Parts Assembly by Throwing Manipulation with a One-Joint Arm

Transcription

1 21 IEEE/RSJ International Conference on Intelligent Robots an Systems, Taipei, Taiwan, October, 21. Parts Assembly by Throwing Manipulation with a One-Joint Arm Hieyuki Miyashita, Tasuku Yamawaki an Masahito Yashima Abstract The present paper proposes the learning control metho for the throwing manipulation which can control not only the position but also the orientation of the polygonal object more accurately an robustly by low-egree-of-freeom robotic arm. We show experimentally the valiity of the propose control metho with the one-egree-freeom robotic arm. We also emonstrate the usefulness of the throwing manipulation by applying it to sorting task an assembly task on experiments. I. INTRODUCTION In the present logistics systems or automate assembly lines, many conveyors an vehicles are use for transporting packages or parts, which make the spee of the physical istribution an the prouction flow be slow an the plant an equipment costs be high. Transportation by throwing objects can prouce the following avantages: (1 mechanical equipments are simplifie or less require; (2 flexibility of the systems is enhance; an (3 the transportation processes are speee up [3]. We consier the applications of the throwing manipulation to transporting, orienting, sorting an assembling various types of objects. Fig. 1 shows one of the application examples, in which a one joint robotic arm throws objects with various orientation carrie by a belt-conveyer an sorts them with appropriate orientation in a storage pallet with an array of nests, each nest holing a single object. The objects in the storage pallet will be then store in a warehouse or will be graspe by a robot gripper in assembly lines. Instea of a manipulation of an object with grasp, the throwing manipulation has rown attention recently in the fiel of robotics [1], [7], [8], [9], [1]. This is because the throwing manipulation cannot only manipulate the object outsie the movable range of the robot, but also manipulate the position an orientation of the object arbitrarily by a robotic arm with fewer control inputs. Lynch et al [5], [6] present the motion planning for the ynamic manipulation such as throwing an object an propose the open-loop controller base on nonlinear optimization techniques. However, they o not aim at throwing the object accurately an robustly. The experimental results show that the controller is subject to the moeling error an sensing error. Aboaf [2] applies a learning control metho to the throwing of a point mass object to overcome these problems. However, constraints on ynamical robot system an actuator s performance limits, etc are not consiere to fin control inputs. The authors [7] propose the control metho for throwing point mass object accurately, which The authors are with Dept. of Mechanical Systems Engineering, National Defense Acaemy of Japan, 1-1-2, Hashirimizu, Yokosuka, JAPAN {g4791, yamawaki, yashima}@na.ac.jp Fig. 1. Object Throwing Arm Storage pallet Application of the throwing manipulation combines the learning control techniques an the nonlinear optimization techniques. As far as I know, there has been no work which takes into consieration the orienting a polygonal object accurately by a throwing manipulation, which is very important in the case of sorting an assembling various types of objects, as shown in Fig. 1. The present paper proposes a learning control metho for the throwing manipulation which can control not only the position but also the orientation of the polygonal object more accurately an robustly to uncertainties. We show experimentally the valiity of the propose control metho an the feasibility of assembly process by the throwing manipulation. The paper is organize as follows: In Section III we stuy the ynamical constraint on the throwing manipulation. Section IV presents the motion planning by optimization, an Section V proposes the learning control metho. Finally, Section VI presents experiments with a one-egree-freeom robotic arm. II. NOTATION AND ASSUMPTIONS We consier a throwing manipulation of a polygonal rigiobject by the rotational one-egree-of-freeom robotic arm in the vertical plane, as shown in Fig. 2. A reference frame x-y is fixe at the pivot point of the robotic arm. The gravitational acceleration g < acts in the y irection. The position an orientation of the object in the x-y frame is escribe as (x, y, φ. The angle of the arm is θ, whichissetto eg when the arm is horizontal. The top surface of the arm intersects with the pivot point. The arm throws the object by counterclockwise rotation. It is assume that the thrown object is capture by a storage pallet fixe at a goal without slipping, rolling or rebouning in orer to simplify the analysis. A frame u-v is locate on the center of mass of the object, where the +v irection is opposite to gravity when the arm angle is θ =. The moment of inertia of the object an its

2 21 IEEE/RSJ International Conference on Intelligent Robots an Systems, Taipei, Taiwan, October, 21. y Fig. 2. x r l θ B MB v N lb la Notation for throwing manipulation g A MA u mass are J an m, respectively. The polygonal object makes line contact with the arm surface, an the vertices of the contacting ege are A an B. The istance in the irection of u between the object center of mass an the vertices A an B are l A an l B, respectively. The height of the center of mass from the arm surface is. The istance between the object center of mass an the pivot point is l. When the arm is horizontal, the x location of the object center of mass in the reference frame is r. The stoppers with a sharp ege are rigily attache to the top surface of the arm to stop the object from sliing on the surface. We assume that there are no friction between the object an the stoppers. III. THROWING MANIPULATION The motion of a throwing manipulation is ivie broaly into two phases before an after an object s release. A. Motion before Object s Release The motion of the object relative to the arm is constraine in orer not to slip, roll or break contacts before the object s release. The motion of the arm is also constraine by the joint actuator performance. 1 Object Constraints: Since the object oes not slie on the arm surface by the stoppers, we consier only the vertical motion with respect to the arm surface an the rolling motion about the vertices A an B. The normal reaction force N acting on the object can be escribe as N = mr θ m θ 2 + mgcθ (1 The reaction moment M A an M B from the arm about each vertex A an B can be escribe as, respectively, M A = J θ ml A r θ + m 2 θ + ml A θ 2 + mr θ 2 ml A gcθ mgsθ (2 M B = J θ + ml B r θ + m 2 θ ml B θ 2 + mr θ 2 + ml B gcθ mgsθ (3 where sθ =sinθ an cθ =cosθ. To achieve the stable throwing, the robotic arm has to accelerate such that there is no relative motion between the object an the arm before the object s release, which is calle the ynamic grasp. We take into account the following constraints to achieve the ynamic grasp. (i Contact Constraints; which make the object maintain the contact with the surface of the arm. N(t > (4 (ii Un-rolling Constraints; which prevent the object from rolling about the vertices A an B. M A (t < (5 M B (t > (6 2 Arm Constraints: The arm is subject to the constraints on the actuator performance an the robot mechanism. (i Joint Angle Constraints θ min <θ(t <θ max (7 To release the object in the right half space, we set θ max = π/2 an θ min = π/2, respectively. (ii Actuator s Torque-Velocity Limits θ min < θ(t < θ max (8 τ min <τ(t <τ max (9 τ max = ξ( θ max (1 where (1 shows the relationship between the maximum torque an maximum velocity of an actuator, which is etermine by the actuator s operation range. B. Motion after Object s Release After the object s release, the object s motion uring a free-flight is etermine by the arm s state (θ t, θ t, a throwing raius l t at a release point an a flight time t f. Since we have four variables (θ t, θt,l t,t f as parameters in the throwing manipulation, the one-joint robotic arm can control four object s state variables in a six-imensional planar state space (x, y, φ, ẋ, ẏ, φ through one throwing. In this paper, in aition to the object s position an orientation (x, y, φ, we choose the vertical velocity ẏ at a goal as the controlle object s variables. By specifying an appropriate velocity ẏ, we can fin object paths with various loci which reach to the same object s position an orientation, which helps to plan the object s path so as to avoi obstacles in front of the goal. We can also ajust an impact velocity at laning. The conition for the velocity at the goal expans the feasibility of various throwing tasks. We efine the object s state variables in a flight time t f, which can be controlle by the one-joint robotic arm, as z =[x(t f, y(t f, φ(t f, ẏ(t f ] T = g(u R 4, (11 which can be expresse by free-flight ballistic equations using parameters u =[θ t, θt, l t, t f ] T R 4. The object s four state variables z is uniquely etermine by specifying the four-imensional parameters (θ t, θt,l t,t f, an vice versa. If we move the robotic arm so as to achieve the arm s state (θ t, θt at a release point,

3 21 IEEE/RSJ International Conference on Intelligent Robots an Systems, Taipei, Taiwan, October, 21. then the object on the arm surface at a throwing raius l t can be thrown to a esire state z in a time t f after the object s release. In the next section, we present the motion planning of the robotic arm. IV. MOTION PLANNING For a given initial state of the object an the robotic arm, the motion planning problem is to fin a robotic arm s trajectory to throw the object to the goal state uner the constraints on the ynamic grasping an robotic arm motion. A. Joint Trajectory Parameterization with B-splines We use uniform cubic B-splines to represent the joint trajectory since the continuity of the trajectory between ajacent B-spline segments is guarantee [4]. The motion time interval [, t t ] from the start of the arm s motion until just before the object s release is ivie into n knots with evenly space in time, t,suchast (= <t 1 < <t n (= t t. The joint trajectory θ i (t (i =, 1,,n on each interval [t i, t i+1 ] is expresse by i+2 θ i (s = θ j B j (s (12 j=i 1 where t(s =t i + s t, ( s 1, an θ j an B j (s are the control points an basis functions of the uniform cubic B-spline. The final joint angle is equivalent to the throwing angle such as θ n = θ t. The release time t t (=t n =n t can be freely chosen. For agivenn, t is free. Since the cubic B-splines lie within the convex hull forme by the control points θ j, the joint trajectory (12 passes near the control points. To obtain an optimal arm trajectory, we optimize the control points θ =[ θ 1, θ,, θ n+1 ] T R n+3 (13 an the time t (< t max incluing some of the joint s initial an final states by formulating the motion planning problem as a constraine optimization programming problem. We show the constraints an objective function of the optimization programming problem below. B. Finite-imensional Constraints To prevent the robotic arm from violating the constraints on the ynamic grasping ue to unknown isturbances which are not shown in the nominal moel an from saturating control inputs, we fin the joint trajectories which satisfy constraints (4 (9 with sufficient tolerance. We introuce margin variables N, M A, M B, θ, θ, τ into each constraints, respectively, an maximize the magnitues of the margin variables in the optimization problem. We rewrite the ynamic grasping constraints (4 (6 by using margin variables N, M B >, M A <. In aition, substituting (12 into them yiels finite-imensional constraints sample at the knot points t,,t n as N ( θ i, θ i, θ i > N, i =,,n (14 ( M A θi, θ i, θ i < MA, i =,,n (15 ( M B θi, θ i, θ i > MB, i =,,n (16 Similarly, the arm s motion constraints (7 (9 can be converte into finite-imensional constraints by using margin variables θ, θ, τ > as follows: θ min + θ<θ i <θ max θ, i=,,n (17 θ min + θ < θ i < θ max θ, i=,,n (18 τ min + τ<τ ( θ i, θ i, θ i <τmax τ, i=,,n (19 where the joint torque τ is expresse by the motion equation of the robotic arm, which is τ ( θ i, θ i, θ i = JR (θ i θ i + h(θ i, θ i (2 where J R is the inertia matrix an h represents the gravity, friction force an reaction force from the object. C. Objective Function Since the throwing manipulation is generally ynamic an fast an nees larger joint riving torques, it is easy for the object to roll on the surface of the arm or to break contacts with the arm uring the manipulation by violating the ynamic grasp constraints. We are greatly concerne with making the throwing manipulation more robust an stable to isturbances an trajectory errors, etc. In orer that the constraints on the ynamic grasping an the arm s motion can be satisfie with sufficient tolerance, we maximize the margin variables = [ N M A M B θ θ τ ] T R 6 (21 in the optimization problem shown below. D. Optimization Programming Problem For a given object s goal state z, we fin an optimal arm trajectory by solving the following optimization problem. max : 1 2 T W (22 subj. to: z = g(u (23 c (24 θ = (25 fin: θ an t Eq. (22 is the objective function to maximize the margin variables, where W is a weight matrix. The free-flight moel is use as the equality constraint (23. The inequality constraints shown in Section. IV-B are combine as shown in (24. As bounary conitions, the initial angular velocity is specifie by (25. In contrast, the initial angle an the final angle an velocity are erive from this optimization. We use sequential quaratic programming (SQP to solve the nonlinear programming. Once we fin the arm trajectory parameters ( θ, t, we can calculate the joint trajectory using (12.

4 21 IEEE/RSJ International Conference on Intelligent Robots an Systems, Taipei, Taiwan, October, 21. y [m] Goal [ra] θ Selecte trajectory θ [ra] π/8 -π/8 Fig. 3. θt =.346 ra -.1 θo = ra -π/4.1.2 Fig Throwing manipulation foun by the optimization. θ [ra/s] E. Simulation Results 1 5 θt = 9.28 ra/s.1.2 θ [ra/s2] tt =.199 sec Arm trajectories foun by the optimization. Consier a throwing manipulation of a rectangular object to a goal. The parameter values of the object an robotic arm are escribe in Section VI. The optimization programming problem is formulate with n =1. Since a local optimum foun by SQP epens on the initial guess, SQP is solve with many ifferent initial guesses given at ranom. Fig. 3 shows the simulation result foun by the optimization which represents the motions of the object an robotic arm. The object is thrown to the goal (x, y, φ = ( m,.4 m, 15 eg with the velocity ẏ =. We assume that the object lans at the storage pallet locate at the goal without slipping an rebouning to simplify the analysis. Fig. 4 shows the corresponing arm trajectories, which have the initial state (θ, θ =(.596 ra, ra/s an final state (θ t, θ t =(.346 ra, 9.28 ra/s at a release time t t =.199 s, respectively. The arm is maximally ecelerate at the release time to set the object free instantaneously. After the release time, the robot arm is move with the joint acceleration at the release time until the arm stops, which is not escribe in Fig. 4. Fig. 5 shows the arm trajectories obtaine for ifferent initial guesses. The arm trajectory in Fig. 4, which has the maximum value of the objective function, is chosen among them. V. LEARNING CONTROL If we have an exact throwing moel, then we can obtain an optimal arm trajectory to throw an object to the goal by using the motion planning metho iscusse in the Fig. 5. Arm trajectories foun for ifferent initial guesses, where t max = 1 msec. previous section. To moel the throwing manipulation system accurately, we are require not only to implement parameter ientification whenever the change of the object, but also to consier the ynamics of the interaction between the robot an the object, as well as the correction of the vision sensor etc. In general, it is impossible to obtain exact moels. In the present paper, instea of raising accuracy of an approximate moel itself, we overcome the problems of the moel with errors by applying an iterative learning control metho to the throwing manipulation. A. Introuction of Virtual Goal To fin an optimal arm trajectory, instea of setting an object s goal state z for the optimization programming problem, we introuce a virtual goal state ẑ, which is upate at each throwing trial in orer that the robotic arm can throw the object to the goal state z as closely as possible. The basic iea of the learning control using the virtual goal is shown in [2]. The virtual goal is obtaine as follows. The error between the goal state z an the actual state z is given by e j = z z j (26 where j enotes the trial number. Accoring to the value of the error, the virtual goal ẑ j+1 of the j+1th throwing is upate as ẑ j+1 = ẑ j + Kei (27 where the iagonal matrix K is gain parameters. Replacing the goal state z in the left-han sie of (23 with the virtual goal state ẑ, we fin an optimal arm trajectory by solving iteratively the optimization programming problem (22-(25 at each throwing trial. B. Learning Control Algorithm The algorithm of the learning control is shown below. 1 The optimization programming problem is solve to obtain the arm trajectory for the first throwing (j =1. In the same manner escribe in step 5 an 6, we make the robotic arm throw the object an measure the actual object state z 1 with a camera. 2 We calculate the error e j of the j th throwing by using (26. If the error norm is greater than the value of the threshol, we move to the next step. Otherwise we

5 21 IEEE/RSJ International Conference on Intelligent Robots an Systems, Taipei, Taiwan, October, 21. assume that the arm succees in throwing the object to the goal, an the learning control is terminate. 3 The virtual goal ẑ j+1 is upate by using (27. 4 The optimization programming problem is solve to obtain the arm trajectory for the j+1 th throwing. 5 To throw the object, the robotic arm is controlle with a PD-compensator so that the arm can track the trajectory obtaine in step 4. 6 We measure several positions an orientations of the flying object with the camera an estimate the object s ballistic trajectory through the least square approximations. We obtain the actual object state z j+1 from the estimate trajectory an return to step 2. VI. EXPERIMENT A. Throwing Robot System We have evelope the throwing robot system which can perform the throwing in the vertical plane as shown in Fig.6. The aluminum arm is centrally-mounte an 62 cm long an 15 cm wie plate. The arm is controlle with PD compensators an is riven by the AC motor with a harmonic rive gear, which has the maximum torque τ max =18Nm an the maximum spee θ max =4πra/s. The throwing raius can be change by ajusting the position of a thin plate on the arm surface. The object on the plate is constraine with 2 mm high stoppers so as not to slie on the arm surface. We use a rectangular parallelepipe wooen block as the object whose mass an size are 29 gan6cm 3 cm 3 cm, respectively. The moment of inertia of the wooen block is estimate on the assumption that the block is homogenous an symmetrical. We measure the positions of markers put on the object uring the flight with a camera at a rate of 1 khz to estimate the object trajectory. The concave storage pallet of woo is fixe at a goal in the vertical plane. B. Experimental Results The goal state z is set to ( m,.4 m, 15 eg, m/s. Fig. 7 shows the transitions of the error norms of the position, orientation an vertical velocity at the goal. The value of the error norm is reuce graually by repeating the learning. The error norm of the 1th throwing is less than 2.5 mm,.1 eg an.1 m/s, respectively. These experimental results show that the propose metho enables the robotic arm to throw the object to the goal accurately. Fig.8 shows the Storage pallet Arm Fig. 6. Motor Object Throwing robot system Position error [m] Orientation error [ra] Velocity error [m/s] θ [ra] θ [ra/s] Trial number i Fig. 7. π/4 π/8 -π/8 Transitions of error norm of object s state at the goal i=1 (final trial i=1 (1st trial -π/ i=1 (1st trial 4 2 i=1 (final trial Time [s] Fig. 8. Arm trajectories foun by the optimization at each trial transitions of arm trajectories foun by the optimization at each iteration. The trajectories are moifie by repeating the learning. Fig. 9 shows snapshots of the successful throwing motion. In this experiment, it is crucial for the stable an robust throwing to keep the un-rolling constraint about the vertex A of the object. In orer that the un-rolling constraint (5 can hol an the object will not start rolling on the arm surface, we fin the robotic arm trajectory so that the margin M A become sufficiently large in the optimization problem. C. Application to Sorting/Assembly Tasks (Attache Vieo The first vieo shows the trial throwing motion in the iterative learning mentione above. By repeating the throwing, the robotic arm succees in throwing the object into the storage pallet with the zero vertical velocity, which is locate on the apex point of the object s ballistic trajectory. The secon vieo shows the sorting task in which the robotic arm throws three objects into storage pallets with an appropriate orientation, as shown in Fig. 1. In the first throwing (Fig. 1(a, the zero vertical velocity is set at the first goal. On the other han, in the secon throwing (Fig. 1(b we set the negative vertical velocity at the secon goal. The object reaches the pallet with escening after passing through the apex point of the ballistic trajectory. In the thir

6 21 IEEE/RSJ International Conference on Intelligent Robots an Systems, Taipei, Taiwan, October, y [m] (a 1st throwing (b 2n throwing (c 3r throwing Fig. 1. Sorting task application on vieo Fig. 9. Snapshots of the throwing of the rectangular object into the pallet y [m].1 throwing (Fig. 1(c, the robotic arm can throw the object to the thir goal avoiing the first an secon pallets by ajusting the negative vertical velocity at the goal. If the negative vertical velocity with a larger (smaller magnitue shoul be set, the object will collie with the upper (lower pallet. The thir vieo shows the parts assembly task with H- an T-shape parts as shown in Fig. 11. First, the robotic arm throws the H-shape part an slots it onto the T-shape part fixe upsie own (Fig. 11(a. Next, the robotic arm throws the other T-shape part an inserts it into the former H-shape part (Fig. 11(b. In orer to prevent the flying part from colliing with the previously-inserte parts from the sie, the vertical velocity at the goal is set so that the linear velocity at the goal may put towar the vertical irection as much as possible. These vieos show the throwing manipulation controlle by the propose learning algorithm can achieve the high positioning performance. VII. CONCLUSION This paper propose the learning control metho for the throwing manipulation which can control not only the position but also the orientation of the polygonal object more accurately an robustly to uncertainties. We showe experimentally the valiity of the propose control metho with the one-egree-freeom robotic arm. We emonstrate the usefulness of the throwing manipulation by applying it to sorting task an assembly task on experiments. In future, we will consier the esign of storage pallets or catching evices which can surely hol the object without rebouning an slipping at laning. Integrating throwing evices an catching evices will help to construct robust flexible manufacturing systems (FMS (a 1st throwing Fig (b 2n throwing Parts assembly task application on vieo REFERENCES [1] H. Arisumi, K. Yokoi an K. Komoriya, Casting Manipulation, Miair Control of a Gripper by Impulsive Force, IEEE Trans. on Robotics, Vol.24, No.2, 28, pp [2] E. W. Aboaf, Task-Level Robot Learning, Technical Report 179, 1988, MIT Artificial Intelligence Laboratory. [3] H. Frank, N. W-Wojtasik, B. Hagebeuker, G. Novak an S. Mahlknecht, Throwing Objects - A bio-inspire Approach for the Transportation of Parts, Proc. of IEEE Int. Conf. on Robotics an Biomimetics, 26, pp [4] Y-C Chen, Solving Robot Trajectory Planning Problems with Uniform Cubic B-Splines, Optimal Control Applications an Methos, Vol.12, 1999, pp [5] K. M. Lynch an M. T. Mason, Dynamic Nonprehensile Manipulation: Controllability, Planning, an Experiments, Int. Journal of Robotics Research, Vol.18, No.1, 1999, pp [6] K. M. Lynch, an C. K. Black, Recurrence, Controllability, an Stabilization of Juggling, IEEE Trans. on Robotics an Automation, Vol.17, No.2, 21, pp [7] H. Miyashita, T. Yamawaki an M. Yashima, Control for Throwing Manipulation by One Joint Robot, Proc. of IEEE Int. Conf. on Robotics an Automation, 29, pp [8] W. Mori, J. Uea an T. Ogasawara, 1-DOF Dynamic Pitching Robot that Inepenently Controls Velocity, Angular Velocity, an Direction of a Ball: Contact Moels an Motion Planning, Proc. of IEEE Int. Conf. on Robotics an Automation, 29, pp [9] T. Senoo, A. Namiki an M. Ishikawa, High-spee Throwing Motion Base on Kinetic Chain Approach, Proc. of IEEE/RSJ Int. Conf. on Intelligent Robotics an Systems, 28, pp [1] T. Tabata, an Y. Aiyama, Passing Manipulation by 1 Degree-of- Freeom Manipulator, Proc. of IEEE Int. Conf. on Intelligent Robotics an Systems, Vol.3, 23, pp