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SPEED-ADAPTED TRAJECTORIES IN THE CASE OF INSUFFICIENT YDRAULIC RESSURE FOR THE FOUR-LEGGED LARGE-SCALE WALKINGVEHICLE ALDURO ABSTRACT When operating walking machines, only a coordinated movement of all cylinders and/or motors can lead to safe, stable walking. The large hydraulically driven walking machine ALDURO, which is investigated in this paper, has no external power supply, and therefore the size of the on-board hydraulic power pack and its diesel engine is limited by its weight. When moving several cylinders with high speed, the hydraulic supply can become insufficient and the resulting trajectories of feet and platform can become unpredictable. When the ALDURO is near its stability limit such behaviour can lead to instability and toppling of the system. The proposed solution under discussion here is to observe the position errors and time derivatives for the cylinders and based on this reduce the speed when necessary. 1. INTRODUCTION The system investigated in this paper is the Anthropomorphically Legged and Wheeled Duisburg Robot (ALDURO). It consists of a platform of 2.0m by 2.2m with a cabin for the operator and four legs, each 1.8m long. The estimated weight is 1200 kg. It can be used as a quadruped walking machine (Fig. 2), and by replacing the two hind feet with wheels, it can also be used as a combined legged and wheeled vehicle (Fig. 1). The latter combines the advantages of a walking machine - high mobility - with the stability and speed of wheeled vehicles [2]. When operating in steep and dangerous terrain, safety plays an important role. It must be guaranteed that the cylinders follow the calc ulated trajectories exactly since a wrong movement might cause the robot to become instable. ALDURO's legs are hydraulically driven, with an open hydraulic system. Normally, when planning hydraulic systems, low weight has no high priority [1]. For the walking machine ALDURO the ratio power per weight was optimized. Fig. 1: Combined Legged Fig. 2: Walking Machine and Wheeled Vehicle While actuating several cylinders simultaneously or moving very fast, the volume flow of the hydraulic supply becomes insufficient and the resulting movements uncontrollable.The proposed solution under discussion here is a decrease in speed of all calculated trajectories. This is admissible as long as the robot is statically stable at any moment. By observing the position errors of the cylinders and their time derivatives, a decision is taken on whether to decrease or increase the speed along the trajectories. To ensure that all movements are influenced simultaneously, the model-time, upon which all calculations depend, is slowed down. Thus we can guarantee that legs and platform can and do follow the prescribed trajectories. This strategy is being tested on a virtual model of the ALDURO and is being tested on a virtual model of the ALDURO and a test stand in the laboratory,consisting of a single leg in scale 1:1, giving very good results. Fig. 3: Experimental setup 2. EXPERIMENTAL SETUP The leg of the ALDURO is anthropomorphic, i.e. it is based on the geometry and function of the human leg. The hip joint is a spherical ball joint with three degrees of freedom (d.o.f.) and the knee a revolute joint with one d.o.f. These joints are actuated with hydraulic cylinders (Fig. 4), whereas the two additional d.o.f. of the foot are passive. To make solution of the explicit inverse kinematics possible we lock one of the hip cylinders. The explicit solution of the direct and inverse kinematics is shown in detail in [4]. To examine the dynamic behaviour of the leg, and to test different control schemes, an experimental setup for a fully sized leg was developed and built (see Fig. 3). It includes all the hydraulic components that will be used on the first prototype ALDURO and is driven by a stationary hydraulic power pack in the laboratory. The experimental set-up is equipped with an open hydraulic system with a 15kW electrical motor and an axial piston pump producing a volume flow of 40 l/min at 200 bar. This is smaller than what will be used on the real system, which will be powerd by a 27kW diesel engine. For tests with an insufficient hydraulic flow, a second set of four hydraulic cylinders (as used to move one leg) is mounted on the floor next to the test stand. An optical fieldbus for the transfer of the sensor and actuator data between the test stand and the control computer (with real-time operating system) is also installed. The hydraulic cylinders include position and pressure sensors that are used as controller inputs. Thus dynamic tests can be carried out to examine the co-operation between the mechanical and hydraulic components and the electronic control system. The hip plate of the experimental set-up is mounted on a pair of linear bearings, and thus is mobile in vertical direction. When combined with the foot, as shown in Fig. 3, or the passive wheel used on the hind leg of the combined legged and wheeled system, this allows loading tests on the leg while performing walking motions. Mechanical stops below the hip plate allow the foot to be lifted in the swing phase of the walking motion. The first (unloaded) tests have shown the mobility of the anthropomorphic leg mechanism to be very good, and the optical fieldbus system has also proven to be reliable. 3. SPEED ADAPTATION 3.1 Insufficient Hydraulic Flow As already mentioned, the weight and size of the onboard power pack is limited and has to be kept low. When moving several cylinders simultaneously or with a high speed the hydraulic supply can become insufficient and the pressure will collapse. In this case the resulting movement depends mainly on the sizes of the proportional valves and external loads on the cylinders. The trajectories of the feet and platform become erratic. When ALDURO is near limit of its stability such behaviors can lead to instability and toppling of the system. As the safety issue is a very important one (people could be harmed) this pressure collapse has to be prevented. To investigate this behavior, a circular movement (radius 0.2 m, velocity 0.5 m/s) of the foot in the horizontal plane 1m below the hip was chosen as a reference. This trajectory has the advantage that it actuates all three unlocked cylinders. With a typical vertical stepping movement of the foot the sideways cylinder (no. 2 in Fig. 4) is nearly stationary. The inverse kinematics for the mapping of the foot co-ordinate into the co-ordinate space for the three cylinders has been developed in the C++ programming environment Ma a a aBILE [3]. When implemented on the test stand the resulting trajectory of the foot degenerated to a rounded rectangle (Fig. 5). To increase the hydraulic consumption, both sets of hydraulic cylinders have been used (i.e. leg cylinders and set of four cylinders on the ground). As we can see in Fig. 6, cylinder no. 2 lags far behind its sinusoidal set-point curve. The linear movement of the cylinder indicates a fully open proportional valve. The available hydraulic volume flow is clearly insufficient. 3.2 Trajectory Speed Reduction The problem of uncontrollable movements can be approached from different directions: Redimensioning of pump and valves, Predict insufficient flow/pressure with detailed model of hydraulics and recalculate critical movements with lower velocities, Detect position errors due to pressure drop and slow down all movements, while maintaining trajectories. The first approach increases the weight of the hydraulic system and is thus undesirable. The second is undesirable because of computing power required and inaccuracy in the hydraulic model. Therefore, detection of the position error and slowing down the movement was chosen and implemented here. Instead of recalculating all trajectories for the case of a necessary deceleration, the time variable on which all trajectories depend is slowed down. To this end a new time variable is introduced. This trajectory time or model time t* can be expressed as a function of realtime t and the error dependent factor for the time increment All trajectories are functions of this t* For the control system running on a real time operating system we need the discrete relationship between t and t* for time step i. As all trajectories are functions of t* they will all be simultaneously slowed down or sped up, depending on k mt; i. As an indicator for insufficient hydraulic flow, a function of the position errors of the cylinders is chosen. The vector s j contains the set-point positions for the cylinders in each leg and i j the vector of measured positions. The difference is the error in meters. Where fl stands for front left, hr for hind right and so forth. With the weight matrix W, the influence of the four cylinders in each leg on the foot position can be adjusted. We take the normalized weighed sum of the squares of the errors for all cylinders e and its first time derivation de and normalize both with the admissible errors (thresholds e thr and de thr). If the sum of e and de exceeds the upper limit, the model time is slowed down. As both e and de are normalized with a threshold, this upper limit can be fixed. In this case it is 2. The third threshold (lower limit) is used to decide when to increase the speed again. To prevent a decrease in error triggering a stop in deceleration or even an acceleration, even if the error is still too large, de is only added if it is positive. where ?k mt,dec is the rate of change of k mt for deceleration and ?k mt,acc is the same for acceleration. K mt has to stay inside the limit [k mt,min…1]. 3.3Results Values for thresholds, acceleration and decelerationrates of model time and cylinder weights are set empirically. The cylinders at the hip joint have a higher influence on the foot position than cylinder no. 4 at the knee. The corresponding weighing is set to 2:1. As the system has to react very quickly to errors,the deceleration rate ?k mt,dec is high. To prevent too much oscillation, the acceleration rate ?k mt,acc is set lower than ?k mt,dec.] Fig. 7: Cylinder positions for speed adapted trajectory In Fig. 7, at t = 2.5 s we can see the set-point value for cylinder 2 rising fast, and that it cannot be followed by the physical system. The resulting normalized error can be seen in Fig. 8, where it oversteps the threshold e thr,2 and triggers the deceleration of the model time t* by decreasing the model time factor k mt (as seen in Fig. 9.) By setting the thresholds e thr and de thr the influence of the absolute error and the change in error can be adjusted. As the latter is a time derivative of a measured signal it is likely to be noisy. Here this threshold is set low. Another possibility would be to filter the signal with the disadvantage that this would produce a larger dead band. 4. CONCLUSIONS Due to high safety demands for the operator driven walking machine ALDURO, the need arises to guarantee actuator movements are executed exactly as calculated by the controller. With all actuators moving simultaneously or at high speed the hydraulic supply can become insufficient and the pressure can collapse.The method described here to prevent this is based on detecting deficiencies in the hydraulic flow by observing the position errors at the actuators and slowing down all movements if necessary. The implementation of an adjustable model time for the calculation of trajectories on the test leg from ALDURO looks promising. The execution of movements at high velocity has been improved drastically. 5. FUTURE WORK At the moment the controller for the leg is a proportional controller with inputs from the inverse kinematics and with one cylinder locked. To be able to use all cylinders and to reduce the tracking errors this controller will be replaced with a force and model based controller. This should reduce the remaining position errors as shown in Fig. 10. Signals from additional pressure sensors in the hydraulic circuit will be included in the evaluation of the hydraulic power supply. 6. ACKNOWLEDGEMENT This work is substantially supported by the Deutsche Forschungs gemeins -chaft DFG (German ResearchCouncil.) 如何為大型的步行機(jī)器人在供能不足的情況下選擇合適的速度軌道 摘 要 在操作步行機(jī)器人時，只有讓所有的油缸協(xié)調(diào)動作，才能使之安全、穩(wěn)定的行走。本文所研究的這個大型液力驅(qū)動步行機(jī)器人ALDURO并沒有外部力量的供給。因此，在機(jī)器人平臺上的液壓裝置及其柴油引擎的尺寸大小都將被它們自身的重量所限制。當(dāng)各個油缸以高速運動時，液壓力的供給就將不足，從而導(dǎo)致機(jī)器人的腳步及其整個平臺的移動軌跡都將變成不可預(yù)知的。當(dāng)此ALDURO在接近其穩(wěn)定邊界的狀態(tài)運作就將導(dǎo)致它的不安定甚至是整個系統(tǒng)的瓦解，下面將討論的解決方案是用于發(fā)現(xiàn)這種油缸時位錯誤并由此在需要的時候降速。 1、介紹 本文中所研究的系統(tǒng)是像人一樣的可以用腳行走但又可以有輪子的機(jī)器人ALDURO。它是由一個其上有一間為操作人員所準(zhǔn)備的小屋的平臺和四只每只長為1.8米的腳所組成。估計它的重量是1200公斤。它可以被用作是像四足動物一樣行走的步行機(jī)器（如圖2），而且當(dāng)將其后面的兩只腳用輪子替換時，它又能被當(dāng)作是一種有腿有輪的，腿和輪相結(jié)合的交通工具（如圖1），現(xiàn)今的這種腿和輪相結(jié)合的步行機(jī)相對于那種全部是輪子的交通工具的穩(wěn)定性和速度而言，其優(yōu)勢在于它高度的機(jī)動性[2]。 Fig. 1: Combined Legged Fig. 2: Walking Machine and Wheeled Vehicle 當(dāng)在險峻及危險的地帶操作時，安全扮演了重要的角色。只要在移動中出現(xiàn)了一個錯誤，機(jī)器人都將可能變的不穩(wěn)定，因此，機(jī)器人的氣缸必須確保要完全地按計劃軌道動作。步行機(jī)器人的腿是籍由一個開啟了的液壓系統(tǒng)所提給的液力驅(qū)動的。正常地情況下，在設(shè)計這種液壓系統(tǒng)時，優(yōu)先考慮重量最大的部分。而對于這個步行機(jī)ALDURO，它每一處的比值都是被最優(yōu)化了的。 一旦同時發(fā)動了多個氣缸或氣缸的運動速度非?？鞎r，由液壓所供給的流量就將變的不足，從而導(dǎo)致機(jī)器人的運動變得無法控制。下面所討論到的解決方案是關(guān)于計算軌道上速度如何減少的，機(jī)器人要隨時都可以穩(wěn)定的停下來。籍由發(fā)現(xiàn)那些氣缸的時位錯誤，并由此作出在軌道上減速或者增速的決定，運動同時被觸發(fā)，所有計算所依賴的采樣頻率將被降低，如此我們便可以保證機(jī)器人的腿以及它的平臺能且確實隨著預(yù)定的軌道運行。這個策略可以被應(yīng)用于ALDURO的虛擬模型的測試中并且正在實驗室中被應(yīng)用。 Fig. 3: Experimental setup 2、實驗的裝備 ALDURO的腿是擬人的，也就是說它是以幾何學(xué)和人類的腿的功能為基礎(chǔ)的。它的股關(guān)節(jié)是一個自由度為三的圓球形的球接頭，它的膝部是一個只有一個自由度的渦形的接頭。這些接頭的動作都是由液壓缸驅(qū)動的（如圖4）。然而，它腳上所附加的兩個自由度對它而言，作用卻是消極的，為了使得到一套明確的翻轉(zhuǎn)運動學(xué)的方案變成可能，我們鎖上了股關(guān)節(jié)處的一個液壓缸。這個直接的翻轉(zhuǎn)運動學(xué)的明確的方案詳細(xì)的顯示在[4]上。 為了調(diào)查腿的動態(tài)行為，并測試不同的控制方案，一種專為一只完全按規(guī)定尺寸制作的腿而設(shè)計的實驗裝置發(fā)展，并被制造出來了（如圖3）。它包括所有將在實驗室中被用于第一臺ALDURO原型的液壓裝置及由液壓力穩(wěn)定驅(qū)動的裝置。這個實驗裝置上由一個擁有一個15kw的電動機(jī)和一個流量為40l/min的柱塞泵的液壓系統(tǒng)所構(gòu)成。這比起真的系統(tǒng)中所使用的27kw的柴油引擎可要小的多了。為了測試液壓流的不夠，另外的四個液壓缸（被看做是一只腳）被安在試驗支架的邊上。 一個用來實現(xiàn)測試支架傳感器內(nèi)容和控制計算機(jī)（有一個即時操作系統(tǒng)）數(shù)據(jù)轉(zhuǎn)換的光學(xué)線路也被安裝使用了。液壓缸就被當(dāng)作是包括了位置和壓力感應(yīng)器的輸入裝置使用。因此，這種動態(tài)的測試就可以在機(jī)械、液壓組件和電子控制系統(tǒng)三者互相結(jié)合的情況下被實行。 實驗裝置的臀部是安裝一對線形軸承上，因此，它在垂直方向上是可移動的。不管是像圖3那樣腳的組合，還是在腿和輪結(jié)合的系統(tǒng)中用輪子裝在兩只后腳上，當(dāng)步行運動進(jìn)行時在其腿上的測試都將被允許載入。在行走過程中那種搖搖擺擺著把臀不以下的腳抬起來的行動被停止下來。這第一次的測試很好的顯示了模擬人腿的機(jī)械裝置的靈活性，并且這里的光學(xué)總線系統(tǒng)也已經(jīng)被證明了是可靠的。 3、加速改編 3.1 不足的液壓流量 正如已經(jīng)提到的，在面板上那些能量供應(yīng)包因為自身的重量和尺寸的被限制而不得不保持底下。當(dāng)多的液壓缸同時運動或當(dāng)它們以高速運行時，液壓力是供給將會不足而且系統(tǒng)的壓力也會匱乏。這種情況產(chǎn)生的運動主要地依靠閥比例的大小和液壓缸的外部的負(fù)荷。它的腳和平臺的軌跡就將是雜亂無章的。當(dāng) ALDURO 接近系統(tǒng)的穩(wěn)定性的界限時這樣的行為能導(dǎo)致不穩(wěn)定甚至推翻原來穩(wěn)定的系統(tǒng)。同樣非常重要的安全問題是必須 防止壓力的崩潰 (人們將會受到傷害)。 為了要研究這種情況, 叁考在水平的狀態(tài)下在臀部下面1 m的腳平面上做一次圓形的運動(半徑 0.2 m,速度 0.5 m/s)。這一個軌道的形成要有三個開啟的液壓缸作用。而腳的一次典型的垂直步進(jìn)運動它旁邊的液壓缸 (圖 4 的 2 號) 幾乎是不動的。像三個液壓缸驅(qū)動腳的旋轉(zhuǎn)映射出的在空間中的旋轉(zhuǎn)的運動學(xué)為已經(jīng)在 C++設(shè)計環(huán)境中得到了發(fā)展[3]。 當(dāng)在試驗支架上進(jìn)行實驗時引起腳的軌道退化到一個封閉矩形時.(圖 5) 為了要增加液壓的消耗,液壓缸的組合就會被使用 (也就是腳上的液壓缸即四個固定組合在地面上的液壓缸). 正如我們能在圖 6 中所看到的那樣,2 號缸遠(yuǎn)遠(yuǎn)滯后于在它的正弦曲線。液壓缸的線性運動顯示出一個完全開啟著的定量閥?？衫玫囊簤毫髁棵黠@不夠。 3.2 軌道速度減少 無法控制其運動的問題可以從不同的幾個方面著手處理: l 改變泵和閥的尺寸。 l 通過詳細(xì)的模型及重新計算臨界的運動來判斷流量/壓力的不足從而以較 低的速度來運轉(zhuǎn)。 l 當(dāng)校對軌道時，通過壓力的下降和整個運轉(zhuǎn)的變慢來發(fā)現(xiàn)位置的錯誤。 第一種方式增加了液壓系統(tǒng)的重量，因此并不受歡迎。而第二種方式不受歡迎的原因是它對計算能力有所要求，而且液壓模型本身也并不很準(zhǔn)確。因此，在這里選擇了通過減慢運行速度來發(fā)現(xiàn)位置誤差的方法并且將它實現(xiàn)。 為了在一次必須的降速之后，不再重新計算所有的軌跡，軌跡計算所依賴的時間變量將被減緩。到此為止一個全新時間變量被引用。這個軌道時間或者說是采樣時間可表示成為即時時間和導(dǎo)致所有誤差產(chǎn)生的時間增量因素的函數(shù)，而所有軌跡都是的函數(shù)。 為了在實時操作系統(tǒng)上控制系統(tǒng)的運行，我們需要在和之間與它們有關(guān)的時間步長。 由于所有的軌跡都是的函數(shù)，因此它們會因為的變化而同時地減緩和加快 液壓缸位置誤差的一個函數(shù)被選擇作為液壓流量不足的指示器。矢量包含了每只腳上的液壓缸的位置作用點，而矢量則是標(biāo)準(zhǔn)的位置矢量，它們的區(qū)別是誤差不同。 fl 代表的是左前方, 而hr代表的是右后方等等。 在重量矩陣W 中，每只腿的四個液壓缸對腳的位置的影響都可以進(jìn)行調(diào)整。 我們把所有的液壓缸的誤差的平方和和液壓缸的最初量進(jìn)行加權(quán)平均并使其標(biāo)準(zhǔn)化，而且要使它在規(guī)格化的誤差范圍內(nèi)（極限和之間） 倘若和的和超出了它們的上界時，采樣頻率就將被減緩。而當(dāng)和的一個極限標(biāo)準(zhǔn)化時，它們的上界將是固定的，這種情況就是所述的第二種極限；而第三種極限（較低的極限）是用來決定何時該再一次加速用的。為了盡量避免在減速甚至是加速過程中的錯誤停機(jī)，如果為正，它也只有加速，即使這樣會讓誤差更大也一樣。 是的減速率而同樣地是的加速率。必須在區(qū)間。 3.3 結(jié)果 對于極限的價值，采樣時間的加速率和減速率和液壓缸的重量都是按經(jīng)驗選擇的。在胯關(guān)節(jié)處的液壓缸對腳的位置比在膝關(guān)節(jié)處的4號液壓缸有著更高的影響力。而它們對應(yīng)的重量比被調(diào)整為2:1。 作為系統(tǒng)必須對誤差產(chǎn)生非常快的反應(yīng), 因此是很高的。而為了避免太多振動，加速率?k mt,acc要比減速率?k mt,dec更低。 Fig. 7: Cylinder positions for speed adapted trajectory 在圖 7 中，在t=2.5 s 我們能看到液壓缸2的點值快速地升起 ，而且它不會被實際的系統(tǒng)跟隨。 產(chǎn)生的標(biāo)準(zhǔn)化誤差可在圖 8 中看到,在它超出e thr,2的極限時,由于減少采樣時間因素k mt而引起采樣時間t*的減短(如圖 9 所示.) 由于設(shè)定極限e thr和de thr，絕對誤差和誤差的變化的影響就可被調(diào)整。 后者是標(biāo)準(zhǔn)的時間信號，但它或許是比較吵雜的。 當(dāng)這里這個極限是置低時。另一種可能性是將會過濾掉那些可能會產(chǎn)生一條比較大的不變波段的信號。 4、結(jié)論 由于對駕駛步行機(jī)器 ALDURO 的操作員有很高的安全性要求，需要保證主動器所有的運動嚴(yán)格的按照控制器所計算的運行。所有主動器同時地或在高速的運行，那么液壓的補(bǔ)給能變成不足并且壓力也會崩潰。這里所描述的方法是避免在對液壓的流動情況檢定不足的基礎(chǔ)上觀察主動器位置誤差以在適當(dāng)?shù)臅r候降低所有運動的速度。通過執(zhí)行調(diào)整采樣時間來計算由測試而得的ALDURO腿上的軌道看起來是蠻有前途的。速度運動的實行徹底被改良。 5、工作前景 現(xiàn)在，機(jī)器人的腿是由一個相應(yīng)的被鎖上的液壓缸的轉(zhuǎn)動來控制的。為了能夠使用所有的液壓缸且減少追蹤的誤差，那種基于壓力和模型基礎(chǔ)上的控制器將會這種控制器取代。如圖10所示，這種控制器將會減少的位置的殘余誤差。 附加在液壓油路上的壓力感應(yīng)器所發(fā)出的信號將會被包含在以液力驅(qū)動的補(bǔ)給評估中。 6、致謝 這項工作得到了Deutsche Forschungs gemeins -chaft DFG （德國研究小組）的充分支持 -２１-