文章目录
一、为什么使用MPC控制器?
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① MPC可以处理多输入耦合控制多输出的问题:
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② MPC可以处理对于控制量有约束的问题
比如，驾驶汽车的时候，汽车速度和转角都有上限
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③ MPC具有预测效果
即MPC考虑的不是根据当前参考值进行控制，而是根据预测时间内的参考去预测当前的控制量，比如在小车转弯时，就避免了可能在过弯处急转弯。
MPC需要的算力高一些，因为MPC控制过程中，每个时间步都要求解一个相同形式不同参数的优化问题。
二、什么是MPC控制器?
- 小车的数学模型:
| item | inputs | outputs |
|---|---|---|
| Plant Model(被控对象的数学模型) | 之后m个控制量 | 之后p个时间步的预测输出 |
- 优化器：
| item | inputs | outputs |
|---|---|---|
| 优化器 | (数学模型, 之后p个时间步的参考轨迹, 约束条件) | 之后m个控制量 |
优化器要考虑两点:
① 预测轨迹与参考轨迹偏差最小
②控制量不突变
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Cost Function \doteq J = \sum_{i=1}^{p} w_e e_{k+i}^2 + \sum_{i=0}^{p-1} w_{\Delta u} {\Delta u_{k+i}}^2
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord mathnormal" style="margin-right: 0.0715em;">C</span><span class="mord mathnormal">os</span><span class="mord mathnormal" style="margin-right: 0.1389em;">tF</span><span class="mord mathnormal">u</span><span class="mord mathnormal">n</span><span class="mord mathnormal">c</span><span class="mord mathnormal">t</span><span class="mord mathnormal">i</span><span class="mord mathnormal">o</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">≐</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord mathnormal" style="margin-right: 0.0962em;">J</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 2.9762em; vertical-align: -1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.6985em;"><span class="" style="top: -1.8723em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span class="" style="top: -3.05em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span><span class="" style="top: -4.3471em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.2777em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0269em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; 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vertical-align: -1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.8482em;"><span class="" style="top: -1.8723em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span class="" style="top: -3.05em;"><span class="pstrut" style="height: 3.05em;"></span><span class=""><span class="mop op-symbol large-op">∑</span></span></span><span class="" style="top: -4.3471em; margin-left: 0em;"><span class="pstrut" style="height: 3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">p</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 1.2777em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0269em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.0269em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Δ</span><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3361em;"><span class="" style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0315em;">k</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2083em;"><span class=""></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8873em;"><span class="" style="top: -3.1362em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></p>
尽管优化器输出的是之后
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仅将这个最有序列的第一步应用到汽车上，忽略其余部分
{\color{Red} 尽管优化器输出的是之后p个时间步的控制量，但是在当前时间步，}\\ {\color{Red} MPC仅将这个最有序列的第一步应用到汽车上，忽略其余部分}
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.8778em; vertical-align: -0.1944em;"></span><span class="mord"><span class="mord cjk_fallback" style="color: red;">尽管优化器输出的是之后</span><span class="mord mathnormal" style="color: red;">p</span><span class="mord cjk_fallback" style="color: red;">个时间步的控制量，但是在当前时间步，</span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0715em; color: red;">MPC</span><span class="mord cjk_fallback" style="color: red;">仅将这个最有序列的第一步应用到汽车上，忽略其余部分</span></span></span></span></span></span></span></p>
三、MPC控制器的参数如何设置?
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参数既会影响控制器的性能，又会影响计算复杂度。
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MPC主要的参数如下:
- ① 采样时间
- ② 预测范围、控制范围
- ③ 约束
- ④ 损失函数的权重
3.1 采样时间
采样时间过大，当干扰发生，控制器无法做出及时的反应。
采样时间过小，会导致过多的计算负载。
可以测量系统的阶跃响应，根据上升时间选择采样时间
3.2 预测范围、控制范围
我们应该选择一个涵盖系统重要动态的预测范围
预测范围过短，无法及时反映，可能会在弯道处急转弯
预测范围过长，之后的不可预测事件可能会白白浪费算力。
我们通常选择 20-30个样本覆盖系统的动态响应，如下图:
通常只有前几个时间步的控制对于预测影响较大(因为前几个要拽向ref轨迹，后面的维持稳定)， 因此，选择大的控制范围只会白白增大计算复杂度
经验法则: 控制范围 = 10%~20% × 预测范围
3.3 约束条件
可以对 控制量、控制量变化率、状态量进行约束
约束分为硬约束和软约束，无法违反的(比如最大车速)叫硬约束
软约束可以违反，但我们不希望违反，可以把它设计到损失函数之中
建议将输出约束设置为软约束，并避免对控制量及其变化率设置硬约束，否则可能会出现无解的情况
3.4 权重
损失函数由多个部分组成，需要设置权重
比如最基本的，我们既希望预测轨迹接近参考轨迹，又希望控制量是平滑的。
四、自适应增益调度和非线性MPC
- 线性系统 + 线性约束 + 时不变约束条件 => 可以使用线性时不变MPC控制器 => 凸优化问题
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但是如果系统是非线性的，就要使用自适应MPC、时间表MPC
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自适应MPC：具体思路是在非线性模型的各个工作点，建立多套线性模型
自适应MPC的线性模型在不同工作点上结构，超参数相同，只是矩阵参数不同而已
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- 时间表MPC: 对于不同工作点，需要设置不同数量超参数(控制量数目、预测步长…)的模型
如果是非线性的，那么优化问题就会变成多极值点的非凸优化问题，求解较复杂
五、Simulink仿真MPC控制器
| item | value |
|---|---|
| 输出量(状态量) | (横向位置Y、偏航角\phi) |
| 控制量 | (前轮转角delta) |
假定
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使用线性化汽车模型
假定V_x恒为15m/s , 使用线性化汽车模型\\
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord cjk_fallback">假定</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord cjk_fallback">恒为</span><span class="mord">15</span><span class="mord mathnormal">m</span><span class="mord">/</span><span class="mord mathnormal">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord cjk_fallback">使用线性化汽车模型</span></span><span class="mspace newline"></span></span></span></span></span></p>
模型下载地址: https://www.mathworks.com/matlabcentral/fileexchange/68992-designing-an-mpc-controller-with-simulink?s_eid=PSM_15028
(一定要记得路径，matlab界面的路径一定得是模型文件路径。路径不对的话，plant和reference的mask无法显示正确，会显示问号，同时直接运行会报错)
步骤: https://ww2.mathworks.cn/help/releases/R2017b/mpc/examples/autonomous-vehicle-steering-using-model-predictive-control.html
5.1 数学分析
下面依次介绍各个模块:
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① Plant: 车辆线性模型
右键Plant模块，选择Block Parameters：
| 变量 | 含义 |
|---|---|
|
小车是自行车模型
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{\color{Red} 小车是自行车模型(bicycle \quad model\quad with \quad two\quad degrees\quad of \quad freedom), 且V_x恒为15m/s:}
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord"><span class="mord cjk_fallback" style="color: red;">小车是自行车模型</span><span class="mopen" style="color: red;">(</span><span class="mord mathnormal" style="color: red;">bi</span><span class="mord mathnormal" style="color: red;">cyc</span><span class="mord mathnormal" style="margin-right: 0.0197em; color: red;">l</span><span class="mord mathnormal" style="color: red;">e</span><span class="mspace" style="color: red; margin-right: 1em;"></span><span class="mord mathnormal" style="color: red;">m</span><span class="mord mathnormal" style="color: red;">o</span><span class="mord mathnormal" style="color: red;">d</span><span class="mord mathnormal" style="color: red;">e</span><span class="mord mathnormal" style="margin-right: 0.0197em; color: red;">l</span><span class="mspace" style="color: red; margin-right: 1em;"></span><span class="mord mathnormal" style="margin-right: 0.0269em; color: red;">w</span><span class="mord mathnormal" style="color: red;">i</span><span class="mord mathnormal" style="color: red;">t</span><span class="mord mathnormal" style="color: red;">h</span><span class="mspace" style="color: red; margin-right: 1em;"></span><span class="mord mathnormal" style="margin-right: 0.0269em; color: red;">tw</span><span class="mord mathnormal" style="color: red;">o</span><span class="mspace" style="color: red; margin-right: 1em;"></span><span class="mord mathnormal" style="color: red;">d</span><span class="mord mathnormal" style="color: red;">e</span><span class="mord mathnormal" style="margin-right: 0.0359em; color: red;">g</span><span class="mord mathnormal" style="color: red;">rees</span><span class="mspace" style="color: red; margin-right: 1em;"></span><span class="mord mathnormal" style="color: red;">o</span><span class="mord mathnormal" style="margin-right: 0.1076em; color: red;">f</span><span class="mspace" style="color: red; margin-right: 1em;"></span><span class="mord mathnormal" style="margin-right: 0.1076em; color: red;">f</span><span class="mord mathnormal" style="color: red;">ree</span><span class="mord mathnormal" style="color: red;">d</span><span class="mord mathnormal" style="color: red;">o</span><span class="mord mathnormal" style="color: red;">m</span><span class="mclose" style="color: red;">)</span><span class="mpunct" style="color: red;">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord cjk_fallback" style="color: red;">且</span><span class="mord" style="color: red;"><span class="mord mathnormal" style="margin-right: 0.2222em; color: red;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight" style="color: red;"><span class="mord mathnormal mtight" style="color: red;">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord cjk_fallback" style="color: red;">恒为</span><span class="mord" style="color: red;">15</span><span class="mord mathnormal" style="color: red;">m</span><span class="mord" style="color: red;">/</span><span class="mord mathnormal" style="color: red;">s</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel" style="color: red;">:</span></span></span></span></span></span></span></p>
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原理(1): 前轮运动 = 前轮相对于车身运动 + 车身相对于质心运动(旋转) + 小车质心相对于地面运动
-
原理(2):
Δ 轮胎受到垂直于车身的力 = C r ( 或者 C l ) Δ 轮胎转角 \Delta 轮胎受到垂直于车身的力 = C_{r}(或者C_{l} ) \Delta 轮胎转角 </span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord">Δ</span><span class="mord cjk_fallback">轮胎受到垂直于车身的力</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.0715em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0278em;">r</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord cjk_fallback">或者</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3361em;"><span class="" style="top: -2.55em; margin-left: -0.0715em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0197em;">l</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord">Δ</span><span class="mord cjk_fallback">轮胎转角</span></span></span></span></span></span></strong></p> <p>(注意: 自行车模型代替小车模型时，四轮变为两轮，因此<span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml"> 自行车的 C f 应该是小车的 2 C f 自行车的C_f应该是 小车的2C_f </span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.9694em; vertical-align: -0.2861em;"></span><span class="mord cjk_fallback">自行车的</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3361em;"><span class="" style="top: -2.55em; margin-left: -0.0715em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1076em;">f</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2861em;"><span class=""></span></span></span></span></span></span><span class="mord cjk_fallback">应该是小车的</span><span class="mord">2</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3361em;"><span class="" style="top: -2.55em; margin-left: -0.0715em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1076em;">f</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2861em;"><span class=""></span></span></span></span></span></span></span></span></span></span></span>)</p> </li></ul>
进行动力学分析：
由于
ψ
较小，简化为线性模型即
:
Y
˙
=
V
x
ψ
+
V
y
[
公式一
]
由于\psi较小，简化为线性模型即: \dot{Y} = V_x \psi + V_y \quad [公式一]\\
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.8889em; vertical-align: -0.1944em;"></span><span class="mord cjk_fallback">由于</span><span class="mord mathnormal" style="margin-right: 0.0359em;">ψ</span><span class="mord cjk_fallback">较小，简化为线性模型即</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 0.9202em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.9202em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord mathnormal" style="margin-right: 0.2222em;">Y</span></span><span class="" style="top: -3.2523em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.1389em;"><span class="mord">˙</span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 0.8889em; vertical-align: -0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right: 0.0359em;">ψ</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 1.0361em; vertical-align: -0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0359em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2861em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 1em;"></span><span class="mopen">[</span><span class="mord cjk_fallback">公式一</span><span class="mclose">]</span></span><span class="mspace newline"></span></span></span></span></span><br> <span class="katex--display"><span class="katex-display"><span class="katex"><span class="katex-mathml">
Δ
前轮相对于车身转角
=
(
V
y
+
ψ
˙
l
f
V
x
)
Δ
后轮相对于车身转角
=
(
−
V
y
−
ψ
˙
l
r
V
x
)
⇒
前轮受到垂直于车身的力
=
2
C
f
(
−
ψ
+
δ
+
Δ
前轮相对于车身转角
)
后轮受到垂直于车身的力
=
2
C
r
(
−
ψ
+
Δ
后轮相对于车身转角
)
⇒
根据
[
公式一
]
:
m
y
¨
=
−
m
V
x
ψ
˙
−
m
V
y
˙
m
V
y
˙
=
前轮受到垂直于车身的力
+
后轮受到垂直于车身的力
⇒
m
y
¨
=
−
m
V
x
ψ
˙
+
2
C
f
[
−
ψ
+
δ
−
V
y
+
ψ
˙
l
f
V
x
]
+
2
C
r
(
−
ψ
−
V
y
−
ψ
˙
l
r
V
x
)
将
[
公式一
]
代入到上式
:
y
¨
=
−
2
C
f
+
2
C
r
m
V
x
y
˙
−
(
V
x
+
2
C
f
l
f
−
2
C
r
l
r
m
V
x
)
ψ
˙
+
2
C
α
δ
m
[
公式二
]
\Delta 前轮相对于车身转角 = (\frac{V_y + \dot{\psi}l_f}{V_x}) \\ \Delta 后轮相对于车身转角 = (- \frac{V_y - \dot{\psi}l_r}{V_x}) \\ \Rightarrow \\ 前轮受到垂直于车身的力 = 2C_{f} (-\psi + \delta + \Delta 前轮相对于车身转角) \\ 后轮受到垂直于车身的力 = 2C_{r} (-\psi + \Delta 后轮相对于车身转角)\\ \Rightarrow \\ 根据[公式一]: \\ m \ddot{y} = -mV_x \dot{\psi} - m\dot{V_y}\\ m\dot{V_y}=前轮受到垂直于车身的力 + 后轮受到垂直于车身的力 \\ \Rightarrow m\ddot{y} = - m V_x \dot{\psi} + 2C_f[-\psi +\delta - \frac{V_y + \dot{\psi}l_f}{V_x}] +2 C_r (-\psi - \frac{V_y - \dot{\psi}l_r}{V_x}) \\ 将 [公式一]代入到上式: \\ {\color{Red}\ddot{y} = -\frac{2C_f+2C_r}{mV_x}\dot{y} -(V_x+\frac{2C_fl_f-2C_rl_r}{mV_x}) \dot{\psi} + \frac{2C_{\alpha} \delta}{m} \quad [公式二] }
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord">Δ</span><span class="mord cjk_fallback">前轮相对于车身转角</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 2.4443em; vertical-align: -0.836em;"></span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.6083em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0359em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2861em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.9313em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">ψ</span></span><span class="" style="top: -3.2634em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.0278em;"><span class="mord">˙</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.1944em;"><span class=""></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0197em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3361em;"><span class="" style="top: -2.55em; margin-left: -0.0197em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1076em;">f</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2861em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.836em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord">Δ</span><span class="mord cjk_fallback">后轮相对于车身转角</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 2.4443em; vertical-align: -0.836em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.6083em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0359em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2861em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.9313em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">ψ</span></span><span class="" style="top: -3.2634em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.0278em;"><span class="mord">˙</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.1944em;"><span class=""></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0197em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.0197em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0278em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.836em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height: 0.3669em;"></span><span class="mrel">⇒</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord cjk_fallback">前轮受到垂直于车身的力</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 1.0361em; vertical-align: -0.2861em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3361em;"><span class="" style="top: -2.55em; margin-left: -0.0715em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1076em;">f</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2861em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">−</span><span class="mord mathnormal" style="margin-right: 0.0359em;">ψ</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 0.7778em; vertical-align: -0.0833em;"></span><span class="mord mathnormal" style="margin-right: 0.0379em;">δ</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord">Δ</span><span class="mord cjk_fallback">前轮相对于车身转角</span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord cjk_fallback">后轮受到垂直于车身的力</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.0715em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0278em;">r</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">−</span><span class="mord mathnormal" style="margin-right: 0.0359em;">ψ</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord">Δ</span><span class="mord cjk_fallback">后轮相对于车身转角</span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height: 0.3669em;"></span><span class="mrel">⇒</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord cjk_fallback">根据</span><span class="mopen">[</span><span class="mord cjk_fallback">公式一</span><span class="mclose">]</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">:</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height: 0.8623em; vertical-align: -0.1944em;"></span><span class="mord mathnormal">m</span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.6679em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">y</span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.1944em;"><span class="mord">¨</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.1944em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 1.1257em; vertical-align: -0.1944em;"></span><span class="mord">−</span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.9313em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">ψ</span></span><span class="" style="top: -3.2634em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.0278em;"><span class="mord">˙</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.1944em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 1.2063em; vertical-align: -0.2861em;"></span><span class="mord mathnormal">m</span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.9202em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0359em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2861em;"><span class=""></span></span></span></span></span></span></span><span class="" style="top: -3.2523em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.1389em;"><span class="mord">˙</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2861em;"><span class=""></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height: 1.2063em; vertical-align: -0.2861em;"></span><span class="mord mathnormal">m</span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.9202em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0359em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2861em;"><span class=""></span></span></span></span></span></span></span><span class="" style="top: -3.2523em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.1389em;"><span class="mord">˙</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2861em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 0.7667em; vertical-align: -0.0833em;"></span><span class="mord cjk_fallback">前轮受到垂直于车身的力</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord cjk_fallback">后轮受到垂直于车身的力</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height: 0.3669em;"></span><span class="mrel">⇒</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 0.8623em; vertical-align: -0.1944em;"></span><span class="mord mathnormal">m</span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.6679em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">y</span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.1944em;"><span class="mord">¨</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.1944em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 1.1257em; vertical-align: -0.1944em;"></span><span class="mord">−</span><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.9313em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">ψ</span></span><span class="" style="top: -3.2634em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.0278em;"><span class="mord">˙</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.1944em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 1.0361em; vertical-align: -0.2861em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3361em;"><span class="" style="top: -2.55em; margin-left: -0.0715em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1076em;">f</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2861em;"><span class=""></span></span></span></span></span></span><span class="mopen">[</span><span class="mord">−</span><span class="mord mathnormal" style="margin-right: 0.0359em;">ψ</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 0.7778em; vertical-align: -0.0833em;"></span><span class="mord mathnormal" style="margin-right: 0.0379em;">δ</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 2.4443em; vertical-align: -0.836em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.6083em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0359em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2861em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.9313em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">ψ</span></span><span class="" style="top: -3.2634em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.0278em;"><span class="mord">˙</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.1944em;"><span class=""></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0197em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3361em;"><span class="" style="top: -2.55em; margin-left: -0.0197em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1076em;">f</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2861em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.836em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">]</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.0715em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0278em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">−</span><span class="mord mathnormal" style="margin-right: 0.0359em;">ψ</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 2.4443em; vertical-align: -0.836em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.6083em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0359em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2861em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.9313em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">ψ</span></span><span class="" style="top: -3.2634em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.0278em;"><span class="mord">˙</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.1944em;"><span class=""></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0197em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.0197em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0278em;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.836em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord cjk_fallback">将</span><span class="mopen">[</span><span class="mord cjk_fallback">公式一</span><span class="mclose">]</span><span class="mord cjk_fallback">代入到上式</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">:</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height: 2.2074em; vertical-align: -0.836em;"></span><span class="mord"><span class="mord accent" style="color: red;"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.6679em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em; color: red;">y</span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.1944em;"><span class="mord" style="color: red;">¨</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.1944em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel" style="color: red;">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord" style="color: red;">−</span><span class="mord" style="color: red;"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.3603em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="color: red;"><span class="mord mathnormal" style="color: red;">m</span><span class="mord" style="color: red;"><span class="mord mathnormal" style="margin-right: 0.2222em; color: red;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight" style="color: red;"><span class="mord mathnormal mtight" style="color: red;">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="color: red; border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="color: red;"><span class="mord" style="color: red;">2</span><span class="mord" style="color: red;"><span class="mord mathnormal" style="margin-right: 0.0715em; color: red;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3361em;"><span class="" style="top: -2.55em; margin-left: -0.0715em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight" style="color: red;"><span class="mord mathnormal mtight" style="margin-right: 0.1076em; color: red;">f</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2861em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="color: red; margin-right: 0.2222em;"></span><span class="mbin" style="color: red;">+</span><span class="mspace" style="color: red; margin-right: 0.2222em;"></span><span class="mord" style="color: red;">2</span><span class="mord" style="color: red;"><span class="mord mathnormal" style="margin-right: 0.0715em; color: red;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.0715em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight" style="color: red;"><span class="mord mathnormal mtight" style="margin-right: 0.0278em; color: red;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.836em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord accent" style="color: red;"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.6679em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em; color: red;">y</span></span><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.0833em;"><span class="mord" style="color: red;">˙</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.1944em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin" style="color: red;">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mopen" style="color: red;">(</span><span class="mord" style="color: red;"><span class="mord mathnormal" style="margin-right: 0.2222em; color: red;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight" style="color: red;"><span class="mord mathnormal mtight" style="color: red;">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin" style="color: red;">+</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord" style="color: red;"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.3714em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="color: red;"><span class="mord mathnormal" style="color: red;">m</span><span class="mord" style="color: red;"><span class="mord mathnormal" style="margin-right: 0.2222em; color: red;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight" style="color: red;"><span class="mord mathnormal mtight" style="color: red;">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="color: red; border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="color: red;"><span class="mord" style="color: red;">2</span><span class="mord" style="color: red;"><span class="mord mathnormal" style="margin-right: 0.0715em; color: red;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3361em;"><span class="" style="top: -2.55em; margin-left: -0.0715em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight" style="color: red;"><span class="mord mathnormal mtight" style="margin-right: 0.1076em; color: red;">f</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2861em;"><span class=""></span></span></span></span></span></span><span class="mord" style="color: red;"><span class="mord mathnormal" style="margin-right: 0.0197em; color: red;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3361em;"><span class="" style="top: -2.55em; margin-left: -0.0197em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight" style="color: red;"><span class="mord mathnormal mtight" style="margin-right: 0.1076em; color: red;">f</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.2861em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="color: red; margin-right: 0.2222em;"></span><span class="mbin" style="color: red;">−</span><span class="mspace" style="color: red; margin-right: 0.2222em;"></span><span class="mord" style="color: red;">2</span><span class="mord" style="color: red;"><span class="mord mathnormal" style="margin-right: 0.0715em; color: red;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.0715em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight" style="color: red;"><span class="mord mathnormal mtight" style="margin-right: 0.0278em; color: red;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord" style="color: red;"><span class="mord mathnormal" style="margin-right: 0.0197em; color: red;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.0197em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight" style="color: red;"><span class="mord mathnormal mtight" style="margin-right: 0.0278em; color: red;">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.836em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose" style="color: red;">)</span><span class="mord accent" style="color: red;"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.9313em;"><span class="" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em; color: red;">ψ</span></span><span class="" style="top: -3.2634em;"><span class="pstrut" style="height: 3em;"></span><span class="accent-body" style="left: -0.0278em;"><span class="mord" style="color: red;">˙</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.1944em;"><span class=""></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin" style="color: red;">+</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord" style="color: red;"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.3714em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="color: red;"><span class="mord mathnormal" style="color: red;">m</span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="color: red; border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="color: red;"><span class="mord" style="color: red;">2</span><span class="mord" style="color: red;"><span class="mord mathnormal" style="margin-right: 0.0715em; color: red;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.1514em;"><span class="" style="top: -2.55em; margin-left: -0.0715em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight" style="color: red;"><span class="mord mtight" style="color: red;"><span class="mord mathnormal mtight" style="margin-right: 0.0037em; color: red;">α</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right: 0.0379em; color: red;">δ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.686em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="color: red; margin-right: 1em;"></span><span class="mopen" style="color: red;">[</span><span class="mord cjk_fallback" style="color: red;">公式二</span><span class="mclose" style="color: red;">]</span></span></span></span></span></span></span></p>
求得小车模型的状态空间表达式为
:
d
d
t
[
y
˙
ψ
ψ
˙
]
=
[
−
2
C
f
+
2
C
r
m
V
x
0
−
(
V
x
+
2
C
f
l
f
−
2
C
r
l
r
m
V
x
)
0
0
1
−
2
l
f
C
f
−
2
l
r
C
r
I
z
V
x
0
−
2
l
f
2
C
f
+
2
l
r
2
C
r
I
z
V
x
]
[
y
˙
ψ
ψ
˙
]
+
[
2
C
f
m
0
2
l
f
C
f
I
z
]
δ
求得小车模型的状态空间表达式为: \qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \\ \frac{d}{dt} <span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display"><span class="MathJax MathJax_FullWidth" id="MathJax-Element-1-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mrow class="MJX-TeXAtom-ORD"><mover><mi>y</mi><mo>&#x02D9;</mo></mover></mrow></mtd></mtr><mtr><mtd><mi>&#x03C8;</mi></mtd></mtr><mtr><mtd><mrow class="MJX-TeXAtom-ORD"><mover><mi>&#x03C8;</mi><mo>&#x02D9;</mo></mover></mrow></mtd></mtr></mtable><mo>]</mo></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1" style="width: 100%; display: inline-block; min-width: 2.382em;"><span style="display: inline-block; position: relative; width: 100%; height: 0px; font-size: 102%;"><span style="position: absolute; clip: rect(2.584em, 1002.03em, 6.837em, -999.997em); top: -4.958em; left: 0em; width: 100%;"><span class="mrow" id="MathJax-Span-2"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.584em, 1002.03em, 6.837em, -999.997em); top: -4.958em; left: 50%; margin-left: -1.162em;"><span class="mrow" id="MathJax-Span-3"><span class="mo" id="MathJax-Span-4" style="vertical-align: 2.23em;"><span style="display: inline-block; position: relative; width: 0.661em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.832em; left: 0em;">⎡<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: -0.706em; left: 0em;">⎣<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.719em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mtable" id="MathJax-Span-5" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 0.711em; height: 0px;"><span style="position: absolute; clip: rect(2.584em, 1000.66em, 6.735em, -999.997em); top: -4.857em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.711em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.141em, 1000.51em, 4.356em, -999.997em); top: -5.414em; left: 0em;"><span class="mtd" id="MathJax-Span-6"><span class="mrow" id="MathJax-Span-7"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.141em, 1000.51em, 4.356em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.301em;"><span class="texatom" id="MathJax-Span-8"><span class="mrow" id="MathJax-Span-9"><span style="display: inline-block; position: relative; width: 0.61em; height: 0px;"><span style="position: absolute; clip: rect(3.141em, 1000.51em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="munderover" id="MathJax-Span-10"><span style="display: inline-block; position: relative; width: 0.61em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.356em, -999.997em); top: -3.997em; left: 0.003em;"><span class="mi" id="MathJax-Span-11" style="font-family: MathJax_Math-italic;">y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.192em, 1000.31em, 3.597em, -999.997em); top: -3.997em; left: 0.104em;"><span class="mo" id="MathJax-Span-12" style="font-family: MathJax_Main;">˙</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.141em, 1000.66em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="mtd" id="MathJax-Span-13"><span class="mrow" id="MathJax-Span-14"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.141em, 1000.66em, 4.356em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.301em;"><span class="mi" id="MathJax-Span-15" style="font-family: MathJax_Math-italic;">ψ</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(2.888em, 1000.66em, 4.356em, -999.997em); top: -2.478em; left: 0em;"><span class="mtd" id="MathJax-Span-16"><span class="mrow" id="MathJax-Span-17"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.888em, 1000.66em, 4.356em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.352em;"><span class="texatom" id="MathJax-Span-18"><span class="mrow" id="MathJax-Span-19"><span style="display: inline-block; position: relative; width: 0.711em; height: 0px;"><span style="position: absolute; clip: rect(2.888em, 1000.66em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="munderover" id="MathJax-Span-20"><span style="display: inline-block; position: relative; width: 0.711em; height: 0px;"><span style="position: absolute; clip: rect(3.141em, 1000.66em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-21" style="font-family: MathJax_Math-italic;">ψ</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.192em, 1000.31em, 3.597em, -999.997em); top: -4.25em; left: 0.205em;"><span class="mo" id="MathJax-Span-22" style="font-family: MathJax_Main;">˙</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.862em;"></span></span></span></span><span class="mo" id="MathJax-Span-23" style="vertical-align: 2.23em;"><span style="display: inline-block; position: relative; width: 0.661em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.832em; left: 0em;">⎤<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: -0.706em; left: 0em;">⎦<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.719em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.964em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.964em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.805em; border-left: 0px solid; width: 0px; height: 4.133em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mrow class="MJX-TeXAtom-ORD"><mover><mi>y</mi><mo>˙</mo></mover></mrow></mtd></mtr><mtr><mtd><mi>ψ</mi></mtd></mtr><mtr><mtd><mrow class="MJX-TeXAtom-ORD"><mover><mi>ψ</mi><mo>˙</mo></mover></mrow></mtd></mtr></mtable><mo>]</mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-1">\begin{bmatrix} \dot{y} \\ \psi \\ \dot{\psi} \end{bmatrix}</script> =<span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display"><span class="MathJax MathJax_FullWidth" id="MathJax-Element-2-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mo>&#x2212;</mo><mfrac><mrow><mn>2</mn><msub><mi>C</mi><mi>f</mi></msub><mo>+</mo><mn>2</mn><msub><mi>C</mi><mi>r</mi></msub></mrow><mrow><mi>m</mi><msub><mi>V</mi><mi>x</mi></msub></mrow></mfrac></mtd><mtd><mn>0</mn></mtd><mtd><mo>&#x2212;</mo><mo stretchy="false">(</mo><msub><mi>V</mi><mi>x</mi></msub><mo>+</mo><mfrac><mrow><mn>2</mn><msub><mi>C</mi><mi>f</mi></msub><msub><mi>l</mi><mi>f</mi></msub><mo>&#x2212;</mo><mn>2</mn><msub><mi>C</mi><mi>r</mi></msub><msub><mi>l</mi><mi>r</mi></msub></mrow><mrow><mi>m</mi><msub><mi>V</mi><mi>x</mi></msub></mrow></mfrac><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>&#x2212;</mo><mfrac><mrow><mn>2</mn><msub><mi>l</mi><mi>f</mi></msub><msub><mi>C</mi><mi>f</mi></msub><mo>&#x2212;</mo><mn>2</mn><msub><mi>l</mi><mi>r</mi></msub><msub><mi>C</mi><mi>r</mi></msub></mrow><mrow><msub><mi>I</mi><mi>z</mi></msub><msub><mi>V</mi><mi>x</mi></msub></mrow></mfrac></mtd><mtd><mn>0</mn></mtd><mtd><mo>&#x2212;</mo><mfrac><mrow><mn>2</mn><msubsup><mi>l</mi><mi>f</mi><mn>2</mn></msubsup><msub><mi>C</mi><mi>f</mi></msub><mo>+</mo><mn>2</mn><msubsup><mi>l</mi><mi>r</mi><mn>2</mn></msubsup><msub><mi>C</mi><mi>r</mi></msub></mrow><mrow><msub><mi>I</mi><mi>z</mi></msub><msub><mi>V</mi><mi>x</mi></msub></mrow></mfrac></mtd></mtr></mtable><mo>]</mo></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-24" style="width: 100%; display: inline-block; min-width: 17.619em;"><span style="display: inline-block; position: relative; width: 100%; height: 0px; font-size: 102%;"><span style="position: absolute; clip: rect(3.192em, 1016.91em, 8.76em, -999.997em); top: -6.224em; left: 0em; width: 100%;"><span class="mrow" id="MathJax-Span-25"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1016.91em, 8.76em, -999.997em); top: -6.224em; left: 50%; margin-left: -8.603em;"><span class="mrow" id="MathJax-Span-26"><span class="mo" id="MathJax-Span-27" style="vertical-align: 2.888em;"><span style="display: inline-block; position: relative; width: 0.661em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.832em; left: 0em;">⎡<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.61em; left: 0em;">⎣<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.617em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.061em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.504em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mtable" id="MathJax-Span-28" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 15.594em; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1005.07em, 8.76em, -999.997em); top: -6.224em; left: 0em;"><span style="display: inline-block; position: relative; width: 5.065em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(2.736em, 1004em, 4.66em, -999.997em); top: -5.768em; left: 0em;"><span class="mtd" id="MathJax-Span-29"><span class="mrow" id="MathJax-Span-30"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.736em, 1004em, 4.66em, -999.997em); top: -3.997em; left: 50%; margin-left: -2.022em;"><span class="mo" id="MathJax-Span-31" style="font-family: MathJax_Main;">−</span><span class="mfrac" id="MathJax-Span-32"><span style="display: inline-block; position: relative; width: 2.989em; height: 0px; margin-right: 0.104em; margin-left: 0.104em;"><span style="position: absolute; clip: rect(3.344em, 1002.89em, 4.356em, -999.997em); top: -4.604em; left: 50%; margin-left: -1.415em;"><span class="mrow" id="MathJax-Span-33"><span style="display: inline-block; position: relative; width: 2.888em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1002.89em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="mn" id="MathJax-Span-34" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span class="msubsup" id="MathJax-Span-35"><span style="display: inline-block; position: relative; width: 0.812em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-36" style="font-size: 70.7%; font-family: MathJax_Math-italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.509em;"><span class="mi" id="MathJax-Span-37" style="font-size: 50%; font-family: MathJax_Math-italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-38" style="font-size: 70.7%; font-family: MathJax_Main;">+</span><span class="mn" id="MathJax-Span-39" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span class="msubsup" id="MathJax-Span-40"><span style="display: inline-block; position: relative; width: 0.762em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-41" style="font-size: 70.7%; font-family: MathJax_Math-italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.509em;"><span class="mi" id="MathJax-Span-42" style="font-size: 50%; font-family: MathJax_Math-italic;">r</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.344em, 1001.37em, 4.255em, -999.997em); top: -3.592em; left: 50%; margin-left: -0.706em;"><span class="mrow" id="MathJax-Span-43"><span style="display: inline-block; position: relative; width: 1.369em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1001.37em, 4.255em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-44" style="font-size: 70.7%; font-family: MathJax_Math-italic;">m</span><span class="msubsup" id="MathJax-Span-45"><span style="display: inline-block; position: relative; width: 0.762em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-46" style="font-size: 70.7%; font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.154em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.408em;"><span class="mi" id="MathJax-Span-47" style="font-size: 50%; font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(0.863em, 1002.99em, 1.217em, -999.997em); top: -1.263em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.989em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.066em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -4.047em; left: 0em;"><span class="mtd" id="MathJax-Span-82"><span class="mrow" id="MathJax-Span-83"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.251em;"><span class="mn" id="MathJax-Span-84" style="font-family: MathJax_Main;">0</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(2.736em, 1005.07em, 4.66em, -999.997em); top: -2.124em; left: 0em;"><span class="mtd" id="MathJax-Span-91"><span class="mrow" id="MathJax-Span-92"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.736em, 1005.07em, 4.66em, -999.997em); top: -3.997em; left: 50%; margin-left: -2.529em;"><span class="mo" id="MathJax-Span-93" style="font-family: MathJax_Main;">−</span><span class="mfrac" id="MathJax-Span-94"><span style="display: inline-block; position: relative; width: 4.002em; height: 0px; margin-right: 0.104em; margin-left: 0.104em;"><span style="position: absolute; clip: rect(3.344em, 1003.9em, 4.356em, -999.997em); top: -4.604em; left: 50%; margin-left: -1.972em;"><span class="mrow" id="MathJax-Span-95"><span style="display: inline-block; position: relative; width: 3.9em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1003.9em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="mn" id="MathJax-Span-96" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span class="msubsup" id="MathJax-Span-97"><span style="display: inline-block; position: relative; width: 0.559em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.21em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-98" style="font-size: 70.7%; font-family: MathJax_Math-italic;">l</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.205em;"><span class="mi" id="MathJax-Span-99" style="font-size: 50%; font-family: MathJax_Math-italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-100"><span style="display: inline-block; position: relative; width: 0.812em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-101" style="font-size: 70.7%; font-family: MathJax_Math-italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.509em;"><span class="mi" id="MathJax-Span-102" style="font-size: 50%; font-family: MathJax_Math-italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-103" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-104" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span class="msubsup" id="MathJax-Span-105"><span style="display: inline-block; position: relative; width: 0.509em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.21em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-106" style="font-size: 70.7%; font-family: MathJax_Math-italic;">l</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.205em;"><span class="mi" id="MathJax-Span-107" style="font-size: 50%; font-family: MathJax_Math-italic;">r</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-108"><span style="display: inline-block; position: relative; width: 0.762em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-109" style="font-size: 70.7%; font-family: MathJax_Math-italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.509em;"><span class="mi" id="MathJax-Span-110" style="font-size: 50%; font-family: MathJax_Math-italic;">r</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.344em, 1001.37em, 4.255em, -999.997em); top: -3.592em; left: 50%; margin-left: -0.656em;"><span class="mrow" id="MathJax-Span-111"><span style="display: inline-block; position: relative; width: 1.369em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1001.37em, 4.255em, -999.997em); top: -3.997em; left: 0em;"><span class="msubsup" id="MathJax-Span-112"><span style="display: inline-block; position: relative; width: 0.61em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.36em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-113" style="font-size: 70.7%; font-family: MathJax_Math-italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.306em;"><span class="mi" id="MathJax-Span-114" style="font-size: 50%; font-family: MathJax_Math-italic;">z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-115"><span style="display: inline-block; position: relative; width: 0.762em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-116" style="font-size: 70.7%; font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.154em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.408em;"><span class="mi" id="MathJax-Span-117" style="font-size: 50%; font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(0.863em, 1004em, 1.217em, -999.997em); top: -1.263em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.002em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.066em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 6.229em;"></span></span><span style="position: absolute; clip: rect(2.736em, 1000.46em, 7.343em, -999.997em); top: -5.313em; left: 6.027em;"><span style="display: inline-block; position: relative; width: 0.509em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -5.768em; left: 0em;"><span class="mtd" id="MathJax-Span-48"><span class="mrow" id="MathJax-Span-49"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.251em;"><span class="mn" id="MathJax-Span-50" style="font-family: MathJax_Main;">0</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -4.047em; left: 0em;"><span class="mtd" id="MathJax-Span-85"><span class="mrow" id="MathJax-Span-86"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.251em;"><span class="mn" id="MathJax-Span-87" style="font-family: MathJax_Main;">0</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -2.124em; left: 0em;"><span class="mtd" id="MathJax-Span-118"><span class="mrow" id="MathJax-Span-119"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.251em;"><span class="mn" id="MathJax-Span-120" style="font-family: MathJax_Main;">0</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.318em;"></span></span><span style="position: absolute; clip: rect(3.192em, 1008em, 8.76em, -999.997em); top: -6.224em; left: 7.545em;"><span style="display: inline-block; position: relative; width: 8.052em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(2.736em, 1008em, 4.66em, -999.997em); top: -5.768em; left: 0em;"><span class="mtd" id="MathJax-Span-51"><span class="mrow" id="MathJax-Span-52"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.736em, 1008em, 4.66em, -999.997em); top: -3.997em; left: 50%; margin-left: -4.047em;"><span class="mo" id="MathJax-Span-53" style="font-family: MathJax_Main;">−</span><span class="mo" id="MathJax-Span-54" style="font-family: MathJax_Main;">(</span><span class="msubsup" id="MathJax-Span-55"><span style="display: inline-block; position: relative; width: 1.066em; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.76em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-56" style="font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.205em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.845em; left: 0.61em;"><span class="mi" id="MathJax-Span-57" style="font-size: 70.7%; font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-58" style="font-family: MathJax_Main; padding-left: 0.205em;">+</span><span class="mfrac" id="MathJax-Span-59" style="padding-left: 0.205em;"><span style="display: inline-block; position: relative; width: 4.002em; height: 0px; margin-right: 0.104em; margin-left: 0.104em;"><span style="position: absolute; clip: rect(3.344em, 1003.9em, 4.356em, -999.997em); top: -4.604em; left: 50%; margin-left: -1.972em;"><span class="mrow" id="MathJax-Span-60"><span style="display: inline-block; position: relative; width: 3.9em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1003.9em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="mn" id="MathJax-Span-61" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span class="msubsup" id="MathJax-Span-62"><span style="display: inline-block; position: relative; width: 0.812em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-63" style="font-size: 70.7%; font-family: MathJax_Math-italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.509em;"><span class="mi" id="MathJax-Span-64" style="font-size: 50%; font-family: MathJax_Math-italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-65"><span style="display: inline-block; position: relative; width: 0.559em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.21em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-66" style="font-size: 70.7%; font-family: MathJax_Math-italic;">l</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.205em;"><span class="mi" id="MathJax-Span-67" style="font-size: 50%; font-family: MathJax_Math-italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-68" style="font-size: 70.7%; font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-69" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span class="msubsup" id="MathJax-Span-70"><span style="display: inline-block; position: relative; width: 0.762em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-71" style="font-size: 70.7%; font-family: MathJax_Math-italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.509em;"><span class="mi" id="MathJax-Span-72" style="font-size: 50%; font-family: MathJax_Math-italic;">r</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-73"><span style="display: inline-block; position: relative; width: 0.509em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.21em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-74" style="font-size: 70.7%; font-family: MathJax_Math-italic;">l</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.205em;"><span class="mi" id="MathJax-Span-75" style="font-size: 50%; font-family: MathJax_Math-italic;">r</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.344em, 1001.37em, 4.255em, -999.997em); top: -3.592em; left: 50%; margin-left: -0.706em;"><span class="mrow" id="MathJax-Span-76"><span style="display: inline-block; position: relative; width: 1.369em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1001.37em, 4.255em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-77" style="font-size: 70.7%; font-family: MathJax_Math-italic;">m</span><span class="msubsup" id="MathJax-Span-78"><span style="display: inline-block; position: relative; width: 0.762em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-79" style="font-size: 70.7%; font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.154em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.408em;"><span class="mi" id="MathJax-Span-80" style="font-size: 50%; font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(0.863em, 1004em, 1.217em, -999.997em); top: -1.263em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.002em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.066em;"></span></span></span></span><span class="mo" id="MathJax-Span-81" style="font-family: MathJax_Main;">)</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.192em, 1000.41em, 4.154em, -999.997em); top: -4.047em; left: 0em;"><span class="mtd" id="MathJax-Span-88"><span class="mrow" id="MathJax-Span-89"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.41em, 4.154em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.251em;"><span class="mn" id="MathJax-Span-90" style="font-family: MathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(2.534em, 1005.07em, 4.66em, -999.997em); top: -2.124em; left: 0em;"><span class="mtd" id="MathJax-Span-121"><span class="mrow" id="MathJax-Span-122"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.534em, 1005.07em, 4.66em, -999.997em); top: -3.997em; left: 50%; margin-left: -2.529em;"><span class="mo" id="MathJax-Span-123" style="font-family: MathJax_Main;">−</span><span class="mfrac" id="MathJax-Span-124"><span style="display: inline-block; position: relative; width: 4.052em; height: 0px; margin-right: 0.104em; margin-left: 0.104em;"><span style="position: absolute; clip: rect(3.242em, 1003.95em, 4.457em, -999.997em); top: -4.705em; left: 50%; margin-left: -1.972em;"><span class="mrow" id="MathJax-Span-125"><span style="display: inline-block; position: relative; width: 3.951em; height: 0px;"><span style="position: absolute; clip: rect(3.242em, 1003.95em, 4.457em, -999.997em); top: -3.997em; left: 0em;"><span class="mn" id="MathJax-Span-126" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span class="msubsup" id="MathJax-Span-127"><span style="display: inline-block; position: relative; width: 0.559em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.21em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-128" style="font-size: 70.7%; font-family: MathJax_Math-italic;">l</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.495em, 1000.31em, 4.154em, -999.997em); top: -4.3em; left: 0.205em;"><span class="mn" id="MathJax-Span-129" style="font-size: 50%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.495em, 1000.31em, 4.255em, -999.997em); top: -3.794em; left: 0.205em;"><span class="mi" id="MathJax-Span-130" style="font-size: 50%; font-family: MathJax_Math-italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-131"><span style="display: inline-block; position: relative; width: 0.812em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-132" style="font-size: 70.7%; font-family: MathJax_Math-italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.509em;"><span class="mi" id="MathJax-Span-133" style="font-size: 50%; font-family: MathJax_Math-italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mo" id="MathJax-Span-134" style="font-size: 70.7%; font-family: MathJax_Main;">+</span><span class="mn" id="MathJax-Span-135" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span class="msubsup" id="MathJax-Span-136"><span style="display: inline-block; position: relative; width: 0.509em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.21em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-137" style="font-size: 70.7%; font-family: MathJax_Math-italic;">l</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.495em, 1000.31em, 4.154em, -999.997em); top: -4.3em; left: 0.205em;"><span class="mn" id="MathJax-Span-138" style="font-size: 50%; font-family: MathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.647em, 1000.31em, 4.154em, -999.997em); top: -3.845em; left: 0.205em;"><span class="mi" id="MathJax-Span-139" style="font-size: 50%; font-family: MathJax_Math-italic;">r</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-140"><span style="display: inline-block; position: relative; width: 0.762em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-141" style="font-size: 70.7%; font-family: MathJax_Math-italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.509em;"><span class="mi" id="MathJax-Span-142" style="font-size: 50%; font-family: MathJax_Math-italic;">r</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.344em, 1001.37em, 4.255em, -999.997em); top: -3.592em; left: 50%; margin-left: -0.656em;"><span class="mrow" id="MathJax-Span-143"><span style="display: inline-block; position: relative; width: 1.369em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1001.37em, 4.255em, -999.997em); top: -3.997em; left: 0em;"><span class="msubsup" id="MathJax-Span-144"><span style="display: inline-block; position: relative; width: 0.61em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.36em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-145" style="font-size: 70.7%; font-family: MathJax_Math-italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.306em;"><span class="mi" id="MathJax-Span-146" style="font-size: 50%; font-family: MathJax_Math-italic;">z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-147"><span style="display: inline-block; position: relative; width: 0.762em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-148" style="font-size: 70.7%; font-family: MathJax_Math-italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.154em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.408em;"><span class="mi" id="MathJax-Span-149" style="font-size: 50%; font-family: MathJax_Math-italic;">x</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(0.863em, 1004.05em, 1.217em, -999.997em); top: -1.263em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.052em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.066em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 6.229em;"></span></span></span></span><span class="mo" id="MathJax-Span-150" style="vertical-align: 2.888em;"><span style="display: inline-block; position: relative; width: 0.661em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.832em; left: 0em;">⎤<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.61em; left: 0em;">⎦<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.617em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.061em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.504em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 6.229em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 6.229em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.476em; border-left: 0px solid; width: 0px; height: 5.424em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mo>−</mo><mfrac><mrow><mn>2</mn><msub><mi>C</mi><mi>f</mi></msub><mo>+</mo><mn>2</mn><msub><mi>C</mi><mi>r</mi></msub></mrow><mrow><mi>m</mi><msub><mi>V</mi><mi>x</mi></msub></mrow></mfrac></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mo stretchy="false">(</mo><msub><mi>V</mi><mi>x</mi></msub><mo>+</mo><mfrac><mrow><mn>2</mn><msub><mi>C</mi><mi>f</mi></msub><msub><mi>l</mi><mi>f</mi></msub><mo>−</mo><mn>2</mn><msub><mi>C</mi><mi>r</mi></msub><msub><mi>l</mi><mi>r</mi></msub></mrow><mrow><mi>m</mi><msub><mi>V</mi><mi>x</mi></msub></mrow></mfrac><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>−</mo><mfrac><mrow><mn>2</mn><msub><mi>l</mi><mi>f</mi></msub><msub><mi>C</mi><mi>f</mi></msub><mo>−</mo><mn>2</mn><msub><mi>l</mi><mi>r</mi></msub><msub><mi>C</mi><mi>r</mi></msub></mrow><mrow><msub><mi>I</mi><mi>z</mi></msub><msub><mi>V</mi><mi>x</mi></msub></mrow></mfrac></mtd><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mfrac><mrow><mn>2</mn><msubsup><mi>l</mi><mi>f</mi><mn>2</mn></msubsup><msub><mi>C</mi><mi>f</mi></msub><mo>+</mo><mn>2</mn><msubsup><mi>l</mi><mi>r</mi><mn>2</mn></msubsup><msub><mi>C</mi><mi>r</mi></msub></mrow><mrow><msub><mi>I</mi><mi>z</mi></msub><msub><mi>V</mi><mi>x</mi></msub></mrow></mfrac></mtd></mtr></mtable><mo>]</mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-2">\begin{bmatrix} -\frac{2C_f+2C_r}{mV_x} & 0 & -(V_x+\frac{2C_fl_f-2C_rl_r}{mV_x}) \\ 0 & 0 & 1\\ -\frac{2l_fC_f-2l_rC_r}{I_z V_x} & 0 & - \frac{2l_f^2C_f+2l_r^2C_r}{I_zV_x} \end{bmatrix}</script> <span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display"><span class="MathJax MathJax_FullWidth" id="MathJax-Element-3-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mrow class="MJX-TeXAtom-ORD"><mover><mi>y</mi><mo>&#x02D9;</mo></mover></mrow></mtd></mtr><mtr><mtd><mi>&#x03C8;</mi></mtd></mtr><mtr><mtd><mrow class="MJX-TeXAtom-ORD"><mover><mi>&#x03C8;</mi><mo>&#x02D9;</mo></mover></mrow></mtd></mtr></mtable><mo>]</mo></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-151" style="width: 100%; display: inline-block; min-width: 2.382em;"><span style="display: inline-block; position: relative; width: 100%; height: 0px; font-size: 102%;"><span style="position: absolute; clip: rect(2.584em, 1002.03em, 6.837em, -999.997em); top: -4.958em; left: 0em; width: 100%;"><span class="mrow" id="MathJax-Span-152"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.584em, 1002.03em, 6.837em, -999.997em); top: -4.958em; left: 50%; margin-left: -1.162em;"><span class="mrow" id="MathJax-Span-153"><span class="mo" id="MathJax-Span-154" style="vertical-align: 2.23em;"><span style="display: inline-block; position: relative; width: 0.661em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.832em; left: 0em;">⎡<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: -0.706em; left: 0em;">⎣<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.719em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mtable" id="MathJax-Span-155" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 0.711em; height: 0px;"><span style="position: absolute; clip: rect(2.584em, 1000.66em, 6.735em, -999.997em); top: -4.857em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.711em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(3.141em, 1000.51em, 4.356em, -999.997em); top: -5.414em; left: 0em;"><span class="mtd" id="MathJax-Span-156"><span class="mrow" id="MathJax-Span-157"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.141em, 1000.51em, 4.356em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.301em;"><span class="texatom" id="MathJax-Span-158"><span class="mrow" id="MathJax-Span-159"><span style="display: inline-block; position: relative; width: 0.61em; height: 0px;"><span style="position: absolute; clip: rect(3.141em, 1000.51em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="munderover" id="MathJax-Span-160"><span style="display: inline-block; position: relative; width: 0.61em; height: 0px;"><span style="position: absolute; clip: rect(3.394em, 1000.51em, 4.356em, -999.997em); top: -3.997em; left: 0.003em;"><span class="mi" id="MathJax-Span-161" style="font-family: MathJax_Math-italic;">y<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.192em, 1000.31em, 3.597em, -999.997em); top: -3.997em; left: 0.104em;"><span class="mo" id="MathJax-Span-162" style="font-family: MathJax_Main;">˙</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.141em, 1000.66em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="mtd" id="MathJax-Span-163"><span class="mrow" id="MathJax-Span-164"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.141em, 1000.66em, 4.356em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.301em;"><span class="mi" id="MathJax-Span-165" style="font-family: MathJax_Math-italic;">ψ</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(2.888em, 1000.66em, 4.356em, -999.997em); top: -2.478em; left: 0em;"><span class="mtd" id="MathJax-Span-166"><span class="mrow" id="MathJax-Span-167"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.888em, 1000.66em, 4.356em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.352em;"><span class="texatom" id="MathJax-Span-168"><span class="mrow" id="MathJax-Span-169"><span style="display: inline-block; position: relative; width: 0.711em; height: 0px;"><span style="position: absolute; clip: rect(2.888em, 1000.66em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="munderover" id="MathJax-Span-170"><span style="display: inline-block; position: relative; width: 0.711em; height: 0px;"><span style="position: absolute; clip: rect(3.141em, 1000.66em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-171" style="font-family: MathJax_Math-italic;">ψ</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.192em, 1000.31em, 3.597em, -999.997em); top: -4.25em; left: 0.205em;"><span class="mo" id="MathJax-Span-172" style="font-family: MathJax_Main;">˙</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.862em;"></span></span></span></span><span class="mo" id="MathJax-Span-173" style="vertical-align: 2.23em;"><span style="display: inline-block; position: relative; width: 0.661em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.832em; left: 0em;">⎤<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: -0.706em; left: 0em;">⎦<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.719em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.964em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.964em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.805em; border-left: 0px solid; width: 0px; height: 4.133em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mrow class="MJX-TeXAtom-ORD"><mover><mi>y</mi><mo>˙</mo></mover></mrow></mtd></mtr><mtr><mtd><mi>ψ</mi></mtd></mtr><mtr><mtd><mrow class="MJX-TeXAtom-ORD"><mover><mi>ψ</mi><mo>˙</mo></mover></mrow></mtd></mtr></mtable><mo>]</mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-3">\begin{bmatrix} \dot{y} \\ \psi \\ \dot{\psi} \end{bmatrix}</script> + <span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display"><span class="MathJax MathJax_FullWidth" id="MathJax-Element-4-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>2</mn><mfrac><msub><mi>C</mi><mi>f</mi></msub><mi>m</mi></mfrac></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mfrac><mrow><mn>2</mn><msub><mi>l</mi><mi>f</mi></msub><msub><mi>C</mi><mi>f</mi></msub></mrow><msub><mi>I</mi><mi>z</mi></msub></mfrac></mtd></mtr></mtable><mo>]</mo></mrow></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-174" style="width: 100%; display: inline-block; min-width: 3.85em;"><span style="display: inline-block; position: relative; width: 100%; height: 0px; font-size: 102%;"><span style="position: absolute; clip: rect(3.04em, 1003.39em, 8.203em, -999.997em); top: -5.87em; left: 0em; width: 100%;"><span class="mrow" id="MathJax-Span-175"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.04em, 1003.39em, 8.203em, -999.997em); top: -5.87em; left: 50%; margin-left: -1.871em;"><span class="mrow" id="MathJax-Span-176"><span class="mo" id="MathJax-Span-177" style="vertical-align: 2.686em;"><span style="display: inline-block; position: relative; width: 0.661em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.832em; left: 0em;">⎡<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.256em; left: 0em;">⎣<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.719em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.263em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.807em; left: 0em;">⎢<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="mtable" id="MathJax-Span-178" style="padding-right: 0.154em; padding-left: 0.154em;"><span style="display: inline-block; position: relative; width: 2.078em; height: 0px;"><span style="position: absolute; clip: rect(3.04em, 1002.08em, 8.203em, -999.997em); top: -5.87em; left: 0em;"><span style="display: inline-block; position: relative; width: 2.078em; height: 0px;"><span style="position: absolute; width: 100%; clip: rect(2.736em, 1001.67em, 4.508em, -999.997em); top: -5.566em; left: 0em;"><span class="mtd" id="MathJax-Span-179"><span class="mrow" id="MathJax-Span-180"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.736em, 1001.67em, 4.508em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.858em;"><span class="mn" id="MathJax-Span-181" style="font-family: MathJax_Main;">2</span><span class="mfrac" id="MathJax-Span-182"><span style="display: inline-block; position: relative; width: 0.964em; height: 0px; margin-right: 0.104em; margin-left: 0.104em;"><span style="position: absolute; clip: rect(3.344em, 1000.81em, 4.356em, -999.997em); top: -4.604em; left: 50%; margin-left: -0.402em;"><span class="msubsup" id="MathJax-Span-183"><span style="display: inline-block; position: relative; width: 0.812em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-184" style="font-size: 70.7%; font-family: MathJax_Math-italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.509em;"><span class="mi" id="MathJax-Span-185" style="font-size: 50%; font-family: MathJax_Math-italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.546em, 1000.61em, 4.154em, -999.997em); top: -3.642em; left: 50%; margin-left: -0.301em;"><span class="mi" id="MathJax-Span-186" style="font-size: 70.7%; font-family: MathJax_Math-italic;">m</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(0.863em, 1000.96em, 1.217em, -999.997em); top: -1.263em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.964em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.066em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mtd" id="MathJax-Span-187"><span class="mrow" id="MathJax-Span-188"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(3.192em, 1000.46em, 4.154em, -999.997em); top: -3.997em; left: 50%; margin-left: -0.251em;"><span class="mn" id="MathJax-Span-189" style="font-family: MathJax_Main;">0</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; width: 100%; clip: rect(2.736em, 1002.08em, 4.66em, -999.997em); top: -2.326em; left: 0em;"><span class="mtd" id="MathJax-Span-190"><span class="mrow" id="MathJax-Span-191"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.736em, 1002.08em, 4.66em, -999.997em); top: -3.997em; left: 50%; margin-left: -1.061em;"><span class="mfrac" id="MathJax-Span-192"><span style="display: inline-block; position: relative; width: 1.825em; height: 0px; margin-right: 0.104em; margin-left: 0.104em;"><span style="position: absolute; clip: rect(3.344em, 1001.72em, 4.356em, -999.997em); top: -4.604em; left: 50%; margin-left: -0.858em;"><span class="mrow" id="MathJax-Span-193"><span style="display: inline-block; position: relative; width: 1.724em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1001.72em, 4.356em, -999.997em); top: -3.997em; left: 0em;"><span class="mn" id="MathJax-Span-194" style="font-size: 70.7%; font-family: MathJax_Main;">2</span><span class="msubsup" id="MathJax-Span-195"><span style="display: inline-block; position: relative; width: 0.559em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.21em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-196" style="font-size: 70.7%; font-family: MathJax_Math-italic;">l</span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.205em;"><span class="mi" id="MathJax-Span-197" style="font-size: 50%; font-family: MathJax_Math-italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-198"><span style="display: inline-block; position: relative; width: 0.812em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.56em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-199" style="font-size: 70.7%; font-family: MathJax_Math-italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.509em;"><span class="mi" id="MathJax-Span-200" style="font-size: 50%; font-family: MathJax_Math-italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(3.344em, 1000.61em, 4.255em, -999.997em); top: -3.592em; left: 50%; margin-left: -0.301em;"><span class="msubsup" id="MathJax-Span-201"><span style="display: inline-block; position: relative; width: 0.61em; height: 0px;"><span style="position: absolute; clip: rect(3.344em, 1000.36em, 4.154em, -999.997em); top: -3.997em; left: 0em;"><span class="mi" id="MathJax-Span-202" style="font-size: 70.7%; font-family: MathJax_Math-italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.053em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; top: -3.895em; left: 0.306em;"><span class="mi" id="MathJax-Span-203" style="font-size: 50%; font-family: MathJax_Math-italic;">z<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; clip: rect(0.863em, 1001.83em, 1.217em, -999.997em); top: -1.263em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.825em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.066em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.875em;"></span></span></span></span><span class="mo" id="MathJax-Span-204" style="vertical-align: 2.686em;"><span style="display: inline-block; position: relative; width: 0.661em; height: 0px;"><span style="position: absolute; font-family: MathJax_Size4; top: -2.832em; left: 0em;">⎤<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="position: absolute; font-family: MathJax_Size4; top: 0.256em; left: 0em;">⎦<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.719em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -1.263em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span><span style="font-family: MathJax_Size4; position: absolute; top: -0.807em; left: 0em;">⎥<span style="display: inline-block; width: 0px; height: 4.002em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 5.875em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 5.875em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.269em; border-left: 0px solid; width: 0px; height: 5.063em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>2</mn><mfrac><msub><mi>C</mi><mi>f</mi></msub><mi>m</mi></mfrac></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mfrac><mrow><mn>2</mn><msub><mi>l</mi><mi>f</mi></msub><msub><mi>C</mi><mi>f</mi></msub></mrow><msub><mi>I</mi><mi>z</mi></msub></mfrac></mtd></mtr></mtable><mo>]</mo></mrow></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-4">\begin{bmatrix} 2 \frac{C_f}{m}\\ 0 \\ \frac{2l_fC_f}{I_z} \end{bmatrix}</script> \delta
</span><span class="katex-html"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord cjk_fallback">求得小车模型的状态空间表达式为</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right: 2em;"></span><span class="mspace" style="margin-right: 2em;"></span><span class="mspace" style="margin-right: 2em;"></span><span class="mspace" style="margin-right: 2em;"></span><span class="mspace" style="margin-right: 2em;"></span><span class="mspace" style="margin-right: 2em;"></span><span class="mspace" style="margin-right: 2em;"></span><span class="mspace" style="margin-right: 2em;"></span><span class="mspace" style="margin-right: 2em;"></span><span class="mspace" style="margin-right: 2em;"></span><span class="mspace" style="margin-right: 2em;"></span><span class="mspace" style="margin-right: 2em;"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height: 3.6913em; vertical-align: -1.5957em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.3714em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height: 0.686em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 2.05em;"><span class="" style="top: -4.05em;"><span class="pstrut" style="height: 5.6em;"></span><span class="" style="width: 0.667em; height: 3.6em;">
<svg width="0.667em" height="3.600em" viewBox="0 0 667 3600">
<path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84
H403z M403 1759 V0 H319 V1759 v0 v1759 h84z">
y˙ψψ˙
=
−mVx2Cf+2Cr0−IzVx2lfCf−2lrCr000−(Vx+mVx2Cflf−2Crlr)1−IzVx2lf2Cf+2lr2Cr
y˙ψψ˙
+
2mCf0Iz2lfCf
δ
正好就是Parameters中的A,B,C,D矩阵
5.2 实验过程
导入矩阵和参考轨迹参数，双击Params.mat即可：
需要更换参考轨迹的话，参考以下方法生成新的posRef和yawRef
打开APP->Driving Scenario designer->选择道路和参考轨迹
开始仿真:
From：https://blog.youkuaiyun.com/weixin_45696231/article/details/126116123?ops_request_misc=&request_id=&biz_id=102&utm_term=%E6%8E%A7%E5%88%B6MPC&utm_medium=distribute.pc_search_result.none-task-blog-2allsobaiduweb~default-0-126116123.142v100pc_search_result_base9&spm=1018.2226.3001.4187
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