Robot kollarda optimum hareket sentezi

Abstract

ÖZET ROBOT KOLLARDA OPTİMUM HAREKET SENTEZİ Bu çalışmada robot kolların noktadan noktaya hareketinde hareketin op timizasyonu problemi ele alınmaktadır. Optimum robot hareketinden beklenmesi gereken niteliklerin,, düşük enerji maliyeti, çabukluk ve yumuşaklık olduğu belirlemesinden yola çıkılarak, bu nitelikleri taşıyan hareketin bulunmasına yönelik yeni bir optimizes yon yak laşımı önerilmektedir. Ya tek başına hareket süresini, ya tek başına enerji har camasını, ya da bunların ağırlıklı ortalamasını minimize etmeye yönelen gele neksel yaklaşımlardan farklı olarak bu yaklaşımda optimum robot hareketi, enerji harcamasını minimize eden hareketlerden (ki verilen her hareket süresine böyle ha re keti erden far klı bir tanesi karşılık gelmektedir) servomotor kapasiteleri sının içinde en hızlı olanı şeklinde tanımlanmakta, bu yoldan bulu nan hareket, hem enerji harcaması hem hareket süresi bakımından `iyi` olduğu kadar enerji bakımından optimum hareketlerin yapısal bir özelliği olan yumuşaklık özelliğine de kendi liginden sahip olmaktadır. Aranan optimum hareketin bulunmasında, problemi bir parametre op timizasyonu problemine dönüştüren bir kabulle, robotun her bir mafsalı ndaki hareket, T toplam hareket süresini göstermek üzere T=t/T şeklinde tanımlanan bir boyutsuz zamanın polinomlarıyla temsil edilmekte ve problem, bu polinom ların bilinmeyen katsayılarının hareketi gerçekleştirmek için gereken toplam enerji harcamasını minimize edecek şekilde, T ha re ket s üresini n işe servo motor kapasitelerinin elverdiği en kısa süre olacak şekilde hesaplanmasıyla çözülmektedir. Çok yönlü bir o pti mi zas yon sağlaması ve hesap tekniği bakımından büyük güçlükler a rzet memesi gibi belirgin avantajlara sahip olan yöntem, robot kol ların `off-line` optimum hareket planlamasında geleneksel yöntemler karşısında güçlü bir seçenek olarak rahatlıkla kullanılabilecektir. Çalışma beş ana bölüm halinde düşünülmüş, bunlardan 1. BÖLÜM konuya ilişkin kavramların tanıtılmasına ve önceki çalışmaların eleştirel bir yakla şımla gözden geçirilerek bu çalışmanın yerinin belirlenmesine ayrılmıştır. 2. BÖLÜM'de robot kinematik ve dinamik hesaplarının esasları, 3. BÖLÜM'de ise robot kollarda optimum hareket sentezi probleminin genel ve soyut bir for mü lasyonu verilmiştir. 4. BÖLÜM'de, sonuncusu bu çalışmanın ana amacını oluştu ran ve yukarıda tanımlanan optimizasyon problemi olmak üzere üç ayrı optimi zes yon probleminin tanım ve çözümleri örneklerle sunulmuştur. Bu bölümde 11 k problem olarak ele alınan `verilmiş hareket süresi için en az enerji harcayan hareketin bulunması` problemi, ana probleme esas oluşturduğundan ayrıntılı olarak incelenmiştir. Son olarak 5. BÖLÜM nihai değerlendirmelere ayrılmıştır.

SUMMARY OPTIMAL TRAJECTORY SYNTHESIS FOR MANIPULATION ROBOTS Studies on optimal trajectory synthesis of manipulation robots can be classified into tvo major groups. To the first group which can be referred to as velocity distribution optimization, belong the studies that assume a predeter mined geometric path for the end-effector (often in parametric form) and that try to find the best velocity profile in the time-velocity plan with regard to 3ome optimization criterion [18-29]. This approach reduces the dimension of the op timization problem and also allows the physical constraints such as collision avoidance conditions or joint mobility limitations to be treated in the geometric path planning stage, simplifying thus considerably the optimization task. But in exchange the overall optimization of the trajectory is less or more sacrificed, because the geometric path itself -when it is not closely dictated by the nature of the manipulation task or the configuration of the workspace- is the most effec tive parameter in the optimization. The second group which can be referred to as point-to-point optimal trajectory synthesis, includes the studies that try to find the optimal time stories of the generalized coordinates of the manipulator or the time sequences of the optimal controls, freely of any prescribed geometric path [17], [30-38]. This approach is obviously superior to the first one in getting true optimal solutions and as such will prove to be more adequate for a large class of optimal trajectory synthesis problems a3 soon a3 some difficulties asso ciated with it be removed. One of the most importants of them is the difficulty one faces in incorporating an obstacle avoidance scheme into the optimization procedure. A first attempt towards the solution of this problem was made by Gil bert and Johnson [33], who introduced the use of distance functions in the for mulation of collision avoidance conditions and implemented them -in an approxi mate sense- by means of interior penalty functions. Whenever an optimization problem is considered, the most natural ques tion arises of selecting a suitable performance criterion. Besides a few ex ceptions, studies on optimal trajectory synthesis of manipulation robots seem to be concentrated on the two following criteria: The most widespreadly used cri terion of minimal motion time, and the minimal energy consumption criterion. These criteria are both of practical importance, because the first is justified by the goal of increasing the productivity, and the second by that of reducing the production cost. However, the individual importance of both criteria, when con sidered in the light of the fact that they lead to radically different solutions, sug gests that any optimization procedure which makes exclusive use of one of them and ignores the other risk3 to become self-defeating. This is especially true for the time-optimal motions because the energy consumption grows higher and higher with the rise of motion speed. A criterion which aims true rational use of the manipulator should therefore incorporate a compromise between the two criteria. viAnother demand -of technological character- from an optimal control is its smoothness. In order to be readily achievable by the actuators, to not cause excessive wear in the mechanisms and to not excite elastic vibrations in the manipulator system, the optimal control must be free of abrupt changes. The time optimization which Is known to lead to bang- bang type controls 1s not suited to this and, while energy optimization which is the most natural optimization for dynamic motion leads to very smooth controls. Kim and Shin [21] and Pfeiffer and Johanni [26] suggested the use of some combined criteria and pointed out the question of smooth control. The for mers used a weighted minimum time-torques criterion in constructing asubop- timal controller for industrial manipulators. Their approach resulted in bang- off- bang controls, as a remedy of which they proposed to switch to a PI control ler in the vicinity of the target point where the elastic vibrations are especially undesired. The latters used a weighted time -squared hand velocity-torques cri terion and deduced that by proper weighting of the torques a smoothing effect on the resulting controls can be achieved. Their illustrative example -though rela tively smooth- still exhibits abrupt changes In the Input torques. Finally, Shin and McKay [24] used a weighted time-energy criterion in a solution by dynamic programming approach. As discussed above, this criterion is highly judicious. However the result was a saw-shaped control function which does not seem to be suited for practical application. This study considers the point-to-point optimal trajectory synthesis problem. An optimization criterion is proposed which both takes adventage of the smoothness of the energy-optimal motions and combines energy and time optimi zations. The analytical task Is defined as: Given an Initial and a target state for the manipulator, find the set of energy-optimal joint trajectories which would give the shortest possible motion time under actuator constraints. The problem is formulated asa two-point boundary value problem and a direct method which makes use of polynomials of a dimensionless time as approximations to the opti mal trajectories, is applied to obtain the solution. This allows the energy con sumed by the actuators in moving the manipulator from the initial to the target state, to be expressed in terms of the coefficients of the polynomials and the traveling time. The coefficients are then selected so as to minimize the consumed energy while the traveling time is selected to be the shortest possible under ac tuator constraints. The procedure gives the speediest possible energy-optimal motion which, as well, has the intrinsic smoothness property of energy-optimal motions. The obstacle avoidance problem is not considered in this study. But this problem can to some extent be covered by the present method by imposing the polynomial trajectories to pass through a number of intermediate points. The approach Is illustrated by means of a 3 degrees-of- freedom anthro pomorphic manipulator driven by DC electric servomotors. The features of the energy-optimal motions which play a crucial role in this approach are also ex amined. * / # The dynamic behaviour of an n degrees-of -freedom manipulator can be viimodeled by n differential equations of the form where T represents the traveling time, and primes denote differentiations with respect to t ; a dimensionless ti me defi ned as T On the other hand, the operation of the low inductance DC electric servo motors which are widespread! y used in robotics practice and which will be con sidered here as actuators can be described as u(t) = Ri(t) + KeN6(t) (3) and w- «2. (4) ' KtN where u=in put voltage, i,R=armature current and resistance, Ke,Kt=voltage and torque constants of the motor, N= reduction ratio of the reductor, and P=total torque produced by the motor. It follows from Eqs. (3) and (4) that the total electrical energy con- s u med i n n se rvo moto r s ca n be ex p r essed as i=lJ° *i T.n2`-.K«i TI?<T/D + -^ei(DFKT,T) KfN2 `` K, dl <5> where P,'s must be substituted from Eq. ( 1 ). Equation 5 constitutes the perfor mance index for energy- optimal motions. In order to transform the infinite dimensional optimization problem into a finite dimensional one, the joint trajectories are approximated by polynomials of the dl menslonless ti me X: efoij,T)= 2 aijTJ-1 (6) where m and p are the numbers of boundary conditions at the starting and st oping poi nts, and each trajectory is given k degrees of freedom that will be used for op- ti mi zati on purposes, k may be regarded as the order of approximation to the op timal trajectory. It is clear that, once the approximation (6) is adopted, any viiiderivative of the joint variable 6j becomes a function of the coefficients ay and the dimensionless time t. As a result, the torques Pj of Eq. (1) become Pj (apq/r,T), and the total consumed energy becomes Et0t(apq,T). In the light of the approximation (6), the m+p boundary condition equa tions give equations for the coefficients 8jj, say fis(aij> = 0, s=l,2,...,m + p (7) As constraints of the servomotors, input voltage and heating limitations are con sidered. These can be expressed by the inequalities ^.(aij/I^Utt. (8) and ^.(a^T)*^. (9) where umj and Umj are, respectively, the peak value (for the motion in question) and the maximum available value of the input voltage, and in^ and Imj are, the actual and the maximum allowable values of the root- mean-square armature current (which is the best measure of the armature heating). At this stage, our problem can be restated as follows: Find the joint trajectories (6) which satisfy the boundary conditions (7), minimize the cost functional (5) and give the shortest possible traveling time T under servomotor constraints (8) and (9). In order to determine the m+p+k unknown coefficients of each polyno mial approximation function given by Eq. (6), in addition to the m+p boundary condition equations (7), we write k minimisation conditions of the total con sumed energy given by Eq. (5), as altoca*,T) I J - = 0, r=l,2,...,n, s = m + p + l,...,m + p+k (10) dars where differentiations with respect to the extra coefficients ars must be calcu lated by using the chain rule and noting that the rest of the coefficients depend on ars's through Eqs.(7). The Eqs. (7) and (10) constitute a system of algebraic equations which enables us to solve the minimum energy trajectories problem. But it should be noted that this solution is function of the traveling time T. Now, recalling our statement of the problem, T is not an arbitrary input but a para meter to be minimized. Hence, the above mentioned solution is not a final solu tion but the locus of the admissible solutions. Our next goal will be to find among all the solutions lying on that locus, the one which gives the shortest T under the servomotor constraints. To do this, let us consider the equations of motion ( 1 ) of ixthe system. By inspection of these equations we can easily conclude that if the traveling time is Infinitely long, no dynamic torque acts on the system and the input torque merely equals the position depending torques. Then suppose that T is gradually shortened. The dynamic torques would appear, become dominant below a certain level of T and go to infinity with T going to zero. In the region where the dynamic torques are dominant, the relation between T and the torque requirement of the system is evident: The shorter is T, the larger is P. Thus, in this region the torque limits and consequently the armature current and supply voltage lim its of the actuators will define the shortest possible traveling time. We may therefore get the minimum time motion as the motion during which at least one of the actuators reaches its limit. Now recalling the servomotor constraint equa tions (8) and (9), the condition for at least one of the actuators reaching one of its limits can be expressed, after some manipulation as sup -1 = 0 (10 Equation (11) which is a single equation with 2n alternatives enables us to find the shortest time motion among the above described locus of minimum energy motions. Thus, that equation together with Eqs. (7) and ( 1 0) constitute a system of n(m+p+k)+1 equations for the unknowns of our problem which are n(m+p+k) coefficients of Eqs. (6) and the traveling time T. It should how ever be noted that the solution of this system requires a somewhat complicated numerical scheme. A special source of difficulty is the Eq.( 1 1 ) which, unless the trajectories are already known, remains undefined. This difficulty can be overcame if an iterative computation method is adopted. The function st/p can than be evaluated at each step of the iteration and the Eq.( 1 1 ) becomes defined for each step. A special computer program has been developed to solve this system (see Annex 1). The above described method is applied to a 3-degrees-of-freedom anthropomorphic manipulator, a schematic of which is shown in Fig. 1. In order to get better Insight into the nature of the problem, a prelimi nary study was carried out on the minimal energy motions and their relation to the traveling time, by solving the energy-optimal trajectories problem for dif ferent T values. Figure 2 shows the results of this study. On that figure, the to tal consumed energies of the minimal energy motions are plotted versus T. The solid lines represent the loci of the first approximation results, the dotted line represents the locus of the non-optimized results, and the dashed line represents the locus of the second approximation results. The general characters of the mo tions corresponding to the three different first approximation loci are also shown on the figure by means of two dimensional sketches illustrating the motions of the second and third arm in the vertical plane rotating with the first arm. Some in teresting features of the minimal energy motions, resulting from Fig. 2 end con firmed by several other case studies can be summarized as follows: a- The energy-optimal motions problem of anthropomorphic manipula tors is a multl -solution problem for long traveling times, but a unique-solution problem for short traveling times. The latter property is useful when the prob lem of finding the speediest possible energy-optimal motion is to be considered.b- While the motions pertaining to other loci are meandring, the motions pertaining to the unique locus of the shorter traveling times region (11 ne A1 ) are - as could be expected- rather direct and therefore suitable for practical appli cation. c- Each locus displays a minimum with respect to T. One of them con stitutes the absolute minimal energy motion for the case study. Further studies shoved that the position depending torques are responsi ble of both the occurence of a multi -solution region and the existance of a mini mum with respect to T. The absolute energy optimal motions are also calculated by using the fol lowing additional equation: aEtot(ajpT)_0 (12) dT where E/0/ must be substituted from Eq. 5. To examine the effectiveness of the energy optimization procedure and that of its different approximation solutions, the unoptlmized motions and the second and third approximations were also considered along the first approxima tion solutions. It was found that for the case study of Fig. 2 and for practically meaningfull T values, the optimization secures an energy saving of about ]0% for the first approximation and about 2Ü& for the second approximation, while the third approximation ha3 only minor contribution. Finally, the main problem of finding the speediest possible minimum energy motion under servomotor constraints is considered. The problem is solved for several sets of values for lmj and Uhj. An inspection of different as pects of the synthesized optimal motions, proved that the proposed optimization approach secures a very smooth motion as predicted. For summing it up: A new approach has been presented for the point-to- point optimal trajectory synthesis of manipulation robots. A well known prop erty of energy-optimal motions of manipulators is that, to every given traveling time corresponds a different energy-optimal motion. The basic idea of the ap proach was to construct speediest possible energy-optimal motions by utilizing that property. Although the approach does not merit the name of combined ener gy-time optimization in the true sense of the concept, it constitutes a powerful alternative to the conventional way of using a weighted criterion in that: a) It guarantees a smooth motion, and b) A weighted energy-time criterion is not readily applicable with any optimization method. Dynamic programming seems to be the only method which can treat such a criterion without trouble. But a dy namic programming approach would suffer of high dimensionality in the case of point-to-point trajectory synthesis. The use of some approximation functions for the joint trajectories was indispensable In order to transform the problem into a finite dimension para meter optimization problem. Polynomials of the dimensionless time have been selected for their easy adaptation to different and arbitrarily high numbers of xiboundary conditions. Possible substitutes are trigonometric functions and spline functions The proposed method can easily be used for the off-line nominal motion synthesis of manipulation robots. xii