Jumping Cat Robot with kicking a Wall

Transcription

1 Jumpin Cat Robot with kickin a Wall Masaki Yamakita Yasuhito Omaari and Yasuaki aniuchi Dept. of Control and Systems Enineerin, okyo Institute of echnoloy Oh-okayama Meuro-ku, okyo 152 Japan, yamakita@ctrl.titech.ac.jp Abstract In this paper we study a robotic system movin in a vertical direction mimickin a cat s behavior as a cat kicks a wall to jump up to a roof, which may be an efficient mechanism for vertical movement and the robot system can be considered as one of the prototypes for Super Mechano Systems (SMS). Concept, modelin, controller desin, simulation and experimental results are discussed. 1. Introduction Cats sometimes jump toward a wall and kick it to et to a hiher-place like a roof (Fi.1), and can move in a vertical direction as a result, and the motion seems to be very skillful and efficient. Considerin the movement from a viewpoint of constraints, a robotic system, which realizes the motion to chane its confiuration accordin to the position (on the round, kickin with one le, in the air, etc), can be considered as one of the prototypes for Super-Mechano Systems. Our purpose in this paper is to analyze and construct a control law for a real machine mimickin the cat s motion. In the considered robotic system, the robotic motion is assumed to be constrained in the saittal plane to make the problem simple. In section 2, the dynamic equation of the system is derived, and in the followin section a control method will be discussed. In section 4, we will show some simulation results. Finally we will show some experimental results and future work will be discussed. 2. Modelin 2.1. System structure Even thouh a real cat twists its body after jumpin to et to a hiher place(the roof), we restrict the jumpinmotion in the saittal plane in this paper so that we can analyze and consider fundamental control problems. In order to realize the motions, jumpin from the crow Fiure 1: Motion of the jumpin cat Actuator2 Actuator1 Actuator3 Actuator5 Fiure 2: Robotic system Actuator6 Actuator4 round to a wall and from the wall to the roof, we consider a 7-link robot as in Fi.2 he feature of the system can be summarized as follows. he robot has 6 rotational actuators at it s joints, when the toe makes contact with the floor or the wall, it is assumed that there is no slip between the toe and the contact point alon the surfaces Modelin he whole system is divided into two 4-link serial links at point X(Fi 3) virtually, and a holonomic body constraint to keep the body as one straiht link is introduced.

2 l / As eneralized coordinate systems, we use!"#$%'&("#)+*, (1) " - # - $(" - &( - )+*, (2) - */ (3) houh these coordinate systems are redundant and the system description becomes complex, the advantaes are as follows: he problem here is how we can jude the mode. he answer lies in the understandin of the constraint force a P (Fi 4). In the case of toe bein on the round, a b is a horizontal constraint force and a c is vertical one, and if the a c is equal to zero and the acceleration upward is positive, the constraint should vanish(a S= ) and toe can move upward. When the toe is on the wall, switchin timin depends on the aed vise versa. Body PSfra replacements 1 1 A 4-link dynamic equation is simpler than a 7- link one and we can use the same equation for each link, PSfra replacements this coordinate system is very useful for judin the timin of switchin the constraint on the toe, which will be mentioned later. round For the followin discussion, Jacobian of the body constraint is defined as follows: "7 '8 :9 6;7 8 < = 9 6;7 '8? / (4) AE IKJ F IKJ F L.N F O x aef aed 2.4. Dynamic equation Fiure 4: Constraint force floor he dynamic equation for the system with the redundant coordinate systems is considered in this section. Inorin the constraint, two 4-link manipulator s dynamic equations are described as 7 '8h ji 4k7 < '8 Miml < 7 %8 Sn (7) = = - l l - nx 4 4 = = 4 - n n - Fiure 3: Robotic system and coordination 2.3. Variable constraints We assume that enouh constraint force is exerted when the toe makes contact with the round or wall, and holonomic constraints 4 P7.QSR 2 8 = are introduced accordin to the state of the system, mode is an index that indicates the state of the toe s contact. For example, when only the hind toe is constrained to the floor or wall, 4 P7 UQSR 2% 8 becomes 4 P%7.QVR 2 8 -XWZY[-\+R!] ^ 3 - W_ - \+R!] ^ 3 (5) and we can calculate the Jacobian,as P%7 '8 :9 P%7 '85< = 9 P`7 8 4 P 7 '8 / (6) In order to chane position constraint to acceleration constraint, we differentiate (4) and (6), and in order to keep the constraint, constraint forces, 9 6 à 6, and 9 P aep are introduced, and followin simultaneous equations are used to express the system includin all constraints: 7 '8h ji 4o7 < '8p< jivl 7 %8 qn W 9 6 ae6 W 9 P aep (8) 9 6 h W 9 < 6 < (9) 9 P h W 9 < P < / (1) From these equations, acceleratin vector h, and constraint forces a 6 and a P can be calculated. herefore, constraint force and acceleration can be used for judin the chane of the mode. Collision with the wall or other thins is assumed to be perfectly inelastic, and the state will shift to that of the under constraint just after the collision which is modeled as effects of impulse forces.

3 = 3. Controller Desin Since the initial confiuration is very important for the robot s jumpin motion, it is determined by stochastic dynamic manipulability measure. For dynamic control of the robot s jump, we pay attention to the motion of the center of mass mainly, and the proposed method is derived as if the center of the ravity is moved by a sprin connected to a virtual wall Stochastic dynamic manipulability measure to 4 t _ 4 i 9 } Ym~x 9 < } l to _ l n t _ n Y :9 } ~x 9 } _ V W 9 } ~ 9 } ~x / For this description of the system, stochastic dynamic manipulability measure is defined as } ˆŠ `ô } ˆ ŒŽ Ž! ~x ô ƒ 7 špœ 9%{;9 { *ž = 8 (13) 7 špœ 9 { 9 { * = 8 Fiure 5: Realizable acceleration for the center of mass here exists a lot of possible postures of the 7-link robot before the jump, and it is not easy to determine what kind of pose is suitable for the jump. herefore, we use a measure to decide the position for jumpin, i.e., stochastic dynamic manipulability measure, which evaluates an expected required torque to realize the desired acceleration. Sub-optimal confiuration for the jump from the round and from the wall based on the stochastic dynamic manipulability measure can be determined by a numerical optimization. In another words, at first we determine the pose which easily achieves the desired acceleration and anular acceleration, and the robot is set to the posture before the jump. Let s assume that, 7 sr r 8 and 6 indicate the coordinates of the center of mass, and anle of the body, respectively. By eliminatin the constraint force a from the system dynamics, we have and n;tu t ji h 4 t Mivl < t (11) xwyz9{ < < swy xr r 6 (12) ƒ : Stochastic dynamic manipulability measure Ÿ : Weiht matrix indicates the direction to accelerate 9%{ : Jacobian matrix. o determine the sub-optimal confiuration with respect to the measure under the constraints, ƒ. is updated by the followin iteration: ƒ ƒ. 7 i '8 ƒ. 7K 8Hi 7 8 (14) ; (15) ƒ ( ª «(16) Uis basis of 7 9 { 8. If we use the iteration and «is a semi-positive constant, Ÿ ƒ. is increased as ƒ. ƒ 7 i '8 ƒ. 7K 8sī 7 7K «± ƒ. 7K 8 / 3.2. Jumpin control (17) Only jumpin with both les or with only the hind le is mentioned here. When the cat is in the air, we just adopt feedback control so that the posture of the les converes to a desired one determined beforehand by the above method, which is ready for the next jump. In order to derive a proposed control alorithm, we re-describe the system, and we pay attention to the center of mass and the body s anle. Usin a coordinate chane from to swk² r. r " 6 *, the system can be expressed as xwy:³ h 7 '8 < n iv 7 '8 < nµz n n - * (18)

4 h h = i t < À Í Ã Ÿ Î Î = ³: 9{ ~x _x 9{ t ~x 7 4 t iml t 8 W 9%{ < < _ µ W 9 } ~x 9 } ~x and 9 { is the tanential map of the coordinate transformation. For the system representation, h w is determined so that the motion of the mass center follows a simple mass-sprin model Jumpin with both les Because of the redundant system, n is selected to minimize the followin criterion function as: 9 jç Ÿ w 7 ³Gn W W h }U.¹ 8 Èi É n ŸªÊ n (21) Ê is an appropri- Ë = = w = Ë = = = Ì and Ë Ì are positive constants, and Ÿ ate positive weiht matrix Jumpin only with the hind le (22) PSfra replacements Wall 7 xr h h r 8 y }!.¹ PSfra replacements Wall ÃÎ 7 sr h h r 8 y Floor x x Floor Fiure 6: Model matchin 1 Fiure 7: Model matchin 2 }U.¹ : Desired anular acceleration }!.¹ : Desired anle : Normal unit directional vector : Coefficient of the virtual sprin º : Lenth of the virtual sprin : Mass of the virtual body he desired acceleration of the body h }!.¹ y»s¼ is, as shown in Fi.6, determined so that the virtual mass concentrated to the mass center moves as if it is pulled by a stron sprin stuck to the wall, and a desired anular acceleration of body h }U.¹»½¼ is desined to rotate in the desired direction as h }U.¹ }U.¹ }U.¹ ¾ }!.¹ W ÂÁ7 < 6 W }! ¹ 8 (19) (2) is a unit vector toward the desired direction for jumpin, and à is a vector from the center of mass to the wall alon to the line from the toe to the center of mass, and º Ä ÃÆÅ indicates the sprin s lenth. Á is an appropriate feedback ain. à : Normal directional vector of the sprin ÃÎ : Modified unit vector for desired direction Í : Modifyin ratio of the direction Î : Normal unit directional vector º : Coefficient of the virtual sprin : Lenth of the virtual sprin : Mass of the virtual body After the motion with the two les, the front le will naturally lift. In order to control the center of mass, only the hind le s actuators are mainly used, and as it is difficult to control all of the three deree of freedom,the sprin-mass model is modified to control the body anle indirectly First, à is determined as a directional vector by connectin the toe and the center of mass, then, it is adjusted to a directional vector ÃeÎ due to the sin of the error of the anular velocity by Í as follows: à Πà W = Í ^UÏ'] 7 }U.¹ W < 6 8 * (23) and, desired acceleration }U.¹ is determined by h }U.¹ ¾ (24)

5 Ÿ Ù º Ç ÈÄ and the weiht Ÿ Ã Î Å Î Ç t is also chaned to tâ Ë = = = Ë = = = = Ã ÎÐ Ã Î / (25) As in the previous method, n - is determined to minimize the followin criterion function: 9 Ç Ÿ t 7 ³Gn W W h }!.¹ 8" i É n Ÿ Ê n (26) Y[m] X[m] Fiure 8: Initial posture: ÚÈÛ ÜGÝmÞß;à á.4 n;ñ is determined locally. 4. Simulation In order to examine the validity of the proposed method, we conducted numerical simulations of the jump for a model of an experimental system. In the optimization and the simulations,it is assumed that the fore le and hind sub-systems have the same parameters Pose optimization We obtained a sub-optimal posture, usin the stochastic dynamic manipulability measure from some of the initial poses by a radient search. One of the results is shown in Fi.8 and Fi.9. By the procedure, the manipulability measure increased by about 2%, from ^ ƒ. SÒ Ó / Ô to ^ ƒ Õ'Ò / Ö Cat jumpin with kickin a wall In the simulation, we assumed that a wall is located at µ W = / Õ [m], and the roof at = / Ö [m]. For the values of control parameters, we used Ó='= [N/m], W Õ'= p e2 Ð ^*, and when the cat jumps from the floor to the wall, we set WMØ ='=5 p e2 Ð ^!*, from the wall to the roof. Since there is no systematic rule to determine the value of parameters, they are determined by trial and error. As shown in Fi.1 the sequence of the jumpin motion is shown. It is shown that the robotic cat jumped up to a roof at y=.5[m] after kickin the wall with a rotation. 5. Experimental Results In order to check the validity of the proposed method, we constructed an experimental system. Y[m] Y[m]Ù X[m] Fiure 9: Sub-optimal posture: ÚÈÛ Ü Ýãầ Þ;à ä X[m] Fiure 1: Simulation result: Jump kickin the wall 5.1. System confiuration We desined a 7-link robotic cat shown in Fi.11. he system confiuration of the whole system is iven in Fi.12. he followin describes the details of the robotic system. Since motors are too heavy to be installed in the robot, power of the actuators is supplied from outside by wires (Fi.13), and the weiht of the wires is compensated for by a counter weiht. he position and the anle of the toe is measured by the CCD camera. In order to measure the anles of the joints, potentiometers are used since encoders of the motor are useless due to the extension of the

6 ñ Motor Joint Fiure 11: Cat robot Fiure 13: Motor and wire system position & anles CCD camera Actuator (motor) P C D/A Wires (controller) Strain ae A/D torques Potentiometer A/D anles Fiure 12: System confiuration CA ROBO Fiure 14: Potentiometer wire. Because of the wires extension, there are a lot of time delays in the wire system. In order to control the anle in such a bad condition, slidin mode control are introduced as, n W æåuç è éëê ^.Ï'] 7 ì 8 (27) ì < { i aå!ç è é ê { (28) { W and a is a positive constant. Furthermore, we introduced strain-aes in the wire system between motors and the body of the cat to measure the equivalent torques exerted to the joints(fi.15), since the power applied by the motors is lost due to friction in the wire system. ension of the wire is measured usin the bendin deformation of a plate stretched from both sides(fi.16). Usin the information from the strain ae, a minor loop compensation is constructed as,, Ê ½ò n W n n W } 8 Ê ì ÂíMî ïð ^UÏ'] 7 ì (29) 2 3 (3) n, n ò is the measured torque Learnin control he problem due to the time loss and disturbance of the wire system is too serious, it was difficult to apply the proposed method directly, therefore the learnin control [6] is applied as follows. In the -th trial, the slidin surface is modified as 3 ì 7 8 < { 3 7 8xi a { 3 7 8xiªó 3 î!7 8 (31) ó î indicates the learnin term which is updated by the followin alorithm. ó îõô ó î W Í î.ö7 W î 8 (32), ö is a learnin filter, î is the experimental data of the th trial, and Í î is positive coefficient. he desired anle is determined based on the simulation data. (See the details in [6].) One of the results is shown in the Fi.17. It is observed that the output of the latter trial is obviously improved. 6. Conclusion and Future Work We proposed a jumpin method of a robotic cat usin sprin-mass model matchin, and we conformed the

7 Anle [deree] Fiure 15: Strain ae system Strain Gae Fiure 16: Principal of measurin torque desired trial 1 trial 15 under Grant COE #9CE24 References [1] D..Greenwood, 1977, Classical dynamics, Prentice Hall. [2] S.adokoro and.akamori, 1993, On Dynamic Evaluation of Dexterity of Robot Manipulators Considerin Deviation of Acceleration, rans. of SICEø Vol.29, No.4, pp [3] S.adokoro and.akamori, 1993, An Evalution Method for Dexterity of Robot Manipulators Considerin Deviation of Direction of Motions, rans. of SICE, Vol.29, No.6, pp [4] Xiaopin Yun and Nilanjan Sarkar, 1998, Unified Formulation of Robotic Systems with Holonomic and Nonholonomic Constraints, Vol. 14, No.4, pp [5] N.Ueno, 1999, A Study on efficient vertical movement of Robotic Systems In spired by Cat s Jumpin, Bachelor thesis, Dept. of Control and Systems En. okyo Inst. of ech. [6] M.Yamakita and K.Furuta, 1992, A Desin of Learnin Control System for Multivariable Systems Asia-Pacific Enineerin Journal(Part A), Vol.2, No.1, pp ime [s] Fiure 17: Learnin result: Anle of the rear toe effectiveness of the control law by simulation, in which the robotic cat could jump towards the wall and land on the wall. For it s initial posture evaluation, we used stochastic dynamic manipulability measure. In order to realize the real robot, VSS control and learnin control were applied and much proress are conformed. But the jumpin has not completed yet, and future work should include: 1 completion of the jumpin with the real robot, 1 learnin control in the task space, 1 realization of a 3D cat robot, and development of the control into the three-dimensional system. Acknowledment his research was partially supported by the Scientific and Research Foundation of the Ministry of Education