Modeling and Analysis of the Dynamic Performance of a High Speed Selective Compliant Assembly Robotic Arm (SCARA) on a Compliant Support

Transcription

1 Modeling and Analysis of the Dynamic Performance of a High Speed Selective Compliant Assembly Robotic Arm (SCARA) on a Compliant Support Migara Liyanage Graduate Student mhl545@mun.ca Geoff Rideout Associate Professor g.rideout@mun.ca Faculty of Engineering and Applied Science Memorial University of Newfoundland St. John s, Newfoundland, Canada Nicholas Krouglicof Associate Professor nickk@mun.ca Abstract A Selective Compliant Assembly Robotic Arm (SCARA) manipulator has been developed with rotary hydraulic actuators for industrial automation. The robotic arm is mounted on a vertical column which has an 'I'-shaped cross section. A bond graph model has been generated for the servo valve, rotary actuator, and links. A lumped-segment torsional vibration model has been generated for the support column. During high speed movement of the manipulator this column vibrates torsionally. This deflects the actual position of the end effector (EE) from the desired position. The rotary joints of the manipulator are driven by double vane rotary hydraulic actuators fitted with rotary encoders to give relative angular positions. Currently, this arm is controlled using simple joint controllers. Since this controller uses the angular displacement of the joints with respect to (w.r.t.) the manipulator links it does not address the issue of external disturbances such as the effects of base vibration. defect removal pick-and-place, brushing, peg-in-hole, circuit board and mechanical assembly operations. The speed and load handling capacity of these manipulators often limit production speeds. A SCARA manipulator with rotary hydraulic actuators has been developed for high speed visual servoing and trajectory following applications. This robot is capable of achieving linear velocities in excess of 2.7 m/s with a 5 kg payload which is well beyond the capabilities of contemporary electro-mechanical manipulators [1]. The proposed visual servoing system is shown in Figure 1. Hydraulic actuators have the capability of producing high power-to-weight ratios (approximately 5 times) and powerto volume ratios (approximately 10 to 20 times) than comparable electric motors [2]. Therefore, the operation of hydraulic systems is often associated with fast starts, stops and speed reversals. When these systems move large payloads the result is very large inertial forces. These are large enough to produce significant deflections in the mounted sturctures. Often, this contributes to large inaccuracies in the EE position. A novel method has been proposed to compensate for EE vibration. This method uses the angle of twist of the vertical column to estimate the deflection and to modify the set point of the joint controller. The results of the analysis show that the proposed method produces a much accurate tracking performance compared to the existing conventional joint control strategy even at high manipulator speeds such as 5.5 m/s. Keywords: robotics, hydraulic SCARA manipulator, visual servoing 1. Introduction support column SCARA robot Modern facilities require large production capacities. Hence, some of the processes are automated and often performed by manipulators. Selective Compliant Assembly Robot Arm (SCARA) type manipulators are used whenever operations are performed in a horizontal plane. These manipulators are widely used to perform tasks such as PSD Camera Figure 1: The proposed high speed visual servoing system

2 Currently, the manipulator is controlled using simple joint controllers. Since the only feedback is the angular displacement of the joints w.r.t. the manipulator links, external disturbances such as the effects of vibration are not addressed. During high speed maneuvering of the manipulator, the supporting column experiences torsional vibration. It deflects the actual position of the EE from the desired position. This phenomenon could pose problems for applications which needs accuracy. The actuators of this manipulator are instrumented with rotary encoders that provide the relative angular position of the joints. Currently, the manipulator is controlled using simple joint controllers. Since the only feedback is the angular displacement of the joints w.r.t. the manipulator links, external disturbances such as the effects of vibration are not addressed. There have been not been many studies over the past which considered EE deflection of a serial manipulator. Fresonke et al. [3] proposed a method for deflection analysis for a general serial manipulator which considers end-effector, gravity, and inertial loads. Vision based feedback is used by Jiang and Eguchi [4] for EE tracking control of a flexible manipulator. This robot is simpler and moves at much slower speeds (40 mm/s). Both these studies did not consider EE vibration due to support column. In this study, a SCARA robot with the supporting column is modelled using bond graph theory. In practice, the absolute position of the is measured with a high speed imaging system and visual servoing techniques are to be applied to compensate for the deflection of the supporting column. The proposed robot controller architecture is based on set point modification to compensate for support column vibration. The rest of this paper is organized as follows. Section 2 provides a description about the development of the bond graph models for various sub components and the development of the control system. In Section 4, some key results of the bond graph simulation are presented. Section 5 provide conclusions of this study. 2. Development of System Models for the High Speed SCARA Type Visual Servoing System The SCARA manipulator consists of two links and two joints. Revolute joints of the manipulator are controlled by double vane rotary hydraulic actuators with integrated hydraulic servo valves. The joints of this robot are instrumented with rotary encoders. The second link incorporates an EE. It is equipped with a high speed image based visual servoing system to estimate its position. The real system is comprised of a camera with a 2D photo sensitive detector (PSD) image sensor. It tracks an infrared light emiting diode which is fixed to the EE. The system is modelled as separate components. a. Modeling of the Hydraulic Actuator The hydraulic actuator used for this robot is a double vane rotary type actuator. A cross section of this actuator is shown in Figure 2. The hydraulic actuator consists of two main compartments which are separated using wedges. Vanes of this actuator are fitted to the shaft so that these are 180 o from each other. These vanes further divide the compartment into two chambers. The actuator housing consists of a series of passageways to channel the oil flow in and out of the actuator. A servovalve is fitted to the actuator in order to channel the oil flow. When the spool of the servo valve opens, one of the chambers is connected to the supply presssure while the other is connected to the reservoir. When one side of the chamber connects to pressure the other side connects to the tank and vice versa. This creates a pressure differential on the vane. In a similar manner the other compatment creates a opposite force resulting in a torque. The torque on the shaft is propotional to the pressure differential between the vanes. Pressure differential could be transformed into torque using a transformer. The transformation ratio is the amount of torque produced per unit of difference in pressure across the vanes per unit of rotation of the shaft. This corresponds to the actuator displacement coeffcient (D M ). Therefore the torque on the shaft is, T = D M.(P 1 -P 2 ) (1) where is T is torque and P 1 and P 2 refer to pressure in chambers 1 and 2. The control volume considered for analysis consists of the fluid volume trapped in the chambers of the actuator. The hydraulic fluid which is used to run the actuator is subjected to compression under high pressure. Hence, the effects of fluid compression cannot be neglected. For a control volume of fluid which is subjected to compression (2), Where P is the pressure of the fluid under compression, V is the control volume and is the effective bulk modulus of the fluid. Considering the compressibility effects of the fluid in both chambers, (2) (3)

3 (4) into the chamber equal to the sum of the fluid into the chamber, change in fluid compressibility and the leakage volume per unit time. Considering fluid continuity principles, (6) where Q in is the fluid flow rate in to the actuator, Q out is the fluid flow rate out of the actuator, Q comp is the volume of fluid compressed and Q L is the leakage flow. The bond graph model for the actuator is given by (Figure 3), Figure 2: Cross section of the double vane rotary hydraulic actuator. where V 1 and V 2 correspond to initial volume of the two chambers of the actuator. Q corresponds to the change in the volume of the fluid. Fluid compressibility is represented as a capacitive element in the bond graph diagram of the actuator. Defects during manufacturing, expansion of the pressurized housing and wear during the operation of the actuator could leave a gap between the actuator housing and the vane. This results in fluid leaks across the vanes as high pressure fluid is moved within the chambers. The fluid leakage is considered as a resistive element. The inverse square law can be used to estimate the leakage flow. It is given by,.sign (5) where C d is the coefficent of fluid discharge, A L is the area of the fluid leakage and is the density of the hydraulic fluid. The principles of fluid continuity apply to the control volume considered in the analysis. Therefore, the rate of fluid flow into the chamber should be same as the total fluid which flows out of the chamber, the fluid which leaks through the vanes and the volume change due to compression. The fluid continuity principle could be applied to both chambers of the actuator giving fluid flow Figure 3: Bond graph for the rotary hydraulic actuator b. Modeling of the Servo Valve An electrohydraulic servo valve is used to regulate the flow of the actuator. The actual valve used for the actautors is a MOOG (R) G761. It consists of the spool, a cylindrical sleeve, flapper and an armature with coils. The spool consists of a set of lands. It moves in a cylindrical sleeve. When a current is applied to the armature coils it creates a force to move the flapper proportional to the current. This regulates the opening of the spool. The movement of the spool progressively changes the exposed aperture size and alters the differential oil flow between two control ports [5]. The armature which moves the flapper consists of two inductance coils that have been placed in series. These coils have a resistance and an inductance. Therefore, the coil of the valve is modelled as simple LR citcuit which has the transfer function shown below.. (7) where I(s) is the current induced in the armature, L is the inductance of the coil, R is the resistance of the coil and U(s) is control signal applied as a voltage.

4 The spool has a mass. Hence, the dynamics of the spool cannot be neglected. The spool dynamics are modelled as a first-order time delay transfer function.. (8) where is the spool valve displacement and is time delay. The directional flow of the hydraulic fluid based on the direction of the spool movement is modelled using bond graphs as shown in Figure 4. In the bond graph shown one of the two bond flows into the left side of the 0-junction will be zero. The half-arrow directions will reverse the sign of Q 1 or Q 2 as necessary. Therefore in the constitutive laws of the modulated resistor, the absolute value of x v is used. Displacement of the spool in a given direction connects the pressure port of the hydraulic supply to a chamber of the actuator while the return port of the hydraulic supply is connected to the opporsite chamber. This results in rotating the actuator in a clockwise or counter clockwise direction. When the spool makes a displacement in the opporsite direction, opposite ports are connected to the actuator chambers resulting in rotation in the reverse direction. Flow rate through the valve is dependent on displacement of the spool. Displacement of the spool will change the exposed aperture size for fluid flow. Fluid flows through a gap from a high pressure side to a low pressure side. This flow could be modelled as a flow through an orifice. This fluid flow rate could be represented as a modulated resistor ("MR" element in Figure 4). The flow rates are given by, If x v > 0,..... (9) 0 (10)..... (11) 0 (12) Figure 4: Bond graph for the hydraulic servo valve If x v moves in the a positive direction then '1' junctions 1 and 3 are activated. Oil will flow at a rate of q 1 from the source to the actuator s first chamber and a flow rate of q 3 will result from the second chamber to the tank. When x v moves in a negative direction '1' junctions 2 and 4 are activated. Oil will flow at a flow rate of q 4 from the source to the second chamber and oil will flow at a rate of q 2 from the first chamber to the tank. c. Modeling of the Support Column The manipulator is mounted on to a vertical beam. It has an 'I' shaped cross section. This will be divided into alternating rotary inertial and torsional spring segments connected by massless shafts. The vertical column was divided into 50 lumped segments. A schematic of the lumped segment model is shown in Figure 5. If x v < 0, 0 (13)..... (14) 0 (15)..... (16) where q 1, q 2, q 3, q 4 correspond to the flow rates through the junctions 1, 2, 3 and 4, k v is the circumference of the cylindrical sleeve, P s is the supply pressure and P T is the tank pressure. Figure 5: Lumped segments of the vertical column

5 An analysis of a single lumped-mass segment develops the incremental solution to the total torsional response. A free body diagram of a single torsion element is shown in Figure 6. The manipulator consists two rigid links fitted with hydraulic actuators. A schematic diagram of the manipulator with the vertical column is shown in Figure 8. These actuators and the EE are considered as point masses in the analysis. The mass of the actuator has to be distributed between two robot links. When the first link is considered, the rotor of the first actuator is assigned at point A and the stator of the second actuator is assigned at point B. In case of the second link the rotor of the second actuator is assinged at point B while the EE is assigned at point C. Hence the center of gravity (COG) needs to be modified for each link. for link AB it will be given by,.. 18 Figure 6: The free body diagram of a single element Each lumped mass segment is considered as a rotational inertia element connected with a torsional spring and a damper. This element transfers torque and relative twist to the elements adjacent with each other. The bond graph for one such element is developed and shown in Figure 7. In the bond graph shown the resistive element corresponds to the material damping factor (R mat ) and the inertia element (J i ) correspond to the mass moment of inertia. The compliance (C i ) is given by, (17) where is the length of a lumped segment, L 1 is the length of the beam, N is the number of segments, G is the shear modulus and J is the polar moment of inertia. where, is the distance to the modified COG, is distance to existing COG, is the mass of link AB, is the mass of the rotor of the actuator, is the mass of the stator of the actuator, is the length of link AB of the robot. For link BC the modified center of gravity will be given by,.. 19 where, is the distance to the modified COG, distance to existing COG, is the mass of link BC, is the length of link BC of the robot. The moment of inertia (MOI) also needs to be modified for each link. It could be calculated for each link using the parallel axis theorem, Modified MOI for link AB is given by, where is the modified MOI inertia about the new COG and is the MOI of the link AB about G where is the modified MOI inertia about the new COG and is the MOI of the link BC about G 3. and for each link was estimated using a CAD model. Figure 7: The bond graph of a single torsional element. d. Modeling of the SCARA Arm Each link is considerd separately to develop the bond graph. Link 2 (denoted by AB) has a modified COG. It rotates about point A at an angular velocity of. Considering the velocity of point A, / (22)

6 Where, is velocity of point A, is the velocity of point and / is relative velocity of point A w.r.t.. Velocity / is given by, / / (23) Where, / is the vector which is defined by the length AG 2..sin. (27). cos. where, is the velocity of point B, is the velocity of point. The bond graph for the link AB is derived from expressions [26] and [27]. Modulated transformers are used to convert the angular velocity to a linear velocity. The mass of the link is conssidered at the '1' junctions corresponding to. The mass of the second actuator which is at B is considered as a point mass acting as an inertia element at G 2. This is connected as an inertia element to the '1' junction corresponding to angular velocity of the link 2. Following a similar procedure a bond graph could be developed for link BC. The two bond graphs will be connected by considering the velocity vector at point B. In order to remove the derivative causality a very low resistance and a stiff spring was attached to point B. Figure 8 shows how the first actuator is attached to the column. The vertical axis through the center of the 'I' beam and the axis of symmetry through the flange makes an angle of with the shaft of the actuator. The angle of twist due to torsional vibration is denoted as 1. The vector defined by the axis of the first revolute joint and the point o' on the flange is given by,. cos (28).sin By taking the derivative w.r.t. time, Figure 8: A schematic diagram of the beam and the robot Vector / is given by, /. cos.sin (24) By substituting / in [23], /.sin. (25). cos. By substituting / in [22],.sin. (26). cos. Similarly, for point B it could be shown that,.sin. (29). cos. The system was modelled as separate components. It includes the valves, actuators, the support beam and the robot. Finally, these components were integrated separately as a total system model. This is shown in Figure 10. e. Modeling of the Control System A schematic diagram of the manipulator with and without vibration is shown in Figure 10. The desired position of the EE is point D. Due to torsional vibration of the support, the origin the first actuator () moves along a circular arc displacing the manipulator from to. This results in considerable deflection of the EE. A novel controller is proposed in this study to account for the effects of compliant support vibration. In order to implement the control system, encoder readings and a high speed camera system are used to obtain the feedback. The encoders provide the relative angles of the joints while the high speed camera system provides the position of the EE w.r.t. a fixed frame of reference. In order to compensate for the

7 effects of vibration set points for the joint angles will be modified. In order to do this angle of twist needs to be estimated for the 'I'-beam. The EE has to arrive at a desired position () which is w.r.t the fixed corordinate frame at O' by /. Due to the controller action if there is no torsional vibration of the beam the EE will be in point C. However, due to the twisting of the 'I'-beam the EE is displaced to point C'. The encoders provide the relative angles of the links and w.r.t. rotating coordinate frame x'-y' that has an angle relative to the inertial frame at O'. The position of C w.r.t. O', in coordinates of the inertial frame x-y, is given by / / + / / 1.cos 1.sin 2.cos 2 3.cos sin 2 3.sin 2 3 The high speed imaging system will measure the position of the EE at C'. The measured position of the EE will be given by /. Since the coordinates of points C and C' are known, and since column deflection rotates line O'C to O'C' through angle, 1 2 have to be modifed. With respect to the inertial coordinate frame at O', / is given by, / / - / where / 1.cos 1 1.sin 1 and r 1 is the distance from O' to A' as shown in Figure 8. The vector / must be expressed in rotated x'-y' components in order to perform inverse kinematics and determine new encoder angle setpoints. The modified desired position coordinate / is given by, /. / where, 1 1 is the rotation matrix. 1 1 The modified desired relative angles for links 2 and 3 are and, respectively. The modified set points for relative angles of links 2 and 3 can be estimated considering inverse kinematics and simple geometry by, 2 1 2, (33) 2, 2.cos,. (34) Figure 9: The bond graph of link AB From the existing position of the EE at, it has to move to point D, which is the desired position of the EE. In order to perform this, the current desired relative angles for the links Figure 10: End effector position with and without vibration

8 The error between the measured relative angle of the joints and modified angle for relative set point is fed through a PD type controller for each valve. The controller signal will be translated into a voltage signal for controlling each valve. These controllers were tuned using a trial and error method. A schematic diagram of the proposed controller is shown in Figure Simulation Results This manipulator has two links of lengths 0.5 m and 0.49 m. The two actuators fixed to its revolute joints are 6.9 kg each. One of these actuators is fixed while the second actuator moves with the first link. It has an EE which carries a pay load of 5 kg. The second actuator and the EE could produce large inertia loads when the robot moves at high speeds. The system was modelled using 20-SIM (TM) bond graph modelling software. The sytsem was solved using a fourth order Runge-Kutta numerical integrator with a step size of 10 s. This study proposes an alternative to the existing method based on online set point modification in order to compensate for the torsional vibration. Therefore, the simulation was carried out considering two cases. The first case considers vibration compensation using set point modification and the second one considers conventional simple joint control. A schematic diagram of the total system modelled is shown in Figure 11. Simulations were carried out for the EE to follow a linear trajectory which has a length of 0.47 m from point (0.3963, ) to (0.7563, ). The torsional vibration of the vertical column is shown in Figure 13. It shows the angle of twist for the two cases. The twisting of the column is higher in the case with no vibration compensation. The beam twists up to 7.2 when there is no vibration compensation while it twists up to an angle of 2.4 with vibration compensation. One interesting inference that could be made from these results is that the beam twists by a lesser amount with vibration compensation because the counter movement of the EE and the second actuator will dampen out the effects of torsional vibration. As a result of this when there is vibration compensation it takes much less time for the manipulator to come to a steady state. Figure 14a and 15a shows the actual angle and the modified desired angles for the revolute joints of the manipulator. The set point shown in the figure is continuously updated based on the torsional vibration of the support column. Figures 14b and 15b show the actual angles and the desired set points for the joint angles when there is no vibration compensation. These set points are estimated assuming that there will be no effects from other forms of vibration and flexing of the manipulator links. In both these cases the manipulator closely follows the desired angles. It also shows that it produces a damping effect when it tries to compensate for vibration. Figure 11: A schematic diagram of the total system

9 The Figures 16 and 17 shows the EE x- and y- position in both cases. It shows the desired and actual EE positions along with the deflection due to vibration. When there is no compensation for vibration the beam would deflect up to 82 mm in the x- direction and 94 mm in y- direction. The same values would be 30 mm in x- direction and 21 mm in y- direction when there is vibration compensation. The Figure 18 shows the desired and actual trajectories of the EE in the robot work profile. It shows that when there is vibration compensation the EE follows closely with original desired trajectory. The average continuous speed of the robot during the trajectory following operation was 5.5 m/s. From the results it clear that set point modification based on EE deflection produce much accurate results in the trajectory following problem even at very higher speeds. 5. Conclusions This study considered a high speed SCARA type manipulator. It uses rotary hydraulic actuators for joint control. During its high speed operation the EE encounters deflection due to the torsional vibration of the compliant support. This creates problems especially in the trajectory following problem. This paper used bond graph analysis to model the system and implement a novel method for controlling the robot EE based on set point modification. the error between the modified set point and the encoder angle is fed as a signal to the valve through a PD type controller. The simulation results demonstrated the effectiveness of proposed method over conventional joint control at speeds of up to 5.5 m/s. References [1] Liyanage, M.H.; Krouglicof, N.; Gosine, R.;, 2011, "Development and testing of a novel high speed SCARA type manipulator for robotic applications", Proceedings of IEEE International Conference on Robotics and Automation (ICRA) 2011, pp , 9-13 May [2] H. Merrit, 1967, Hydraulic control systems. John Wiley & Sons, Inc., New York. [3] Fresonke, D.A.; Hernandez, E.; Tesar, D.;, 1988, "Deflection prediction for serial manipulators", Proceedings of IEEE International Conference on Robotics and Automation 1988, pp vol.1, Apr [4] Zhao-Hui Jiang; Eguchi, T.;, "Vision feedback based end-effector motion control of a flexible robot arm," IEEE International Conference on Systems, Man and Cybernetics, 2007., pp , 7-10 Oct [5] R. Poley, 2005, Dsp control of electro-hydraulic servo actuators, Texas Instruments, Application Report SPRAA76. The total system that was considered in the study is comprised of a SCARA type manipulator, hydraulic actuators, servo valves and a support column. The SCARA links were controlled by rotary hydraulic actuators with servo valves. These servo valves are connected to a 20 MPa (3000 psi) hydraulic supply. The SCARA manipulator system is supported by a support column which has an 'I' shaped cross section. A lumped-segment torsional vibration model was developed for the support column. The hydraulic actuators are instrumented with rotary encoders. These provide the relative joint angles. In addition to this the EE position was used measured using a high speed imaging system. Bond graph theory facilitated the construction of component submodels and the integration of hydraulic, mechanical, and controller elements into a total controlled system model. Currently, the this manipulator is controlled using simple joint control. It only uses relative joint angles from the rotary encoders as feedback. This fails to capture effects due to external disturbances. The proposed method uses the position of the EE along with the encoder readings as feedback. Using these the deflection of the first actuator due to torsional vibration is estimated. Considering the position of the EE and the position of the first actuator a modified set of desired relative angles are calculated. Then

10 Figure 12: A schematic diagram of the proposed control architecture (a) (b) Figure 13: Angle of twist of the support column (a) (b) Figure 14: Actual and desired angle of link 2

11 (a) (b) Figure 15: Actual and desired angle of link 3 (a) (b) Figure 16: EE -x position (a) (b) Figure 17: EE -y position

12 Figure 18: EE position in work profile