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Chapter 1                    Three wheelsomnidirectional mobile robot kinematicsThe notationand most background information in this section are cited from  7 unless stated otherwise.

 The robot under study in this project is a three-wheelomnidirectional robot, all of the wheels are driven and are Swedish 90degreewheels arranged as shown in Figure  2.1.This section introduce notation used to present the robot motion in both globalframe and local frame of the robot, and develop a kinematic representation ofthe whole robot. ?Figure 2.1 Three wheels omnidirectional robot. Adapted from 7.

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1.1      Robot position The model of the robot chassis is referred to as a rigidbody, ignoring all joints and degrees of freedom internal to the robot and itswheels.1.1.1   Global and local framesDescribing the position of the robot needs a steadyreference frame for the plane, this frame is called global reference frame, itstwo axes are  and . The frame refereed to the robot itself iscalled  local reference frame, the originof this frame  is a reference pointchosen on the robot chassis (P), and the two axes  and  arerelative to this point, in this case (P) is chosen at the center of the robotand  isaligned with the axis of the second wheel. For motion description It’s alsoimportant to find a relationship between these two frames, the angulardifference between the local and global frames is given by ?. This notation is shown in Figure2.

2.  ?Figure 2.2 Global and local frames. Adapted from 7.

1.1.2   Robot poseThe pose of the robot at any instant is described by threeelements; its x and y coordinates, and the angle between global and localreferences. The pose can be written as vector with three elements and it isreferred to the global frame: And the robot motion at a certain posereferred to global frame is given by the velocities ,  and : 1.1.3   Robot motion mappingTo map the motion from a global reference to the robot localframe, an orthogonal rotation matrix (R( )) is used: The robot motion at a certain posereferred to local frame is given by: 1.2      Forward kinematicsForward kinematics predicts the overall motion of the roboton global frame given the speed and geometry of the wheels. The three wheelsomnidirectional robot has three wheels, each radius is r.

The distance betweenthe center (P) and any of the three wheels is . The wheel rotational position is given by  and the wheels spinning speed is given by . Figure 2.3 shows the previousparameters. ?Figure2.3 Robot and wheel geometry and speed.

Adapted from 7.The forward kinematics model represents the motion ofthe  robot as function of r, , , ,  and : To map the motion from local frame toglobal frame the inverse orthogonal rotation matrix  is used:   1.3      Wheels kinematic constraintsIn order to find the kinematic model for the mobile robot,the wheels motion constraints should be presented. The kinematic properties andconstrains varies for different wheels types. Since the robot in this projecthas 90 degrees Swedish wheels, this section will evaluate a presentation forthe Swedish wheels constrains. At the begging, it’s important to make three assumptionsthat simplify wheels motion constraints presentation, and present pure rollingand rotation for the wheel without sliding content: 1.     The wheel’s plane remainsvertical.2.

     There is only one point ofcontact between the wheel and the ground at the single moment.3.     No sliding occurs at thepoint of contact. Under the previous assumptions,there are two general motion constrains every wheel type:1.     The concept of rollingcontact: the wheel should roll in appropriate amount based on the motion alongthe direction of wheel plane to achieve pure rolling.2.

     The concept of no lateralslippage: orthogonal sliding must not happen for the wheel on its plane. ? Figure2.4 Swedish  90degrees wheel and its parameters. Adapted from 7. The Swedish wheel as shown inFigure 2.8 has a radius r.

The position A is expressed by the distance between the center P and the wheel  and the angle . The wheel rotational position around thehorizontal axis is .  isthe angle from the wheel plane to the robot chassis and it is fixed for theSwedish wheel.

 isthe angle between the rollers axes and the wheels main axis, in case of 90degrees Swedish wheel .  The rolling constraint for the wheel toachieve pure rolling along the wheel plane is given by: But for the Swedish wheel, the smallrollers spin around the zero component of the velocity at the contact point. Thus, moving in the direction of thiszero velocity component is impossible without sliding. For that, the rollingconstraint formula is represented along the velocity zero component instead ofthe wheel plane : Forthe robot in this project,  is zero because the wheels are tangential torobot chassis ,   for the Swedish 90 wheel is zero the rollingconstraint formula along the velocity zero component:  (*)The sliding orthogonal to the wheel planeis not constrained for the Swedish wheel because of  the free rotation of the rollers  .

This is presented : Forthe robot in this project :        1.4      Complete kinematic model  A full model for the three wheels omnidirectional robot  shown in Figure  2.3 can be built by rewriting the rollingconstrain equation (*):   (**)Such that    isa constant diagonal 3×3 matrix that consists the robot wheels radiuses. In thisproject all radiuses are equal (r ): And    isa 3×3 matrix that relates to the rolling constrains: Rewriting equation (**):   Representing the robot motion as linear velocities in x and y direction  and , and the rotational velocity  .   The robot motion can be described as a linear velocity V andan angular velocity  as shown in Figure2.

3. The relationship between ,  ,  in the globalframe and  V ,  isrepresented in the following equations: 1.5      The values of  the angle for the three wheelsAs a convention  isalong the axis of the first wheel as shown in Figure 2.5. Thus, the anglebetween  and the wheel ( ) value for each wheel is constant andtheir values are (  , (  and ( .  Figure ?2.

5 Robot axis along wheel 1 axis and value As the robot changes its orientation, the angle between theglobal frame and the local frame ?shown in Figure 2.2 changes. The value of angle  atthe initial position shown in Figure 2.5 is zero.

 Chapter 1                    Three wheelsomnidirectional mobile robot kinematicsThe notationand most background information in this section are cited from  7 unless stated otherwise. The robot under study in this project is a three-wheelomnidirectional robot, all of the wheels are driven and are Swedish 90degreewheels arranged as shown in Figure  2.1.

This section introduce notation used to present the robot motion in both globalframe and local frame of the robot, and develop a kinematic representation ofthe whole robot. ?Figure 2.1 Three wheels omnidirectional robot. Adapted from 7.1.

1      Robot position The model of the robot chassis is referred to as a rigidbody, ignoring all joints and degrees of freedom internal to the robot and itswheels.1.1.1   Global and local framesDescribing the position of the robot needs a steadyreference frame for the plane, this frame is called global reference frame, itstwo axes are  and . The frame refereed to the robot itself iscalled  local reference frame, the originof this frame  is a reference pointchosen on the robot chassis (P), and the two axes  and  arerelative to this point, in this case (P) is chosen at the center of the robotand  isaligned with the axis of the second wheel. For motion description It’s alsoimportant to find a relationship between these two frames, the angulardifference between the local and global frames is given by ?.

This notation is shown in Figure2.2.  ?Figure 2.

2 Global and local frames. Adapted from 7.1.

1.2   Robot poseThe pose of the robot at any instant is described by threeelements; its x and y coordinates, and the angle between global and localreferences. The pose can be written as vector with three elements and it isreferred to the global frame: And the robot motion at a certain posereferred to global frame is given by the velocities ,  and : 1.1.

3   Robot motion mappingTo map the motion from a global reference to the robot localframe, an orthogonal rotation matrix (R( )) is used: The robot motion at a certain posereferred to local frame is given by: 1.2      Forward kinematicsForward kinematics predicts the overall motion of the roboton global frame given the speed and geometry of the wheels. The three wheelsomnidirectional robot has three wheels, each radius is r. The distance betweenthe center (P) and any of the three wheels is . The wheel rotational position is given by  and the wheels spinning speed is given by . Figure 2.3 shows the previousparameters. ?Figure2.

3 Robot and wheel geometry and speed. Adapted from 7.The forward kinematics model represents the motion ofthe  robot as function of r, , , ,  and : To map the motion from local frame toglobal frame the inverse orthogonal rotation matrix  is used:   1.3      Wheels kinematic constraintsIn order to find the kinematic model for the mobile robot,the wheels motion constraints should be presented.

The kinematic properties andconstrains varies for different wheels types. Since the robot in this projecthas 90 degrees Swedish wheels, this section will evaluate a presentation forthe Swedish wheels constrains. At the begging, it’s important to make three assumptionsthat simplify wheels motion constraints presentation, and present pure rollingand rotation for the wheel without sliding content: 1.

     The wheel’s plane remainsvertical.2.     There is only one point ofcontact between the wheel and the ground at the single moment.3.

     No sliding occurs at thepoint of contact. Under the previous assumptions,there are two general motion constrains every wheel type:1.     The concept of rollingcontact: the wheel should roll in appropriate amount based on the motion alongthe direction of wheel plane to achieve pure rolling.2.     The concept of no lateralslippage: orthogonal sliding must not happen for the wheel on its plane.

? Figure2.4 Swedish  90degrees wheel and its parameters. Adapted from 7. The Swedish wheel as shown inFigure 2.8 has a radius r.

The position A is expressed by the distance between the center P and the wheel  and the angle . The wheel rotational position around thehorizontal axis is .  isthe angle from the wheel plane to the robot chassis and it is fixed for theSwedish wheel.  isthe angle between the rollers axes and the wheels main axis, in case of 90degrees Swedish wheel .

  The rolling constraint for the wheel toachieve pure rolling along the wheel plane is given by: But for the Swedish wheel, the smallrollers spin around the zero component of the velocity at the contact point. Thus, moving in the direction of thiszero velocity component is impossible without sliding. For that, the rollingconstraint formula is represented along the velocity zero component instead ofthe wheel plane : Forthe robot in this project,  is zero because the wheels are tangential torobot chassis ,   for the Swedish 90 wheel is zero the rollingconstraint formula along the velocity zero component:  (*)The sliding orthogonal to the wheel planeis not constrained for the Swedish wheel because of  the free rotation of the rollers  .This is presented : Forthe robot in this project :        1.

4      Complete kinematic model  A full model for the three wheels omnidirectional robot  shown in Figure  2.3 can be built by rewriting the rollingconstrain equation (*):   (**)Such that    isa constant diagonal 3×3 matrix that consists the robot wheels radiuses. In thisproject all radiuses are equal (r ): And    isa 3×3 matrix that relates to the rolling constrains: Rewriting equation (**):   Representing the robot motion as linear velocities in x and y direction  and , and the rotational velocity  .   The robot motion can be described as a linear velocity V andan angular velocity  as shown in Figure2.

3. The relationship between ,  ,  in the globalframe and  V ,  isrepresented in the following equations: 1.5      The values of  the angle for the three wheelsAs a convention  isalong the axis of the first wheel as shown in Figure 2.

5. Thus, the anglebetween  and the wheel ( ) value for each wheel is constant andtheir values are (  , (  and ( .  Figure ?2.5 Robot axis along wheel 1 axis and value As the robot changes its orientation, the angle between theglobal frame and the local frame ?shown in Figure 2.

2 changes. The value of angle  atthe initial position shown in Figure 2.5 is zero. 

x

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