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Analysis of three common kinematic models of mobile chassis

2022-07-27 04:09:00 【K. Fire is covered with liver】

Catalog

Preface

One 、 Four wheel differential motion model

Two 、 Kinematic model of mcnamu wheel

3、 ... and 、 Two wheel differential kinematics model

summary

Preface

Now junior summer vacation , School is about to start preparing to finish , Took over the laser of the senior brother in the Laboratory SLAM Trolley project , Start with the lower computer , Try to accept the whole project , Gain something and innovate .

This article is in the process of learning code , For the four-wheel differential car 、 Study and summary of kinematics model of mcnamu wheel car and two wheel differential car .

One 、 Four wheel differential motion model

Symbol description ：

In the four wheel differential model , The speeds of the front and rear wheels are synchronized , Here is the geometric center of the chassis COG Along the y Point in the axis direction ICR As the center of the circle when the whole chassis moves in a circular motion ,ICR and COG The distance is related to the angular velocity of circular motion .

Formula derivation ：

An object that moves in a circle around the center of a circle , Its linear speed v、 angular velocity w And circumferential radius d Satisfy w=v/d. Therefore, the constraint relationship in the chassis can be established ：

$\omega _c=\frac{v_c}{d_c}$

Suppose the line $d_{c}$ And y The included angle of the shaft is , You can deform the above formula , The results are shown below ：

$\omega _c=\frac{v_c}{d_c}=\frac{v_c\cos \alpha _c}{d_c\cos \alpha _c}=\frac{v_{cx}}{d_{cy}}$

$\omega _c=\frac{v_c}{d_c}=\frac{v_c\sin \alpha _c}{d_c\sin \alpha _c}=\frac{v_{cy}}{d_{cx}}$

When the rigid body rotates , The angular velocity at each position is the same as that at the center of mass , That is, four wheels around ICR The angular velocity of rotation is also $\omega _{c}$ . Infer according to the above formula , The following constraint relation can also be obtained ：

$\omega _c=\frac{v_i}{d_i}=\frac{v_i\cos \alpha _i}{d_i\cos \alpha _i}=\frac{v_{ix}}{d_{ix}}$

$\omega _c=\frac{v_i}{d_i}=\frac{v_i\sin \alpha _i}{d_i\sin \alpha _i}=\frac{v_{iy}}{d_{iy}}$

meanwhile , $d_{i}$ And $d_{c}$ stay x Axis and y The projection length on the axis satisfies the following equation ：

$d_{1y}=d_{2y}=d_{cy}-\frac{c}{2}$

$d_{3y}=d_{4y}=d_{cy}+\frac{c}{2}$

When the four-wheel differential chassis is set to the left wheel 、 The speed of the cruise ship is $v_{l}$ and $v_{r}$ , And the front wheels 、 When the rear wheel speed is strictly synchronized , There is the relationship shown in the following formula :

$v_l=v_{1x}=v_{2x}$

$v_r=v_{3x}=v_{4x}$

Combining the above formula, we can get the following results ：

$v_l=v_{1x}=v_{2x}=\omega _c\left( d_{cy}-\frac{c}{2} \right) =v_{cx}-\omega _c\cdot \frac{c}{2}$

$v_r=v_{3x}=v_{4x}=\omega _c\left( d_{cy}+\frac{c}{2} \right) =v_{cx}+\omega _c\cdot \frac{c}{2}$

Sort out the above formula , Get the forward kinematics relationship of the four-wheel differential chassis ：

$\left[ \begin{array}{c} v_{cx}\\ \omega _z\\ \end{array} \right] =\left[ \begin{matrix} \frac{1}{2}& \frac{1}{2}\\ -\frac{1}{c}& \frac{1}{c}\\ \end{matrix} \right] \left[ \begin{array}{c} v_l\\ v_r\\ \end{array} \right]$

Reference blog ： Mobile robot motion model （ Two rounds 、 Four rounds 、 Mcnaim wheel and probabilistic motion ）_anthony-36 The blog of -CSDN Blog _ Mobile robot motion model

Two 、 Kinematic model of mcnamu wheel

Known Car Center xy Direction linear velocity $v_{tx}$ 、 $v_{ty}$ And angular velocity $\omega$ , You can calculate the center speed of each wheel ：

$\vec{v}=\vec{v}_t+\vec{v}_r=\vec{v}_t+\vec{\omega}\times \vec{r}$

Then the center of the wheel xy The directional velocity component is ：

$v_x=v_{tx}-v_{rx}=v_{tx}-v_r\cos \theta =v_{tx}-v_r\frac{r_y}{r}=v_{tx}-\omega r_y$

$v_y=v_{ty}+v_{ry}=v_{ty}+v_r\sin \theta =v_{ty}+v_r\frac{r_x}{r}=v_{ty}+\omega r_x$

According to the speed of the wheel center , It can be divided into the speed along the roller direction and the speed perpendicular to the roller direction , The speed in the vertical roller direction can be omitted , $\underset{u}{\rightarrow}$ Is the unit vector in the direction of the roller （ Projection ）, You can get ：

$\vec{v}_{||}=\vec{v}\cdot \vec{u}$

Roller direction and x The included angle in the axis direction is 45°, Then the unit vector can be expressed as ：

$\vec{u}=-\frac{1}{\sqrt{2}}\vec{i}+\frac{1}{\sqrt{2}}\vec{j}$

Then there are ：

$v_{||}=\vec{v}\cdot \vec{u}=\left( \vec{v}_x\vec{i}+\vec{v}_y\vec{j} \right) \cdot \left( -\frac{1}{\sqrt{2}}\vec{i}+\frac{1}{\sqrt{2}}\vec{j} \right)$

After simplification, we get ：

$\vec{v}_{||}=-\frac{1}{\sqrt{2}}v_x+\frac{1}{\sqrt{2}}v_y$

Then from the speed of the roller in contact with the ground to the linear speed of the wheel ：

$v_v=v_{||}\cdot \sqrt{2}=-v_x+v_y$

Substitute the above $v_{x}$ 、 $v_{y}$ The calculation formula is ：

$v_v=-\left( v_{tx}-\omega r_y \right) +\left( v_{ty}+\omega r_x \right) =v_{ty}-v_{tx}+\omega \left( r_x+r_y \right)$

The calculation formula of the four wheels can be obtained in turn ：

$\left\{ \begin{array}{l} v_{v1}=v_{ty}-v_{tx}+\omega \left( r_x+r_b \right)\\ v_{v2}=v_{ty}+v_{tx}-\omega \left( r_x+r_b \right)\\ v_{v3}=v_{ty}-v_{tx}-\omega \left( r_x+r_b \right)\\ v_{v4}=v_{ty}+v_{tx}+\omega \left( r_x+r_b \right)\\ \end{array} \right.$

In style $r_{x}$ 、 $r_{y}$ It is half of the transverse distance and longitudinal distance of the car body

Reference blog ： Understand the principle and analysis of omni-directional movement of mcnamu wheel _ Su Shoukun's blog -CSDN Blog _ Principle of mcnamu wheel

When the transverse distance and longitudinal distance of the trolley are close , The kinematic model of wheat wheel will degenerate into the kinematic model of four-wheel differential car ,y The speed in the direction is 0, $r_{x}$ 、 $r_{y}$ equal , Look at the formula and it comes out .

3、 ... and 、 Two wheel differential kinematics model

The motion characteristic is two wheel differential drive , The rotation of two isomorphic driving wheels behind its bottom provides power , The follower in front plays a supporting role and does not promote its movement , The two wheel differential drive is shown in the figure below .