Synthesis features of the space manipulator control system

A.P.Alpatov, P.A.Byelonozhko, P.P.Byelonozhko, S.V.Tarasov, A.A.Fokov

Institute of Technical Mechanics of the National Sciences Academy of Ukraine

and National Space Agency of Ukraine

Ukraine, 49600, Dnipropetrovs'k, st. Leshko-Popelya 15

While there appeared essentially new technical devices in a space robotics, expansion of orbital application area of manipulators is proceeding. Functional requirements both to devices, and to control systems become stricter. Perspective projects of orbital complexes, bases on the Moon and planets of Solar system, assume in addition to the traditional problems solved with use of manipulators (particularly moving of the loads, the special equipment, astronauts, etc. in a vicinity of orbital stations and the reusable transport spaceships), occurrence of new problems, for example, installation of space techniques on special robotized orbital platforms.

The proposed synthesis technique of an executive control system has been developed in research, that is devoted to the influence of manipulator elements non-rigidity of reusable spaceship on dynamics of system [1-11].

Control subject is a objects system (orbital spaceship (OSS) and payload (PL)), connected by six-linked anthropomorphous handling device (manipulator), formed by rotary kinematic pairs of the fifth class. In the assumption of an opportunity to neglect disturbingљ exposure (aerodynamic, gravitational, etc.) and at the switched - off stabilization system of angular position OSS, and the set entry conditions, mutual position of elements of system OSS-manipulator-PL is determined by action of the torques created in joints of the manipulator by mobility degrees drives. An input of an executive control system is programs of an angle change in joints as functions of time, an output is changes in time of joint angles.

The basic features of control object are:

- Mobility OSS (the basis of the manipulator) in inertial space;

- Small mass of the handling device in comparison with OSS and PL mass;

- Final rigidity of links and reducing unit;

- Low engine power of mobility degree drives.

At the initial stage the simplified design model of the handling device - two rigid bodies connected by the manipulator with rigid mass less links is considered. According to accepted assumptions the kinetic momentum and angular momentum of system OSS-manipulator-PL are constant:

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (1)

where љare OSS and PL mass accordingly; љare inertia tensor relatively to the center of OSS and PL mass accordingly; љare radius-vectors of the OSS and PL mass centers accordingly relatively to inertial basis with the beginning in the center of OSS-manipulator-PL system mass ; љare linear velocities of the OSS and PL weight centers accordingly relatively to the same basis; љare angular of OSS and PL velocities accordingly relatively to the same basis; љis the constant vector determined by initial position and initial OSS and PL velocities, љis a zero vector of corresponding dimension.

The manipulator root link of an examined design joints to OSS, further there follow shoulder and elbow links and three wrist links. The tip equipped link is motionless relatively to PL, i.e. the load is grasped without slippage. A vector of the generalized coordinates , where љis a transposing symbol which elements are mutual turned angles of kinematics' circuit adjacent parts in rolling shoulder joints, shoulder, elbow and wrist pitches, wrist yaws and rolling accordingly, unequivocally determines a configuration of the manipulator. In the accepted generalized coordinates the dynamic equation of OSS-manipulator-PL system are:

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (2)

where љare vectors, which elements are the first and second derivatives on time from elements of a vector љaccordingly; љand љare matrixes, which elements depend on elements of corresponding vectors; љis a vector of the torques created by drives in corresponding joints. Distinctive feature of system (2) is the account of the manipulator basis mobility without increase an order of the differential equations. For the known vector љunequivocally determined by starting conditions, the system (2) of six ordinary differential equations of the second order relatively to six independent generalized coordinates љcompletely describes movement of mechanical system OSS-manipulator-PL in inertial basis. Decrease the differential equations order of system within the framework of the accepted assumptions is achieved by use of the first integrals of movement (1).

Without restriction of a generality for the further reasoning, we put . For rather small capacity of joint drives angular moving velocity of kinematic circuit elements of OSS-manipulator-PL are insignificant, both their squares and mutual products at the first approximation can be neglected. Then the system of the equations (2) can be transformed:

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (3)

The equations (3) are initial for synthesis of an executive control system at the accepted simplified design model of the handling device.

Without stopping on determined by a problem specificity features of system reception of the handling device dynamics equations (2) and its transformations to a form (3), analyzed in [1-11], we consider further procedure of an executive control system synthesis on the basis of the concept of the allocated systems with use of a known "frozen" parameter method [12].

At movement of considered mechanical system on some set basic trajectory determined by known changing in time laws of generalized coordinates , elements of a matrix also are known functions of time. For all that changing in time laws of the control torques are supposed to provide the set movement. The noted low power of mobility degrees electromechanical drives is the use precondition of the "frozen" parameter method: matrix љelements are considered to change slowly in comparison with transient duration in drives. Then the system (3) can be described as stationary on separate time intervals - in a vicinity of corresponding points of a basic trajectory. The system (3) made linear in a vicinity of these points as balance positions has the form:

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (4)

where the matrix љelements are constant and are determined by the chosen point of a basic trajectory, i.e. some configuration of system OSS-manipulator-PL; љis a generalized coordinate deviation vector from the values corresponding to the considered point of a basic trajectory.

The equations (4) added with known [12] equations of drives on the basis of electric direct current motors, are describing control object system of the linear differential equations with constant factors. For synthesis of executive system well developed methods of the linear control theory can be used. However, reduction system of the equations(3) by the indicated way to the form (4) demands the aprioristic assignment of a basic trajectory, while in a considered case it involves difficulties over potential variety of movement modes. Therefore at a stage of regulator synthesis it was offered to choose a configuration of system OSS-manipulator-PL for definition of a matrix љnot as the fixed point of a basic trajectory, but with the regards for the below-mentioned reasons.

In the equations (4) describing movement in a vicinity of corresponding basic configuration, it is possible to give elements of a matrix љthe certain physical sense. Let the configuration of the manipulator changes for the - degrees of mobility, i.e. all joints except љone are motionless. Then the diagonal matrix element љmeans the inertia moment of the "hardened" parts of system OSS-manipulator-PL developed by the љjoint drive. This movement, obviously, is described by the equation . Rest - column elements of љmatrix determine value of the torques arising in motionless joints according to ratio . Synthesis of an executive control system for one joints at the motionless others corresponds to known [12] concepts of the allocated systems which application validity at the described stage of an examined problem solution is confirmed with further research results. Feedback parameters are determined independently for each joint; the analysis of joint cross influence through control object is made at the next stage of synthesis. Thus, without taking into account interference of joints, at the set parameters of drives and the fixed configuration of the handling device the control object is characterized by diagonal elements of a matrix . It is natural to choose from examined set as the least favorable according to - allocated system a configuration in which the element љis maximal. Then as the "average" configurations named further basic, it is expedient to choose such ones in which values of matrix љdiagonal elements the least differ from maximal, achievable in any other configurations from set examined. Let's designate љas the maximal values of the matrixes љdiagonal elements achievable on some set of examined configurations. At the initial stage two configurations are chosen as basic, for the first of which a condition љis satisfied for , and for the second one a condition љis satisfied for . The top index in brackets in a design-nation of a diagonal element љcorresponds to number of a configuration, љ(tab.1).

Table 1. Parameters of basic configurations

Configuration	x1	x2	x3	x4	x5	x6	Diagonal elements of matrix A
1	90њ	5њ	5њ	5њ	0њ	90њ	, ,
2	0њ	5њ	105њ	5њ	0њ	90њ	, ,

The researched manipulator is equipped with sensors of relative angular coordinates љand velocities , . As it was already mentioned, taking into account an unequivocal kinematic dependence of the parameters, describing mutual position of coordinate systems, connected with PL and OSS, from values of the accepted generalized coordinates, we suppose that inputs of the close-looped executive control system are preset angles change laws in joints , and outputs are values of the generalized coordinates . At the initial stage the influence of non-rigidity of the reducer elements is not taken into account. Thus, the executive control system represents set of the servo systems working on the general loading - the handling device, and it is multiply connected due to channel interference through control object. The block diagram of an executive control system for one of joints is submitted in figure 1, љ- is the differential operator, љis a reducer transfer number.

 

 

 

 

 

 

 

 

 

Fig. 1

 

Transfer factors of an angle and angular velocity sensor are designated as љand љaccordingly. In the considered constructive circuit the signal on angular velocity is got from a shaft of the engine, and a signal on an angle - from a reducer output shaft. To the system input there is given the control voltage , for simplification the joint number index record is omitted. Nonlinearity provides accordingly restriction of a control signal level and the torque size on a shaft of loading, љand љis regulator parameters. On fig. 1 the variant of the executive system is represented as much as possible simplified according to a described technique of stage-by-stage control synthesis. It's attractive from the point of view of technical simplicity and reliability as researches have shown, its realization provides comprehensible parameters of the manipulator controlled movement. For all that the result analysis of dynamics modelling allows offering effective variants of adjusting devices in future. Thus, at the initial stage the problem of control synthesis for each joint is reduced to a choice of feedback factors on an angle and angular velocity љand љaccordingly.

There has been established that the stability requirement of the close-looped allocated system determined above as an executive control system for the separate joint at the motionless others does not impose rigid restrictions on parameter values љand . The stability area on a plane of these parameters is open, and its position does not depend on corresponding value of a matrix љdiagonal element. In other words within the accepted mathematical description the allocated system stability does not depend on a configuration of the manipulator and mass-inertial parameters PL (parameters OSS is supposed set). Thus, at a choice of feedback parameters additional reasons can be considered. In particular, the variant of a choice of factor љwhich provides the torque component caused by own mistake of the sensor, which isn't more 10 % of the nominal engine torque is considered. The factor љchoice provides minimization of the maximal transient time value in all an examined range of PL mass change.

Stability of the synthesized close-looped control for each joint at the mobile others, i.e. with the regard of multiconnectivity influence of multivariate control system through the control object is confirmed with both frequency methods and direct integration.

Mathematical modelling of the manipulator controlled movement dynamics is carried out within the accepted simplified mathematical description of the handling device and mobility degrees drives with the purpose of confirmation of the concept correctness and the assumptions, based on an executive control system synthesis. Check of the synthesized control system serviceability is executed. The quantitative parameter estimation of transients and mutual influence of control channels is carried out. In figures 2-6 as an example modelling results of two controlled movement modes of system OSS-manipulator-PL are shown: execution of initial deviations from a basic configuration and a turn of the manipulator link under the set program. Calculated control object parameters with sufficient researching accuracy correspond to the data resulted in the technical literature [13-15] for manipulators of reusable transport systems and "Buran".

On fig. 2 work process by an executive control system of an initial deviation љon a shoulder yawing angle is represented from a basic configuration 1. Deviation change diagram on shoulder Dx₁ yawing angle, a wrist Dx₅ yawing angle and a wrist Dx₆ roll, corresponding to mobility degrees for maximal values of the matrix A diagonal elements in a configuration 1 are resulted (tab. 1).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

On fig. 3 executive process of an initial deviation љon an elbow pitch angle from a basic configuration 2 is represented. Deviations change diagram on a shoulder Dx₂ pitch angles, an elbow Dx₃ and a wrist Dx₄ pitch, corresponding to mobility degrees for maximal values of the matrix A diagonal elements in a configuration 2 are resulted (tab. 1).

On fig. 4 executive process of an initial deviation љon a elbow pitch angle from a basic configuration 2 similar to fig. 3 is represented but mass PL is reduced in 10 times. Both diagrams are shown for the same generalized coordinates corresponding to a configuration 2.

Similar results are received for other joint deviations. As it was expected, the strongest influence the deviation joint exerts on the joints working close to its planes. For example, the initial deviation on one of pitch joints causes the most essential "induced" deviations in the others <joints (fig. 2 and fig. 3). As a whole the maximal values of the "induced" deviations make 10-20 % of an initial "stimulating" joint deviation size. The regulator parameters are chosen so that mass PL reduction in 10 times does not change essentially time dependence character of joint mistakes.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The fig.5 and fig.6 illustrate work process by an executive cont-rol system of a pro-gram turn on one of mobility degrees. On fig. 5 program change function љof a joint elbow pitch angle on љand real coor-dinate x₃(t) change are represented. The gene-ralized coordinate values in the turn beginning correspond to a configuration 2. Program functions of other joints angles change are accepted , i=1,2,4,5,6, where љis constant value of the corres-ponding coordinate for a configuration 2 (tab. 1).

On fig. 6 diagrams of mistakes (deviations from constant program values) for a shoulder and wrist pitch joints are resulted.

Thus, at a initial syn-thesis stage of an executive control system:

-         the simplified design model of control object (the handling device and drive mobility degrees) is offered and the assumptions are accepted, allowing to lower the order of the dynamic equation system;

-         the concept of the basic configuration is formulated, allowing to use a method of the "frozen" parameters without the aprioristic definition of a basic trajectory;

-         the structure of an executive control system as set of servo systems on each joint is determined;

-         on the basis of the allocated system concept the feedback parameters are chosen;

-         an executive system stability with consideration of channel interrelation through the control object is confirmed by frequency methods and numerical integration;

-         the resulted diagrams has confirmed serviceability of the suggested executive system providing comprehensible quality of controlled movement dynamics;

-         the received results have formed a basis of the further researches of influence of the handling device final link rigidity and mobility degrees of reducer on dynamics of controlled manipulator movement.

Problems have been formulated:

-         more full account of link and reducer elastic properties getting importance first of all for control object features of dynamics;

-         the program movement synthesis providing the solution of manipulation problems and preferable from the requirements to controlled movement parameters (position accuracy, to transient characteristics);

-         the analysis of an executive control system, synthesis of adjusting devices;

-         the solution of these problems according to the suggested methodical approach is expedient for carrying out stage by stage.

As an example let's consider the executive system synthesized at a preliminary stage with reference to the manipulator with massless rigid links. And it will be used for problem of initial deviations in joints for the manipulator with massless non-rigid links.

In connection with that shoulder and elbow manipulator links are essentially longer than the others, the model (fig.7), taking into account elastic properties of these links is offered. Other links are considered as earlier not deformable. On fig.7 it is designated: , i=1,2,:,6, is the coordinate system connected with i-th the manipulator link, with the start in the beginning of a link, considering from OSS root joint, one of axes is combined with an axis of the joint, the second one is directed on a tangent to an link axis, the third one adds system up to right; љare the coordinate systems connected with elastic shoulder and elbow links with the beginning at the link end with axes, located similarly; l_i,љ i=1,2,:,6, are the vectors, connecting the start and the end of a corresponding link.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

As the joint coordinates (measured by sensor) in shoulder and wrist joint pitch act the angles between an undeformable link axis and a tangent to an axis deformable, in the elbow pitch joint act a angle between tangents to deformable link axes. As an additional used generalized coordinates describing elastic deformation of a shoulder can be spherical coordinates of an l₂ vector in љcoordinate system and the Euler angles determining angular system љposition relative to , or equivalent parameters are accepted. Elastic coordinates for an elbow are similarly inserted. Thus, the amount of the generalized coordinates arises from 6 in the design model with undeformable links up to 16 in the design model which is taking into account elastic shoulder and elbow deformations.

In the accepted generalized coordinates the system of OSS-manipulator-PL dynamics equation can be written down as:

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (5)

where љis a vector of the generalized coordinates, which first six elements are mutual turning angles of adjacent manipulator links (for elastic links they are determined above), and other ten are the elastic coordinates describing a shoulder and an elbow deformation, Q is a matrix of rigidity with constant elements, љis a vector, which first six elements are the torques created by joint drives and other elements are equal zero.

The system (5) linearized in a vicinity of a basic configuration (in which manipulator is suggested not to be deformed and the corresponding generalized coordinates are zero) looks like:

љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљљ (6)

where elements of a constant matrix љare determined by parameters of a basic configuration; љis a deviation vector of the generalized coordinates from the values corresponding to a basic configuration.

Consequence of an assumption about the mass less manipulator links is singularity of systems (5) and (6). The constant matrix љdegeneracy at the senior derivatives demands use special methods for integration (6) [5, 16].

Let's note, that the execution of an initial deviation on an elbow pitch angle (in the joint between its elastic deformable links) at motionless other joints for the set basic configuration corresponds to movement of system OSS-manipulator-PL represented by the simplified design model with two rigid bodies, connected by means of two joint connected elastic mass less links [3]. On fig. 8 the executive case of an initial deviation on 0,5^o in the pitch joint is resulted.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

On fig. 9 elastic angular coordinate changes љand љdescribing tangent deviation in joints to shoulder axis from a vector direction љare represented.

 

 

 

 

 

 

 

 

 

 

 

 

 

On fig. 10 changes љand of angular tangent deviations in joints to an elbow axis from a direction of a vector љshown.

It is necessary to note, that the executive control system synthesized for the manipulator with rigid links provides attenuation of the elastic fluctuations caused by an initial deviation from a basic configuration. The analysis of transients has formed a basis of the further researches on adjusting devices development. The given example illustrates the offered stage-by-stage approach to synthesis of an executive control system.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

As a result of all research complex hierarchical mathematical model set of the manipulating device having a various complexity level has been developed [1-7]. In addition to describe design model in which the link mass (not deformable and deformable) is taken into account were examined above. Combinations of system parameters at which use of the simplified design models is fair have been determined. On the basis of the developed methodical approaches the structure is offered and parameters of control algorithms providing comprehensible quality of controlled movement dynamics (including the indemnification of dynamic mistakes) are determined. Formation ways of program movements for angles in joints as functions of time are offered and realized in settlement variants. Estimations of influence on controlled movement dynamics of system element final rigidity (links of the handling device and drive reducers of mobility degrees) are made, recommendations on an expedient choice of design parameters are formulated.

The suggested approach to the an executive control system synthesis problem solution of the space manipulator, based on use of dynamic model set, can be recommended as practically effective analysis tool of the modern complex spatially advanced space objects. At dynamic research of their controlled movement there is a necessity of elastic properties account of design elements.

Particularly let's note, that at a choice stage of the design model it's expediently to take into account opportunities of modern computer modelling means of rigid bodies system dynamics of the considerably simplifying bulky procedure of an equation conclusion and equipped with well advanced means of visualization and the complex analysis of researched dynamic features process. For their effective use at the similar problems solution the original approach to representation is offered for elastic deformable elements by finite-dimensional design model [17].

 

References

(in Russian version of article)

 

 

 

Anatoly Petrovich Alpatov, Graduated from Kazan Aviation Institute, the Faculty of Aircraft. Doctor of Sciences (the second academic degree). Head of the Department of Systems Analysis and Control Problems of the Institute of Technical Mechanics of National Academy of Sciences of Ukraine and National Space Agency of Ukraine. Scientific interests: systems analysis, dynamics of controlled motion of complex space systems, dynamics of spacecraft with magnet control systems, movable control over mechanical systems.

Peter Alexeyevich Byelonozhko, Graduated from Dnipropetrovs'k State University, Physical and Technical Faculty. Candidate of Sciences (the first academic degree, comparable with PhD degree). Senior scientific worker of the Department of Systems Analysis and Control Problems of the Institute of Technical Mechanics of National Academy of Sciences of Ukraine and National Space Agency of Ukraine. Scientific interests: dynamics of large transformable mechanical systems, control over complex space systems.

Pavel Petrovich Byelonozhko, Graduated from Dnipropetrovs'k State University, Physical and Technical Faculty. Candidate of Sciences (the first academic degree, comparable with PhD degree). Senior scientific worker of the Department of Systems Analysis and Control Problems of the Institute of Technical Mechanics of National Academy of Sciences of Ukraine and National Space Agency of Ukraine. Scientific interests: dynamics of large transformable mechanical systems, control over complex space systems.

Sergey Vasilyevich Tarasov, Graduated from Dnipropetrovs'k State University, Physical and Technical Faculty. Candidate of Sciences (the first academic degree, comparable with PhD degree). Leading research worker of the Institute of Transport Systems and Technologies of National Academy of Sciences of Ukraine. Scientific interests: dynamics of large transformable mechanical systems, control over complex space systems.

Alexander Anatolyevich Fokov, Graduated from Dnipropetrovs'k State University, Physical and Technical Faculty. Candidate of Sciences (the first academic degree, comparable with PhD degree). Senior scientific worker of the Department of Systems Analysis and Control Problems of the Institute of Technical Mechanics of National Academy of Sciences of Ukraine and National Space Agency of Ukraine. Scientific interests: dynamics of large transformable mechanical systems, control over complex space systems.

 

 

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