Gelöste Aufgaben/JUMP

Problem Statement

This example from Mechatronics combines mathematical models of mechanical and electrical components to simulate an RC-car-model. We've taken some reference-parameters from commercial vendors and adopted them as appropriate.

Jump!

Create a mathematical Model for an RC-Model Car.

The model shall account for the basic car-functions in a 2D-driving environment including vehicle dynamics, drive-train, battery and driver.

A 2D-test-parkour shall be provided to assess basic climbing and jumping capabilities of the car.

Overall Scope

We are tasked to provide a mathematical performance model for an RC-car model with a given reference-configuration. The mathematical model shall account for different operational envelopes and test for e.g. climbing capabilities, typical loads on springs and car-agility.

We follow a generic approach in modelling and simulation which consists of three overall steps:

• Structuring
• Modelling
• Simulating

Structure

The function of “structuring” is to break down this complex task - modelling and simulating the performance of our RC-car - into work-packages. We aim at reducing the complexity of the model until the remaining entities appears to be rather trivial to manage.

The structure we are aiming at is a pre-requisite for step 2: building the mathematical model. This model is the projection of the physical processes into mathematical equations. We’ll be implementing our model as an Initial-Value-Problem (IVP) , written in mathematical terms as

${\dot {\underline {Q}}}={\underline {f}}({\underline {Q}},t);{\dot {(.)}}={\frac {\displaystyle d}{\displaystyle dt}}(.)$

The left-hand side of the equation, q, are the state variables of our system. They are the minimum-set of values to describe the physical state of our model at a certain time t. The state variables stem from the car's sub-systems. We choose four workpackages to contribute to the state-variables:

• Battery:the power-source for the car
• Car Body: consisting of the car's chassis and its wheels
• Driver-Controls: processes that control acceleration and deceleration of the car
• E-Motor and Drive-Train: the mechanisms that transfer battery power into mechanical energy.

State Variables

Each workpackage contributes state variables to the column vector q:

The IVP establishes mathematical equations describing how these state variables change over time, e.g. how a driver actuates the "gas"-pedal, which increases the motor-voltage, thus increasing the motor-torque and hence accelerating the car.

Block Diagram.

Our first approach is to depict each workpackage as graphical block and connect information / property exchange between blocks as in a diagramming tool:

The driver chooses a "gas"-pedal state "p" depending on information form the car's location, speed and battery temperature "T_B".

At the E-motor, "p" sets the nominal voltage "U_M" of the E-motor, this implies a motor current "I_M" and wheel torque "M_W".

The motor current "I_M" then drives both the battery temperature and the available battery voltage "U_B".

And the wheel torque propels the car, delivering new locations and velocities.

These are the submodels to be developed in each workpackage:

Battery

Car-Body

Driver Controls

E-Motor and Drive-Train

Model

The mathematical model consists of XX differential equations with coordinates for each subsystem:

${\underline {q}}_{B}(t):=\left({\begin{array}{c}\color {orange}Q_{B}(t)\\\color {orange}U_{B2}(t)\\\color {orange}T_{B}(t)\\\end{array}}\right)$ for the battery,

${\underline {q}}_{C}(t):=\left({\begin{array}{c}\color {blue}u_{1}(t)\\\color {blue}u_{2}(t)\\\color {blue}u_{3}(t)\\\color {blue}u_{4}(t)\\\color {blue}\varphi (t)\\\color {blue}\psi (t)\\\end{array}}\right)$ for the car-body

${\underline {q}}_{D}(t):=\left({\begin{array}{c}\color {green}E(t)\\\end{array}}\right)$ for the driver and

${\underline {q}}_{E}(t):=\left({\begin{array}{c}\color {red}I_{M}(t)\\\color {red}\psi _{M}(t)\end{array}}\right)$ for the E-motor.

However, we have no elasticity accounted for between the E-motor and the wheel, thus

${\dot {\tilde {\psi }}}(t)=n_{G}\cdot {\dot {\tilde {\psi }}}_{M}(t)$

and so the remaining coordinates are

${\underline {q}}(t):=\left({\begin{array}{c}\color {orange}Q_{B}(t)\\\color {orange}U_{B2}(t)\\\color {orange}T_{B}(t)\\\color {blue}u_{1}(t)\\\color {blue}u_{2}(t)\\\color {blue}u_{3}(t)\\\color {blue}u_{4}(t)\\\color {blue}\varphi (t)\\\color {blue}\psi (t)\\\color {green}E(t)\\\color {red}I_{M}(t)\\\end{array}}\right)$.

We have to deal with first and second order time derivatives of coordinates from q. All mechanical coordinates (e.g. u_i, φ) are represented with their second-order time-derivatives that come from their D'Alembert's Inertia Forces. In order to transfer the equations of motion into a uniform set of differential equation of first order, we employ e.g.

$\color {lightblue}{\frac {\displaystyle du_{1}(t)}{\displaystyle dt}}\color {black}=\color {blue}v_{1}(t)\,,\,\color {lightblue}{\frac {\displaystyle d\varphi (t)}{\displaystyle dt}}\color {black}=\color {blue}\Omega (t){\text{ and }}\color {lightblue}{\frac {\displaystyle d\psi (t)}{\displaystyle dt}}\color {black}=\color {blue}\omega (t)$

so that the column matrix of state variables of the system is

${\underline {Q}}(t):=\left({\begin{array}{c}\color {orange}Q_{B}(t)\\\color {orange}U_{B2}(t)\\\color {orange}T_{B}(t)\\\color {blue}v_{1}(t)\\\color {blue}v_{2}(t)\\\color {blue}v_{3}(t)\\\color {blue}v_{4}(t)\\\color {blue}\Omega (t)\\\color {blue}\omega (t)\\\color {lightblue}u_{1}(t)\\\color {lightblue}u_{2}(t)\\\color {lightblue}u_{3}(t)\\\color {lightblue}u_{4}(t)\\\color {lightblue}\varphi (t)\\\color {blue}\psi (t)\color {black}\!\!\!\!\!\!\!\!\!\!\!--\\\color {green}E(t)\\\color {red}I_{M}(t)\\\end{array}}\right)$.

But as indicated above the rotation angle of the front wheel ψ_M never appears in our equation - we remove it from the list of state variables.

Now in matrix-notation, the resultant equations of motion are

${\underline {\underline {M}}}\cdot {\dot {\underline {Q}}}={\underline {f}}({\underline {Q}})$

with

${\underline {\underline {M}}}=\left({\begin{array}{cccccccccccccccc}\color {orange}1&&&\\&\color {orange}1&&\\&&\color {orange}1&\\&&&\color {blue}2{m_{2}}+{m_{1}}&\color {blue}0&\color {blue}0&\color {blue}0&\color {blue}2{m_{2}}\,{a_{3}}&\color {blue}0\\&&&\color {blue}0&\color {blue}2{m_{2}}+{m_{1}}&\color {blue}{m_{2}}&\color {blue}{m_{2}}&\color {blue}{a_{1}}\,{m_{2}}-{a_{2}}\,{m_{2}}&\color {blue}0\\&&&\color {blue}0&\color {blue}{m_{2}}&\color {blue}{m_{2}}&\color {blue}0&\color {blue}{a_{1}}\,{m_{2}}&\color {blue}0\\&&&\color {blue}0&\color {blue}{m_{2}}&\color {blue}0&\color {blue}{m_{2}}&\color {blue}-{a_{2}}\,{m_{2}}&\color {blue}0\\&&&\color {blue}2{m_{2}}\,{a_{3}}&\color {blue}{a_{1}}\,{m_{2}}-{a_{2}}\,{m_{2}}&\color {blue}{a_{1}}\,{m_{2}}&\color {blue}-{a_{2}}\,{m_{2}}&\color {blue}{m_{2}}\left(2{{a}_{3}^{2}}+{{a}_{2}^{2}}+{{a}_{1}^{2}}\right)+{J_{1}}\color {black}+\color {red}\left(n_{G}^{2}+2\,n_{G}+1\right)\,J_{3}&\color {blue}0\color {red}-\left(n_{G}^{2}+n_{G}\right)\,J_{3}\\&&&\color {blue}0&\color {blue}0&\color {blue}0&\color {blue}0&\color {blue}0\color {red}-\left(n_{G}^{2}+n_{G}\right)\,J_{3}&\color {blue}{J_{2}}\color {black}+\color {red}n_{G}^{2}\,J_{3}\\&&&&&&&&&\color {lightblue}1&&&&\\&&&&&&&&&&\color {lightblue}1&&&\\&&&&&&&&&&&\color {lightblue}1&&\\&&&&&&&&&&&&\color {lightblue}1&\\&&&&&&&&&&&&&\color {lightblue}1\\&&&&&&&&&&&&&&\color {green}1\\&&&&&&&&&&&&&&&\color {red}1\end{array}}\right)$

and

${\underline {f}}({\underline {Q}})=\left({\begin{array}{l}\color {orange}-\alpha (I_{B})\cdot \beta (T_{B})\cdot I_{B}\\\color {orange}{\frac {\displaystyle I_{C}}{\displaystyle C}},\\\color {orange}{\frac {\displaystyle 1}{\displaystyle m_{B}\;c_{B}}}\cdot \left({\dot {Q}}_{H}-{\dot {Q}}_{D}\right)\\\color {blue}\cos(\gamma _{A})\,F_{T}-\sin(\gamma _{2})\,F_{N2}-\sin(\gamma _{A})\,F_{N1}\\\color {blue}\left(-2\,m_{2}-m_{1}\right)g+\sin(\gamma _{A})\,F_{T}+\cos(\gamma _{2})\,F_{N2}+\cos(\gamma _{A})\,F_{N1}\\\color {blue}-F_{T}\sin \left(\varphi (t)-\gamma _{A}\right)+F_{N1}\cos \left(\varphi (t)-\gamma _{A}\right)-m_{2}g\cos \left(\varphi (t)\right)-b_{1}\,{\dot {u}}_{3}-k_{1}\,u_{3}(t)\\\color {blue}\sin(\gamma _{2})\,F_{N2}\sin \left(\varphi (t)\right)+\left(\cos(\gamma _{2})\,F_{N2}-m_{2}g\right)\cos \left(\varphi (t)\right)-b_{2}\,{\dot {u}}_{4}-k_{2}\,u_{4}(t)\\\color {blue}\left(a_{3}\,F_{N1}-a_{1}\,F_{T}\right)\sin \left(\varphi (t)-\gamma _{A}\right)+\left(a_{3}\,F_{T}+a_{1}\,F_{N1}\right)\cos \left(\varphi (t)-\gamma _{A}\right)+\sin \left(\varphi (t)\right)+\left(\sin(\gamma _{2}(t))\,F_{N2}u_{4}(t)+\left(\sin(\gamma _{1}(t))\,F_{N1}-\cos(\gamma _{2}(t))\,F_{T}\right)u_{3}(t)+\left(a_{2}-a_{1}\right)\,m_{2}g+\left(-a_{3}\sin(\gamma _{2}(t))-a_{2}\cos(\gamma _{2}(t))\right)\,F_{N2}\right)\cos(\varphi (t))+\left(\sin(\gamma _{1}(t))\,R\,F_{T}+\cos(\gamma _{2}(t))F_{N1}\,R\right)\sin(\gamma _{1}(t))+\left(\cos(\gamma _{2}(t))\,R\,F_{T}-\sin(\gamma _{1}(t))\,F_{N1}\,R\right)\cos(\gamma _{2}(t))+B\,\left(\omega (t)+\Omega (t)\right)+M_{W}\\\color {blue}\left(-\sin(\gamma _{A})\,R\,F_{T}-\cos(\gamma _{A})\,F_{N1}\,R\right)\sin(\gamma _{A}(t)+\left(\sin(\gamma _{A})\,F_{N1}\,R-\cos(\gamma _{A})\,R\,F_{T}\right)\cos {\left({{\gamma }_{A}}(t)\right)}+B\,\left(\omega (t)+\Omega (t)\right)+M_{W}\\\color {lightblue}v_{1}\\\color {lightblue}v_{2}\\\color {lightblue}v_{3}\\\color {lightblue}v_{4}\\\color {lightblue}\Omega \\\color {green}e\\\color {red}{\frac {\displaystyle 1}{\displaystyle L}}\left(U_{B}\cdot p(t)-R\,I_{M}(t)-e\right)\\\end{array}}\right)$

Simulation

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