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Observer-Based Force Control of Robots
OBSERVER-BASED FORCE CONTROL OF ROBOTS
J. SUCIIY SLOYAK ACADEMY OF SCIENCES. IrutilJlU of C:mtrol TIwory and RobotiCI,
s.v.ma 5,
974 ()() BansJai
BystrlC4, SLOYAKIA
AIIItnIet. A model of CJDOodesreo-of &eedam robot CClGIacting 1be caviroomaIl iI dGveloped. To a11cviatc 1be problem ofDODCOlocatioo 1be IIruaural propcrtieI of1be arm arc also modelod md the Ilates ofmcdgnica11)'1tcm betweal ooloc:aled md DODOOloadcd . . . . Ire I'CCXIIIItnIdcd with 1be recJuccd..order obacrYcr md ueI with a robuIt QOIIIro) t«.baiquo bacd CD clefiDitioo of ..,.mJ md modal popertiCII ofthe nvdumica11)'1tcm.
in essence an adaptive observer which also served as an identifier monitoring the changes in the arm as well as in the environment The main disadvantage of this observer was its high complexity making the control algorithm hardly implementable in real time as well as the high volume of work which has to be done in advance. The control was based on the exact model. In this article, a simpler modelling technique is used. This model is derived analytically, it should be, however, experimentally verified in the case of real implementation. The reduced order observer is then designed to deliver the nonmeasured states to the control algorithm. This algorithm is essentially the same as proposed in Kazerooni et. al. (1986b). Here it has somewhat shifted meaning. It is not merely conventional pole-placement but also determination of spectral and/or modal properties of the mechanical system (beam) by control.
1. INTRODUCI'ION The modem robots in the not veIY far future can hardly be imagined without the equipment, algorithms and software for handling the contact with their working environment. That is the reason why so much research effort is paid to the force control of robots. However, in spite of this effort there are probably no ripe solutions for the factory floor. This situation is caused by the overall complexity of the problem. In comparison
with position control of robots the following aspects have to be taken into account with the force control: noncolocation of the fo~orque sensor and actuators, compliance/stiffness of the mechanical parts, friction, dynamical properties of the force torque sensor and of the contacted
environment, changing properties of the contacted environment In this contribution the attention will be paid to the problem of noncolocation. Robots are usually driven by actuators placed in or near the joints, while fo~rque sensors are commonly situated in the wrist of the robot There is mechanical dynamical system between both of these members. As Eppinger and Seering (1992) pointed out, ignoring this part in the model may be the cause of instability. They propose that sophisticated model of vibratory modes and coupling of input forces to output motions could be used along with more sensors to give reliable information about the response of these modes. The aim of this article is to contribute to the goal formulated above. In the former work, SucbY (1992) used the so called modal observer. It was
2. SYSTEM MODELLING For the sake of simplicity a one link manipulator is considered according to Fig. I. This manipulator is equipped with the following sensors: colocated position (and velocity) sensor, nearly colocated strain-gauge sensor and force sensor at the tip of the arm or in the environment This possibility is given by the construction of the sensor, the transmitting part of which may be easily positioned anywhere on the beam and the receiving part may be positioned opposite to the sending part. This arrangement corresponds to 57
the implemented ex:perimcnta1 device. Not all possibilities of this device will be exploited here.
The beam part of the arm will be modelled according to Pota (1992). The beam will be divided into n equally long elements numbered sequentially with i = 1,2,... ,n. The length of one single elements is 1. The number of beam elements has to be chosen aa:ording to the number of eigenforms taken into account The strain-gage sensor is connected with the first element on the actuator side (see Fig.2 and Fig.3) while the force torque sensor is connected with the last element Then the mass and stiffness matrices M; and K;, i = 1,2 ... , n may be computed, boundary conditions accounted for and mass condesation performed to come to the final form of mass and stiffness matrices M and K. Now it is possible to express the kinetic and potential energy of the beam, form Lagrangian L and apply Langrange-Euler equations to get differential equations of motion. Let us denote qT= [Q.2,'b... ,Q.r+-d the n x 1 vector of the single elements deflections in nodes 2,3, ... , n + 1, where the nodes are numerated from the beginning of the beam on the actuator side, ql = 0 from boundary condition because the first node is fixed to the screwfec:der. T denotes the transposition of the vector. Let us further denote IT= [1,1...1), d the position of the arm on the screwfeeder. Then under the assumption of small displacements qi the expresions for the kinetic and potential energies may by written as
_0
~
M01'O&
91 1~--===--'VVV\-'~f-JVWv---=11
Fig. 1. Layout of the experimental device According to SucbY (1992) the force sensor and the environment is appropriately modeled as spring-damper-masslspring-damper respectively. In Fig. 2 the more general case may be seen when force sensor is placed at the tip of the arm whereas in Fig. 3 the particular implementation of experimental device is reflected when the sensor is firmly connected with the environment. The index s pertains to the sensor, e to the environment
The conesponding Lagrangian is L
Fig.2. Model of the experimental device with tip force sensor
=f[meci2 +qTMq+2qT Mid_qT Kq)
(3)
where m. =m + ITMl Using Lagrange - Euler equtions
;(;.)- : . =Q.. ;=1.2.3•...•• +1
(4)
the system of dynamic equations is obtained meii+1T Mq=rT= Fm
(S)
Mii+M1ii-Kq =11F
(6)
where Fm is the applied motor force, Fm = r.T, T is the motor torque, r is the gearing constant and F is the measured contact force. The further part to model is the actuator. As in Kazerooni et.al. (1986)
Fig. 3. Model of the experimental device with force sensor in the environment
a -ll(t) + 1'(t) 58
= ,u(t)
(7)
is the actuator bandwidth and ~t) is the actuator input force. If for example the Dec:essity of dividing the beam into four elements is proved then the following state vector is defined: XT =[X., X2 ••• XIl1, where XI= d, X:z= ~ X? Q3, xr Q4, xs= 'Is. xr ci,
rearrangement or by ttansformation (11) in obvious way. Starek (1990) writes the equation (12) for the case of controlled vibration, what is also our case, in the form
x,= cb. x.==lh xr«i",
where D is the matrix of controlled variables in physical space of appropriate dimensions (11x11) and J.L (t) = Cl J.L (t). In the sequel the whole system will be modelled by equations (12b) and
Cl
XuF
x= AX+D.J.l(t)+bF
«is
and XII= T. With this state vector the state space equations of the whole system are
(12b)
(13).
3. ROBUST COMPLIANT CONTROL
Y{t) =CX){t)
(9) The aim of this section is to develop the compliant control of the modeled arm. It is desired that by using the state feedback the behaviour of the arm tip will correspond to the target impedance and at the same the spectral andIor modal properties of the beam can be shaped. The approach employed here is the combination of techniques found in Kazerooni et. al. (1986) and Starek (1990). The first author uses the pole-placemcnt technique for achieving the prescribed target impedance while the second author uses the same pole-placement technique for modifying the spectral (eigenvalues) and modal (eigenvectors) properties of mechanical system However, both use different models. Model (12b), (13) developed here includes both modcis and this is prerequisite for achieving the stated aim. Let the control is chosen so that
Here AI is a Sx5 matrix with zero first column, • is vector Sx1, b) is vector Sx1, 0, are zero matrices or vectors of appropriate dimensions. The output vector yet) is constructed in such a way as to have all possible information from the disposal. Thus YI is the sensors at translational position from position sensor, Y2 is the velocity of translational motion, Y3 is the output from the strain-gage sensor, Y.. is the tip
mr+ds+k
position With this in view
c=[~o : : : : ~ : : : : : I
~
0
0
0
0
0
0
0
0
0
0
~
0
0
0
000
0
(10)
0
where Cl, ~ are constants of the strain-gage sensor and the optical sensor. ~ may be placed in any column excluding the first one according to actual placement of its transmitter. Here it is placed in the fifth column to measure the tip deflection. In the next section another form of equations (8) and (9) is needed. This form better corresponds to the algorithm. The states are rearranged so that first two states express the global dynamics ( d and ci), the third state is T and the last eight states express the beam dynamics. These are also rearranged so that the first here are the measured states. Then the new state vector is given by the following transformation
J.l(t) = -GX + qd.F
Here G is the state feedback gain matrix (11xll) and Id feedforward gain vector (11x1). Substituting (14) into (12b) yields
X=(A-DG)X+(Dgd+b)F
mr
where the entries of matrix T are ones and zeros. Using the transformation (11) the equations (8) and (9)
Y=H.X
(1S)
G and Id are designed so that two matrices and one vector, i.e. A-DG, H, and Did + b result in target impedance + ds + k and at the same time the desired spectral and modal properties of the beam structure are reached. The technique for ful1illing the first goal was published in Kazerooni (1986) while the technique for the second goal is in St.8rek (1990) The first technique is applied to the subsystem described by the first three state equations, the second technique for the remaining subsystem. Both are based on the results of Moore (1976). As the model (12b) and (13) includes both the subsystem developped for robot impedance control and the subsystem descnbing the dynamic behaviour of the mechanical part (beam) between the actuator
(11)
x= AX+a,u(t)+bF
(14)
(12) (13)
where A (llxll), H (4xll), b (llxl) and Cl (11x1) are matrices and vectors gained from original matrices and vectors in (8) and (9) by 59
S. REFERENCES
and forceltorque sensor both above goals may be reached under certain circnmst;nas. It is not possible to present all equation here in detail
Ackermann, J (1977). Design by pole placement (in German). Regelungstcchnik, pp. 173-179 Bruce-Boye, C. (1991). Design of a fWl-ordcr
because of limited space. To implement the control Jaw (14) all states should be accessible. However, this is not the case here. States X3 and Xcs to XII are not acces5Ible, therefore it is necessary to introduce the reduced order observer for these states. If the partition of the state vector is defined as X = (ZT, yT)T, where Z includes the inacce5Slble states (x], X6, ... Xll) and Y (X., X2", "-4, xs) accessible states, than by appropriate transformation or rearranging (see Ackcrman (1977), it follows from (12b):
Z =PZ+QY +EJ1+eF
(16)
Y = RZ+SY +LJ1+1F
(17)
observer for flexible robot arm (in German). AutomatisienmgstecJrni/c, 39, pp. 5-10 Eppinger, SD., W:P. Seering (1992) Three dynamic problems in robot force control. IEEE Trans. Rob. Automation. 8, pp.7S1-7S8 Kazcrooni, T. B. et 01 (1986). Robust compliant motion for manipulators, Part I: The fundamental concepts of compliant motion. IEEE J. Robotics and Automation. 2, pp.83-92 Kazcrooni, T. B. et al.. (1986). Robust compliant motion for manipulators, Part n: Design method. IEEE J. Robotics and Automation, 2, pp. 93-10S Moore, B.C. (1976). On the flexibility oJrered by state feedback in multivariable systems beyond closed-loop eigeovalue assignment IEEE Trans. on Aut . Control. 21, pp. 689-692 Pota, H. R (1991). A prototype flexible robot arm-an interdisciplinary undergraduate project. IEEE Trans. Education, 35, pp. 83-89 Schmidt, R (1988). Application of state observers for monitoring and failure detection of mechanical constructions (in German). FortscJrrittbericJrte VDI. Reihe 11, No. 109, VDI Verlag St8rek, L. (1990). Eigenstructure assignment for vibrating mechanical system with control. (In Slovak) StrojnJcky I!asopis, 41, pp.S37-SS8 Suchy, J. (1992). Robot force control using modal obsevers. Proceedings of the Second Int. Symp. Measuremnent and Control in Robotics (lSCMJt92), pp. 199-206 SucbY,J. et.al.. (1993) Identification and force control of one-degree-of-feedom robot Proc. of the Int. Workshop Software Engineering for Parallel Real-Time Systems, pp. 80-86
where P, Q, E, R, S, L are matrices and e, I vectors of appropriate dimensions. The observer is constructed only for the subsystem (16) as
W= KW +NY +QJ1+qF
(18)
(19) where matrices Kt N, Q, J, and vector q are chosen to bring the reconstruction error Z - Z to zero.
4. CONCLUSION In the present article an attemp has been made to
work out the concept of impedance control of one-degree of freedom robot while taking into account the dynamic properties of mechanical system between the actuator and the force sensor. These dynamic properties may be shaped by statefeedback so that some recommendations of Eppinger and Seering (1987) can be satisfied by control rather than by mechanical hardware. Howerer, the algorithmic overhead is still rather high although the measures were adopted to lower it The main overhead is in oftline computing. The future work will be oriented toward refinment of the presented algorithms and experimental verifying of them on more perfect device then that dcscn"bed here. The aim of future work will also be generalizing the algorithm to robots with more degrees of freedom.
60
J. SUCIIY SLOYAK ACADEMY OF SCIENCES. IrutilJlU of C:mtrol TIwory and RobotiCI,
s.v.ma 5,
974 ()() BansJai
BystrlC4, SLOYAKIA
AIIItnIet. A model of CJDOodesreo-of &eedam robot CClGIacting 1be caviroomaIl iI dGveloped. To a11cviatc 1be problem ofDODCOlocatioo 1be IIruaural propcrtieI of1be arm arc also modelod md the Ilates ofmcdgnica11)'1tcm betweal ooloc:aled md DODOOloadcd . . . . Ire I'CCXIIIItnIdcd with 1be recJuccd..order obacrYcr md ueI with a robuIt QOIIIro) t«.baiquo bacd CD clefiDitioo of ..,.mJ md modal popertiCII ofthe nvdumica11)'1tcm.
in essence an adaptive observer which also served as an identifier monitoring the changes in the arm as well as in the environment The main disadvantage of this observer was its high complexity making the control algorithm hardly implementable in real time as well as the high volume of work which has to be done in advance. The control was based on the exact model. In this article, a simpler modelling technique is used. This model is derived analytically, it should be, however, experimentally verified in the case of real implementation. The reduced order observer is then designed to deliver the nonmeasured states to the control algorithm. This algorithm is essentially the same as proposed in Kazerooni et. al. (1986b). Here it has somewhat shifted meaning. It is not merely conventional pole-placement but also determination of spectral and/or modal properties of the mechanical system (beam) by control.
1. INTRODUCI'ION The modem robots in the not veIY far future can hardly be imagined without the equipment, algorithms and software for handling the contact with their working environment. That is the reason why so much research effort is paid to the force control of robots. However, in spite of this effort there are probably no ripe solutions for the factory floor. This situation is caused by the overall complexity of the problem. In comparison
with position control of robots the following aspects have to be taken into account with the force control: noncolocation of the fo~orque sensor and actuators, compliance/stiffness of the mechanical parts, friction, dynamical properties of the force torque sensor and of the contacted
environment, changing properties of the contacted environment In this contribution the attention will be paid to the problem of noncolocation. Robots are usually driven by actuators placed in or near the joints, while fo~rque sensors are commonly situated in the wrist of the robot There is mechanical dynamical system between both of these members. As Eppinger and Seering (1992) pointed out, ignoring this part in the model may be the cause of instability. They propose that sophisticated model of vibratory modes and coupling of input forces to output motions could be used along with more sensors to give reliable information about the response of these modes. The aim of this article is to contribute to the goal formulated above. In the former work, SucbY (1992) used the so called modal observer. It was
2. SYSTEM MODELLING For the sake of simplicity a one link manipulator is considered according to Fig. I. This manipulator is equipped with the following sensors: colocated position (and velocity) sensor, nearly colocated strain-gauge sensor and force sensor at the tip of the arm or in the environment This possibility is given by the construction of the sensor, the transmitting part of which may be easily positioned anywhere on the beam and the receiving part may be positioned opposite to the sending part. This arrangement corresponds to 57
the implemented ex:perimcnta1 device. Not all possibilities of this device will be exploited here.
The beam part of the arm will be modelled according to Pota (1992). The beam will be divided into n equally long elements numbered sequentially with i = 1,2,... ,n. The length of one single elements is 1. The number of beam elements has to be chosen aa:ording to the number of eigenforms taken into account The strain-gage sensor is connected with the first element on the actuator side (see Fig.2 and Fig.3) while the force torque sensor is connected with the last element Then the mass and stiffness matrices M; and K;, i = 1,2 ... , n may be computed, boundary conditions accounted for and mass condesation performed to come to the final form of mass and stiffness matrices M and K. Now it is possible to express the kinetic and potential energy of the beam, form Lagrangian L and apply Langrange-Euler equations to get differential equations of motion. Let us denote qT= [Q.2,'b... ,Q.r+-d the n x 1 vector of the single elements deflections in nodes 2,3, ... , n + 1, where the nodes are numerated from the beginning of the beam on the actuator side, ql = 0 from boundary condition because the first node is fixed to the screwfec:der. T denotes the transposition of the vector. Let us further denote IT= [1,1...1), d the position of the arm on the screwfeeder. Then under the assumption of small displacements qi the expresions for the kinetic and potential energies may by written as
_0
~
M01'O&
91 1~--===--'VVV\-'~f-JVWv---=11
Fig. 1. Layout of the experimental device According to SucbY (1992) the force sensor and the environment is appropriately modeled as spring-damper-masslspring-damper respectively. In Fig. 2 the more general case may be seen when force sensor is placed at the tip of the arm whereas in Fig. 3 the particular implementation of experimental device is reflected when the sensor is firmly connected with the environment. The index s pertains to the sensor, e to the environment
The conesponding Lagrangian is L
Fig.2. Model of the experimental device with tip force sensor
=f[meci2 +qTMq+2qT Mid_qT Kq)
(3)
where m. =m + ITMl Using Lagrange - Euler equtions
;(;.)- : . =Q.. ;=1.2.3•...•• +1
(4)
the system of dynamic equations is obtained meii+1T Mq=rT= Fm
(S)
Mii+M1ii-Kq =11F
(6)
where Fm is the applied motor force, Fm = r.T, T is the motor torque, r is the gearing constant and F is the measured contact force. The further part to model is the actuator. As in Kazerooni et.al. (1986)
Fig. 3. Model of the experimental device with force sensor in the environment
a -ll(t) + 1'(t) 58
= ,u(t)
(7)
is the actuator bandwidth and ~t) is the actuator input force. If for example the Dec:essity of dividing the beam into four elements is proved then the following state vector is defined: XT =[X., X2 ••• XIl1, where XI= d, X:z= ~ X? Q3, xr Q4, xs= 'Is. xr ci,
rearrangement or by ttansformation (11) in obvious way. Starek (1990) writes the equation (12) for the case of controlled vibration, what is also our case, in the form
x,= cb. x.==lh xr«i",
where D is the matrix of controlled variables in physical space of appropriate dimensions (11x11) and J.L (t) = Cl J.L (t). In the sequel the whole system will be modelled by equations (12b) and
Cl
XuF
x= AX+D.J.l(t)+bF
«is
and XII= T. With this state vector the state space equations of the whole system are
(12b)
(13).
3. ROBUST COMPLIANT CONTROL
Y{t) =CX){t)
(9) The aim of this section is to develop the compliant control of the modeled arm. It is desired that by using the state feedback the behaviour of the arm tip will correspond to the target impedance and at the same the spectral andIor modal properties of the beam can be shaped. The approach employed here is the combination of techniques found in Kazerooni et. al. (1986) and Starek (1990). The first author uses the pole-placemcnt technique for achieving the prescribed target impedance while the second author uses the same pole-placement technique for modifying the spectral (eigenvalues) and modal (eigenvectors) properties of mechanical system However, both use different models. Model (12b), (13) developed here includes both modcis and this is prerequisite for achieving the stated aim. Let the control is chosen so that
Here AI is a Sx5 matrix with zero first column, • is vector Sx1, b) is vector Sx1, 0, are zero matrices or vectors of appropriate dimensions. The output vector yet) is constructed in such a way as to have all possible information from the disposal. Thus YI is the sensors at translational position from position sensor, Y2 is the velocity of translational motion, Y3 is the output from the strain-gage sensor, Y.. is the tip
mr+ds+k
position With this in view
c=[~o : : : : ~ : : : : : I
~
0
0
0
0
0
0
0
0
0
0
~
0
0
0
000
0
(10)
0
where Cl, ~ are constants of the strain-gage sensor and the optical sensor. ~ may be placed in any column excluding the first one according to actual placement of its transmitter. Here it is placed in the fifth column to measure the tip deflection. In the next section another form of equations (8) and (9) is needed. This form better corresponds to the algorithm. The states are rearranged so that first two states express the global dynamics ( d and ci), the third state is T and the last eight states express the beam dynamics. These are also rearranged so that the first here are the measured states. Then the new state vector is given by the following transformation
J.l(t) = -GX + qd.F
Here G is the state feedback gain matrix (11xll) and Id feedforward gain vector (11x1). Substituting (14) into (12b) yields
X=(A-DG)X+(Dgd+b)F
mr
where the entries of matrix T are ones and zeros. Using the transformation (11) the equations (8) and (9)
Y=H.X
(1S)
G and Id are designed so that two matrices and one vector, i.e. A-DG, H, and Did + b result in target impedance + ds + k and at the same time the desired spectral and modal properties of the beam structure are reached. The technique for ful1illing the first goal was published in Kazerooni (1986) while the technique for the second goal is in St.8rek (1990) The first technique is applied to the subsystem described by the first three state equations, the second technique for the remaining subsystem. Both are based on the results of Moore (1976). As the model (12b) and (13) includes both the subsystem developped for robot impedance control and the subsystem descnbing the dynamic behaviour of the mechanical part (beam) between the actuator
(11)
x= AX+a,u(t)+bF
(14)
(12) (13)
where A (llxll), H (4xll), b (llxl) and Cl (11x1) are matrices and vectors gained from original matrices and vectors in (8) and (9) by 59
S. REFERENCES
and forceltorque sensor both above goals may be reached under certain circnmst;nas. It is not possible to present all equation here in detail
Ackermann, J (1977). Design by pole placement (in German). Regelungstcchnik, pp. 173-179 Bruce-Boye, C. (1991). Design of a fWl-ordcr
because of limited space. To implement the control Jaw (14) all states should be accessible. However, this is not the case here. States X3 and Xcs to XII are not acces5Ible, therefore it is necessary to introduce the reduced order observer for these states. If the partition of the state vector is defined as X = (ZT, yT)T, where Z includes the inacce5Slble states (x], X6, ... Xll) and Y (X., X2", "-4, xs) accessible states, than by appropriate transformation or rearranging (see Ackcrman (1977), it follows from (12b):
Z =PZ+QY +EJ1+eF
(16)
Y = RZ+SY +LJ1+1F
(17)
observer for flexible robot arm (in German). AutomatisienmgstecJrni/c, 39, pp. 5-10 Eppinger, SD., W:P. Seering (1992) Three dynamic problems in robot force control. IEEE Trans. Rob. Automation. 8, pp.7S1-7S8 Kazcrooni, T. B. et 01 (1986). Robust compliant motion for manipulators, Part I: The fundamental concepts of compliant motion. IEEE J. Robotics and Automation. 2, pp.83-92 Kazcrooni, T. B. et al.. (1986). Robust compliant motion for manipulators, Part n: Design method. IEEE J. Robotics and Automation, 2, pp. 93-10S Moore, B.C. (1976). On the flexibility oJrered by state feedback in multivariable systems beyond closed-loop eigeovalue assignment IEEE Trans. on Aut . Control. 21, pp. 689-692 Pota, H. R (1991). A prototype flexible robot arm-an interdisciplinary undergraduate project. IEEE Trans. Education, 35, pp. 83-89 Schmidt, R (1988). Application of state observers for monitoring and failure detection of mechanical constructions (in German). FortscJrrittbericJrte VDI. Reihe 11, No. 109, VDI Verlag St8rek, L. (1990). Eigenstructure assignment for vibrating mechanical system with control. (In Slovak) StrojnJcky I!asopis, 41, pp.S37-SS8 Suchy, J. (1992). Robot force control using modal obsevers. Proceedings of the Second Int. Symp. Measuremnent and Control in Robotics (lSCMJt92), pp. 199-206 SucbY,J. et.al.. (1993) Identification and force control of one-degree-of-feedom robot Proc. of the Int. Workshop Software Engineering for Parallel Real-Time Systems, pp. 80-86
where P, Q, E, R, S, L are matrices and e, I vectors of appropriate dimensions. The observer is constructed only for the subsystem (16) as
W= KW +NY +QJ1+qF
(18)
(19) where matrices Kt N, Q, J, and vector q are chosen to bring the reconstruction error Z - Z to zero.
4. CONCLUSION In the present article an attemp has been made to
work out the concept of impedance control of one-degree of freedom robot while taking into account the dynamic properties of mechanical system between the actuator and the force sensor. These dynamic properties may be shaped by statefeedback so that some recommendations of Eppinger and Seering (1987) can be satisfied by control rather than by mechanical hardware. Howerer, the algorithmic overhead is still rather high although the measures were adopted to lower it The main overhead is in oftline computing. The future work will be oriented toward refinment of the presented algorithms and experimental verifying of them on more perfect device then that dcscn"bed here. The aim of future work will also be generalizing the algorithm to robots with more degrees of freedom.
60