A QUADRATIC PROGRAMMING APPROACH TO THE MINIMUM ENERGY PROBLEM OF A MOBILE ROBOT

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1 A QUADRAIC PROGRAMMIG APPROACH O HE MIIMUM EERGY PROBLEM OF A MOBILE ROBO Alain Segundo Potts*, José Jaie da Cruz* Reinaldo Bernardi** *Departent o elecounications Control, São Paulo University, São Paulo, Brazil **Genius Institute o echnology, São Paulo, Dept. o elecounications Control, São Paulo University, Brazil alain_do@yahoo.co, jaie@lac.usp.br, rbernardi@genius.org.br Keywords: Abstract: Mobile robots, quadratic prograing, iniu energy, optial control. As a consequence o physical constraints o dynaical nonlinearities, optial control probles involving obile robots are generally diicult ones. Many algoriths have been developed to solve such probles, the ore coon being related to trajectory planning, iniu-tie control or any speciic perorance index. evertheless optial control probles associated to obile robots have not been reported. Miniu energy probles subject to both equality inequality constraints are generally intricate ones to be solved using classical ethods. In this paper we present an algorith to solve it using a Quadratic Prograing approach. In order to illustrate the application o the algorith, one practical proble was solved. IRODUCIO. Preliinaries Mobile robotics is an iportant research area or its study any researchers have dedicated a lot o tie to it. here are any probles in obile robots that are not present in industrial robots. Probles with posture aintenance, localization, equilibriu energy consuption are coon both at design operation ties. Libed robots can be considered an iportant engineering conquest due to the act that they have larger obility, lexibility reedo o oveents than any other autoatic achine (Dude, ). Research in this area requires strong nowledge o echanics, electronics, coputation, eventually bioechanics. Libed robots are capable o waling clibing have been developed around the world. (Arada et al., 3; Vir, 5). Soe o the have been used to inspect bridges (Abderrahi et al., 999) pipelines (Galves; Santos; Peier, ). Presently the literature reports just two robots with the ability o tree clibing. he irst one was developed at the Waseda University, Japan, the second one is the RiSE robots (Robots in Scansorial Enviroents) (Saundersa et al., 6) developed at the United States. he RiSE robot is a eber o a new class o clibing robots whose design is based on anials. For all o the, the energy consuption is a big proble. Generally the battery is their heaviest part since a considerable aount o energy is necessary to drive the legs otors. ypically a leg has the or o a serial echanis with a highly nonlinear dynaics. his is one o the reasons that ae to ind an optial control law a diicult proble. A second reason is the presence o both equality inequality constraints iposed on the syste e.g., the actuator o each joint is subject to saturation (Spont et al., 989). Many authors have wored in the optiization o robots operation in the context o trajectory planning. (Lin et al., 983; Garg et al., ; Luo et al., 4). he probles considered in general aied to iniize the tie or soe quadratic perorance index. evertheless in the ajority o the both the Coriolis Centriugal ters were oitted. Optial trajectory control systes usually can be built by solving two associated sub-probles: i) the optial trajectory planning (OP); ii) the trajectory tracing control (C). Since there are any coplex constraints conditions concerning robot ineatics dynaics, the corresponding 45

2 ICICO 8 - International Conerence on Inoratics in Control, Autoation Robotics algoriths or solving the OP proble have been iproved in recent years (Luo et al., 4). Even though good results have been obtained with the OP proble or both anipulators industrial robots, this is not the case or obile robots. A iniu energy orulation ay be an interesting approach or obile robots, particularly in applications where the battery weight is a critical issue. Miniu energy probles ay be diicult to solve by classical ethods since they involve both the nonlinear dynaics o the robot a set o constraints. In this paper a Quadratic Prograing approach to the iniu energy proble o a obile robot is proposed. he ethod is based on the discretization o the proble. uerical tests were perored or Kaanbaré, a robot currently under developent at the Autoation Control Laboratory (LAC), University o São Paulo. he prototype o the Kaanbaré plator presented in this wor was developed considering certain capabilities (abilities), such as: locootion in irregular environents (unpredictability o the branch coplexity that copose a tree), surounting obstacles (nodes sall twigs), tree clibing descent without rising stability, eeping low structural weight (echanics + electronics + batteries). (Bernardi et al., 6) Each leg has three rigid lins connected by two rotational joints with one degree-o-reedo (d.o..) each. he irst lin is connected to the plator by a two d.o.. rotational joint (Fig. ).. he Robotic Plator Kaabaré is a bioietic robotic plator, i.e., a robotic plator inspired in nature, with the purpose o clibing trees or environental research applications (see Fig. ). More speciically the plator locootion is inspired in the or lizards clib trees. he ain characteristics sought in the deinition o the Kaanbaré plator were: locootion in irregular environents (unpredictability o the branch coplexity that copose a tree), surounting obstacles (nodes sall twigs), tree clibing descending without rising stability, eeping low structural weight (echanics + electronics + batteries). (Bernardi et al., 6) Figure : Robot Kaanbaré. Figure : Lib o the Kaanbaré Plator. All joins are controlled by DC otors. he control proble or obile robot is the proble o deterining the tie history o join required to cause the end-eectors (the gripper) executed a coed otion. here are any control techniques ethodologies that can be applied to the control o libed robots. he particular control ethod chosen as well as the anner in which it is ipleented can have a signiicant ipact on the perorance o the robot consequently on the range o its possible applications. Optial control o energy can be an interesting policy when the robot ust operate autonoously or a long tie. he control proble or the tree clibing robot considered here is to deterine the tie history o each lib joint required to cause the leg to ove ro an initial angle to a inal one in such a way that the energy loss in the otors is inial. Kaanbaré is the word in the upi indian language or chaeleon. 46

3 A QUADRAIC PROGRAMMIG APPROACH O HE MIIMUM EERGY PROBLEM OF A MOBILE ROBO FORMULAIO AD SOLUIO OF HE OPIMAL COROL PROBLEM. he State Space Model his wor consider the control strategy naed independent joint control. In this type o control each axis o the lib is controlled as a single input /single output syste. Any coupling eects due to the otion o the others lins is either ignored or treated as a disturbance. (Spont et al., 989) he space state odel that describes the dynaics o the DC otor located at the i-th joint ( i n ) o the leg can be expressed by the equations: where x = Ax + Bu () y = Cx + Du + Eτ () (3) A =, b / J (4) B = K /, J C = [ K b / R a ], (5) D = /, (6) R a ( ) E = r / R K, (7) a θ (8) x =, θ r u = V τ. (9) K J is the oent o inertia o the rotor, b is the viscous daping coeicient o the echanical syste, V is the arature voltage (control variable), R a is the arature resistance, Y = ia is the arature current, K is the torque coeicient o the otor, K is the counter-electrootive coeicient, b τ is the load torque, r < is the gear reduction actor, θ is the angular position o the rotor θ is the angular speed o the rotor. otice that the input variable u depends on both the control variable V the load torque τ, which also depends on V through θ θ. he load torque τ, which depends on the robot dynaics, is given by : (Spong, 989) ( ) ( )[ ] ( ) ( ) τ = M q q + B q qq i z + C q q + G q () where ( ) B q is the Coriolis torque atrix, C( q ) is the centriugal torque atrix, G( q ) is the gravitational torque M q is the leg inertia atrix, ( ) vector, [ qq i z] = [ qq qq 3 q n q n], i z: < i< n,< z< n q = q q q n q = [ q q q ] n, where q i, i n, is the generalized coordinate o joint i n is the nuber o joints o the leg. Soe dynaical eects lie riction were not included in () although they ay be signiicant or soe libs. In addition, a ore detailed odel o the leg dynaics could include various sources o lexibility, deection o the lins under load vibrations (Borrow et al., 4). evertheless, this odel is suiciently accurate or our purposes since these eects are not signiicant or the leg under consideration. One o the characteristics o the independent joint odel is that τ is ultiplied by the gear reduction r. hus eect o the gear ratio is to reducing the coupling nonlinearities presents in dynaics o the libs. he solution to equation () is given by: At () ( ) t At ( ς ) ( ς ) x t = e x + e Bu dς (). he Optial Control Proble In this section it is assued that τ(t) is nown or all t in the interval [ t ]. he perorance index adopted is the Joule loss in the arature resistance o each otor during the otion: 47

4 ICICO 8 - International Conerence on Inoratics in Control, Autoation Robotics t in ε = P dt () where t is the tie required to ove the joint ro the initial to the inal position, P is the power dissipated: where Y was deined in (). he optial control proble is subject to the ollowing constraints: Kb r x () t u() t () t I R + a R + a RaK τ, (4) Kb r I x () t u() t () t R + R + R K τ, (5) a a a ( ) V u t V, (6) ( ) x t = q r, (7) ( ) x t =, (8) where I V are, respectively, the axiu arature current axiu arature voltage o the otor t t. With no loss o generality we tae x () =. Considering the otor initially at rest, x () =. his type o proble is hard to solve generally involve a great coputational eort (Kir, 998). Mobile robots require a quic solution to solve it in real-tie is practically ipossible. hese are aong the reasons or which we decided to loo or another ind o solution..3 Discretization Let us to deine the tie-step as: t Δ t = = t t ( ) [ ' ' ' ' τ ' '][ τ] P = Ri = RY = R xc+ u D+ E Cx+ Du+ E (3) aa a a < (9) where can be chosen suiciently large to discretize t. hen it is assued that u( t ) is a stepwise constant unction u is used to denote the value o u( t ) or all t in the -th tie interval [ t, t ). aing into account equations (), (), (3) (9), the unctional ε can be rewritten as: = = in ε = P Δ t = R i Δt a a () the solution o syste () as: At x( t) = e Γ IU () where, t t t Aς Aς Aς Γ= e Bdς + e Bdς + + e Bdς, () t [ ] U u u u I = (3) e e en = (4) Where e i = or i e i = or < i. Matrix Γ can be calculated oline since it depends only on the otor paraeters. Fro equations () () it is possible rewrite () as: in ε τ σ ( U) = U QU + K U + (5) where atrices Q K depend only on the otor paraeters σ is a constant. he constraints or the proble are: K r I x u I R + R + R K τ b a a a (6) V u V (7) x = q r (8) x = (9) where constraints (6) (7) apply or all,. Since σ is a constant, it is not relevant or the optiization. 48

5 A QUADRAIC PROGRAMMIG APPROACH O HE MIIMUM EERGY PROBLEM OF A MOBILE ROBO he proble given by equations (5)-(9) has thus the or o a Quadratic Prograing proble which is certainly the ind o nonlinear prograing proble closest to linear prograing ro analytical coputational point o view. Solution or this ind o proble can be eiciently ound by nuerical ethods. (Avriel, 976) (Winston, 995)..4 he Algorith An iterative algorith to solve the iniu energy proble based in equations (3) () is proposed in this section. o start the algorith it is assued that τ : = τ or all,. hen Quadratic Prograing proble o iniizing the unction () subject to constraints (6) to (9) is * solved. Denote by U the optial solution or this proble. Using equation (7) recalling equation (), both the otor angular position θ ( ) angular speed θ ( ) can be evaluated or all,. he second step o the algorith begins by using the leg dynaical equations () to evaluate a new torque tie history τ or all,. he new Quadratic Prograing proble is then solved * U is obtained. θ ( ) θ ( ) are evaluated or all,. τ τ ε j j he process is repeated until + < τ or a given accuracy ε τ two consecutive steps j j+. When convergence is attained the optial j+* vector o arature voltages V can be evaluated using equation (9). he algorith is expected to converge since the gear reduction ratio r is generally sall the eect o the torque on the otor dynaics is correspondingly sall too. he algorith ay thus be suarized as: Algorith τ = j = repeat j = j + j in ε( U) = U QU + τ K U subject to constrains : equations (6) to (9) j j j* x = b x + K U J J j j [ q q q ] = r[ x x x] j τ + ( ) ( )[ ] ( ) = M q q + B q qq + C q q + G( q) j j ax τ + τ ε until ( ( ) τ ) 3 SIMULAIO RESULS his section presents the results obtained ro the application o the algorith above to a leg siilar to that o the Kaanbaré plator. he algorith code was written in MatLab. he ollowing data were used: t = s, =, ετ = π θ θ θ = π 4 Algorith convergence occurred in 5 iterations. able shows the algorith steps until the optial solution is reached. he overall processing tie was quite sall. 5 Figure 3 show the optial solution U or each otor. Voltage [volts] Voltage [volts] Voltage 3 [volts].5 Optial Control Optial Control Optial Control ie [s] Figure 3: Optial Solution. * 49

6 ICICO 8 - International Conerence on Inoratics in Control, Autoation Robotics Iteration (j) able : Miniu Energy Consuption. Energy Consuption P t [watts] = ε = Δ Joint (Motor ) Joint (Motor ) Joint 3 (Motor 3) Figures 4 5 show ineatics o each joint; the angular position the speed acceleration achieved or the optial controlu. 5 * τ [n] orque rad rad 5 Join Position Join Position Join Position 3 τ [n] τ [n] orque orque ie[s] rad ie [s] Figure 4: Joints Positions. Figure 6: orques or each otor. In this particularly case the otor s 3 have the sae torque. [rad/s] [rad/s] [rad/s] -.5 Angular Speed ie [s] [rad/s ] [rad/s ] [rad/s ].. Angular Acceleration ie [s] Figure 5: Speed s Joints Acceleration s Joints. 4 COCLUSIOS his paper discussed the orulation solution o an iportant proble related to obile robotics: the iniu energy loss proble. An optial control proble was orulated to represent this case. Ater discretization in tie the optial control proble was rewritten in the or o a Quadratic Prograing proble whose solution could be obtained eiciently. he algorith was tested or a leg siilar to that o the Kaanbaré plator. Although the whole proble is nonlinear quite coplex the algorith converged quicly in all the 5

7 A QUADRAIC PROGRAMMIG APPROACH O HE MIIMUM EERGY PROBLEM OF A MOBILE ROBO tests perored by now. It is thus expected that the it can be used to operate in real-tie. REFERECES Vir, G. S. 5. he CLAWAR Project: Developents in he Oldest Robotics heatic etwor. IEEE Robotics & Autoation Magazine, pp. 4-. Winston, W Introduction to Matheatical Prograing. Applications Algoriths. Duxbury Press. Caliornia, nd edition. Abderrahi, M., Balaguer, C., Gienez, A., Pastor, J. M., Padrón, V. M., 999. ROMA: A Clibing Robot or Inspection Operations. In, Proceedings o the IEEE International Conerence on Robotics Autoattion, Vol 3, pp -5. Arada, M., Santos, P. G. de, Jiénez, M. A., Prieto, M. 3. Application o CLAWAR Machines. In, he International Journal o Robotics Research. Vol., nº3 4, pp Avriel, M, 976. onlinear Prograing. Analysis Methods. Prentice-Hall. ew Jersey, st edition. Bernardi, R., Cruz, J.J. 6. KAMABARÉ. A treeclibing bioietic robotic plator or environental research. In, International Conerence on Inoratics in Control, Autoation Robotics. Bobrow, J.E., Par, F.C., Sideris, A. 4. Recent Advances on the Algorithic Optiization o Robot Motion. Departaent o Mechanical Aerospace Engineering, University o Caliornia. Dude, G., Jenin, M.,. Coputational Principles o obile Robotics. Cabridge Press. st edition. Galvez, J. A., Santos, P. G. de, Peier, F.,. Intrinsic actile Sensing or the Optiization o Force Distribution in a Pipe Crawling Robot. In, IEEE/ASME ransactions on Mechatronics, Vol. 6, n. Garg, D. P., Kuar, M.,. Optiization hecniques applied to ultiple anipulators or path planning torque iniization. In Eng. Applications Art, Int., Vol 5, pp 4-5. Kir, D., 998. Optial Control heory an Introduction. Dover Publication, ew Yor, nd edition. Lin, Ch., Chang P., Luh, J.Y.S. 983 Forulation Optiization o Cubic Polynoial Joint rajectories or Industrial Robots. In IEEE ransactions on Autoatic Control, Vol AC-8, o, Deceber. Lu, B., Cooe, D., Galt, S., Collie, A., Chen, S. 5. Intelligent legged clibing service robot or reote Maintenance applications in hazardous environents. In Robotics Autonoous Systes. Accepted June 5. Luo, X., Fan, X., Zhang, H., Chen., 4. Integrated Optiization o rajectories Planning or Robots Manipulators Based on Intensiied Evolutionary Prograing. Proceedings o the 4 IEEE, International Conerence on Robotics Bioietics. Saundersa, A., Goldanb, D., Fullb,R., Buehlera, M. 6. he RiSE Clibing Robot: Body Leg Design. In, Proceeding o SPIE, Vol. 63, 637. Spong, M., Vidyasagar, M., 989. Robot Dynaics Control. John Wiley & Sons, Canada, st edition. 5