JP5591185B2 - Vehicle position and orientation measurement device - Google Patents

Vehicle position and orientation measurement device Download PDF

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JP5591185B2
JP5591185B2 JP2011128036A JP2011128036A JP5591185B2 JP 5591185 B2 JP5591185 B2 JP 5591185B2 JP 2011128036 A JP2011128036 A JP 2011128036A JP 2011128036 A JP2011128036 A JP 2011128036A JP 5591185 B2 JP5591185 B2 JP 5591185B2
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幸泰 明見
涼太郎 鈴木
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Mitsubishi Electric Corp
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Description

この発明は、任意の慣性座標系における車両の位置・姿勢を測定するための車両の位置姿勢測定装置に関するものである。   The present invention relates to a vehicle position / orientation measuring apparatus for measuring the position / orientation of a vehicle in an arbitrary inertial coordinate system.

従来の車両の位置姿勢測定装置では、車輪径が変化しないと仮定して非操舵左右輪の回転数から求めた移動距離差および平均移動距離と、さらにトレッドも変化しないとする仮定により、車両の位置および姿勢を測定していた(例えば特許文献1、非特許文献1)。
さらに、車輪の回転数の合計および/または二つの車輪の回転数差および/または車輪の所定の回転に対する時間ないし回転状態もしくは回転状態に達するまでの時間に依存した補正係数を導入することにより、カーブ走行中などに生じるトレッドの変化に対応した位置姿勢測定方式も提案されている(例えば特許文献2)。
In the conventional vehicle position and orientation measurement apparatus, it is assumed that the wheel diameter does not change, the moving distance difference and the average moving distance obtained from the rotation speeds of the non-steered left and right wheels, and the tread does not change. The position and orientation were measured (for example, Patent Document 1 and Non-Patent Document 1).
Furthermore, by introducing a correction factor depending on the sum of the wheel speeds and / or the difference between the speeds of the two wheels and / or the time for a given rotation of the wheel or the time to reach or reach the rotational state, A position / orientation measurement method that responds to tread changes that occur during running on a curve has also been proposed (for example, Patent Document 2).

特開昭61−274214号公報JP 61-274214 A 特表昭59−502153号公報JP-T 59-502153

C. Ming Wang著、”Location Estimation and Uncertainty Analysis for Mobile Robots”、IEEE International Conference on Robotics and Automation、pp.1230-1235, 1988(第2節)C. Ming Wang, “Location Estimation and Uncertainty Analysis for Mobile Robots”, IEEE International Conference on Robotics and Automation, pp. 1230-1235, 1988 (Section 2) 馬屋原将明 他著、”非線形逐次最小2乗法に基づく耐クロストークノイズキャンセラ”、電子情報通信学会論文誌A、pp.162−169、2002」Masaaki Masaya et al., “Anti-Crosstalk Noise Canceller Based on Nonlinear Sequential Least Squares Method”, IEICE Transactions A, pp. 162-169, 2002 " 谷萩隆嗣 著、”カルマンフィルタと適応信号処理”、pp.42−51、コロナ社、2005年12月Takashi Tanizaki, “Kalman filter and adaptive signal processing”, pp. 42-51, Corona, December 2005

このような車両の位置姿勢測定装置では、タイヤ空気圧の左右差などを要因として、非操舵左右輪の有効径に差異がある場合に、左右輪の移動距離演算に誤差が生じ、その結果位置・姿勢の測定精度が悪化するという問題点があった。   In such a vehicle position / orientation measurement device, when there is a difference in the effective diameter of the left and right non-steering wheels due to differences in the tire air pressure between the left and right, an error occurs in the movement distance calculation of the left and right wheels. There was a problem that posture measurement accuracy deteriorated.

この発明は、上記のような問題点を解決するためになされたものであり、車両の位置姿勢を精度良く測定する車両の位置姿勢測定装置を提供することを目的とする。   The present invention has been made to solve the above-described problems, and an object of the present invention is to provide a vehicle position / orientation measurement apparatus that accurately measures the position and orientation of a vehicle.

この発明は、前輪を操舵する車両の位置および姿勢を検出するための車両の位置姿勢測定装置であって、ステアリング角を検出するステアリング角検出手段と、車両の前後左右四輪の車輪速を検出する車輪速検出手段と、前記ステアリング角検出手段からの出力に基づいて実舵角を演算する実舵角演算手段と、前記実舵角と前記車輪速に基づきリアトレッドおよび後輪車輪半径のそれぞれの推定値を演算するトレッド・車輪半径推定手段と、前記推定値と後輪の前記車輪速に基づき車両の位置および姿勢を演算する車両位置姿勢演算手段と、を備え、任意の慣性座標系に対する車両の位置および姿勢を測定することを特徴とする車両の位置姿勢測定装置等にある。   The present invention is a vehicle position / orientation measurement device for detecting the position and orientation of a vehicle that steers front wheels, and detects a steering angle detection means for detecting a steering angle and wheel speeds of front, rear, left, and right wheels of the vehicle. Wheel speed detecting means, an actual steering angle calculating means for calculating an actual steering angle based on an output from the steering angle detecting means, a rear tread and a rear wheel wheel radius based on the actual steering angle and the wheel speed, respectively. Tread and wheel radius estimating means for calculating the estimated value of the vehicle, and vehicle position and orientation calculating means for calculating the position and orientation of the vehicle based on the estimated value and the wheel speed of the rear wheel, and for any inertial coordinate system A vehicle position / orientation measurement apparatus or the like that measures the position and orientation of a vehicle.

この発明によれば、前記トレッド・車輪半径推定手段を備えたことにより、走行中にリアトレッドが変化した場合だけでなく、タイヤ空気圧の減少などによって車輪径が変化した場合でも、車両の位置および姿勢を高精度に測定できる。   According to this invention, since the tread / wheel radius estimating means is provided, not only when the rear tread changes during traveling, but also when the wheel diameter changes due to a decrease in tire air pressure, etc. The posture can be measured with high accuracy.

この発明の実施の形態1による車両の位置姿勢測定装置の構成図である。1 is a configuration diagram of a vehicle position / orientation measurement apparatus according to Embodiment 1 of the present invention; FIG. この発明の原理を説明するための説明図である。It is explanatory drawing for demonstrating the principle of this invention. この発明の実施の形態2による車両の位置姿勢測定装置の構成図である。It is a block diagram of the vehicle position and orientation measurement apparatus according to Embodiment 2 of the present invention.

以下、この発明による車両の位置姿勢測定装置を各実施の形態に従って図面を用いて説明する。なお、各実施の形態においては、前輪を操舵する車両の位置姿勢測定を行う装置の例を示しており、また同一もしくは相当部分は同一符号で示し、重複する説明は省略する。   DESCRIPTION OF EMBODIMENTS Hereinafter, a vehicle position / orientation measuring apparatus according to the present invention will be described with reference to the drawings according to each embodiment. In each embodiment, an example of a device that measures the position and orientation of a vehicle that steers the front wheels is shown, and the same or corresponding parts are denoted by the same reference numerals, and redundant description is omitted.

実施の形態1.
図1はこの発明の実施の形態1による車両の位置姿勢測定装置の構成図である。この車両の位置姿勢測定装置は、ステアリング角θを検出するためのステアリング角センサ1と、車両の各車輪の車輪速を検出するために車輪速センサ2(左前輪),車輪速センサ3(右前輪),車輪速センサ4(左後輪),車輪速センサ5(右後輪)を有している。車輪速センサ2,3,4,5は、それぞれ対応する車輪が所定角度回転する毎に検出パルスを出力する公知の構成の車輪速センサである。
Embodiment 1 FIG.
1 is a block diagram of a vehicle position and orientation measurement apparatus according to Embodiment 1 of the present invention. This vehicle position / orientation measuring apparatus includes a steering angle sensor 1 for detecting a steering angle θ, a wheel speed sensor 2 (front left wheel), and a wheel speed sensor 3 (right) for detecting the wheel speed of each wheel of the vehicle. Front wheel), wheel speed sensor 4 (left rear wheel), and wheel speed sensor 5 (right rear wheel). The wheel speed sensors 2, 3, 4, and 5 are wheel speed sensors having a known configuration that outputs a detection pulse each time a corresponding wheel rotates by a predetermined angle.

実舵角演算器6は、ステアリング角センサ1により検出されたステアリング角θを予め実験によって求めた関係式(1)に代入し、実舵角δFl,δFrを演算する。 The actual steering angle calculator 6 substitutes the steering angle θ detected by the steering angle sensor 1 into the relational expression (1) obtained in advance through experiments, and calculates actual steering angles δ Fl and δ Fr.

Figure 0005591185
Figure 0005591185

トレッド・車輪半径推定器7は、実舵角演算器6から出力される実舵角δFl,δFrと車輪速センサ2,3,4,5からそれぞれ出力される各パルス信号PFl,PFr,PRl,PRrから、フロントトレッドT,リアトレッドT,左前車輪半径rFl,右前車輪半径rFr,左後車輪半径rRl,右後車輪半径rRrを推定する。図2を参照しながら、その原理を説明する。車両の左前車輪11,右前車輪12,左後車輪13,右後車輪14の横滑り角が全て0であると仮定すると、 The tread / wheel radius estimator 7 outputs the actual rudder angles δ Fl , δ Fr output from the actual rudder angle calculator 6 and the respective pulse signals P Fl , P output from the wheel speed sensors 2, 3, 4, 5. fr, P Rl, from P Rr, estimates front tread T R, the rear tread T R, the left front wheel radius r Fl, right front wheel radius r fr, the rear left wheel radius r Rl, the right rear wheel radius r Rr. The principle will be described with reference to FIG. Assuming that the sideslip angles of the left front wheel 11, the right front wheel 12, the left rear wheel 13, and the right rear wheel 14 of the vehicle are all 0,

Figure 0005591185
Figure 0005591185

が成り立つ。ただし、
Fl,vFr,vRl,vRr:それぞれ車輪11,12,13,14の移動速度
,v:それぞれ重心速度のx軸成分,y軸成分(慣性座標系は任意)
(ドット)γ:車両の姿勢角(ヨー角:基準座標xy上の車両方向のx軸とのなす角度)γの単位時間当たりの変化量(角速度)
:前輪軸中心(左前車輪11と右前車輪12を結ぶ線の中心)と車両重心c.g.との間の長さ
:後輪軸中心(左後車輪13と右後車輪14を結ぶ線の中心)と車両重心c.g.との間の長さ
である。例えば、後退駐車時のように、車両が低速で定常運動を行なっていると考えられる場合においては、このような仮定が成立する。
Holds. However,
v Fl , v Fr , v Rl , v Rr : movement speeds of the wheels 11, 12, 13, 14 respectively v x , v y : x-axis component and y-axis component of the center-of-gravity velocity, respectively (the inertial coordinate system is arbitrary)
(Dot) γ: attitude angle of the vehicle (yaw angle: angle formed with the x-axis of the vehicle direction on the reference coordinates xy) change amount of γ per unit time (angular velocity)
L F : Length between front wheel axis center (center of line connecting left front wheel 11 and right front wheel 12) and vehicle center of gravity c.g. L R : Rear wheel axis center (left rear wheel 13 and right rear wheel 14 It is the length between the center of the connecting line) and the vehicle center of gravity c.g. For example, when it is considered that the vehicle is performing a steady motion at a low speed as in reverse parking, such an assumption is established.

さらに、上式を整理すると、   Furthermore, if we organize the above formula,

Figure 0005591185
Figure 0005591185

が得られる。Lはホイルベースを示す。ところで、各車輪速センサ2,3,4,5からの出力パルス数PFl,PFr,PRl,PRrと各車輪11,12,13,14の移動速度vFl,vFr,vRl,vRrとの間には、関係式 Is obtained. L indicates a foil base. By the way, the number of output pulses P F1 , P Fr , P R1 , P Rr from the wheel speed sensors 2, 3, 4, 5 and the moving speeds v F1 , v Fr , v Rl of the wheels 11, 12, 13, 14 , V Rr , the relational expression

Figure 0005591185
Figure 0005591185

が成り立つ。ただし、NFl,NFr,NRl,NRrは各車輪が1回転したときのパルス数であり、Tはサンプリング時間、nは正の整数、である。したがって、ここで考えている定常運動状態では、操舵速度は十分小さいので、 Holds. However, N Fl , N Fr , N Rl , and N Rr are the number of pulses when each wheel makes one rotation, T is the sampling time, and n is a positive integer. Therefore, in the steady motion state considered here, the steering speed is sufficiently low,

Figure 0005591185
Figure 0005591185

と仮定できるので((ドット)δFl,(ドット)δFrはそれぞれ実舵角δFl,δFrの単位時間当たりの変化量(実舵角速度))、(9),(10)より、 ((Dot) δ Fl , (dot) δ Fr is the change amount per unit time (actual rudder angular velocity) of the actual rudder angle δ Fl , δ Fr ) (9), (10),

Figure 0005591185
Figure 0005591185

が成り立つ。そこで、この関係式(12)から、車輪半径およびトレッドの走行中の実際の値である真値とメーカが公表しているノミナル値(公称値)との誤差量が満たす関係式を導出し、その関係式に対して非線形逐次最小二乗法、拡張カルマンフィルタなどを用いたパラメータ推定手法を適用することで、実舵角δFl,δFrとパルス数PFl,PFr,PRl,PRrから、ノミナル値に誤差補正を加えた車輪半径およびトレッドの推定値を得ることができる。 Holds. Therefore, from this relational expression (12), a relational expression that satisfies the error amount between the wheel radius and the true value, which is the actual value during running of the tread, and the nominal value (nominal value) published by the manufacturer is derived. By applying a parameter estimation method using a non-linear sequential least square method, an extended Kalman filter or the like to the relational expression, from the actual steering angles δ Fl , δ Fr and the number of pulses P Fl , P Fr , PR Rl , PR Rr Thus, it is possible to obtain an estimated value of the wheel radius and tread obtained by adding an error correction to the nominal value.

例えば、車輪半径およびトレッドの真値rFl,rFr,rRl,rRr,T,Tは、ノミナル値r Fl,r Fr,r Rl,r Rr,T ,T とスケールファクタ誤差ΔrFl,ΔrFr,ΔrRl,ΔrRr,ΔT,ΔTを用いて、 For example, the wheel radii and the true values of the treads r Fl , r Fr , r Rl , r Rr , T F , and T R are nominal values r * Fl , r * Fr , r * Rl , r * Rr , T * F , Using T * R and scale factor errors Δr Fl , Δr Fr , Δr Rl , Δr Rr , ΔT F , ΔT R ,

Figure 0005591185
Figure 0005591185

と表現できるので、関係式(13)を関係式(12)に代入することにより、誤差量ΔrFl,ΔrFr,ΔrRl,ΔrRr,ΔT,ΔTが満たすべき関係式が得られる。なお、非線形逐次最小二乗法および拡張カルマンフィルタについては、それぞれ上記非特許文献2,3などを参照のこと。 Since can be expressed as, by substituting equation (13) into equation (12), the error amount Δr Fl, Δr Fr, Δr Rl , Δr Rr, ΔT F, the relationship to be satisfied by the [Delta] T R obtained. For the non-linear sequential least square method and the extended Kalman filter, see Non-Patent Documents 2 and 3 above, respectively.

車両位置姿勢演算器8は、トレッド・車輪半径推定器7から出力される後輪(左後、右後)車輪半径およびリアトレッドの推定値rRl ,rRr ,T と、後輪(左後、右後)の車輪速センサ4,5からそれぞれ出力される各パルス信号PRl,PRrから、車両の平面位置および姿勢を演算する。図2を参照しながら、その原理を説明する。車両の運動学的モデルより、後輪軸中心の位置x,yは関係式 The vehicle position / orientation calculator 8 includes rear wheel (left rear, right rear) wheel radius and rear tread estimated values r Rl , r Rr , T R output from the tread / wheel radius estimator 7, The plane position and attitude of the vehicle are calculated from the pulse signals P Rl and P Rr output from the wheel speed sensors 4 and 5 of the wheels (left rear and right rear), respectively. The principle will be described with reference to FIG. From the kinematic model of the vehicle, the position x, y of the rear wheel axle center is a relational expression.

Figure 0005591185
Figure 0005591185

を満たす((ドット)x,(ドット)yはx,y座標の値の単位時間当たりの変化量)。そこで、関係式(6),(7)を関係式(14)に代入して得られる関係式および関係式(6)をサンプリング時間T(0<T<<1)で離散近似すると、後輪軸中心の位置x(nT),y(nT)および姿勢角(ヨー角)γ(nT)が満たすべき差分方程式 ((Dot) x, (dot) y is the amount of change per unit time of the x and y coordinate values). Therefore, when the relational expression obtained by substituting the relational expressions (6) and (7) into the relational expression (14) and the relational expression (6) are discretely approximated with the sampling time T (0 <T << 1), the rear wheel shaft is obtained. Difference equation to be satisfied by center position x (nT), y (nT) and posture angle (yaw angle) γ (nT)

Figure 0005591185
Figure 0005591185

が得られる(上記非特許文献1参照)。ただし、 Is obtained (see Non-Patent Document 1 above). However,

Figure 0005591185
Figure 0005591185

である。さらに、ここで考えている定常運動状態では、車両加速度は十分小さいので、 It is. Furthermore, in the steady motion state considered here, the vehicle acceleration is sufficiently small,

Figure 0005591185
Figure 0005591185

と近似できるので、関係式(10)より、 From the relational expression (10),

Figure 0005591185
Figure 0005591185

が成り立つが、後輪車輪半径およびリアトレッドの真値は未知であるので、それらをトレッド・車輪半径推定器7からの出力値、すなわち後輪車輪半径およびリアトレッドの推定値rRl ,rRr ,T に置き換えると、関係式(19)が得られる。 However, since the rear wheel wheel radius and the true value of the rear tread are unknown, they are output from the tread / wheel radius estimator 7, that is, the rear wheel wheel radius and the estimated value of the rear tread r Rl , r. If replaced with Rr and T R , the relational expression (19) is obtained.

Figure 0005591185
Figure 0005591185

よって、車両運動を記述する慣性座標系の設定に応じて、初期値を適切に設定することにより、任意の慣性座標系における位置x(nT),y(nT)および姿勢角(ヨー角)γ(nT)は、後輪車輪半径およびリアトレッドの推定値rRl ,rRr ,T とパルス信号PRl,PRrから、関係式(15)と(19)に基づいて演算できる。 Accordingly, by appropriately setting the initial values according to the setting of the inertial coordinate system describing the vehicle motion, the positions x (nT), y (nT) and the posture angle (yaw angle) γ in an arbitrary inertial coordinate system. (nT) can be calculated based on the relational expressions (15) and (19) from the rear wheel wheel radius and rear tread estimated values r Rl , r Rr , T R and pulse signals P Rl , P Rr. .

以上のような構成によれば、トレッド・車輪半径推定器7がオンラインで車輪半径およびトレッドを推定しつつ、その結果に基づき車両位置姿勢演算器8が車両の位置および姿勢を演算するので、リアトレッドあるいは後輪車輪径が変化しノミナル値と異なるといった場合でも、車両の位置および姿勢を高精度に測定できる。   According to the above configuration, the tread / wheel radius estimator 7 estimates the wheel radius and tread online, and the vehicle position / orientation calculator 8 calculates the position and orientation of the vehicle based on the result. Even when the tread or rear wheel diameter changes and differs from the nominal value, the position and posture of the vehicle can be measured with high accuracy.

実施の形態2.
図3はこの発明の実施の形態2による車両の位置姿勢測定装置の構成図である。この装置は、車両の各車輪の車輪速を検出するための車輪速センサ(左前輪)2,車輪速センサ(右前輪)3,車輪速センサ(左後輪)4,車輪速センサ(右後輪)5以外のセンサを必要としない。この構成によれば、図1に示したステアリング角センサ1と実舵角演算器6が不要となるほか、ステアリング機構に基づく検出誤差による車両の位置姿勢測定精度の悪化が生じない。
Embodiment 2. FIG.
FIG. 3 is a block diagram of a vehicle position / orientation measurement apparatus according to Embodiment 2 of the present invention. This device has a wheel speed sensor (left front wheel) 2, a wheel speed sensor (right front wheel) 3, a wheel speed sensor (left rear wheel) 4, a wheel speed sensor (right rear wheel) for detecting the wheel speed of each wheel of the vehicle. No sensors other than 5) are required. According to this configuration, the steering angle sensor 1 and the actual steering angle calculator 6 shown in FIG. 1 are not required, and the vehicle position and orientation measurement accuracy is not deteriorated due to a detection error based on the steering mechanism.

図3のトレッド・車輪半径推定器7aの原理を説明する。関係式(12)よりδFl,δFrを消去すると、 The principle of the tread / wheel radius estimator 7a shown in FIG. 3 will be described. If δ Fl and δ Fr are eliminated from the relational expression (12),

Figure 0005591185
Figure 0005591185

が成り立つ。この関係式(20)から導出できる車輪半径およびトレッドの真値とノミナル値との誤差量が満たす関係式に対して、非線形逐次最小二乗法、拡張カルマンフィルタなどのパラメータ推定手法を適用すると、パルス数PFl,PFr,PRl,PRrのみから、ノミナル値に誤差補正を加えた車輪半径、またはトレッド、または車輪半径およびトレッドの両方のそれぞれ推定値を得ることができる。 Holds. When a parameter estimation method such as a non-linear sequential least square method or an extended Kalman filter is applied to the relational expression that satisfies the error amount between the wheel radius and the true value of the tread and the nominal value that can be derived from this relational expression (20), From only P Fl , P Fr , P Rl , and P Rr , it is possible to obtain the estimated values of the wheel radius obtained by adding error correction to the nominal value, or the tread, or both the wheel radius and the tread.

1 ステアリング角センサ(ステアリング角検出手段)、2,3,4,5 車輪速センサ(車輪速検出手段)、6 実舵角演算器(実舵角演算手段)、7,7a トレッド・車輪半径推定器(トレッド・車輪半径推定手段)、8 車両位置姿勢演算器(車両位置姿勢演算手段)、11,12,13,14 車輪。   1 Steering angle sensor (steering angle detection means), 2, 3, 4, 5 Wheel speed sensor (wheel speed detection means), 6 Actual steering angle calculator (actual steering angle calculation means), 7, 7a Tread / wheel radius estimation (Tread / wheel radius estimating means), 8 vehicle position / orientation calculator (vehicle position / orientation calculating means), 11, 12, 13, 14 wheels.

Claims (2)

前輪を操舵する車両の位置および姿勢を検出するための車両の位置姿勢測定装置であって、
ステアリング角を検出するステアリング角検出手段と、
車両の前後左右四輪の車輪速を検出する車輪速検出手段と、
前記ステアリング角検出手段からの出力に基づいて実舵角を演算する実舵角演算手段と、
前記実舵角と前記車輪速に基づきリアトレッドおよび後輪車輪半径のそれぞれの推定値を演算するトレッド・車輪半径推定手段と、
前記推定値と後輪の前記車輪速に基づき車両の位置および姿勢を演算する車両位置姿勢演算手段と、
を備え、任意の慣性座標系に対する車両の位置および姿勢を測定することを特徴とする車両の位置姿勢測定装置。
A vehicle position and orientation measurement device for detecting the position and orientation of a vehicle that steers front wheels,
Steering angle detection means for detecting the steering angle;
Wheel speed detection means for detecting the wheel speed of the front, rear, left and right four wheels of the vehicle;
Actual steering angle calculation means for calculating an actual steering angle based on the output from the steering angle detection means;
A tread / wheel radius estimating means for calculating an estimated value of each of the rear tread and the rear wheel radius based on the actual steering angle and the wheel speed;
Vehicle position and orientation calculation means for calculating the position and orientation of the vehicle based on the estimated value and the wheel speed of the rear wheel;
A vehicle position / orientation measurement apparatus for measuring a position and an attitude of a vehicle with respect to an arbitrary inertial coordinate system.
前輪を操舵する車両の位置および姿勢を検出するための車両の位置姿勢測定装置であって、
車両の前後左右四輪の車輪速を検出する車輪速検出手段と、
前記車輪速に基づきリアトレッドおよび後輪車輪半径のそれぞれの推定値を演算するトレッド・車輪半径推定手段と、
前記推定値と後輪の前記車輪速に基づき車両の位置および姿勢を演算する車両位置姿勢演算手段と、
を備え、任意の慣性座標系に対する車両の位置および姿勢を測定することを特徴とする車両の位置姿勢測定装置。
A vehicle position and orientation measurement device for detecting the position and orientation of a vehicle that steers front wheels,
Wheel speed detection means for detecting the wheel speed of the front, rear, left and right four wheels of the vehicle;
A tread / wheel radius estimating means for calculating respective estimated values of the rear tread and the rear wheel radius based on the wheel speed;
Vehicle position and orientation calculation means for calculating the position and orientation of the vehicle based on the estimated value and the wheel speed of the rear wheel;
A vehicle position / orientation measurement apparatus for measuring a position and an attitude of a vehicle with respect to an arbitrary inertial coordinate system.
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