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Quasi-exact linear spring countergravity system for robotic manipulators
Mech. Mach. Theo O" Vol. 33, No. 1/2, pp. 59-70, 1998 ~, 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain P l h S0094-114X(97)00036-0 0094-114X/98 $19.00 + 0.00
Pergamon
QUASI-EXACT
LINEAR
SPRING COUNTERGRAVITY
SYSTEM FOR ROBOTIC MANIPULATORS
J. L. PONS, R. CERES and A. R. JIMI~NEZ lnstituto de Automfitica Industrial, Consejo Superior de Investigaciones Cientificas, Ctra. Campo Real, Km. 0.200 La Poveda, 28500 Arganda del Rey, Madrid, Spain (Received 18 November 1996)
Abstract--A new quasi-exact linear spring counterbalancing system for general manipulators is
theoretically stated and experimentally tested on a parallelogram manipulator. The effect of configuration parameters on developed balancing torque is studied, which gives insight into possible active application of the presented configuration. Quasi-exact balancing of gravitational torque is achieved without any assumption on initial spring length thus avoiding classical shortcomings arising from perfect spring equilibrator theory. © 1998 Elsevier Science Ltd
1. I N T R O D U C T I O N .
REVIEW ON GRAVITY BALANCING TECHNIQUES
The total torque needed to drive a manipulator for a given motion consists of dynamic and gravity-induced terms. Dynamic terms are non-linearly dependent on position, velocity and acceleration, and as a result of them, there is a coupling effect among different joints [1]. Coupling and non-linear effects result in a more complex control requirement since the action of the control law on a given shaft also affects the motion in all the other joints. Nevertheless, as it is well known, [2], the relative importance of these dynamic terms as compared to gravity-induced ones is low, and if no counterbalancing is provided, motor-transmission units have to be selected to cope with this high torque. Additional problems related to gravity terms come out of the need of safety systems which preserve the structure against collapsing in case of power failure. Of course, the design of these safety systems depends on the relative size of gravity terms, thus again a counterbalancing is useful. Even when a specific design of counterbalancing systems could be done for any given manipulator, it is obvious that the best approach should take into account the selection of the appropriate manipulator structure, the kind of work that the manipulator is going to perform and the selection of the counterbalancing system. As far as the selection of structure is concerned, in [3], the authors showed how the selection of an appropriate structure enables the counterbalancing for all the joints of the manipulator. In this case, the choice of an inverse parallelogram structure made it possible to use spring compensation on both joints of the manipulator, while the selection of a classical parallelogram structure would have made it necessary to use mass counterbalancing with the corresponding increase in the overall weight of the structure. On the other hand, the kind of work to be performed affects the choice of counterbalancing technique. Gravity compensation systems can be further classified in passive and active systems. Since the manipulator will, in general, work with different payloads, a passive compensation system will only work well when, during operation, light changes in payload are expected. For instance, if the manipulator is scheduled to perform pick-and-place motion, large variation in payload will occur and an active gravity counterbalancing system has to be designed. The most common passive systems are mass and spring counterbalancing. Mass compensation is basically achieved by moving the centre of mass of the link which is going to be balanced by adding mass until the final position falls on the shaft. This kind of compensation, which is the one used in the forearm of the Kuka 160, has been widely used in jointed manipulators and has the 59
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J.L. Pons et al.
advantage of linearizing the equations of motion of the structure. The main disadvantage comes from its passive nature and from the difficult exact adjustment of the counterbalancing weight, besides inertial characteristics are increased thus reducing dynamic behaviour. Since compensation is only achieved for a given payload the structure has to be equipped with safety brakes. Several works, [4], address the general problem of determining the minimum number of counterweights necessary for exact force balancing. To better understand the possibilities of spring gravity counterbalancing we note that the general form of gravity torque is derived from gravitational potential energy as given in equation (1), where ae, be and ~ are structure dependent constants and 0i defines the position of a given link. In spring counterbalancing systems, springs are used to produce a counteracting torque which is a function of the position, 0. When torsion springs are used, the counteracting torque results to be linear with the link position. Therefore, when the spring rate is properly chosen, good fit between gravity and counterbalancing torque is achieved for the first angle range. However, since gravity torque will continuously reduce its slope relative to the counteracting torque, we end up with a practical limitation to the workspace of the manipulator, [3]. Vg = ~ (ae sin 0e + be cos 0i) + ~
(1)
/=1
Linear springs have been widely applied in gravity balancing systems. Linear springs usually act through a lever thus producing a counterbalancing torque shape which depends on specific configuration. In [5], even when balancing torque shape does not fit gravity torque shape, an overall reduction to approx. 30% the initial gravity terms is achieved. In order to get a better fit between counteracting and gravity torque some special mechanisms have been designed based on linear springs. As shown in [6], theory of perfect spring equilibrators seeks for optimum parameters for linear counteracting springs, in such a way that elastic potential energy exactly matches gravitational energy, and thus several configurations have been suggested. One of them, known as sine mechanism, is presented in [2], and allows, at least theoretically, to shape the balancing torque to a sine function which exactly equals the gravity terms. However a general practical shortcoming for this mechanism arises from the need of obtaining a practical initial length of the spring equal to zero. As pointed out in [7], this could be achieved by twisting the wire while the spring is being coiled. A general advantage of spring balancing is that no safety systems are required since there is always an equilibrium position. Up to now, only passive techniques have been discussed and all of them have the shortcoming of its inability to cope with varying payloads. Active mechanisms use external power sources to produce a counteracting torque which can be easily made dependent on external loading, i.e. changing payloads. Both electrical and pneumo-hydraulic active mechanisms are used. in industrial manipulators. This imposes the use of additional power sources as in the case of electrically driven robots with pneumo-hydraulic gravity compensators. Moreover, when using electric torque motors in the balancing systems, it is necessary again to provide a safety system to cope with power failure. Throughout this work, we will present a new gravity balancing configuration based on linear springs. It makes no assumption about initial length of linear springs therefore it needs no special manufacturing process. The specific choice of mechanism configuration allows to shape in a practical way the balancing torque of the springs for a wide position range of the manipulator. Since spring compensation is used, no safety systems are required. Moreover, the specific configuration allows us, by modifying its parameters, firstly to adjust precisely the balancing torque to our manipulator, and secondly to use it as an active balancing system. 2. ANALYSIS OF THE NEW BALANCING STRATEGY The new configuration for our linear spring countergravity system is based on a linear spring acting through a lever to produce the balancing torque. In order to better understand the fundamentals of this system we first analyse the background on theory of perfect spring equilibrators. Then, we introduce the basic configuration schematically shown in Fig. 2 to see how we can use it to construct our counterbalancing system. Throughout this section we obtain the basic
Quasi-exact linear spring countergravitysystem
61
equations governing the system. These equations show us four basic parameters that can be used to precisely adjust and tune our configuration for a given manipulator, and direct us to the possibility of actively compensating for varying payloads. 2. I. Background on spring equilibrators Here, we recall equation (1) defining gravitational potential energy for a given manipulator as a function of structure dependent constants and generalized co-ordinates. Perfect countergravity balancing might be reached only if exact match between elastic potential and gravity one is achieved. In order to do so, dependence of spring potential on position should be through sine and cosine functions: Vk = ~ (c, sin 0r + dr cos 0t) + fl
(2)
i=1
For spring force to be dependent on link position, it has to be attached either between link and fixed points or between consecutive links. Regarding Fig. 1 showing typical configuration for spring equilibrator, we write the expression of spring length as a function of link position: '~ = x / g + ~ - 2t, t2 cos(~ - 0) - ,~0
(3)
Perfect equilibrator theory assumes initial spring length equal to zero, 60 = 0, thus potential energy can be written as: 1
1
V~ = ~/£62 = ~ (K, + K: cos 0)
(4)
Now, by properly choosing spring configuration and rate, exact static matching between both energies can be achieved. A practical shortcoming comes out of the need of obtaining practical zero initial length for the springs as pointed in [2]. 2.2. Bas& configuration The basic configuration of a linear spring used in this new quasi-exact gravity balancing system is shown schematically in Fig. 2. The lever that is used to create the balancing torque has a length r~r while the spring whose initial length is 10i is mounted between the lever end and a fixed point, O,. All the forces that the spring creates as the link position varies, 0j, are represented in the figure. The spring force, Fsr, acting through the lever, creates counterbalancing torque Tsr, which in turn compensates for the gravity term, Tg. It is important to note that for a given link position, it is
11
Fig. I. Basic configurationof a linear spring equilibrator.
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J . L . Pons et al.
•
]~Link j
',
loi + ~xi
F,i ' ~ X y
IN
Fig. 2. Basic configuration of a single counterbalancing spring.
possible either to increase or decrease the spring length by changing lever angel ~, we will use this property later on to shape the balancing torque. In all the equations subscript i refers to the spring number, (if a set of linear springs is used), while subscript j does to the link whose gravity effects are being compensated. In order to see how we can shape the balancing torque we will derive the expression for this torque as a function of all the parameters given in the figure plus the spring rate, K~. To do this, we first note that the balancing torque depends only on the spring force vector, F~i and lever vector r~, and can be calculated as their inner product accordingly to the coordinate frame shown in the figure: Ts, = Fs~ x rsi =
/~i,
J k F~¢ 0
rsi,-
rsl,
(5)
0
For this configuration, we can work out the expression defining the lever vector, equation (6), and the vector in the direction of the force created by the spring, equation (7). r~ = rs~ (cos 0r + ai)i - r~ sin (0r + ~)j
(6)
vr~, = [r,,(cos(0j + ~,) - 1) - (10i+ 6x)]i - r,, sin(0j + ~,)j
(7)
In order to reach the complete expression for the force in the spring we need to relate its deformation, 6s~ to all the parameters defining the configuration: lever length, rs~, lever angle, ~, and initial deformation, fix,. With this deformation we express the force in the spring as the product of deformation and spring rate in the direction given by vector vr~,: 6s, = xfr~, + (lo, + fix, + r,,) 2 - 2r,,(lo, + 6,~ + r~,)cos(0j + a,) - (lo, + fix,)
(8)
and finally the expression for the force in the spring:
YF~
F~ = g~c5~, vr~,l =Ks,[~r~i + (loi + fix, + r~) z - 2r,,(10, + 6x~+ rs,)cos(0~ + ¢,) -(lo, + 6,,)] ×
[r,,(cos(O/+ ~,) - 1) - (10,+ 6x)]i - r~ sin(0j + ~,)j ~f[rs,(cos(0r + ~,) - 1) - (10,+ 6x)] 2 + [r~ sin(0r + ~,)]2
(9)
Quasi-exact linear spring countergravity system
63
According to equation (5) we can now calculate the total counterbalancing torque that a set of compensating linear springs will create on link j. In order to do this, we just sum up a series of terms of the form of equation (5), which is the partial torque created by every spring. For simplicity, we just express in equation (10) how to sum up the effect of every spring in the total balancing torque on link j, Tsj; the final expression can be worked out substituting equations (6), (8) and (9) into equation (10) Fsi
Ts~ =
~
x
rsi&,Ks,
(10)
i=1
Potential elastic energy for this case, in which no assumption has been made regarding initial length of the springs, results:
'L •=, K~[G + (rs~+ loe+ 6,.,)2 _ 2r~i(r~, + lo~ + 6,.~)cos(n - 0j) + 66,
-- 260,x/r~i + (r~, + 10, + 6,,.)2 -- 2rs,(r~i + lo, + &,)cos(n -- 0,)]
(11)
Equation (10) gives us the total counterbalancing torque of a set of linear springs. Every term in the summation depends on particular configuration of every single compensating spring. This configuration is defined by four parameters which we can use to shape every single balancing torque, and as a result, the total torque on shaft j, Ts/. Our interest focuses now on how to get this, this is to say, what particular value of every parameter will give us the right compensation torque in the entire manipulator range. In the next section we study how to alter the final torque by changing the configuration parameters. 2.3. Effect o f configuration parameters
In the last section we worked out the complete balancing function for a single compensating linear spring and pointed to the expression of the total torque created by a set of linear springs. Equation (10) clearly shows that parameter Ks, modifies the final torque as a constant that multiplies a basic compensating function. This suggests to us to use the spring rate to fit the compensating torque to every particular manipulator provided a good shaping of the compensating function. In order to see this, in Fig. 3, we represent different counterbalancing functions as generated with different values of the spring rate. A more interesting point of view can be obtained from the study of the influence of the other parameters ~, r~ and 6,,. on the balancing torque of a single linear spring. Unlike Ksi, the other parameters are comprised in the expression of Fsi and rs~, thus not only altering the magnitude of the final torque but also modifying the value of their inner product as given in equation (5). The effect of changing the value of :¢i can be better understood by looking at Fig. 2. The deformation of every spring is a function of link position, 0r and lever angle ~. If we keep all the other parameters unchanged, the balancing torque for a given link position, 0j, and a given lever angle, ~, will be equal to the balancing torque for a link position 0r + ~ with lever angle equal to zero. Therefore the effect of the lever angle is to shift the balancing torque left or right depending on the particular value of ~. We represent this graphically in Fig. 4. Different balancing torques have been computed for different lever angles ~ = - 2 0 . . . 2 0 °. As we can see the shape of the counterbalancing torque is kept unchanged by this parameter when we use just a linear spring. However, as we will see in the next section this parameter is responsible for the final shaping of the new linear spring configuration presented in this paper. The case of 6,,. is not as straightforward as the two previous ones. While K~ is a constant in the final compensating function, and ~i just shifts the basic function, 6,.~acts in a more tricky way on the final function. Again, we will better understand its influence by having a look at Fig. 2. On the one hand, when we increase 6,.i we get more deformation in the spring and, as a result, the M M T 331-2
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Fig. 3. Increasing the balancing torque by changing the spring rate.
force is increased, which will in turn raise the balancing torque. On the other hand, when the angle between F~ and r~j is less than 90 ° an increase in 6,.~will raise the value of the inner product between these two vectors while in the opposite case the same increase in ~.,.~will lower this inner product. These two effects are mixed together in the final expression of the counterbalancing torque and can be better seen in Fig. 5. When we compare the effect of changing Ksi and 6,.i represented in Figs 3 and 5, respectively, we notice that the influence of altering the angle between Fs, and r~, discussed in the previous paragraph is not negligible. In fact, this helps to keep the amount in which the balancing torque is increased almost constant in all the link position range, unlike in the case of altering the spring rate. Something similar occurs when studying the effect of varying the lever length, r~. However, for the lever length, the change in the angle between force and lever is a second order effect as can be seen in Fig. 6. Therefore, it can be assumed that by altering the lever length we act on the final
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counterbalancing function in a quite similar way to the case of altering the spring rate, this is just multiplying the function by a constant. From this discussion we can conclude that when using a single compensating linear spring with this configuration it is not possible to shape the counterbalancing function to a sine function (gravity torque) as desired and this is mainly due to the use of a non-zero initial spring length. The next section will show us how a combination of linear springs enables us to successfully shape the balancing torque giving us the opportunity of using spring compensation for a wide range of the link motion. Moreover, we will see how to use the new spring configuration to actively compensate for torque caused by varying payloads. 2.4.
New
linear
spring
configuration
Let us consider the new linear spring configuration presented in Fig. 7. The goal of this new configuration is to create an appropriate gravity counterbalancing torque by adding up different 40
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Fig. 7. Configuration of the new linear spring counterbalancing system. single counterbalancing functions as presented in the previous sections. An appropriate counterbalancing function will present the following characteristics: sine-like function resembling gravity induced torque, capacity of being easily tuned or adjusted to the particular gravity terms of a given manipulator and possibility of being used for active balancing to account for varying payloads. The new linear spring configuration as presented in Fig. 7 comprises a couple of linear springs with the following configuration parameters: ~, = - ~ 2 = c~, r~, = r~2 = r~, K~ = Ks2 = Ks and 6,.~ = 6.,.2= fix. Since both springs have opposite lever angles the corresponding single balancing torque are shifted left and right the same amount as plotted in Fig. 8. The total balancing torque of the set of linear springs is obtained by adding T~] and T~2 as shown in the figure. Now we realize that depending mainly upon lever angle although also on the other parameters, the total torque can be shaped to a sine function in a wide link position range. The use of opposite lever angles for both springs has the advantage of keeping a symmetric balancing torque function. This enables to use the countergravity configuration for a wider link position range which extends to both positive and negative positions. Since gravity induced terms 60
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Fig. 8. Shaping of the compensating torque by modifying ~c
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60
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Fig. 9. Effect o f varying ~ in the final configuration.
are always symmetrical functions, this suggests to us this configuration as the most appropriate for counterbalancing manipulators. Even when opposite lever angles ensures symmetric functions, the particular value of this angle has to be chosen from the shaping point of view. The offset created by one of the springs, when the link position is 0j = 0, is exactly compensated by the other spring that creates an offset of the same magnitude but opposite sign. Thus, since for this position there is no gravity torque, this is an effective equilibrium position. As the link position increases either to positive or negative angles, the countergravity torque in one of the springs goes up while the other goes down. If correct configuration parameters are chosen the total countergravity torque will match the gravity terms. The choice of optimum particular design parameters for a given manipulator is a highly non-linear problem. The balancing torque depends on lever angle, spring rate, lever length and initial deformation of every spring. The assumptions we have made in our configuration (equal parameters for both springs) reduces the number of parameters to four. The optimization problem could be more easily solved by stating a least squares problem in the terms given in equation (12). [~s, Ks, r,, 6~] p[min{~,ll;~'[T~(0,) - Ts(Oj, c~,, Ks, r~, 6,.)]260,}
(12)
The optimization process described in the previous equation is defined between the limits of motion of the link, 0,,, and 0M. The proper choice of configuration parameters makes minimum the value of the integral in equation (12). The term inside the integral represents the error between compensating and link gravity functions. The effect of configuration parameters on the final function was described in the previous section for a single linear spring. For this new configuration all the remarks we made for the single spring are still applicable. However, it is interesting to have a look at the special effect of lever angle on the shaping of the balancing function. This can be seen in Fig. 9 in which only positive values of the link position are plotted since, as shown before, it is a symmetric function. In order to see the influence of ~, all the other parameters are kept unchanged while the lever angle is increased from 27 to 42 °. From lower to higher values of the angle we see that the concavity of the function changes from positive to negative values. The sine Table 1, Parameters of balancing system for Agribot Parameter
Process 1
Process 2
~t (deg) r~ (mm) K, (N/mm) 6,. (mm)
37.14 50 24.86 0
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Fig. 10. Gravity induced torque (--), theoretical balancing torque (...) and actual balancing torque (- - -).
function is also represented in the figure with a dotted line, and all the balancing functions have been normalized so they can be easily compared to the sine function. As we will present in the next section, an optimum value for the lever angle is obtained from the least squares problem stated in equation (12). This optimum lever angle, which in general will be different according to the particular values of all the other parameters, allows us to obtain a quasi-sine counterbalancing function. In spite of all the theoretical advantages of this new configuration, during the manufacturing process it is not always possible to obtain linear springs whose rates exactly match those drawn from the optimization process in the design stage. Therefore, the balancing system should be somehow adjustable to our particular manipulator. As we have shown, by altering the initial deformation of our springs we can easily raise or lower the level of compensation of the system. Thus, it seems to be a quite straightforward tuning method.
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Fig. I ]. Imbalance between gravity induced torque and actual quasi-exact balancing torque after tuning process.
Quasi-exact linear spring countergravitysystem
69
Another interesting aspect of the gravity counterbalancing system is the possibility of actively compensating for varying payloads. Even when this is not thoroughly treated in this paper, let us now look at the capabilities of our passive balancing system to be actively used. Once the configuration parameters are selected, based on the above mentioned method, the shape of the balancing function is defined. The level of compensation, as we saw before, can be altered by changing either the spring rate, Ks or the initial deformation, 6,. Automatic compensation for varying payloads is more easily understood by using as an input to the active balancing system the initial deformation of the springs. This imposes actively modifying the position at which the spring is attached to a link or fixed points. 3. CASE STUDY: QUASI-STATIC EQUILIBRATOR FOR THE AGRIBOT In order to experimentally check out the applicability of the quasi-exact equilibrator system, let us consider the Agribot Manipulator. Agribot (Agricultural Robot), has been developed at the IAI for aiding fruit picking and is a parallelogram structure manipulator. Since motion is powered by autonomous batteries, a critical aspect of its design is reduction of energy consumption, thus a countergravity system is needed. Gravitational potential energy for the first link of the robot is measured to be V~ = 8.71 g cos 0 mN, where 0 defines position of the link with respect to an absolute vertical line. Corresponding gravity induced torque on the shaft results to be 7"8 = 85.44 sin 0 mN. When using quasi-exact balancing, four parameters have to be determined from the optimization problem stated in equation (12). For this case, the range of optimization is selected to be the same as the practical range of motion of the manipulator. For our case, motion extends from - 4 5 to 60 deg thus these are entered as optimization parameters in equation (12). We are also interested in exact tuning of our equilibrator system. As shown in previous sections, this can be easily done by altering the value of 6, once the equilibrator system has been manufactured and some disagreement is found between designed parameters and actual ones. This suggests to us to set an initial 6, = 0 and run the optimization process for the remaining three parameters. Once our system is manufactured and mounted, exact actual values are measured and then a second optimization process is started, this time the only parameter will be &, and lever length, spring rate and lever angle are set to be the measured ones. Table 1 shows values produced by both optimization processes. It is interesting to have a look at the counterbalancing torque as obtained by the theoretical optimization process compared to gravitational torque and to actual counterbalancing torque after manufacturing and mounting the system. Since matching is quite good, Fig. 10 shows a zoom of all these curves. The dotted line represents theoretical balancing torque while the dashed line shows actual balancing torque. It is found that theoretical balancing function quasi-matches gravitational torque throughout the entire motion range of the manipulator. Disagreement between theoretical and actual balancing functions is mainly due to the difference between designed and finally obtained spring rate. For this case the spring rate mismatch is about 3% of the designed one. Following our design procedure, balancing function error is partially corrected by tuning the system by using parameter 6~ drawn from the second optimization process. Due to scale problems it is difficult to check in Fig. 10 the real unbalance between final balancing function and gravity induced torque. Figure l l shows the balancing mismatch at every angle in the entire range of motion after correction of balancing torque, 6, = 1.2 mm. As we can see, reduction to less than 1% the initial gravity torque is achieved throughout the whole range of motion of the manipulator. 4. CONCLUSIONS This paper presents an easy to implement quasi-exact counterbalancing system based on linear springs for all kind of angular manipulators. The new equilibrator matches elastic energy to gravitational one by optimizing a set of configuration parameters. Unlike previous equilibrators, no assumption is made on initial length of springs therefore no special manufacturing process is needed, and standard springs can be used.
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N o limit to m o t i o n range is imposed to the manipulator and an overall reduction o f gravity induced terms is achieved t h r o u g h o u t the entire r o b o t range. A practical implementation o f the new equilibrator system is presented for an Agricultural r o b o t developed at IAI. Overall reduction to less than 1% the initial gravity torque is reached. The effect o f varying configuration parameters is thoroughly studied in the paper, thus getting insight into how to use this new configuration for actively compensating varying payloads during dynamic operation. Acknowledgements--The authors sincerely acknowledge the CYCIT that funded this research under project Agribot
(TAP93-0583).
REFERENCES
1. Pons, J. L., Ceres, R. and Jimenez, A. R., J. Intelligent Robotic Systems, 1997, 18, 277. 2. Rivin, E. I., Mechanical Design of Robots. McGraw-Hill, New York, 1987, p. 51. 3. Gopalswami, A., Gupta, P. and Vidyasagar, M., Proc. of the 1992 IEEE Int. Conference on Robotics and Automation, p. 664. 4. Yao, J. and Smith, M. R., Mech. Mach. Theoo', 1993, 28, 583. 5. Pons, J. L., Ceres, R. and Jimenez, A. R., Proc. of the 1996 IEEE Int. Conference on Robotics and Automation, 1996. 6. Shim, E. and Streit, D. A., Mech. Mach. Theory, 1991, 26, 645. 7. Petrov, B. A., Manipulators. Mashinostroenie Publishing House, Leningrad, 1984, p. 51 (cited in [2]).
Pergamon
QUASI-EXACT
LINEAR
SPRING COUNTERGRAVITY
SYSTEM FOR ROBOTIC MANIPULATORS
J. L. PONS, R. CERES and A. R. JIMI~NEZ lnstituto de Automfitica Industrial, Consejo Superior de Investigaciones Cientificas, Ctra. Campo Real, Km. 0.200 La Poveda, 28500 Arganda del Rey, Madrid, Spain (Received 18 November 1996)
Abstract--A new quasi-exact linear spring counterbalancing system for general manipulators is
theoretically stated and experimentally tested on a parallelogram manipulator. The effect of configuration parameters on developed balancing torque is studied, which gives insight into possible active application of the presented configuration. Quasi-exact balancing of gravitational torque is achieved without any assumption on initial spring length thus avoiding classical shortcomings arising from perfect spring equilibrator theory. © 1998 Elsevier Science Ltd
1. I N T R O D U C T I O N .
REVIEW ON GRAVITY BALANCING TECHNIQUES
The total torque needed to drive a manipulator for a given motion consists of dynamic and gravity-induced terms. Dynamic terms are non-linearly dependent on position, velocity and acceleration, and as a result of them, there is a coupling effect among different joints [1]. Coupling and non-linear effects result in a more complex control requirement since the action of the control law on a given shaft also affects the motion in all the other joints. Nevertheless, as it is well known, [2], the relative importance of these dynamic terms as compared to gravity-induced ones is low, and if no counterbalancing is provided, motor-transmission units have to be selected to cope with this high torque. Additional problems related to gravity terms come out of the need of safety systems which preserve the structure against collapsing in case of power failure. Of course, the design of these safety systems depends on the relative size of gravity terms, thus again a counterbalancing is useful. Even when a specific design of counterbalancing systems could be done for any given manipulator, it is obvious that the best approach should take into account the selection of the appropriate manipulator structure, the kind of work that the manipulator is going to perform and the selection of the counterbalancing system. As far as the selection of structure is concerned, in [3], the authors showed how the selection of an appropriate structure enables the counterbalancing for all the joints of the manipulator. In this case, the choice of an inverse parallelogram structure made it possible to use spring compensation on both joints of the manipulator, while the selection of a classical parallelogram structure would have made it necessary to use mass counterbalancing with the corresponding increase in the overall weight of the structure. On the other hand, the kind of work to be performed affects the choice of counterbalancing technique. Gravity compensation systems can be further classified in passive and active systems. Since the manipulator will, in general, work with different payloads, a passive compensation system will only work well when, during operation, light changes in payload are expected. For instance, if the manipulator is scheduled to perform pick-and-place motion, large variation in payload will occur and an active gravity counterbalancing system has to be designed. The most common passive systems are mass and spring counterbalancing. Mass compensation is basically achieved by moving the centre of mass of the link which is going to be balanced by adding mass until the final position falls on the shaft. This kind of compensation, which is the one used in the forearm of the Kuka 160, has been widely used in jointed manipulators and has the 59
60
J.L. Pons et al.
advantage of linearizing the equations of motion of the structure. The main disadvantage comes from its passive nature and from the difficult exact adjustment of the counterbalancing weight, besides inertial characteristics are increased thus reducing dynamic behaviour. Since compensation is only achieved for a given payload the structure has to be equipped with safety brakes. Several works, [4], address the general problem of determining the minimum number of counterweights necessary for exact force balancing. To better understand the possibilities of spring gravity counterbalancing we note that the general form of gravity torque is derived from gravitational potential energy as given in equation (1), where ae, be and ~ are structure dependent constants and 0i defines the position of a given link. In spring counterbalancing systems, springs are used to produce a counteracting torque which is a function of the position, 0. When torsion springs are used, the counteracting torque results to be linear with the link position. Therefore, when the spring rate is properly chosen, good fit between gravity and counterbalancing torque is achieved for the first angle range. However, since gravity torque will continuously reduce its slope relative to the counteracting torque, we end up with a practical limitation to the workspace of the manipulator, [3]. Vg = ~ (ae sin 0e + be cos 0i) + ~
(1)
/=1
Linear springs have been widely applied in gravity balancing systems. Linear springs usually act through a lever thus producing a counterbalancing torque shape which depends on specific configuration. In [5], even when balancing torque shape does not fit gravity torque shape, an overall reduction to approx. 30% the initial gravity terms is achieved. In order to get a better fit between counteracting and gravity torque some special mechanisms have been designed based on linear springs. As shown in [6], theory of perfect spring equilibrators seeks for optimum parameters for linear counteracting springs, in such a way that elastic potential energy exactly matches gravitational energy, and thus several configurations have been suggested. One of them, known as sine mechanism, is presented in [2], and allows, at least theoretically, to shape the balancing torque to a sine function which exactly equals the gravity terms. However a general practical shortcoming for this mechanism arises from the need of obtaining a practical initial length of the spring equal to zero. As pointed out in [7], this could be achieved by twisting the wire while the spring is being coiled. A general advantage of spring balancing is that no safety systems are required since there is always an equilibrium position. Up to now, only passive techniques have been discussed and all of them have the shortcoming of its inability to cope with varying payloads. Active mechanisms use external power sources to produce a counteracting torque which can be easily made dependent on external loading, i.e. changing payloads. Both electrical and pneumo-hydraulic active mechanisms are used. in industrial manipulators. This imposes the use of additional power sources as in the case of electrically driven robots with pneumo-hydraulic gravity compensators. Moreover, when using electric torque motors in the balancing systems, it is necessary again to provide a safety system to cope with power failure. Throughout this work, we will present a new gravity balancing configuration based on linear springs. It makes no assumption about initial length of linear springs therefore it needs no special manufacturing process. The specific choice of mechanism configuration allows to shape in a practical way the balancing torque of the springs for a wide position range of the manipulator. Since spring compensation is used, no safety systems are required. Moreover, the specific configuration allows us, by modifying its parameters, firstly to adjust precisely the balancing torque to our manipulator, and secondly to use it as an active balancing system. 2. ANALYSIS OF THE NEW BALANCING STRATEGY The new configuration for our linear spring countergravity system is based on a linear spring acting through a lever to produce the balancing torque. In order to better understand the fundamentals of this system we first analyse the background on theory of perfect spring equilibrators. Then, we introduce the basic configuration schematically shown in Fig. 2 to see how we can use it to construct our counterbalancing system. Throughout this section we obtain the basic
Quasi-exact linear spring countergravitysystem
61
equations governing the system. These equations show us four basic parameters that can be used to precisely adjust and tune our configuration for a given manipulator, and direct us to the possibility of actively compensating for varying payloads. 2. I. Background on spring equilibrators Here, we recall equation (1) defining gravitational potential energy for a given manipulator as a function of structure dependent constants and generalized co-ordinates. Perfect countergravity balancing might be reached only if exact match between elastic potential and gravity one is achieved. In order to do so, dependence of spring potential on position should be through sine and cosine functions: Vk = ~ (c, sin 0r + dr cos 0t) + fl
(2)
i=1
For spring force to be dependent on link position, it has to be attached either between link and fixed points or between consecutive links. Regarding Fig. 1 showing typical configuration for spring equilibrator, we write the expression of spring length as a function of link position: '~ = x / g + ~ - 2t, t2 cos(~ - 0) - ,~0
(3)
Perfect equilibrator theory assumes initial spring length equal to zero, 60 = 0, thus potential energy can be written as: 1
1
V~ = ~/£62 = ~ (K, + K: cos 0)
(4)
Now, by properly choosing spring configuration and rate, exact static matching between both energies can be achieved. A practical shortcoming comes out of the need of obtaining practical zero initial length for the springs as pointed in [2]. 2.2. Bas& configuration The basic configuration of a linear spring used in this new quasi-exact gravity balancing system is shown schematically in Fig. 2. The lever that is used to create the balancing torque has a length r~r while the spring whose initial length is 10i is mounted between the lever end and a fixed point, O,. All the forces that the spring creates as the link position varies, 0j, are represented in the figure. The spring force, Fsr, acting through the lever, creates counterbalancing torque Tsr, which in turn compensates for the gravity term, Tg. It is important to note that for a given link position, it is
11
Fig. I. Basic configurationof a linear spring equilibrator.
62
J . L . Pons et al.
•
]~Link j
',
loi + ~xi
F,i ' ~ X y
IN
Fig. 2. Basic configuration of a single counterbalancing spring.
possible either to increase or decrease the spring length by changing lever angel ~, we will use this property later on to shape the balancing torque. In all the equations subscript i refers to the spring number, (if a set of linear springs is used), while subscript j does to the link whose gravity effects are being compensated. In order to see how we can shape the balancing torque we will derive the expression for this torque as a function of all the parameters given in the figure plus the spring rate, K~. To do this, we first note that the balancing torque depends only on the spring force vector, F~i and lever vector r~, and can be calculated as their inner product accordingly to the coordinate frame shown in the figure: Ts, = Fs~ x rsi =
/~i,
J k F~¢ 0
rsi,-
rsl,
(5)
0
For this configuration, we can work out the expression defining the lever vector, equation (6), and the vector in the direction of the force created by the spring, equation (7). r~ = rs~ (cos 0r + ai)i - r~ sin (0r + ~)j
(6)
vr~, = [r,,(cos(0j + ~,) - 1) - (10i+ 6x)]i - r,, sin(0j + ~,)j
(7)
In order to reach the complete expression for the force in the spring we need to relate its deformation, 6s~ to all the parameters defining the configuration: lever length, rs~, lever angle, ~, and initial deformation, fix,. With this deformation we express the force in the spring as the product of deformation and spring rate in the direction given by vector vr~,: 6s, = xfr~, + (lo, + fix, + r,,) 2 - 2r,,(lo, + 6,~ + r~,)cos(0j + a,) - (lo, + fix,)
(8)
and finally the expression for the force in the spring:
YF~
F~ = g~c5~, vr~,l =Ks,[~r~i + (loi + fix, + r~) z - 2r,,(10, + 6x~+ rs,)cos(0~ + ¢,) -(lo, + 6,,)] ×
[r,,(cos(O/+ ~,) - 1) - (10,+ 6x)]i - r~ sin(0j + ~,)j ~f[rs,(cos(0r + ~,) - 1) - (10,+ 6x)] 2 + [r~ sin(0r + ~,)]2
(9)
Quasi-exact linear spring countergravity system
63
According to equation (5) we can now calculate the total counterbalancing torque that a set of compensating linear springs will create on link j. In order to do this, we just sum up a series of terms of the form of equation (5), which is the partial torque created by every spring. For simplicity, we just express in equation (10) how to sum up the effect of every spring in the total balancing torque on link j, Tsj; the final expression can be worked out substituting equations (6), (8) and (9) into equation (10) Fsi
Ts~ =
~
x
rsi&,Ks,
(10)
i=1
Potential elastic energy for this case, in which no assumption has been made regarding initial length of the springs, results:
'L •=, K~[G + (rs~+ loe+ 6,.,)2 _ 2r~i(r~, + lo~ + 6,.~)cos(n - 0j) + 66,
-- 260,x/r~i + (r~, + 10, + 6,,.)2 -- 2rs,(r~i + lo, + &,)cos(n -- 0,)]
(11)
Equation (10) gives us the total counterbalancing torque of a set of linear springs. Every term in the summation depends on particular configuration of every single compensating spring. This configuration is defined by four parameters which we can use to shape every single balancing torque, and as a result, the total torque on shaft j, Ts/. Our interest focuses now on how to get this, this is to say, what particular value of every parameter will give us the right compensation torque in the entire manipulator range. In the next section we study how to alter the final torque by changing the configuration parameters. 2.3. Effect o f configuration parameters
In the last section we worked out the complete balancing function for a single compensating linear spring and pointed to the expression of the total torque created by a set of linear springs. Equation (10) clearly shows that parameter Ks, modifies the final torque as a constant that multiplies a basic compensating function. This suggests to us to use the spring rate to fit the compensating torque to every particular manipulator provided a good shaping of the compensating function. In order to see this, in Fig. 3, we represent different counterbalancing functions as generated with different values of the spring rate. A more interesting point of view can be obtained from the study of the influence of the other parameters ~, r~ and 6,,. on the balancing torque of a single linear spring. Unlike Ksi, the other parameters are comprised in the expression of Fsi and rs~, thus not only altering the magnitude of the final torque but also modifying the value of their inner product as given in equation (5). The effect of changing the value of :¢i can be better understood by looking at Fig. 2. The deformation of every spring is a function of link position, 0r and lever angle ~. If we keep all the other parameters unchanged, the balancing torque for a given link position, 0j, and a given lever angle, ~, will be equal to the balancing torque for a link position 0r + ~ with lever angle equal to zero. Therefore the effect of the lever angle is to shift the balancing torque left or right depending on the particular value of ~. We represent this graphically in Fig. 4. Different balancing torques have been computed for different lever angles ~ = - 2 0 . . . 2 0 °. As we can see the shape of the counterbalancing torque is kept unchanged by this parameter when we use just a linear spring. However, as we will see in the next section this parameter is responsible for the final shaping of the new linear spring configuration presented in this paper. The case of 6,,. is not as straightforward as the two previous ones. While K~ is a constant in the final compensating function, and ~i just shifts the basic function, 6,.~acts in a more tricky way on the final function. Again, we will better understand its influence by having a look at Fig. 2. On the one hand, when we increase 6,.i we get more deformation in the spring and, as a result, the M M T 331-2
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64
J.L. Pons et
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Fig. 3. Increasing the balancing torque by changing the spring rate.
force is increased, which will in turn raise the balancing torque. On the other hand, when the angle between F~ and r~j is less than 90 ° an increase in 6,.~will raise the value of the inner product between these two vectors while in the opposite case the same increase in ~.,.~will lower this inner product. These two effects are mixed together in the final expression of the counterbalancing torque and can be better seen in Fig. 5. When we compare the effect of changing Ksi and 6,.i represented in Figs 3 and 5, respectively, we notice that the influence of altering the angle between Fs, and r~, discussed in the previous paragraph is not negligible. In fact, this helps to keep the amount in which the balancing torque is increased almost constant in all the link position range, unlike in the case of altering the spring rate. Something similar occurs when studying the effect of varying the lever length, r~. However, for the lever length, the change in the angle between force and lever is a second order effect as can be seen in Fig. 6. Therefore, it can be assumed that by altering the lever length we act on the final
40
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Quasi-exact
linear spring countergravity
system
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60
counterbalancing function in a quite similar way to the case of altering the spring rate, this is just multiplying the function by a constant. From this discussion we can conclude that when using a single compensating linear spring with this configuration it is not possible to shape the counterbalancing function to a sine function (gravity torque) as desired and this is mainly due to the use of a non-zero initial spring length. The next section will show us how a combination of linear springs enables us to successfully shape the balancing torque giving us the opportunity of using spring compensation for a wide range of the link motion. Moreover, we will see how to use the new spring configuration to actively compensate for torque caused by varying payloads. 2.4.
New
linear
spring
configuration
Let us consider the new linear spring configuration presented in Fig. 7. The goal of this new configuration is to create an appropriate gravity counterbalancing torque by adding up different 40
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60
J.L. Pons et al.
66
1Jl~ j
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Fig. 7. Configuration of the new linear spring counterbalancing system. single counterbalancing functions as presented in the previous sections. An appropriate counterbalancing function will present the following characteristics: sine-like function resembling gravity induced torque, capacity of being easily tuned or adjusted to the particular gravity terms of a given manipulator and possibility of being used for active balancing to account for varying payloads. The new linear spring configuration as presented in Fig. 7 comprises a couple of linear springs with the following configuration parameters: ~, = - ~ 2 = c~, r~, = r~2 = r~, K~ = Ks2 = Ks and 6,.~ = 6.,.2= fix. Since both springs have opposite lever angles the corresponding single balancing torque are shifted left and right the same amount as plotted in Fig. 8. The total balancing torque of the set of linear springs is obtained by adding T~] and T~2 as shown in the figure. Now we realize that depending mainly upon lever angle although also on the other parameters, the total torque can be shaped to a sine function in a wide link position range. The use of opposite lever angles for both springs has the advantage of keeping a symmetric balancing torque function. This enables to use the countergravity configuration for a wider link position range which extends to both positive and negative positions. Since gravity induced terms 60
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,,
60
Quasi-exact linear spring countergravity system O.g
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40
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Fig. 9. Effect o f varying ~ in the final configuration.
are always symmetrical functions, this suggests to us this configuration as the most appropriate for counterbalancing manipulators. Even when opposite lever angles ensures symmetric functions, the particular value of this angle has to be chosen from the shaping point of view. The offset created by one of the springs, when the link position is 0j = 0, is exactly compensated by the other spring that creates an offset of the same magnitude but opposite sign. Thus, since for this position there is no gravity torque, this is an effective equilibrium position. As the link position increases either to positive or negative angles, the countergravity torque in one of the springs goes up while the other goes down. If correct configuration parameters are chosen the total countergravity torque will match the gravity terms. The choice of optimum particular design parameters for a given manipulator is a highly non-linear problem. The balancing torque depends on lever angle, spring rate, lever length and initial deformation of every spring. The assumptions we have made in our configuration (equal parameters for both springs) reduces the number of parameters to four. The optimization problem could be more easily solved by stating a least squares problem in the terms given in equation (12). [~s, Ks, r,, 6~] p[min{~,ll;~'[T~(0,) - Ts(Oj, c~,, Ks, r~, 6,.)]260,}
(12)
The optimization process described in the previous equation is defined between the limits of motion of the link, 0,,, and 0M. The proper choice of configuration parameters makes minimum the value of the integral in equation (12). The term inside the integral represents the error between compensating and link gravity functions. The effect of configuration parameters on the final function was described in the previous section for a single linear spring. For this new configuration all the remarks we made for the single spring are still applicable. However, it is interesting to have a look at the special effect of lever angle on the shaping of the balancing function. This can be seen in Fig. 9 in which only positive values of the link position are plotted since, as shown before, it is a symmetric function. In order to see the influence of ~, all the other parameters are kept unchanged while the lever angle is increased from 27 to 42 °. From lower to higher values of the angle we see that the concavity of the function changes from positive to negative values. The sine Table 1, Parameters of balancing system for Agribot Parameter
Process 1
Process 2
~t (deg) r~ (mm) K, (N/mm) 6,. (mm)
37.14 50 24.86 0
37.14 50 24.86 1.2
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50
Fig. 10. Gravity induced torque (--), theoretical balancing torque (...) and actual balancing torque (- - -).
function is also represented in the figure with a dotted line, and all the balancing functions have been normalized so they can be easily compared to the sine function. As we will present in the next section, an optimum value for the lever angle is obtained from the least squares problem stated in equation (12). This optimum lever angle, which in general will be different according to the particular values of all the other parameters, allows us to obtain a quasi-sine counterbalancing function. In spite of all the theoretical advantages of this new configuration, during the manufacturing process it is not always possible to obtain linear springs whose rates exactly match those drawn from the optimization process in the design stage. Therefore, the balancing system should be somehow adjustable to our particular manipulator. As we have shown, by altering the initial deformation of our springs we can easily raise or lower the level of compensation of the system. Thus, it seems to be a quite straightforward tuning method.
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Fig. I ]. Imbalance between gravity induced torque and actual quasi-exact balancing torque after tuning process.
Quasi-exact linear spring countergravitysystem
69
Another interesting aspect of the gravity counterbalancing system is the possibility of actively compensating for varying payloads. Even when this is not thoroughly treated in this paper, let us now look at the capabilities of our passive balancing system to be actively used. Once the configuration parameters are selected, based on the above mentioned method, the shape of the balancing function is defined. The level of compensation, as we saw before, can be altered by changing either the spring rate, Ks or the initial deformation, 6,. Automatic compensation for varying payloads is more easily understood by using as an input to the active balancing system the initial deformation of the springs. This imposes actively modifying the position at which the spring is attached to a link or fixed points. 3. CASE STUDY: QUASI-STATIC EQUILIBRATOR FOR THE AGRIBOT In order to experimentally check out the applicability of the quasi-exact equilibrator system, let us consider the Agribot Manipulator. Agribot (Agricultural Robot), has been developed at the IAI for aiding fruit picking and is a parallelogram structure manipulator. Since motion is powered by autonomous batteries, a critical aspect of its design is reduction of energy consumption, thus a countergravity system is needed. Gravitational potential energy for the first link of the robot is measured to be V~ = 8.71 g cos 0 mN, where 0 defines position of the link with respect to an absolute vertical line. Corresponding gravity induced torque on the shaft results to be 7"8 = 85.44 sin 0 mN. When using quasi-exact balancing, four parameters have to be determined from the optimization problem stated in equation (12). For this case, the range of optimization is selected to be the same as the practical range of motion of the manipulator. For our case, motion extends from - 4 5 to 60 deg thus these are entered as optimization parameters in equation (12). We are also interested in exact tuning of our equilibrator system. As shown in previous sections, this can be easily done by altering the value of 6, once the equilibrator system has been manufactured and some disagreement is found between designed parameters and actual ones. This suggests to us to set an initial 6, = 0 and run the optimization process for the remaining three parameters. Once our system is manufactured and mounted, exact actual values are measured and then a second optimization process is started, this time the only parameter will be &, and lever length, spring rate and lever angle are set to be the measured ones. Table 1 shows values produced by both optimization processes. It is interesting to have a look at the counterbalancing torque as obtained by the theoretical optimization process compared to gravitational torque and to actual counterbalancing torque after manufacturing and mounting the system. Since matching is quite good, Fig. 10 shows a zoom of all these curves. The dotted line represents theoretical balancing torque while the dashed line shows actual balancing torque. It is found that theoretical balancing function quasi-matches gravitational torque throughout the entire motion range of the manipulator. Disagreement between theoretical and actual balancing functions is mainly due to the difference between designed and finally obtained spring rate. For this case the spring rate mismatch is about 3% of the designed one. Following our design procedure, balancing function error is partially corrected by tuning the system by using parameter 6~ drawn from the second optimization process. Due to scale problems it is difficult to check in Fig. 10 the real unbalance between final balancing function and gravity induced torque. Figure l l shows the balancing mismatch at every angle in the entire range of motion after correction of balancing torque, 6, = 1.2 mm. As we can see, reduction to less than 1% the initial gravity torque is achieved throughout the whole range of motion of the manipulator. 4. CONCLUSIONS This paper presents an easy to implement quasi-exact counterbalancing system based on linear springs for all kind of angular manipulators. The new equilibrator matches elastic energy to gravitational one by optimizing a set of configuration parameters. Unlike previous equilibrators, no assumption is made on initial length of springs therefore no special manufacturing process is needed, and standard springs can be used.
70
J.L. Pons et al.
N o limit to m o t i o n range is imposed to the manipulator and an overall reduction o f gravity induced terms is achieved t h r o u g h o u t the entire r o b o t range. A practical implementation o f the new equilibrator system is presented for an Agricultural r o b o t developed at IAI. Overall reduction to less than 1% the initial gravity torque is reached. The effect o f varying configuration parameters is thoroughly studied in the paper, thus getting insight into how to use this new configuration for actively compensating varying payloads during dynamic operation. Acknowledgements--The authors sincerely acknowledge the CYCIT that funded this research under project Agribot
(TAP93-0583).
REFERENCES
1. Pons, J. L., Ceres, R. and Jimenez, A. R., J. Intelligent Robotic Systems, 1997, 18, 277. 2. Rivin, E. I., Mechanical Design of Robots. McGraw-Hill, New York, 1987, p. 51. 3. Gopalswami, A., Gupta, P. and Vidyasagar, M., Proc. of the 1992 IEEE Int. Conference on Robotics and Automation, p. 664. 4. Yao, J. and Smith, M. R., Mech. Mach. Theoo', 1993, 28, 583. 5. Pons, J. L., Ceres, R. and Jimenez, A. R., Proc. of the 1996 IEEE Int. Conference on Robotics and Automation, 1996. 6. Shim, E. and Streit, D. A., Mech. Mach. Theory, 1991, 26, 645. 7. Petrov, B. A., Manipulators. Mashinostroenie Publishing House, Leningrad, 1984, p. 51 (cited in [2]).