Advance Modeling of a Skid-Steering Mobile Robot for Remote ...

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6.1 Simulink Model 82Figure 6.3: Block scheme inside the ¨Wheel¨ block.˙ω i = K w (ω ire f− ω i ) (6.1)It is also well known that the motor dynamics can be described by the equation:τ i = I m ˙ω i (6.2)where I m represents the motor inertia.By combining equation (6.1) and (6.2), we obtain the following relation defining the torquecontrol input:τ i = I m K w (ω ire f− ω i ) (6.3)If we do not take into account any gear-transmission and electrical issue typical of DCmotors, we have K w = 1I m, therefore the torque control torque is simply provided by the differenceof the desired and measured wheel angular velocity. The block in yellow implementsthe motor model defined in (6.1) and provides both the wheel angular velocity and torque asoutput. It is worth noticing that such a model holds only if the motor can always provide the
6.1 Simulink Model 83desired torque. Such an assumption can be considered verified, except for the fact that, whenthe motor can not provide the required torque, the robot safe mode makes the motor stall.Moreover, the model does not take into account any elasticity effect due to the motor belt.However, the model is still meaningful as the elasticity-damping effects of the motor belt arealready take into account in the model for the tire longitudinal reaction force, as explainedin Section 4.2.The block in magenta computes the wheel center accelerations from the robot accelerationby employing (2.55), considering ˙V, ˙V i instead of V, V i , then ˙V i is integrated to provide V i .The blocks in light blue, red and green represent respectively the longitudinal, lateral andvertical reaction force block and implement (4.13), (4.9) and (4.15). In particular, while theblock in green directly implements (4.15), the blocks for the longitudinal and lateral reactionforces present some differences with respect to (4.13), (4.9) due to simulation issues whichare further discussed in the following description of the ¨Flongitudinal¨ block. However, weprecise that, as such a description goes into details of the implementation of (4.13),(4.9), thereader, if not interested in, may go directly to Section 6.2 without any loss of comprehension.Figure 6.4(a) depicts the block scheme for the longitudinal reaction force. The same blockscheme is provided for the lateral reaction force, by replacing the µ ix with µ iy and ∆v ix withv iy as driving velocity.The block in cyan implements the piece-wise linear function in (4.12a), representing thelongitudinal dynamic friction coefficient when the wheels are slipping, and it multiplies itby the normal reaction force N i . In such a way, the output of this block differs from thestatic and kinetic friction force only for the constant terms µ smax , µ kmax , which is useful for theimplementation of the hysteresis block determining the switch of the contact point dynamics,as it is explained in the following.The block in gray provides the contact point velocity by implementing (4.13), as depictedin Figure 6.4. However, (4.13) presents an important issue from an implementation point ofview, that is the complex behavior of the normal reaction force N ic perceived by the contactpoint when moving. As already introduced in Section 4.1, the normal reaction force N ic is notgiven only by the tire mass but it increases as the contact point gets closer to its equilibriumposition, thereby perceiving also the robot weight. This increase of the normal reaction forceperceived from the contact point, i.e. the increasing of the kinetic friction force, makes thecontact point stops instead of following the wheel center velocity, under certain conditions
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Università degli Studi di GenovaRo
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IINothing great in the World has be
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CONTENTSIV6 Simulation 786.1 Simuli
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LIST OF FIGURESVII6.4 (a) Block sch
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List of Tables5.1 Table of the geom
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Introduction 2boDynamics, VGo Commu
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Introduction 4The robot employed, i
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Chapter 1Hardware and SoftwareIn th
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1.1 The Pioneer 3-AT Robot 8four ho
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1.2 The sensors 10control resolutio
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2.1 State of the Art 12not negligib
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2.1 State of the Art 14velocity vec
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2.1 State of the Art 16Figure 2.3:
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2.1 State of the Art 18Finally, by
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2.1 State of the Art 20constraint (
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2.1 State of the Art 22approximates
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2.2 Generalized Modeling 24⎡1 0¯
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2.2 Generalized Modeling 26V = [v T
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2.2 Generalized Modeling 28where T(
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2.2 Generalized Modeling 30⎡g R(J
Page 42 and 43: 2.2 Generalized Modeling 32where we
Page 44 and 45: Chapter 3Experimental DataIn this c
Page 46 and 47: 3.1 The accelerometer 36(a)(b)Figur
Page 48 and 49: 3.2 Characterization of the Vibrati
Page 54 and 55: 3.3 Characterization of the Tire Fo
Page 66 and 67: 4.1 Tire Lateral Force 56Figure 4.1
Page 68 and 69: 4.1 Tire Lateral Force 58velocity b
Page 70 and 71: 4.1 Tire Lateral Force 600 ≤ µ i
Page 72 and 73: 4.3 Tire Vertical Force 62f ix (t)
Page 74 and 75: Chapter 5Identification of the Robo
Page 76 and 77: 5.1 Identification of the Geometric
Page 78 and 79: 5.2 Identification of the Tire Dyna
Page 88 and 89: Chapter 6SimulationIn this chapter,
Page 90 and 91: 6.1 Simulink Model 80On the right h
Page 94 and 95: 6.1 Simulink Model 84(a)(b)Figure 6
Page 96 and 97: 6.1 Simulink Model 86that v icx can
Page 98 and 99: 6.2 Simulation Results 88(a)(b)Figu
Page 100 and 101: 6.2 Simulation Results 90enough to
Page 102 and 103: 6.2 Simulation Results 92hand side
Page 104 and 105: 6.2 Simulation Results 94(a)(b)Figu
Page 106 and 107: 6.2 Simulation Results 96• For λ
Page 108 and 109: AppendixesAppendix A1 %% Main progr
Page 110 and 111: Appendixes 10080 title([conf,’; A
Page 112 and 113: Appendixes 10224 Hd = design(h,filt
Page 114 and 115: Appendixes 1043031 % Filter the dat
Page 116 and 117: Appendixes 106(a)(b)(c)Figure 6.10:
Page 130 and 131: Appendixes 120(a)(b)Figure 6.24: Pe
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Appendixes 132(a)(b)(c)Figure 6.35:
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Bibliography[1] G. Oriolo, A. De Lu
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