the Mitsubishi PA-10 robot arm

Proceedings of the 2003 IEEERSI Inti. Conference on Intelligent Robots and Systems Las Vegas. Nevada . October 2003 Estimation and modeling of the harmonic drive transmission in the Mitsubishi PA-IO robot arm* Christopher W. Kennedy and Jaydev P. Desai’ Program for Robotics, Intelligent Sensing and Mechatronics (PRISM) Laboratory 3141 Chestnut Street, MEM Department, Room 2-115 Drexel University, Philadelphia, PA 19104, USA Email: {cwk,desai}@coe.drexel.edu Abstract The purpose of this paper is to present our results in developing a dynamic model of the Mitsubishi PA-IO robot arm for the purpose of low velocity trajectory tracking using very low feedback gains. The novelty of this research is therefore the development of a systematic algorithm to extract the model parameters of a harmonic drive transmission in the robot arm to facilitate model-based control. We have chosen the elbow pitch joint (joint 4J of the PA- I O robot arm for estimation and modeling purposes. We have done several experiments to identify the various parameters of the harmonic drive system. We conclude with a sample trajectory tracking task whereby the feedback torque required to do trajectory tracking with and without the parameter identification of the HDT is significantly different. 1 Introduction Accurate modeling of the inherent dynamics of a robot manipulator is essential in many manipulation tasks. Through careful modeling, low feedback gains can be used along with a feedforward model to accurately follow a desired trajectory for tasks requiring low interaction forces while manipulating and interacting with objects in the environment. The purpose of this paper is to present our results in developing a dynamic model while taking into account the inherent dynamics of the Mitsubishi PA- 10 robot arm for the purpose of low velocity trajectory tracking using very low feedback gains. The PA-IO is ideal for precision manipulation tasks due to the backdrivability, precise positioning capabilities and zero backlash afforded by its harmonic drive transmission. However, the compliance and oscillations inherent in harmonic ’This material is based upon the wolk supported by the National Science Foundation grants EIA 0079830. and CAREER award IIS 0133471 and the American Healt Association grant m60368u. ‘Corresponding author drive systems make the development of an accurate dynamic model of the robot extremely challenging. The novelty of this research is therefore the development of a systematic algorithm to extract the model parameters of a harmonic gear driven transmission in the Mitsubishi PA-IO robot arm. This robot is significantly used in research laboratories [ l - 31 and we believe that this paper is the first of its kind to address the transmission modeling and implementation on the Mitsubishi PA-IO robot arm in a research environment. There is a substantial body of previous research in the area of modeling harmonic drive systems 14-61, Tuttle [4] presents an excellent overview of modeling and parameter identification of harmonic drive systems and Kircanski 151 provides a detailed analysis of the nonlinear behavior of harmonic gears due to compliance, friction and hysteresis. Although these research efforts provide significant insight into the physical phenomena that characterize harmonic drive behavior, all of their experimental work has been periormed on customdesigned. elaborate test-beds that allow direct measurement of many important system parameters such as compliance and kinematic transmission error. In addition, many control schemes for harmonic drive systems rely on torque sensors mounted directly to the transmission components [7-91. To our knowledge, there is no published work to date describing an efficient means of modeling, parameter identification, and control of the HDT in the Mitsubishi PA-IO robot arm. This paper is organized as follows: in section 2, the PA-IO system is described, including the control system architecture, section 3 offers a brief overview of harmonic drive transmissions along with a model of harmonic gearing, section 4 outlines our methods for experimentally determining the necessary parameters for model-based control of the PA-IO robot arm, section 5 details our experiments designed to verify the effectiveness of our model, and finally in section 6 concluding remarks are presented. 0-7803-7860-1/03/$17.00 @ 2003 IEEE 3331 Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on August 03,2010 at 02:03:11 UTC from IEEE Xplore. Restrictions apply. 2 The Mitusubishi PA-IO Robot The Mitsubishi PA-10 robot arm is a 7 degree+f- freedom robot arm with an open control architecture and is manufactured by Mitsubishi Heavy Industries (see Figure la). The four layer control architecture is made up of the robot arm, servo controller, motion control card, and the upper control computer. The host computer runs the QNX real- time operating system and achieves communication rates of up to 700 Hz with the robot servo driver through the ARCNET motion control card. ARCNET is a token passing LAN protocol developed by Datapoint Corporation. The robot joints are actuated through three-phase AC servo motors and harmonic drive gear transmissions. Joint positions are measured through resolvers at the joint output axis, with a resolution of 0.000439'' over +/- 3 output revolutions. Control of the robot can be achieved in either 'Velocity mode' or 'Torque mode'. In 'Velocity mode', the desired velocity for each joint is sent to the servo driver from the host computer. A high- gain digital PI feedback loop running at 1538 Hz on the servo driver then controls the joint velocity. In Torque mode' the desired joint torque (in this case the motor toque constant times the motor current, before conversion by three-phase), is sent to the servo driver. The expression for communicating a desired torque to the robot arm through the servo driver in this system is given by: T m =kT *I=Td'(O.OOINm/dig) (1 ) where T,,, is the motor torque, kr is the motor torque constant. I is the motor current, and Td is the desired torque written to the servo driver in the form of a 2 byte integer. 3 Harmonic Drive Systems Harmonic drive gears, also called strain-wave gearing, were developed by C. Walton Musser in the 1950's primarily for aerospace applications. They are compact, light-weight, and have torque ratios between 30:l and 320:1, making them ideal for robotics applications. Harmonic drives are composed of three components: the wave generator, the flexspline and the circular spline (see Figure 1). The wave generator is an elliptical ball bearing assembly and is nested inside the flexspline. The teeth on the nonrigid flexspline and the rigid circular spline are in continuous engagement. Since the flexspline has two teeth fewer than the circular spline, one revolution of the input causes relative motion between the flexspline and the circular spline equal to two teeth. With the circular spline rotationally fixed, the flexspline rotates in the opposite direction to the input at a reduction ratio equal to one-half the number of teeth on the flexspline. The displacement, velocity and torque relationships between the transmission elements in the ideal case are therefore given by: 1 ( N + l ) - - N T --Tfs 1 Twg =- where N is the transmission ratio, e,, is the wave generator angle, 9- is the circular spline angle, BtS is the flex spline angle, w,, is the wave generator angular velocity, wFI is the circular spline angular velocity, wtS is the flexspline angular velocity, Ty is the wave generator torque, T, is the circular spline torque, and Te is the flexspline torque. In general either the flexspline or the circular spline can be fixed. However, in our robot the circular spline is fixed while the flexspline rotates with the joint. WLe-m Fdlpl" cm!Lsplm Figure 1. Harmonic Drive Components. The flexibility inherent in the harmonic drive systems provides advantages such as zero backlash due to natural preloading. However, there are also several disadvantages such as nonlinearity due to friction, alignment error of the components, and transmission losses due to compliance in the system. All of these were found to be critical in the modeling of the Mitsubishi PA-10 robot arm. In the following subsections we will describe in detail our methodology for estimation and modeling of the: a) friction in a HDT, b) torsional stiffness, and c) the kinematic transmission error. This will be followed by a model of the HDT for the Mitsubishi PA-10 robot arm. 3332 Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on August 03,2010 at 02:03:11 UTC from IEEE Xplore. Restrictions apply. 3.1 Friction In Harmonic Drives Because we are primarily concerned with controlling the robot at low velocities, we model friction in the robot joints using a Stribeck curve at low velocities and a linear relationship at high velocities. The expression for this model is given by: Td(6) = Fo +be + Fce -(6N,)2 (5) where Tvl is the viscous friction torque, F., F, and V, are the Stribeck coefficients, 0 is the rotational velocity, and b is the viscous friction term. Friction in harmonic drives is strongly position dependent due to the kinematic error. Therefore, we model coulomb friction as the torque required to maintain a very slow velocity over the entire range of motion of the robot. 3.2 Torsional Stiffness in Harmonic Drives Harmonic drives exhibit significant compliance when externally loaded. Experimental tests in [4, 51 indicate that stiffness in harmonic drives increases with increasing load. In addition, harmonic drives display a behavior known as 'soft wind-up' which is characterized by very low stiffness at low applied loads. The reader is referred to Kircanski's paper 151 for an excellent overview of harmonic drive torsional stiffness behavior. Our experimental tests on the Mitsubishi PA-IO robot arm reveal that wave generator compliance has a significant effect on the robot dynamics. The effect of the non-linear stiffness profile of the wave generator is prevalent when the load on the wave generator is below a critical point where a dramatic decrease in stiffness occurs. 3.3 Klnematlc Transmission Error Harmonic drives display kinematic error that causes the torque transmission characteristics of the drive to deviate from the ideal transmission model. This error is caused by a number of factors such as tooth-placement errors on both the circular spline and flexspline. out-of-roundness in the three transmission components, and misalignment during assembly (41. The resulting error signature can display frequency components at two cycles per wave-generator revolution and several subsequent harmonics. Based on the above. the error function including two harmonics of wave generator rotation is given by: eem = Alsin(ew + 'pl )+A2sin(29wg + 'p2 ) where are the amplitudes of the sinusoids, pi are the phase shifts and e,, is the wave generator position. This expression is of limited use in our case, because it is not possible to measure the wave generator rotation. Furthermore, although kinematic error has a significant effect on the torque transmission characteristics of HDTs, we found that compensating for coulomb friction using the toque required to maintain slow velocity eliminated almost all the effects of kinematic error. Therefore, we neglect the effect of kinematic error in the feedforward implementation of our model. 3.4 Harmonic Drive Model We consider the model of the harmonic drive to be composed only of friction, gravity, externally applied torque, and stiffness. Because we are concerned primarily with low-velocity applications, we are neglecting acceleration terms in our model (thus the only torque component from the dynamics will be the gravitational torque). We have modeled the non-linear expression for torque transmission in harmonic drives in our robot arm as: (6) J,N = Td (e) + Td (8) + TgP) + Tc (Td, Td, Tg) (7) where TI. is the input torque, N is the transmission ratio, T@ is the coulomb friction, Td is the viscous friction torque calculated using equation (9, T, is the gravity toque, and T, is the torque used to deform the wave generator. A schematic for the proposed control system is shown in Figure 2. ... . . . . . . . "'"""?.??? ....... 0 w I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I / \ Figure 2. Proposed control system including model of friction, gravity and compliance of the wave generator. 3333 Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on August 03,2010 at 02:03:11 UTC from IEEE Xplore. Restrictions apply. 4 Experimental Methods We have chosen the elbow pitch joint (joint 4) for analysis because it not only is strongly influenced by gravity and friction, but also is the most affected by the oscillations induced by kinematic error. During normal operation with the high gain controllers, the joint emits a chattering noise due to vibration. This joint will therefore benefit most by accurate modeling. Although we present results for the elbow pitch joint only, we have obsenred similar behavior in the other joints with the exception of the chattering noise. 4.1 Friction Characterization . Friction is typically characterized by its relationship with velocity. To determine the friction-velocity relationship, the robot was commanded to move at a constant velocity and the mean toque required to maintain the velocity was taken to be the friction for that value of velocity. After collecting data for velocities from 0.005 radls to 0.5 radls in increments of 0.05 radls, the data was fit to the friction model in equation (5). To collect this data, the robot was mounted on the wall as shown in Figure 3 to negate the effect of gravity. The results for the dynamic friction are shown in Figure 4. friction experiments. To obtain our model of coulomb friction, torque data was collected in 'Velocity mode' to maintain a velocity of 0.005 rad/% This data was then fed- forward in the form of a look-up table to represent coulomb friction. L p.6 : A zo4p--_.,. :"r"; I 020 Roll8nalVelocd&dfs) Figure 4. Mitsubishi PA-IO robot arm. Friction data for elbow pitch joint o 4.2 Estlmatlon and modeling of Stiffness, Gravlty Compensation, and Kinematic Error To determine the effect of stiffness and gravity, we commanded the elbow pitch joint to move at a constant velocity of 0.01 radls from -90' to + 90' with weights from 0 to 10 Ibs attached to apply torque to the drive. The output torque required to track this trajectory for weights of 0, 5, and 10 Ibs is shown in Figure 5. 100 E z ?fa b 50 a v = n - -50 Posffion @d) ' ' -1 -0.5 Figure 5. Constant velocity experiments with varying weights. The effects of gravity and stiffness are clearly evident from these plots. Note the sharp decrease in torque when the output torque approaches 0 Nm. This is due to the 'soft wind-up' behavior of the harmonic drive system. As the load on the wave generator decreases, the stiffness if the wave generator decreases, hence more torque output is required to sustain motion. We determined the torque used to deform the wave generator during the motion in Figure 5 by equating the output torque during the motion with the right side of equation (7). The friction torque for these motions is nearly identical to the coulomb friction, hence we can eliminate friction from equation (7) by subtracting the coulomb friction torque from the total toque (as noted previously, coulomb friction torque was computed from the arm motion at low velocity while it was attached to the wall). The gravity torque is also known from the estimated masses of the robot arm links and the known weights attached to the robot. We can therefore subtract the gravity torque from the right side of equation (7). This leaves the remaining torque equal to the torque used to deform the wave generator. The successive plots of each step of this process are shown in Figure 6. 3334 Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on August 03,2010 at 02:03:11 UTC from IEEE Xplore. Restrictions apply. 80, I . . Figure 6. Steps in model identification process. To verify that this approach is valid, the stiffness torque calculated as described above was plotted as a function of the external torque (gravity plus mean coulomb friction) for each of the three weight trials in both the positive and negative directions (Figure 7). Figure 7. Stiffness torque as a function of external torque for three weight trials in positive and negative directions. As can be observed from Figure 7, the model of stiffness torque as a function of external load appears to be fairly repeatable. The region where the torque decreases or increases sharply due to a rapidly changing stiffness coefficient seems to be modeled effectively. We model this function as composed of three linear regions for each direction. For the feedforward implementation, we use the data collected in the previous experiments to generate a relationship between the external load and the input torque required to generate the necessary output torque to compensate for the load. 5 Results To verify our model, we fed forward the computed torques to track a trapezoidal velocity profile at 0.1 rad/s. The gains were set to 0.5 N and 0.15 Ns for the proportional and derivative gains respectively. Velocity measurements were taken by differentiating the position measurements and filtered using a fifth order Bunerworth filter with a cutoff frequency of 200 Hz. This filter was chosen because it was empirically determined to give the best results. The feedback toques generated while tracking the desired trajectory are shown in Figure 8 for the low gain implementation with the feedforward model, and for the high gain implementation with no feedforward model. The magnitude of the feedback torques is approximately 1 Nm in the region of high wave generator stiffness and approximately 6 Nm in the region of low wave generator stiffness for the low gain feedforward case. - 0 5 0 0.5 1 1.5 ’ PwIion [nd) Figure 8. Feedback torque results for low gain controller with model based control and high gain controller without model based control. The tracking error for the low gain feedforward case is shown in Figure 9. The mean tracking error is approximately 0.02 radians in the region of high wave generator stiffness, and 0.2 radians in the region of low wave generator stiffness. 1.51 I Figure 9. Desired trajectovand actual trajectory of robot during trajectory tracking experiment. 5 Discussion Our approach in developing a dynamic model for the HDT in the Mitsubishi PA-10 robot arm can be described as follows: 1) we developed a mathematical model for the HDT accounting for friction, transmission compliance and gravity, resulting in equation (6), 2) we characterized friction in the transmission using constant velocity experiments, 3) we commanded the to robot move 3335 Authorized licensed use limited to: Harbin Institute of Technology. Downloaded on August 03,2010 at 02:03:11 UTC from IEEE Xplore. Restrictions apply. along a test trajectory with weights of known magnitude attached to determine the effect of transmission compliance, and 4) we determined the model parameters based on the experimental data and then fed forward our model to the robot for the purpose of tr