...EEL201 Term Paper: Serial Communication Over Parallel 1 Serial Communication Over Parallel Sachin Garg Indian Institute of Technology, Delhi-110016 E-mail:  Abstract—Communication is the essence of today’s technologically advanced world. The two most basic methods of digital communication are: Serial and Parallel. Parallel communication involves multiple wires running parallel to each other and can transmit data on all the wires simultaneously. However, Serial communication requires a single wire for transferring data bits one at a time. A natural question arises out of human curiosity: which one of the two methods is better. This term paper discusses these two methods, their various applications and the advantages of the Serial Communication method over the Parallel one. Index Terms—Parallel Communication, Serial Communication UART, RS-232 DTE, Synchronous, Asynchronous. Parallel counterpart have been resolved. Although the Parallel Transmission Method is still used within internal buses and for devices such as printers, but the latest devices, ports and technologies are mostly based on Serial Techniques. For example - Universal Serial Bus and WAN, both make use of the Serial Type Data Transmission. II. TERMS USED IN DATA COMMUNICATION A. Transmission Distance The transmission distance of a communication signal is the length of path the signal travels as it goes from transmitter to the receiver. B. Data Terminal Equipment (DTE) Data......

Words: 2963 - Pages: 12

Free Essay

Parallel and Series Ports

...Video Summary 1 Topics covered in the videos 1.07, 1.09 and 1.11 * Ports * Connectors * Monitors * Video buses * Troubleshooting * Printers Ports * Serial ports transmit data 1 bit at a time. They are usually male and have 9-25 pins. * Parallel ports transmit data 8 bits at a time and are generally female with 25 pins. * Game ports have 15 pins laid out in two rows (female). * Video monitors have 15 pins but are laid in three rows (male) Connectors * Early mice used a serial connector and keyboards used a 5 pin DIN connector. * Modern mice and keyboards use a PS/2 technology which is a much smaller 6 pin DIN connector or a USB or wireless connection. * NIC- Network Interface card uses a RJ-45 connector. * The NIC is responsible for connecting a computer to local and distant networks or the internet. * It also converts parallel communication used by the motherboard system into serial communication used by the network cable. Modem, Video and USB * It is the slowest possible way of connecting to the internet. * Its standard speed is 56Kbps although v.92 standard supports faster speed. * They are responsible for converting the analog signal from the phone line into a digital format a computer can use. * It also converts the digital signal back to analog when sending a transmission. They connect using a RJ-11 connector. * Modern computers come with two types of video ports , the VGA (analog......

Words: 906 - Pages: 4

Free Essay


...UNIT 2 D.C MACHINES 1. Give methods adopted to reduce the effect of armature reaction in a DC machine. i) By increasing the length of air gap at pole tips ii) By increasing the reluctance at pole tips iii) Providing compensating winding and interpoles. 2. List the advantages of lap winding in a DC machine. i) Reduction of weight of armature core ii) Cost of armature and field conductors iii) Overall length and diameter of machine. 3. State the relationship between number of armature coils and number of commutator segments in a dc machine. The relationship between number of armature coils and number of commutator segments in a dc machine is where - commutator segment pitch C- no. of coils D- diameter of commutator 4. State different losses in a dc generator. a) Copper losses: (i) armature copper loss (Ia2 Ra) ii) field copper loss (I2snRsh, I2seRse) b) Magnetic losses ( iron or core loss): i) hysteresis loss (Whα B1.6maxf) ii) eddy current loss(Weα B2maxf) iii) mechanical losses 5. What are the main dimensions of a rotating machine? The main dimensions of a rotating machine are armature diameter and stator core length. 6. Define specific magnetic loading. It is defined as the average flux density over the air gap of the machine. Specific magnetic loading = 7. Define specific electric loading. It is defined as the no of armature conductors per meter of armature periphery at the air gap. Specific electric loading = 8. What is output......

Words: 1035 - Pages: 5

Premium Essay

Kinematics - Velocity and Acceleration

...Lab #4: Kinematics - Velocity and Acceleration Introduction: The purpose of this lab is to discover and understand the relationships between position, velocity, and acceleration. Additionally, constant/uniform acceleration due to the force of gravity will be examined to find possible mathematical relationships to position and velocity. Velocity and acceleration are changes in position and velocity, respectively, with regards to time. This change can be shown mathematically in calculus derivatives: EQ 1. EQ 2. As dt decreases in value, the instantaneous velocity and acceleration can also be found. Furthermore, if constant acceleration is established, two basic relations between distance, velocity, and the constant acceleration can be found: EQ 3. EQ 4. In any environment near Earth, the acceleration in the vertical direction is constant at a value of g=9.8m/s2 towards the center of Earth or often written as g=-9.8m/s2. In such an environment there is no natural acceleration in the horizontal direction, thus the horizontal motion is analyzed independently of the vertical motion. Thus it can be established that the general form of a position curve for a projectile would follow an inverse parabola shape and the maximum height occurs when vertical velocity is zero. By calculus derivation, it can also be found that the velocity graph would display a linear line with a negative slope. Procedure: This lab consists of two separate but related......

Words: 1974 - Pages: 8

Free Essay

Dynamic Modelling of Parallel Manipulator

... “Newton Euler Strategy for the Dynamic Analysis of 3-DOF Parallel UPU Manipulator” Abstract This projects aims to describe the dynamic analysis of the 3-DOF Parallel UPU manipulator. The 3 DOF UPU is a parallel manipulator in which the U and P notations are for universal and prismatic pairs respectively, is a well-known manipulator that provides the platform with three degrees of freedom of pure translation, pure rotation or mixed translation and rotation with respect to the base, according to the relative directions of the revolute pair axes. The pure translational parallel 3-UPU manipulators have received a great lot of attention. Over the years many studies have been reported in the literature on the kinematic analysis, workspace optimization, singularities and joint clearance influence on the platform accuracy of this manipulator. However, the evaluation of the dynamic modelling of the parallel manipulators based on the two most famous methods i.e. Newton- Euler and Lagrange’s method have not been clearly discussed so far. Therefore, this project tries to throw light on this area and covers the Newton Euler Strategy for the calculation of the dynamic model of the 3 DOF UPU Parallel Manipulator. 1. Introduction Over the last few decades the parallel manipulators have attracted the attention of lot of researchers because of their complementary characteristic with serial manipulator. The word 'Parallel’ which is used here is to point out is not in the......

Words: 2803 - Pages: 12

Premium Essay


...Should machines be used to do good and services instead humans? I believe they should not for many reasons. Machines can not be smart unless a human is controling it. So it would not matter if its an intelligent machine or not a human is still controlling it to do everything. When using a machine it could easily malfunction and it could be hard to fix the problem or it will just take a while to fix it. If a human is taking over instead of the machine there may be fewer problems. Machines have so many problems that it would not be worth having. Also, the more machines you have the less jobs there are for people because everyone thinks it would be better to have machines instead of people. When less people are out of work that means less money for those people and sometimes they will lose their homes or cars because they can not afford anything. Sometimes working with machines can be very stressful because they may not work at times or they could be running extremely slow and won’t get anything done. Machines are not smart at all, only when people are controlling them they are but not all the time. It may seem smart but its really not. In conclusion, I think machines should not be used to take over a human job because machines can not think only humans can think and make right or wrong decisions. Machines do not have brains, their not wired to think so why have them do stuff that we can do ourselves....

Words: 266 - Pages: 2

Premium Essay

Machine Design

...conditions are very unstable, and any small reduction in speed or viscosity or increase in bearing pressure will break down the film and cause the coefficient of friction to vary considerably. * Full film lubrication A film of oil will adhere to metal surfaces. As a result when two surfaces move relative to one another, and with a wedge configuration which allows oil to be forced between the surfaces, a pressure will develop to separate the surfaces and form a thicker film. The temperature is related to the rate of shear (speed) and in turn, oil viscosity and load. 9) Absolute viscosity-is defined as the force necessary to move a plane surface of one square centimeter in area past another plane surface at a rate of 1cm per second. Kinematic viscosity-is the accepted standard viscosity measured throughout the world and is the Absolute viscosity/Density at the same temperature. 10) SAE-Society of Automotive Engineers 11) Refining processes of oil * Asphalt removal-the solvent is liquid propane which dissolves out everything dissolves out everything bar the asphalt. The solvent is then distilled off. * Aromatic removal-furfural, which is an aldehyde produced from the carbohydrates in mealie cobs and sugar cane, has the property of dissolving out aromatics, leaving the god oil and waxes. * Wax removal-methyl ethyl ketone has the property of agglomerating the waxes in oil. After the addition of this material , the oil is chilled to a set temperature and......

Words: 1658 - Pages: 7

Premium Essay

Parallel Justice for Victims

...Susan Herman’s idea of “parallel justice” for victims is a good idea. Criminal justice is extremely offender-oriented and the criminal justice system doesn’t focus much on helping the victim adjust back to normal life after whatever traumatic event they go through. Our system focuses on the offender and making sure the offender can adjust to life after whatever punishment they get and we make sure that they do not end up back in the criminal justice system. Susan Herman’s idea of parallel justice for victims has many potential strengths having the safety of the victim be the primary concern and focusing on the victim is something that the criminal justice system definitely needs to work on as soon as possible. I also believe that every victim of a crime should be offered immediate support, compensation for losses, and practical assistance. The third point is the most important I believe, having the victim explain what happened to them and the impact of the crime had on their lives and what resources that they are going to need to get back into their everyday life. Finally, victim needs should always be met and they should be met as quickly as possible. Focusing our criminal justice system and its resources on the victim a little more rather than the offender is a step that must be taken soon. The victim deserves to be able to get back into normal life and feel safe and be compensated for any damages, psychically and emotionally, that they have received from the offender, and......

Words: 408 - Pages: 2

Free Essay


...Standard Signaling Method for a Bidirectional Parallel Peripheral Interface for Personal Computers IEEE Computer Society .- Sponsored by the Microprocessor and Microcomputer Standards Committee h Published by the Institute of Electrical and Electronics Engineers, Inc., 345East 4Tth Street, New York, N y lwlz USA. lEEE December 2, 1994 SH 1 7335 Authorized licensed use limited to: UNIVERSITY OF VICTORIA. Downloaded on July 29,2010 at 20:21:17 UTC from IEEE Xplore. Restrictions apply. THIS PAGE WAS BLANK IN THE ORIGINAL Authorized licensed use limited to: UNIVERSITY OF VICTORIA. Downloaded on July 29,2010 at 20:21:17 UTC from IEEE Xplore. Restrictions apply. Recognized as an American National Standard (ANSI) IEEE Std 1284-1994 IEEE Standard Signaling Method for a Bidirectional Parallel Peripheral Interface for Personal Computers Sponsor Microprocessor and Microcomputer Standards Committee of the IEEE Computer Society Approved March 30, 1994 IEEE Standards Board Approved September 2, 1994 American National Standards Institute Abstract: A signaling method for asynchronous, fully interlocked, bidirectional parallel communications between hosts and printers or other peripherals is defined. A format for a peripheral identification string and a method of returning this string to the host outside of the bidirectionaldata stream is also specified. Keywords: bidirectional parallel communications, computers, interfaces,......

Words: 34986 - Pages: 140

Premium Essay

Parallel Journeys

...Book Review Parallel Journeys Parallel Journeys, by Eleanor Ayer with Helen Waterford and Alfons Heck, compares the life of two young people in Germany during the rise of Hitler and the outbreak of World War II. They were born within sixty miles of each other, but their lives took dramatically different paths. Alfons, is a boy who grew up on a farm in Germany. When Hitler came to power he became involved with the Hitler Youth. Alfons grew up on a farm but was a Nazi boy who became a commander at age sixteen. This book truly depicts Hitler's impact and influence on the youth of Germany. Alfons was brainwashed by Hitler's promises, a full pledged Nazi youth is how it is presented in the book he participated in all rallies. He started his training to become the future of Germany and eventually of the world. There he learned discipline and order. He trained to become a soldier and he learned the ways of warfare. He was taught to be a follower of anti-Semitism and was told to hate the Jewish people. He along with millions of Nazi’s believed that they were the reason for Germany's problems. As years went by, Alfons rose through the ranks of the military until the end of the war where realized what atrocities had been committed, he than took off his uniform and went on to help people. On the other side of the spectrum we have Helen Waterford a young Jewish girl who grew up in Frankfurt, Germany. She married Siegfried Wohlfarth and moved to Amsterdam because of tension......

Words: 2271 - Pages: 10

Free Essay

Parallel Postulate

...Elliptic Geometry The parallel postulate (a.k.a. Euclid’s fifth postulate) in Euclidean Geometry states that if one cuts a line segment with 2 lines, and the interior angles add up to less than 180 degrees, then the two lines will eventually meet if extended infinitely (Euclid’s parallel postulate exact words: if a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles). Mathematicians such as Aristotle tried to prove this prove the parallel postulates, but they always ended up using the postulate itself in proving it, causing great controversies. In the late 19th century, mathematicians began to question if the postulate was true, which gave rise to the Non-Euclidean geometries. Elliptic geometry is a form of Non-Euclidean Geometry. It is different from Euclidean Geometry because it replaces the parallel postulate with the statement, "through any point in the plane, there exist no lines parallel to a given line” (there are basically no parallel lines), and because it modifies Euclid’s second postulate (there are no infinate lines in elliptical geometry). Bernard Riemmannian, the man who founded Elliptical (a.k.a. Riemmanian Geometry) Geometry had a lecture in 1854 discussing the ideas currently called manifolds, Riemmanian metric, and curvature. He also made many Non-Euclidean......

Words: 422 - Pages: 2

Free Essay


...  • FREE OPERATIONS TRAININ • FREE TECHNICAL TRAINING   FREE DELIVERY                                                                                18,000 Location 132 V. LUNA ROAD EXTENSION SIKATUNA VILLAGE QUEZON,CITY   NEAR CAMP CARINGAL LOOK FOR: MR. RICMAR YTING AND MS. DINAH GOROSPE MS. RHIZA BAGSIK SALES MANAGER     CALL US NOW! EXCITING PROMOS AND FREEBIES AWAIT YOU! TEL. NOS. 433-5671 • 433-3413 • 435-3533 MOBILE NOS. • 0906-328-4737 (GLOBE) • 0922-207-4051 (SUN) • 0947-177-4969 (SMART) #09277507384 (GLOBE) #09292688228 (SMART) *09236575658 (SUN) *09154997653 (GLOBE) PHOTO DIGITAL PRINTING MACHINE If you are interested in setting-up a DIGITAL PHOTO PRINTING BUSINESS commercial or home based and cannot afford the machines used by other digital printing companies or don’t have the knowledge w/ this kind of business. We can help you start your own digital photo printing business for a complete package price of P21,000.00 ONLY. ALL EQUIPMENTS ARE BRAND NEW & w/ WARRANTY. We used high quality non-fading ink that would last for 100 years. (Super vivid colors ang quality at di po sya kumukupas kahit mabasa or maarawan) Digital photo printing business is now emerging as profitable market boosted by the growing number of affordable camera phones and digital cameras. Low capital but higher profit compared to Internet Café Business. ROI: 4 to 6 mos. (depending upon your shop location) With...

Words: 1437 - Pages: 6