Modeling, Full Identication and Control of the Mitsubishi PA-10 Robot Arm
Nikolaos A. Bompos Panagiotis K. Artemiadis Apollon S. Oikonomopoulos Control Systems Lab, School of Mechanical Eng. National Technical University of Athens, 9 Heroon Polytechniou Str, Athens, 157 80, Greece {mc00123,partem,apollon,kkyria}@mail.ntua.gr corresponding author Kostas J. Kyriakopoulos
Abstract This paper presents the modeling, identication and control of the 7 degrees of freedom (DoFs) Mitsubishi PA-10 robot arm. The backdrivability, high accuracy positioning capabilities and zero backlash afforded by its harmonic drive transmission, make the PA-10 ideal for precise manipulation tasks. However, the lack of any technical knowledge on the dynamic parameters of its links and the non linear characteristics of friction at its joints, make the development of an accurate dynamic model of the robot extremely challenging. The innovation of this research focuses on the development of the full dynamic model of the PA10 robot arm, the development of a new non linear model for the friction at its joints, the estimation of the stiffness characteristics of its joints and nally the full identication of the dynamic parameters of the robot arm. The accuracy of the full dynamic model identied is proved by an end-effector trajectory tracking task using a model-based inverse dynamic controller. Index Terms Robot parameter identication, robot friction, dynamics
I. INTRODUCTION The accurate modeling of the dynamics of robot manipulators has received increased attention, as dexterous manipulation tasks and interaction with the environment demand the knowledge of the full dynamic model of the robot. Dynamic models of robot arms used in model-based control schemes are designed in terms of various inertial and friction parameters that must be either measured directly or determined experimentally. However, direct measurements of such characteristics are rather impractical or even impossible in many cases. Inertial parameters of robot links can not be measured without dismantling the robot arm, while highly nonlinear inherent phenomena at robot joints can not be directly quantied. Therefore, models describing nonlinear effects such as friction, should be addressed in conjunction with methods of determining parameters of the dynamic model of the arm based on experiments, in order to fully identify the dynamic model of the robot arm. The Mitsubishi PA-10 is a 7 degrees of freedom (DoFs) robot arm, widely used due to the backdrivability, accurate positioning and zero backlash afforded by its harmonic drive transmission. Its relatively small volume along with the dexterity provided by the redundancy in DoFs, entail its usage in demanding tasks such as surgical tool placement or teleoperated soft tissue manipulation. However, the inherent compliance and non-linear friction in its harmonic drive system make the
development of an accurate dynamic model of the robot an extremely challenging issue. Modeling of the PA-10 transmission system has been addressed in low velocity, low impedance implementation in [1]. In that approach, the dynamic model of the harmonic drive transmission was developed, for low velocity applications. The friction effects were modeled using a Stribeck model, while the stiffness of the harmonic drives was modeled using a combination of linear functions for each joint. However, the inertial effects of the robot arm were neglected due to low velocity implementation, while the redundant DoF (joint 3) was neglected during the end-effector tracking experiments. Focusing on the area of modeling of harmonic drive systems, an overview of modeling and parameter identication of such systems in [2] is worth noticing. Kircanski [3] provided a detailed analysis of the nonlinear behavior of harmonic gears due to compliance, friction and hysteresis, while Taghirad [4] presented a control scheme for harmonic drive systems based on experimentally identied models for the inherent dynamics of those. However, most of these works are tested in custom-designed experimental platforms, that allow direct measurement of system parameters such as compliance and kinematic transmission error. In commercial robotic manipulators though, like the PA-10, those measurements are not available. Friction modeling in robotic manipulators has been widely analyzed in [5]. A presentation of methodologies developed for experimentally determining accurate models of non-linear friction is addressed in [6], while an analysis of various friction models has been implemented in [7]. Dynamic parameter identication of robotic manipulators has been widely analyzed in [8], while a method for determining parameters of a dynamic robot model based on experiments is presented in [9]. In this paper, the modeling, full identication and control of the Mitsubishi PA-10 robot arm are implemented. A new non-linear model for the friction phenomenon at the PA-10 harmonic drive system is introduced and experimentally veried, while a model for the effect of stiffness of the robot joints is developed. Then, after transforming the dynamic model of the robot arm to an identiable form, the full set of dynamic parameters is identied through experiments. The accuracy of the identied model is veried through end-effector trajectory tracking experiments, using a model-based inverse dynamic

TABLE I PA-10 MODIFIED D-H PARAMETERS i 7 iaidi 0.0.0.0.07 i q1 q2 q3 q4 q5 q6 q7

Fig. 1.

Mounting and conguration of the PA-10.
starts from 0. The dynamic model of the robot arm is constructed using the Lagrange equations expressed by L d L = i dt qi qi , i = 1,. , 7 (2)
Fig. 2. Links and frames assignment of the PA-10. Dimensions are given in m.
where qi the joint angles, qi the joint velocities, i the joint torques and L is dened by L=T U (3)
controller. Consequently, the novelty of this paper focuses on the methodology consisting of the full identication of inertial, friction and stiffness characteristics of the PA-10 robot arm, at a variety of velocity and load conditions. The rest of the paper is organized as follows: Section II gives a description of the modeling methodology proposed, with the appropriate segregation of the distributed sub-problems of dynamic modeling and nonlinear friction modeling and stiffness identication. Section III illustrates the experimental results of the full model identication of the robot arm through the implementation of a model-based inverse dynamic controller, while section IV concludes the paper. II. M ODELING A. Dynamic Model The Mitsubishi PA-10 robot arm has 7 rotational DoFs, arranged in an anthropomorphic way: 2 DoFs at the shoulder, 2 DoFs at the elbow, and 3 DoFs at the wrist. The robot servo controller communicates with a personal computer (PC) via the ARCNET protocol. The robot arm can be controlled either at velocity mode where the desired joint velocity is commanded from the PC, or at torque mode where the desired joint torque is commanded. For the identication of the full robot arm model, the velocity mode is used. The PA-10 has been mounted in an horizontal way as shown in Fig. 1. The links of the robot arm along with their frames are shown in Fig. 2. The frames are assigned using the modied Denavit-Hartenberg (D-H) notation [10]. The modied D-H parameters are listed in Table I, while the relation between frame i 1 and i is given by ci si 0 ai1 s c ci ci1 si1 si1 di i1 T = i i1 i si si1 ci si1 ci1 ci1 di (1) where c, s correspond to cos and sin respectively. In Fig. 2, each link is denoted with different color, and the numbering
where T and U are the total kinetic energy and potential energy of the system respectively, dened by
where Ti , Ui the kinetic and potential energy of link i respectively. If the augmented link i is dened as the combination of link i and motor i+1, then the kinetic energy of the augmented link i is given by
1 iT i Ti = 2 mi piT pi + piT S i mi ri i + 1 i i i i i i Ii i i,C i iT +kr,i+1 qi+1 Imi+1 zmi+1 i + 2 kr,i+1 qi+1 Imi+1

where mi : the overall mass of the augmented link i, pi : the linear velocity of the link i referred to frame i i i i : the angular velocity of link i expressed with reference to frame i ri i : the vector with start the reference frame i and end the i,C center of mass of the augmented link i, referred to frame i, kr,i+1 : the gear reduction ratio of motor i + 1, Imi+1 : the inertia tensor of the rotor i + 1 relative to its center of mass, zi i+1 : the unit vector along the rotor axis i + 1 referred to m frame i, S () : a matrix operator dened for a vector r = T rx ry rz by 0 rz ry 0 rx S (r) = rz (6) ry rx 0 i : the inertia tensor of the augmented link i with respect to Ii the origin of frame i, given by Iixx Iixy Iixz i = Iixy Iiyy Iiyz (7) Ii Iizz Iixz Iiyz
The potential energy of the augmented link i is given by
i iT Ui = g0 mi pi + mi ri,Ci i
parameters are given by (8) IjxxR = Ijxx Ijyy I(j1)xxR = I(j1)xx + Ijyy + 2dj Mjz + d2 Mj j I(j1)xyR = I(j1)xy + aj1 Sj1 Mjz + aj1 dj Sj1 Mj I(j1)xzR = I(j1)xz aj1 Cj1 Mjz aj1 dj Cj1 Mj I(j1)yyR = I(j1)yy + CCj1 Ijyy + 2dj CCj1 Mjz + a2 + d2 CCj1 Mj j1 j I(j1)yzR = I(j1)yz + CSj1 Ijyy + 2dj CSj1 Mjz +d2 CSj1 Mj j I(j1)zzR = I(j1)zz + SSj1 Ijyy + 2dj SSj1 Mjz + a2 + d2 SSj1 Mj j1 j M(j1)xR = M(j1)x + aj1 Mj M(j1)yR = M(j1)y Sj1 Mjz dj Sj1 Mj M(j1)zR = M(j1)z + Cj1 Njz + dj Cj1 Mj M(j1)R = M(j1) + Mj (15) where: SS () = sin () sin (), CC () = cos () cos (), CS () = cos () sin (), IjxxR : the grouped inertial parameter corresponding to link j with respect to the x axis, Ijxx : the moment of inertia of link j with respect to the x axis, Mjz : the rst moment of inertia of link j with respect to the z axis, corresponding to mi Ci z in (11) Mj : the mass of the link j, , corresponding to mi in (11) M(j1)xR : the grouped parameter corresponding to the rst moment of inertia of link j 1 with respect to the x axis M(j1)R : the grouped parameter corresponding to the mass of the link j 1. All the other symbol denitions can be resulted from the above. Considering augmented links used in (5) and using (15), the nal base parameter vector of the PA-10 robot arm is given by: = [ I1zzR M2yR I2xxR I2yzR I2zzR Im3 I3xxR 3zzR Im4 M4yR I4xxR I4yzR I4zzR Im5 I I5xxR I5zzR Im6 M6yR I6xxR I6zzR Im7 ]T (16) where the relation between the 21 base parameters with the PA-10 augmented link parameters of (11) is dened in Appendix. The matrix multiplication of the 211 base parameter vector with the corresponding 721 Y matrix was tested to result to the same expression, as this resulted from (12) in symbolic form, using the Mathematica R software package. Moreover, using the grouped parameter Y matrix, the dynamic model equation was written in the following form: B (q) q + C (q, q) q + G (q) + F = (17)

where pi the position vector of link i referred to frame i and i i g0 the gravity acceleration vector referred to frame i [11]. If i ri,Ci is dened by

i ri,Ci =

then the Lagrangian of the system can be expressed by

T T Ti Ui i

where i is the 111 vector of dynamic parameters dened by i = [ mi Iixx mi C i x mi C i y mi C i z Iixy Iixz Iiyy Iiyz Iizz

Imi+1 ]T

(11) and Ti , Ui 111 vectors that are dependent on joint positions qi and joint velocities qi. The derivation required by the Lagranges equation (2) does not alter the property of linearity in the parameters. Therefore, for the total links of the robot arm, it is = Y (q, q, q) (12)
where a 71 vector of joint torques, a p1 vector of constant parameters and Y a 7p matrix which is a function of joint positions, velocities and accelerations. Considering the PA-10 robot arm and (11), is initially a 771 vector, omitting friction parameters. For the purpose of the identication of the parameter vector , the base parameters should be determined, since they constitute the only identiable parameters. These base parameters can be deduced from the standard parameters by eliminating those that have no effect on the dynamic model and by grouping some others. The center of mass for each augmented link i is located on the axis passing through the center of the frames i and i + 1 as reported by mass distribution drawings provided by Mitsubishi Heavy Industries [12]. Due to this fact, the following simplication can be made: C1 x = C1 y = C2 x = C2 z = C3 x = C3 y = C4 x = C4 z = C5 x = C5 y = C6 x = C6 z = 0 (13)
Moreover, due to specic symmetries of the links, the following simplications can be made: I1xz = I1xy = I1yz = I2xy = I2xz = I3xz = I3xy = I3yz = I4xy = I4xz = I5xz = I5xy = I5yz = I6xz = I6xy = I6yz = 0
where B (q) the 77 inertia matrix, C (q, q) the 77 Coriolis Centrifugal matrix, G (q) the gravity vector and F the vector of friction analyzed in following section. B. Friction Model For the identication of the friction forces, a large number of constant velocity experiments were conducted for each of the seven joints, controlling the arm in velocity mode, in such a conguration that gravity did not affect joint motion.

For the determination of the base parameters, a straightforward closed-loop form method is used [13]. If j a revolute joint, between links j and j 1, then the resulting grouped
TABLE II F RICTION C OEFFICIENTS Positive Velocity f3 f4 0.4469 0.0622 0.5445 0.0202 0.2499 0.0537 -0.0954 0.1767 0.1045 0.1060 0.0373 0.0220 0.0895 0.1605 Negative Velocity 0.6203 0.0583 0.4624 0.0215 0.2451 0.1073 -0.0626 0.4307 0.0378 1.2697 0.0286 0.0411 0.0841 0.1683

Joint 6 7

f1 0.4362 0.2896 0.1434 0.0640 0.0231 0.0551 0.0304 0.3952 0.1960 0.1391 0.0690 0.0337 0.0390 0.0273
f2 0.6632 0.7659 0.3302 0.0922 0.1198 0.0829 0.1102 0.8550 0.7482 0.3576 0.1122 0.0796 0.0846 0.1070
f5 0.5612 0.4498 0.2991 -0.1283 0.1005 0.0 0.1073 0.7244 0.3744 0.3389 -0.0919 0.1047 0.0 0.1008
f6 16.9787 48.4405 16.6124 39.4975 11.4160 0.0 8.3338 19.1778 38.7804 10.0974 -101.92 1.0697 0.0 8.0266

Fig. 3.

Joints 1, 2: Friction experimental data and tted curve.
It was noticed that the friction torque at the most joints had the above characteristics: 1) Static friction was noticed less than kinetic friction and the rise from static does not occur instantly. 2) Kinetic friction was noticed more than viscous friction within the low velocity region, while the drop from kinetic to low-velocity viscous does not occur instantly. It must be noted that static friction was dened as the torque needed to initiate a motion with constant velocity of 0.001rad/sec. From the above characteristics of friction it can be concluded that a Stribeck model is inadequate for the case of PA-10 [5]. Therefore, a new term has been added to the Stribeck model, concluding to a new model for friction given by: F = f1 q + f2 sign (q) f3 sign (q) e f4 f5 sign (q) e

Fig. 4.

Joints 3, 4, 5: Friction experimental data and tted curve.
(18) where fi , i = 1,., 6 the model coefcients. It must be noted 1 that the new term f5 sign (q) e f6 |q| was added to account for the second specic characteristic of friction discussed above, i.e. the drop in friction force from kinetic to low-velocity viscous friction. Experiments of constant velocity conducted for velocities ranging from 0.001rad/sec to 60% of maximum value for each joint. Experimental results along with the tted model of form dened in (18) are shown in Fig. 3-5. Torque values correspond to motor axis, before transmission. The friction model coefcients, computed by applying the nonlinear least squared method, are listed in Table II. Friction was noticed to be non-symmetric at positive and negative velocities. It must be noted that each joint was forced to one minute of moderate velocity activity spanning the workspace, before the identication of the friction model, as suggested in [5]. C. Stiffness Model Harmonic drives exhibit signicant compliance when externally loaded [1]. Experimental tests in [2] indicate that stiffness in harmonic drives increases with increasing load. Direct measurement of stiffness entails the measurement of position in both motor side and joint side. However, this is

Fig. 5.

Joint 6, 7: Friction experimental data and tted curve.
impossible in commercial robots. Therefore the determination of the torque used to deform the compliant elements of the joint transmissions can only be accomplished through the torque measurements during loaded conditions. The experiments conducted on the PA-10 robot arm using as load for each joint the weight of its following links, revealed that joint transmission compliance has a signicant effect on the robot arm dynamics. Each joint was commanded to move at constant velocity, in such conguration that gravity due to its following links affected its motion. The torque due to joint transmission stiffness was computed by subtracting friction and gravitational terms from the total torque measured. The gravitational term of torque for each joint is computed using the G vector in (17) and the data provided by Mitsubishi [12]. Experimental results revealed that stiffness effect is present at joints 1-4, while joint 5-7 are not considerably affected. In Fig. 6, the stiffness experimental data and tted linear models are depicted, while the torque values given correspond to the
Fig. 6. Stiffness experimental data and tted models for Joints 1-4. Experimental values are depicted as dots while tted model as lines of the same color.

Fig. 7.

End-effector trajectory tracking performance.
motor axis. The tted linear models are given by: S1 S2 S3 S4 = 0.1036L1 2.= 0.1054L2 0.01256 = 0.1199L3 + 0.00192 = 0.1161L4 0.00488
C. Experimental Verication of the Full Identied Model The verication of the full model identied is performed through an end-effector trajectory tracking experiment. The trajectory of the end-effector was restricted to the xz plane of the base reference frame, while the orientation was kept constant. The trajectory is given by x = 0.085 (cos (t) + sin (2t) 1) z = 0.075 (sin (t) + cos (2t) 1) + 0.7248 y = 0.7
where Si the stiffness torque, Li the load torque, while the coefcients are calculated using least squares method. for each joint i = 1,., 4.
III. E XPERIMENTAL R ESULTS A. System Components The experimental setup consisted of the PA-10, its servocontroller and a PC running GNU/Linux connected to it via the ARCNET interface. The control of the robotic arm was realized by means of a custom multi-threaded C library, based on the Linux kernels ARCNET driver and providing an Application Programming Interface (API) featuring almost all of the capabilities of the PA-10. The library acquires its timing through a High Resolution POSIX Timer, with a resolution of 10 sec, which, combined with the preemption features of the modern Linux kernels, offers a 99.9% steady communication cycle of 2.5 ms, even under heavy CPU and I/O stress.

where t denotes time. This trajectory results to joint motions that are of different form of those used during identication experiments, thus providing an excellent mean of verication of the identied model. The control law implemented was chosen to have the simplest form in order to verify the identied model. An inverse dynamic controller was selected, given by u = B (d + KV (qd q) + KP (qd q)) + q C (q, q) + G (q) + F + S (21)
B. Dynamic Parameter Identication For the purpose of dynamic parameter identication, each joint was controlled in velocity mode. Sinusoidal as well as polynomial velocity proles were sent to the joints, in order to form exciting trajectories for the identication of the base parameters [8]. The friction and stiffness terms were computed in advance and therefore known during the dynamic parameter identication. Computed values of the PA-10 base parameters are listed in Appendix. The values 2 , 10 and 18 corresponding to rst moment of inertia can be compared to values calculated by manufacturers provided data [12]. The accuracy of the estimated values with respect to the real ones, is 99.3, 98.2 and 97% respectively.
where B, C, matrices of the dynamic model equation dened in (17), G, F, S the gravity, friction and stiffness vector respectively, KV , KP 77 diagonal positive denite matrices of gains and qd the desired joint trajectories computed from the desired end-effector trajectory given in (20), using an inverse kinematics algorithm. The algorithm used is the Jacobian pseudo-inverse algorithm for redundant manipulators described in [11], with an added term to avoid joint limits. In Fig. 7 the trajectory followed by the end-effector is shown along with the desired, as well as the error in end-effector position in Cartesian space. As it can be seen, the end-effector moved along the desired trajectory with high accuracy. The maximum error in joint space was 0.15 deg, while the mean position error in Cartesian space was limited to 0.8 mm while the smaller value found in literature for the same experiment was at 2 cm in [1]. In Fig. 8 the torque sent to each joint during the trajectory tracking experiment is plotted. As it can been seen the feedback torque is very low with respect to the total torque u given in (21), which proves the high accuracy of the identied full model of the PA-10 robot arm.

1 = I1zzR = I1zz + I2yy + kr Im1 = 6.3376 Kgm= M2yR = C2 y m2 C3 z m3 d3 (m3 + m4 + m5 + m6 ) = 7.3424 Kgm 3 = I2xxR = I2xx + I3yy + 2d3 C3 z m3 + 2 d3 (m3 + m4 + m5 + m6 ) I2yy = 2.9419 Kgm2 2yzR = I2yz = 3.7152 Kgm= I 5 = I2zzR = I2zz + I3yy + 2d3 C3 z m3 + 2 d2 (m3 + m4 + m5 + m6 ) + kr Im2 = 4.2823 Kgm6 = Im3 = 4.Kgm 7 = I3xxR = I3xx + I4yy I3yy = 1.5248 Kgm2 3zzR = I3zz + I4yy = 3.8092 Kgm2 8 = I 9 = Im4 = 2.Kgm= M4yR = C4 y m4 C5 z m5 d5 (m5 + m6 ) = 3.2317 Kgm 11 = I4xxR = I4xx + I5yy + 2d5 C5 z m5 + 2 4yy = 0.7116 Kgm2 d5 (m5 + m6 ) I 12 = I4yzR = I4yz = 0.4189 Kgm2 4zzR = I4zz + I5yy + 2d5 C5 z m5 + 13 = I 2 d5 (m5 + m6 ) = 0.4961 Kgm= Im5 = 3.Kgm2 15 = I5xxR = I5xx + I6yy I5yy = 0.2917 Kgm2 5zzR = I5zz + I6yy = 0.1309 Kgm2 16 = I 17 = Im6 = 2.Kgm= M6yR = C6 y m6 = 0.05377 Kgm 19 = I6xxR = I6xx I6yy = 0.1861 Kgm2 6zzR = I6zz = 0.0747 Kgm= I 21 = Im7 = 9.Kgm2

Fig. 8.

Joints Torques during tracking experiment.
IV. C ONCLUSION In this paper the modeling and full identication and control of the Mitshibishi PA-10 robot arm was developed. A new non-linear friction model for the robot joints was constructed and experimentally identied, while the stiffness effect of the joints was also identied through experimental procedure in loaded conditions. The parameters of the dynamic model were grouped to an identiable form, and identied through experiments. A model-based inverse dynamic controller was nally used to evaluate the identied model within a trajectory tracking experiment. In this experiment, the low feedback torque measured at the robot joints proved the models accuracy. Consequently, as the Mitsubishi PA-10 robot arm is widely used in research laboratories world wide, the full model identied here is of high importance as it allows the implementation of any model-based controller, incorporating accurate models for friction and stiffness effects at the robot joints. ACKNOWLEDGMENTS The authors want to acknowledge the contribution of the European Commission through contract NEUROBOTICS (FP6IST-001917) project. This research project is co-nanced by E.U.-European Social Fund (75%) and the Greek Ministry of Development-GSRT (25%). APPENDIX Base Parameters of the PA-10 robot arm The base parameters of the PA-10 robot arm and their identied values are given by:

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